Download Free Field Approach to 2-Dimensional Conformal Field Theories

Progress of Theoretical Physics Supplement No. 102, 1990
67
Free Field Approach to 2-Dimensional Conformal Field Theories
Peter BOUWKNEGT,*'#l Jim MCCARTHY**·tl and Krzysztof PILCH***·t n
*Center for Theoretical Physics, Massachusetts Institute of Technology
Cambridge, MA 02139, U.S.A.
and
Institute for Theoretical Physics, University of California
Santa Barbara, CA 93106, U. S. A.
**Department of Physics, Brandeis University, Waltham, MA 02254, U.S.A.
***Department of Physics, University of Southern California
Los Angeles, CA 90089-0484, U.S.A.
We review various aspects of the free field approach to (rational) conformal field theories.
In particular, we will discuss resolutions of irreducible modules in terms of free field Fock
spaces for WZNW-models and their coset models, as well as the free field realization of chiral
vertex operators. We provide a host of clarifying examples and detailed proofs of results
that were announced elsewhere.
Contents
§L
Ll
§ 2.
2.1
Introduction
Notations
Fock space realizations and resolutions for finite dimensional Lie algebras
Fock space realizations
2.2. Intertwiners
2.3. The EGG-resolution
2.4. Twisted Verma modules, realizations and resolutions
2.5. Finite dimensional coset models
2.6. Finite dimensional vertex operators
§ 3. Free field approach to affine Kac-Moody algebras
3.L The Fock space realization
3.2. Intertwining operators
3.3. The resolution
Supported in part by the U.S. Department of Energy under Contract #DE-AC02-76ER03069 and by NSF
grant #PHY-89-04035, supplemented by funds from NASA.
Address after Nov. 1; CERN-TH, CH-1211 Geneve 23, Switzerland.
tJ Supported by the NSF Grant #PHY-88-04561.
t t > Supported in part by the USC Faculty Research and Innovation Fund.
#)
68
P. Bouwknegt, J. McCarthy and K. Pilch
3.4. Chiral vertex operators and fusion rules
3.5. Coset conformal field theories
Appendices
A. Some lemmas on the Weyl group
B. Quantum group identities
C. Cohomology of double complexes
D. Proof of Theorems 2.12 and 2.12'
E. Restricted quantum group Verma modules
References
§ 1.
Introduction
Free field realizations, so widely used in the early works on string theory (see e.g.
Ref. 1) and references therein), have naturally found their way into the study of two
dimensional conformal field theories. 2>-s> Their power is illustrated in the work of
Dotsenko and Fateev4 >,s> who, using ideas of Feigin and Fuchs, 2>managed to compute
the correlators of the Virasoro minimal models. 6>
A complete free field description of a conformal field theory has three major
ingredients :
i) a realization of the chiral algebra Jl by free fields,
ii) a projection from the free field Fock spaces to the irreducible representations
of Jl ("null-state decoupling"),
iii) a realization of the chiral vertex operators.
Although these ingredients are implicitly present in the work of Dotsenko and
Fateev, the full underlying structure has only been realized and appreciated recently
through the work of Felder. 7 > This work not only shows why the Dotsenko-Fateev
prescription works, but also immediately suggests how to compute the higher genus
conformal blocks. 8 >-IZ> The structure revealed by Felder in the case of the Virasoro
minimal models is expected to exist for generic 2D conformal field theories ; i.e. the
projection can be enforced by a "ERST-like" procedure, and the chiral vertex operators can be naturally realized as "ERST-invariant" operators. Specifically, the
projection to the irreducible representations LA of Jl can be achieved by a "two-sided
resolution" of LA in terms of Fock space modules FAu>, i.e. a complex (FA, d)={(FAu>,
d<i>), iEZ}, with a differential d<i> that intertwines with Jl
(1·1)
and such that the cohomology of this complex is given by
Hd(i)(FA)={LA
0
if i=O
otherwise.
0
(1·2)
Second, the chiral vertex operators are realized as equivalence classes of chain maps
V(z):(FA, d)-+(FA', d'), i.e. a collection of maps v<i): FA<i>-+FW satisfying d'u>v<i)
= v<•+l>du>, iEZ,
Free Field Approach to 2-Dimensional Conformal Field Theories
···~
FA(- 1 )
.1-
...
d(-1)
~
FA(O)
.1-
v<-1>
v<•>
d'(-1)
~
F1-; 1 >
~
d(O)
~
F1g>
FA( 1 )
.1-
v(l>
d'(O)
~
d(1)
~
FA(Z)
.1-
~···
v<2>
d'(l)
F1V
~
69
F1~>
~
...
(1· 3)
modulo trivial chain maps v<il=d'u- 1>j<il+ jU+ 1>du> where f={fu>: FAu>~ F1~- 1 >}.
A particularly useful tool that will be used throughout the paper is the algebraic
Lefschetz theorem
(1·4)
for a chain map {Cl<il: FA<il~FAUl} such that o<o>ILA=O.
An important parallel development was the discovery of free field realizations for
more general chiral algebras, such as affine Kac-Moody algebras/ 3 >- 16 >parafermion
algebras/ 7 >-21 >CW-algebras. 22 > Proposals of free field parametrizations that could be
relevant to other coset conformal field theories have also been made. 18 > For other
related papers see Refs. 23)---....35).
The corresponding resolutions in terms of those free field Fock spaces were
constructed in Refs. 36)---....40) for affine Kac-Moody algebras, Ref. 37) for the CWalgebras (through the quantum Hamiltonian reduction 41 >-43 >), in Refs. 44) and 45) for
parafermion algebras and in Ref. 44) for generic coset conformal field theories.
The investigation of chiral vertex operators in this context, i.e. of the chain maps
between two resolutions, was initiated in Refs. 36) and 46).
The above formulation puts the free field approach to 2D conformal field theories
in a convenient setting, namely that of homological algebra.47l' 48 > Here, in general,
one tries to deal with a module over a ring fR by "replacing" the module by a
resolution in terms of modules with certain suitable properties. A relevant example
in our context is the Bernstein-Gel'fand-Gel'fand (BGG) 49 > resolution of g-modules in
terms of Verma modules. Verma modules have the special property that they are
free over CU(n-) on a single generator, the highest weight vector (in fact they are
uniquely characterized by this property). However, the relevant Fock space modules
of chiral algebras Jl are neither isomorphic to a Verma module nor to a dual Verma
module, but rather "intermediate" between a Verma module and a dual Verma
module. This has the important consequence that the corresponding resolutions are
two-sided. The proofs of standard textbook theorems on homological algebra, which
all assume one-sided resolutions, must then be suitably modified-in particular, the
use of induction techniques has to be adapted. Such developments may be critical in
supplying a rigorous formulation of the work described here.
This paper is in large part a review of our work, 39 >' 40 >' 44 >' 46 >but there is a significant
fraction which is new. In particular the proofs of several results, as promised in Refs.
44) and 46), are included in the relevant sections. Throughout, we will use an
instructive finite dimensional analogue-the free oscillator approach to finite dimensional Lie algebras-as a guide. Therefore, we will start (in § 2) by developing the
concept of free field realizations, resolutions, coset models and vertex operators in this
finite dimensional context. We will not review the geometric interpretation of the
70
P. Bouwknegt, ]. McCarthy and K. Pilch
finite dimensional algebra realization, but present instead a completely algebraic
approach. We include a discussion of the "twisted Verma modules" of Feigin and
Frenkel, 50>which are in some respects the closest finite dimensional analogues, since
their corresponding resolutions turn out to be two-sided as well. In § 3 we will treat
the infinite dimensional case along the same lines.
Specifically, the results that we will discuss in § 3 include:
i) For any Kac-Moody algebra fj, at arbitrary level k, a free field realization may
be constructed from t (=rank g) scalar fields ¢;(z), i=1, ···, t and ILI+I pairs of
conjugate spin (1, 0) bosonic first order fields (/3a(z), ra(z)), aE.d+. The Sugawara energy momentum tensor in this realization reduces to the free field tensor
for !3r¢, with an additional background charge (Feigin-Fuchs) term for the scalar
fields ¢;(z).
ii) There are screening operators s;(z), of conformal dimension one, whose operator
products with the Kac-Moody currents contain at most a total derivative singular
term. Thus, to construct intertwiners with the full Kac-Moody algebra we look
at the class of integrated products of screening currents. Since the operator
product algebra of screening currents is nonlocal this is a nontrivial problem. In
fact there exists a set of "basis contours" within which products of screenings
obey the algebra of the positive root generators of the quantum group CfJ q(g ), q
=exp(i7r/(k+hv)). It follows then quite easily that given any singular vector in
an associated quantum group Verma module we can construct an intertwiner
between two Fock space modules.
iii) For A an integrable {j-weight, there is a complex of Fock modules (1·1), where
each FA <iJ is the direct sum of an (in general) infinite set of Fock space modules
F w*A characterized by affine W eyl group elements w of a certain "twisted length"
lr(w)=i. It is conjectured that this complex (FA, d) provides a resolution of the
irreducible highest weight module LA.
iv) For integrable {j-weight A;,, i=1, 2, 3, the chiral vertex operators mapping LA, to
LA., transforming according to LA,, may be realized as chain maps between the
resolutions (FA,, d) and (FA., d'). Clearly each v<o is itself a collection { Vw',w:
Fw*A•-+Fw'*A•, lr(w)=lr(w')=i}. The components Vw',w are built with integrated screenings, and can be identified with elements of the quantum group Verma
modules. We have proven that a chain map exists iff there is v<o> and v< 1>such
that d'(O) v<o)= vn>d(O). We conjecture-as can be proven in the finite dimensional analogue problem-that the dimension of the vector space of chain maps
modulo trivial chain maps is precisely given by the fusion rules of the WZNWmodels.
v) We have also begun to extend the above results i)"""'iv) to G/H coset models. 50
Of course, since the chiral algebra is not known in general for these models, we
must take a more "extrinsic" approach. As a first step, we have shown that for
any irreducible highest weight module LA of fj, and LA' of ii, there exists a
subcomplex of the resolution (FA, d) which provides a resolution of the coset
module whose character is the appropriate branching function. The subcomplex
is just the restriction of (FA, d) to the subspaces of fi-singular vectors-i.e. those
vectors annihilated by the generators in ii+ H-as may be enforced by a "usual"
Free Field Approach to 2-Dimensional Conformal Field Theories
71
BRST procedure.
That the quantum group CU q(g) plays an important role should perhaps not be
surprising. In fact, it has been conjectured that the problem of classification of
rational conformal field theories is intimately related to the theory of quantum
groups. 52 >-ss> For example, the braiding matrices corresponding to exchange of
chiral vertex operators56 >' 57 > are related to quantum group 6j-symbols. It has also
been suggested that the fusion rules are related to truncated tensor product rules of
irreducible quantum group representations, 55 > as is known to be true empirically for
(2). 58> The apparent relation of conformal field theories to quantum group theory
has not been easy to understand. We feel that the results outlined above indicate that
the free field approach to 2D conformal field theory is a natural arena to address this
issue.
Unfortunately, lack of space and time has forced us to restrict the discussions in
this paper mainly to our own work on this subject. For closely related interesting
developments the reader may want to consult for example Refs. 59)""-'66).
su
1.1.
Notations
Throughout the paper we will use the following notations (see, e.g. Ref. 67)):
g a semi -simple Lie algebra
t a Cartan subalgebra with dual t*
CU( ·) the universal enveloping algebra functor
g=n-ffitffin+ a Cartan (triangular) decomposition
b±= n±ffit the two Borel sub algebras
G, T, N±, B± the corresponding groups
.t the rank of g
L1± a system of positive/negative roots
av=(2a)/(a, a) the co-roots
M = Z · Lh v the co-root lattice of g
W the Weyl group of g
ra the reflection in the root aEL1+
r, the reflection in a simple root a;, i = 1, · · ·, .t
(,) the bilinear form on t or t*, sometimes also denoted by ·
p the element of t* such that (p, a/)=1, i=1, ···, .t
co*tl=w(tl+p)-p for wE W, t\Et* a shifted Weyl group action
Z+={O, 1, 2, ···}
h v the dual Coxeter number of g
P, P+ the set of integral, and integral dominant weights, respectively
C[t] the set of polynomials in the variable t with coefficients in C.
Throughout this paper we will use the Chevalley basis of g, which we recall is
defined by the following commutators :
i, j=1, ···, .t
(1·5)
72
P. Bouwknegt,]. McCarthy and K. Pilch
as well as the Chevalley-Serre relations
for
ai.isO,
(1·6)
where ai.i=(ai, a;v) is the Cartan matrix of g. For convenience we will also sometimes use the notations ea for the generator corresponding to a root a ELl, and Ia= e-a
for aE.L/+. We fix a normalization of (,) such that (8, 8)=2 for the highest root
and put d;= -Ha;, a;) such that the matrix d;ai.i is symmetric.
We will distinguish between quantities of the affine Kac-Moody algebra fj and its
underlying finite dimensional Lie algebra g by putting hats on the former. Furthermore we will implicitly identify the affine Weyl group W of fj with its projection Waff
onto t*. Also we will identify an affine weight AEP with its components (Jf, k) in
tEe Cc, and use the same notation for quantum group weights through the correspondence q=exp{in/(k+ hv)}.
e
§ 2.
2.1.
Fock space realizations and resolutions for finite dimensional Lie algebras
Fock space realizations
A finite dimensional analogue of the free field realizations of affine Kac-Moody
algebras that we are interested in, is provided by realizations of finite dimensional
(semi-simple) Lie algebras in terms of differential operators on polynomial spaces.
These arise naturally from the group action on (local holomorphic) sections of a line
bundle over the flag manifold cc /B, where B is a Borel subgroup of the complexified
group cc. This is known as Borel-Weil theory, 68 > and has previously entered physics
in discussions of coherent states (see Ref. 69) and references therein). The algebra is
then realized in terms of differential operators of degree one on the space of
polynomials in Za, aE.LI+/0> (see also Ref. 71) and references therein), giving a
description naturally isomorphic to a dual Verma module (see e.g. Ref. 40) for more
details).
In this section we will instead discuss these free field realizations in the closely
related algebraic setting of Verma modules, thereby of course losing track of the
global properties of the Borel-Weil theory. To make the transition to the infinite
dimensional case as natural as possible we will give the realization in terms of a set
of bosonic oscillators ya, {r, aE.L/+, satisfying [ra, pa']=oaa' on a Fock space built on
a vacuum lA> satisfying .BaiA>=O. One may, of course, always keep the specific
realization r=z, .8= -(o/oz) in mind. To keep track of the highest weight A we will
use coordinate momentum pairs (pi, q i) with commutators [pi, qi] = - ioi.i such that
P;IA>=A;IA>, where A' are the components of A with respect to some orthonormal
basis. In particular we will have translation operators TAA' =exp(i(A'- A)· q) such
that lA'>= TAA'IA>.
Now suppose we want to obtain a realization of g on such a Fock space and such
Free Field Approach to 2-Dimensional Conformal Field Theories
73
that the resulting module is isomorphic to a Verma module MA of highest weight A.
We proceed as follows. By the Poincare-Birkhoff-Witt (PBW) theorem72 > we choose
a certain basis of MA =CU(n-)vA. Then, through the replacement Ia~ ra we identify
every basis element of MA with a Fock space monomial (i.e. we choose a map from
CU(n-) to the symmetric algebra S (n_) on n-). The action of g on MA can be
represented by an action of g on these Fock space monomials in terms of /3roscillators. This leads to the required free field realization, as is perhaps best
illustrated in an example :
Example 2.1.
Let g=su(3). We choose the PEW-basis of MA as follows (/3= Ia.)
(2·1)
Using the commutators [A lz]=- /3, [/I, /3]=0, [/z, /3]=0, we find
(2·2)
and with a little more work, e.g.
hiVAp,q,r=( -2p-q+ r+(A, ai))vAp,q,r,
eiVAp,q,r =((r- q )p- p(p-1)+ p(A, ai))VAP-I,q,r- QVAp,q-I,r+I.
(2·3)
Upon identifying VAp,q,r with the monomial (ri)P(y 3)q(r2YIA>, we find the following
realization of g
ei = -(ai· p)f3I-( ri 13 I_ r2 132+ r3 /33)/3I + r2 133,
e2= -(a2· P)/32_ rz /32 /32_ ri /33,
e3= _ (ai. p)/33- (a2•p)(/33 +pi /32) _ ( ri pi+ r2 132+ r3 /33)/33 _ r2 pi 132132,
hi= ai. p+(2ri /3I_ f /32+ r3 /33)'
h2= az· p+(- ri pi +2r2 /32+ r3 /33)'
!I=ri,
/2= r 2 - r 3/3I,
/3=r3 .
(2·4)
Obviously, for generic g, the generators of the Cartan subalgebra in this realization are independent of the choice of PEW-basis. Indeed,
(2·5)
The form of the other generators of course does depend on the specific choice of basis
and although every choice of PEW-basis of MA would suffice, the following choice
74
P. Bouwknegt, ]. McCarthy and K. Pilch
seems to be a convenient one. Choose a presentation of the longest Weyl group
element Wo of g in terms of simple reflections
(2·6)
and take the basis
(2·7)
Let us illustrate this choice in several examples.
Example 2.2. For su(n) we could take
(2·8)
such that
(2·9)
where li.i denotes Ia,; for the root ai.i= E;- EJ. This gives rise to the following realization (for the sake of simplicity we only give the simple root generators)
e;=- (a;. p)pu+l_ "2J rji/3ji+l + "2J
j<i
ri+lj
pi.i- ("'2, ri.i pi.i- "2J ri+lj!3i+lj)/3ii+l
i>t
i>i+l
i>t+l
'
(2·10)
Example 2.3. Let g=so(5). Taking wo=r1r2r1r2, where a2 is the short root, leads to
the PBW-basis (we use the shorthand notation /122= !a,+2a., etc.)
!/!1~f{2d2 8 VA ~( r 1)P( 'Y12 )q( r 122 Y( r 2)8
\A>.
(2·11)
We find the following realization
e1= _ (a1v. p)/31-( r1 13 1+ r12 1312_ r2 /32)/31 +
f 1312_ ~ r122p12 1312 ,
e2= _ (a2v. p)/32- r2 132132_ 2 r1 1312_ 2 r12 13 122,
h1 =a1 v. P+(2r1 /31 + r12 /312_ r2 /32)'
h2=a2 v. P+2(- r1 /31 + r122 !3122+ r2 /32),
/1=r1 ,
(2·12)
For specific applications other choices of basis might be more convenient. For
instance, in the discussion of coset models described by some embedding H c G it is
convenient to have a realization of n- 8 in terms of oscillators pa, ra for aELh 8 only.
This can be achieved by choosing a PBW-basis of CU(n- G) respecting the decomposition CU(n_ G)=CU(n~ 18 )®CU(n- 8 ).
Free Field Approach to 2-Dimensional Conformal Field Theories
75
Example 2.4. Consider the diagonal embedding SU(3)cSU(3) X SU(3). We distinguish the groups by superscripts or subscripts in round brackets. The appropriate
PEW-basis places the generators of the diagonal subgroup c<D> (1/D>= f/ 1>+ f/ 2>, I
= 1, 2, 3) on the left,
(2·13)
Identifying this with the monomial
( r/)P•( ri)P•( ri)P•( r_/)q•( r.L3)q•( Y.L 2)q•IA (1>, A <2 >> ,
(2·14)
one can find the explicit realization as in the other examples. The fact the f/D> are
realized purely in terms of the r/ is manifest by construction since they act freely on
(2·13) from the left, not disturbing the perpendicular coordinates. In contrast, note
that the subgroup generators corresponding to positive roots, e/D>, will contain both
parallel and perpendicular coordinates since their action must be pulled all the way
through (2 ·13) onto the highest weight state. As a final comment, we note that the
realization obtained by the direct product of that in Example 2.1 is related to the
present one by a simple coordinate transformation.
What characterizes the above realizations is that they are free over CU(n_) on one
generator, the highest weight vector (vacuum) lA>. By taking the Fock space adjoint
-i.e. rat=pa, pat=ra, aELI+-as well as relabelling e;~/; in the realizations described above, we obtain another highest weight realization of g on the same Fock
space. Clearly this corresponds to a realization of g in terms of differential operators ([3~o/oz) of order one-exactly that on sections of a line bundle over cc /B
alluded to at the beginning of this section. One easily observes that this realization
is co-free over CU(n+) on one generator, hence it is isomorphic to the dual Verma
module. In fact, there exists a whole spectrum of modules which are intermediate
between a Verma module and a dual Verma module in the sense that they are free
over CU(m-), co-free over CU(m+) where m= m+ffim- is a nilpotent algebra isomorphic
to n+. These modules were introduced by Feigin and Frenkel, 50> where they were
called "twisted Verma modules". In a lot of respects they resemble the Fock space
modules of affine Kac-Moody algebras which we will introduce later, so we will
discuss them separately in § 2.4.
2.2. Intertwiners
The aim of this section is to discuss intertwiners between Verma modules.
Suppose we have two Verma modules MA and MA', and we want to determine the set
Homvcu>(MA, MA'). We will do this in several steps. Requiring the invariance under
CU(n_) leads to the following lemma:
Lemma 2.1. Given A, A'E t*, there exists a 1-1 correspondence between CU(n_) and
Homvcn-l(MA, MA').
Proof Since Verma modules are free over CU(n-) on one generator VA (the highest
weight vector), the maps ¢EHomvcn-lMA, MA') are completely determined by their
action on VA. Explicitly, suppose ¢EHomvcn-lMA, MA') and that
P. Bouwknegt, J. McCarthy and K. Pilch
76
(2·15)
¢(vA)=xvA'
for some xECU(n-), then
if>(YVA)=yif>(VA)=yXVA',
\;:fyECU(n-).
(2·16)
Conversely, every xECU(n_) determines a map ¢EHomu<niMA, MA') by (2·16).
Suppose we define a representation p of CU(n_) on MA by right multiplication
i=1, ···, f'
(2·17)
and a "translation operator" TAA' : MA ~ MA' by
TAA'(yvA)=yvA',
(2·18)
then the above reasoning shows that there exists a 1-1 correspondence between CU
(n-) and Homv<niMA, MA') given by
xH p(x)Tf'.
(2·19)
0
Of course, not all xECU(n-) will give rise to an element ¢EHomv<ulMA, MA')
through (2·19). The condition that the map ¢should intertwine with CU(t) is however easily incorporated-it just requires ¢ to preserve isospin. This precisely means
that we should restrict ourselves to elements in CU(n_) of isospin ..l=A'- A, and thus
every p(j;) should be accompanied by a change in highest weight A~ A'= A+ a;.
Clearly, at this point it is convenient to introduce the operators
(2·20)
Then every polynomial in the s;'s will give rise to a map that intertwines with
CU(t)EBCU(n-). Because of their role in the infinite dimensional case we will refer to
the operators s; as the screening operators of g. It is obvious, but worth noting, that
the screening operators generate an algebra isomorphic to n-.
Requiring finally the invariance under CU(n+) leads to
2.2. There is a 1-1 correspondence between Homv<u>(MA, MA') and singular
vectors in MA' of weight A. Moreover, every such intertwiner is injective, i.e. defines
an embedding MA "-+ MA'·
THEOREM
Proof Requiring that ¢ of (2·16) intertwines with CU(n+) simply amounts to the
requirement
(2·21)
i.e. XVA' should be a singular vector in MA' of weight A. Injectivity follows immedi0
ately from (2 ·16).
Now let us make the above discussion of intertwiners somewhat more concrete by
making explicit use of the Fock space realizations of Verma modules introduced in
§ 2.1. In the Fock space realization of MA we have of course an explicit representation of TAA' in terms of the coordinate-momentum operators (p, q) as
77
Free Field Approach to 2-Dimensional Conformal Field Theories
(2·22)
TAA'=ei<A'-AH,
while the operators p(f;) are represented by polynomials in /:1 and
abuse of notation we will denote these also by p(/;)).
Example 2.5.
MA for su(3).
r
(with a slight
Let us determine the operators p(f;) in the Fock space realization of
We use the same notation as in Example 2.1. We have e.g.
p(f1)v~·q,r=- f/fsqf2r/1VA
= -(f/+1fsqf2r + rf/fsq+l/2r- 1)VA
(2·23)
and in the same way
(2·24)
In the Fock space realization these are thus represented by
p(/1)= -(r1_ rs /:12),
p(/2)=- r2 ,
p(/s)=-r3,
(2·25)
which clearly satisfy the commutators of n-, e.g. [p(/1), p(/2)]=-p(/s).
Similarly, for the other examples of § 2.1
su(n)
(2·26)
so(5)
p(/1) = _ ( r1_ r12 fF + ~ r122 p2 p2) ,
p(/2)= -r2 •
(2·27)
The subset of polynomials in the screening operators that intertwine with the
action of CU(g) could in principle be determined by explicit evaluation of commutators. Let us first evaluate the commutator [e,, sJ].
Lemma 2.3.
We have
- - uij
s- (p ,
[ e;, Sj ] -
Proof
a; v) e ia,.q .
(2·28)
By acting on a generic vector YVAEMA we obtain
([e;, sJ)YVA = -e;(yfiVA+a;)-sie;YVA).
(2·29)
Because [e;, y]ECU(t)EBCU(n_) (we took a simple root generator), the two terms
combine into
78
P. Bouwknegt, J. McCarthy and K. Pilch
(2·30)
which proves the lemma.
0
(Note that the commutator [e;, p(/;)] would have been considerably more difficult
to calculate.)
Let us now explicitly exhibit the prototype of intertwining operator. Suppose A
EP is such that (A, a;v)EZ+, then we claim that s;<A+p,a,v>EHomucg>CMn*A, MA). The
proof is a straightforward calculation. Let vEMnM then
[e;, S;(A+p,a,v>]v=-
~
O:<:j:<:(A, a,')
slA,a,V)-J(p, a;V)eia,.qs/v
=0.
(2. 31)
Note that in the context of Theorem 2.2 the example above corresponds to the
singular vector j/A+p,a,v>vA of weight r; *A in MA.
The other intertwiners can, in principle, be found by making use of the basic
commutator (2·28) to obtain (on MA)
(2·32)
where aj=a;;,+l +··· +a;;n and -denotes omission.
An alternative way to find explicit formulas for the intertwiners that correspond
to reflections in other than simple roots is to make repeated use of Lemma B.1 (for q
=1) (see Example 2.7).
The singular vector structure of MA was investigated in detail by BernsteinGel'fand-Gel'fand.73' Let us recall their results as far as relevant for the construction
of the resolution of an irreducible highest weight module for dominant integral A.
Recall thereto that the length l(w) of a Weyl group element wE W is defined as
the minimal number of reflections r; in simple roots a; such that w = r., ... r,k (see also
Appendix A). Then define the following partial ordering, the so-called Bruhat
ordering, on W. For w', w" E W we write w'--> w" if w' =raw" for some aELh as well
as l(w')=l(w")+l. Let w'sw" iff there exist W1, ... , whEW such that w'->w1_.. ..
--> Wk-> w".
THEOREM
2.4. 73 >
For AEP+ we have
(2·33)
Moreover, for w s w', every such intertwiner is a multiple of the canonical embedding
lw',w: Mw*A ~Mw'*A·
Example 2.6. For g=su(3) and AEP+ the theorem above gives rise to the following
directed graph of submodules of MA, and intertwiners between them (see also Ref. 74))
Free Field Approach to 2-Dimensional Conformal Field Theories
/'
Mnr•*A
--+
Mrt*A
'\.
MA .
~
Mr1r2rz*A
79
'\.
/'
Mr•rt*A
--+
Mr.*A
(2·34)
Due to the uniqueness of the intertwiners the diagram is commutative (up to proportionality) and every other intertwiner between modules of the form Mw*A is a composite of the ones depicted in this diagram (see Refs. 75) and 71) for an elucidation of this
fact*>). The Verma modules Mw*A occurring along a vertical line in this diagram
correspond to Weyl group elements w of fixed length, while between consecutive
vertical lines the length differs by one.
Example 2. 7. Consider the intertwiners of Example 2.6. Denote l;=(A + p, a;), z
= 1, 2, 3, then we have already seen that ( ¢w',w : Mw*A--+ Mw'*A)
(2·35)
Lemma B.l applied to A=s1, B=s2 (for q=l) now gives
(2·36)
where s3=[s2, sd and
b(m, n; j)
m!n!
j!(m- j) !(n-j)!
(2·37)
It follows that
(2·38)
Similarly we have
(2·39)
2.3.
The EGG-resolution
In this section we will discuss how to obtain resolutions of the irreducible modules
LA (for A dominant integral) in terms of Verma modules MA, the so-called BernsteinGel'fand-Gel'fand resolutions, by suitably combining the intertwiners of § 2.2. Define
thereto
(2·40)
We have
*l We thank V Dobrev for sending his papers74 >'75 >' 71 > and correspondence on this issue. 76 >
80
P. Bouwknegt, ]. McCarthy and K. Pilch
2.5. 49 ) For AEP+ we have a resolution of LA in terms of Verma modules,
i.e. a complex (MA, d)
THEOREM
(2·41)
with cohomology
H~il(MA)~{Lo A
for i=O
otherwise,
(2·42)
where t=IL1+I=l(wo), M1il is given by (2·40).
Proof (sketch) Consider the collection of modules MA ={Mw*AiwE W}. According to
Theorem 2.4 there exist intertwiners cf>w.,w, : Mw,M ~ Mwz*A for WI~ Wz, and they are
unique up to a multiplicative constant. Every other intertwiner within MA is a
composite of the "elementary intertwiners" c/>w,,w. for WI~ wz. To get control over
the "composite intertwiners" we need some results regarding the structure of the Weyl
group. 49 l
The first result is that given WI, wzE W such that l(wi)= /(wz)+2, the number of
elements w'E W such that WI~ w' ~ Wz is either zero or two.
Let us call the quadruple (wi, w', w", wz) a square if there are two elements w'
=I= w" E W such that WI~ w' ~ Wz and WI~ w" ~ Wz.
The idea is now to construct the differential d of a complex by combining the
intertwiners {cf>w.,w., WI~ wz} and forcing d 2 =0 by cancellation of the maps within a
square of the Weyl group. This requires the following result: To each arrow WI
~ W2 we can assign a number s(wi, w2)= ±1 such that for every square (wi, w2, Wg, w4)
the product of numbers assigned to the four arrows occurring in it is equal to -1.
Now, recalling that every intertwiner cf>w.,w, is a multiple of the canonical embedding tw.,w, we can define an intertwiner d<il: M1il~ M1i+Il by its components
l(wi)=-i, l(w2)=-(i+1)
(2·43)
The above mentioned results ensure that d<i+IldUl=O.
The proof of (2·42), which we will not reproduce here, can be given directly 49 l or,
more easily, by establishing an isomorphism to the so-called weak resolution.77l D
Let us illustrate the usefulness of this resolution by giving a derivation of the
Weyl character formula for an irreducible highest weight module with dominant
integral highest weight A. Using the algebraic Lefschetz theorem (1·4) we obtain
=
~
~
WEW
(- 1)t<wl TrM... e2>riB·h=
~
~
WEW
21ri8•(W*A)
(- 1)t<wl II e (1 e 21rr8•a)
.
.
aeA+
-
Using the Kostant partition function K( · ), defined by
(2·44)
Free Field Approach to 2-Dimensional Conformal Field Theories
81
the Weyl character formula immediately leads to Kostant's formula for mA(A), the
degeneracy of a weight A in the irreducible representation LA,
(2·46)
Twisted Verma modules, realizations and resolutions
Twisted Verma modules were introduced by Feigin and Frenkel 50l as slightly
better finite dimensional analogues of the Fock space modules (Wakimoto modules)
of affine Kac-Moody algebras to be discussed later. In this section we will discuss
their free field realizations, intertwiners and resolutions. The discussion closely
parallels the one of the untwisted case as given in the previous sections.
Instead of defining twisted Verma modules through their cohomological properties50l we will define a twisted Verma module MAw of highest weight A, for any given
Weyl group element wE W, by the following properties:
i) MAW is free overCU(n- wn n-) and co-free overCU(n- wn n+), where n- w= w· n-· w- 1•
2.4.
ii)
(2·47)
The second property is essentially the statement that MAw is generated by one
vector, namely the highest weight vector VA. Note that with these conventions MA 1
~ MA and M:t 0 ~(MA)*, where Wo again denotes the longest Weyl group element. For
convenience we will also introduce the notation FAw=M:two (FA 1 =FA), and refer to
these modules as twisted Fock space modules.
Realizations of these modules on polynomial spaces are obtained as follows.
First, we construct a module which is free over CU(n- w) by Weyl rotation of the Verma
module. Thus, choose the vector VwA satisfying n+ wVwA =0 and build the g-module
CU(n- w)VwA. Identify the elements in a PEW-basis with monomials in rwa, aELh
through the replacement Ia w= w · Ia • w- 1 ~ rwa. Then determine the induced action on
these Fock space monomials, replacing wA....., pw in the result. (In practice, since all
n- w are isomorphic we obtain the results by renaming the realization computed for
one case, say w=l.) Of course, this is not yet a realization with highest weight. To
achieve that we simply replace rwaH /3-wa, /3waH- r-wa for aE w- 1(LL) n Lh, and also
shift pw~p=pw+(wp-p). Now consider the new result acting on the Fock space
generated by the ra's on the vacuum lA> such that P;IA>=kiA>. As a consequence
of the replacements above this is a realization on a g-module with highest weight lA>
which is no longer free over CU(n- w), but instead free over CU(n- wn n-) and co-free
over CU(n- wn n+). The shift pw ~ p is required so that the highest weight of this
module will be equal to A. To see this explicitly, note that on CU(n_ w)VwA we have
(2·48)
After changing rwa ~ /3-wa, /3wa ~- r-wa,
h;=a;V•PW-
~
aew-•(4-)n4+
vaE w- (LL) n Ll+ we have
(wa, a/)/3-war-wa+
1
~
aew-•(4+)n4+
(wa, a;Y)rwa/3wa
82
P. Bouwknegt, ]. McCarthy and K. Pilch
(2·49)
where we used78 >
L:
aew(Ll+)nil-
a=wp-p.
(2·50)
Thus, to make it into a realization with highest weight A, we simply have to shift as
above-of course not only in the generators h,, but also in the generators of n- w.
Note that after this procedure the realization of h becomes independent of the
particular twist w. Thus the character of the constructed module is clearly given by
(2·47), so we have indeed constructed a realization of MAw as required.
Example 2.8. Let us illustrate the above for the twisted module MJi'r
=su(3). We have nr 2 r'={e3, A e2}, and thus (see Example 2.1)
2
~ FJ,t
of g
e1 = -(a1. p)(:J1- r1 (:Jl (:11 + r2 (:13,
/1 = r1 + r3 (:12 ,
J2= _ (a2•P)r2+ (a1•p)( r2 + r3 (:11) + r2( _ r1 p1- r2 (:12- r3 (:13) + r1r3 p1 p1 ,
Ia= -(a3· P)r3_ r1r2_ r3(r1 (:11+ r2fJ2+ r3fJ3)'
(2· 51)
while the generators h; are the same as in Example 2.1.
The construction of the intertwiners between twisted Verma modules MAw again
proceeds in several steps. First, using exactly the arguments of the untwisted case,
one establishes a 1-1 correspondence between Homv<n-•)(CU(n- w)VwA, CU(n- w)VwA') and
CU(n- w). The intertwining properties with CU(t) then imply that every such intertwiner is a polynomial in the screening operators
(2·52)
To go to the free field realization of intertwiners between twisted modules we now
make the replacements ywa H p-wa, (:Jwa H - y-wa for aE W- 1(L1-) n L/+ and pw----+p= pw
+(wp-p). Then with the same notation after replacement, we have
(2·53)
which commute with n- wffit by construction, and further,
[e;w, Sjw]=-8.:;(p-(wp-p), wa;V)e'wa,.q.
(2·54)
The proof of (2·54) parallels that given for the Verma module case, up to the
replacements described above. The replacements can easily be effected as a similarity transformation between operators, thus it is clear from the above discussion that
all intertwiners between twisted modules may be found as polynomials in the s,w.
In fact, we have
Free Field Approach to 2-Dimensional Conformal Field Theories
2.6. There exists an
Homv(u)(Mw-I*A, Mw-I*A').
THEOREM
isomorphism
83
between Homv<ulMAw, M:f,)
and
Proof Through the similarity transformation on the screening operators that compose the intertwiners Homv(u)(CU(n- w)VwA, CU(n- w)VwA') and similar reasoning as in the
untwisted case (see Theorem 2.2), one establishes a 1-1 correspondence between
Homv<ulMAw, M:f,) and n+ w-singular vectors in CU(n- w)VA'-<wP-P> of weight A -(wp- p)
(recall the shift in A!). These, in turn, are in 1-1 correspondence with n+-singular
vectors in the Verma module CU(n-)vw-l*A' of weight w- 1 *A by a Weyl reflection. D
Again some intertwiners are easily constructed explicitly. We claim that for m
=(A+ p, wa/)EZ+ we have (s,w)mEHom 'U(u)(M J!:,.,M, MAw). Clearly (s,w)m maps
MJ!:,.,M to MAw since
(2·55)
so the statement follows from
=0.
(2·56)
To formulate the analogue of Theorem 2.4 we introduce a "twisted length lw" on
Wby
(2·57)
where the first term on the r.h.s. is suggested by Theorem 2.6 above-in particular, the
elementary intertwiners (s,w)m decrease the twisted length by one unit. The second
term is chosen to insure the "normalization" lw(1)=0, V wE W. Note in particular
that Mw)=l(w) and lw(w)= -l(w).
Moreover, it follows from woLI±=Lh: and Lemma A.2 that lwwo(w')= -lw(w'), thus
relating twisted lengths relevant for twisted Verma modules and twisted dual Verma
modules. We proceed with the definition of a "twisted Bruhat ordering" on W.
First define w'~ww" iff there exists an aEL/+ such that w'=rwaw" as well as lw(w')
=lw(w")+l. Then define w'sww" iff there exist W1, ···, WkE W such that w'->wwl~w
···~wwk~ww". Then noting that w'sw" iff w- 1 w's_w- 1 w" we find
THEOREM
2.7.
For AEP+ we have
if w'sww"
otherwise.
(2·58)
We should emphasize however that, contrary to the untwisted case, the inter-
84
P. Bouwknegt, ]. McCarthy and K. Pilch
twiners that exist for w' s ww" are no longer injective as a consequence of interchanging the role of ra and /3a for various roots a.
Example 2.9. As an example we take g=su(3), AEP+ and a twisting by w= rzr1 (see
also Example 2.8). We find the following directed graph of intertwiners
MJ.•n
MT2T!
rt*A
~
?
Mr2r1
r2*A
\,
X:
\,
?
Mi~~!*A
~
M~~~i*A'
(2·59)
M~~~!r1*A
where again the diagram is commutative up to proportionality and every intertwiner
between modules of the form M:;,•n is proportional to a composite of the ones depicted
above. The twisted Verma modules M:;,•,n occurring along a vertical line all have the
same twisted length lr.r/w) and the length differs by one unit for consecutive vertical
lines.
Again, the intertwiners corresponding to reflections in other than simple roots can
be found by making repeated use of Lemma B.1, which simply amounts to replacing
s, by s,w in the expressions for the intertwiners in the untwisted case.
Finally, a resolution of LA for AEP+ in terms of twisted Verma modules MAw is
obtained by combining the intertwiners of Theorem 2.7 exactly as in the untwisted
case. Let
(2·60)
We have
2.8. For AEP+ we have a resolution of LA zn terms of twisted Verma
modules, i.e. a complex (MAw, d)
THEOREM
(2·61)
with
if i=O
otherwise,
(2·62)
Proof By using the fact that W1 ~ wWz iff w- 1W1-> w- 1wz we directly obtain from the
previous section that for lw( w1) = lw( wz) + 2 the number of elements w' E W such that
W1 .... ww' .... wWz is either zero or two. In case there are two such elements, w' and w"
say, let us call the quadruple (w1, w', w", wz) a w-square. To every arrow W1 ~ wWz we
can now assign a number sw( W1, wz) = ± 1 such that for every w-square the product
assigned to the four arrows occurring in it is equal to -1, simply by choosing sw( w1,
wz)=s(w- 1w1, w- 1wz), where the s(·, ·)are given in§ 2.3. Now for W1->wwz let
Free Field Approach to 2-Dimensional Conformal Field Theories
85
Zw-•w,,w-•w, be the canonical embedding of Mw-•w•*A into Mw-•w•*A written in terms of
screening operators s,. We have seen that by replacing s, by s,w we obtain the
corresponding element r/Jw,,w, in Homv<u>(M~•*A, M:g•*A). Now choose
lw(WI)=- i, lw(wz)= -(i + 1)
(2·63)
Then d<•+I>d<i>=o. Given the weak resolution in terms of twisted Verma modules 50>
the cohomology (2 · 62) can be proved by establishing an isomorphism of the weak
resolution to (2·61), exactly as in the untwisted case (see Ref. 77) for details).
0
Finite dimensional coset models
In this section we will discuss finite dimensional G/H coset theories, which simply
correspond to the decomposition of finite dimensional irreducible representations of a
semisimple Lie group G with respect to its subgroup H. The goal is to illustrate how
the free field resolutions introduced in § 2.4 can be used to calculate the branching rule
multiplicities for finite dimensional algebras. Our approach is an extension of a
rather well-established method for computing these multiplicities (as explained e.g. in
Ref. 79), Ch. 18), and provides a cohomological interpretation for the corresponding
"branching function" formula. 80 >
Suppose G is a semisimple Lie group and LAG a finite dimensional irrep with
highest weight A. We assume that the embedding of H in G is regular, and choose
the triangular decomposition of algebra h=n_HffitHffin+H to agree with that of g,
i.e. tH c t G and n± Hc n± G. Denote by Z the centralizer of H in G. As a Z X H
module,
2.5.
(2·64)
where the sum runs over a finite set of dominant integral weights of H. The problem
is to determine the multiplicities bA,A'-defined as dimension of the space L~~~,-with
which a given H irrep VJ, occurs in the decomposition (2·64). Clearly, L~~~, is the
subspace of H-singular states (annihilated by n+ H) in LAG, with the H-weight A'.
The following theorem gives a characterization of L~~~, in terms of a (twisted-)
resolution of LAG· [Recall that a twisted Fock space FAw corresponds to the twisted
Verma module MX'wo.]
2.9. Let (FAw, d) be a resolution of the irrep LAG of G given in Theorem
2.8. For each FX'(i>, denote by S:f.Sil the subsPace of H-singular states with the
H-weight A'. Then
1. d: S:f.Si.l-> S:f.Sit 1 >, i.e., (S:f,A', d) is a subcomplex of (FAw, d).
2. The cohomology of the subcomplex (S:f,A', d), with A' a dominant integral
weight of H, is
THEOREM
)- s-z,OLGIH
H d<il(Sw
A,A' -u
A,A'.
(2·65)
We will often refer to complexes (FAw, d) and (S:f,A', d) simply as the complex
and the subcomplex, respectively.
86
P. Bouwknegt, J. McCarthy and K. Pilch
In§ 2.4 we saw that an important property required from any resolution was that
the cohomology of the corresponding complex was nontrivial only in one dimension.
This allowed us for instance to calculate the dimension of LAc using the algebraic
Lefschetz theorem. Part 2 of Theorem 2.9 asserts that the same holds for the
subcomplex (S~.A', d), which thus provides a resolution of the "coset module" L~~;t•.
Proof The first part of the theorem is quite obvious, since the generators of g-and
so, in particular those of n+Hcn+c-commute with d. Thus for any H-singular
vector¢, n+ H· ¢=0, we haven+ H · d¢=0, and d¢ has the same H-weight as¢. In fact
this shows that for any H-weight A' (not necessarily dominant!) the subspace S~.~~ is
mapped by d into S~.~t 1 l, or, equivalently, that (S~.A', d) is a subcomplex. Also, we
have
s~.~~=
{W'E
EB
s:g.*A,A' ,
Wllw(W')=i}
(2·66)
where s:g'*A,A' is the subspace of H-singular states with the H-weight A' in the twisted
Fock space F:g'*A·
A priori, unlike in the case of the complex, the cohomology of the subcomplex
need not be concentrated in the 0-th dimension. For, although it is true that any
closed element tj;ES~.~~. d¢=0, of the subcomplex is also a closed element of the
complex, its cohomology class in the subcomplex may be nontrivial-even if it is
trivial in the complex. That is, even if tf;=dx, xEF:t<•-1), we may not be able to find
fES~.~;-ll such that tf;=di.
But, before we prove part 2 of the theorem in general, let us consider an example.
Example 2.10. rank H=rank G, H=SU(2)X U(l)N, A'=(ja, a)
We will show that for j-::::.0 the cohomology of the subcomplex can be nontrivial
only in the 0-th dimension. Let (e, h, /) denotes the standard basis in su(2)ch.
Thereto consider ¢ES~.~~ whose cohomology class in the complex is trivial, i.e. tf;=dx
and e¢=0. The question is whether one can find a deformation ox such that dox=O
and i=x+ox is H-singular. Assume thus ex=#=O and observe that, since eEn+H
C n+ c has positive G-weight and the set of weights in F:t<il is bounded from above,
there exists a smallest n"2::.1 such that enx*O and en+lx=O. For such x we can
construct ox as follows : Consider a one parameter family of vectors of the form
xr(t)= x+ t/ex, tER. Clearly, dxr(t)=¢ because d/ex= /e¢=0. Moreover,
(2·67)
where we used the identity [en,!]=nen- 1(h+2), and hx=2jx. We see that for j-::::.0
one can always choose t=tl=-l/2n(j+l), such that enxl(tr)=O. Repeating this
process n times we obtain
(2·68)
This shows that for j-::::.0 the cohomology class of¢ in the subcomplex is nontrivial if
and only if it is nontrivial in the complex, thus proving that H~il(S~.A·)CH1P(FAw).
87
Free Field Approach to 2-Dimensional Conformal Field Theories
One may note that the r.h.s. of (2·68) coincides with the projection of x onto the
eigenspace of the quadratic Casimir operator Cz=(l/2)h(h+2)+2/e of su(2). This
suggests a general proof along these lines.
To streamline the proof in the general case it is convenient to introduce a notion
of projective modules. 47> We say that a module Pis projective in some category of
modules if, given any two modules N and N and a surjective morphism 6: N--> N, we
can "lift" any homomorphism rp: P--> N to a homomorphism ip : P--> N such that rp
= 6° ip, i.e. the following diagram is commutative :
(2·69)
p -----. N
We will extensively use the following standard fact about the category (') of
modules. 49 >
g
2.10. For any finite dimensional semi-simple Lie algebra g, Verma modules
MA, where A is a dominant g-weight, are projective in the category (') of g-modules.
THEOREM
Proof 81 > Consider arbitrary modules N, NE ('), an epimorphism (i.e. a surjective
morphism) 6EHomv(ul(N, N) and a homomorphism rpEHomv<ulMA, N). The image
of the highest weight state VA EMA under rp is a singular vector v= rp(vA) in the module
Nat weight A, and thus an eigenvector of the universal quadratic Casimir operator
CzECU(g) with an eigenvalue cz(A)=(A, A +2p). By diagonalizing Cz on N, we can
find a vector fJEN such that 6( v)=v and Cz v =cz(A) v. Consider the set {CU(n+)· v}
eN. Note that because the set of weights in N is bounded from above (N is a
module in the category (')) this set must have at least one singular vector, v, with the
weight A+ ,8, where ,8= z;.;n;a; is a linear combination of simple roots with nonnegative integral coefficients. Clearly, we have Cz v = cz(A) v because v is in the
submodule generated by v ; but also, Cz v = cz(A + ,8) v because v is singular. We
obtain
(A, A +2p)=(A + ,8, A+ ,8+2p)
=(A, A +2p)+2z;.n;(p,+a,)+ z;.bi,jn;nJ,
Z
(2·70)
Z,J
where p;=(p, a;) >0, a;=(A, a;) are non-negative when A is dominant, while the
matrix (bii)=((a;, aJ), related to the Cartan matrix (aii) by bii= -}(a;, a;)ai,j, is
positive definite. Thus we see that all n,'s must vanish, i.e. ,8=0, which implies that
in fact v = v. Having shown that vis singular, we define ip by setting ip(vA)= v and
extending ip to the entire MA so that it is an CU(g)-homomorphism. It is obvious that
ip is a sought after lift of rp.
0
We now return to the proof of the theorem. First we will show that H~0 (S'i,A·)
We proceed as in the example. Let <P=dx, <f!ES'f(,<j,) and xEF:t<•- 1>.
cH~0 (FAw).
88
P. Bouwknegt,]. McCarthy and K. Pilch
Consider H-modules Nand N generated by x in Ff<•- 1>and rf; in Ff(i>, respectively.
Then d restricted toN maps onto N. Also it is clear that Nand N belong to the
category () of H-modules. Since rf; is H-singular, there exists a unique homomorphism cp from the Verma module MA' of H onto N, which, using Theorem 2.10, can be
lifted to a homomorphism cp: MA'--+ NcF:<•- 1>. Then i= cp(vA') is H-singular and
di=dcp(vA')=cp(vA')=rf;, which shows that cohomology class of rf; vanishes in the
subcomplex.
To prove the equality in (2·65) we must still show that any H-singular element
of LAG has an H-singular representative in Ff<0 >. In general, one can only expect that
if vELAG isH-singular then its representative rf; in Ff<0 >, drf;=O, v=[rf;], satisfies earp
=dx, aEL1+ 8 , where xEF:f<•- 1>will in general depend on a and rf;. [Of course, this
problem does not arise in the case of the untwisted resolution!] Let us now takeN
to be an H-module which consists of elements in Ker(d< 0 >) that project onto the
H-submodule N in LAG generated by v. One easily verifies that once more we are in
a position to use Theorem 2.10 to deduce that there exist ¢ E S~f,<J.l such that [ ¢] = [rf;].
This concludes the proof of Theorem 2.9.
D
We have shown that the complex (SJ:,A', d) provides a resolution of the coset
module L~~;[,. The reader will observe that if His abelian the difficulties discussed
above do not arise, because in this case the subcomplex is simply a restriction of the
complex to a given H-weight (or better, charge). The cohomology of the subcomplex
is then nonvanishing only for i=O and coincides with a subspace of LAG which consists
of states with the H-weight A'. Also, it follows from the above remarks that if we
restrict the Ff(i>•s to the states with a given H-weight A', not necessarily dominant,
we will obtain a subcomplex (F;f.A', d) with a nontrivial cohomology only at i=O. It
is only the further restriction of this subcomplex to H-singular subspaces that the
dominance of A' becomes important.
When His abelian, the spaces SJ:,A' occurring at each step of the resolution are
simply a restriction of the free field Fock spaces to a subspace of definite charge.
However, when H is nonabelian, a further projection onto H-singular states yields
subspaces for which a parametrization in terms of free fields is less obvious. In the
present case of finite dimensional coset models there are two methods to accomplish
the projection onto the subcomplex of H-singular states within the context of free
fields. The first one is to use the well-known BRST techniques, and further extend
the complex by introducing ghost oscillators. The second one, which seems to work
only for the resolution in terms of non-twisted Fock spaces, is to explicitly solve for
the H-singular states, and show that they correspond to free Fock spaces of certain
new fields. As we discuss at the end of this section, in the finite dimensional models
the two methods are equivalent. However, it seems that only the first method has an
infinite dimensional generalization. Thus we will discuss it in more detail now.
Introduce a set of IL1+ 8 ! conjugate pairs of free fermionic "ghost" oscillators
(c-a, ba), {c-a, bP}=8a.o, a, j3EL1+ 8 • The corresponding Fock space Fgh is generated by
the c-a•s acting on the vacuum !O> which is annihilated by all the ba's, ba!O>=O. Of
course, for a finite number of ghost pairs, Fgh is finite dimensional. One can introduce
a gradation in the ghost Fock space Fgh=E.BnF~~>, where F~~> is the eigenspace of the
89
Free Field Approach to 2-Dimensional Conformal Field Theories
ghost number operator Ncb= ~aELI•• c-aba. For an arbitrary H-module M we consider
a module M®Fgh which is graded according to the cb-ghost number, (M®F8 h)<n>
=M®F~~>, in particular M®F~g>~M. Then we may construct a nilpotent operator,
Q: M®F~~>~ M®F~~+l>,
(2·71)
where ea acts on M®Fgh as ea®l.
It is clear that the cohomology H~0 >(M®Fgh) is isomorphic to the space of
H-singular vectors in M, in fact Q is the BRST operator associated with the constraints ea=O, aELhH. To see how this works let us consider an example.
Example 2.11.
h=su(2), M=LJa, j>O.
In the above example we have one set of ghosts (c, b) and Fgh is spanned by two
states, IO> and ID=ciO>. The BRST operator is simply Q=ce. The closed states
are of the form lj>®IO> and LJa®ID. From those only lj>®IO> and 1- j>®ID are not
in the image of Q. Thus we find that both H~0 >(Lj®Fgh) and H~1 >(Lj®F8h) are
!-dimensional.
Note that if we replace LJ by the Fock space F., the second cohomology class
"disappears" because the state 1- j) in F., is in the image of e and thus of Q.
The idea now is to use Q to project onto the subcomplex of the resolution.
Consider first the resolution (FA, d) of an irrep LAG, and (FA,A', d) the subcomplex
with an H-weight A'. Given the H-ghost Fock space we may form a double complex
(FA,A'®Fgh, d, Q) in which d acts as d®l and Q is given by (2·71). The spaces in
this complex are labelled by two degrees, i and n, F~~~,tg;F~~>, where i=O, ···, l(wo)
and n=l, ···, 2 LJ+H As discussed in Appendix C, to such a double complex we can
associate a single complex
1
1•
(KA,A', D),
D=Q+(-l)Ncod.
(2·72)
The main result of this section is
2.11. Let (FA, d) be a non-twisted resolution of an irrep LAG, and A' a
dominant H-weight. Then the cohomology of the complex (KA,A', D) associated with
the double complex (FA,A'®Fgh, d, Q) is
THEOREM
(2·73)
Proof The proof follows from Theorem C.l in Appendix C and the following technical result which is proved in Appendix D.
2.12. Let FA be a non-twisted Fock sPace G-module with the highest weight
A (not necessarily dominant), and SA the sPace of all H-singular states in FA. Then
the ERST cohomology of FA considered as an H-module is given by
THEOREM
P. Bouwknegt, J. McCarthy and K. Pilch
90
(2·74)
Observe that in the double complex (FA,A'®Fgh, d, Q) all the columns complexes,
labelled by n, are identical and their cohomology simply reproduces the H-weight
space L~,A' of LAG· On the other hand the cohomology of the row complexes, labelled
by i, using the above theorem reproduces the subspaces of H singular vectors. In the
notation of Appendix C we can summarize this as follows :
tO.F, )- N,OLGA,A'\CI
tO.F,(n)
H ·(z,n>(F
d
A,A'\01 gh - U
gh
,
Vn,
(2·75)
(2·76)
Let us restrict A' to be a dominant H-weight. On introducing the associated
single complex (K, D) we can use Theorem C.1 in Appendix C to determine its
cohomology. In fact we can compute it in two ways, first with respect to Q and then
d, or other way round. The result is that the nontrivial cohomology can only occur
in degree leSS than min(l(wo), 214 1), and iS given by
•H
(2·77)
or, equivalently,
(2·78)
The first result (2·77) proves that the complex (K, d) is a resolution for the coset
module L~~;[,. This completes the proof of Theorem 2.11.
D
For a twisted resolution (FAw, d), we introduce a double complex (FAW.A'®Fgh, d,
Q), where the (i, n) space, i=O, ·--, l(wo) and n=1, ···, 2w 1, is equal to FJ,~;-t<w»rg;p~~>.
[We shifted the labelling in i by -l(w) to bring the double complex to the first
quadrant.] Let (Kf.A', D) be the associated single complex. We then have a
generalization of Theorem 2.11.
2.13. For any twisted resolution of LAG and a dominant H-weight A', the
cohomology of the single complex (Kf.A', D) associated with the double complex (FX:A'
®Fgh, d, Q) is
THEOREM
(2·79)
Proof First note that the argument leading to (2·77) in the proof of Theorem 2.11 is
not valid if we take instead a twisted resolution. The reason is that the cohomology
of Q on a twisted Fock space need not be concentrated in a single dimension.
However, for the double complex (FAW.A'®Fgh, d, Q) we still have an analogue of
(2·75),
(2·80)
Moreover, (2·78) suggests that if we take another resolution of LAG then the cohomology of the corresponding double complex should not change. In fact using
Theorem C.1 we obtain
Free Field Approach to 2-Dimensional Conformal Field Theories
if p< l(w),
91
(2. 81)
and otherwise
(2·82)
But then comparing with (2·78) and (2·77) we see that (2·82) is nontrivial only when
p=l(w) and is equal to the coset module as stated in Theorem 2.13.
D
In the non-twisted case there is an explicit parametrization of the resolution
(SA,A', d) in terms of "perpendicular" Fock spaces. In § 2.1 we explained that there
exists a natural parametrization of the non-twisted Fock spaces Fw*A in the resolution
of LAG, such that the generators of n+ 8 are realized only in terms of r,,a, !3 11 a, aEL1+ 8 •
[Theorem 2.14 below constitutes the general discussion of this parametrization for the
case G x G/G.] With this choice all states independent of r,/, aELl+ 8 -i.e. the subspace Fil,M generated on lw *A> by r a, aELJrtH-are manifestly H-singular. In
Appendix D we are then able to prove the following sharpening of Theorem 2.12.
j_
THEOREM
2.12'.
If the generators ea of
n+ 8 are realized only in terms of parallel
variables ra, !3a, a ELl+ H then
(2·83)
Thus SA,A' coincides with the subspace with the H-weight A' of the Fock space
FA.l_ of perpendicular variables, and (Fl.A', d) is thus a resolution of the coset module
L~~;f,.
We conclude this discussion with a simple application of the resolution (FA,A', d,
Q) and/or (Fl.A', d) of L~~~, to compute multiplicities (branching functions) bA,A'·
Example 2.12. rank H=rank G
Let us decompose the set of roots of G with respect to Has Ll+ G=Ll+ 8 UL1r 18 , Ll+ 8
nLJrtH=0. The calculation of multiplicities is most easily performed in two steps.
First we use the Lefschetz formula in the complex (FA®Fgh, Q), where A is an
arbitrary G-weight, to derive the character of SA, the space of all H-singular vectors
in FA. Identifying weights of G with those of H, we obtain using (2·47)
21.oi+HI
chsA= L: ( -1)n chFA(
n=O
L:
npe-P)
{n,=O,ll:2::,e4,.n,= n)
(2·84)
where KG 1u( ·) is the Kostant partition function on the lattice spanned by the coset
roots LJrtH. We read off from (2·84) that
92
P. Bouwknegt, ]. McCarthy and K. Pilch
dim SA,A'=KctH(A- A').
(2·85)
Then from the resolution (SA,A', d) we obtain for the multiplicity
bA,A'=dim L~~X,=dim H~0 >(sA,A')
(2·86)
which is the branching function formula derived in Ref. 80) in the case of equal ranks
and a regular embedding.
Observe that we could have also derived (2·85) directly using the realization of SA
as a Fock space of perpendicular variables FAl.·
We will now show how to choose the "perpendicular" variables for the coset
G X G/G, G simple, and explore more carefully the consequences of doing so. As
usual we will distinguish between the groups by subscripts or superscripts in round
brackets. As discussed in § 2.1 for arbitrary G/H coset, the appropriate PBW -basis
of CU(n- G) in the untwisted Verma module realization is that respecting the decomposition CU(n-H)®CU(n9.'H). Specifically for G X G/G we will take the basis
(2·87)
where Pa, Qa are non-negative integers and the order of the roots in the products is
specified as in (2 · 7). Identify this with a Fock space basis via
(2·88)
One may now derive the realization of the generators in these coordinates via the
action on this basis. There are several salient features of the result for arbitrary G
which should be emphasized. One is just that the diagonal subgroup generators are
realized in terms of parallel variables only-which is of course their raison d'etre
-since the action of these generators from the left obviously does not disturb the
perpendicular variables. Similarly, since the screening charges correspond to the
action of n- from the right, those of c<o are purely expressed in terms of perpendicular variables.
Both of these remarks apply immediately to the untwisted Fock space realization
obtained by conjugation of the above as described below Example 2.4 in § 2.1. Our
final observation is specific to the untwisted Fock space resolution. When acting on
states in the subspace F}m,A'"'' the screening charges of G< 2 > can be identified with the
negative root generators of c< 1>; i.e. s~2 >=- /J1> on this subspace. This follows from
the discussion above since the action of /J2> from the right can be commuted past the
V(n9.'H) factor in the PEW-basis until it is to the right of the U(n-H) factor. Here
we may write it as /J2 >=(1J1>+ /J2 >)-/J1>. The first term goes to the left and increases
the degree of the subgroup generators in the basis, giving terms with non zero powers
of r11 in the realization. The second term can be taken to the right, and of course is
just the action of - /J1> on purely perp states. On conjugation the first set of terms
have non-zero powers of /3 11 and thus vanish on the subspace Ff"',A'"'• leaving the
Free Field Approach to 2-Dimensional Conformal Field Theories
93
desired result
The following criterium is a consequence of our discussion.
THEOREM
LA/~LA.
2.14. 79 > Let LA 1 and LA. be two irreps of G. The degeneracy of LA. in
is equal to the number of independent solution to the equations
(2·89)
i=1, ···, f,
where vELA1 has weight Aa- A2.
Proof The proof is left as an exercise to the reader.
2.6.
D
Finite dimensional vertex operators
This section contains a study of the finite dimensional analogue for the free field
representation of chiral vertex operators, which provides an algebraic setting for
computing the Clebsch-Gordon (C-G) coefficients of Lie algebras.
Given three dominant integral weights A,, i = 1, 2, 3, of a semisimple Lie algebra
g we introduce the formal "chiral vertex operator" as a set of operators {<l>v: LA1
~LA., vELA.}, which transform according to the irrep LA. under g, i.e.,
[x, <Z>v]= <Z>xv,
xEg.
(2·90)
It is clear that, given the standard inner product, matrix elements <vai<Z>v.lvl>, V;ELA.,
are proportional to the corresponding C-G coefficients. Further, {<Z>vlvELA.} is
specified by any of its components <Z>v, and any non-zero matrix element of this
component fixes the normalization of the vertex. One choice of the component used
to represent the vertex might be more convenient depending on the method in which
the irreps of g are constructed. For instance, in the context of twisted modules, we
note that the component <Z>w<vA,h where wE Wand VA. is the highest weight vector of
LA., will be particularly convenient since it commutes with all the generators of n+ w.
Suppose irreps LA1 and LA. are given in terms of free field resolutions (FAW.., d) and
(FAW., d'), respectively, in the class of twisted modules defined by the choice of wE W.
In the following we will study the free field representation of <l>w<vA•h which for
convenience we will simply denote by a>. Given a mapping between cohomology
classes of two complexes, it is a natural problem to look for a representative as a
mapping between the complexes. 48 > In the present case such a representative is given
by V: (FJl,., d)~(F)f., d'), defined as a collection of maps v<n: F~;{~F~;;, i=-l(w),
···, l(wo)-l(w), with components v<il={ Vwa,Wi EHom 'll(n.•)(Fwi*Ai, Fw.*A.), lw(wl)
=lw(wa)=i}, satisfying
d'(i) vu>= v<•+l)d(i)'
i= -l(w), ···, l(wo)-l(w)'
(2 ·91)
such that the map induced on the cohomology classes agrees with a>, i.e. v<o>iL•. =a>.
It is an easy exercise to check that (2·91) guarantees that V induces a well-defined
mapping on the cohomology classes. Clearly not all mappings between complexes
which satisfy (2·91) induce a nontrivial mapping on the cohomologies. For that
reason we say that a vertex V is trivial if there exits f={fu>EHom 'll(n.•)(F :flu>,
F:f.<•-l>)} such that
94
P. Bouwknegt,]. McCarthy and K. Pilch
(2·92)
In particular one is interested in computing the dimension N:l[A. of the vector space
of vertices V, modulo trivial vertices, for three given weights as above. Finally, the
representation of other components of {(].)v} can be obtained by the action of the
algebra.
From §§ 2.2 and 2.4 we know that the components Vws,w of the vertex, Vw.,w,
EHomvcn.wlF~.M., F~•*A.), are exactly the operators which can be built as
polynomials in screening operators, s;w. To ensure the correct weight, wAz, under
transformation (2·90) we must also have an appropriate translation factor. Thus we
are led to consider operators of the form
1
e•wAz·q X
[polynomial in screenings],
(2·93)
*
*
where the polynomial is homogeneous of degree W1 A1 + wAz- W3 A3 in the screenings. We will call them "screened vertex operators". Further, from the discussion
in § 2.4 it is clear that the screened vertex operators for one class of twisted modules
are operationally constructed from those of another by a set of invertible transformations. Thus without loss of generality it is sufficient to develop the theory for
untwisted modules, which we will do unless otherwise stated.
Example 2.13. su(2) vertices
Consider untwisted modules, w=1, giving resolutions of su(2) irreps with highest
weights ha and ha. The representation of a vertex (])is is V = {v<o>, v(l>} such that
the following diagram is commutative
8 zh+I
0-----+
Fjz
-----+
J. v<•J
F-jz-1-----+
0
(2·94)
J. voJ
s2is+l
0-----+
Fia
-----+
F-js-1
-----+ 0 .
A priori,
(2·95)
which requires that
If h + jz- js-:?.2j1 + 1, we can factor s 2M 1 from v<o> which is then trivial.
Thus we must
have
jz- j1~h.
(2·97)
However, in order to solve (2 · 91) for v<l) we need (2h + 1) + U1 + jz- j3) 2 2j1 + 1, i.e.
(2·98)
Clearly (2·96), (2·97) and (2·98) reproduce the tensor product rules for su(2).
By their explicit form and the fact that the algebra of screening operators is
Free Field Approach to 2-Dimensional Conformal Field Theories
95
isomorphic with n-,*> there is clearly a natural identification of components of
screened vertex operators with elements of Verma modules, via
(2·99)
Using this identification, we are able to study the free field representations of the
vertex <P using the BGG resolution for an irrep which we discussed in § 2.3. In
particular we have the "extension property" as embodied in
THEOREM 2.15.
Given v<o> and V<1l such that
(2·100)
and
(2·101)
the remaining components v(i>, i=2, ···, l(wo), are uniquely determined by (2·91), up
to addition of trivial pieces of the form (2·92).
Proof The identification (2·99) restates (2·100) as a relation between states in MA,,
namely that d'<o> v<o> is an element of the Verma submodule M1;:1l(see (2 ·40)).
Similarly, if we act on (2·100) with d'< 1>from the left we obtain
(2·102)
which can be interpreted as the statement that d'< 1>v<I> is closed in M~~~> of the BGG
resolution, and thus is exact. That is, there exists a v<z>EM~~z> such that
(2·103)
In the same way we may compute all of v<n, i=3, ···, l(wo). Clearly, solving (2·91)
using the EGG-resolution is unique only up to "trivial elements", which add to a total
ambiguity of the form (2·92).
D
The question of constructing a nontrivial vertex then reduces to the problem of
finding V(O) and V(l) Obeying (2•100) and (2•101), and then COmputing V(i)' i=2, ···,
l(wo) using (2·91). The following theorem characterizes the space of nontrivial
vertices, v:I~A •.
THEOREM 2.16. A nontrivial vertex operator VE v:I~A. exists precisely when the tensor
product rules are satisfied, i.e., irrep LA, occurs zn the decomposition of the tensor
product LA,0LA •.
Proof Recall that the irrep LA, is the quotient of the Verma module MA, by its
maximal submodule M1;: 1>, and thus we may restate (2·100) and (2·101) as requiring
*>
We will only be interested in the algebraic structure behind the screening operators, and thus the
identification with n+ or n_ is a matter of convenience since both algebras are isomorphic.
P. Bouwknegt, J. McCarthy and K. Pilch
96
that v<o> corresponds to a vector cjJEMA1 which is non-zero as a vector in LA1 at
weight A3- Az, and satisfies the following equations
i=1, ... , f.
(2·104)
Under the action of Wo on LA1 these equations are equivalent to
(2·105)
i=1, ···, f'
where cjJ*=woc/J is an element of LA1 at weight Az*-Ag*, where Az*=-woAz and A3*
= - woA3 are the highest weights of the conjugate representations. We recognize
(2 ·105) as the condition stated in Theorem 2.14 79> under which L~. occurs in the
decomposition of LA/i9L~ •• or equivalently that LA. occurs in the decomposition of
LA/3SJLA..
0
This shows that the dimension N:l:A. of the space of nontrivial vertices is equal
to the multiplicity N:l:A. given by the nontrivial C-G coefficients.
One might expect that it is possible to compute multiplicities N:l~"Az from some
cohomological setup without solving for the vertices explicitly. Indeed, we observe
that that vertices satisfying (2·91) can be considered as elements of a double complex
that arises by the following standard construction :48>
Given two resolutions (FA~> d) and (FA., d') we introduce the double complex
(X, a, a') of maps of the form (2·93),
X ={X<i,j>ixu,j>3x.:;: F5!{-+ F~ij,
i, j=1, ···, t},
(2·106)
The differentials a and a' have degrees (0, -1) and (1, 0), respectively, and they
anticommute, aa'+a'a=O. As discussed in Appendix C we can define the associated
single complex (K, D), where K ={K<P>iK<P>=EBi-j=PX(i·j), P=- t, ... , t} and D: K<P>
-+K<P+l>, D=a+a'.
In terms of the complex K one can interpret Eqs. (2 · 91) and (2 · 92) for the vertex
as follows: Consider Vas an element of K< 0 >. Then (2·91) simply says that DV =0,
whilst (2·92) implies that Vis a nontrivial element of the cohomology of K. The
question is whether (K, D) provides a resolution of the space of nontrivial vertices.
This is answered in affirmative by the following theorem.
THEOREM
2.17.
The cohomology of the complex (K, D) is
Hlfl(K) = oP.Ovj~Az .
(2·107)
Moreover,
(2·108)
where
97
Free Field Approach to 2-Dimensional Conformal Field Theories
Proof Once more by the algebra of screenings we may identify a component Xw,w'
EX<l(w),l<w'))' w, w'E W, as an element of the Verma module Mw'*A• of the weight
w*A3-A2. Then, for fixed w, the column complex in (X, a, a') coincides with the
BGG resolution for the weight space w A3- A2 in the irrep LA, and we obtain
*
(2·110)
The spaces in the complex (L1~.A., a') on the r.h.s. are (recall that we denote by LA(A)
the subspace of LA with the weight A)
(2·111)
The differential a' is well defined on this quotient and is given explicitly in terms of
polynomials in the generators of n- acting on the irrep LA,. Note that complex
(L1~A., a') can also be obtained from the complex of the Fock space resolution of the
irrep LA. by the following formal procedure: Take (FA,, d) and substitute
(2·112)
To compute the cohomology of this complex we will relate it to the double
complex of the Fock space resolution of LA,®LAs* that arises in the G X G/G finite
dimensional coset model, as discussed at the end of § 2.5. Acting with the Weyl
element wo on LA, we obtain the following isomorphism
(L A.
At,A•,
a')~(
- Wo
LA•
A,,A.,
Wo
a' Wo-1) .
(2·113)
The complex on the r.h.s. is obtained from the resolution (FA,*, d') by
(2·114)
We can now represent LA, at each point of this complex as the cohomology of the
corresponding resolution (FA, d). This yields a new double complex in which the
two differentials acting on Fock spaces are constructed as polynomials in the generators, e,, of n+ and polynomials in the screenings, s,, respectively. However, by
Theorem 2.14 and the discussion above it such a double complex is then equivalent to
the subcomplex of G-singular states in the double complex of the tensor product
(FA,®F A,*, d, d') at the weight A2*. For a dominant weight A2, or equivalently A2*,
we know by Theorem 2.9 that the cohomology of the associated single complex must
be concentrated in 0-th dimension. Using Theorem C.1 we can go backwards to the
original complex to obtain (2·107). The dimension formula (2·108) follows then the
algebraic Lefschetz theorem (1·4) and (2·46).
D
Equation (2 ·109) is also called Steinberg formula. 72 ) Nate that the last line in
(2·109) can also be obtained directly provided one knows that the cohomology of the
complex (L1~A., a') is concentrated in a single dimension. In fact comparing the
latter complex with (2 ·100) and (2 ·101) we see that its cohomology corresponds to the
space of the v<o) components of nontrivial vertices.
98
P. Bouwknegt, J. McCarthy and K. Pilch
§ 3.
3.1.
Free field approach to affine Kac- Moody algebras
The Fock sj;ace realization
After this rather lengthy exposition of free field realizations and their resolutions
for finite dimensional Lie algebras we will now turn to the case of affine Kac-Moody
algebras. There probably exist a huge set of potentially interesting realizations on
free field Fock spaces but the specific realization that we are looking for, with the
application to conformal field theory in mind, is a realization where the Kac-Moody
currents themselves can be expressed in terms of a set of conformal fields. The most
straightforward way to obtain such a realization is to attempt to "affinize" the
realizations obtained in the previous chapter.
In principle one can affinize all the twisted realizations. However, since the
associated Fock space modules turn out to be of the same complexity we will try to
avoid confusion by restricting ourselves to affinizing the dual Verma module only.
Thus, the realization will be in terms of a set of conjugate first order bosonic fields
(/3a(z), ra(z)) of conformal dimension (1, 0),82 > one such pair for every positive root a
ELl+, and a set of scalar fields ¢/(z) as many as the rank .t of g. We will take the
following convention for their (nonvanishing) operator product expansions
aaa'
ra(z)(3a'(w)= z- w '
(3·1)
¢i(z)¢/(w)=- oij ln(z- w).
(3·2)
Their mode expansions are given by
ra(z)= ~ rnaZ-n' !3a(z)= ~ !3naZ-n-1'
nEZ
nEZ
¢i(z)=q'-ipi In z+i~ ani z-n.
n*O
n
(3·3)
Let us denote the Lie algebra of oscillators by a. The algebra a admits a Cartan
decomposition a=a-EBaoEBa+, where a-={/3na, n<O, aEL1+}U{rna, n:::;;;o, aEL1+}
U{ani, n<O, i=l, ···, .t}, ao={pi, i=l, ···, .t} and a+={/3na, n20, aEL1+}U{rna, n>O, a
ELJ+}U{an;, n>O, i=l, ···, .t}. For a weight A let FA be the CZ.l(a)-module induced
from a vector lA> satisfying a+IA>=O, pijA>=a+A;IA>. (The factor a+ is inserted for
convenience, and will be determined later.) We will refer to this module as a Fock
space module.
The affinization of the finite dimensional realization consists of the following
steps. Replace 13a~ /3a(z), ra~ ra(z), p;~(I/a+)ia¢;(z) and normal order the expression that one obtains. In the negative root currents e-a(z) add terms of the form
: r··· ror: (z) (of the correct isospin) which have vanishing zero mode piece, such that
the commutators work out, e.g. by requiring the correct central charge term in
e;(z)/;(w). One can prove that this procedure is always possible and moreover
unique. 14 >
The affinization of the Cartan subalgebra generators (2·5) is easily accomplished
Free Field Approach to 2-Dimensional Conformal Field Theories
99
These do satisfy the correct OPE, with central charge k, provided we set
1
(3·5)
The expressions for the other generators of course depend on the basis we have
chosen, so let us give the resulting realizations in some examples (compare Examples
2.2 and 2.3)
Example 3.1.
su (n)
(3·6)
Example 3.2.
so (5)
(3·7)
To be able to make use of the above realization in solving a specific conformal
field theory we should of course compute the stress energy tensor in the above
realization. For the WZNW-models the stress energy tensor is given by the Sugawara
construction, which can be shown40> to reduce to the free field form.
THEOREM
3.1. Let xa, a=l, ···, dim G be an orthonormal basis of
g,
then
P. Bouwknegt, J. McCarthy and K. Pilch
100
The central charge of the corresponding Virasoro algebra is given by
kdim G
k+hv '
(3·9)
Finally, observe that by affinizing the finite dimensional realization we have
constructed a module which is free over CU(iii niL) and co-free over CU(iii n ii+),
where iii=(n+®C[t, t- 1 ])EB(t®tC[t]). This shows already that these Fock space
modules resemble the twisted modules introduced in § 2.4 in a lot of respects. This,
ultimately, was our main motivation for studying twisted Verma modules for finite
dimensional Lie algebras. In fact, the analogy can be pushed further by observing
that translation operators tr, rEM in the affine Weyl group act on the generators of
ii as
(3·10)
So, if we choose some rEM such that (r, a)>O, \laELh then we formally have50 >
iii= lim ii~·.
N-oo
(3·11)
This suggests that we may, as in the finite dimensional case, analyze the Fock space
modules by starting from known results for the Verma modules and "performing an
infinite twist" limN-ootNr· Though this point of view seems to work to some extent (as
confirmed by our results) it does not seem to be helpful in, for instance, the derivation
of the realizations above. The reason is, as one easily convinces oneself, that the
generators in the Verma module realization obtained similarly to § 2.1 will have terms
of arbitrarily large order in the /3r-modes.
3.2.
Intertwining operators
The next step is the determination of all possible intertwiners between the Fock
space modules. Intuitively, one might expect that the set of intertwiners is described
by a condition similar to Theorem 2.7, by making use of the infinite twist limN-ootNr.
This result would be in complete agreement with the analysis that we will present
below, but for similar reasons to those alluded to in the previous section we have not
been able to prove this rigorously. So, the approach we will take here is rather to
study the affinization of the finite dimensional screening operators directly as natural
candidates for the building blocks of the intertwiners. The intertwining property of
certain combinations of screening operators will be established by an explicit evaluation of the commutators as we gave for illustrational purposes in the previous section.
The affinization of the screening operators (2·53) is given by*>
*>
This result, as well as the other results in this section, are also valid for non-simply-laced Lie algebras.
In particular one does not need to introduce additional fermions as is claimed in Ref. 38).
Free Field Approach to 2-Dimensional Conformal Field Theories
s;(z)=p(e;): e-ia.ad: (z),
101
(3·12)
where p(e;)(z) is some polynomial in /3, r-fields, e.g. for su (n) (compare Example 2.5)
(3·13)
One readily checks that the screening operators (3·12) are primary fields of conformal
dimension one with respect to the stress energy tensor (3·8). Contrary to the finite
dimensional case these operators do not generate the algebra ii+, due to the nonlocality of s;(z) with respect to s;(w) for (a;, ai)=I=O_
To evaluate the OPE's of the screening operators with the Kac-Moody currents
one first observes that the OPE of s;(z) with e;(w), h;(w) and /H;(w) at most contains
a term of order 1/(z-w). This follows from the fact that all generators have
conformal dimension one and the fact that there exist no spin-0 operators of the
correct isospin to allow for an order 1/(z-w)Z-term. However, the C/(1/(z-w))
term only receives contributions from single contractions, i.e. ordinary commutator
pieces, and hence vanishes due to the finite dimensional result. Similarly, the only
nonvanishing OPE can be argued to equal
(3·14)
for some c-number a. The c-number piece can be calculated.
Example 3.3. Consider su (n). Since the c-number piece a does not depend on the
choice of basis it suffices to do the calculation for one particular value of i. For i
=n-1(=1) we have (see (3·13))
(3·15)
In this case the C/(1/(z-wY) term arises from the piece -(k+n-2)orn-ln(z)
-a+ 1rn-ln(an-1' io¢)(z) in /;(z) (Example 3.1). One easily computes that this gives a
=(k+n-2)+2=k+n.
The result (in all known examples) is that the two terms nicely combine into
- Vij
i'
2(k+hv)
f(i Z ) Sj ( W ) (
a;, a; )
aw(-1-e -ia.a,•(J(w)) .
z-w
(3·16)
This suggests to try intertwiners of the form*>
iiE{1, ... , f}
*)
(3·17)
Conventionally,•> one likes to think of the "screening charges" Q,=fs,(z) as some operators that can be
arbitrarily inserted into the conformal blocks without destroying the chiral algebra Ward identities.
Though, for practical purposes, this may be a convenient way of thinking about the problem, it is,
strictly speaking, incorrect. The "operators" Q, do not, in general, exist as operators with a welldefined action on the Fock spaces. It is only for certain combinations of screenings that one can find
integration contours that give rise to operators acting on the Fock spaces.
102
P. Bouwknegt, J. McCarthy and K. Pilch
for some suitably chosen contour r. This expression can be written somewhat more
explicitly by the Campbell-Baker-Hausdorff formula
(3·18)
where
"'
~
'l'<<>l ( z ) = - z. "-'
anz -n .
n
(3·19)
n<(>)O
Hence, by acting with (3·17) on a vector vEFA, we obtain a vector given by some
integral whose integrand has monodromy properties determined by the factor
IT
IT
(zk- Zt)a+•(a,.,a,,)
l~k<l~n
Zk -a+•(A,a,.) .
(3·20)
l~k~n
Let us introduce contours Fzo as a set of nested contours taken counterclockwise from
a basepoint Zo to Zo around 0 and nested according to lz1l > ··· > lzNI. The integrand is
defined by the analytic continuation from the line passing through 0 and Zo where it
is taken to be real (see Refs. 7) and 40)). If F=Fzo in (3·17) we will use the shorthand
notation [s;,···s;Jzo- For the discussion of the intertwiners it suffices to restrict
ourselves to say Zo=1 in which case we will omit the subscript Zo.
Having fixed the contour and a set of simple roots a;1 , j=1, ···, N such that /3
= ~iai1 we may still obtain different operators [s;,···sd: FA~ FA-P by permuting the
a;/s.
There are three important lemmas that we will need in the determination of the
subset of intertwiners. The first expresses the fact that not all of the permutations
of screenings give rise to independent operators. It is the analogue of the statement
that in the finite dimensional case screening operators s,, i = 1, · · ·, .t generate an
algebra isomorphic to n+.
Lemma 3.2. Within the contour integrals I: the screening operators s, satisfy the
Serre relations of the quantum group CUq(n+) (see Appendix B) for the value q
=exp(iTC/(k + hv)), z.e.
[[
··· (
a··]
~o ( -1)" [ 1-K
I-a,J
u
qd,
) ]] =0
s,J-aii-"SiS/ ···
for
av~O,
(3·21)
where the dots stand for arbitrary combinations of screenings, and d;= i-(a;, a;).
The proof, as explained in detail in Ref. 40) for the simply-laced case, can be given
by writing out the integrals in terms of integrals over the unit circle with a fixed
ordering of the variables. The vanishing is then a consequence of the cancellations
among the various phase factors one picks up by going around the other variables.
The second lemma expresses the fact that not all polynomials in screenings give
rise to non-trivial operators, but that there are some vanishing relations.
Free Field Approach to 2-Dimensional Conformal Field Theories
103
Lemma 3.3.
(3·22)
Proof We can write
(3·23)
The phase factor vanishes for N=2(k+hv)/(a;, a;).
D
The third lemma, the analogue of the commutator (2 · 31), can be proved by using
(3·16) and evaluating the boundary terms at arg z;=O and arg z;=27r, while being
careful with the phase factors that one picks up by crossing the variables.
Lemma 3.4.
We have the following commutator while acting on a vector vEFA
(3·24)
;;z;=i
where
(3·25)
is independent of j, and as before
(3·26)
One observes that the combinatorics which makes the commutator vanish for
specific polynomials in the screenings is similar to that in the finite dimensional case,
the only difference being that one works with q-numbers instead of ordinary numbers.
We therefore arrive at the following characterization of the intertwiners between
Fock space modules
3.5. There exists a map from Homv,<ul(MJ, MAq) to Homv<ul(FA, FA') where
the restricted quantum group Verma modules MA q are defined in Appendix B, and the
quantum group parameter q and the central charge k of ii are related through q
=exp(i7r/(k+ hv)). Explicitly, if we characterize an intertwiner in Homv,<ul(MJ, MAq)
by the singular vector P[/]vA, of weight A', where P[!] is some polynomial in the
generators f,, then the map is given by
THEOREM
P[!]vA H
[P[k-~s;]].
(3·27)
(Note that this map is well-defined because of Lemmas 3.2 and 3.3.)
We would like to believe that this map is bijective. To prove surjectivity, we would
have to show that all intertwiners between these Fock space modules are of the form
(3 ·17), i.e. can be built from the screening operators s;(z) by taking appropriate
contour integrals. To prove injectivity, we clearly need to understand the vanishing
properties of integrals of the type (3·17). In the examples to be discussed below the
nonvanishing of the Fock space intertwiners could be established by the explicit
104
P. Bouwknegt, ]. McCarthy and K. Pilch
evaluation of the image of a certain vector.
Let us explicitly exhibit some singular vectors in MAq, i.e. those that lie along the
simple root directions. Let thereto wE Wand write w= taw for some aEM, wE W.
Define
if (w(A+p), a;v)>O,
if (w(A +p), a/)<0.
(3·28)
Then by a straightforward calculation one checks that (/;) 1vw*A is a singular vector
in M~*A of weight w'*A, i.e., [(s;) 1]EHomvculFw*A, Fw'*A), 40 l where
' {tardiJ = wr iiF'a;
w=
fa-a,YtW = t -ur'a,r ur'a,
if (w(A+p), a/)>0
if (w(A + p), a;v)< o.
(3·29)
The successive application of this result would yield an intertwiner
: F w*A--> Ft-a,•W*A, which however vanishes due to Lemma 3.3.
Summarizing, we have reduced the problem of the determination of the Fock
space intertwiners to finding the singular vectors in a certain (restricted) quantum
group Verma module. The analysis of the singular vector structure for Vq(su(2))
was given in Ref. 58) (where also, for su(2), a direct relation to the Fock space
intertwiners of Refs. 36) and 37) is suggested). For CfJ isu(3)) the singular vector
structure was determined in Ref. 83). The structure that emerges (which is expected
to hold for arbitrary groups g) is the following. For generic q the singular vectors
are just q-deformations bf the singular vectors that exist for q=1, and are thus
completely described by Theorem 3.5. For q a root of unity parametrized by q=exp
(7ri/(k+hv)) (and integrable weight A) the singular vectors are in 1-1 correspondence with elements of the affine Weyl group of g. The directed graph given by the
set of intertwiners is a tiling of .e'-dimensional space by the directed graph for q=1,
i.e. translations of the q = 1 graph along the root directions.
Let us formulate this more precisely. Inspired by (2·57) we introduce thereto a
twisted length lr on the affine Weyl group W by
[s~(k+hV)f(a,,a,)]
(3·30)
where rEM is chosen such that (r, a)>O, VaEL/+. Note that lr is well-defined
because the limit is already obtained at a finite value of N since there are at most a
finite number of cancellations (at most l(w)) that can occur between t-Nr and w.
Also, one can show that lr does not depend on the choice of r (see Lemma A.3).
su
Example 3.4. For
(2) we can take r=a, i.e., tr=ron. For w=(rori)m we have
f-N,w=(riro)N-m provided N";?.m. Thus, l(t-N,w)-l(t-N,)=2(N -m)-2N= -2m for
N";?.m, i.e.
lr((rori)m)= -2m.
Similarly
lr((rori)mro)= -2m-1,
(3·31)
Free Field Approach to 2-Dimensional Conformal Field Theories
105
lr((rlro)m)=2m,
lr((rlro)mr1)=2m+ 1.
Example 3.5.
(3·32)
For su (3) we can take r= as, i.e. tr= ror1 rzr1.
One easily verifies, e.g.
lr(rlro)=O,
(3·33)
Using the twisted length we now define a twisted Bruhat ordering on W. Define
w ~ r w' if there exists an aE .J+ such that w =raw' and lr( w) = lr( w') + 1. Then define
WSr w' iff there exist W1, ···, WkE W such that w~r WI~r· .. ~r wk~r w'. Then we
conjecture
CONJECTURE
3.6.
For AEl\ (integrable weight) we have
- -
{1
if w < r w'
dim Hom'U,(ulMZ,*A, MZ,'*A)= 0
h -.
ot erwzse.
(3·34)
As mentioned above, this conjecture has been proved for g=su(n), n=2, 3.
Through the correspondence in Theorem 3.5, which was conjectured to be a bijection,
this theorem gives a complete description of the Fock space intertwiners.
Example 3.6.
given by
The directed graph of intertwiners for V q(su(2)) 58 > (or su (2) 36 >· 37>) is
(3·35)
and for V;(su(3)) 83 >· 40 > (or su (3) 40 >· 37 >)
r1
----+
/'
'\.
X
ro
r1
'\.
r2
ro
X
X
r1
r2
X
/'
rt
'\.
X
r2
'\.
X
ro
r1
r2
X
X
ro
/'
r1
'\.
X
r2
'\.
X
ro
r1
/'
r2
r1
----+
/'
r2
----+
ro
r2
----+
/'
ro
----+
'\.
ro
X
'\.
/'
r1
----+
/'
T2
----+
ro
r2
----+
/'
'\.
ro
ro
ro
----+*
/'
r1
----+
r1
•----+
X
'\.
'\.
/'
r2
----+
ro
r2
----+
/'
ro
----+
X
r1
----+
X
'\.
X
ro
(3·36)
where the arrows continue in all directions. For a similar hexagonal structure in the
case of su (3) Verma modules we refer to Ref. 74).
106
P. Bouwknegt, }. McCarthy and K. Pilch
The vertices of these diagrams, i.e. MZ,*A or Fw*A, are in 1-1 correspondence with
elements of the affine Weyl group. If one associates simple reflections r; with the
edges as in the picture, then the Weyl group element associated to a vertex can be
obtained by combining the reflections that one encounters by taking a path along the
edges from the middle vertex ( *) to the required vertex (e.g. the vertex ( •) is
associated with w = r1 r2ro). The Weyl group elements associated to vertices along a
vertical line have a fixed twisted length (compare Example 2.9) and the depicted
intertwiners increase the twisted length by one unit.
The intertwiners along simple root directions are explicitly given by (3·28). The
other intertwiners can in principle be obtained by a successive application of Lemmas
B.1 and B.2, exactly as in the finite dimensional case.
Example 3. 7.
d1,ro~
For su (3) we have e.g. (see Ref. 40) for more details)
'£,_bq(T2, Ts;j)[(s1) 1d(ss)i(s2)l.-j],
Q5;.j5;.[,
(3·37)
where l;=(A+p, a;), [;=(k+3)-l,, ss=-s1s2+q- 1s2s1, and
i+Cm-i)(n-i)
[m]q![n]q!
bq ( m, n ., J")- q
[J.] q.1[ m _ J.] q.1[ n _
.]
J q.1
(3·38)
A last remark concerns the affinization of the other twisted realizations of§ 2.4.
One can go through the same steps as described above for the "affinized dual Verma
module" with the result that the structure of the intertwiners now becomes the
affinization of Theorem 2.7, e.g. for su (3), .M;•r, it becomes a tiling of the 2dimensional plane by the hexagon of Example 2.9.
The resolution
Collecting the material on intertwiners we come to a final conjecture on the
existence of resolutions of irreducible highest weight modules LA, AEJ\ in terms of
Fock space modules.
3.3.
3.7. For every AEP+ we have a resolution of LA in terms of Fock space
modules, i.e. we have a complex (F, d)
CONJECTURE
(3·39)
where
(3·40)
whose cohomology is given by
(3·41)
Free Field Approach to 2-Dimensional Conformal Field Theories
107
Let us make some remarks on the status of this conjecture. The fact that we can
combine the intertwiners of Conjecture 3.6 into a complex has been proved for
(2)
in Refs. 36) and 37) and for
(3) in Ref. 40). There is an important comment that
we should make namely that unlike in the finite dimensional case, there exist elements
w, w"E W such that the number of w'E W satisfying w-+r w'-+r w" equals one,
namely w"= f-a,W. The nilpotency of the differential along these su(2) directions is
ensured by Lemma 3.3. This feature does not have an analog in Verma modules but
is purely a "quantum group" effect. This is one reason it cannot be completely trivial
to relate resolutions through an "infinite twist". To get cancellations among the
maps that form a (twisted) square in the affine Weyl group one has to make a clever
choice of signs as in the finite dimensional case (see Ref. 40) for details on Sit (3)).
Although the signs can be chosen periodically, this is not necessarily the same
periodicity as present in the intertwiners themselves. We regard the mere fact that
we have discovered enough intertwiners to build a complex as very suggestive that we
may have found all, as conjectured in the previous section.
As for the cohomology this has, to our knowledge, only been proved rigorously
for
(2). 36 >'37> Sample calculations for
(3) have not revealed any contradiction.
The most promising approach to prove Conjecture 3.7 seems to be to show the
equivalence to the weak resolution of Feigin and Frenkel 50 > as can be done in the finite
dimensional case, 77> and the infinite dimensional case for Verma modules.m This
resolution, where the differential is the de Rham operator on the tangent bundle of a
semi-infinite dimensional flag manifold, has a similar structure as the one described
above, i.e. its terms are of the form (3·40). However, it has not been shown that this
differential only maps between the Fock space modules as supposed in Conjecture 3.6,
which would probably require a proof of the characterization of Homv<n>(FwM,
Fw'*A). Also, the heart of the Rocha-Caridi's proof, which is the construction of a
map between the two complexes is established by induction to i. This would have to
be suitably modified due to the two-sidedness of the above complexes.
One might wonder whether the analogous complex in terms of restricted quantum
group Verma modules gives a resolution of the irreducible representation. This is
not the case, however. In fact, it appears that the complex has trivial cohomology
everywhere. This, we have been able to prove for CU q(su(n)), n=2, 3 (see Appendix
E). In fact, triviality of the complex is crucial for the construction of the screened
vertex operators. As far as we understand the construction of the BGG-type resolution for QG irreducible highest weight modules is still an important open problem.
su
su
su
3.4.
su
Chiral vertex operators and fusion rules
In this section we outline the construction of chiral vertex operators of the
WZNW models in the framework of the free field resolution (3·39), in complete
analogy with the finite dimensional case discussed in § 2.6. For three given irreducible representations LA., i = 1, 2, 3 of a chiral algebra Jl, we will introduce the chiral
vertex operator84 H 6> as a set of (multivalued) conformal fields, <P"'(z): LA,-+ LA., ¢
ELA., such that if¢ is a primary state then <Prp(z) behaves as a primary field of Jl. In
particular it is completely specified by its highest weight component, <P(z), and other
components are obtained by the action of Jl. The "fusion rule", N1,"A., is simply the
108
P. Bouwknegt, ]. McCarthy and K. Pilch
dimension of the vector space of such chiral vertex operator sets for the three given
representations. It should be stressed that, unlike in the discussion in § 2.6, in the
infinite dimensional case we cannot prove that homomorphisms between Fock spaces
are always expressed in terms of screening operators. But since this is physically
well motivated, we develop the theory below with this assumption.
Thus, based on the finite dimensional case discussed in§ 2.6, the Virasoro minimal
models, 7 > and the su(2) WZNW model, 36 > we now seek a representative of the highest
weight component (J)(z) as a chain map between two resolutions (FA,, d) and (FA., d')
defined by a collection of screened vertex operators { v< 0 (z), iEZ} whose components
{Vw,w'(z)ll7 (w')=l 7 (w)=i} are of the form
Vw,w'(z)=
r dzr···dzn VA.(z) {ii"··in}
~ a;,. .. ;nS;,(zr)···s;n(Zn)
Jr.
= VA.(z) x [polynomial in screenings].., ,
(3·42)
where
(3·43)
and the degree of each term in the polynomial is a;,+···+a;n=w'*Ar+Az-w*Aa.
One verifies that Vw,w'(z) is a conformal primary field of the chiral algebra of the
WZNW model with the weight Az, and the conformal dimension (Az, Az+2p)
/2(k+hv). Because of the conformal invariance we may restrict our analysis to a
specific value of the coordinate z. In the remainder of this section we set z=1, and
do not write it explicitly.
The requirement that the vertex V = {v< 0 } defines a chain map between two
resolutions is given by a condition similar to (2·91) in the finite dimensional case, i.e.
that the following diagram is commutative
···~
FA"";_ I)
~
···~
v<-I>
F1-;,IJ
d(O)
d(-1)
~
F1°[
~
v<•>
d'(-1)
~
~
d'(O)
F1°i
~
F1![
~ v<''
F11i
d(l)
~
F12[
~
d'(l)
~
~···
(3·44)
v<•>
F12j
~···
Similarly, we say that a vertex is trivial if there exists f={!Ci>, iEZ} with
components Uw.w', lr(w')=lr(w)+1=i} of the form (3·42) such that
(3·45)
We will refer to the r.h.s. of (3·45) as a gauge transformation of the vertex. The
space of screened vertex operators satisfying (3·44), modulo trivial vertices of the
form (3·45), will be denoted by v1~,A., and N1,",A.=dimv1~,A •.
The same arguments as in the construction of intertwiners in § 3.2 show that the
screening currents inside [ ·] satisfy the algebra of the negative root generators of
quantum group CU q(g ). Thus it is natural to identify components Vw,w' of a vertex
with elements of the restricted QG Verma modules llfz,,M,,
(3·46)
Free Field Approach to 2-Dimensional Conformal Field Theories
109
or with elements of the transposed Verma modules MZll:A.,
(3·47)
The transposed Verma module M:F is spanned by elements of the form VA/;,/;,_,···/;.,
and is isomorphic with the Verma module Mj. via VA/;J;,_,··-/;,~/;~···/;f_J[*vA•, where
/;*= f-wo(«t)·
These identifications, together with triviality of the complex (MAq, d), discussed
in Appendix E, suggest that an analogue of Theorem 2.15 should hold. Indeed, we
have
3.8. A nontrivial vertex
component v<o> such that
THEOREM
VEv2~.A.
is uniquely characterized by its 0-th
(3·48)
and
(3·49)
The remaining components vu>, i-=1=0, are completely determined by (3·44), up to
addition of trivial pieces of the form (3·45).
Proof First we will prove that given v<o> satisfying (3·48) we can solve (3·44) for the
remaining components. Identifying {(d'<o> v<o>)w,w'llr(w')=O, w fixed} with an element
of M:il 0 > we can interpret (3·48) as the condition that this element is closed (see,
Appendix E). Since by Theorem E.1 the cohomology of (MAq, d) is trivial we conclude that there exists v< 1> such that
(3·50)
which satisfies
(3·51)
Repeating the same argument we determine all the vertices v<il, i?:.2. On the other
hand if we identify {(V< 0 >dH>)w,w'llr(w)=O, w' fixed} with an element of Mi{, we can
use the triviality of (Mi{, d), which obviously follows from Theorem E.1, to calculate
v<-ll, and then the remaining vertices vu>, i-::;;. -2. It is also clear that the ambiguity
one has at each step of this procedure adds up to the total ambiguity of the form
(3·49). In particular, if we start with v<o> given by the r.h.s. in (3·49) we will end up
D
with a trivial vertex.
Obviously, we could have chosen any other component v(i>, satisfying
d'u> v<ildU-l>=o, and prove that it determines the vertex completely. In practice
solving (3·48) for v<o> presents a formidable task because, in principle, one must still
determine an infinite number of components Vw,w', lr(w)=lr(w')=O. This difficulty
does not arise in the case of the su(2) WZNW model, which we will now discuss in
some detail.
Example 3.8. Screened vertex operators and fusion rules in the su(2) WZNW model.
P. Bouwknegt, J. McCarthy and K. Pilch
110
su
Recall from § 3.3 that a resolution of an
(2) representation with a highest
weight A= ja at level k (A is integrable for O~j ~ k/2) is given by a complex in which,
for all i, Fjil is just a single space,
p!il={
Fi-n(k+2),
F-J-1-n(k+2l ,
J
i=2n,
i = 2n + 1 .
(3·52)
In terms of screening current s(z), we have
d(i)=
{
[ s 2i+l]
'
[sk+ 2-< 2i+1l],
i=2n
'
i=2n+ 1.
(3·53)
Given three integrable weights A;=j;a, i=1, 2, 3, the highest weight component
of the vertex is specified by a set of screened vertex operators v<o : FJ\i>-> FX> of the
form (3·42) which in this case simply gives
(3·54)
where a, are normalization constants determined by (3·44). By the general argument
given above we only need to determine the screened vertex v<o>, which satisfies
conditions (3·48) and (3·49). Clearly, to define v<o> we must have
(3·55)
Then (3·48) becomes (ao=I=O)
(3·56)
or, equivalently,
[sk+2+Ua+i•-il>]=O,
(3·57)
which is satisfied provided
(3·58)
On the other hand (3·49) is given by
(3·59)
where j<o> and j<1l are of the form V,. times a power of the screening current, which
must be positive if the corresponding I exists. Thus (3·59) simply requires that these
powers of screenings are negative, which gives
h+h+h~k.
(3·60)
h?:.h-h.
(3·61)
The solution to (3·55), (3·58), (3·60) and (3·61) is
lh- hl~h~min(k-(jl +h), jl + j2),
(3·62)
Free Field Approach to 2-Dimensional Conformal Field Theories
111
which shows that a nontrivial vertex exists if and only if the su(2) WZNW model
fusion rules87 > are satisfied.
In the case of finite dimensional vertices it was convenient to relate the problem
of constructing vertices to that of finding special states in an irrep of the algebra. We
will now rederive Eqs. (3·55)'"'-'(3·61) from this point of view.
Example 3.8. cont.
Thereto, let us identify a nontrivial vertex, via its v<o> component, with the
element rf; at weight h-h in the quotient, LJ,, of the restricted Verma module .zCt; by
its submodule M!.j,-1, generated by the singular vector JZil+lvj,. For an integrable
weight j1a, L'J, is an irreducible module of Vq(su(2)). We can now rewrite (3·48),
which is equivalent to (3·50) and (3·49) as the following conditions for the state rf;
(3·63)
Since rf; = ajil+iz-i•vj,, this approach leads to the following interpretation of the inequalities above :
h+j2-h~O,
so that
-h+h+h~O,
so that ¢$.M!.J,-1,
h-h+h~O,
so that
rf;EKer JZi•+l,
h+h+hsk,
so that
rf;$.Im jk+2-(2i•+l>.
rf;EMJ;,
(3·64)
Solving (3·50) we find a1=q- 2iz<2i•+l>ao, where the phase arises from moving s 2i•+l
through Vj,. The remaining normalization constants are determined from (3·44) and
are given by
_ ( - 1)2iznao , a2n+1_ ( - 1)2jzna1 , n E
a2n-
z
.
(3·65)
This shows that we can always choose a representative of a vertex satisfying (3·44)
to be "periodic" in the sense of (3·65). Heuristically, we could have anticipated this
result as follows: An element C =sk+ 2 lies in the center of CUq (su(2)), and C Vj,
=( -1) 2jz Vj. c. Thus c v<O>=( -1)2i• v<O) c. On the other hand formally c v<O>
= V( 2) c' which gives (3·65) for n=l.
Once again we would like to derive (3·62) by a counting argument, not requiring
an explicit construction of the vertex. We will now present two such calculations:
the first one based on a double complex argument similar to that discussed in § 2.6, and
the second one based on a cohomological interpretation of conditions (3·63) for the
v<o> component of a nontrivial vertex.
Example 3. 8. cont.
First consider the double complex (X, a, o') build exactly as in the finite dimensional case (see (2·106)). It is easy to verify by counting powers of screenings that
the cohomology of the associated complex (K, D) is trivial for P=l=-0. However, each
112
P. Bouwknegt, ]. McCarthy and K. Pilch
K<P> is infinite dimensional, and the Lefschetz formula now simply gives an undefined
alternating sum where each term is infinite. Thus, to make this procedure meaningful we must "cut the complex down to size". This is achieved by restricting to the
space of "periodic" maps, xU,j>=( -1)2j•x<i+ 2 ,j+2>, Vi, jEZ, which clearly form a
subcomplex since D preserves the symmetry. As we observed above the vertices we
are interested in, i.e. elements of K' 0 >, always have such symmetric representatives.
Moreover, it is clear that if f=Dg for a periodic collection of maps f, then we may
choose g periodic (again each component of g lies in a one dimensional space). Thus
the cohomology in this subcomplex is precisely equal to that in the complex (K, D).
Now applying the Lefschetz formula to the periodic subcomplex we obtain
ih•,i2= nEZ
~ [8(h + j2- h+n(k+2))- 8(h +h+ h+ 1 + n(k+2))
- 8(- j1-1 +h- h+ n(k+2))+ 8(- j1 + j2- ja+ n(k+2))],
(3·66)
which is known to reproduce fusion rules given in (3·62).
The second method is to observe that the last two lines in (3·64) characterize ¢
as a nontrivial cohomology class in the following complex arising from the resolution
with respect to the weight ha,
(3·67)
where all the spaces L'J,' 0 are isomorphic with LJ., and the action of differentials is
simply that of the generators on the quotient. This complex has a weight space
decomposition, and for each weight the corresponding subcomplex has non-zero
spaces at most at two points. The latter follows from the integrability of weight ha
and Jk+ 2 =0. Once more a straightforward power counting shows that a nontrivial
cohomology of (3·67) at weight h-h can occur only at the 0-th degree. Thus, we can
use the Lefschetz formula to compute the dimension of the space of nontrivial yco>•s,
and the result agrees with (3·66).
In the remaining part of this section we will discuss the general structure of
vertices when f ~ 2. We formulate and illustrate on examples some conjectures
which generalize properties established in the finite dimensional case in § 2.6 and in
the
(2) case above.
One may try to calculate the dimension N:l,",A• using a double complex as above.
In this case the "periodicity" condition should be
su
(3·68)
We use obvious notation for the components of the vertex, e.g. Vlff~As,w'*A• = VJ,~~'- In
the case of Sit (3) we have verified on several examples88 >that one can always choose
a representative of a vertex in the class of gauge equivalent vertices, such that it is
periodic in the sense of (3 · 68). However, a general proof of this property, that should
follow from the basic equation (3·44) of the vertex, is not known.
Assuming that we can restrict the double complex to periodic vertices, and that
Free Field Approach to 2-Dimensional Conformal Field Theories
113
the cohomology of the restricted complex is concentrated in K<o> we obtain from the
Lefschetz formula the dimension of the space of nontrivial vertices :
(3·69)
The last equality, in which we used (2·46), gives precisely the formula derived by the
algorithm which was proven to reproduce the fusion rule coefficients_ 89 H 4 >
In the above derivation we considered the vertex as a chain map between two
complexes. On the other hand by Theorem 3.8 we know that the vertex is completely
characterized by its v<o> component. In principle the characterization of the vertex
given by Theorem 3.8 requires that we specify all of v<o>, i.e. an infinite number of
components. One would certainly expect this to be unnecessary, since the cohomology of the Fock space resolution of LA can be concentrated at FA cF1°>, and
only the Vi~~> component gives rise to a nontrivial mapping when we pass to the
cohomology space. Thus one should be able to derive (3·69) by counting properly
defined nontrivial Vi~~> components.
We say that a vertex is diagonal if the off-diagonal components of v<o> vanish, i.e.
V~~lo,=O for w=Fw'. For such diagonal vertex, equations in (3·50) which involve the
Vi~~> component are
(3·70)
In order that the mapping induced by Vi~~> on cohomology classes was nontrivial we
require that
Vi~~)=F
{wE
_~
W[/,(w)~-1}
df.--.J>j~~i +
-~
{wE W[l,(w)~I}
JlYod1J.i,
(3·71)
which is just the (1, 1) component of (3·49).
3.9. Any vertex is gauge equivalent to a diagonal vertex, i.e. given v<o>
Which satisfies (3•48) there eXiStS a diagonal lf(O) SUCh that V(O)_ lf(O)=d'(-I)j<Ol
+J<ll d<o> for some po> and J< 1>.
CONJECTURE
su
This conjecture has been proven88 > in the case of
(3) under a restriction that the
weights Ar, Az and As satisfy the finite dimensional (!) tensor product rules, i.e., that
the irrep of su(3) with the highest weight As occurs in the tensor product of irreps
given by Ar and Az. Using a convenient basis in CU q(n~u<S>) (see Appendix E) we show
by an explicit calculation that given v<o> which satisfies (3·48) there exists a gauge
transformation such that the new (1, 1) component satisfies (3·70). Similarly, given
w~) satisfying (3·70) one can use (3·44) to solve explicitly for the remaining components of a diagonal v<o>, and show that the solution is unique up to a gauge transformation. Combining these two facts we verify the conjecture in the restricted case.
One can also check that in this case (3·71) simplifies and in fact is equivalent to
114
P. Bouwknegt, J. McCarthy and K. Pilch
(3·72)
because the remaining terms in the gauge transformation (3·71) can be recast into the
form above.
If we identify Vi~~> with an element ¢ in L1., the quotient of the restricted QG
Verma module M:l, and the submodule generated by singular vectors j,<A•+p,a,v>vA., we
may interpret (3·70) and (3·71) as the following conditions on ¢
f,(As+P,a;V)¢=0'
¢=I=
_~
{wE Wll,(w)= -1)
i=1, ···, f'
dl,-;}>[/,]xw ,
(3·73)
(3·74)
By analogy with Theorem 2.14 we will call this a generalized Zhelobenko condition.
One can try to count the diagonal vertices by constructing a complex similar to
(3·68) in the su (2) case. For this we would need to prove the following:
CONJECTURE 3.10.
Let
(L~~~A.,
d) be the complex obtained from the resolution (FA., d)
by the substitution
(3·75)
If A1, A2 and A3 are dominant integrable weights of ii then the cohomology of
(Lq:l~.A.,
d) is concentrated in the zeroth dimension, z.e.
(3·76)
It is easy to verify that in most cases complex (Lq:l~.A., d) has only one nontrivial
space, and thus the conjecture is trivially satisfied. For instance this happens when
the labels (A1, a; v) are small in comparison with (A3, a; v) and k + h v- (A3, a; v). In
some sense, this situation corresponds to the classical limit, when we know that the
analogous theorem is true (see remarks at the end of § 2.6).
We can now calculate the dimension of the space of nontrivial Vi~~> vertices from
the Lefschetz formula and, using (2·46), we obtain
(3·77)
This agrees with the previous calculation (3·69) and suggests that indeed all the
information about the vertex is contained in its Vi~~> component.
The explicit form of the Vi~~> component of the vertex, which satisfies (3·70) and
(3·71), is needed for computing tree level correlation functions in the WZNW model.
On the other hand, (3·70) and (3·71), or equivalently (3·73) and (3·74), provide a new
algorithm for verifying whether for given three integrable weights A1, A2 and A3 of ii
the fusion rules are satisfied. Let us illustrate some of this on examples for su (3).
Example 3.9.
is
The relevant part of the resolution (FA, d) for su (3) (see Example 3.7)
Free Field Approach to 2-Dimensional Conformal Field Theories
Qf Fr,*A
/'
Qf'
'\. Frz*A
115
/;=(A+p, a;v), i=l, 2, 3,
l;=k+3-l;' i=l, 2, 3'
Q/=[s/], i=l, 2,
Qg 1 see (3·37) .
(3·78)
The operator Qg 1 is a homogenous polynomial in s1 and S2 of order la3 whose explicit
form is given in (3·37).
I.
A1=8, A2=6, A3=3, k';;?:2
The most general screened vertex
"VI~~>
in this case is
(3·79)
where S12= -s1s2+q- 1S2SJ.
Equations (3·70) and (3·71) are explicitly given by
(3·80)
(3·81)
and
(3·82)
By the counting of screenings the most general form of the components of v<I> is
(3·83)
while, for k';;:::2 (as required by integrability of 6 and 8), the only non-vanishing
component of the gauge transformation is
(3·84)
We can use this freedom to set c1=0, i.e. to diagonalize v<I> in (3·80). Substituting
(3·83) in (3·80) we then obtain
[si 2][a1S1S22+ a2S12S2] = [b1S1Sl+ b2S12S2][s1 2] ,
+ a2S12S2]=[d1S1S2+ d2S12][s22],
q- 2[s2][a1S1S22
(3·85)
where the phase on the l.h.s. of the second equation arises from pulling S2 across VAz.
Using the algebra of the generators of CfJ q(ntu< 3>), summarized in Appendix B, we find
that there is a one parameter family of solutions
a1=[j;qa,
a2=a,
b1=[~J:a, b2=-[ 3~qa,
d1=[ 3ta,
d2=q- 4 ~~~:a
(3·86)
with arbitrary a. In particular this proves that .N~.s=l.
116
II.
P. Bouwknegt,]. McCarthy and K. Pilch
Ar=8, A2=8, Aa=8, k=2,
k~3
In this example the most general form of
Vi~~>=
Vi~~>
is
VA2[arsrs2+a2sd,
(3·87)
While the Only non-Vanishing COmponentS Of V(l) in (3·70) are vg!n and vg!r 2, Which
are in form the same as the r.h.s. in (3·87), but with different coefficients which are
uniquely expressed in terms of ar and az by solving (3·70). Moreover, the most
general form of the r.h.s. in (3 · 71) for this case is
,c(O)
+ Qk+I ,C(O)
+ Qk-l ,C(O) + ,C(l) Q 2+ ,C(2) Q 2
Q1k+I Jr2rorz,l
(3·88)
2
Jrtrort,l
3
Jro,l
Jl,rt 1
Jl,rz 2 ,
and we see that for k~3 none of these terms contribute a gauge transformation to
Vi~~>. This gives a 2-parameter family of vertices, in agreement with .N~.8 =2. On the
other hand, for k=2 we have
Qa 1 = q 3 [srs2] + q[3]q[sd ,
which allows one linear combination of the terms in
shows that .N~.s=l for k=2.
3.5.
(3 · 89)
W~>
to be gauged away. This
Coset conformal field theories
The coset construction51 >associates a RCFT to every pair (jj, ii) of Kac Moody
algebras, where h is a subalgebra of g, with central charge kH related to kc by the
Dynkin index j of the embedding hCg, kH= jkc. In this section we will only consider
regular embeddings, i.e. j = 1. The energy momentum tensor
(3·90)
where Tc(z) and TH(z) denote the Sugawara tensors of G and H respectively,
commutes with the generators of ii. It thus generates a "coset" Virasoro algebra
with central charge c=cc-cH, and we may decompose a given irreducible highest
weight module LAc of jj as
(3·91)
The sum runs over a finite set of integrable weights of ii. The modules L~:~, are of
course highest weight modules of the coset chiral algebra-but not necessarily irreducible. The branching function is defined as (q = e 2 m)
etH =Tr
qa/H-c/24
bA,A'(3·92)
•
L~;f(.
In line with our previous discussions, we might try to treat the coset model as a
minimal model of some chiral algebra Jl. We would then construct a free field
realization of Jl and a resolution of irreducible Jl-modules in terms of these free field
Fock spaces. But in fact the chiral algebra of a generic coset model is not known.
Thus we find it necessary to approach the problem more "extrinsically", starting from
the known free field description of jj and projecting out the G/H model. As a first
step we generalize the results of Theorem 2.9 and obtain a resolution of L~:~'·
Clearly L~:~, can be identified with the set of vectors in L~ which are ii-singular
Free Field Approach to 2-Dimensional Conformal Field Theories
117
(annihilated by ii+ H) with integrable ii-weight A'. This leads to the following
resolution of L~~;[, in terms of a subcomplex of the resolution of L~.
3.11. Let (FA, d) be a resolution of the irrep L~, AEJ\, of {i given in
Theorem 3.7. For each F~il, denote by S~?A' the subspace of ii-singular states with the
ii-weight A'. Then
1. d: S~?A'-4 sy,-;:Y, i.e. (SA,A', d) is a subcomplex of (FA, d).
2. The cohomology of the subcomplex (SA,A', d), with A' an integrable weight of ii, is
THEOREM
H d(il(S A,A')-u.N,OLGIH
A,A'.
(3·93)
The reader will notice the similarity to Theorem 2.9, and indeed the proof is
almost identical. In particular the first part of the theorem follows from the fact that
the generators of iii_f_c ii~ commute with d. Moreover, part two hinges on the
following generalization of Theorem 2.10.
3.12. For any affine Kac-Moody algebra {i, Verma modules MA, where A is
a dominant {i-weight, are projective in the category LJ of {i-modules.
THEOREM
Proof The proof is essentially identical to the one of Theorem 2.10 for Verma
modules of finite dimensional Lie algebras. The only difference is that the Cartan
matrix of a Kac-Moody algebra is positive semi-definite. Thus we can conclude that
the last term in (2·70) is non-negative, but it may vanish even if some n, are non-zero.
However, for a dominant weight A the coefficients p,+a, in the second term in (2·70)
are strictly positive, and this enforces vanishing of all n;'s. A more complete
discussion can be found, e.g. in Ref. 44).
D
We may now return to the proof of the theorem, which follows from the result
above by the same reasoning as the one used in § 2.5 to obtain the twisted resolution
of finite dimensional coset models. For the sake of completeness we recall the main
steps. From Theorem 3.12 it follows that any ii-singular element of L~ has an
ii-singular representative in F~0 >. It further follows that any state which is trivial in
the complex is also trivial in the subcomplex, and thus the cohomology of the
subcomplex vanishes, except at the restriction of F~0 >.
There remains the problem of actually performing this projection onto the
resolution of L~~;[,. The procedure which seems to generalize from our finite dimensional analysis in § 2.5 is the ERST projection. Introduce conjugate pairs of free
fermionic ghost oscillators, one pair for each generator in ii+ H. Thus for the positive
modes we have the set of conjugate pairs {(c~n, bnb), n >0} (a denotes adjoint indices
of hand rab is the Killing metric of h) with {c~m, bnb}= Omn"Yab, and for the zero modes
just {(coa, boa), aE.d+ H) with {coa, bl}= oaP. Denote by Fgh the Fock space generated
by the c~n, n2:0, acting on the vacuum IO> annihilated by all the boa and bna. The
ghost Fock space is graded by the cb-ghost number-i.e., Fgh=EBnEz.Fi~>. For an
arbitrary ii-module M, we introduce the module M@Fgh, graded as (M@Fgh)<n>
=M@Fi~>, with M@Fig>=.M. Then we may construct a nilpotent operator Q: M
@Fi~>-4 M@F~~+l> via
P. Bouwknegt, J. McCarthy and K. Pilch
118
where xE ii+ H acts on M@Fgh as x®l.
That Q is nilpotent follows from associativity of ii+ H by a straightforward
calculation. Further, it is clear that the cohomology H¢0 >(M@Fgh) is isomorphic to
the space of ii-singular vectors in M. Given the resolution (FA, d) of a fi irrep LAG,
denote the subcomplex with fixed ii-weight A' by (FA,A', d). Then we may form a
double complex (FA,A'®F8 h, d, Q), where Q is given in (3·94) and d acts as d®l.
The spaces in this complex are labelled by two degrees, i and n, Ft~,ti!)F~~>, where i
EZ and nEZ+. As discussed in Appendix C, to such a double complex we can
associate a single complex (K, D). The following is required if the BRST construction is to be applicable.
Let (FA, d) be the resolution of an irrep L~, AEP+, and A' an
integrable h-weight. Then the cohomology of the complex (KA,A', D) associated with
the double complex (FA,A'®Fgh, d, Q) is
CONJECTURE 3.13.
(3·95)
In § 2.4 we argued that a good finite dimensional analogue of the resolution
(FA, d) is the one in terms of twisted Fock spaces. Conjecture 3.13 should then be
viewed as an infinte dimensional generalization of Theorem 2.13. Recall that the
proof of the latter required an indirect argument. More precisely, we showed that
the cohomology of the complex (KX'.A', d) could be computed using a more convenient
resolution of L~ via a complex of dual Verma modules, which are characterized by the
co-free action of n+ H. Generalization of this proof to the infinite dimensional case is
not straightforward. There exists a resolution of fi irrep L~ in terms of dual Verma
modules 95 > for which, after restricting to a fixed fi-weight, one should be able to prove
the analogue of Theorem 2.11. However, in relating the cohomology of this complex
to that of (FA,A'®Fgh, d, Q) we are interested in, one encounters the same difficulties
as in relating the Fock space resolution to the EGG-resolution. For this reason we do
not pretend to have a rigorous understanding of the proof.
This construction allows us to compute the branching functions. That can be
seen from the following set of examples, which all use the Lefschetz sum to express
bA,A'=q -h.·-c/24"'
~
"'
~(-
zEZ n<oO
1)'( - 1)n T rFt>·®F,i::' ( Q fo)
(3·96)
where
(3·97)
projects onto the appropriate isospin, hA' is the eigenvalue of LoH on ii-singular states
with h-weight A', and we have used explicitly that LgiH=LoG-LoH. Moreover, hoi
and La are just the D-invariant extensions of hoi and LoG, respectively,
io=LoG+ ~( ~ (nc:::bna+nc~nb;;-a)+ ~ nc!-nbni),
n>O
aELJ+H
I:::;;:iS::l
Free Field Approach to 2-Dimensional Conformal Field Theories
119
The calculations use the identity96 l-99 '
1
II (1-qne2~ri6•a)(1-qn-le-2"i6•a)
_ _____;1c____ ~ e-27rtn6•a¢n.
II (1- qn)2 nEZ
(3·99)
n;;;,l
n;;;,J
Here we have introduced the quantity
</Jn=
( -1)mq(l/2)m(m+I)+nm.
~
(3·100)
m;;;,o
It is simple algebra to check that this may also be written
</Jn= _
~
( -1)mq(112)m(m+I)+nm,
(3·101)
m<O
and thus, as a consequence, we obtain
(3·102)
</J-n=qn</Jn.
Example 3.10. G simple, simply-laced, H = U(1)1
These are the "(generalized) parafermion" theories/ 00 >- 102 ' and hA'=(A', A')/2k.
The ghost contribution in (3·96) exactly cancels that from the non-zero modes of ¢',
i=1, ···, f. Thus
_ q L1A,A·-c/24 w~c
~ (-1)1(W) [
bA,A,-
1
)
de ( ----.=II.,----.=II.--(,-::1---q-=-n
e 2Jri6•(w*A-A')
e'2c:;;",rr.6:-;;·"")(7::1---q-=-n'1-e-2""~r::oo;6""'·""')
aEL1+c n;;;,J
'
(3·103)
where
LJA,A'
(A, A+2pc)
2(k+ hc v)
(3·104)
The integral over B-i.e. the projection onto the h-isospin A'-is easily computed by
using (3 · 99), and we find
Note that by construction this expression must agree with the more familiar
expression for the parafermionic characters 101 > r/c,A in terms of the Kac-Peterson
string functions. 103 '
There is an interesting subtlety arising in this example; namely, it would be
incorrect to interchange the summation over naEZ, and the summation that defines
<Pn•• due to an infinite degeneracy in the energy. The interchange may be made after
splitting the sum and using (3 ·102).
Example 3.11. G simple, H=®H,, rank(G)=rank(H)=f
For this case hA'=~;(A'(il, A'(i'+2pH,)/(2(k+h'k)), where A'<il=A'IH,. The ghost
contribution in (3·96) is now seen to exactly cancel that from the /3", r", aELJ+H, and
the nonzero modes of¢\ i=1, ···, f. The projection onto the h-isospin A' gives
120
P. Bouwknegt, J. McCarthy and K. Pilch
where
LJA,A'
(A, A+2pc)
2(k+ he v)
(3·107)
The example of G x G/G can be found in Ref. 44). Although we have not worked
out the most general coset model in all detail, there are no conceptual difficulties in
applying this procedure to obtain the character of any coset model. There is one
subtlety, however, that it may be necessary to introduce a character valued partition
function in the intermediate steps to make all the sums well-defined.*> This typically
occurs when the centralizer of H in G is nonzero.
It is well-known that the conformal dimension hA,A' of a coset primary field ([)A,A'
is, in general, not just the difference between the conformal dimensions of G and H,
but differs from this by the addition of a positive integer. One might wonder how this
comes about in our derivation of the character since, e.g. Eqs. (3·106) and (3·107) seem
to suggest otherwise. The answer is that if negative coefficients na, aELi+ are needed
to satisfy A- A'= L: naa then, due to the identity cP-n = qnc/Jn, the ground state energy
will be raised by lnal.
Example 3.12.
Concretely, in the example of G/H above
(3·108)
where
LJA,A'
is given in (3 ·107).
However, to prove that the representation of conformal dimension hA,A' is present
in the coset, one should show that the branching function bA,A' is nonzero. It is
conjectured (and proved in several cases) in Ref. 104) that bA,A' is nonzero as soon as
there exist na, aELJ;IH such that A-A'=L:naa.
We have presented a procedure to obtain free field resolutions that works for
arbitrary coset models. It can be a complicated problem to go further for a specific
model in this way. However, there is certainly one case where we can give a
complete treatment already, and that is the generalized parafermions. The reason is
that in this case-and only this case-the projection may be achieved by a separation
into parallel and perpendicular fields exactly as for the untwisted resolution in the
finite dimensional problem. Then for a given irreducible coset module L~;If the
required resolution is just the restriction of an appropriate "perpendicular subcomplex" of (FA, d) to H-isospin A. For this purpose it is convenient to bosonize the
,By-systems in the realization of {j. Following Ref. 82), we have
*l
As of course is also true in the WZNW-model itself.
Free Field Approach to 2-Dimensional Conformal Field Theories
/3a=aee•wa=a(e-;xa)e'wa,
121
(3·109)
are conjugate spin (1, 0) fermionic first order fields, x bosonizes the
and w is a symplectic boson. As is well-known, however, the Fock space
of the xw-fields is not isomorphic to the /3r-Fock space. The bosonization process
introduces additional degrees of freedom, namely ~oa. Equivalence of the Fock
spaces is achieved by removing the zero modes ~oa and the identification of the xa and
wa momentum operators. 82 >'96 >-99> The problem of the removal of ~oa, or equivalently
the projection on nKerr;oa, has an elegant solution in the context of "resolutions".
One can extend the intertwining operator r;oa=fe•xa to a 1-dimensionalline of Fock
spaces. This 1-dimensionalline obviously defines a complex (due to the fermionic
character of r;oa) which has trivial cohomology, and whose intertwiners commute with
the intertwiners coming from {j (as they are independent of ~oa). Hence, the projection onto nKer r;oa is achieved by taking a half-infinite alternating sum. 21 > This will
be made more explicit now.
For G simple, the Cartan subalgebra currents of {j always are realized as h;(z)
=Jk+hv ia¢;+~ae.:~.(a;, a)ra(z)/3a(z), and thus after bosonizing we obtain
where
r;~
r;~-system
(3·110)
Hence we may realize fi in terms of
=!ka;ji(z), by the redefinition
.t "parallel" bosonic fields
;p simply as
h;(z)
(3·111)
A convenient set of "perpendicular" free fields,
vanishing 2-point functions are
w and i,
such that the only non
<¢i(z)¢i(w)>=-(a;, ai)ln(z-w),
<ia(z)ia'(w)>= -saa' ln(z- w),
<wa(z)wa'(w )>= saa' ln(z- w)'
(3·112)
(and thus w and x are trivially fi-invariant) is conveniently given by 18 >
(3·113)
In these fields the energy-momentum tensor for the G/H parafermion theory is
indeed "free-field" form
~
T GtH = £..J
ae<l+
(l(
·a ~a)2 _ _l_ z·a2 x~a_lt
·a2 ~a-( J k+
hv -If)(
) ·a2 ~a)
2'z·a w~a)2+_l_
v Jk
v
p, a z w .
2 zx
2
2z w
h
+h
(3·114)
Further, the {j-screening currents s; are independent of ¢-that is obvious since they
commute with the generators of {j, in particular h,-and thus the intertwiners in the
122
P. Bouwknegt, ]. McCarthy and K. Pilch
resolution (FA, d) of L~ simply reduce to "purely perpendicular" operators. This
shows explicitly that the Fock spaces FA.L of the perpendicular fields indeed form a
subcomplex. Denoting the value of the zero modes Pwa=- Pxa of wa, xa by na we have
FA.L=EBn,ezF~·l.L. Then from (3·113) the projection onto a fixed H-isospin ,.\ is
achieved by restricting the sum over naEZ to those satisfying A-t!=~ae.:~+naa.
Thus we are led to the complex
(3·115)
where
(3·116)
This complex is not yet a resolution of the irreducible coset representation since
by the bosonization we have introduced additional degrees of freedom l;oa. As
explained above these can be removed by considering a (half-infinite) extension of the
complex by the intertwining operators r;oa=feix',
0
0
,1,
,1,
d(-1)
···~FtP.L~
J,
···~
d(O)
~
d(l)
F1~~.L ~
,1,
7]oa
~
~
,1,
,1,
F1o>.L
,A
J,
7]oa
0
,1,
···
7Joa
~···
(3·117)
,1,
By construction the cohomology of this resulting (multi-dimensional) complex is
concentrated in F1~~.L, where it is isomorphic to the irreducible module L~~lf.
We may, of course, apply this resolution to rederive the formula for the character
given in (3·105). Using Lox=(1/2)PiPx+1), it is clear that the Fock space one
reaches from F~:}.L by taking m steps with the intertwining operator r;oa has a highest
weight vector whose conformal dimension differs from that of F~:J..L by the amount
1
1
1
-f<m+na)(m+na+ 1)---zna(na+ 1)=--zm(m+ 1)+nam.
(3·118)
Hence, taking the alternating sum of the zero modes contributing to the x trace over
the 7Joa-directions of the complex, produces the factors of ¢n in the result. The
alternating sum over the part of the complex coming from ii results, in the usual way,
in the sum over the affine Weyl group. As an aside, note that the result (3·102) can
now be understood as a consequence of the trivial 7Joa-cohomology! Also, the above
derivation of the character constitutes another proof of the identity (3 · 99).
The point of the reformulation was not the computation of characters. Rather,
it allows the construction of representatives of chiral vertices between irreducible
parafermion modules in the reduced free field space, as we now briefly indicate.
Take first a representative of the highest weight state in LA•·••· This is easiest when
A2 is in the finite dimensional irrep with highest weight A2, since the representative can
Free Field Approach to 2-Dimensional Conformal Field Theories
123
then be taken to be an exponential in free fields-otherwise there will be derivative
prefactors as discussed above. Now, since the screening operators only depend on
the perpendicular fields, the construction continues precisely as discussed in § 3.4.
For a generic coset model this problem seems difficult to attack using the BRST
projection technique we have presented here, and there is a clear need for further
investigation. It seems likely that progress might be made if the BRST projection
was modified along the lines of Refs. 105)'"'"'107).
Acknowledgments
We would like to thank the following: V. Dobrev, V. Dotsenko, B. Feigin, G.
Felder, M. Frau, E. Frenkel, K. Gaw~dzki, A. Lerda, G. Lusztig, N. Reshetikhin, S.
Sciuto, ]. Sidenius for discussions; D. Nemeschansky for collaboration on Ref. 46).
The Aspen Center for Physics where the writing of this work began. K. P. thank:;
M. I. T. for hospitality during various stages of this work.
Appendix A
- - Some Lemmas on the Weyl Group - -
In this appendix we establish some lemmas regarding the concept of twisted
length of Weyl group elements introduced in §§ 2.4 and 3.2. to make contact with
previous approaches. 40 J,soJ
We recall that the usual length of a Weyl group element wE W (for both finite
dimensional Lie algebras g and affine Kac-Moody algebras ii) is defined as the least
number of simple reflections that are required to write w as a string of these simple
reflections. An elementary result is the following :
Lemma A.1.73l
l(w)=i(J)(w)l,
(A·1)
where
(A·2)
We have the following generalization of Lemma A.1, which shows that definition
(2·57) of the twisted length lw coincides with the one given in Ref. 50)
Lemma A.2.
(A·3)
where
(J)w±(w')=(w(Ll±) nLI+) n w'(LJ_)
= w(Ll±) n (J)(w').
Proof We have
l(w- 1 w') -l(w- 1)= ILl+ n w- 1 w'(LI-)I-ILI+ n w- 1(L1-)I
= lw(Ll+) n w'(LJ_)I-Iw(Ll+) n LI-1
(A·4)
124
P. Bouwknegt, J. McCarthy and K. Pilch
=I w(L1+) n w'(LJ_) n .a+l +I w(L1+) n w'(LJ_) n L1-l
-lw(L1+) n LJ_ n w'(L1+)1-Iw(L1+) n LJ_ n w'(L1-)I
=I (/)w +( w')l-1 (/)w -( w')l .
D
For affine Weyl groups W we have the additional possibility of taking (at least
formally) infinite twists limN-ootNr characterized by some vector r, as explained in§ 3.
For these a similar lemma holds. Define thereto
.J~+>={a=n8+a, aEL1+, n~O},
.J~->={a=n8-a, aEL1+, n>O}.
(A·5)
Note that formally .J~±>=(limN-oofNr.J±) n .J+. Introduce furthermore
(J)±(w)=.J~±>n w(.J_).
(A·6)
Then we have
Lemma A.3.
(A·7)
where the twisted length lr is defined in (3·30).
The proof is exactly the same as that of Lemma A.2 in the finite dimensional case. In
particular it shows, as mentioned in § 3.2, that lr does not depend on the choice of r
EM provided (r, a) >0, V aEL1+, and that the definition of lr coincides with that given
in Refs. 40) (where it was called modified length) and 50).
Appendix B
- - Quantum Group Identities-In this appendix we collect some notations and lemmas which are used throughout the paper. Let qE C be such that q 2 =Fl. We use the following definitions from
q-number analysis:
qn-q-n
q -ql,
n
[n]q!=IT[k]q,
k=l
[m]
n
q
[n]q ! [m- n]q ! '
(B·1)
known as the q-number, q-factorial and q-binomial, respectively.
Let us review briefly the definition of the quantum group 'l.Jq(g). 108H 10> Suppose
g is a finite dimensional Lie algebra with Cartan matrix aii=(aj, a;v) of rank .f. Fix
a normalization such that (8, 8)=2 for the highest root 8 of g and defined;= {-(a;, a;)
such that d;aij=djaj;. Fix a complex number q such that q 2d'=F1 (1~i~.f).
Then, 'lJ q(g) is the associative C -algebra with generators e,, /,, MI, (1 ~ i ~f) (k,
~qa'), and relations (we use the conventions of Ref. 111))
Free Field Approach to 2-Dimensional Conformal Field Theories
l~ij -1)"[1- az:;J
j,l-ao-K /;/," =0
(
K=O
K
'
qdi
125
(B·2)
This algebra is endowed with a co-multiplication, co-unit and antipode which makes
it into a Hopf algebra. We refrain from giving their definitions as we will not need
them here.
We now describe the definition of the "quantum group Verma module" MAq. 112 >
We define MAq=CfJq(g)vA where VA is a (highest weight) vector satisfying,
i=1, ···,f.
e;VA =0,
(B·3)
The space MAq has an (overcomplete) basis consisting of monomials j,···/,nVA, and is
a CfJ q(g) module under the action
j;(/,···f,nVA)= j;j,···f,nVA,
V )=q(A,at)q-dt(a<t,+···+aunlj ···ftnA,
V
k z·(fl1 ···ftnA
it1
(B·4)
where a> is defined in (3·26). This module is integrable for AEP+ and reduces to the
conventional Verma module for gin the limit q-+1, i.e., is a deformation of MA. 112 >
For q a root of unity, say q=exp(7ri/(k+hv)), the algebra CfJq(n_) contains an
ideal .5 q generated by the elements Ia 2 <k+hVJt<a,a>, a ELl+. Let us define q] q(n_)
=CfJin-)/ Sq. Then we might also consider the "restricted quantum group Verma
module" MAq=q] in-)vA, where VA satisfies the identities (B·3). This becomes a
CfJq(g) module under the action (B·4).
The following lemma proves to be useful for the explicit determination of the
intertwiners :
Lemma B.1. 113 > Consider the associative algebra with two generators A, B and
defining relations
A 2 B-(q+q- 1 )ABA+BA2 =0,
(B·5)
AB2 -(q+q- 1 )BAB+B2 A=O.
(B·6)
Define C=-AB+q- 1 BA, then
(i)
AC=qCA,
qBC=CB;
(B·7)
At
At->
C>
Bk-J
Bk
2: q>+(k-j)(l-j)
•
(ii) - - - - - - =
[k]q ! [l]q ! o,;;,,;;mm(k,/)
[!- J ]q ! [j]q ! [k- j]q !
(B·8)
(iii)
(B·9)
(iv)
(B·10)
126
P. Bouwknegt, ]. McCarthy and K. Pilch
If in addition q is a root of unity, then there are additional relations.
Lemma B.2. 40 l Let A, B and C be as in Lemma B.l.
= M- k for 0 < k < M, then for 0 < l < k < M we have
Let qM = -1, and define
Ai;.+IBI=(-1)1 ~ qH<I-j)(k-j)
[l]q ![k]q!
Al-jCjBI-JAk.
[j]q ![/-j]q ![k-j]q!
'
(B·ll)
(ii) AIB.k+t=( -1)1 ~ qH<I-j)(k-j)
[l]q ! [k]q !
Bk Al-jCjBI-j
[j]q![/-j]q![k-j]q!
.
(B·12)
(i)
O~j~/
O~j~/
k
Appendix C
- - Cohomology of Double Complexes - -
In this appendix we summarize briefly elementary facts about the cohomology of
double complexes of the type discussed in §§ 2.5 and 2.6. For a more detailed
exposition and proofs the reader may consult standard textbooks on homological
algebra, e.g., Refs. 48) and 114).
C.l. A double complex (X, d', d") is a bigraded module X={X<Mllp,
qEZ} with differentials d': X<Ml--> x<P+l,q) and d": X<Ml--> X<M+ll such that
DEFINITION
d'd'=O,
d"d"=O
and
d'd"-d"d'=O.
(C·1)
If we represent a double complex as a family of modules in the p-q plane then the
first two conditions d' d'=O and d" d"=O say that each row and each column is a
complex. Denote the corresponding cohomology groups by H~f.ql(X) and H~f,.ql(X),
respectively. [The reader should not be mislead by the notation. Depending on the
case, p and q label the P-th cohomology group in the q-th complex in H~f.ql(X), or the
other way round in H~f,.ql(X).] Using the las,t condition d' d"- d" d'=O one verifies
easily that d": H~f.ql(X)--> H~f.q+ll(X) and similarly d': H~f,.ql(X)--> H~+l,ql(X) are
well-defined and nilpotent. The cohomology groups of these two complexes are
denoted by H~f,.qJ Hd'(X) and H~f.qJ Hd"(X), respectively.
With any double complex one can also associate a single complex (K, D), where
K={K<nl=EJjp+q=nX<Ml} and D=d'+( -1)Pd" on x<Ml. Properties (C·l) then guarantee that DD=O. A natural question that arises is to what extent can one determine
the cohomology of the complex (K, D) given some information about the cohomologies of the row and column complexes in the double complex. The answer
can usually be obtained by examining the so-called spectral sequence, and the results
we review in the remainder of this appendix are in fact the most elementary consequences of such analysis.
Let us restrict to a double complex in the first quadrant, i.e., such that x<Ml=O
if P<O or q<O. In §§ 2.5 and 2.6 we make frequent use of the following theorem.
THEOREM
C.l. Let (X, d', d") be a double complex such that the cohomology groups
H~f,.ql(X)=O
for q=t=qo.
Then
n<qo,
n?:::.qo.
(C·2)
Free Field Approach to 2-Dimensional Conformal Field Theories
Similarly,
if
H~f.q>=o
H}r>(K)
127
for P=I=Po, then
={H~t;,o.n-Po>Hd·(X~
:
n<Po,
n?:Po.
(C·3)
Proof We use the notation of Ref. 114), Ch. III). Since H~t;,.q>=t=o only for q=qo we
have mM>(X)=H~f.q>Hd"(X)=O for q=l=qo. But then d2=0 on E£*·*>(X), which
implies that in fact Ei*·*>(X)=E2(*-*>(x). Then (C·2) follows from the fact that the
cohomology of the single complex is given by (Ref. 114), Theorem 14.14, p. 165)
(C·4)
D
The proof of (C·3) is analogous.
Remark The double complexes that are discussed in§§ 2.5 and 2.6 can be brought to
the first quadrant by a suitable relabelling.
An interesting consequence of Theorem C.1 is
Lemma C.2.
If
H~f.q>(X)=O
for P=I=Po and
H~·q>=o
for q=l=qo then
(C·5)
Proof
Take n=p+qo in (C·2) and (C·3).
Appendix D
--Proof of Theorems 2.12 and 2.12' - -
THEOREM 2.12. Let FA be a non-twisted Fock space G-module with the highest weight
A (not necessarily dominant), and SA the space of all H-singular states in FA. Then
the ERST cohomology of FA considered as an H-module is given by
(D·1)
Proof In this proof we make use of standard techniques as elaborated for a similar
problem in Ref. 43). Since we identified the positive root generators of h with those
of g, we see that the general form of the representation of ea, aELl+ H in FA, which we
discussed in § 3.1, is
ea =,a a+ higher degree terms .
(D·2)
On the space FA®Fgh we can introduce additional grading according to the "total
ghost number", i.e. we define a degree (deg) such that
deg(ya)=-deg(,Ba)=1,
aELJ+G,
deg(c-a)= -deg(ba)=1, aELl+ H.
(D·3)
(D·4)
Then the "higher degree terms" in (D·2) have deg> -1. Also, on defining
(D·5)
we can split Q as
128
P. Bouwknegt, J. McCarthy and K. Pilch
(D·6)
Let NH="'2.aE.tJ.•(c-aba-ra/3a) be the total "H-ghost number" operator.
that all eigenvalues of this operator are non-negative. Then we have
Lemma D.l.
Note
The cohomology of Qo on FA@Fgh is
H~~'(FA@Fgh)-::::<on,oFA.l.,
(D·7)
where FA.l. is the subsPace of FA annihilated by /3a, aE.Ll+ H.
Proof Using the identity
{Qo,- "'2, bara}=NH'
(D·8)
aELJ+H
we see that any closed state ¢, NH¢=N¢, N=i=O, is also exact. Indeed,
(D·9)
Thus the cohomology of Qo must be concentrated at n=O on the states which are
annihilated by all /3a, aELJ+H· These states are in 1-1 correspondence with FA.l.. D
To proceed further we must introduce yet another gradation in the space FA@Fgh.
Since the embedding is regular we can consistently assign G-weights (!) +a and -a
to the ghosts ba and c-a, respectively. The complex has then G-weight direct sum
decomposition, FA@Fgh="'i},;.F;., and each weight space F;. is mapped by Q and Qo into
itself. Thus we can consider each subcomplex (F;., Q) separately. Moreover, since
Fgh, and similarly each weight space in FA, are finite dimensional, we see that the F/s
must also be finite dimensional.
We will now prove that for n > 0 the cohomology of Q is trivial. Observe that ¢
EF}n> is of the form¢= rf;t+ ··· + ¢L, rf;t=i=O, f>O, where vectors¢, have definite degree,
deg(rf;i)=i, i=f, , L, and Lis the maximal degree occurring in F}n>. Call ord(¢)
=deg(rf;t)=f. Since
(D·10)
Q¢=0 implies Qor/Jt=O. But then using Lemma D.1 for n>O we find r/Jt=QoXt, where
deg(xt) =deg( rf;t). Consider now ¢- Qx~. which represents the same cohomology
class. Since ord(¢-Qxt)>ord(¢), it is clear that repeating the same reasoning L-f
more times we will find vectors Xm, ···,XL such that rf;=Q(xt+···+xL). In the last
step one uses that Q>xL=O, because ord(Q>xJ>L. Thus we have shown that the
cohomology of Q is concentrated at n=O, and, since
(D·ll)
it is given by (D·1). This concludes the proof of Theorem 2.12.
D
In the above proof we used only a part of the result in Lemma D.1, i.e. the
triviality of the Qo cohomology for n >O (which implied N>O). If we parametrize the
Fock space FA such that the generators of n+ H are realized using only the parallel
variables, ra, /3a, aELJ+H then we can prove the following sharpening of Theorem 2.12.
Free Field Approach to 2-Dimensional Conformal Field Theories
THEOREM 2.12'.
129
If the generators ea, aELh H are realized only in terms of parallel
variables then
(D·12)
Introduce an H-degree (deg') and the corresponding H-order (ord') by restricting a in (D·3) to lie in LJ+ H. [Obviously, this H-degree coincides with the H-ghost
number.] We can repeat the entire proof of Theorem 2.12 with deg and ord replaced
with deg' and ord'. We find that a nontrivial cohomology of Q must be concentrated
on states with ord'=O. The cohomology states of Qo are in 1-1 correspondence with
vectors ¢EFA such that deg'(¢)=0 (Lemma D.1). Clearly all such vectors define also
cohomology states of Q. On the other hand if ¢1 and ¢z, Q¢1 = Q¢z=O, have the same
component of deg' = 0, then ord'( ¢1- ¢z) > 0, which implies [ ¢1] = [ ¢z] in H~o). This
shows that both cohomology of Q and Qo are realized by the same vectors which have
vanishing total H-ghost number.
D
Proof
Appendix E
--Restricted QG Verma Modules--
In this appendix we argue that the cohomology of a naive generalization of the
BGG complex to restricted QG Verma modules is trivial. We will consider only the
simply-laced case.
To define a BGG-like complex for restricted QG Verma modules we observe that
the standard BGG complex, (MA, d), can be obtained from the Fock space resolution,
(FA, d), of an irrep LA of a semisimple Lie algebra as follows: Given (FA, d) each
component dw',w: Fw*A ~ Fw'M of the differential d corresponds to a singular vector,
dw',w[s,Hj,]vw*A• in the Verma module Mw*A (Ref. 40) Theorem 3.6), and thus defines
an embedding dw,w': Mw'*A ~ Mw*A· The BGG resolution is constructed by simply
replacing Fock spaces with Verma modules (with the same highest weight) and
reversing all the arrows in the complex.
Similarly, given the Fock space resolution of an irrep LA of Kac-Moody algebra
fj we have found (Theorem 3.5) that each component dw',w of the differential corresponds to a singular vector in the restricted Verma module MZ,*A of the quantum
group, CU q(g ). By the same procedure as in the finite dimensional case we then obtain
a complex (MA q, d) of CU q(g) restricted Verma modules. Our discussion of the
CU isu(2)), CU isu(3)) and CU isu(N)) cases, which is presented below, suggests that the
following is true :
THEOREM E.l.
The cohomology of (liiAq, d) is trivial.
By definition the restricted Verma module MAq, AEP+, is isomorphic with the
quotient CfJq(n_)=CUin-)/ .J Aq, where .J Aq is the (double-sided) ideal generated by the
elements t:+hV, aELl+. For g=su(2), q] in-) is generated by a single element, /,
satisfying jk+ 2=0, and thus, as a vector space, is spanned by 1, /, ···, jk+I. By
inspection we obtain
Proof
Lemma E.2.
For ¢EV q(n~u<z)),
130
P. Bouwknegt, ]. McCarthy and K. Pilch
cJ;r=o iff
cf;=xJk+Z-n'
n:::;;:k+1
(E·l)
for some xEq) q(n~u<z>).
The proof of the theorem in the CfJ q(su(2)) case follows directly from this lemma.
The algebra Cf)q(n'!!< 3 >) is generated by/,, i=1, 2, which satisfy
(E·2)
together with the Serre relations (B·2). The PBW theorem gives then two convenient bases in CfJ q(n~u< 3 >), 115>' 116 > which consist of elements of the form
Jr!t.f/,
i=l=j, O:::;;:m, n, p:::;;:k+2,
(E·3)
one with i=1, j=2 and the other with i=2, j=l. The analogue of Lemma E.2 in this
case is
cf;//=0 iff
cf;=x/.r 3-n, j=1, 2,
n:::;;:k+2
(E·4)
for some xEq) q(n~u<s>).
To prove it one expands cf; in one of the bases (E·3) and explicitly constructs x so that
(E·4) is satisfied.
In the context of the oomplex (MAq, d) the result of Lemma E.3 can be interpreted
that the restriction to the line along a simple root direction yields a complex with
trivial cohomology. More precisely, given MZ,*A, wE W, and a;, the restricted
complex is of the form
(E·5)
where t;=(w *A+ p, a;)mod k+3 and T;=k+3- t;, with the normalization of the
embeddings determined by that of d, as in (3·39) (see also Ref. 40)).
Introduce a filtration · · · 2 Kz 2 K12 Ko 2 K-12 ·· · of the complex (MA q' d) such that
factor complexes Kn+l/Kn coincide with complexes (E·5). Explicitly, we may construct spaces Kn, nEZ as follows. Let A, be a fundamental weight in P+. Then
K21-1=
lEZ.
(E·6)
The complex (MAq, d) has a weight space decomposition. As usual let (MA,A, d)
denotes the subcomplex with weight A. Using vanishing conditions (E·2) one deduces
that there can only be a finite number of spaces in this complex which are non-zero.
Thus the filtration (E·6) induces a finite filtration {Kn(A), nEZ} of (MA,A, d). Moreover, we have just shown that for each n the quotient complex associated with this
filtration has a trivial cohomology. A standard way to proceed (see, e.g. Ref. 114), pp.
157-158) is to consider a long exact sequence in cohomology that arises from the short
Free Field Approach to 2-Dimensional Conformal Field Theories
131
exact sequence
0 ~ Kn(.-1)
Since
~
Kn+l(.-1)
~
H~i>(Kn+l(.-1)/Kn(.-1))=0,
Kn+l(.-1)/Kn(.-1)
~
0.
(E·7)
Vi, we find that
H~i)(Kn+l(..t))~H~i>(Kn(.-1)),
Vi, nEZ.
(E·8)
However, for a fixed .-1, Kn(.-1)={0} provided n is large enough, which together with
(E·8) implies that in fact H~;>(Kn(.-1))=0 for all i and n. Thus the cohomology of
(MA,;., d), and consequently of (MA, d) is trivial.
It is straightforward to extend the above arguments to CU isu(N)) using the
induction on N. Choosing N -1 simple roots one may define a filtration such that the
quotient complex corresponds to a complex of CU isu(N -1)). The reminder of the
proof proceeds then as above. In fact it is clear that the same result should hold for
any quantum group CU q(g).
D
In§ 3.4 we consider a quotient module LAq=MAq/MJH>, where as usual ifx<-I> is
the submodule generated by the singular vectors flA+p,a,v>vA. In fact one can introduce such a quotient for an arbitrary value of q=I=O (when q is not a root of unity one
takes the quotient of the full Verma module by its maximal submodule). For AEP+,
LAq is an irreducible module of the quantum group/ 17> and it is clear that for a generic
q the structure of LAq should be exactly the same as of the irreducible representation
of the underlying finite dimensional Lie algebra (see e.g. Ref. 112)). However, it is
less clear when q is a root of unity, because the overlaps between submodules
generated by different singular vectors may in principle change due to some peculiar
identities between q-numbers. On the other hand if we consider weights which are
integrable, i.e. (A+ p, 8)< k+ hv, it is reasonable to expect that the structure of LAq
will be similar to its finite dimensional counterpart, because in this case only "low
powers" of the generators are present. In fact the latter observation is true. We
will now demonstrate this in the case of CU q(su(2)) and CU q(su(3)).
E.4. Let A be a dominant integrable weight of CU q(g). Then there is a 1-1
correspondence between the states in LAq and in the corresponding representation LA of
the finite dimensional Lie algebra g.
THEOREM
Proof We will present a somewhat pedestrian approach in the case of CU q(su(2)) and
CUq(su(3)).
In the case of CUq(su(2)) the states of L]a, 2j~k, are spanned by
(E·9)
so the theorem is obvious.
In the case of CU q(su(3)) we will construct explicitly the basis in LAq using the
bases (E·3) in the restricted QG Verma module. We will see that in comparison with
the finite dimensional case the only complication is to keep track of vanishing
conditions (E·2).
Let t1=n1a1 +n2a2 be a weight, ;\~A, with the corresponding weight spaces MAq(.-1)
and MA(.-1) in the Verma modules MAq and MA, respectively. Let l;=(A+p, a;). We
132
P. Bouwknegt, ]. McCarthy and K. Pilch
need to consider four cases : 1. n1 < /1 and n2 < l2 ; 2. n1 < /1 and n2;;::.: !2 ; 3. n1;;::.: /1 and n2
< !2 ; 4. n1zl1 and n2 z/2. In the first case all the states
-rn•-J112
-{J f:nz-J
· 0, ···,m1n
· ( n1, n2 ) ,
11
2 VA, ;=
(E·10)
that span MAq(ll) and/or MA(Il) are linearly independent in LAq and/or LA. In the
second case we find the same correspondence, except that n2 -!2 < j s. n1, where the
lower range restriction removes the states in (E·10) which are descendants of the
singular vector j.f•vA. Clearly if n1S.n2-l2 there will be no states in the quotient.
Similarly, using the other basis in (E·3), we show the 1-1 correspondence between the
states in LAq(ll) and LA(Il) in Case 3. In Case 4 let us assume that n1S.n2. Then in
the Verma modules we have bases:
(E·ll)
(E·12)
Vectors Vj in (E·ll) with jS.n2-l2 span the submodule MMA(Il) of MA, while Vj
in (E·12) with j S.minCn2-l2, k+2) span the submodule M~•*A(Il) of MAq. The submodules MnM(I!) and M~•*A(Il) are spanned by a set (possibly overcomplete in the
case of M~.*A(Il)) of vectors
(E·13)
max(O, n2- k-2)S.j sminCn1-l1, k+2).
Let us first consider MMA(Il).
(E·13) into the basis (E·10)
(E·14)
Using identity (B·8) with q=1 we can expand
(E·15)
Note that all coefficients in the sum on the r.h.s. are nonvanishing, and the highest
term in the expansion of iJJ is vJ+li· If /1 S. n2-l2+ 1, we can use (E·15) to express all
basis vectors (E·ll) as a linear combination of iJ/s and vectors in Mrz*A(Il). Thus in
this case LA (I!) is trivial. If /1 > n2 -12 + 1, we first observe, by the highest term in the
expansion (E·15), that none of iJJ lies in Mrz*A(Il). Thus MAHl(l!)= MMA(Il)
ffiMrz*A(Il). Then it is clear that the vectors Vnz-lz+l, ···, Vz,-~ form a basis in LA(Il).
In M~•*A(Il) we obtain using (B·8)
(E·16)
Once more all the coefficients on the r.h.s. are non-zero if the j's are within the
indicated range (E·14). Thus, if /1S. n2-l2+ 1 the same argument as above shows
that LAq(ll) is trivial. If l1>n2-l2+1, the integrability of A implies that n1sk+2.
Indeed, if we had n1 > k + 2 then also n2 > k + 2 and therefore I!:::; woA. Since woA is
the lowest weight in LA, we deduce that LA(Il) is trivial and thus we must have /1
sn2-l2+1, which is a contradiction. Since n1sk+2, all iJ/s in (E·14) are linearly
Free Field Approach to 2-Dimensional Conformal Field Theories
133
independent, and, exactly as in the finite dimensional case, we show that Vnz-tz+l, ···,
V1,-1 form a basis in LAq(t\).
In the case nz~n1 the proof is the same using the other basis (E·3). This
concludes the proof of the theorem in the case of CU q(su(3)).
D
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