Download Superconformal field theories in six dimensions

SCFTs in 6D
David R. Morrison
Superconformal field theories
in six dimensions
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
David R. Morrison
Examples
F-theory
University of California, Santa Barbara
Recent Progress in String Theory and Mirror Symmetry
Brandeis University
7 March 2015
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
David R. Morrison
Superconformal field theories
in six dimensions
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
David R. Morrison
Examples
F-theory
University of California, Santa Barbara
Recent Progress in String Theory and Mirror Symmetry
Brandeis University
7 March 2015
Based on work done with M. Bertolini, M. Del Zotto,
J. J. Heckman, P. Merkx, D. Park, T. Rudelius, and C. Vafa
arXiv:1312.5746, arXiv:1412.6526, arXiv:1502.05405,
arXiv:1503.?????
Quivers
Classification
Finite subgroups of
E8
Introduction
I
I
I
The maximum spacetime dimension in which a
superconformal field theory is possible is six (Nahm).
The degrees of freedom in such a theory are not
described by particles, but the theory is a local quantum
field theory (Seiberg and others).
The worldvolume quantum field theory for a (stack of)
M5-branes is a six dimensional superconformal field
theory (with maximal supersymmetry).
I
Compactification of the maximally supersymmetric
theory has led to a host of interesting theories in lower
dimesions (very active area of research since 2009).
I
In this talk, we will focus instead on the minimally
supersymmetric theories, that is, theories with
N = (1, 0) supersymmetry.
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
N = (1, 0) superconformal field theories
SCFTs in 6D
David R. Morrison
Introduction
I
The conformal symmetry of these theories is so(6, 2).
I
The superconformal algebra is described with 8
supersymmetry generators Qi and 8 superconformal
generators Sj .
I
The theory has an su(2) R-symmetry.
I
These theories typically have nontrivial global (flavor)
symmetries.
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
N = (1, 0) superconformal field theories
SCFTs in 6D
David R. Morrison
Introduction
I
The conformal symmetry of these theories is so(6, 2).
I
The superconformal algebra is described with 8
supersymmetry generators Qi and 8 superconformal
generators Sj .
I
The theory has an su(2) R-symmetry.
I
These theories typically have nontrivial global (flavor)
symmetries.
Multiplets in a 6D supersymmetric theory:
I
+ , fermions)
Gravity multiplet (gµν , Bµν
I
− , fermions)
Tensor multiplet(s) (S, Bµν
I
Vector multiplet(s) (Aµ , fermions)
I
Hypermultiplet(s) (φR , fermions)
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Strings and their tension
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
I
I
I
I
Having Bµν ’s implies that there are strings in these
theories, and a BPS lattice Λ ⊂ R1,T .
The scalars S1 , . . . ST are naturally parameterized by a
space whose universal cover is SO(1, T )/SO(T ).
The expectation values hSi i control the string tensions.
When hSi i = 0 we get a tensionless string, and we
expect a local superconformal field theory to describe it.
Non-zero expectation values of Si parameterize the Coulomb
branch of the theory, and this is where the theory is typically
studied in detail.
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Anomaly-free gauge fields
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
I
I
I
On the Coulomb branch, the gauge fields are potentially
subject to an anomaly.
The Green-Schwarz-West-Sagnotti mechanism specifies
modified Bianchi identities for gauge fields in these
theories, which gives the possibility of an anomaly-free
theory: potential anomalies from fermions in matter
multiplets are cancelled by the gauge field anomaly
which follows from the modified Bianchi identity.
In practice, this requirement is a severe constraint on
the gauge groups and matter representations.
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Anomaly-free gauge fields
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
I
I
For example, one solution of the anomaly constraints
allows an arbitrary number of copies of su(N), labeled
by an integer j = 1, . . . , p, together with (N, N)
bi-fundamental matter charged under su(N)j and
su(N)j+1 , with an extra N fundamentals for su(N)1 and
an extra N fundamentals for su(N)p .
In fact, these theories also have a global symmetry
SU(N) × SU(N) which can be thought of as occupying
the 0th and (p+1)st positions in the chain, so that all of
the matter sits in bi-fundamentals. (Sometimes it’s
gauge-global bi-fundamentals.)
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Other anomalies
I
I
I
The global symmetry group G of such a theory is
typically anomalous, and cannot be directly gauged.
However, in many cases one can add additional (free)
hypermultiplets charged under G to render the
combined theory anomaly-free.
For example, in the su(N)⊕p theory described above,
the global symmetry SU(N)p+1 is anomalous unless an
additional N fundamentals for this group are added to
the theory. Then the combined theory has an
anomaly-free global symmetry, which can be gauged
(using the GSWS Bianchi identity), extending the chain
by 1.
One can also ask if the theory can be coupled to
gravity, a necessary condition for which is that the
gauge-gravity anomaly vanish (using the modified
Bianchi identity for the graviton as well).
This was studied in some nontrivial cases in [Del Zotto,
Heckman, Morrison, Park].
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Examples
SCFTs in 6D
David R. Morrison
Introduction
I
I
I
The N = (2, 0) theories have an ADE classification.
N=(1, 0) SCFTs
A stack p of M5-branes gives one model for the
N = (2, 0) theory of type Ap−1 .
Strings
The heterotic e8 ⊕ e8 string compactified on a local K3
surface with a point-like instanton with instanton
number k provides a family of examples of N = (1, 0)
theory.
I
Using the Hořava–Witten model of the heterotic string,
the previous example can be viewed as p M5-branes
dissolved into the boundary M9-brane, creating a
point-like instanton in real codimension 4.
I
More generally, one can put p point-like instantons at
the singular point of an ALE space C2 /Γ.
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
1
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
1
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
These examples can be studied in F-theory, using
F-theory/heterotic duality [Aspinwall-Morrison ’97].
6D SCFTs from F-theory
I
I
F-theory is a quantum gravity theory which allows a
nontrivial axio-dilaton profile, exploiting the SL(2, Z)
symmetry of the type IIB string.
The data of an F-theory compactification is provided by
a base space B, a complex line bundle L on B, and
sections f and g of L⊗4 and L⊗6 , respectively,
determining a Weierstrass equation
y 2 = x 3 + fx + g .
I
I
If B has complex dimension 2 and if the total space of
the elliptic fibration can be blown up to a Calabi–Yau
threefold, we get a 6D supersymmetric theory.
The strings arise from wrapping D3-branes along
complex curves in B, and their tensions are supplied by
the areas of these curves. Thus, to get a tensionless
string limit, we need a configuration of algebraic curves
which are contractible to a point.
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
I
More generally, we can study a local model B which is a
neighborhood of a curve collection {Σj }. The key
condition for contractibility is that the intersection
matrix (Σj · Σk ) be negative definite. This matrix also
defines the lattice of BPS string charges.
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
I
The key quantity for understanding these models is the
discriminant locus
∆ := {4f 3 + 27g 2 = 0}.
This typically contains some or all of the Σj ’s as
components with some multiplicity. The Kodaira
classification determines the type of singular fibers and
also (when supplemented by Tate’s algorithm) the
gauge algebra associated with each Σj .
I
There can be additional, non-compact components of
∆ in B. These are associated with global symmetries.
Detailed investigation: [Bertolini, Merkx, Morrison].
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Pointlike instantons on C2 /Γ, as studied by [Aspinwall-Morrison
’97].
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
12 + n
II∗
C∞
Strings
No anomalies
Examples
∗
II
12 − n
F-theory
C0
Figure 2: Point-like E8 instantons in the simplest case.
Quivers
Classification
Finite subgroups of
E8
re the curly line represents the locus of I1 fibres. This will be the case in all subsequent
rams. The overall shape of this curve is meant to be only schematic. (In particular, we
e omitted the cusps which this curve invariably has.) The important aspect is the local
metry of the collisions between this curve and the other components of the discriminant
ch we try to represent accurately.
This is the F-theory picture of the physics discussed in [6] that each point-like instanton
s to a massless tensor in six dimensions (here represented as a blowup of the original
e Fn ). We also see that 12 − n of the instantons are associated to one of the E8 factors
the other 12 + n are tied to the other E8 [14]. After blowing up the base however, one
blow down in a different way to change n. Thus after blowing up, it is not a well-defined
n this section we are going to force a “vertical” line of bad fibres (along an f direction)
the discriminant
so that it has aon
transverse
“horizontal” line of II∗ SCFTs in 6D
Pointlike instantons
C2 /Γ, intersection
as studiedwith
by the
[Aspinwall-Morrison
along C0 without any additional local contributions to the collision from the rest of the David R. Morrison
’97].
minant. One may show [34] that such intersections of curves within the discriminant
correspond to fibres with the same J-invariant. In this section we require J = 0 which Introduction
sponds to Kodaira types II, IV, I∗0 , IV∗ , and II∗ . In each case, the order of vanishing N=(1, 0) SCFTs
12 +
s twice the order of vanishing of b, with
an
playing no significant
II∗ rôle. Thus, to" analyze Strings
∞ divisor B .
= 0 cases we need only concern ourselves with the geometry of C
the
No anomalies
or example, let us consider the case illustrated in figure 4 in which we add a vertical
of II∗ fibres along the f direction. To do this, we must subtract 5f from B " which Examples
es that what remains can only produce 7 − n and 7 + n simple
point-like instantons of F-theory
II∗
ype we discussed above. It is therefore clear that, whatever else C
we
0 may have done to Quivers
− ten
n of the instantons. Note
uce this extra line of II∗ fibres, we have had to “use 12
up”
Classification
B " intersects f twice, producing collisions between the I1 part of the discriminant and
∗
Finite subgroups of
ertical line of II fibres as shown.
E8
Figure 2: Point-like E8 instantons in the simplest case.
7+n
II∗ C∞
re the curly line represents the locus of I1 fibres. This will be the
case in all subsequent
rams. The overall shape of this curve is meant to be only schematic. (In particular, we
e omitted the cusps which this curve invariably has.) The important aspect is the local
metry of the collisions between this curve and the other components
of the discriminant
II∗
C0
ch we try to represent accurately.
7 −[6]n that each point-like instanton
This is the F-theory picture of ∗the physics discussed in
s to a massless tensor in sixIIdimensions (here represented as a blowup of the original
e Fn ). We also see that 12 − n of the instantons are associated to one of the E8 factors
the other 12 + n are
tied 4:to 10
theinstantons
other E8 [14].
blowing up the base however, one
Figure
on anAfter
E8 singularity.
blow down in a different way to change n. Thus after blowing up, it is not a well-defined
SCFTs in 6D
David R. Morrison
7 + n II∗
Introduction
C∞
N=(1, 0) SCFTs
Strings
II∗
6−n
∗
II
No anomalies
C0
Examples
F-theory
Blow up
Quivers
#
II∗
!
!
!
!
!
"
"
!
! "
"
"
"
" II∗
Classification
7 + n II∗
II∗
6−n
C∞
C0
Figure 5: 11 instantons on an E8 singularity.
Finite subgroups of
E8
SCFTs in 6D
The intersection of two E8 branes (Kodaira type II ∗ ) is
David R. Morrison
associated to a Weierstrass model whose blowup is a
Calabi–Yau threefold X whose map to the base B has a
Introduction
(complex) two-dimensional fiber over the intersection
N=(1, 0) SCFTs
point.
Strings
I In other words, when we compactify on a circle to get
No anomalies
Examples
an M-theory model, we find an infinite tower of light
F-theory
states (from wrapping an M2-brane over arbitrary
Quivers
algebraic curves within the two-dimensional fiber). This
Classification
is another signal of conformality in the parent
The elliptic fibration of figure 4 is quite singular and requires many blowups in the base
Finite subgroups of
six-dimensional
before it becomes
smooth. Fortheory.
example, the degrees of (a, b, δ) for II∗ fibres are (4, 5,E10)
8
espectively.
two such
curves
and we
up the point of
I OnThus,
the ifother
hand,
by intersect
blowingtransversely
up the base
B, blow
we can
ntersection, the exceptional divisor will
contain
degrees (8, 10, 20). As in section 3, this
e →
e are
ensure that all fibers X
B
one dimensional. This
ndicates a non-minimal Weierstrass model, and when passing to a minimal model, L is
the that
Coulomb
branch
thethese
theory.
adjusted inisa way
subtracts
(4, 6, 12)offrom
degrees and restores KX to 0. We are
hus leftIwith
an exceptional
of degrees
(4, 4, in
8), detail
which isfor
a curve
of IV∗ fibres. This
Aspinwall
and Icurve
worked
this
out
the collision
new curve will intersect
the old curves of II∗ fibres and these points of intersection also need
∗
of two II fibers.
blowing up. Iterating this process we finally arrive at smooth model (i.e., no further blowups
I
need to be done) when we have the chain
!
! II∗
!
I0
!
"
!
"
!
"
!
"
!
" ! I∗0
" ! IV∗ " ! I∗0
" ! II
! II
! "IV
! "II
! "II
! "IV
!
I0
!
! II∗ .
!
Quivers
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
I
The su(N)⊕p field theory examples can also be realized
by constructions in heterotic M-theory and in F-theory.
I
They are given by a stack of p M5-branes at an AN−1
singularity.
Quivers
I
They also have a quiver description as above.
Finite subgroups of
E8
I
They are realized in F-theory by a chain of −2 curves
with Kodaira fibers of type IN over each one.
Examples
F-theory
Classification
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
Variants
David R. Morrison
Introduction
I
Other types of quivers can realized by variants of this
construction.
N=(1, 0) SCFTs
Strings
No anomalies
I
For example, putting flavor branes IN and IN+pk
through an Ap−1 singularity leads to:
Examples
F-theory
Quivers
Classification
IN
IN
D1
D2
A(p-1)
(p-1)
IN+pk
IN+pk
with gauge algebra
Finite subgroups of
E8
D(p-2)
...
D
Lp−1
j=1
su(N + jk).
Classification of 6D SCFTs from F-theory
I
I
I
I
A key ingredient is non-Higgsable clusters [Morrison-Vafa
’96, Morrison-Taylor ’12]: curve configurations with a
minimal gauge algebra. They are: 3, 4, 5, 6, 7, 8, (12),
32, 232, 322.
The classification is largely bottom up, relying on field
theory and anomaly cancellation. One hopes to
eventually understand it purely in those terms.
Classification result #1: contracting {Σj } leads to a B
of the form C2 /G with G ⊂ U(2).
Classification result #2: all 6D SCFTs in F-theory can
be described by quivers, whose links may themselves be
SCFTs. (Nodes and links are explicitly classified.)
SCFTs in 6D
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
A test: finite subgroups of E8
The basic heterotic M-theory example
SCFTs in 6D
David R. Morrison
Introduction
1 N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
has an important variant: the instanton solution may be
twisted by a Wilson line. Such Wilson lines correspond to
homomorphisms Γ → E8 up to conjugacy.
A test: finite subgroups of E8
The basic heterotic M-theory example
SCFTs in 6D
David R. Morrison
Introduction
1 N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
has an important variant: the instanton solution may be
twisted by a Wilson line. Such Wilson lines correspond to
homomorphisms Γ → E8 up to conjugacy. The E8 global
symmetry breaks to the commutant of the image of Γ.
SCFTs in 6D
David R. Morrison
I
I
Cyclic subgroups of E8 (up to conjugacy) were classified
by Victor Kac. For example, there are two cases for Z2 ,
with commutants (E7 × SU(2))/Z2 and Spin(16)/Z2 ,
respectively.
Certain other subgroups of E8 (up to conjugacy) were
classified by D. D. Frey. This includes some dihedral
groups as well as SL(2, F5 ) and A5 . All of these are
relevant for ADE subgroups of SU(2).
I
On the other hand, we can use the F-theory
classification to ask what are the infinite chains, and
how an infinite chain can end.
I
There is an almost perfect match!
Introduction
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
SCFTs in 6D
David R. Morrison
Group
ΓAn−1 = Zn
ΓDn = Dn−2
ΓE6 = T
∼
= SL(2, F3 )
ΓE7 = O
ΓE8 = I
∼
= SL(2, F5 )
Order
n
4(n−2)
Generators
ζn
ζ2n−4 , δ
24
ζ4 , δ, τ
Quotient groups
Zk if k | n
Z2 , Dih2k if k | (n−2),
D` if ` | (n−2) but 2` 6 | (n−2)
Z3 , A4
48
120
ζ8 , δ, τ
−(ζ5 )3 , ι
Z2 , S3 , S4
A5
e 2πi/n
N=(1, 0) SCFTs
Strings
No anomalies
Examples
F-theory
Quivers
Classification
Finite subgroups of
E8
Finite subgroups of SU(2), where
ζn ≡
Introduction
, δ≡
−1
e −2πi/n
2πi/5
1
e
+ e −2πi/5
ι ≡ 4πi/5
1
e
− e 6πi/5
e −2πi/8
e 10πi/8
1
.
2πi/5
−2πi/5
−e
−e
1
1
, τ ≡ √
2
e −2πi/8
e 2πi/8
SCFTs in 6D
E7 × SU(2) cases
David R. Morrison
Introduction
N=(1, 0) SCFTs
Strings
su2
[E7 ] 1 2
su4
su4
2
[SU(2)]
No anomalies
... 2 [SU(4)]
Examples
F-theory
su2 su4
[E7 ] 1 2 2
su2
[E7 ] 1 2
su6
[SU(2)]
so7
3
[SU(2)]
Quivers
su6
2
... 2 [SU(6)]
Classification
Finite subgroups of
E8
so9 sp1 so11 sp2 so13 sp3 so15
1 4 1
4
1
4
1
spn−4
4 ... 1
[SO(2n)]