Download Gauge theory and knots

CHERN-SIMONS THEORY AND TOPOLOGICAL INVARIANT
FRANCESCO DILDA
1. Introduction
In this note, we review the first part of the renowned Edward Witten’s paper “Quantum
Field Theory and the Jones Polynomial” ([4]). The aim of this seminar for the QFT2 exam is
to highlight some introductory aspects of how, through quantum field theory, it is possible to
compute topological invariants of manifolds. Specifically, I consider a Chern-Simons action on
a 3-dimensional manifold and demonstrate that computations using its path integral partition
function (whether with or without certain gauge-invariant observables) lead to the topological
invariance of the fiber bundle structure. For basic concepts on connections in a vector fiber
bundle, I refer to [1], as well as, for a more rigorous treatment to [2] or some times to Wikipedia.
For some more explicit computations I sometimes gave a look to [3].
Introduction of framework and some notation. We will work on an oriented 3-dimensional
boundaryless manifold M , which is the base space of a trivial G-fiber bundle with fiber F .
On F we have an action of G through a representation R (with an induced representation
ρ := de R of his Lie algebra g) as shown in Figure 1. We will pick a compact simple Lie group
G. Due to triviality, I have a preferred choice of a (flat) connection
d0 : Ωp (M ) ⊗ F −→ Ωp+1 (M ) ⊗ F
ω a ⊗ ea 7−→ dω a ⊗ ea
where {ea } ⊂ F is a chosen basis of the fiber (and {ea } ⊂ F ∗ dual basis). Then, we can write
any connection on M as DA = d + A where A ∈ Ω1 (M, ρ(g) ⊂ End(F )) (connection 1-form),
so that it is possible to expand A = Aab ⊗ ea ⊗ eb , with Aab ∈ Ω1 (M ), which acts as
A : Ωp (M ) ⊗ F −→ Ωp+1 (M ) ⊗ F
sa ⊗ ea 7−→ (Aab ∧ sb ) ⊗ ea
and all these definitions are intended to be extended to the whole domain space by R-linearity.
Given a connection 1-form we can define the corresponding curvature 2-form
FA := d0 A + A ∧ A
where
∀A ∈ Ωp (M, End(F )), B ∈ Ωq (M, End(F ))
A ∧ B := (Aab ∧ Bcb ) ⊗ ea ⊗ ec
Finally, we can act on our sistem with a gauge transformation, which for a trivial bundle
coincide with a smooth function
g : M −→ G
We will call the group of gauge transformations G.
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2
FRANCESCO DILDA
R
End(F)
M × F
G
ρ
π
g
M
Figure 1. A diagram of the set up
The Chern-Simons action functional. On M we can consider the Chern-Simons invariant
(with some modifications, it will be our action functional), depending only on the choice of A
Z
2
I[A] :=
Tr A ∧ d0 A + A ∧ A ∧ A .
3
M
It can be easily shown that this action is invariant under infinitesimal gauge transformations;
however, under large gauge transformations (not homotopic to identity) it varies by a constant
times an integer. In fact, given the transformed connection A0 = Adg (A) + gdg −1 , defining
As := A + s(A0 − A) with s ∈ [0, 1], then we can extend the structure of trivial fiber bundle
on M × [0, 1], choosing As as connection on it and so
SCS (A0 ) − SCS (A) =
Z
d Tr(As ∧ d0 As + As ∧ As ∧ As )
M ×[0,1]
Z
Tr(Fs ∧ Fs ) = 8π 2 N
=
M ×[0,1]
for some N ∈ Z. In the last equality, we have used the fact that Chern’s forms, when
integrated, give integer value up to a constant. Ultimately, we can see that the equations of
motion are
Z
∀δA
!
Tr(δA ∧ F ) ===⇒ F = 0
0 = δSCS =
M
so that the space of physical solutions of the classical theory coincides with the space of all
flat connections modulo gauge transformation (we consider equivalent two connections linked
by a gauge transformation).
2. Quantization and some computations
In this section, we will quantize the CS theory via path integral and compute the partition
function in a weak coupling limit. We will see that these computation, once suitably regularized, results in a product of topological invariants. Such an outcome is expected because we
have incorporated in our theory only topological data concerning the fiber bundle structure,
in the sense that we haven’t picked a metric on our manifold. Firstly, we introduce a suitable
constant in front of the integral of the Chern-Simons form so that the action we will consider
CHERN-SIMONS THEORY AND TOPOLOGICAL INVARIANT
3
from now on is
k
SCS [A] :=
4π
2
Tr A ∧ d0 A + A ∧ A ∧ A
3
M
Z
where k is our coupling constant. In fact, if we make the kinetic term independent of k
(as it’s usual in Quantum field theories) absorbing it in the definition of the field A, it will
appear k −1/2 in front of the interacting term A3 . From now on, we will consider the adjoint
representation of the gauge group and so F ≡ g.
Now, if we compute the variation of the action under gauge transformations we see that it
varies by
Z
k
δSCS [A] =
Tr g −1 dg ∧ g −1 dg ∧ g −1 dg =: 2kπw(g)
12π M
where w(g) is the winding number of the gauge transformation g, which must necessarily be an
integer, as shown in the previous section. This means that classically the Chern-Simons action
is not gauge invariant, but it’s not all over because, when we will quantize, this shift will turn
to be a phase shift. Thus, this classical action integer variation under gauge transformations
implies a quantization condition on k, which must necessarily take value in Z. To ensure that
the partition function
Z
Z=
DA exp{iSCS [A]}
A
G
is a gauge invariant. Here, we are formally integrating on the physical space.
Weak coupling limit. Now, let’s consider the limit of large k, where the integrand functional
in the path integral becomes highly oscillating. In this case, it is possible to use the stationary
phase approximation. To realize it, we should know well the space of classical physical solutions
(stationary point of the action modulo gauge transformations), so that we can integrate/sum
over its various contributions. It can be shown that this space is in correspondence with
equivalence classes of group morphisms φ : π1 (M ) → G up to conjugation. For our present
purpose, we will consider the simplest case in which this space is the union of isolated points
{Aα } ⊂ AG . Thus, we can write the partition function as
Z
Z
X
X
k
I[Aα ]
DB exp i
Tr(B ∧ Dα B) =:
µα
Z=
exp
A
4π
M
α
α
G
where the linear terms in B have given null contribute since we are expanding around stationary points and we have neglected the cubic term. Here I’ve defined also Dα := d0 + Aα .
Going through the computation. Making a shift of the integration variable, we integrate
over B in each µα term. To explicitly implement the computation we need to fix the gauge
(something like the choice of a section of the projection A → AG ). This is done using the
Faddeev-Popov method, i.e. choosing a gauge fixing fuctional F [B], and writing
Z
Z
ik
δF [B β ]
µα = exp
I[Aα ]
DB δ(F [B]) det
exp i
Tr(B ∧ Dα B)
4π
δβ
A
M
where ∀β : M → g is a small gauge transformation and B β = B − Dα β. Now, we need to pick
an F [B] but the problem is that we can’t do it without introducing a metric. Specifically,
we will choose the Lorenz gauge F [B] = Tr(∗Dα ∗ B) (in coordinate Tr(Dµ B µ )). Here, we
have chosen a metric on M , adding a datum which is not invariant under diffeomorphisms. A
consequence of this is that now Dα is constructed by the connection on F and the Levi-Civita
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FRANCESCO DILDA
connection on T M . Thus, the various parts become (since now we will omit the label α untill
necessary)
δF [B β ]
= ∗D ∗ D = ∆
δβ
Z
Z
δ(T r(∗D ∗ B)) = Dφ exp i
T r(φ ∗ D ∗ B)
M
3
where φ ∈ Ω (M, ρ(G)). What is left to take care of is the path integral
Z
Z
k
B ∧ d0 B + φ ∗ D ∗ B
DB Dφ exp i
Tr
4π
H
M
where H = Ω1 (M, ρ(g)) ⊕ Ω3 (M, ρ(g)).
L
L
Now, we can define the operator L : p Ωp (M, ρ(g)) → p Ωp (M, ρ(g)) as L := D ∗ + ∗ D,
which isLalso self-adjoint with respect
to our scalar product. Furthermore, we can easily see
L
that L( p odd Ωp (M, ρ(g))) ⊂ p odd Ωp (M, ρ(g)). Therefore, from now on we can consider
p
p
only L− , the restriction to odd forms. If we define H := k/4πB + 1/2 4π/kφ (where the
sum between different graded forms is formal), we can rewrite the path integral above as
Z
Z
DH exp i
Tr(H ∧ ∗(L− H))
H
M
and solve it as a gaussian integral obtaining the final expression
det(∆)
ik
I[Aα ] p
.
µα = exp
4π
det(L− )
The first factor is clearly a topological invariant. Instead, in general, the second factor is not
and this has been caused by choice of a metric in the gauge fixing procedure. However, it is
known that this failure is fault of a phase introduced by det(L− ). In fact, the absolute value
of the ratio can be shown to be the Ray-Singer torsion (Tα ), which is a topological invariant,
and ∆ is self-adjoint and positive definite, so its determinant will be real and positive (once
suitably regularized/defined).
The phase of the determinant. Now, we are (path) integrating over the infinite
diR
mensional
space
H
on
which
L
acts
and
we
have
an
euclidean
scalar
product
h·,
·i
:=
−
M
R
Tr(·
∧
∗·).
Since
L
is
also
self-adjoint,
we
can
find
an
ortonormal
basis
of
his
eigenvectors
−
M
χj ∈ H
s.t. L− χj = λj χj
with λj ∈ R
and expanding each element in H as a linear combination of χj s, we can write the measure as
an infinite product of one dimensional onces


Y Z ∞ dxj
X
iπ
1
√ exp iλj x2j =
exp
sign(λj )
|
det(L
)|
4
π
−
−∞
j
j
where the last factor is clearly not well defined without regularization process. An appropriate
way to regularize it is via the operator’s eta-function. We define
1X
sign(λj )|λj |−s
ηAα (s) :=
2 j
that in a certain domain of s converges and we can continue analitically it in 0. Now we
can substitute the expression above with ηL− (0). In fact, in the finite dimensional case they
coincide but the latter retains meanings even in the infinite dimensional case. Eta depends
CHERN-SIMONS THEORY AND TOPOLOGICAL INVARIANT
5
explicitly by the operator considered, but here I labelled it with Aα because the operator in
turn depends by the connection 1 form. So substituting we obtain
1
iπ
exp
ηA (0)
| det(L− )|
2 α
Now, using the Atyah-Patodi-Singer theorem it can be found that
C2 (adj)
1
I[Aα ].
(ηAα (0) − η0 (0)) =
2
8π 2
Here η0 (0) is the eta-function associated with the operator d0 ∗ + ∗ d0 whose spectrum is of
the form
σ(d0 ∗ + ∗ d0 ) = {χ ⊗ ea ⊗ eb ∈ Ω(M ) ⊗ ρ(g) : χ ∈ σ(d ∗ + ∗ d)}
where d ∗ + ∗ d acts on differential forms and d is the exterior derivatve taking into account
the spin connection. So if we define ηgrav (s) as the eta-function of d ∗ + ∗ d then we can easily
see that
η0 (s) = d ηgrav
with d = dim(g)
then
C2 (adj)
ηAα (0) =
I[Aα ] + d ηgrav
4π 2
and so in the end, putting everything together
X
idπ
C2 (adj)
i
Z = exp
ηgrav
k+
I[Aα ] Tα .
exp
2
4π
2
α
Thus, we were able to isolate the dependence on the choice of metric (non-topological data)
in a global factor of the partition function. The rest of it is a topological invariant.
Regularization of metric dependent term. Untill now, we tried to compute the partition
function via gauge fixing and then eta-function regularization. Doing this, we introduced a
non topological invariant term. Despict of this we still have the freedom, in our regularization
method, to introduce a local term. In particular, it can be shown (thanks to Atiyah-PatodiSinger theorem) that
1
1 I[g]
ηgrav +
2
12 8π 2
is independent by the metric tensor picked. Here I[g] stands as
Z
2
I[g] :=
Tr ω ∧ dω + ω ∧ ω ∧ ω
3
M
where ω is the Levi-Civita (compatible with g) spin connection. By this modification during
the regularization process we can obtain
X
idπ
1
i
C2 (adj)
Z = exp
ηgrav + I[g]
exp
k+
I[Aα ] Tα
2
6
4π
2
α
an espression of the Z completely independent by the metric chosen for the gauge fixing. It
remains only a residue dependence on the framing, i.e. on the ortonormal frame choosen.
This dependence is the same thing as the variation of the classical CS action under large
gauge transformation. If we pick two frame not homotope, the difference between the two
gravitational CS actions is of the form I 0 [g] − I[g] = 8π 2 s for some integer s. So the partition
function is well defined module
d
.
Z −→ Z + exp 2πis
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FRANCESCO DILDA
We can think of what we have obtained as a class of results (one for every homotopy class of
framing) that topologically characterize the fiber bundle on which we are working.
3. Conclusions
In this work we have seen a first example of the power of topological field theory (like it is CS
in 3 dimensions) to compute topological invariants of the manifold on which we are working.
After this, in the paper [4] it is considered how to relax the assumption on the discreteness
of the classical solutions space. This introduce the problem of knowing the geometry of such
space (called the moduli spaces of flat connections). After that, to face the problem in general
(out of the weak coupling limit) he approaches the problem through canonical quantization.
References
[1] J. Baez and J. P. Muniain. Gauge fields, knots and gravity. 1995.
[2] D. Bleecker. Gauge Theory and Variational Priciples. 1981.
[3] David Grabovsky. “Chern-Simons Theory in a Knotshell”. In: (2022).
[4] Edward Witten. “Quantum Field Theory and the Jones Polynomial”. In: Commun. Math.
Phys. 121 (1989). Ed. by Asoke N. Mitra, pp. 351–399.