$Z=\frac{1}{N!}\int\prod_{i=1}^{N}\left(\frac{d\lambda_{i}}{2\pi}e^{-\frac{8\pi^{2}N}{t}\lambda_{i}^{2}}\right)\Delta(\lambda)^{2}.$
(5.161)

Introducing the $L^{2}(\mathbb{R})$ measure

$d\mu(x):=dx\ e^{-\frac{8\pi^{2}N}{t}x^{2}},$ (5.162)

we can write the partition function as

$Z=\frac{1}{N!}\int\prod_{i=1}^{N}d\mu(\lambda_{i})\Delta(\lambda)^{2}.$
(5.163)

Recalling that the Vandermonde determinant is evaluated from the matrix
(5.142), expressed in terms of the polynomials $\\{1,x,x^{2},\cdots\\}$, we
notice that we can equivalently express it in terms of another set of _monic_
polynomials,

$p_{k}(x)=x^{k}+\sum_{j=0}^{k-1}a_{j}^{(k)}x^{j}$ (5.164)

since by elementary row operations

$\Delta(\lambda)=\det||\lambda_{i}^{j-1}||=\det||p_{j-1}(\lambda_{i})||.$
(5.165)

It is useful to chose the set $\\{p_{k}\\}_{k\geq 0}$ to be _orthogonal_ with
respect to the matrix model measure,

$\int d\mu(\lambda)p_{n}(\lambda)p_{m}(\lambda)=h_{n}\delta_{nm}$ (5.166)

since the knowledge of this set, and in particular of the normalization
constants $h_{n}$, allows to compute the partition function. Writing the
determinant as

$\Delta(\lambda)=\sum_{\sigma\in
S_{N}}(-1)^{\mbox{sign}(\sigma)}\prod_{k=1}^{N}p_{\sigma(k)-1}(\lambda_{k}),$

then (5.163) reduces to

$Z=\prod_{k=0}^{N-1}h_{k}.$ (5.167)

In our case the matrix model is Gaussian, and the corresponding set of
orthogonal polynomials are the _Hermite polynomials_ ,

$H_{n}(x):=e^{x^{2}}\left(-\frac{d}{dx}\right)^{n}e^{-x^{2}},\qquad\int_{-\infty}^{+\infty}dx\
e^{-x^{2}}H_{n}(x)H_{m}(x)=\delta_{nm}2^{n}n!\sqrt{\pi}$ (5.168)

so, normalizing $h_{n}=1$ and inserting the correct prefactors, we consider
the set of _orthonormal_ polynomials with respect to the measure
$d\mu(\lambda)$

$P_{n}(\lambda):=\sqrt{\sqrt{\frac{8\pi N}{t}}\frac{1}{2^{n}n!}}\
H_{n}\left(\frac{\sqrt{8\pi^{2}N}}{t}\lambda\right).$ (5.169)

The expectation value of any observable of the type
$\mathrm{Tr}(f(X))=\sum_{k}f(\lambda_{k})$ can be simplified as

$\displaystyle\langle\mathrm{Tr}f(X)\rangle$
$\displaystyle=\frac{1}{N!Z}\int\left(\prod_{i=1}^{N}d\mu(\lambda_{i})\right)\Delta(\lambda)^{2}\sum_{k=1}^{N}f(\lambda_{k})$
(5.170) $\displaystyle\begin{aligned} =\frac{1}{N!}\sum_{k}\sum_{\sigma\in
S_{N}}\int d\mu(\lambda_{1})\ P_{\sigma(1)-1}(\lambda_{1})^{2}\cdots\int
d\mu(\lambda_{k})\ P_{\sigma(k)-1}(\lambda_{k})^{2}f(\lambda_{k})\cdots&\\\
\cdots\int d\mu(\lambda_{N})\ P_{\sigma(N)-1}(\lambda_{N})^{2}&\end{aligned}$
$\displaystyle=\sum_{j=0}^{N-1}\int d\mu(\lambda)\
P_{j}(\lambda)^{2}f(\lambda).$

Applying this formula to the expectation value of the circular Wilson loop in
the fundamental representation we have

$\displaystyle\langle W_{\mathbf{N}}(C)\rangle$
$\displaystyle=\frac{1}{N}\left\langle\mathrm{Tr}\exp(2\pi X)\right\rangle$
(5.171) $\displaystyle=\frac{1}{N}\sum_{j=0}^{N-1}\int d\lambda\
P_{j}(\lambda)^{2}e^{-\frac{8\pi^{2}N}{t}\lambda^{2}+2\pi\lambda}.$

A useful formula to simplify this integral is

$\int_{-\infty}^{+\infty}dx\
H_{n}(x)^{2}e^{-(x-c)^{2}}=2^{n}n!\sqrt{\pi}L_{n}(-2c^{2})$ (5.172)

where $c$ is a constant and $L_{n}(x)$ are the _Laguerre polynomials_ ,
satisfying the properties

$\displaystyle L_{n}^{(m)}(x)$
$\displaystyle=\frac{1}{n!}e^{x}x^{m}\left(\frac{d}{dx}\right)^{n}\left(e^{-x}x^{n+m}\right),$
(5.173) $\displaystyle L_{n}(x)$ $\displaystyle\equiv L_{n}^{(0)}(x),$ (5.174)
$\displaystyle L_{n}^{(m+1)}(x)$ $\displaystyle=\sum_{j=0}^{n}L_{j}^{(m)}(x),$
(5.175) $\displaystyle L_{n}^{(m)}(x)$
$\displaystyle=\sum_{k=0}^{n}\binom{n+m}{n-k}\frac{(-x)^{k}}{k!}.$ (5.176)

Substituting (5.172) in (5.171), and expanding in series we have

$\displaystyle\langle W_{\mathbf{N}}(C)\rangle$
$\displaystyle=\frac{1}{N}e^{c^{2}}L_{N-1}^{(1)}(-2c^{2})\qquad\text{with}\
c:=\sqrt{\frac{t}{8N}}$ (5.177) $\displaystyle\begin{aligned}
=&\frac{1}{N}\sum_{k=0}^{\infty}\frac{1}{k!}\left(\frac{t}{8N}\right)^{k}\sum_{j=0}^{N-1}\frac{N!}{(j+1)!(N-1-j)!}\frac{1}{j!}\left(\frac{t}{4N}\right)^{j}-\\\
&-\frac{1}{N}\sum_{j=1}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}\frac{j(j+1)}{2}+\frac{1}{N}\sum_{j=0}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}\frac{1}{2}+O(1/N^{2})\end{aligned}$
$\displaystyle=\sum_{j=0}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}+O\left(1/N^{2}\right)$

where we expanded the first terms with respect to powers of $1/N$, and already
noticed that for $N\gg 1$ the odd-power terms cancel. We can thus examine the
large-$N$ limit, and inserting the definition of the modified Bessel function
$I_{n}(2x)=\sum_{k=0}^{\infty}\frac{x^{n+2k}}{k!(n+k)!}$ the expectation value
gives

$\langle
W_{\mathbf{N}}(C)\rangle=\frac{2}{\sqrt{t}}I_{1}\left(\sqrt{t}\right)+O(1/N^{2}),$
(5.178)

matching the result obtained with the saddle point technique in (5.158). In
general, the expansion is in powers of $1/N^{2}$ rather than $1/N$, as
expected from the analogy “$N^{2}\leftrightarrow 1/\hbar$” that we noticed in
(5.145). Solutions to the matrix model for higher representations have also
been found, see [85, 95, 96].

###  5.5 Localization of $\mathcal{N}=2$ Chern-Simons theory on the 3-sphere

In this section we review another example of supersymmetric localization
applied to the computation of Wilson loop expectation values, in an
$\mathcal{N}=2$ matter-coupled Euclidean Super Chern-Simons (SCS) theory on
the 3-sphere $\mathbb{S}^{3}$. We follow the derivation of Kapustin-Willet-
Yaakov [49], and Mariño [50], inspired in part by the work discussed in the
previous section. We consider a generic compact Lie group $G$ as the gauge
group, with Lie algebra $\mathfrak{g}$.

####  5.5.1 Matter-coupled $\mathcal{N}=2$ Euclidean SCS theory on
$\mathbb{S}^{3}$

The case of $\mathcal{N}=2$ Euclidean supersymmetry on $\mathbb{S}^{3}$ was
discussed as an example in Sections 4.3.6 and 4.4.3 for the gauge sector. We
report the action for the SCS theory

$S_{CS}=\frac{k}{4\pi}\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\
\mathrm{Tr}\left\\{\frac{\varepsilon^{\mu\nu\rho}}{\sqrt{g}}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}\right)-\tilde{\lambda}\lambda+2\sigma
D\right\\}$ (5.179)

and the supersymmetry variations, already considered in curved space

$\displaystyle\delta
A_{\mu}=\frac{i}{2}(\tilde{\epsilon}\gamma_{\mu}\lambda-\tilde{\lambda}\gamma_{\mu}\epsilon)$
(5.180)
$\displaystyle\delta\sigma=\frac{1}{2}(\tilde{\epsilon}\lambda-\tilde{\lambda}\epsilon)$
$\displaystyle\delta\lambda=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}-D+i(D_{\mu}\sigma)\gamma^{\mu}+\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\epsilon$
$\displaystyle\delta\tilde{\lambda}=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}+D-i(D_{\mu}\sigma)\gamma^{\mu}-\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\tilde{\epsilon}$
$\displaystyle\delta
D=-\frac{i}{2}\left(\tilde{\epsilon}\gamma^{\mu}D_{\mu}\lambda-(D_{\mu}\tilde{\lambda})\gamma^{\mu}\epsilon\right)+\frac{i}{2}\left([\tilde{\epsilon}\lambda,\sigma]-[\tilde{\lambda}\epsilon,\sigma]\right)-\frac{i}{6}\left(\tilde{\lambda}\gamma^{\mu}D_{\mu}\epsilon+(D_{\mu}\tilde{\epsilon})\gamma^{\mu}\lambda\right).$

where the $D_{\mu}$ are gauge-covariant derivatives with respect to the metric
and spin connection induced by the round metric (4.154), that in stereographic
coordinates $x^{\mu=1,2,3}$ is given by

$g_{\mu\nu}=e^{2\Omega(x)}\delta_{\mu\nu}\qquad
e^{2\Omega(x)}=\left(1+\frac{x^{2}}{4r^{2}}\right)^{-2}$ (5.181)

with $r$ being the radius of the embedding
$\mathbb{S}^{3}\hookrightarrow\mathbb{R}^{4}$. We remark again that this
supersymmetric action is actually superconformal, thus can preserve
supersymmetry on this conformally flat background, even with positive scalar
curvature. The new background preserves all the original $\mathcal{N}=2$
algebra, generated by conformal Killing spinors $\epsilon,\tilde{\epsilon}$,
taken to satisfy

$\nabla_{\mu}\epsilon=\frac{i}{2r}\gamma_{\mu}\epsilon,\qquad\nabla_{\mu}\tilde{\epsilon}=\frac{i}{2r}\gamma_{\mu},\tilde{\epsilon}$
(5.182)

where every equation has two possible solutions.

We consider also coupling the theory to matter fields, adding them in chiral
multiplets in a representation $R$ of the gauge group, to preserve
supersymmetry. The 3-dimensional $\mathcal{N}=2$ chiral multiplet (or
hypermultiplet) is, as for the gauge multiplet, given by dimensional reduction
of the $\mathcal{N}=1$ chiral multiplet in 4 dimensions: a complex scalar
$\phi$, a 2-component Dirac spinor222222Recall that $Spin(3)=SU(2)$ has no
Majorana spinors. We consider the reduced 4-dimensional Majorana spinor $\psi$
as a 3-dimensional Dirac (complex) spinor, since they have the same number of
real components. $\psi$ and an auxiliary complex scalar $F$. Every field comes
with its complex conjugate from the corresponding anti-chiral multiplet. The
supersymmetric action for the matter multiplet coupled to the gauge multiplet
is given by

$S_{m}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\
\left(D_{\mu}\tilde{\phi}D^{\mu}\phi+\frac{3}{4r^{2}}\tilde{\phi}\phi+i\tilde{\psi}\not{D}\psi+\tilde{F}F+\tilde{\phi}\sigma^{2}\phi+i\tilde{\phi}D\phi+i\tilde{\psi}\sigma\psi+i\tilde{\phi}\tilde{\lambda}\psi-i\tilde{\psi}\lambda\phi\right)$
(5.183)

where the $\mathfrak{g}$-valued fields in the gauge multiplets act on the
chiral multiplet in the representation $R$. This is the “covariantization” of
the flat space action for the matter multiplet (see for example [97]), with
the addition of the conformal coupling of the scalar field to the curvature,
$\frac{3}{4r^{2}}\tilde{\phi}\phi$. The supersymmetry transformations for the
chiral multiplet, with respect to the conformal Killing spinors
$\epsilon,\tilde{\epsilon}$, are

$\displaystyle\delta\phi$
$\displaystyle=\tilde{\epsilon}\psi\qquad\delta\tilde{\phi}=\tilde{\psi}\epsilon$
(5.184) $\displaystyle\delta\psi$
$\displaystyle=(-i\gamma^{\mu}D_{\mu}\phi-i\sigma\phi)\epsilon-\frac{i}{3}\gamma^{\mu}(\nabla_{\mu}\epsilon)\phi+\tilde{\epsilon}F$
$\displaystyle\delta\tilde{\psi}$
$\displaystyle=\tilde{\epsilon}(i\gamma^{\mu}D_{\mu}\tilde{\phi}+i\sigma\tilde{\phi})+\frac{i}{3}(\nabla_{\mu}\tilde{\epsilon})\gamma^{\mu}\tilde{\phi}+\epsilon\tilde{F}$
$\displaystyle\delta F$
$\displaystyle=\epsilon(-i\gamma^{\mu}D_{\mu}\psi+i\lambda\phi+i\sigma\psi)$
$\displaystyle\delta\tilde{F}$
$\displaystyle=(iD_{\mu}\tilde{\psi}\gamma^{\mu}-i\tilde{\lambda}\tilde{\phi}+i\sigma\tilde{\psi})\tilde{\epsilon}.$

The above variations generates a superconformal algebra that closes off-shell:

$[\delta_{\epsilon},\delta_{\tilde{\epsilon}}]=-i(\mathcal{L}_{v}+G_{\Lambda}+R_{\alpha}+\Omega_{f})$
(5.185)

where $\mathcal{L}_{v}$ is the Lie derivative (translation) along the Killing
vector field $v=(\tilde{\epsilon}\gamma^{\mu}\epsilon)\partial_{\mu}$, acting
on one forms as
$\mathcal{L}_{v}(A)_{\mu}=v^{\nu}\partial_{\nu}A_{\mu}+A_{\nu}\partial_{\mu}v^{\nu}$,
and on spinors as
$\mathcal{L}_{v}\psi=\nabla_{\nu}\psi-\frac{1}{4}(\nabla_{\mu}v_{\nu})\gamma^{\mu\nu}\psi$.
$G_{\Lambda}$ is a gauge transformation with respect to the parameter
$\Lambda:=A(v)+\sigma(\tilde{\epsilon}\epsilon)$. $R_{\alpha}$ is a
$U(1)^{\mathcal{R}}$ R-symmetry transformation, and $\Omega_{f}$ is a
dilatation [50]. The matter coupled action $S_{CS}+S_{m}$ is known to be
superconformal at quantum level, but one could also add a superpotential for
the matter multiplet. This choice is restricted by the condition of unbroken
superconformal symmetry both at classical and at quantum level, since the
localization principle works only if the supersymmetry algebra closes off-
shell. It turns out that the localization locus is at trivial configurations
of the matter sector, thus the precise choice of superpotential does not
influence the computation.

####  5.5.2 The supersymmetric Wilson loop

The Wilson loop under consideration, in the representation $R$ of the gauge
group, is defined as [97]

$W_{R}(C)=\frac{1}{\dim{R}}\mathrm{Tr}_{R}\left(\mathcal{P}\exp{\oint_{C}dt\
(iA_{\mu}\dot{C}^{\mu}+\sigma)}\right)$ (5.186)

with $C:\mathbb{S}^{1}\to\mathbb{S}^{3}$ a closed curve of tangent vector
$\dot{C}$, normalized such that $|\dot{C}|=1$. In order to localize its
expectation value, we have to consider those curves such that this operator
preserves some supersymmetry on the 3-sphere. Its variation under (5.180) is
proportional to

$\delta
W_{R}(C)\propto-\tilde{\epsilon}(\gamma_{\mu}\dot{C}^{\mu}+1)\lambda+\tilde{\lambda}(\gamma_{\mu}\dot{C}^{\mu}-1)\epsilon.$
(5.187)

Imposing the vanishing of this expression for all gauginos, we get the
following conditions on the conformal Killing spinors,

$\tilde{\epsilon}(\gamma_{\mu}\dot{C}^{\mu}+1)=0,\qquad(\gamma_{\mu}\dot{C}^{\mu}-1)\epsilon=0.$
(5.188)

We have two more conditions on the conformal Killing spinors, thus the maximum
number of solutions is reduced by half. The Wilson loop can at most be
invariant under two of the four possible supersymmetry variations, and for
that it is called _1/2-BPS_.

We can find explicitly one family of supersymmetric Wilson loops and one
supersymmetry variation with respect to which we are going to perform the
localization procedure. In order to solve the conformal Killing equations and
the conditions (5.188), we chose explicitly an orthonormal basis and a
corresponding vielbein on $\mathbb{S}^{3}$. Since as a manifold
$\mathbb{S}^{3}\cong SU(2)$, we can use Lie theory to describe the geometry on
the 3-sphere. In particular, the vielbein can be chosen proportional to the
Maureer-Cartan form $\Theta\in
T^{*}(SU(2))\otimes\mathfrak{su}(2)$,232323Again, we use Roman letters as
“flat” indices, and Greek letters as “curved” indices.

$e^{i}_{\mu}:=\frac{r}{2}e^{i}(\Theta(\partial_{\mu}))$ (5.189)

where $\\{e^{i}\\}$ is a basis of $\mathfrak{su}(2)^{*}$, dual to a basis
$\\{T_{i}\\}$ of $\mathfrak{su}(2)$.242424Say, the standard basis given by the
Pauli matrices, $T_{i}:=\sigma_{i}/\sqrt{2}$. One can check that this vielbein
is consistent with the round metric, giving
$g_{\mu\nu}=e_{\mu}^{i}e_{\nu}^{j}\delta_{ij}$ (see [50]). Using this
orthonormal basis, the spin connection components are

$(\omega_{\mu})_{ij}=\frac{1}{r}e^{k}_{\mu}\varepsilon_{ijk}$ (5.190)

where $\varepsilon_{ijk}$ is the Levi-Civita symbol. In this basis the
conformal Killing spinor equation for $\epsilon$ looks particularly simple,

$\left(\partial_{\mu}+\frac{1}{8}(\omega_{\mu})_{ij}[\gamma^{i},\gamma^{j}]\right)\epsilon=\frac{i}{2r}\gamma_{\mu}\epsilon\quad\Leftrightarrow\quad\partial_{\mu}\epsilon=0$
(5.191)

where we used the commutator
$[\gamma^{i},\gamma^{j}]=2i\left.\varepsilon^{ij}\right._{k}\gamma^{k}$. We
see that the components of $\epsilon$ are constants. The corresponding
condition for the supersymmetry of the Wilson loop then requires
$\gamma_{\mu}\dot{C}^{\mu}$ to be constant too, as the components of the
vector field $\dot{C}^{i}$ in the orthonormal frame. This means that the
Wilson loop has to describe grat circles on $\mathbb{S}^{3}$. Following [49],
we take $\dot{C}$ parallel to one of the $e^{i}$, say $e^{3}$, and the
conformal Killing spinor to satisfy

$(\gamma_{3}-1)\epsilon=0.$ (5.192)

We will consider the one dimensional subalgebra generated by such restricted
spinor, and put $\tilde{\epsilon}=0$.

####  5.5.3 Localization: gauge sector

We focus now on the localization of the Chern-Simons path integral, without
coupling to the matter multiplet. Ignoring the issue of gauge fixing, we would
add to the action the localizing term $tS_{loc}=t\delta\mathcal{V}$, with
$t\in\mathbb{R}^{+}$ a parameter, $\delta$ being the supersymmetry
transformation generated by the conformal Killing spinor $\epsilon$ described
in the last section, and $\mathcal{V}$ some fermionic functional whose bosonic
part is positive semi-definite. At the end of Section 4.4, we pointed out that
the Super Yang-Mills Lagrangian is an example of $\delta$-exact term, so we
put

$\displaystyle S_{loc}:=2S_{YM}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\
\mathrm{Tr}\left(i\tilde{\lambda}\gamma^{\mu}D_{\mu}\lambda+\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+D_{\mu}\sigma
D^{\mu}\sigma+i\tilde{\lambda}[\sigma,\lambda]+\right.$ (5.193)
$\displaystyle+\left.\left(D+\frac{\sigma}{r}\right)^{2}-\frac{1}{2r}\tilde{\lambda}\lambda\right)$

whose bosonic part is indeed positive semi-definite. This localizing term can
be derived also from the functional [49]

$\mathcal{V}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\
\mathrm{Tr}\left((\delta\tilde{\lambda})\lambda\right)$ (5.194)

analogously to the one used in the previous chapter for the gauge multiplet.
$S_{YM}$ being supersymmetric means that $\delta^{2}=0$ on $\mathcal{V}$,
making the localization principle applicable. As usual, the limit $t\to\infty$
localizes the path integral on the configurations that make this term vanish:
the terms involving bosonic fields are separately non-negative, while the
gaugino and its conjugate have to vanish identically. Summarizing, the
localization locus is given by

$\left\\{\begin{aligned} &\lambda=\tilde{\lambda}=0\\\ &F=0\Rightarrow A=0\
\text{(up to a gauge transformation)}\\\ &\sigma=a\in\mathfrak{g}\
(\mathrm{constant})\\\ &D=-\frac{1}{r}a\end{aligned}\right.$ (5.195)

Keeping into account the gauge-fixing procedure (as we should), the ghost $c$,
anti-ghost $\tilde{c}$ and Lagrange multiplier $b$ are added to the theory,
taking value in the Lie algebra $\mathfrak{g}$, together with the BRST
differential $\delta_{B}$ that acts as

$\delta_{B}X=-[c,X]\qquad\delta_{B}c=-\frac{1}{2}[c,c]\qquad\delta_{B}\tilde{c}=b\qquad\delta_{B}b=0$
(5.196)

where $X$ is any field in the original theory, acted by a gauge transformation
parametrized by $c$. The BRST differential is nilpotent, $\delta_{B}^{2}=0$.
The total differential

$Q:=\delta_{\epsilon}+\delta_{B}$ (5.197)

acts now as the equivariant differential for the $(U(1)\rtimes G)$-equivariant
cohomology in the BRST-augmented field space. The original CS action is
automatically $Q$-closed since it is gauge invariant, so we can combine the
localization principle with the gauge-fixing procedure adding to the
Lagrangian the term

$Q\left((\delta\tilde{\lambda})\lambda-\tilde{c}\left(\frac{\xi}{2}b-\nabla^{\mu}A_{\mu}\right)\right)$
(5.198)

where we suppressed the Lie algebra bilinear $\mathrm{Tr}$ for notational
convenience. Since the first term is gauge invariant,
$\delta_{B}\left((\delta\tilde{\lambda})\lambda\right)=0$, this gives the same
localization term as before. If $\delta[\mbox{ghosts}]=0$ on the gauge-fixing
subcomplex, the second term gives

$Q\left(\tilde{c}\left(\frac{\xi}{2}b-\nabla^{\mu}A_{\mu}\right)\right)=\frac{\xi}{2}b^{2}-b\nabla^{\mu}A_{\mu}+\tilde{c}\nabla^{\mu}D_{\mu}c+\tilde{c}\nabla^{\mu}\delta
A_{\mu}.$ (5.199)

The first two terms give, upon path integration over $b$, the usual gauge-
fixing Lagrangian in the $R_{\xi}$-gauge; the third term is the ghost
Lagrangian. The fourth term
$\propto\left(\tilde{c}\nabla^{\mu}\tilde{\lambda}\gamma_{\mu}\right)$ does
not change the partition function: if we see this term as a perturbation of
the gauge-fixed action, all diagrams with insertion of
$\left(\tilde{c}\nabla^{\mu}\tilde{\lambda}\gamma_{\mu}\right)$ will vanish,
since $\tilde{c}$ is coupled only to $c$ via the propagator but there are no
vertices containing $c$. In other words, the fermionic determinant arising
from the path integration over ghosts is not changed by this term. The
modified localizing term (5.199) is $Q$-closed: the old localizing term
because of gauge invariance and supersymmetry, while the gauge-fixing and
ghost terms follows by $Q^{2}A_{\mu}=0$ that is easy to check. After path
integration over the auxiliary $b$ the limit $t\to\infty$ finally localizes
the theory to the same locus (5.195), with ghosts put to zero.

Evaluating the classical action at the saddle point configuration, we get

$S_{CS}[a]=\frac{k}{4\pi}\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\
\mathrm{Tr}\left(-\frac{2}{r}a^{2}\right)=-k\pi r^{2}\mathrm{Tr}(a^{2})$
(5.200)

where we used $\mbox{vol}(\mathbb{S}^{3})=2\pi^{2}r^{3}$. The supersymmetric
Wilson loop observable (5.186) localizes to

$W_{R}(C)=\frac{1}{\dim{R}}\mathrm{Tr}_{R}\left(e^{2\pi ra}\right)$ (5.201)

since the curve $C$ is a great circle of radius $r$. Integrating as usual the
rescaled fluctuations above the localization configuration, and taking the
limit $t\to\infty$ as in (5.76), the partition function and the Wilson loop
expectation value are thus given by a finite-dimensional integral over
$\mathfrak{g}$ with Gaussian measure, the “matrix model”

$\displaystyle Z$ $\displaystyle=\int_{\mathfrak{g}}da\ e^{-k\pi
r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]$ (5.202) $\displaystyle\langle
W_{R}(C)\rangle$ $\displaystyle=\frac{1}{Z\dim{R}}\int_{\mathfrak{g}}da\
e^{-k\pi r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]\mathrm{Tr}_{R}\left(e^{2\pi
ra}\right).$

As we pointed out in the last section, the integration over the Lie algebra
$\mathfrak{g}$ can be reduced over its Cartan subalgebra $\mathfrak{h}$,
exploiting the gauge invariance of the matrix model under the adjoint action
of $\mathfrak{g}$ itself. This for example means, in the case of a matrix
gauge group, that we integrate over the diagonalized matrices “fixing the
gauge” of the matrix model. The corresponding Faddeev-Popov determinant is
also called _Vandermonde determinant_ ,

$\prod_{\alpha}\left(\rho_{\alpha}(a)\right)$ (5.203)

where the product runs over the roots of $\mathfrak{g}$. There is left an
overcounting given by the possible permutations of the roots, the action of
the _Weyl group_ $\mathcal{W}$ of $\mathfrak{g}$, cured dividing by its order
$|\mathcal{W}|$. The path integrals are thus rewritten as

$\displaystyle Z$
$\displaystyle=\frac{1}{|\mathcal{W}|}\int_{\mathfrak{\mathfrak{h}}}da\
\prod_{\alpha}\left(\rho_{\alpha}(a)\right)e^{-k\pi
r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]$ (5.204) $\displaystyle\langle
W_{R}(C)\rangle$
$\displaystyle=\frac{1}{Z|\mathcal{W}|\dim{R}}\int_{\mathfrak{h}}da\
\prod_{\alpha}\left(\rho_{\alpha}(a)\right)e^{-k\pi
r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]\mathrm{Tr}_{R}\left(e^{2\pi
ra}\right).$

Here we summarize the computation of the 1-loop determinant from [49]. For
convenience, we put $r=1$ and $\xi=1$. Inserting the contribution of ghosts,
the Lagrangian for the localizing term is given by (suppressing the
$\mathrm{Tr}$)

$\mathcal{L}_{loc}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+D_{\mu}\sigma
D^{\mu}\sigma+\left(D+\sigma\right)^{2}+i\tilde{\lambda}\not{D}\lambda+i[\tilde{\lambda},\sigma]\lambda-\frac{1}{2}\tilde{\lambda}\lambda+\partial_{\mu}\tilde{c}D^{\mu}c-\frac{1}{2}b^{2}+b\nabla^{\mu}A_{\mu}.$
(5.205)

Considering the limit $t\to\infty$, we rescale as usual the fields around the
configuration (5.195):

$\sigma=a+\sigma^{\prime}/\sqrt{t},\qquad D=-a+D^{\prime}/\sqrt{t},\qquad
X=X^{\prime}/\sqrt{t},$ (5.206)

where $X$ are all the fields without zero modes, and then rename
$\sigma^{\prime}\to\sigma$, $D^{\prime}\to D$, $X^{\prime}\to X$. In the
limit, only quadratic terms in the fluctuations survive,

$\mathcal{L}_{loc}\sim\frac{1}{2}\partial_{[\mu}A_{\nu]}\partial^{[\mu}A^{\nu]}-[A_{\mu},a]^{2}+(\partial\sigma)^{2}+(D+\sigma)^{2}+i\tilde{\lambda}\not{\nabla}\lambda+i[\tilde{\lambda},a]\lambda-\frac{1}{2}\tilde{\lambda}\lambda+|\partial\tilde{c}|^{2}-\frac{1}{2}b^{2}+b\nabla^{\mu}A_{\mu}.$
(5.207)

The resulting theory is free, and we can integrate it giving the corresponding
1-loop determinant. We will neglect all overall normalization constant from
the Gaussian integrations. The integral over the auxiliary field $b$ gives the
gauge fixing term $-\frac{1}{2}(\nabla^{\mu}A_{\mu})^{2}$. The contribution
from $D$ is purely Gaussian and can be integrated out removing the
corresponding term. The integration over $\sigma$ gives a determinant
$\det{(\nabla^{2})}^{-1/2}$, and the (Grassman) integral over the ghosts gives
$\det{(\nabla^{2})}$. It is useful to separate the gauge field as (Helmolz-
Hodge decomposition)

$A_{\mu}=B_{\mu}+\partial_{\mu}\phi$

with $\phi$ scalar and $B_{\mu}$ divergenceless, $\nabla^{\mu}B_{\mu}=0$. With
this decomposition, the Lorentz gauge condition becomes $\nabla^{2}\phi=0$,
and we can integrate $\phi$ giving a determinant $\det{(\nabla^{2})}^{-1/2}$,
that cancels the above two other contributions. We are left with

$-B_{\mu}\Delta
B^{\mu}-[a,B_{\mu}]^{2}+i\tilde{\lambda}\not{\nabla}\lambda+i[\tilde{\lambda},a]\lambda-\frac{1}{2}\tilde{\lambda}\lambda$
(5.208)

where $\Delta$ is the vector Laplacian. Now we use the fact that the path
integral can be reduced over the Cartan subalgebra of $\mathfrak{g}$,
considering $a\in\mathfrak{h}$, and

$B_{\mu}=B_{\mu}^{(\mathfrak{h})}+B_{\mu}^{\alpha}e_{\alpha}$ (5.209)

where $B_{\mu}^{(\mathfrak{h})}$ is the component of $B_{\mu}$ along
$\mathfrak{h}$, and similarly for the gaugino. This component does not enter
in the Lie brackets with $a$, so its contribution to the path integral is
independent of $a$, and we drop it. The remaining interesting terms are

$\sum_{\alpha}\left(B_{\mu}^{-\alpha}(-\Delta+\rho_{\alpha}(a)^{2})B_{\mu}^{\alpha}+\tilde{\lambda}^{-\alpha}\left(i\not{\nabla}+i\rho_{\alpha}(a)-\frac{1}{2}\right)\lambda^{\alpha}\right)$
(5.210)

where the $a$-dependent kinetic terms are clearly identified, and the
component fields appearing are real or complex valued scalars and spinors. The
Gaussian integration over these fields lead to the determinant factors

$Z^{g}_{1-loop}[a]=\prod_{\alpha}\frac{\det\left(i\not{\nabla}+i\rho_{\alpha}(a)-\frac{1}{2}\right)}{\det{\left(-\Delta+\rho_{\alpha}(a)^{2}\right)}^{1/2}}.$
(5.211)

Now, using the fact that the eigenvalues of the Laplacian on divergenceless
vectors are $(l+1)^{2}$ with degeneracy $2l(l+2)$, and the eigenvalues of
$i\not{\nabla}$ are $\pm\left(l+\frac{1}{2}\right)$ with degeneracy $l(l+1)$,
where $l\in\mathbb{Z}^{+}$, the corresponding determinants can be written as
infinite products

$\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l+i\rho_{\alpha}(a))^{l(l+1)}(-l-1+i\rho_{\alpha}(a))^{l(l+1)}}{((l+1)^{2}+\rho_{\alpha}(a)^{2})^{l(l+2)}}=\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l+i\rho_{\alpha}(a))^{(l+1)}}{(l-i\rho_{\alpha}(a))^{(l-1)}}$
(5.212)

where the equality follows after some simplifications. Since roots come in
pairs $(\rho_{\alpha},-\rho_{\alpha})$, taking the square of this one gets

$\left(Z^{g}_{1-loop}[a]\right)^{2}=\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l^{2}+\rho_{\alpha}(a)^{2})^{(l+1)}}{(l^{2}+\rho_{\alpha}(a)^{2})^{(l-1)}}=\prod_{\alpha}\prod_{l=1}^{\infty}\left(l^{2}+\rho_{\alpha}(a)^{2}\right)^{2}.$
(5.213)

Collecting a factor $l^{4}$ the product splits in the factorization formula
for the hyperbolic sine,

$\frac{\sinh(\pi z)}{\pi
z}=\prod_{l=1}^{\infty}\left(1+\frac{z^{2}}{l^{2}}\right)$ (5.214)

and an $a$-independent divergent part that can be regularized with the zeta-
function method,

$\prod_{l=1}^{\infty}l^{4}=e^{4\sum_{l=1}^{\infty}\log(l)}=e^{-4\zeta^{\prime}(0)}=e^{2\log(2\pi)}.$
(5.215)

Up to an overall normalization constant, the $a$-dependence of the 1-loop
determinant is finally given by

$Z_{1-loop}^{g}[a]=\prod_{\alpha}\left(\frac{2\sinh(\pi\rho_{\alpha}(a))}{\pi\rho_{\alpha}(a)}\right)$
(5.216)

where we see cancellation between the denominator and the Vandermonde
determinant (5.203).

Collecting the above results, the localization formulas for the partition
function and the expectation value of the supersymmetric Wilson loop in the
pure CS theory are

$\displaystyle Z$
$\displaystyle\sim\frac{1}{|\mathcal{W}|}\int_{\mathfrak{\mathfrak{h}}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)$
(5.217) $\displaystyle\langle W_{R}(C)\rangle$
$\displaystyle=\frac{1}{Z|\mathcal{W}|\dim{R}}\int_{\mathfrak{h}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi
a}\right)\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)$
$\displaystyle=\frac{1}{\dim{R}}\frac{\int_{\mathfrak{h}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi
a}\right)\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)}{\int_{\mathfrak{\mathfrak{h}}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)}.$

These general localization formulas can be tested comparing their results for
specific choices of $G$ to perturbative calculations, for example. In the case
of $U(N)$ gauge group, the integral over the Cartan subalgebra is an integral
over diagonal matrices $a=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{N})$, and
the roots are given by $\rho_{ij}(a)=\lambda_{i}-\lambda_{j}$ for $i\neq j$.
The Weyl group is $S_{N}$, thus $|\mathcal{W}|=N!$. If we take the Wilson loop
in the fundamental representation, from (5.217) we get

$\displaystyle Z$
$\displaystyle\sim\frac{1}{N!}\int\left(\prod_{i}d\lambda_{i}\
e^{-k\pi\lambda_{i}^{2}}\right)\prod_{i\neq
j}2\sinh(\pi(\lambda_{i}-\lambda_{j})),$ (5.218) $\displaystyle\langle
W_{\mathbf{N}}(C)\rangle$
$\displaystyle=\frac{1}{ZN!N}\int\left(\prod_{i}d\lambda_{i}\
e^{-k\pi\lambda_{i}^{2}}\right)\left(e^{2\pi\lambda_{1}}+\cdots+e^{2\pi\lambda_{N}}\right)\prod_{i\neq
j}2\sinh(\pi(\lambda_{i}-\lambda_{j})),$

that are sums of Gaussian integrals, and can be computed exactly. The result
for the Wilson loop expectation value is

$\langle
W_{\mathbf{N}}(C)\rangle=\frac{1}{N}e^{-Ni\pi/k}\frac{\sin\left(\frac{\pi
N}{k}\right)}{\sin\left(\frac{\pi}{k}\right)},$ (5.219)

which is known as the exact result [83], up to the overall phase factor
$e^{-Ni\pi/k}$. This kind of phase factors arise in perturbative calculations
in the so-called _framing_ of the Wilson loop. A perturbative calculation of
the Wilson loop involves computations of correlators of the type $\langle
A_{\mu_{1}}(x_{1})A_{\mu_{2}}(x_{2})\cdots\rangle$, where $x_{1},x_{2},\cdots$
are coordinates of points on the image of the curve $C$. This contribution
diverges when $x_{1}=x_{2}$, so it is necessary to choose some regularization
scheme to perform the computations. For example, considering the 2-point
function $\langle A_{\mu_{1}}(x)A_{\mu_{2}}(y)\rangle$, this clashing of
points can be avoided requiring that $y$ is integrated over a shifted curve
$C_{f}$ such that

$C^{\mu}_{f}(\tau)=C^{\mu}(\tau)+\alpha\ n^{\mu}(\tau)$ (5.220)

where $n$ is orthogonal to $\dot{C}$. The choice of such an orthogonal
component (frame) at every point on the curve is called framing. Even if at
the end of the calculation one takes $\alpha\to 0$, this procedure leaves a
deformation-dependent term, that in pure $U(N)$ CS is

$e^{\frac{i\pi N}{k}\chi(C,C_{f})}$ (5.221)

where $\chi(C,C_{f})$ is a topological invariant that takes integer values
corresponding to the number of times the path $C_{f}$ winds around $C$. We see
that localization produces an expectation value at framing -1 (see also [98]
for a detailed discussion about framing).

####  5.5.4 Localization: matter sector

We turn now to the result for the localization of the matter-coupled theory.
This is of course gauge invariant, so the equivariant differential acts
effectively as $Q\sim\delta$, since the ghost sector has been already
considered in the previous paragraph. This means that, following the
localization principle, we have to extend the matter action with a
$\delta$-exact term. We are free to consider the canonical choice (5.81) as in
[49], or using the fact that [50] the matter action (5.183) is actually given
by a supersymmetry variation, as the case of the YM action. This means that we
can consider the localizing terms

$tS_{m}+tS_{YM+ghosts}$

or, schematically

$t\int\delta\left((\delta\psi)^{\dagger}\psi+\tilde{\psi}(\delta\tilde{\psi})^{\dagger}\right)+tS_{YM+ghosts}$

that have positive semi-definite bosonic parts. The second term in both
choices is the one analyzed in the previous paragraph, and gives the same
localization locus for the gauge and ghost sector, while both the first terms
vanishes for the field configurations

$\psi=0,\qquad\phi=0,\qquad F=0.$ (5.222)

This means that the classical action of the matter sector does not contribute
to the partition function, but only in the 1-loop determinant. Expanding the
fields around this configuration and scaling the fluctuations with the usual
$1/\sqrt{t}$ factor, we see that there are no couplings to the gauge sector
fluctuations that survive in the $t\to\infty$ limit, but only to the zero mode
$a$ of $\sigma$. Thus the determinant factorizes as

$Z_{1-loop}[a]=Z_{1-loop}^{g}[a]Z_{1-loop}^{m}[a].$ (5.223)

If matter is present in different copies of chiral multiplets, in maybe
different representations of the gauge group, the determinant factorizes in
the same way for each multiplet.

The determinant for the matter sector can be computed diagonalizing the the
kinetic operators acting on the scalar and the fermion field, after having
integrated out the auxiliary $F$, and considering the path integration over
the Cartan subalgebra with $a\in\mathfrak{h}$. In particular, the relevant
kinetic operators that have to be diagonalized are

$K_{b}^{(\rho)}=\left(-\nabla^{2}+\rho(a)^{2}-i\rho(a)+\frac{3}{4}\right),\qquad
K_{f}^{(\rho)}=\left(i\not{\nabla}+i\rho(a)\right),$ (5.224)

for the (complex) bosonic and fermionic parts, where $a$ is regarded as acting
on the representation $R$ with weights $\\{\rho\\}$. The eigenvalues of
$-\nabla^{2}$ are $4j(j+1)$ with $j=0,\frac{1}{2},\cdots$ with degeneracy
$(2j+1)^{2}$, that we can rewrite as $l(l+2)$ with degeneracy $(l+1)^{2}$ and
$l=0,1,\cdots$. The eigenvalues of $i\not{\nabla}$ are
$\pm\left(l+\frac{1}{2}\right)$ with degeneracy $l(l+1)$, with $l=1,2,\cdots$.
Thus the one loop determinant results, after a change of dummy index and some
simplifications

$\displaystyle Z_{1-loop}^{m}[a]$
$\displaystyle=\prod_{\rho}\frac{\det(K_{f})}{\det(K_{b})}$ (5.225)
$\displaystyle=\prod_{\rho}\prod_{l=1}^{\infty}\frac{\left(l+\frac{1}{2}+i\rho(a)\right)^{l(l+1)}\left(l+\frac{1}{2}-i\rho(a)\right)^{l(l+1)}}{\left(l+\frac{1}{2}+i\rho(a)\right)^{l^{2}}\left(l-\frac{1}{2}-i\rho(a)\right)^{l^{2}}}$
$\displaystyle=\prod_{\rho}\prod_{l=1}^{\infty}\left(\frac{l+\frac{1}{2}+i\rho(a)}{l-\frac{1}{2}-i\rho(a)}\right)^{l}$

This product can be regularized using the zeta-function. We refer to [50] for
the details of the computation, and report here the result in the case the
fields take value in a self-conjugate representation $R$ of the gauge
group:252525For example, if $R=S\oplus S^{*}$.

$Z_{1-loop}^{m}[a]=\prod_{\rho}\left(2\cosh{(\pi\rho(a))}\right)^{-1/2}$
(5.226)

where now $a\in R(\mathfrak{h})$ and $\rho(a)$ is the weight of the Cartan
element in the representation $R$.

Summarizing, we have seen that the application of the supersymmetric
localization principle to the matter-coupled SCS theory on $\mathbb{S}^{3}$
reduces the path integral to a finite-dimensional integral describing a matrix
model over the Lie algebra of the theory. Using the notation (5.128), the
localization formulas for the partition function and the supersymmetric Wilson
loop expectation value, with matter multiplets coming in self-conjugate
representations $R_{1}\oplus R_{1}^{*},R_{2}\oplus R_{2}^{*},\cdots$ are

$\displaystyle Z$
$\displaystyle=\frac{1}{|\mathcal{W}|}\int_{\mathfrak{\mathfrak{h}}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\frac{\det_{ad}2\sinh(\pi
a)}{\left(\det_{R_{1}}2\cosh{(\pi a)}\right)\left(\det_{R_{2}}2\cosh{(\pi
a)}\right)\cdots}$ (5.227) $\displaystyle\langle W_{R}(C)\rangle$
$\displaystyle=\frac{1}{Z|\mathcal{W}|\dim{R}}\int_{\mathfrak{h}}da\
e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi
a}\right)\frac{\det_{ad}2\sinh(\pi a)}{\left(\det_{R_{1}}2\cosh{(\pi
a)}\right)\left(\det_{R_{2}}2\cosh{(\pi a)}\right)\cdots}$

####  5.5.5 The ABJM matrix model

ABJM theory is a special type of matter-coupled SCS theory in 3-dimensions
constructed in [99], that has the interesting property to be dual under the
AdS/CFT conjecture to a certain orbifold background in M-theory. It consists
of two copies of $\mathcal{N}=2$ SCS theory, each one with gauge group $U(N)$,
and opposite levels $k,-k$. In addition, the are four matter (chiral and anti-
chiral) supermultiplets $\Phi_{i},\tilde{\Phi}_{i}$, with $i=1,2$, in the bi-
fundamental representation of $U(N)\times U(N)$,
$(\mathbf{N},\bar{\mathbf{N}})$ and $(\bar{\mathbf{N}},\mathbf{N})$. This
field content can be represented as the _quiver_ in Fig. 5.1.

Figure 5.1: The quiver for ABJM theory. The two nodes represent the gauge
multiplets, with the convention of specifying the level of the CS term. The
oriented links represent the matter multiplets in the bi-fundamental and anti-
bi-fundamental representations.

The superpotential for the matter part is given by

$W=\frac{4\pi}{k}(\Phi_{1}\tilde{\Phi}_{1}\Phi_{2}\tilde{\Phi}_{2}-\Phi_{1}\tilde{\Phi}_{2}\Phi_{2}\tilde{\Phi}_{1}),$
(5.228)

and this structure actually enhance the supersymmetry of the resulting theory
to $\mathcal{N}=6$.262626This is not apparent from the original action, but
can be realized noticing that the superpotential has an $SU(2)\times SU(2)$
symmetry that rotates separately the $\Phi_{i}$ and the $\tilde{\Phi}_{i}$.
This, combined with the original $SU(2)^{\mathcal{R}}$ symmetry of the theory,
gives an $SU(4)\cong Spin(6)$ symmetry that acts non-trivially on the
supercharges. Thus the final theory has to have an enhanced $\mathcal{N}=6$
supersymmetry. If now
$a=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{N},\hat{\lambda}_{1},\cdots,\hat{\lambda}_{N})$,
the weights in the bi-fundamental representations are

$\rho_{i,j}^{(N,\bar{N})}(a)=\lambda_{i}-\hat{\lambda}_{j},\qquad\rho_{i,j}^{(\bar{N},N)}(a)=\hat{\lambda}_{j}-\lambda_{i}.$
(5.229)

Plugging this information into (5.227), the partition function in this case
localizes to the following matrix model,

$Z\sim\frac{1}{N!N!}\int\left(\prod_{i}d\lambda_{i}d\hat{\lambda}_{i}\
e^{-k\pi(\lambda_{i}^{2}-\hat{\lambda}_{i}^{2})}\right)\frac{\prod_{i\neq
j}\left(2\sinh(\pi(\lambda_{i}-\lambda_{j}))2\sinh(\pi(\hat{\lambda}_{i}-\hat{\lambda}_{j}))\right)}{\prod_{i,j}\left(2\cosh(\pi(\lambda_{i}-\hat{\lambda}_{j}))\right)}.$
(5.230)

The circular Wilson loop under consideration can be called now _1/6 BPS_ with
respect to the enhanced supersymmetry of the model. Its expectation value in
the fundamental representation is obtained by plugging a factor
$(1/N)\sum_{i}e^{2\pi\lambda_{i}}$ as before. This matrix model cannot be
solved exactly as in the case of the pure CS discussed above, but can be
studied in the $N\to\infty$ limit with the saddle-point technique showed in
Section 5.4.5 [50, 94]. We also mention that, in this particular theory with
enhanced $\mathcal{N}=6$ supersymmetry, it was possible to construct a _1/2
BPS_ Wilson loop (so invariant under half of the $\mathcal{N}=6$ supersymmetry
algebra). The latter can be solved applying the same localization scheme that
brings to the matrix model describing the 1/6 BPS Wilson loop presented above
[100, 101]. A compact review introducing the state of the art on recent
results about supersymmetric Wilson loops in ABJM and related theories can be
found in [84].

##  Chapter 6 Non-Abelian localization and 2d YM theory

In this chapter we are going to summarize the result obtained mainly in [15]
by Witten. This was the first attempt in the physics literature of extending
the equivariant localization formalism to possibly non-Abelian group actions.
In that work, a modified definition of equivariant integration was defined,
and this allowed for an extension of the same procedure discussed in Chapter 3
to show the localization property of integrals computed over spaces with
generic symmetry group $G$. This new formalism was applied to the study of
2-dimensional Yang-Mills (YM) theory over a Riemann surface, a relatively
simple model from the physical point of view, but with a very rich underlying
mathematical structure. In the following, we are going first to review the
geometry of this special model, in connection with the symplectic geometry
introduced in Section 3.3, as a motivation for the more mathematical
discussion about the Witten’s equivariant integration and non-Abelian
localization principle that will follow. Next, we will review the ideas
underlying the application of this new localization principle to the YM
theory, and how this application results in a “mapping” between this model and
a suitable topological theory, establishing the topological nature of the YM
theory in the weak coupling limit. In the final section, we will summarize the
interpretation given by the localization framework to the already existing
solution for the partition function of this model.

As we pointed out in the Introduction, other generalizations of the
Duistermaat-Heckman theorem to non-Abelian Hamiltonian systems also appeared
in the mathematical literature, as the result obtained by Jeffrey and Kirwan
in [16]. Other applications of this extended formalism followed, and Witten’s
approach was used for example more recently to describe Chern-Simons theories
over a special class of 3-manifolds in [102].

###  6.1 Prelude: moment maps and YM theory

In the next section we are going to review Witten’s extension of the
equivariant localization principle to possibly non-Abelian group actions, and
a generalization of the DH formula in this direction. In [15] this was applied
to reinterpret the weak coupling limit of pure YM theory on a Riemann surface.
This theory is exactly solvable, in the sense that its partition function can
be expressed in closed form, and its zero-coupling limit is known to describe
a topological field theory. These features make 2-dimensional YM theory very
appealing from the mathematical structure it carries, and make it possible to
compare results or interpretations obtained via this “new” localization method
with already existing solutions of the problem.

We are going to discuss more about the topological interpretation of 2d YM
theory later, while in this section we review some results introduced by
Atiyah and Bott [103] about the symplectic structure underlying this special
QFT. This can be useful to contextualize the generic discussion of the next
section, and it prepares the ground for the formal application of the non-
Abelian localization principle.

We start by considering the partition function of YM theory on a compact
orientable Riemannian manifold $\Sigma$ of arbitrary dimension,

$\displaystyle Z(\epsilon)$
$\displaystyle=\frac{1}{\mathrm{vol}(\mathcal{G}(P))}\left(\frac{1}{2\pi\epsilon}\right)^{\dim(\mathcal{G})/2}\int_{\mathcal{A}(P)}DA\
e^{-S[A]},$ (6.1) $\displaystyle S[A]$
$\displaystyle=-\frac{1}{2\epsilon}\int_{\Sigma}\mathrm{Tr}(F^{A}\wedge\star
F^{A}).$

Here $\epsilon:=g_{YM}^{2}$ is the square of the YM coupling constant. To
describe the rest of the ingredients, let us recall the geometry underlying
the gauge theory (to fill some of the details, see Appendix A.1). The
dynamical field here is the _connection_ $A\in\Omega(P;\mathfrak{g})$ on a
principal $G$-bundle $P\xrightarrow{\pi}\Sigma$, where $G$ is a compact
connected Lie group with Lie algebra $\mathfrak{g}$. The path integral is thus
taken over the space $\mathcal{A}(P)$ of $G$-equivariant vertical 1-forms with
values in $\mathfrak{g}$, that is naturally an _affine space_ modeled on the
infinite-dimensional vector space $\mathfrak{a}$ of $G$-equivariant horizontal
1-forms with values in $\mathfrak{g}$. This gives to $\mathcal{A}(P)$ the
structure of an infinite-dimensional manifold, whose tangent spaces are
$T_{A}\mathcal{A}(P)\cong\mathfrak{a}\cong\Omega^{1}(\Sigma;\mathrm{ad}(P))$,
where we identified horizontal forms over $P$ with forms over the base
$\Sigma$.111Recall that horizontality means essentially to have components
only in the “directions” of the base space, and the $G$-equivariance ensures
the right transformation behavior as forms valued in the _adjoint bundle_
$\mathrm{ad}(P)$, the associated bundle to $P$ that has $\mathfrak{g}$ as
typical fiber. Thus $\mathfrak{a}\cong\Omega^{1}(\Sigma;\mathrm{ad}(P))$. In
other words, any vector field $\alpha\in\Gamma(T\mathcal{A}(P))$ can be
expanded locally as

$\alpha=\alpha_{\mu}^{a}T_{a}\otimes dx^{\mu},\qquad\alpha_{\mu}^{a}\in
C^{\infty}(\Sigma\times\mathcal{A}(P)),$ (6.2)

with coefficients that depend on the point $A\in\mathcal{A}(P)$ and
$p\in\Sigma$. The curvature
$F^{A}=dA+\frac{1}{2}[A\stackrel{{\scriptstyle\wedge}}{{,}}A]$ of the
connection $A$ is a horizontal 2-form over $P$, so we can identify it as a
2-form on the adjoint bundle without loss of information,
$F^{A}\in\Omega^{2}(\Sigma;\mathrm{ad}(P))$. As such, it can be integrated as
a differential form over $\Sigma$. In the action $S[A]$, “$\mathrm{Tr}$”
represents a (negative definite) invariant inner product on $\mathfrak{g}$,
and $\star$ is the Hodge dual operation, that is identified by the presence of
a metric on $\Sigma$.222The definition of the Hodge star is, implicitly,
$\alpha\wedge\star\beta=g^{-1}(\alpha,\beta)\omega$ for any
$\alpha,\beta\in\Omega^{k}(\Sigma)$. Here $g^{-1}$ is the “inverse” metric on
$\Sigma$, that extends multi-linearly its action on every tangent space as
$g^{-1}(\alpha,\beta)=g^{\mu_{1}\nu_{1}}\cdots
g^{\mu_{k}\nu_{k}}\alpha_{\mu_{1}\cdots\mu_{k}}\beta_{\nu_{1}\cdots\nu_{k}}$.
$\omega$ is a volume form (that can be induced by $g$, for example). The Hodge
star satisfies the property $\star^{2}\alpha=(-1)^{k(\dim(\Sigma)-k)}\alpha$.

$\mathcal{G}(P)\cong\Omega^{0}(\Sigma;\mathrm{Ad}(P)$ is the group of gauge
transformations, that is locally equivalent to the space of $G$-valued
functions over $\Sigma$, and acts naturally on $\mathcal{A}(P)$. If $\phi\in
Lie(\mathcal{G}(P))\cong\Omega^{0}(\Sigma;\mathrm{ad}(P))$ is an element of
the Lie algebra of infinitesimal gauge transformations, its associated
fundamental vector field at the point $A\in\mathcal{A}(P)$ is

$\underline{\phi}_{A}\equiv\delta_{\phi}A=\nabla^{A}\phi=d\phi+[A,\phi].$
(6.3)

The path integral measure $DA$ can be defined formally as the Riemannian
measure induced by a metric on the affine space $\mathcal{A}(P)$. The latter
can be induced by the metrics on $\Sigma$ and on $\mathfrak{g}$, and defined
pointwise in $\mathcal{A}(P)$ as

$(\alpha,\beta)_{A}:=-\int_{\Sigma}\mathrm{Tr}(\alpha^{A}\wedge\star\beta^{A})$
(6.4)

for every $\alpha^{A},\beta^{A}\in\Omega^{1}(\Sigma;\mathrm{ad}(P))$. With
this definition, the YM action can be rewritten as

$S[A]=\frac{1}{2\epsilon}(F,F)_{A}.$ (6.5)

We can now specialize the discussion to the case in which $\dim(\Sigma)=2$,
i.e. the base space is a Riemann surface. It is a well-known fact in geometry
that any Riemann surface is a _Kähler manifold_ : it admits a Riemannian
metric $g$, a symplectic form $\omega$ (that can be a choice of volume form),
and a complex structure $J$ such that the compatibility condition
$g(\cdot,\cdot)=\omega(\cdot,J(\cdot))$ is satisfied.333A complex structure on
a vector space $V$ is an isomorphism $J:V\to V$ such that $J^{2}=-id_{V}$. It
intuitively plays the role of “multiplication by $i$” when one considers the
complexified $V^{\mathbb{C}}:=V\otimes\mathbb{C}$, allowing for a
decomposition of $V^{\mathbb{C}}$ in a holomorphic subspace (generated by the
eigenvectors with eigenvalue $+i$) and anti-holomorphic subspace (generated by
the eigenvectors with eigenvalue $-i$). A manifold $M$ has _almost complex
structure_ if there is a tensor $J\in\Gamma(T^{1}_{1}M)$ that acts as a
complex structure in every tangent space. If the holomorphic decomposition can
be extended on an entire neighborhood of every point by a suitable choice of
coordinates, $M$ has _complex structure_ , and admits an atlas of holomorphic
coordinates. Riemann surfaces can thus be thought as 2-dimensional real
manifolds, or 1-dimensional complex manifolds. This special property holds
also for $\mathcal{A}(P)$, since in addition to the metric (6.4) we can define
the symplectic form $\Omega\in\Omega^{2}(\mathcal{A}(P))$ such that

$\Omega_{A}(\alpha,\beta):=-\int_{\Sigma}\mathrm{Tr}(\alpha^{A}\wedge\beta^{A}),$
(6.6)

and the complex structure on $T\mathcal{A}(P)$ is provided by the Hodge
duality,
$\star:\Omega^{1}(\Sigma;\mathrm{ad}(P))\to\Omega^{1}(\Sigma;\mathrm{ad}(P))$
such that $\star^{2}=-1$. Then the compatibility condition is immediately
satisfied, since $(\cdot,\cdot)=\Omega(\cdot,\star(\cdot))$. The fact that
$\Omega$ is symplectic can be seen by noticing that, in any basis, it has
constant components (i.e. independent from $A\in\mathcal{A}(P)$):

$\Omega_{ab}^{\mu\nu}(A)=\Omega_{A}(T_{a}\otimes dx^{\mu},T_{b}\otimes
dx^{\nu})=-\mathrm{Tr}(T_{a}T_{b})\varepsilon^{\mu\nu}\left(\int_{\Sigma}dx^{1}dx^{2}\right)\quad\in\mathbb{R}.$
(6.7)

The non-degeneracy follows from the non-degeneracy of $\mathrm{Tr}$ and of
$\int_{\Sigma}$, and the skew-symmetry is obvious from the definition. Thus
$\mathcal{A}(P)$ is Kähler.

For our applications, we focus on the fact that $\mathcal{A}(P)$ has now a
canonical symplectic structure. It is natural to wonder if it possible to
extend all the machinery that we introduced in Section 3.3 also to this case,
and in particular if the $\mathcal{G}(P)$-action on $\mathcal{A}(P)$ results
to be symplectic or Hamiltonian with respect to $\Omega$. The answer was given
by in [103], and we state it in the following theorem.

######  Theorem 6.1.1 (Atiyah-Bott).

In 2-dimensions, the group $\mathcal{G}(P)$ of gauge transformations acts in
an Hamiltonian way on $\mathcal{A}(P)$, with a moment map identified by the
curvature $F$.

###### Proof.

To see this, let us introduce the moment map as $\mu:Lie(\mathcal{G}(P))\to
C^{\infty}(\mathcal{A}(P))$ such that

$\mu_{\phi}(A):=\langle
F^{A},\phi\rangle=-\int_{\Sigma}\mathrm{Tr}(F^{A}\phi),$ (6.8)

and check that the Hamiltonian property is satisfied. For every
$\alpha\in\Gamma(T\mathcal{A}(P))$ and $\phi\in Lie(\mathcal{G}(P))$, we
compute

$\displaystyle(\iota_{\phi}\Omega_{A})(\alpha)$
$\displaystyle=\Omega_{A}(\underline{\phi},\alpha)=-\int_{\Sigma}\mathrm{Tr}(\nabla^{A}\phi\wedge\alpha^{A})=\int_{\Sigma}\mathrm{Tr}(\phi\nabla^{A}\alpha^{A}),$
(6.9) $\displaystyle\left.\delta\mu_{\phi}\right|_{A}(\alpha)$
$\displaystyle=-\int_{\Sigma}\mathrm{Tr}\left(F^{A+\alpha}\phi-F^{A}\phi\right)=-\int_{\Sigma}\mathrm{Tr}\left(\phi\nabla^{A}\alpha^{A}\right),$
(6.10)

where $\delta$ is the de Rham differential on $\mathcal{A}(P)$, that acts in
the usual sense of variational calculus. We see that
$\iota_{\phi}\Omega=-\delta\mu_{\phi}$, thus $\mu$ provides a correct moment
map for the $\mathcal{G}(P)$-action. If we identify $Lie(\mathcal{G}(P))$ with
$Lie(\mathcal{G}(P))^{*}$ through the pairing $\langle\cdot,\cdot\rangle$
introduced above, and regard the curvature
$F:\mathcal{A}(P)\to\Omega^{2}(\Sigma;\mathrm{ad}(P))$ as an element of
$C^{\infty}(\mathcal{A}(P))\otimes Lie(\mathcal{G}(P))^{*}$, we can simply
write that $\mu\equiv F$. ∎

Another corollary of $\mathcal{A}(P)$ being Kähler is that the path integral
measure $DA$ is formally equivalent to the Liouville measure induced from
$\Omega$, since by compatibility of the structures the latter is equivalent to
the Riemannian measure induced by $(\cdot,\cdot)$. Since we are working on an
infinite-dimensional space, we can write this measure formally as

$DA=\exp(\Omega),$ (6.11)

as we did in (3.43) but with $n=\infty$. With this identification, we see that
the path integral of the 2-dimensional YM theory acquires the very suggestive
form

$Z(\epsilon)\propto\int_{\mathcal{A}(P)}\exp\left(\Omega-\frac{1}{2\epsilon}(\mu,\mu)\right).$
(6.12)

This path integral resembles very much an infinite-dimensional version of the
type of integrals we treated when discussing the Duistermaat-Heckman
localization formula in Section 3.3, but with the fundamental difference that
now the exponent of the integrand is not the moment map, but its square. We
will return to this point in the next section.

Here we notice that in the weak coupling limit $\epsilon\to 0$ the path
integral will receive contributions from the saddle points of the action
$S=\frac{1}{2\epsilon}(\mu,\mu)$, that is the space of solutions of the
classical equations of motion $\nabla^{A}\star F^{A}=0$. Every one of these
contributions brings roughly a term that decays as
$\sim\exp\left(-1/\epsilon\right)$ to the partition function, the main one
being determined by the absolute minimum at $\mu=0$, the subspace of flat
connections $\mu^{-1}(0)\subset\mathcal{A}(P)$. Eliminating the redundancy
from the gauge freedom of the theory, the most interesting piece of the
_physical_ field space, especially in the weak coupling limit, is thus
determined by the quotient

$\mathcal{A}_{0}:=\faktor{\mu^{-1}(0)}{\mathcal{G}(P)},$ (6.13)

or in other words, when computing the path integral one is interested in the
$\mathcal{G}(P)$-equivariant cohomology of $\mu^{-1}(0)$,
$H_{\mathcal{G}}^{*}(\mu^{-1}(0))\cong H^{*}(\mathcal{A}_{0})$. The quotient
$\mathcal{A}_{0}$ is the _moduli space of flat connections_. It turns out that
this space has a nice interpretation in symplectic geometry in terms of
_symplectic reduction_. A theorem by Marsden-Weinstein-Meyer (MWM) [104, 105,
34] in fact states that, in a generic Hamiltonian $G$-space
$(M,\omega,G,\mu)$, if the zero-section of the moment map $\mu^{-1}(0)\subset
M$ is acted on _freely_ by $G$, then the base space $M_{0}:=\mu^{-1}(0)/G$ of
the principal $G$-bundle $\mu^{-1}(0)\xrightarrow{\pi}M_{0}$ is a symplectic
manifold, with symplectic form $\omega_{0}\in\Omega^{2}(M_{0})$ satisfying

$\left.\omega\right|_{\mu^{-1}(0)}=\pi^{*}(\omega_{0}).$ (6.14)

In other words, the restriction of $\omega$ to $\mu^{-1}(0)$ is a basic form,
completely determined by a symplectic form $\omega_{0}$ on the base space. The
space $(M_{0},\omega_{0})$ is called Marsden–Weinstein quotient, symplectic
quotient or symplectic reduction of $M$ by $G$.444Symplectic reduction in
classical mechanics on $M=\mathbb{R}^{2n}$ occurs when one of the momenta is
an integral of motion, $0=\dot{p}_{n}=-\partial_{n}H$. In that case, one can
solve the system in the reduced coordinates
$(q^{1},\cdots,q^{n-1},p_{1},\cdots,p_{n-1})$ and then solve for the $n^{th}$
coordinate separately. The MWM theorem essentially generalizes this process in
a fully covariant setting. Returning to the case of YM theory, this means that
in the limit $\epsilon\to 0$, when the path integral is reduced to
$\mathcal{A}_{0}$ by gauge fixing, the symplectic form can be reduced without
loss of information on this base space. In the next section we will see that,
applying localization, this is extended to the whole exponential.

###  6.2 A localization formula for non-Abelian actions

In the last section we found an Hamiltonian interpretation of the system
$(\mathcal{A}(P),\Omega,\mathcal{G}(P),\mu\equiv F)$ for YM theory on a
2-dimensional Riemann surface $\Sigma$. Here we would like to make contact
with the DH formula, that we described for analogous systems in finite-
dimensional geometry. We notice that the main differences with the case
treated in Section 3.3 are essentially two: $\mathcal{G}(P)$ is non-Abelian in
general for non-Abelian gauge groups $G$, and the path integral is not in the
form of an oscillatory integral of the DH type. Indeed, schematically we have

$\text{DH}\to\int\exp{\left(\omega+i\mu\right)},\qquad\quad\text{YM}\to\int\exp{\left(\Omega-\frac{1}{2}|\mu|^{2}\right)}.$
(6.15)

In the following, we will describe the solution proposed in [15] to generalize
the DH formula to the non-Abelian case starting from the first integral in
(6.15), and how this procedure can be used to recover the second one, of the
YM type. We consider a generic Hamiltonian system $(M,\omega,G,\mu)$ with
compact semisimple Lie group $G$ of dimension $\dim(G)=s$, and the associated
Cartan model defined by the space
$\Omega_{G}(M)=(S(\mathfrak{g}^{*})\otimes\Omega(M))^{G}$ of equivariant
forms, on which we defined the action of the extended operators555We adopt
Witten’s conventions and substitute $\phi^{a}\mapsto i\phi^{a}$ in the
definition of the Cartan differential, analogously to the DH case of Section
3.3.

$\displaystyle d_{C}$ $\displaystyle=1\otimes d-i\phi^{a}\otimes\iota_{a},$
(6.16) $\displaystyle\mathcal{L}_{a}$ $\displaystyle\equiv
1\otimes\mathcal{L}_{a}+\mathcal{L}_{a}\otimes
1\qquad\text{with}\quad\mathcal{L}_{a}\phi^{b}=f^{b}_{ac}\phi^{c},$
$\displaystyle\iota_{a}$ $\displaystyle\equiv 1\otimes\iota_{a}.$

An element $\alpha\in\Omega_{G}(M)$ is an invariant polynomial in the
generators $\phi^{a}$ of $S(\mathfrak{g}^{*})$, with differential forms on $M$
as coefficients. This means that integration over $M$ provides a map in
equivariant cohomology of the type

$\int_{M}:H^{*}_{G}(M)\to S(\mathfrak{g}^{*})^{G},$ (6.17)

or in other words that the integral of an equivariant form is in general a
polynomial in the $\phi^{a}$. This is not quite satisfactory, as we would like
an integration that generalizes the standard de Rham case, giving a map
$H^{*}_{G}(M)\to\mathbb{R}$ (or $\mathbb{C}$). In the case of $G=U(1)$ we
often solved this problem by setting the unique generator $\phi=-1$ (or
$\phi=i$ in this conventions), thus constructing a map in the localized
cohomology, $\int_{M}:H_{U(1)}^{*}(M)_{\phi}\to\mathbb{R}$. Here in the non-
Abelian case, the trick of algebraic localization is not so trivial in
practice, and we avoid it.

An alternative and fruitful idea to saturate the $\phi$-dependence is to make
them dynamical variables, and integrate over them too. Since $\phi^{a}$ can be
regarded as an Euclidean coordinate over $\mathfrak{g}$, this means defining
an integration over $M\times\mathfrak{g}$. As a vector space, the Lie algebra
has a natural measure $d^{s}\phi$ that is unique up to a multiplicative
factor. We fix that factor by choosing a (positive-definite) inner product
$(\cdot,\cdot)$ on $\mathfrak{g}^{*}$ and setting666The inner product on
$\mathfrak{g}^{*}$ is induced from an inner product on $\mathfrak{g}$, and
when we write $(\phi,\phi)$ we really mean $\sum_{a}(\phi^{a},\phi^{a})$. This
can be stated more formally defining $\phi:=\phi^{a}\otimes
T_{a}\in\mathfrak{g}^{*}\otimes\mathfrak{g}$, and then letting act the inner-
product on the $T_{a}$’s, normalized in order to produce a Kronecker delta. We
avoid this cumbersome notation, since the action of the various inner products
is always clear from the context.

$\int_{\mathfrak{g}}d^{s}\phi\
e^{-\frac{\epsilon}{2}(\phi,\phi)}=\left(\frac{2\pi}{\epsilon}\right)^{s/2},$
(6.18)

essentially as we did in (5.132). Since our goal is to integrate equivariant
forms that have polynomial dependence on the $\phi^{a}$, or at most
expressions of the form $\exp(\phi^{a}\otimes\mu_{a})$ that have exponential
dependence, integrating over $\mathfrak{g}$ with the bare measure $d^{s}\phi$
would produce possible divergences. To ensure convergence of these class of
functions, the _equivariant integration_ is defined as [15]

$\boxed{\int_{M\times\mathfrak{g}}\alpha:=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}e^{-\frac{\epsilon}{2}(\phi,\phi)}\alpha}$
(6.19)

where $\epsilon$ is inserted as a regulator. Notice that in general the limit
$\epsilon\to 0$ is not well-defined, for what we said above.

With this enhanced definition of equivariant integration of elements of
$\Omega_{G}(M)$, we can apply the equivariant localization principle to the
present case. Let $\alpha\in\Omega_{G}(M)$ be an equivariantly closed form, so
that $d_{C}\alpha=0$, and choose an equivariant 1-form
$\beta\in\Omega_{G}^{1}(M)=\Omega^{1}(M)^{G}$. The latter is independent on
$\phi^{a}$, and plays the role of the localization 1-form. By the same
arguments of Section 3.1, $\alpha$ and $\alpha e^{d_{C}\beta}$ are
representatives of the same equivariant cohomology class in $H^{*}_{G}(M)$,
and we can deform the integral of $\alpha$ as

$I[\alpha;\epsilon]:=\int_{M\times\mathfrak{g}}\alpha=\int_{M\times\mathfrak{g}}\alpha
e^{td_{C}\beta}\qquad\forall t\in\mathbb{R}.$ (6.20)

In particular, taking the limit $t\to\infty$, this integral localizes on the
critical point set of the localization 1-form $\beta$. This can be seen simply
by expanding the definition of equivariant integration from (6.20),

$\displaystyle I[\alpha;\epsilon]$
$\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\
\alpha\exp\left(-\frac{\epsilon}{2}(\phi,\phi)-it\phi^{a}(\iota_{a}\beta)+td\beta\right)$
(6.21)
$\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\
\alpha\exp\left(-\frac{\epsilon}{2}(\phi,\phi)-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+td\beta\right),$

where in the second line we completed the square and shifted variable in the
$\phi$-integral. Since the term $td\beta$ gives a polynomial dependence on $t$
(by degree reasons, it is expanded up to a finite order), the limit
$t\to\infty$ converges and makes the integral localize on the critical points
of $\iota_{a}\beta$. This shows the localization property of equivariant
integrals in the non-Abelian setting. If for example we suppose $\alpha$ to be
independent on the $\phi^{a}$, we can perform the Gaussian integration to
further simplify $I[\alpha;\epsilon]$,

$I[\alpha;\epsilon]=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\alpha\exp\left(-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+td\beta\right).$
(6.22)

We now apply the above non-Abelian localization principle to the special case
in which $\alpha=\exp(\omega-i\phi^{a}\otimes\mu_{a})$, i.e. generalizing the
DH formula of Section 3.3. We will suppress tensor products in the following,
for notational convenience. First of all, it is straightforward to see that
this form is equivariantly closed,

$d_{C}e^{\omega-i\phi^{a}\mu_{a}}\propto
d_{C}(\omega-i\phi^{a}\mu_{a})=-i\phi^{a}\iota_{a}\omega-i\phi^{a}d\mu_{a}=-i\phi^{a}\iota_{a}\omega+i\phi^{a}\iota_{a}\omega=0.$
(6.23)

The DH oscillatory integral becomes, following the same steps of (6.20) and
(6.21),

$\displaystyle Z(\epsilon)$
$\displaystyle=\int_{M\times\mathfrak{g}}\frac{\omega^{n}}{n!}e^{-i\phi^{a}\mu_{a}}$
(6.24)
$\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(\omega-i\phi^{a}\mu_{a}-\frac{\epsilon}{2}(\phi,\phi)+t(d\beta-i\phi^{a}(\iota_{a}\beta))\right)$
$\displaystyle=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\exp\left(\omega-\frac{1}{2\epsilon}(\mu,\mu)-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+\frac{it}{\epsilon}\sum_{a}\mu_{a}(\iota_{a}\beta)\right),$

and it is independent of $t$. Specializing to the case $t=0$, we get

$\boxed{\int_{M\times\mathfrak{g}}\frac{\omega^{n}}{n!}e^{-i\phi^{a}\mu_{a}}=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\exp\left(\omega-\frac{1}{2\epsilon}(\mu,\mu)\right)},$
(6.25)

that shows the equivalence of the YM type partition function and the
equivariant integral of the DH type! If instead we take the limit
$t\to\infty$, we see that the integral localizes on the critical points of
$(\iota_{a}\beta)$. With a smart choice of localization 1-form, we can show
that this localization locus coincides with the critical point set of the
function $S:=\frac{1}{2}(\mu,\mu)$.

###### Proof.

Since $(M,\omega)$ is symplectic, it admits an almost complex structure
$J\in\Gamma(T^{1}_{1}M)$ and a Riemannian metric $G\in\Gamma(T^{0}_{2}M)$ such
that $\omega(\cdot,J(\cdot))=G(\cdot,\cdot)$ (see [34], proposition
12.6).777In the case of 2-dimensional YM theory, we recall that $J\equiv\star$
is the Hodge duality operator. We pick the localization 1-form

$\beta:=dS\circ J=J^{\sigma}_{\nu}\partial_{\sigma}S\
dx^{\nu}=J^{\sigma}_{\nu}\sum_{a}\mu_{a}(\partial_{\sigma}\mu_{a})\ dx^{\nu},$

and the localization condition $\iota_{a}\beta=0$. Now we use the compatible
metric $G$, that has components
$G_{\mu\nu}=\omega(\partial_{\mu},J(\partial_{n}u))=\omega_{\mu\sigma}J^{\sigma}_{\nu}$.
We consider its “inverse” $G^{-1}$ acting on $T^{*}M$ with components
$G^{\mu\nu}=J^{\mu}_{\sigma}\omega^{\nu\sigma}$, where $\omega^{\nu\sigma}$
are the components of the “inverse” symplectic form, and compute the norm of
the 1-form $dS$,

$G^{-1}(dS,dS)=(\partial_{\mu}SJ^{\mu}_{\sigma})\omega^{\nu\sigma}\partial_{\nu}S=\beta_{\sigma}\sum_{a}\mu_{a}(\omega^{\nu\sigma}\partial_{\nu}\mu_{a})=-\sum_{a}\mu_{a}\beta_{\sigma}(T_{a})^{\sigma}=-\sum_{a}\mu_{a}(\iota_{a}\beta)=0,$

where we used the Hamiltonian equation $d\mu_{a}=-\iota_{a}\omega$ and the
localization condition $\iota_{a}\beta=0$. By the non-degeneracy of $G$, this
condition is equivalent to $dS=0$, that precisely identifies the critical
points of $S$. ∎

Rephrasing the above result in the language of the last section, we just
showed in general terms that the 2-dimensional YM partition function localizes
on the moduli space of solutions of the EoM, meaning that this theory is
essentially classical. We remark again that this localization locus consists
of two qualitatively different types of points: those that minimize absolutely
$S$, that is $\mu^{-1}(0)\subset M$, and the higher extrema with $\mu\neq 0$.
The former ones in the gauge theory are the flat connections, and they give
the dominant contribution to the partition function. The latter ones decays
exponentially in the limit $\epsilon\to 0$ as $\sim\exp(-S/\epsilon)$. In
general thus the partition function can be written as a sum of terms coming
from all these disconnected regions of $M$,

$Z(\epsilon)=\sum_{n}Z_{n}(\epsilon).$ (6.26)

Let us consider the dominant piece $Z_{0}(\epsilon)$ coming from
$\mu^{-1}(0)$, that we interpret in the gauge theory as the rough answer in
the weak coupling limit, and that we can select by restricting the integration
over $M$ to a suitable neighborhood $N$ of $\mu^{-1}(0)$,

$Z_{0}(\epsilon)=\frac{1}{\mathrm{vol}(G)}\int_{N}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(\omega-i\phi^{a}\mu_{a}-\frac{\epsilon}{2}(\phi,\phi)+td_{C}(dS\circ
J)\right),$ (6.27)

where we inserted the localization 1-form such that the $t\to\infty$ limit
identifies the critical locus $\mu^{-1}(0)$. Cohomological arguments show
that, if $G$ acts freely on $\mu^{-1}(0)$, this integral retracts on the
symplectic quotient $M_{0}:=\mu^{-1}(0)/G$, giving

$Z_{0}(\epsilon)=\int_{M_{0}}\exp\left(\omega_{0}+\epsilon\Theta\right)$
(6.28)

for some 4-form $\Theta\in\Omega^{4}(M_{0})$. In particular, we see that in
the weak coupling limit $\epsilon\to 0$, $Z_{0}(0)$ gives the volume of the
symplectic quotient $M_{0}$.

###### Argument for (6.28).

The precise proof is technical and it can be found in [15], we only sketch the
main instructive ideas here. The neighborhood $N$ is chosen small enough to be
preserved by the $G$-action, and represents the split with respect to the
normal bundle we used in Section 4.2. Thus it retracts equivariantly onto
$\mu^{-1}(0)$, meaning that it is homotopic to $\mu^{-1}(0)$ and that the
homotopy commutes with the $G$-action.

First of all we recall what we noticed at the end of the last section: if $G$
acts freely on $\mu^{-1}(0)$ the MWM theorem tells us that the symplectic form
retracts on the symplectic quotient $M_{0}:=\mu^{-1}(0)/G$, so it does not
contribute to the integration over the “normal directions” to $M_{0}$ in $N$.
Here we are not considering a simple integration over $N$, but an equivariant
integration, that provides a map $H^{*}_{G}(N)\to\mathbb{C}$. So in this case
we consider the equivariantly closed extension
$\tilde{\omega}=\omega-i\phi^{a}\mu_{a}$ as representative of a cohomology
class $[\tilde{\omega}]\in H^{2}_{G}(M)$. When restricted over $N$, this class
is the pull-back of a cohomology class $[\omega_{0}]\in H^{2}(M_{0})$, since
$H^{*}_{G}(N)\cong H^{*}_{G}(\mu^{-1}(0))\cong H^{*}(M_{0})$, the first
equivalence following from the retraction of $N$ onto $\mu^{-1}(0)$ and the
second from the fact that the $G$-action is free on $\mu^{-1}(0)$ (these
properties were explained in Chapter 2). Thus we can substitute in the
integral $\tilde{\omega}\mapsto\omega_{0}$ without changing the final result.

The same kind of argument works for the term $\frac{1}{2}(\phi,\phi)$. It is
easy to check that this is both $G$-invariant and equivariantly closed, so it
represents an element $\left[\frac{1}{2}(\phi,\phi)\right]\in H^{4}_{G}(M)$.
When we restrict it to $N$, as above, this class is the pull-back of some
class $[\Theta]\in H^{4}(M_{0})$, and we can make the substitution
$\frac{1}{2}(\phi,\phi)\mapsto\Theta$ in the integral without changing the
final result.

Since both $\omega_{0}$ and $\Theta$ are standard differential forms over
$M_{0}$ and thus independent of $\phi$, the integration over $\mathfrak{g}$
goes along only with the remaining term $d_{C}(dS\circ J)$. We already know
that on $\mu^{-1}(0)\subset N$ this term is zero, so one has to show that its
integral over the normal directions to $\mu^{-1}(0)$ in $N$ produces a trivial
factor of 1. In [15] it is proven that

$\frac{1}{\mathrm{vol}(G)}\int_{F}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(td_{C}(dS\circ
J)\right)=1,$

where $F$ is any fiber of the normal bundle to $\mu^{-1}(0)$ in $N$. From this
(6.28) follows. ∎

######  Example 6.2.1 (The height function on the 2-sphere, again).

Beside the main application of Witten’s localization principle to non-Abelian
gauge theories, we try now to apply this new formalism to the old and simple
example of the height function on the 2-sphere, to compare it with the results
obtained in Chapter 3. Setting $G=U(1)$, $M=\mathbb{S}^{2}$,
$\mu=\cos(\theta)$ and $\omega=d\cos(\theta)\wedge d\varphi$, the equivariant
integration (6.19) of the DH oscillatory integral gives

$\displaystyle Z(\epsilon)$
$\displaystyle=\int_{\mathbb{S}^{2}\times\mathfrak{g}}\omega
e^{-i\phi\mu}=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\int_{-1}^{+1}d\cos\theta\int_{-\infty}^{+\infty}\frac{d\phi}{2\pi}\exp\left(-i\phi\cos\theta-\frac{\epsilon}{2}\phi^{2}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi\epsilon}}\int_{-1}^{+1}dx\
e^{-\frac{x^{2}}{2\epsilon}}=1-2I(\epsilon),$

where
$I(\epsilon):=\int_{1}^{\infty}\frac{dx}{\sqrt{2\pi\epsilon}}\exp(-x^{2}/2\epsilon)$
is a trascendental error function. The three terms in the final result for
$Z(\epsilon)$ (two of which are equal to $-I(\epsilon)$) correspond to the
contributions of the extrema of $(\cos\theta)^{2}$: the two maxima at
$\theta=0,\pi$ contribute with $-I(\epsilon)$ and the minimum at
$\theta=\pi/2$ contributes with $1$. The latter is the dominant piece when
$\epsilon\to 0$, since $I(0)=0$. We see that in general the modified
equivariant integration of this new formalism gives an incredibly complicated
answer, when compared to the simple result of Example 3.2.1 obtained via the
usual equivariant localization principle.

######  Remark.

We notice that we could have expressed equivalently the whole dissertation
above in supergeometric language, since we discussed in Section 4.1 that
integration over $M$ is equivalent to integration over $\Pi TM$. In these
terms, maybe more common in QFT, we can introduce coordinates
$(x^{\mu},\psi^{\mu},\phi^{a})$ over $\Pi TM\times\mathfrak{g}$, where
$\psi^{\mu}:=dx^{\mu}$ are Grassmann-odd, and interpret elements of
$\Omega_{G}(M)$ as elements of $C^{\infty}(\Pi TM\times\mathfrak{g})^{G}$. An
equivariant form $\alpha$ is thus (locally) a $G$-invariant function of
$(x,\psi,\phi)$. For example, the Cartan differential and the definition of
equivariant integration become

$\displaystyle d_{C}$ $\displaystyle=\psi^{\mu}\frac{\partial}{\partial
x^{\mu}}-i\phi^{a}T_{a}^{\mu}\frac{\partial}{\partial\psi^{\mu}},$ (6.29)
$\displaystyle\int_{M\times\mathfrak{g}}\alpha$
$\displaystyle:=\frac{1}{\mathrm{vol}(G)}\int
d^{2n}xd^{2n}\psi\frac{d^{s}\phi}{(2\pi)^{s}}\
\alpha(x,\psi,\phi)e^{-\frac{\epsilon}{2}(\phi,\phi)}.$

###  6.3 “Cohomological” and “physical” YM theory

In this section we are going to review the relation between 2-dimensional YM
theory that we described in Section 6.1 and a topological field theory (TFT)
that can be viewed as its “cohomological” counterpart. We can translate almost
verbatim the general principles that we discussed in the last section, setting

$M\mapsto\mathcal{A}(P),\quad\omega\mapsto\Omega,\quad
G\mapsto\mathcal{G}(P),\quad\mu\mapsto F,$ (6.30)

while we regard $G$ as a compact connected Lie group that acts as the gauge
group on the principal bundle $P\to\Sigma$ over a Riemann surface $\Sigma$.
The moment map is formally equivalent to the curvature $F$ if we identify
$Lie(\mathcal{G}(P))\cong Lie(\mathcal{G}(P))^{*}$ through an inner product on
$\mathfrak{g}$, as in (6.8).

The non-Abelian localization principle of the last section, if used in
reverse, already showed that an equivalent way to express the standard YM
theory is through a “first-order formulation”

$S[A,\phi]=-\int_{\Sigma}\mathrm{Tr}\left(i\phi
F^{A}+\frac{\epsilon}{2}\phi\star\phi\right)$ (6.31)

where we consider $\phi\in\Omega^{0}(\Sigma;\mathrm{ad}(P))\cong
Lie(\mathcal{G}(P))$, and $F^{A}\in\Omega^{2}(\Sigma;\mathrm{ad}(P))$ is the
curvature of $A$.888More precisely, we should say that $A^{a}_{\mu}$ and
$\phi^{a}$ are _coordinates_ functions on $\mathcal{A}(P)\times
Lie(\mathcal{G}(P))$, so they effectively are elements of
$C^{\infty}(\mathcal{A}(P))\otimes Lie(\mathcal{G}(P))^{*}$. This caveat will
be logically important in the following, and it goes along with the _functor
of points_ approach we used in the supergeometric discussion of Chapter 4.
This is essentially what is written in (6.25), where on the LHS we have the
first-order action (the $\epsilon$ dependence is contained in the equivariant
integration), and on the RHS we have the standard YM action
$S[A]=\frac{1}{2\epsilon}(F,F)_{A}$. The first-order formulation has the
quality of showing very clearly the weak coupling limit behavior when
$\epsilon\to 0$, that is less obvious in the standard formulation. In this
limit, the theory becomes topological, in the sense that the action does not
depend on the metric anymore (the metric appears in the Hodge duality
$\star$),

$S_{\epsilon\to 0}[A,\phi]=-\int_{\Sigma}\mathrm{Tr}(i\phi F^{A}).$ (6.32)

This theory is called “BF model”, and it is the prototype of a TFT of Schwarz-
type. The YM theory can thus be seen as a “regulated version” of a truly
topological field theory.999Notice that, although YM theory is clearly
dependent on the metric of $\Sigma$, in 2 dimensions it shows a very “weak”
dependence to it. In fact, in this dimensionality the action can be simplified
as (suppressing the constants and the Lie algebra inner product)
$\displaystyle\int F\wedge\star F$ $\displaystyle=\int
g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\sqrt{|g|}d^{2}x=\int(F_{01})^{2}g^{\mu\rho}g^{\nu\sigma}\varepsilon_{\mu\nu}\varepsilon_{\rho\sigma}\sqrt{|g|}d^{2}x=$
$\displaystyle=\int(F_{01})^{2}|g^{-1}|\sqrt{|g|}d^{2}x=\int(F_{01})^{2}|g|^{-1/2}d^{2}x,$
so the metric does not appear through its components, but only in the
invariant quantity $\det(g)$. At least classically, it is intuitive from the
EoM with respect to $\phi$ that the only contribution to the classical
solutions comes from the moduli space of flat connections, where $F^{A}=0$ up
to gauge transformations. It is not trivial, though, to infer that this is all
the theory has to offer also at the quantum level, that is essentially the
result we showed in general terms in the last section, via the localization
principle applied to the path integral $Z(\epsilon\to 0)$.

In this section we discuss, following [15], how the localization principle can
be translated in the language of TFT, in order to give a more physical
interpretation of the abstract mathematical results that we discussed in
finite dimensions. In particular, we will see that the BF model partition
function can be recovered as an expectation value in a TFT, and that this
ensures its localization properties onto the moduli space of flat connections.
The regulated version at $\epsilon\neq 0$ will not follow precisely this
behavior, as we already know that higher extrema of the YM action contribute
to the partition function $Z(\epsilon)$, but these will have a nice
interpretation in terms of the moduli space.

#### Intermezzo: TFT

This is a good moment to explain briefly in more general terms what one
usually means by TFT, and how localization enters in this subject.
Traditionally, TFT borrows the language of BRST formalism for quantization of
gauge theories, as many examples of topological theories arise from that
context. Recall that the standard BRST quantization procedure is based on the
definition of a differential, the “BRST charge” $Q$, that acts on an extended
graded field space, whose grading counts the “ghost number” (in other words,
$Q$ is an operator of degree $\mathrm{gh}(Q)=+1$). The BRST charge represents
an infinitesimal supersymmetry transformation, that squares to zero in the
gauge-fixed theory. On the Hilbert space, physical states are those of ghost
number zero, and that are annihilated by the BRST charge,

$Q|\text{phys}\rangle=0,\quad\mathrm{gh}|\text{phys}\rangle=0.$ (6.33)

The action of $Q$ on the field space is often denoted with a Poisson bracket-
like notation, $\Phi\mapsto\delta_{Q}\Phi:=-\\{Q,\Phi\\}$ for any field
$\Phi$. By gauge invariance of the vacuum, $Q|0\rangle=0$, and so for any
operator $\mathcal{O}$ one has that $\langle
0|\\{Q,\mathcal{O}\\}|0\rangle=0$.

For a QFT being “topological”, in physics one usually means that all its
quantum properties are independent from a choice of a metric on the base space
$M$.101010Here the word “topological” is somewhat overused. Mathematically, a
topological space consists of a set $M$ and a _topology_ $\mathcal{O}_{M}$,
that is roughly the set of all “open neighborhoods” in $M$. In QFT one almost
always works on base spaces that have the structure of a _manifold_ of some
kind (smooth, complex, …), so that it allows for the presence of an _atlas_
$\mathcal{A}_{M}$ of charts that identifies it locally as $\mathbb{R}^{n}$ for
some (constant) $n$. A manifold is thus a triple
$(M,\mathcal{O}_{M},\mathcal{A}_{M})$, and the choice of a metric is only on
top of this structure. So metric-independence does not generically mean that
the QFT describes only the topology of $M$, but it can depend on the choice of
a smooth (or complex, …) structure on $M$. This is rephrased in the
requirement that the partition function of the theory should be metric-
independent. Assuming the path integral measure to be metric-independent and
$Q$-invariant (so that the BRST symmetry is non anomalous), the variation with
respect to the metric of the partition function is

$\delta_{g}Z\propto\int_{\mathcal{F}}[D\Phi]e^{-S[\Phi]}\delta_{g}S[\Phi]=\int_{\mathcal{F}}[D\Phi]e^{-S[\Phi]}\left(\int_{M}d^{n}x\sqrt{|g|}\delta
g^{\mu\nu}T_{\mu\nu}\right)\propto\langle 0|T_{\mu\nu}|0\rangle,$ (6.34)

so a suitable definition of TFT is the one that requires the energy-momentum
tensor $T_{\mu\nu}$ to be a BRST variation, $T_{\mu\nu}=\\{Q,V_{\mu\nu}\\}$
for some operator $V_{\mu\nu}$. This would ensure $\delta_{g}Z=0$ for what we
said above.

Collecting the above remarks, we can give the following “working definition”
[106]. A _Topological Field Theory_ is a QFT defined over a
$\mathbb{Z}$-graded field space $\mathcal{F}$, with a nilpotent operator $Q$
(i.e. a _cohomological vector field_ on $\mathcal{F}$), and a $Q$-exact
energy-momentum tensor $T_{\mu\nu}=\\{Q,V_{\mu\nu}\\}$, for some
$V_{\mu\nu}\in C^{\infty}(\mathcal{F})$. _Physical states_ are defined to be
elements of the $Q$-cohomology of $\mathcal{F}$ in degree zero,
$|\text{phys}\rangle\in H^{0}(\mathcal{F},Q)$.111111We stress that, if we see
$\mathcal{F}$ as a graded extension of an original field space
$\mathcal{F}_{0}$ acted upon by gauge transformations, the $Q$-cohomology of
$\mathcal{F}$ is exactly the analogous of the gauge-equivariant cohomology
$H_{gauge}^{*}(\mathcal{F}_{0})$, computed in the Cartan model with Cartan
differential $Q$.

######  Remark.

  * •

$Q$ is called “BRST charge” or “operator”, but in general it can be every
supersymmetry charge (as it is a cohomological vector field). We saw examples
in the last chapter where $Q=\delta_{susy}+\delta_{BRST}$, where
$\delta_{BRST}$ is the actual gauge-supersymmetry, and $\delta_{susy}$ is a
Poincaré-supersymmetry.

  * •

The one above is a good “working definition” for most examples, but it is not
completely adequate in all cases. Indeed, there are examples of QFT where
$T_{\mu\nu}$ fails to be BRST-exact, but nonetheless one can still establish
the topological nature of the model. We do not need to treat any example of
this kind, so we refer to [106] for more details.

  * •

We already encountered an example of TFT in Section 5.2, i.e. supersymmetric
QM. There, we called it topological because its partition function was
determined by topological invariants of the base space, here we point out that
it indeed fits into this general discussion. In fact, in (5.69) we saw that
its action can be expressed as $S=\\{Q,\Sigma\\}$, where $Q\equiv Q_{\dot{x}}$
was the $U(1)$-Cartan differential on the loop space, and $\Sigma$ some other
loop space functional. The $Q$-exactness of the energy-momentum tensor follows
from this.

  * •

If the energy-momentum tensor is $Q$-exact, in particular the Hamiltonian
satisfies

$\langle\text{phys}|H|\text{phys}\rangle=\langle\text{phys}|T_{00}|\text{phys}\rangle=0.$
(6.35)

This means that the energy of any physical state is zero, and thus the TFT
does not contain propagating degrees of freedom. It can only describe
“topological” properties of the base space.

TFTs fall in two main broad categories. The first one is constituted by the
so-called TFT of _Witten-type_ (or also _cohomological TFT_). Their defining
property is to have a $Q$-exact quantum action,

$S_{q}=\\{Q,V\\}$ (6.36)

for some operator $V$. As the case of supersymmetric QM, the energy-momentum
tensor is automatically $Q$-exact too,

$T_{\mu\nu}=\frac{2}{\sqrt{|g|}}\left\\{Q,\delta V/\delta
g_{\mu\nu}\right\\}.$ (6.37)

From the equivariant point of view, these theories have a very simple
localization property. The action is representative of the trivial
$Q$-equivariant cohomology class, so the partition function can be in fact
written as

$Z\propto\int_{\mathcal{F}}[D\Phi]e^{-tS_{q}[\Phi]}$ (6.38)

for any $t\in\mathbb{R}$, by the standard argument of supersymmetric
localization. In this case $\mathcal{F}$ is non-compact and so simply taking
$t=0$ is not really allowed, but the limit $t\to\infty$ is perfectly defined,
producing the path integral localization onto the space of solutions of the
classical EoM of $S_{q}$. This means that all TFT of Witten-type are
completely determined by their semiclassical approximation!

The second main class of TFT is called of _Schwarz-type_ (or also _quantum
TFT_). In this case one starts with a classical action $S$ that is metric-
independent, so that the classical energy-momentum tensor is zero. The usual
BRST quantization of this action produces a quantum action of the type
$S_{q}=S+\\{Q,V\\}$ for some $V$, and again

$T_{\mu\nu}=\frac{2}{\sqrt{|g|}}\left\\{Q,\delta V/\delta
g_{\mu\nu}\right\\},$ (6.39)

since the classical piece does not contribute. Chern-Simons theory and BF
theory are of this type. Regarding the second remark above, we mention that
some Schwarz-type TFTs fail to respect the $Q$-exactness property of
$T_{\mu\nu}$, for example in the case of non-Abelian BF theories in dimension
$d>3$ (this happens essentially because one can ensure nilpotency of the BRST
charge only on-shell, and moreover $Q$ has to be defined in a metric-dependent
way).

As we anticipated before we are interested in BF theories, an example of
Schwarz-type TFT, and in particular in their 2-dimensional realization. On a
base space $\Sigma$ of generic dimension $d$, the classical action of the BF
theory is

$S_{BF}[A,B]:=-\int_{\Sigma}\mathrm{Tr}(B\wedge F^{A}),$ (6.40)

where $F^{A}$ is the curvature 2-form of a connection $A$, and $B$ is a
$(d-2)$-form with values in the adjoint bundle. The BRST charge is the usual
gauge-supersymmetry. In the 2-dimensional case, $B\equiv
i\phi\in\Omega^{0}(\Sigma;\mathrm{ad}(P))$, and the naive BRST quantization of
this theory has no problems (as we mentioned, in more than 3 dimensions the
definition of $Q$ have to be modified and things complicate). The quantum
action is

$\displaystyle S_{q}[A,B,\pi,b,c]$
$\displaystyle=-\int_{\Sigma}\mathrm{Tr}\left(BF^{A}+\pi(\nabla^{A}\star
A)+b(\nabla^{A}\star\nabla^{A})c\right)$ (6.41)
$\displaystyle=S_{BF}[A,B]+\\{Q,V\\},$ $\displaystyle\text{with}\quad V$
$\displaystyle:=-\int_{\Sigma}\mathrm{Tr}\left(b(\nabla^{A}\star A)\right),$

where we introduced a ghost $c$, an anti-ghost $b$ and an auxiliary field
$\pi$ with the BRST transformations properties

$\delta_{Q}A=\nabla^{A}c,\quad\delta_{Q}c=-\frac{1}{2}[c,c],\quad\delta_{Q}b=\pi,\quad\delta_{Q}\pi=0,\quad\delta_{Q}B=-[B,c].$
(6.42)

Even though this theory is not of Witten-type, so it has no direct
localization onto the subspace of classical solutions, it turns out that this
is still the case. Traditionally, this is shown finding a suitable
redefinition of coordinates in field space that trivializes the bosonic sector
of the action (meaning that there are no derivatives acting on bosonic
fields), and whose Jacobian cancels in the path integral with the 1-loop
determinant obtained by integrating out the fermions (in this case, the ghosts
fields). This is called _Nicolai map_ , and for the 2-dimensional BF model is
given by the redefinitions [106, 107]

$\xi(A):=F^{A},\qquad\eta(A):=\nabla^{A_{c}}\star A_{q},$ (6.43)

where $A:=A_{c}+A_{q}$ is the expansion of the gauge field around a classical
(on-shell) solution $A_{c}$. Assuming the fermions being integrated out, the
path integrals over $B$ and $\pi$ exactly identify the space of zeros of
$\xi,\eta$, that is the moduli space of solutions to $F^{A}=0$ up to gauge
transformations, $\mathcal{A}_{0}$. We do not pursue this direction further,
but summarize the approach taken in [15], more related to localization.

#### Cohomological approach

Thinking in equivariant cohomological terms, one can expect to show the
localization of the BF model (and of its “regulated” version, _i.e._ YM
theory) finding a suitable “localization 1-form” $V^{\prime}$. This has to be
such that the deformation of the action given by $t\\{Q,V^{\prime}\\}$ induces
the path integral to localize in this subspace when $t\to\infty$, analogously
to what we did for example in Section 5.1, and also to the discussion of the
last section in the finite-dimensional case. This is exactly what is shown in
[15], where the partition function of the model of interest is found as an
expectation value inside a cohomological TFT, proving automatically its
localization behavior.

This cohomological TFT is constructed in a way such that the action of the
BRST operator $Q$ coincides with the Cartan differential $d_{C}$ arising in
the symplectic formulation of 2-dimensional YM theory. Using the more common
supergeometric language of Sections 5.1, 5.2 and recalled in (6.29) (in finite
dimensions), we move from the field space $\mathcal{A}(P)\times
Lie(\mathcal{G}(P))$ to $\mathcal{F}:=\Pi T\mathcal{A}(P)\times
Lie(\mathcal{G}(P))$, introducing the graded coordinates
$(A_{\mu}^{a},\psi_{\mu}^{a},\phi^{a})$ and regarding $d_{C}$ as a
supersymmetry (BRST) transformation,

$d_{C}\equiv-\\{Q,\cdot\\}=\int_{\Sigma}d\Sigma\left(\psi_{\mu}^{a}\frac{\delta}{\delta
A^{a}_{\mu}}-i(\underline{\phi}^{a})^{\mu}\frac{\delta}{\delta\psi_{\mu}^{a}}\right),$
(6.44)

where we recall from (6.3) that the fundamental vector field associated to the
action of a Lie algebra element $\phi\in Lie(\mathcal{G}(P))$ is
$\underline{\phi}=\nabla\phi$. The BRST transformation of every field follows
the rule $\delta_{Q}\Phi=-\\{Q,\Phi\\}\equiv d_{C}\Phi$, and on the
coordinates we have

$\delta_{Q}A=\psi,\qquad\delta_{Q}\psi=-i\nabla\phi,\qquad\delta_{Q}\phi=0.$
(6.45)

The ghost numbers of the elementary fields are
$\mathrm{gh}(A,\psi,\phi)=(0,1,2)$. Other multiplets could be added as well,
but this is the basic one we start with.

Before describing in more detail the particular cohomological theory and its
relation with the physical YM theory, we summarize the strategy that has been
followed. One starts with a suitable cohomological TFT with action

$S_{c}=\\{Q,V\\},$ (6.46)

for some operator $V$, that ensures the localization onto the moduli space
$\mathcal{A}_{0}$ of flat connections. Then the mapping to the physical theory
is done by the common equivariant localization procedure. The TFT is deformed
adding a cohomologically trivial localizing action,

$S(t)=S_{c}+t\\{Q,V^{\prime}\\}=\\{Q,V+tV^{\prime}\\},$ (6.47)

for some gauge invariant operator $V^{\prime}$ that forces only the
interesting YM multiplet $(A,\psi,\phi)$ to survive. Since the field space
here is non-compact, some additional care must be taken in claiming the
$t$-independence of the deformed theory. In particular, the new term must not
introduce new fixed points of the $Q$-symmetry, that would contribute to the
localization locus of the resulting theory. These fixed points, if presents in
the theory at $t\neq 0$, can be interpreted as “flowing from infinity” in the
moduli space (since this is, as just remarked, non-compact).121212This will be
exactly the case in going to the YM theory with $\epsilon\neq 0$, where the
new fixed points are just the higher extrema of the action
$S=\frac{1}{2}(F,F)$. If this is not the case, one can infer properties of the
“physical” theory at $t=\infty$ by making computations in the cohomological
one at $t=0$.131313Notice that this is logically the opposite of what one does
usually in using localization. Here the “easy” theory is the cohomological one
at $t=0$, while the more difficult (but more interesting) is the one at $t\neq
0$.

##### The cohomological theory

The cohomological theory considered in [15] makes use of the additional
multiplets $(\lambda,\eta)$ and $(\chi,H)$ with the transformation properties

$\delta_{Q}\lambda=\eta,\qquad\delta_{Q}\eta=-i[\phi,\lambda],\qquad\delta_{Q}\chi=-iH,\qquad\delta_{Q}H=[\phi,\chi].$
(6.48)

The extended field space and ghost numbers are

$\begin{array}[]{rccccc}\mathcal{F}=&\underbrace{\Pi T\mathcal{A}(P)\times
Lie(\mathcal{G}(P))}&\times&\underbrace{\left(\Omega^{0}(\Sigma;\mathrm{ad}(P))\right)^{2}}&\times&\underbrace{\left(\Omega^{0}(\Sigma;\mathrm{ad}(P))\right)^{2}}\\\
&(A,\psi,\phi)&&(\lambda,\eta)&&(\chi,H)\\\
\mathrm{gh}=&(0,1,2)&&(-2,-1)&&(-1,0).\end{array}$ (6.49)

Of course the definition of the BRST operator will be extended from (6.44),
but we do not need it explicitly. The operator $V$ defining the TFT is chosen
to be

$V=\frac{1}{h^{2}}\int_{\Sigma}d\Sigma\
\mathrm{Tr}\left(\frac{1}{2}\chi(H-2(\star
F^{A}))+g^{\mu\nu}(\nabla^{A}_{\mu}\lambda)\psi_{\nu}\right),$ (6.50)

where $h\in\mathbb{R}$ is a parameter from which the theory is completely
independent (it is analogous to $1/t$ appearing in (6.38)), that can be
interpreted as the “coupling constant” of the TFT. Computing the cohomological
action $S_{c}$ one sees that the $H$ field plays an auxiliary role, and can be
eliminated setting $H=\star F$. Analyzing then the theory in the $h\to 0$
limit (its semiclassical “exact” approximation), the localization locus is
identified by the BRST-fixed point (we refer to [15] for the details)

$\delta_{Q}\chi=0\quad(\Rightarrow
F=0),\qquad\delta_{Q}\psi=0\quad(\Rightarrow\nabla\phi=0).$ (6.51)

This is analogous to the usual situation in Poincaré-supersymmetric theories,
as discussed in Chapter 5, where the localization locus was always identified
by the subcomplex of BPS configurations, given by the vanishing of the
variation of the fermions. Here the “fermions” are the fields with odd ghost
number. This means that the final moduli space contains $\mathcal{A}_{0}$,
plus maybe some contributions from the zero-modes of the other bosonic (even
ghost number) fields, $\lambda$ and $\phi$.

#####  $t\neq 0$ deformation

The deformation (6.47) can be made in order to reduce effectively the field
content of the theory to the YM multiplet only, $(A,\psi,\phi)$. In
particular, to eliminate the non-trivial presence of the field $\lambda$ from
the contributions to the localization locus one can consider

$V^{\prime}=-\frac{1}{h^{2}}\int_{\Sigma}d\Sigma\
\mathrm{Tr}\left(\chi\lambda\right).$ (6.52)

Computing the deformed action $S(t)$, one sees that for $t\neq 0$ all the
additional fields $H,\chi,\lambda,\eta$ can be integrated out (again, some
more details on the technical passages can be found in [15]). In particular,
the EoM for $H$ and $\lambda$ are

$H=0,\qquad\lambda=-\frac{1}{t}(\star F).$ (6.53)

This is already the sign that the localizing term $V^{\prime}$ qualitatively
changed the localization property of the theory. In fact, we see that for
$t=0$ we do not have an algebraic equation for $\lambda$, but (6.53) reduces
to the solution $F=0,H=0$ of the cohomological theory. The theories defined by
$S(t)$ and $S_{c}$ may be thus different, but the failure of their equivalence
can only come from new components of the moduli space that flow in from
infinity for $t\neq 0$; the contribution of the “old” component must be
independent of $t$. Taking the limit $t\gg 1$, the dominant contribution to
the deformed action is (suppressing the $A$-dependence)

$\displaystyle S(t\gg 1)$
$\displaystyle=-\frac{1}{t}\left\\{Q,\int_{\Sigma}d\Sigma\
\mathrm{Tr}\left(\psi^{\mu}\nabla_{\mu}(\star F)\right)\right\\}$ (6.54)
$\displaystyle=\frac{1}{t}\int_{\Sigma}d\Sigma\
\mathrm{Tr}\Bigl{(}i\nabla_{\mu}\phi\nabla^{\mu}(\star F)-(\star
F)[\psi_{\mu},\psi^{\mu}]+\nabla_{\mu}\psi^{\mu}\epsilon^{\nu\sigma}\nabla_{\nu}\psi_{\sigma}\Bigr{)}.$

The main point is that the $\phi$-EoM is actually equivalent to the YM
equation $\nabla\star F=0$. This means that the moduli space of the deformed
theory contains the moduli space of the standard YM theory, and indeed
includes all the higher extrema corresponding to non-flat connections. These
solutions with $F\neq 0$ have $\lambda\sim-1/t$, and thus their contribution
to the path integral goes roughly as $\exp(-1/t)$, as expected. When $t=0$,
the cohomological theory is recovered and the only contribution to the moduli
space is given by the flat connections.

##### Connection with 2-dimensional YM theory

We already argued that the deformed TFT gained all the YM spectrum “flowing
from infinity in the moduli space”, but it remains to see how one can get
practically the YM (and BF) partition function from the theory defined by
$S(t)$. This is simply obtained by another deformation of the exponential in
the partition function: we notice that the YM action is gauge invariant, so it
is meaningful to compute the expectation value of $e^{S_{YM}}$ in the TFT.
Thus we consider an exponential operator of the form

$\begin{gathered}\exp\left(\omega_{0}+\epsilon\Theta\right)\\\
\text{with}\quad\omega_{0}:=\int_{\Sigma}\mathrm{Tr}\left(i\phi
F+\frac{1}{2}\psi\wedge\psi\right),\qquad\Theta:=\frac{1}{2}\int_{\Sigma}\mathrm{Tr}(\phi\star\phi).\end{gathered}$
(6.55)

Since the quantity

$\left\langle\exp\left(\omega_{0}+\epsilon\Theta\right)\right\rangle_{t}\propto\int_{\Pi
T\mathcal{A}(P)\times Lie(\mathcal{G}(P))}DAD\psi D\phi\
\exp\left(\omega_{0}+\epsilon\Theta-S(t)\right)$ (6.56)

is well defined for $t\to\infty$, we can actually take $t=\infty$ and drop
$S(\infty)=0$ (recall that the path integral is independent on the actual
value of $t$), getting exactly the YM partition function

$\left\langle\exp\left(\omega_{0}+\epsilon\Theta\right)\right\rangle_{t}\propto\int_{\Pi
T\mathcal{A}(P)\times Lie(\mathcal{G}(P))}DAD\psi D\phi\
\exp\left(\omega_{0}+\epsilon\Theta\right)\propto Z(\epsilon),$ (6.57)

up to some normalization constant.

In the limit $\epsilon\to 0$, when only the BF model survives, the path
integral over $\phi$ produces the constraint $\delta(F)$, so localizing the
expectation value onto the space of flat connections. This means that,
although we started from different theories $S_{c}\cong S(0)$ and $S(t)$, this
particular expectation value satisfies

$\left\langle\exp\left(\omega_{0}\right)\right\rangle_{t}=\left\langle\exp\left(\omega_{0}\right)\right\rangle_{t=0}$
(6.58)

and the BF model is recovered as an expectation value in the cohomological
theory. This gives another interpretation to the topological behavior of the
BF model, and a measure of the failure of 2-dimensional YM theory in being
topological.

Concluding, we only point out that the the operators $\omega_{0}$ and $\Theta$
are precisely the infinite-dimensional realization in this example of the
general expressions in (6.28). In fact, the symplectic 2-form on
$\mathcal{A}(P)$

$\Omega=\int_{\Sigma}\mathrm{Tr}(\psi\wedge\psi)$ (6.59)

only serves to have a formal interpretation of the measure $DAD\psi
e^{\Omega}$, since the field $\psi$ is really a spectator in the action

$S[A,\psi,\phi]=-\int_{\Sigma}\mathrm{Tr}\left(i\phi
F+\frac{\epsilon}{2}\phi\star\phi+\frac{1}{2}\psi\wedge\psi\right).$ (6.60)

###  6.4 Localization of 2-dimensional YM theory

As we said at the beginning of the chapter, 2-dimensional YM theory is an
exactly solvable theory, whose partition function can be expressed in closed
form, for example by group characters expansion methods [20, 15]. This makes
it possible to compare results from the localization formalism, and obtain a
new geometric interpretation of the already present solution of the theory. In
general, its partition function on a Riemann surface $\Sigma$ of genus $g$,
with a simply-connected gauge group $G$, is given as

$Z(\epsilon)=(\mathrm{vol}(G))^{2g-2}\sum_{R}\frac{1}{\dim(R)^{2g-2}}e^{-\epsilon\tilde{C}_{2}(R)},$
(6.61)

where the sum runs over the representations $R$ of $G$, and $\tilde{C}_{2}(R)$
is related to the quadratic Casimir
$C_{2}(R):=\sum_{a}\mathrm{Tr}_{R}(T_{a}T_{a})$ of the representation $R$ by
some normalization constant. For not simply-connected $G$ this formula has to
be slightly modified (see [15]).141414Simply-connectedness implies that the
principal $G$-bundle $P\to\Sigma$ has to be trivial. When one drops this
condition, the triviality is not ensured and contributions to the formula
appear due to singular points in $\Sigma$ for the connection. We are not
interested in reviewing the proof of (6.61) in general, but we present a quick
argument for a simple example, that already contains the logic behind it.

######  Example 6.4.1 (YM theory with genus $g=1$).

We quickly motivate the result for the YM partition function on a genus 1
surface. We can think of this surface as a disk with boundary, that is
homeomorphic to a sphere with one hole. The radial direction is identified
with the interval $[0,T]$, and the angular coordinate as $[0,L]$ with the
edges identified.

It is more natural to compute the partition function of the theory in the
Hamiltonian formulation,

$Z=\mathrm{Tr}_{\mathcal{H}}\mathcal{P}\exp\left(-\int_{0}^{T}dt\ H(t)\right)$
(6.62)

up to possible normalization factors, where $\mathcal{H}$ is the Hilbert space
of the system. To this end, let us consider the canonical quantization of the
YM action. To make sense of the partition function we must fix a gauge, and we
do this setting $A_{t}=0$ (temporal gauge). In this gauge the action
simplifies as

$S[A]=-\frac{1}{2\epsilon}\int dxdt\
\mathrm{Tr}(F_{01}^{2})=\frac{1}{2\epsilon}\int dxdt\
(\partial_{t}A^{a}_{x})^{2},$ (6.63)

where we expanded $A_{\mu}=A^{a}_{\mu}T_{a}$ with respect to the generators of
$\mathfrak{g}$, and suppressed the inner product implicitly summing over the
Lie algebra indices. We see that the only non-zero canonical momentum is
$\Pi^{a}_{x}=(1/\epsilon)\partial_{t}A^{a}_{x}$, acting on the Hilbert space
as $\Pi^{a}_{x}(t,x)\mapsto\frac{\delta}{\delta A_{x}^{a}(t,x)}$. The
canonical Hamiltonian in temporal gauge is thus

$H(t)=\epsilon\int_{0}^{L}dx\
(\Pi^{a}_{x}(t,x))^{2}\mapsto\epsilon\int_{0}^{L}dx\ \frac{\delta}{\delta
A_{x}^{a}(t,x)}\frac{\delta}{\delta A_{x}^{a}(t,x)}.$ (6.64)

The Hilbert space $\mathcal{H}$ can be considered to consist of gauge
invariant functions $\Psi(A)$. The only gauge invariant data obtained from the
gauge field at any point $p\in\Sigma$ is its holonomy,

$U_{p}[A]:=\mathcal{P}\exp\left(\oint_{C(p)}A\right)\quad\in G,$ (6.65)

where $C(p)$ is a loop about $p$. In the case of the one holed-sphere, all
loops are homotopic to the one on the boundary, so $\Psi\in\mathcal{H}$ must
be an invariant function of

$U=\mathcal{P}\exp\int_{0}^{L}dxA_{1},$ (6.66)

and independent of $t\in[0,T]$. Any invariant function must be expandible in
characters of representations of $G$, so
$\Psi(U)=\sum_{R}\Psi_{R}\mathrm{Tr}_{R}(U)$, where $\Psi_{R}\in\mathbb{C}$
and $\mathrm{Tr}_{R}(U)$ is the Wilson loop in the representation $R$ of $G$.
We notice that the basis functions $\chi_{R}(U):=\mathrm{Tr}_{R}(U)$
diagonalize the Hamiltonian, since

$H\chi_{R}(U)=\epsilon\mathrm{Tr}_{R}\int_{0}^{1}dx\ T_{a}T_{a}\
\mathcal{P}\exp\int_{0}^{L}dxA_{1}=\epsilon LC_{2}(R)\chi_{R}(U),$ (6.67)

where $C_{2}(R):=\mathrm{Tr}_{R}(T_{a}T_{a})$ is the quadratic Casimir in the
representation $R$, a time-independent eigenvalue of $H$. Via this
diagonalization the partition function is easily computed,

$Z=\sum_{R}e^{-TL\epsilon c(R)},$ (6.68)

matching (6.61) for $g=1$. All the geometric information about $\Sigma$ that
enters in $Z$ is its total area $TL$, and any other local property. Notice
that in the topological limit $\epsilon\to 0$, the Hamiltonian vanishes (as
the theory has no propagating degrees of freedom) and the partition function
simplifies further.

From the localization formalism discussed in the last sections, we expect the
partition function to be of the type

$Z(\epsilon)=Z_{0}(\epsilon)+\sum_{n}Z_{n}(\epsilon),$ (6.69)

with $Z_{0}(\epsilon)$ representing the contribution from the moduli space
$\mathcal{A}_{0}$ of flat connections, such that
$Z_{0}(0)\sim\mathrm{vol}(\mathcal{A}_{0})$, and the other $Z_{n}(\epsilon)$
coming from contributions of the higher extrema of the YM action, such that
$Z_{n}(\epsilon)\sim\exp(-1/\epsilon)$ in the weak coupling limit $\epsilon\ll
1$. Using cohomological arguments, in [15] (also nicely reviewed in [102]) it
was shown how to recover the general features of (6.61), and in particular how
to interpret it in terms of an $\epsilon$-expansion at weak coupling, in
relation to the expected form (6.69). The detailed derivation is cumbersome
and requires some more technical background, so we refer to the article for
it, but the logic is essentially the same as for the discussion at the end of
Section 6.2. The strategy is the following. Any solution to the YM EoM
identifies a disconnected region $\mathcal{S}_{n}\subset\mathcal{A}(P)$. For
every such region, one fixes a small neighborhood $N_{n}$ around
$\mathcal{S}_{n}$ that equivariantly retracts onto it. The technically
difficult passage is to perform the integral over the “normal directions” to
$\mathcal{S}_{n}$ in $N_{n}$, and then reduce it on the moduli space
$\mathcal{A}_{n}:=\mathcal{S}_{n}/\mathcal{G}(P)$. The main difficulty is that
in general the MWM theorem (or its equivariant counterpart) does not work,
since the action of $\mathcal{G}(P)$ is not generally free on
$\mathcal{S}_{n}$ (also for $n=0$), as we assumed in writing down (6.28) for
the $\mathcal{S}_{0}\equiv\mu^{-1}(0)$ component. For the higher extrema, this
is readily seen by the fact that the equation

$\nabla(\star F)=0\qquad\text{with}\ F\neq 0$ (6.70)

identifies a vacuum $f:=(\star F)$ as a preferred element of $\mathfrak{g}$
(being it covariantly constant over $\Sigma$), and thus the gauge group is
spontaneously broken to a subgroup $G_{f}\subseteq G$. The action of the whole
gauge group thus cannot be free on this subspace, and the quotient
$\mathcal{S}_{n}/\mathcal{G}(P)$ is singular. Via a suitable choice of
localization 1-form one is still able to extract information by this integral
over the normal directions, and in particular to compute the
$\epsilon$-dependence of the higher extrema contributions.

We limit ourselves now to the comparison of the exact result (6.61) applied to
the case $G=SU(2)$, with the expectation (6.69) obtained by cohomological
arguments. For this gauge group, the character expansion of the partition
function results

$Z(\epsilon)=\frac{1}{(2\pi^{2})^{g-1}}\sum_{n=1}^{\infty}\frac{\exp(-\epsilon\pi^{2}n^{2})}{n^{2g-2}}.$
(6.71)

Simply taking $\epsilon=0$, we see that this is finite and proportional to a
Riemann zeta-function, but to explore better the $\epsilon$-dependence it is
convenient to consider

$\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\left(-\frac{1}{2}\right)^{g-1}\sum_{n=1}^{\infty}\exp\left(-\epsilon\pi^{2}n^{2}\right)=\frac{(-1)^{g-1}}{2^{g}}\left(-1+\sum_{n\in\mathbb{Z}}\exp\left(-\epsilon\pi^{2}n^{2}\right)\right).$
(6.72)

This is not quite in the expected form, since the exponentials in the sum go
to zero for $\epsilon\to 0$ but not as $\exp(-1/\epsilon)$. We can bring this
expression closer to the desired result using the Poisson summation formula

$\sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\hat{f}(k)\qquad\text{with}\
\hat{f}(k):=\int_{-\infty}^{+\infty}f(x)e^{-2\pi ikx},$ (6.73)

where $f$ is a function and $\hat{f}$ its Fourier transform, and rewriting the
sum of exponentials in (6.72) as

$\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g-1}}{2^{g}}\left(-1+\sqrt{\frac{1}{\pi\epsilon}}\sum_{k\in\mathbb{Z}}\exp\left(-\frac{k^{2}}{\epsilon}\right)\right).$
(6.74)

This is exactly the result that could be obtained via integration over normal
coordinates in the localization framework (see [102], eq. (4.102)), but
fundamentally differs from our expectation, since for $\epsilon\to 0$ the
contribution from the flat connections (with $k=0$) is singular for the
presence of the square root. This means that the partition function is not
really a polynomial in $\epsilon$ for small couplings, but an expression of
the form

$Z(\epsilon)=\sum_{m=0}^{g-2}a_{m}\epsilon^{m}+a_{g-3/2}\epsilon^{g-3/2}+\text{exponentially
small terms}.$ (6.75)

The singularity in $Z(\epsilon\to 0)$ arises because, for gauge group $SU(2)$,
the subspace $\mu^{-1}(0)$ is singular and the MWM theorem does not apply.

A simpler situation would occur considering the gauge group $SO(3)$ (which is
not simply connected) and a non-trivial principal bundle over $\Sigma$. In
this case, the character expansion of the partition function requires some
modifications with respect to (6.61), the result being

$Z(\epsilon)=\frac{1}{(8\pi^{2})^{g-1}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\exp(-\pi^{2}\epsilon
n^{2})}{n^{2g-2}}.$ (6.76)

Following the same idea as above, we look at the $(g-1)^{th}$ derivative

$\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g}}{8^{g-1}}\sum_{n=1}^{\infty}(-1)^{n}\exp(-\pi^{2}\epsilon
n^{2})=\frac{(-1)^{g}}{8^{g-1}}\frac{1}{2}\left(-1+\sum_{n\in\mathbb{Z}}(-1)^{n}\exp(-\pi^{2}\epsilon
n^{2})\right),$ (6.77)

and we rewrite the sum using the Poisson summation formula, getting

$\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g}}{2\cdot
8^{g-1}}\left(-1+\sqrt{\frac{1}{\pi\epsilon}}\sum_{k\in\mathbb{Z}}\exp\left(-\frac{\left(k+1/2\right)^{2}}{\epsilon}\right)\right).$
(6.78)

This time we see that the contribution for $k=0$ from the moduli space of flat
connections is finite for $\epsilon\to 0$, and the whole
$\partial^{g-1}Z/\partial\epsilon^{g-1}$ is constant up to exponentially small
terms. This means that the partition function at weak coupling $Z(\epsilon\to
0)$ is a regular polynomial of degree $g-1$ in $\epsilon$, up to exponentially
decaying terms,

$Z(\epsilon)=\sum_{m=0}^{g-2}a_{m}\epsilon^{m}+O(\epsilon^{g-1}),$ (6.79)

and it reflects the fact that, for a non-trivial $SO(3)$-bundle, $\mu^{-1}(0)$
is smooth and acted on freely by $G$. These two quick examples capture the way
this localization framework can give a very geometric interpretation to the
$\epsilon$-expansion of the partition function, and its dependence on the
classical geometry of the moduli space.

Finally, we point out that an analogous treatment was done more recently in
[102] to analyze in this cohomological framework Chern-Simons theory on
3-dimensional _Seifert manifolds_. A Seifert manifold is a smooth object that
can be described as an $\mathbb{S}^{1}$-bundle over a 2-dimensional orbifold,
and this feature makes it possible to dimensionally reduce the Chern-Simons
theory along the direction of the circle $\mathbb{S}^{1}$ to a 2-dimensional
YM theory over a singular base space. It turns out that the localization locus
of the resulting theory receives contributions only from the flat connections
over the total space. This is in accordance with the fact that Chern-Simons
theories are by themselves TFT (of Schwarz-type). 2-dimensional YM theories
have been studied extensively in the past years, and many interesting results
were obtained thanks to their non-perturbative solvability. For example, exact
results for Wilson loops expectation values and their relation with higher
dimensional supersymmetric theories were studied in [108, 109, 110]. A
relation with certain topological string theories and supersymmetric black
hole entropy computations were analyzed in [111]. A duality between higher-
dimensional supersymmetric gauge theories and deformations of 2-dimensional YM
theory was revisited in [112, 113]. Localization techniques play an important
role in all those cases.

##  Chapter 7 Conclusion

In this thesis we reviewed and summarized the main features of the formalism
of equivariant cohomology, the powerful localization theorems first introduced
by Atiyah-Bott and Berline-Vergne, and the principles that allow to formally
apply these integration formulas to QFT. From the physical point of view, the
equivariant (or supersymmetric) localization principle gives a systematic
approach to understand when the “semiclassical” approximation of the path
integral, describing the partition function or an expectation value in QFT,
can give an exact result for the full quantum dynamics. We discussed the
applicability of these techniques in the context of _supersymmetric_ theories.
These are characterized by a space of fields that is endowed with a graded
structure and the presence of some symmetry operator whose “square” gives a
standard “bosonic” symmetry of the action functional. This supersymmetry
operator is interpreted as a differential acting on the subspace of symmetric
configurations in field space, and its cohomology describes the field
theoretical analog of the $G$-equivariant cohomology of a $G$-manifold.

After having introduced the general features of the mathematical theory of
equivariant cohomology and equivariant localization, we reviewed the concepts
in supergeometry that allow for the construction of supersymmetric QFT, and
that constitute the correct framework to translate the mathematical theory in
the common physical language. Since many recent applications of the
localization principle aimed at the computations of path integrals in
supersymmetric QFT on curved spaces, we included a discussion of the main
tools needed to define supersymmetry in such instances. Then we collected some
examples from the literature of application of the supersymmetric localization
principle to path integrals in QFT of diverse dimensions. The common feature
of these examples is that, via a suitable “cohomological” deformation of the
action functional, it is possible to reduce the infinite-dimensional path
integral to a finite-dimensional one that represents its semiclassical limit,
as stressed above. We described cases in which this reduction relates the
partition function to topological invariants of the geometric structure
underlying the theory, namely the cases of supersymmetric QM (a 1-dimensional
QFT) and the weak coupling limit of 2-dimensional Yang-Mills theory (its
“topological” limit). We also reviewed the more recent applications to the
computations of the expectation values of supersymmetric Wilson loops in 3-
and 4-dimensional gauge theories, namely Supersymmetric Chern-Simons theory
and Supersymmetric Yang-Mills theory defined on the 3- and the 4-sphere. In
these cases, the path integral results to be equivalently described by some
0-dimensional QFT with a Lie algebra as target space, called “matrix model”.

In the last few decades, the literature concerning the applications of
supersymmetric localization has grown exponentially, and many other advanced
examples of its use in the physics context have been found. From the point of
view of supersymmetric QFT, a consistent slice of the state-of-the-art on the
subject can be found in [12], including computations analogue to the one we
showed for Wilson loop expectation values or topological invariants over more
complicated geometries. From the point of view of Quantum (Super)Gravity,
these techniques have found applications in the computations of the Black Hole
quantum entropy [114, 115]. In many circumstances, localization allows for the
analysis of properties of QFT at strong coupling, an otherwise prohibited
region of study with conventional perturbative techniques. This feature can be
used also to test a class of conjectural dualities between some types of gauge
theories and string theories, the so-called AdS/CFT correspondences [96].
Concerning the subject of Wilson loops in 3-dimensional Chern-Simons theories
and their relations to matrix models and holography, for which localization
has played an important role, a recent reference that concisely reviews the
state-of-the-art is [84].

##  Appendix A Some differential geometry

###  A.1 Principal bundles, basic forms and connections

Here we recall some notions about principal bundles that can be useful to
follow the discussion, especially of the first chapters of this thesis.
Principal bundles are the geometric construction behind the concepts of
_covariant derivatives_ and _connections_ in gauge theory or General
Relativity, for example. If $G$ is a Lie group, a principal $G$-bundle is a
smooth bundle $P\xrightarrow{\pi}M$ such that

  1. (i)

$P$ is a (right) $G$-manifold;

  2. (ii)

the $G$-action on $P$ is _free_ ;

  3. (iii)

as a bundle, $P\to M$ is isomorphic to $P\to P/G$, where the projection map is
canonically defined as $p\mapsto[p]$.

Notice that since the $G$-action is free, a principal $G$-bundle is a fiber
bundle with typical fiber $G$, and by the third property it is at least
_locally trivial_ , i.e. over every open set $U\subseteq M$ it looks like
$G\times U$. Morphisms of principal bundles are naturally defined as maps
between bundles that preserve the $G$-structure, so $G$-equivariant maps. A
principal bundle is _trivial_ if it is isomorphic through a principal bundle
isomorphism to the trivial product bundle $G\times M$. A useful fact is that
the triviality of a principal bundle is completely captured by the existence
of a _global_ section $\sigma:M\to P$ such that $\pi\circ\sigma=id_{M}$. Since
every principal bundle is locally trivial, than local sections can always be
chosen and they constitute a so-called _local trivialization of the bundle_.

The main example of principal bundle that occurs in the geometric construction
of spacetime is the _frame bundle_ $LM$ over some $n$-dimensional smooth
manifold $M$. At every point $p\in M$, the elements of the fiber $L_{p}M$ are
the _frames_ at $p$, i.e. all the possible bases $e=(e_{1},\cdots,e_{n})$ for
the tangent space $T_{p}M$. $LM$ has a natural $GL(n,\mathbb{R})$ right action
that corresponds to the rotation of the basis, $e\cdot
g:=(e_{k}g^{k}_{1},\cdots,e_{k}g^{k}_{n})$ for $g\in GL(n,\mathbb{R})$. In
gauge theories, the _structure group_ (or sometimes _gauge group_) $G$ of the
theory is the Lie group acting on the right on a principal $G$-bundle.

Since the fibers of the principal bundle are essentially the Lie group $G$,
tangent vectors on $P$ can come from its Lie algebra $\mathfrak{g}$. This
leads to the following definition.

######  Definition A.1.1.

The _vertical sub-bundle_ $VP$ of the tangent bundle $TP$ is the disjoint
union

$VP:=\bigsqcup_{p\in P}V_{p}P,\qquad\text{with}\
V_{p}P:=\mathrm{Ker}(\pi_{*p})=\\{X\in T_{p}P|\pi_{*}(X)=0\\}\subset T_{p}P.$

Analogously but for differential forms, the _basic forms_ inside $\Omega(P)$
are those forms $\omega\in\mathrm{Im}(\pi^{*})$, so that it exists an
$\alpha\in\Omega(M)$ such that $\omega=\pi^{*}\alpha$. The space of basic
forms is denoted $\Omega(P)_{bas}$.

The vertical vectors in every $V_{p}P$ are in one to one correspondence with
the Lie algebra elements in $\mathfrak{g}$, through the Lie algebra
homomorphism

$X\in\mathfrak{g}\mapsto\underline{X}:=\left.\frac{d}{dt}\right|_{t=0}\left(\cdot
e^{tX}\right)^{*}$ (A.1)

that maps Lie algebra elements to the corresponding _fundamental vector
fields_.111The choice of the sign at the exponential differs from the one in
(2.19) because here we are considering a right action. Fundamental vector
fields satisfy the following properties:

  1. (i)

$[\underline{X},\underline{Y}]=\underline{[X,Y]}$;

  2. (ii)

the integral curve of $\underline{X}$ through $p\in M$ is

$\displaystyle\gamma_{p}:\mathbb{R}$ $\displaystyle\to M$ (A.2) $\displaystyle
t$ $\displaystyle\mapsto\gamma_{p}(t)=p\cdot e^{tX};$

  3. (iii)

denoting with $r_{g}$ the right action of $g\in G$,

$\left(r_{g}\right)_{*}(\underline{X}_{p})=\left(\underline{Ad_{g^{-1}*}(X)}\right)_{r_{g}(p)}.$
(A.3)

As for the vertical vector fields being encoded in the Lie algebra
$\mathfrak{g}$, also the basic forms can be characterized in terms on the
(infinitesimal) action of $\mathfrak{g}$ on $\Omega(P)$. This can be seen
introducing the following definitions.

######  Definition A.1.2.

A differential form $\omega\in\Omega(P)$ is said to be _$G$ -invariant_ if it
is preserved by the $G$ action:

$\omega=(r_{g})^{*}\omega\qquad\forall g\in G.$

The space of $G$-invariant forms is commonly denoted $\Omega(P)^{G}$. A
differential form is called _horizontal_ if it is annihilated by vertical
vector fields,

$\iota_{X}\omega=0\qquad\forall X\in\Gamma(VP).$

The properties of being invariant and horizontal can be also stated
infinitesimally with respect to the action of the Lie algebra $\mathfrak{g}$.
If we define

$\mathcal{L}_{X}:=\mathcal{L}_{\underline{X}},\qquad\iota_{X}:=\iota_{\underline{X}},\qquad\forall
X\in\mathfrak{g},$ (A.4)

then an invariant form is characterized by $\mathcal{L}_{X}\omega=0$ for every
$X\in\mathfrak{g}$, and a horizontal form by $\iota_{X}\omega=0$ for every
$X\in\mathfrak{g}$. This makes the concepts of invariant and horizontal
elements independent from the principal bundle structure, so that they can be
defined by this characterization for every $\mathfrak{g}$-dg algebra, as in
Section 2.3. Also basic forms can be defined for every $\mathfrak{g}$-dg
algebra, combining the definitions of invariant and horizontal forms, thanks
to the following theorem:

######  Theorem A.1.1 (Characterization of basic forms).

Schematically,

$\text{invariant}+\text{horizontal}\Leftrightarrow\text{basic}.$

###### Proof.

For notational convenience only, let us consider 1-forms.

  * $(\Leftarrow)$

If $\omega=\pi^{*}\alpha$ is basic, then

$(r_{g})^{*}\omega=(r_{g})^{*}\pi^{*}\alpha=(\pi\circ
r_{g})^{*}\alpha=\omega,$

since the principal bundle is locally trivial. So $\omega$ is also invariant.
For a vertical vector $X\in VP$,

$\iota_{X}\omega=(\pi^{*}\alpha)(X)=\alpha(\pi_{*}(X))=0.$

So $\omega$ is also horizontal.

  * $(\Rightarrow)$

Let $\omega\in\Omega^{1}(P)$ be horizontal and invariant. Since $\pi$ is
surjective, for every vector $X\in T_{p}P$ there exists $Y=\pi_{*}(X)\in
T_{\pi(p)}M$. We can define $\alpha\in\Omega^{1}(M)$ such that, at every $x\in
M$

$\alpha_{x}(Y):=\omega_{p}(X)\qquad\text{for}\ p\in\pi^{-1}(x),$

and thanks to the horizontality and invariance of $\omega$ we can check that
this form is well defined, i.e. independent from the choice of point $p$ in
the fiber $\pi^{-1}(x)$ and from the choice of vector $X$ such that
$\pi_{*}(X)=Y$. In fact, if $X^{\prime}\in T_{p}P$ is another vector such that
$\pi_{*}(X^{\prime})=Y$, then $\pi_{*}(X-X^{\prime})=0$ so $(X-X^{\prime})\in
V_{p}P$. By horizontality,
$\omega(X-X^{\prime})=0\Rightarrow\omega(X)=\omega(X^{\prime})$, so $\alpha$
is independent from the choice of vector. Moreover, if
$p^{\prime}\in\pi^{-1}(x)$ is another point in the fiber, there exists a $g\in
G$ such that $r_{g}(p)=p^{\prime}$, so by $G$-invariance
$\omega_{p^{\prime}}=\omega_{p}$ and thus $\alpha$ is independent from the
choice of point in the fiber.

∎

######  Proposition A.1.1.

The differential $d$ closes on the subspace $\Omega(P)_{bas}$ of basic forms,
defining a proper subcomplex. This extends to any $\mathfrak{g}$-dg algebra.

###### Proof.

Consider $\omega\in\Omega(P)_{bas}$, and its differential $d\omega$. We
characterize basic forms by being horizontal and $G$-invariant. By Cartan’s
magic formula the Lie derivative commutes with the differential, so for every
$X\in\mathfrak{g}$, $\mathcal{L}_{X}d\omega=d(\mathcal{L}_{X}\omega)=0$. Thus
$d\omega$ is still $G$-invariant. Also,
$\iota_{X}d\omega=\mathcal{L}_{X}\omega-d\iota_{X}\omega=0$. Thus $d\omega$ is
still horizontal, and so basic. ∎

We recall now the definition of connection and curvature on principal bundles,
from which one inherits covariant derivatives on associated vector bundles.

######  Definition A.1.3.

An _(Ehresmann) connection_ on a principal $G$-bundle $P\to M$ is an
_horizontal distribution_ $HP$, i.e. a smooth choice at every point $p\in P$
of vector subspaces $H_{p}P\subset T_{p}P$ such that

  1. (i)

$T_{p}P=V_{p}P\oplus H_{p}P$;

  2. (ii)

$(r_{g})_{*}(H_{p}P)=H_{p\cdot g}P$ ($G$-equivariance of the horizontal
projection).

Given a horizontal distribution $HP$, every vector $X\in T_{p}P$ decomposes
into an _horizontal_ and a _vertical_ part,

$X=hor(X)+ver(X).$

A _connection 1-form_ on $P\to M$ is Lie algebra-valued 1-form
$A\in\Omega^{1}(P)\otimes\mathfrak{g})$ such that

  1. (i)

for any $X\in\mathfrak{g}$, $\iota_{X}A=A(\underline{X})=X$ (_vertical_
1-form);

  2. (ii)

for any $g\in G$, $(r_{g})^{*}A=(Ad_{g^{-1}*}\circ A)$ ($G$-equivariance).

The choice of a horizontal distribution is equivalent to the choice of a
connection 1-form on $P$, since at every $p\in P$ one can use $A$ as a
projection onto the vertical subspace $V_{p}P\cong\mathfrak{g}$, and $\pi_{*}$
as a projection onto the horizontal subspace, identifying
$V_{p}P:=\mathrm{Ker}(\pi_{*p})$ and $H_{p}P:=\mathrm{Ker}(A_{p})$. This
choice is smooth and $G$-equivariant since $A$ is, by definition. Notice that
the splitting $HP\oplus VP$ induces a splitting
$\Omega^{1}(P)=\Omega^{1}_{hor}(P)\oplus\Omega^{1}_{ver}(P)$, and that we can
identify the “space of connection 1-forms” as

$\mathcal{A}(P):=\\{A\in(\Omega^{1}_{ver}(P)\otimes\mathfrak{g})|A\ \text{is
}G\text{-equivariant}\\}.$ (A.5)

It is easy to see that for every $A,A^{\prime}\in\mathcal{A}(P)$, their
difference is not a connection, and in fact it is an _horizontal_
$\mathfrak{g}$-valued 1-form,

$(A-A^{\prime})\in\mathfrak{a}:=\\{a\in(\Omega^{1}_{hor}(P)\otimes\mathfrak{g})|a\
\text{is }G\text{-equivariant}\\}.$ (A.6)

This means that every connection $A$ can be written as another connection
$A^{\prime}$ plus a horizontal form, or in other words that
$(\mathcal{A}(P),\mathfrak{a})$ can be seen as a natural _affine space_ ,
modeled on the infinite-dimensional vector space $\mathfrak{a}$. As for any
affine space, one can think of the space of connections as an infinite-
dimensional smooth manifold, with tangent spaces
$T_{A}\mathcal{A}(P)\cong\mathfrak{a}$ at every $A\in\mathcal{A}(P)$.

######  Definition A.1.4.

The _covariant exterior derivative_ on $\Omega(P)$ is $D:=d\circ hor^{*}$. The
_curvature_ of a connection 1-form $A$ is

$F:=DA=dA(hor(\cdot),hor(\cdot))\in\Omega^{2}(P)\otimes\mathfrak{g}.$

The curvature $F$ satisfies the following properties:

  1. (i)

by definition, $F$ is horizontal: $\iota_{X}F=0$ for every $X\in\mathfrak{g}$;

  2. (ii)

by $G$-equivariance of $A$, $F$ is $G$-equivariant too;

  3. (iii)

it obeys the structural equation

$F=dA+\frac{1}{2}[A\stackrel{{\scriptstyle\wedge}}{{,}}A]$ (A.7)

where $[A\stackrel{{\scriptstyle\wedge}}{{,}}A]=f^{a}_{bc}A^{b}\wedge
A^{c}\otimes T_{a}$ with respect to a basis $\\{T_{a}\\}$ of $\mathfrak{g}$
and the structure constants $f^{a}_{bc}$;

  4. (iv)

it obeys the second Bianchi identity,

$DF=0\qquad\text{or}\qquad dF=[F\stackrel{{\scriptstyle\wedge}}{{,}}A].$ (A.8)

One can consider the very trivial construction of a principal $G$-bundle as
$G\to pt$, where $P=G\times pt\cong G$. Here the right $G$-action is simply
the diagonal action (trivial on $pt$, induced by the natural action on $G$).
On this bundle there is a canonical choice of connection 1-form, the _Maureer-
Cartan (MC) form_ $\Theta\in\Omega^{1}(G)\otimes\mathfrak{g}$. For every
vector $X\in T_{g}G$ at some $g\in G$, there is a Lie algebra element $A\in
T_{e}G\cong\mathfrak{g}$ such that $X=l_{g*}(A)$, and the MC form is defined
by

$\Theta_{g}(X):=l_{g^{-1}*}(X)=A.$ (A.9)

One can check that this form is indeed $G$-equivariant, and it is obviously
vertical, giving a connection 1-form. Moreover it satisfies the _Maurer-Cartan
equation_

$d\Theta+\frac{1}{2}[\Theta\stackrel{{\scriptstyle\wedge}}{{,}}\Theta]=0,$
(A.10)

so that by (A.7) we see that its curvature is zero. On the trivial principal
$G$-bundle $G\times M\to M$ one can always define a connection 1-form by
pulling back the MC connection along the projection $\pi_{1}:G\times M\to G$.
In the general case, the principal bundle $P$ is locally trivial, so in any
local patch $G\times U_{\alpha}\to U_{\alpha}$ one can pull back the MC
connection and use a suitable partition of unity to glue together the local
pieces to a global connection 1-form on $P$. This shows that any principal
bundle allows for a connection. The curvature $F$ of the chosen connection $A$
measures, in a sense, the deviation of $A$ from being the Maurer-Cartan
connection.

As said before, a connection on a principal $G$-bundle allows for the
definition of a covariant derivative on _associated vector bundles_. An
associated vector bundle to the principal $G$-bundle $P\xrightarrow{\pi}M$ is
a vector bundle constructed over $M$ with some typical fiber $V$ (a vector
space) that has a (left) $G$-action compatible with the one on $P$. Precisely,
the associated bundle is $P_{V}\xrightarrow{\pi_{V}}M$, where

$\displaystyle P_{V}:=\faktor{(P\times V)}{\sim_{G}}\qquad\text{with}\
(p,v)\sim_{G}(p\cdot g,g^{-1}\cdot v)\ \forall(p,v)\in P\times V,g\in G,$
(A.11) $\displaystyle\pi_{V}([p,v]):=\pi(p),$

and it has indeed typical fiber $V$. In the case of the frame bundle $P=LM$,
one can construct the tangent bundle $TM$, the cotangent bundle $T^{*}M$ and
all the tensor bundles as associated to $LM$. In fact, for the tangent bundle
for example, the typical fiber is $V:=\mathbb{R}^{n}$ and the
$GL(n,\mathbb{R})$-action is $(g\cdot v)^{k}=g^{k}_{j}v^{j}$. This encodes the
change of basis rule if we see vectors as elements $[e,v]\in
LM_{\mathbb{R}^{n}}$,

$[e,v]\equiv e_{k}v^{k}\sim_{GL}[e\cdot g,g^{-1}\cdot v]\equiv
e_{j}g^{j}_{k}(g^{-1})^{k}_{j}v^{j}=e_{k}v^{k}.$ (A.12)

On the associated vector bundle, a _field_ (in physics terms) is a (local, at
least) section $\phi:M\to P_{V}$, that can be always seen locally as a
$V$-valued function on every $U\subseteq M$, $\tilde{\phi}:U\to V$, so that
$\phi(x)=[p,\tilde{\phi}(x)]$ for some chosen $p\in\pi^{-1}(x),x\in U$.
Another example of this concept that came up in Chapter 2 is the _homotopy
quotient_ $M_{G}:=(M\times EG)/G$ of a $G$-manifold $M$. This is precisely the
associated bundle with fiber $V=M$ to the principal $G$-bundle $EG\to BG$