A_{\mathrm{res}}+A_{\mathrm{fl}}.$

Inside $\mathcal{Y}^{\prime}$, we have the gauge-fixing lagrangian
$\mathcal{L}\subset\mathcal{Y}^{\prime}$ given by forms of degree 0 in $I$ —
i.e., $\mathcal{L}$ is given by $A_{\mathrm{fl}}=0$ — and we write for
$\psi_{\mathrm{fl}}\in\mathcal{L}$

$\psi_{\mathrm{fl}}=\psi_{\mathrm{fl}}^{[>0]}+\psi^{-}_{\mathrm{fl}}+\psi_{\mathrm{fl}}^{+}+\psi_{\mathrm{fl}}^{[<0]}.$

Recollecting, for a field
$\psi+A\in\mathcal{B}\times\mathcal{V}\times\mathcal{L}$ we obtain

(44) $\displaystyle\psi$
$\displaystyle=\widetilde{\psi}+\psi_{\mathrm{res}}+\psi_{\mathrm{fl}}$
$\displaystyle=\widetilde{\psi^{[>0]}_{\mathrm{in}}}+\widetilde{\psi_{\mathrm{in}}^{-}}+\widetilde{\psi^{[>0]}_{\mathrm{out}}}+\widetilde{\psi_{\mathrm{out}}^{+}}+\psi_{\mathrm{res}}+\psi_{\mathrm{fl}}^{[>0]}+\psi^{-}_{\mathrm{fl}}+\psi_{\mathrm{fl}}^{+}+\psi_{\mathrm{fl}}^{[<0]},$
$\displaystyle A$ $\displaystyle=dt\cdot A_{\mathrm{res}}.$

The gauge-fixed polarized action is then computed as follows:

######  Lemma 5.1.

Restricted to the gauge-fixing lagrangian, the polarized action can be written
as

(45)
$S^{f}[\mathsf{A}]=S_{\mathrm{source}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{fl}}]+S_{0}[\psi_{\mathrm{fl}}]+S_{\mathrm{int}}[\psi_{\mathrm{res}},\psi_{\mathrm{fl}},A_{\mathrm{res}}]+S_{\mathrm{back}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},A_{\mathrm{res}}],$

where

$\displaystyle
S_{\mathrm{source}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},\psi_{\mathrm{fl}}]$
$\displaystyle=(\psi^{+}_{\mathrm{out}},\psi_{\mathrm{fl}}^{-}(1))-(\psi^{-}_{\mathrm{in}},\psi_{\mathrm{fl}}^{+}(0))+(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{fl}}(1))-(\psi^{[>0]}_{\mathrm{in}},\psi_{\mathrm{fl}}^{[<0]}(0)),$
$\displaystyle S_{0}[\psi_{\mathrm{fl}}]$
$\displaystyle=\int_{I}(\psi_{\mathrm{fl}}^{+},d_{I}\psi_{\mathrm{fl}}^{-})+\int_{I}(\psi^{[<0]}_{\mathrm{fl}},d_{I}\psi_{\mathrm{fl}}^{[>0]}),$
$\displaystyle S_{\mathrm{int}}[\psi_{\mathrm{fl}},A_{\mathrm{res}}]$
$\displaystyle=-\int_{I}\,dt(d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{fl}}^{-})-\int_{I}\,dt(d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi^{+}_{\mathrm{fl}})+\int_{I}dt(A_{\mathrm{res}}^{[>1]},d_{\mathfrak{g}}\psi_{\mathrm{fl}}^{[<0]}),$
$\displaystyle
S_{\mathrm{back}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},A_{\mathrm{res}}]$
$\displaystyle=(\psi^{[>0]}_{\mathrm{out}}-\psi^{[>0]}_{\mathrm{in}},\psi_{\mathrm{res}})+(A_{\mathrm{res}}^{[>1]},d_{\mathfrak{g}}\psi_{\mathrm{res}}).$

Here we have introduced the notation
$A^{[1]}_{\mathrm{res}},A^{[>1]}_{\mathrm{res}}$ for the components of
$A_{\mathrm{res}}$ valued in $X^{[1]}$ and in $X^{[>1]}$,
respectively.343434We understand $A^{[k]}_{\mathrm{res}}$ as valued in
$X^{[k]}$ but with ghost number $k-1$; the shift is due to the fact that
$A^{[k]}_{\mathrm{res}}$ is a coefficient of a $1$-form on the source. This
shift is a standard feature of the AKSZ construction. In particular,
$A^{[1]}_{\mathrm{res}}$ is an object of ghost number zero.

###### Proof.

The polarized action is

$S^{f}[\psi+A]=\frac{1}{2}\int_{I}(\psi,d_{I}\psi)+\int_{I}(A,d_{\mathfrak{g}}\psi)+f(\psi),$

where

$\displaystyle f(\psi)$
$\displaystyle=f_{\mathrm{out}}(\psi(1))-f_{\mathrm{in}}(\psi(0))$
$\displaystyle=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{fl}}(1))+\frac{1}{2}(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{fl}}(1))-\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{fl}}(0))+\frac{1}{2}(\psi^{+}_{\mathrm{fl}}(0),\psi^{-}_{\mathrm{in}}).$

Splitting $\psi$ as in (44) and letting the support of $\widetilde{\psi}$ go
towards $\partial I$ we obtain that

$\displaystyle\frac{1}{2}\int_{I}(\psi,d_{I}\psi)$
$\displaystyle=\frac{1}{2}\int_{I}(\widetilde{\psi}+\psi_{\mathrm{res}}+\psi_{\mathrm{fl}},d_{I}(\widetilde{\psi}+\psi_{\mathrm{res}}+\psi_{\mathrm{fl}}))$
$\displaystyle=\frac{1}{2}(\widetilde{\psi},\psi_{\mathrm{res}})\bigg{|}^{t=1}_{t=0}+\frac{1}{2}(\widetilde{\psi},\psi_{\mathrm{fl}})\bigg{|}^{t=1}_{t=0}+\frac{1}{2}\int_{I}(\psi_{\mathrm{fl}},d_{I}\psi_{\mathrm{fl}})$
$\displaystyle=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}}-\psi^{[>0]}_{\mathrm{in}},\psi_{\mathrm{res}})+\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi_{\mathrm{fl}}^{[<0]}(1))+\frac{1}{2}(\psi_{\mathrm{out}}^{+},\psi_{\mathrm{fl}}^{-}(1))$
$\displaystyle-\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{fl}}(0))-\frac{1}{2}(\psi_{\mathrm{in}}^{-},\psi_{\mathrm{fl}}^{+}(0))+\int_{I}(\psi_{\mathrm{fl}}^{+},d_{I}\psi_{\mathrm{fl}}^{-})+\int_{I}(\psi_{\mathrm{fl}}^{[<0]},d_{I}\psi_{\mathrm{fl}}^{[>0]})$

and

$\displaystyle\int_{I}(dt\cdot A_{\mathrm{res}},d_{\mathfrak{g}}\psi)=$
$\displaystyle-\int_{I}\,dt(d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{fl}}^{-})-\int_{I}\,dt(d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi^{+}_{\mathrm{fl}})$
$\displaystyle+\int_{I}dt(A_{\mathrm{res}}^{[>1]},d_{\mathfrak{g}}\psi_{\mathrm{fl}}^{[<0]})+(A_{\mathrm{res}}^{[>1]},d_{\mathfrak{g}}\psi_{\mathrm{res}}).$

Collecting the various terms, we obtain (45). ∎

Notice that adding $f$ has the effect of doubling the boundary source terms.

####  5.1.4. Effective action

The effective action is defined by

$\displaystyle Z$
$\displaystyle=e^{\frac{i}{\hbar}S_{\mathrm{eff}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},A_{\mathrm{res}}]}=\int\mathcal{D}\psi_{\mathrm{fl}}\,e^{\frac{i}{\hbar}S^{f}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},\psi_{\mathrm{fl}},A_{\mathrm{res}}]}$
$\displaystyle=e^{\frac{i}{\hbar}S_{\mathrm{back}}}\int\mathcal{D}\psi_{\mathrm{fl}}\,e^{\frac{i}{\hbar}(S_{\mathrm{source}}+S_{0}+S_{\mathrm{int}})}$

where the integral is defined in terms of Feynman diagrams.

######  Proposition 5.2.

The effective action is given by

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=S_{\mathrm{ph}}+S_{\mathrm{gh}}\quad\mbox{with}$ $\displaystyle
S_{\mathrm{ph}}$
$\displaystyle=(\psi^{+}_{\mathrm{out}},\psi^{-}_{\mathrm{in}})+(\psi^{-}_{\mathrm{in}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})+(\psi_{\mathrm{out}}^{+},d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})+\frac{1}{2}(d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}),$
$\displaystyle S_{\mathrm{gh}}$
$\displaystyle=(\psi^{[>0]}_{\mathrm{out}}-\psi^{[>0]}_{\mathrm{in}},\psi_{\mathrm{res}})+(A^{[>1]}_{\mathrm{res}},d_{\mathfrak{g}}\psi_{\mathrm{res}}).$

###### Proof.

In terms of Feynman diagrams, $S_{\mathrm{source}}$ generates boundary
vertices (figure 2(a)), $S_{\mathrm{int}}$ generates bulk vertices (Figure
2(b)).353535We are ignoring here the ghost vertices that in this case do not
contribute to the effective action since no vertex carries as $\psi^{[>0]}$.

$\psi_{\mathrm{in}}^{-}$$\psi_{\mathrm{out}}^{+}$ (a) Boundary vertices

$d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}$

$d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}$

(b) Bulk vertices

Figure 2. Vertices in 1D AKSZ theories with linear polarizations

The term $S_{0}$ generates the propagators

$\displaystyle\eta_{\mathrm{ph}}(t,t^{\prime})=\frac{i}{\hbar}\langle\psi^{+}_{\mathrm{fl}}(t)\psi^{-}_{\mathrm{fl}}(t^{\prime})\rangle$
$\displaystyle=(\theta(t^{\prime}-t))\cdot\omega^{-1}$
$\displaystyle\eta_{\mathrm{gh}}(t,t^{\prime})=\frac{i}{\hbar}\langle\psi_{\mathrm{fl}}^{[>0]}(t)\psi_{\mathrm{fl}}^{[<0]}(t^{\prime})\rangle$
$\displaystyle=(\theta(t-t^{\prime})-t)\cdot\omega^{-1}$

with $\theta(t)$ the Heaviside function and $\omega^{-1}$ the inverse of the
pairing on $\mathfrak{g}$. There are three types of connected Feynman diagrams
contributing to $S_{\mathrm{eff}}$ (see Figure 3):

  1. (1)

A single edge connecting the two boundary vertices (Figure 3(a)). That diagram
evaluates to $(\psi_{\mathrm{in}}^{-},\psi_{\mathrm{out}}^{+})$.

  2. (2)

A single edge connecting a boundary vertex to a bulk vertex (Figure 3(b)) .
Those diagrams yield
$(\psi^{-}_{\mathrm{in}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})+(\psi^{+}_{\mathrm{out}},d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}).$

  3. (3)

A single edge connecting the two bulk vertices (Figure 3(c)) . This diagram
gives, using $\int_{I\times I}\eta_{\mathrm{ph}}(t,t^{\prime})=\frac{1}{2}$,
the contribution
$\frac{1}{2}(d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})$.

$\psi^{-}_{\mathrm{in}}$$\psi^{+}_{\mathrm{out}}$ (a) Single edge connecting
two boundaries

$\psi^{-}_{\mathrm{in}}$$\psi^{+}_{\mathrm{out}}$$d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}$$d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}$
(b) Single edge connecting bulk and boundary

$d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}$$d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}$
(c) Single edge connecting two bulk vertices

Figure 3. Connected Feynman diagrams in effective action

In total we obtain the effective action

(46)
$S_{\mathrm{eff}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},A_{\mathrm{res}}]=(\psi^{[>0]}_{\mathrm{out}}-\psi^{[>0]}_{\mathrm{in}},\psi_{\mathrm{res}})-(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})\\\
+(\psi^{-}_{\mathrm{in}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})+(\psi_{\mathrm{out}}^{+},d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})+\frac{1}{2}(d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})+(A^{[>1]}_{\mathrm{res}},d_{\mathfrak{g}}\psi_{\mathrm{res}}).$

Separating the term depending only on ghost number 0 fields from the rest, we
obtain the proof. ∎

######  Proposition 5.3.

The lagrangian generated by the $\mathrm{gh}=0$ part of the action is the
evolution relation in $X^{[0]}_{\mathrm{in}}\times X^{[0]}_{\mathrm{out}}$.

###### Proof.

The Euler-Lagrange equations of the theory in ghost number 0 are

$\displaystyle d_{I}\psi^{[0]}$ $\displaystyle=-d_{\mathfrak{g}}A^{[1]},$
$\displaystyle d_{\mathfrak{g}}\psi^{[0]}$ $\displaystyle=0.$

Projecting to boundary values $(\psi_{\mathrm{in}},\psi_{\mathrm{out}})$ we
obtain the equations

(47) $\displaystyle
d^{-}_{\mathfrak{g}}\psi^{+}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}\psi^{-}_{\mathrm{in}}$
$\displaystyle=d^{+}_{\mathfrak{g}}\psi^{-}_{\mathrm{out}}+d_{\mathfrak{g}}^{-}\psi^{+}_{\mathrm{out}}=0,$
(48) $\displaystyle\psi^{+}_{\mathrm{out}}-\psi^{+}_{\mathrm{in}}$
$\displaystyle=d^{+}_{\mathfrak{g}}a,$ (49)
$\displaystyle\psi^{-}_{\mathrm{out}}-\psi^{-}_{\mathrm{in}}$
$\displaystyle=d^{-}_{\mathfrak{g}}b,$

for some $a,b\in X^{[1]}$, and the first equation forces $a=b$ (up to a
$d_{\mathfrak{g}}^{+}$ and $d_{\mathfrak{g}}^{-}$-closed term). On the other
hand, the lagrangian generated by $S_{\mathrm{ph}}$ is given by

$\displaystyle\psi^{+}_{\mathrm{in}}$ $\displaystyle=-\frac{\partial
S_{\mathrm{ph}}}{\partial\psi^{-}_{\mathrm{in}}}=\psi^{+}_{\mathrm{out}}-d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},$
$\displaystyle\psi^{-}_{\mathrm{out}}$ $\displaystyle=\frac{\partial
S_{\mathrm{ph}}}{\partial\psi^{+}_{\mathrm{out}}}=\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},$
$\displaystyle 0$ $\displaystyle=\frac{\partial S_{\mathrm{ph}}}{\partial
A^{[1]}_{\mathrm{res}}}=d_{\mathfrak{g}}^{+}\psi_{\mathrm{in}}^{-}+d_{\mathfrak{g}}^{-}\psi^{+}_{\mathrm{out}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}.$

The first two equations are equivalent to equations (48),(49), while the last
equation enforces the constraint (47). ∎

####  5.1.5. Quantum master equation

The modified quantum master equation

$(-\hbar^{2}\Delta_{\mathrm{res}}+\Omega)e^{\frac{i}{\hbar}S_{\mathrm{eff}}}$

is equivalent to

(50)
$\left(\frac{1}{2}\left\\{S_{\mathrm{eff}},S_{\mathrm{eff}}\right\\}_{\mathrm{res}}-i\hbar\Delta_{\mathrm{res}}S_{\mathrm{eff}}+\Omega\right)Z=0.$

Here we denote by $\\{\cdot,\cdot\\}_{\mathrm{res}}$ the BV ($+1$-shifted
Poisson) bracket on $\mathcal{V}$.

######  Proposition 5.4.

The effective action $S_{\mathrm{eff}}$ given by (46) satisfies the mQME (50)
with boundary BFV operator $\Omega$ given by the standard quantization of

$\Theta(\psi)=\frac{1}{2}(\psi,d_{\mathfrak{g}}\psi)=(\psi^{+},d_{\mathfrak{g}}^{-}\psi^{[1]})+(\psi^{-},d_{\mathfrak{g}}^{+}\psi^{[1]})+(\psi^{[<0]},d_{\mathfrak{g}}\psi^{[>1]}).$

###### Proof.

Expanding degree-wise as a differential operator, we obtain
$\Omega=\Omega^{(1)}+\Omega^{(0)}=\Omega^{(1)}_{\mathrm{out}}+\Omega^{(0)}_{\mathrm{out}}+\Omega^{(1)}_{\mathrm{in}}+\Omega^{(0)}_{\mathrm{in}}$
with

$\displaystyle\Omega^{(0)}_{\mathrm{out}}$
$\displaystyle=\left(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{out}},\psi^{+}_{\mathrm{out}}\right),$
$\displaystyle\Omega^{(0)}_{\mathrm{in}}$
$\displaystyle=-\left(d_{\mathfrak{g}}^{+}\psi^{[1]}_{\mathrm{in}},\psi^{-}_{\mathrm{in}}\right),$
$\displaystyle\frac{i}{\hbar}\Omega^{(1)}_{\mathrm{out}}$
$\displaystyle=\left(d_{\mathfrak{g}}^{+}\psi^{[1]}_{\mathrm{out}},\frac{\delta}{\delta\psi_{\mathrm{out}}^{+}}\right)+\left(d_{\mathfrak{g}}\psi_{\mathrm{out}}^{[>1]},\frac{\delta}{\delta\psi_{\mathrm{out}}^{[>0]}}\right),$
$\displaystyle\frac{i}{\hbar}\Omega^{(1)}_{\mathrm{in}}$
$\displaystyle=\left(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{in}},\frac{\delta}{\delta\psi_{\mathrm{in}}^{-}}\right)+\left(d_{\mathfrak{g}}\psi^{[>1]}_{\mathrm{in}},\frac{\delta}{\delta\psi^{[>0]}_{\mathrm{in}}}\right).$

First of all, notice that $\Delta_{\mathcal{V}}S_{\mathrm{eff}}=0$ since in
the only possibly nonvanishing term
$(A^{[1]}_{\mathrm{res}},d_{\mathfrak{g}}\psi_{\mathrm{res}})$ fields are not
paired with their antifields because of the degree shift by the differential.
Computing the BV bracket we obtain

(51) $\frac{1}{2}\\{S_{\mathrm{eff}},S_{\mathrm{eff}}\\}_{\mathrm{res}}=\\\
=-(\psi^{[1]}_{\mathrm{out}}-\psi^{[1]}_{\mathrm{in}},d_{\mathfrak{g}}^{-}\psi^{+}_{\mathrm{out}}+d_{\mathfrak{g}}^{+}\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})-(\psi^{[>1]}_{\mathrm{out}}-\psi^{[>1]}_{\mathrm{in}},d_{\mathfrak{g}}\psi_{\mathrm{res}})$

(only terms of opposite ghost number survive in the pairing). On the other
hand, since $\Omega^{(0)}$ is a multiplication operator and $\Omega^{(1)}$
contains only derivatives of first order, we have $\Omega
Z=\Omega^{(0)}Z+\frac{i}{\hbar}\Omega^{(1)}(S_{\mathrm{eff}})Z$ and

$\displaystyle\frac{i}{\hbar}\Omega^{(1)}_{\mathrm{out}}S_{\mathrm{eff}}$
$\displaystyle=(\psi^{-}_{\mathrm{in}},d_{\mathfrak{g}}^{+}\psi^{[1]}_{\mathrm{out}})+(d_{\mathfrak{g}}^{+}\psi^{[1]}_{\mathrm{out}},d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})+(d_{\mathfrak{g}}\psi^{[>1]}_{\mathrm{out}},\psi_{\mathrm{res}}),$
$\displaystyle\frac{i}{\hbar}\Omega^{(1)}_{\mathrm{in}}S_{\mathrm{eff}}$
$\displaystyle=-(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{in}},\psi^{+}_{\mathrm{out}})+(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{in}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})-(d_{\mathfrak{g}}\psi^{[>1]}_{\mathrm{in}},\psi_{\mathrm{res}}).$

A straightforward computation shows that
$\Omega^{(0)}+\frac{i}{\hbar}\Omega^{(1)}S_{\mathrm{eff}}$ coincides with
(51), thus completing the proof. ∎

###  5.2. General polarizations

Next, we will consider the case where $X^{[0]}_{\mathbb{C}}$ is equipped with
another polarization $\mathcal{P}_{\mathrm{nl}}$ which is not necessarily
linear (see [16, Section 12] for the corresponding toy model). Let the base
$\mathcal{B}^{\mathcal{P}_{\mathrm{nl}}}$ be locally parametrized by a
coordinate $\psi^{Q}$, and the fibers by a coordinate $\psi^{P}$. Suppose
$G(\psi^{-},\psi^{Q})$ is a generating function of the canonical
transformation363636I.e., the function $G(\psi^{-},\psi^{Q})$ satisfying
$\delta G=\psi^{+}\delta\psi^{-}-\psi^{P}\delta\psi^{Q}$.
$(\psi^{-},\psi^{+})\to(\psi^{Q},\psi^{P})$. Then we have that
$\psi^{+}=F(\psi^{-},\psi^{Q})=\frac{\partial G}{\partial\psi^{-}}$. We assume
that $G$ is analytic in $\psi^{-}$ in a neighborhood $U$ of
$\\{0\\}\times\mathcal{B}^{\mathcal{P}_{\mathrm{nl}}}$.

We now consider again the 1D AKSZ theory on the interval, where we choose the
polarizations at the two endpoints to be parallel in the ghost sector. In the
physical sector we choose the $\psi^{-}$-representation on the in-boundary and
the $\psi^{Q}$-representation on the out-boundary:

$\displaystyle\mathcal{P}$
$\displaystyle=\mathcal{P}_{\mathrm{in}}\times\mathcal{P}_{\mathrm{out}}\quad\mbox{with}$
$\displaystyle\mathcal{P}_{\mathrm{in}}$
$\displaystyle=\left\\{\frac{\delta}{\delta\psi^{[<0]}_{\mathrm{in}}},\frac{\delta}{\delta\psi_{\mathrm{in}}^{+}}\right\\},$
$\displaystyle\mathcal{P}_{\mathrm{out}}$
$\displaystyle=\left\\{\frac{\delta}{\delta\psi^{[<0]}_{\mathrm{out}}},\frac{\delta}{\delta\psi_{\mathrm{out}}^{P}}\right\\}.$

The base is

$\mathcal{B}=\mathcal{B}_{\mathrm{in}}\times\mathcal{B}_{\mathrm{out}},\;\;\mathcal{B}_{\mathrm{in}}=X^{[>0]}_{\mathbb{C}}\oplus
X^{-},\;\;\mathcal{B}_{\mathrm{out}}=X^{[>0]}_{\mathbb{C}}\times\mathcal{B}^{\mathcal{P}_{\mathrm{nl}}}.$

The polarized 1-form is $\alpha^{f}_{\partial
I}=\alpha^{\partial,\mathcal{P}_{\mathrm{out}}}+\alpha^{\partial,\mathcal{P}_{\mathrm{in}}}$
with

$\displaystyle\alpha^{\partial,\mathcal{P}_{\mathrm{in}}}$
$\displaystyle=-(\psi^{[<0]}_{\mathrm{in}},\delta\psi^{[>0]}_{\mathrm{in}})-(\psi_{\mathrm{in}}^{+},\delta\psi_{\mathrm{in}}^{-})=\alpha^{\partial}_{\mathrm{in}}-\delta
f_{\mathrm{in}},$ $\displaystyle\alpha^{\partial,\mathcal{P}_{\mathrm{out}}}$
$\displaystyle=(\psi^{[<0]}_{\mathrm{out}},\delta\psi^{[>0]}_{\mathrm{out}})+(\psi_{\mathrm{out}}^{P},\delta\psi_{\mathrm{out}}^{Q})=\alpha^{\partial}_{\mathrm{out}}+\delta
f_{\mathrm{out}},$

where

$\displaystyle f_{\mathrm{in}}$
$\displaystyle=\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{in}})-\frac{1}{2}(\psi^{+}_{\mathrm{in}},\psi^{-}_{\mathrm{in}}),$
$\displaystyle f_{\mathrm{out}}$
$\displaystyle=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{out}})-\frac{1}{2}(\psi_{\mathrm{out}}^{+},\psi_{\mathrm{out}}^{-})-G(\psi^{-}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}}).$

Thus, $\alpha^{f}_{\partial I}=\alpha_{\partial I}+\delta f$ with

$f(\psi_{\mathrm{out}},\psi_{\mathrm{in}})=f_{\mathrm{out}}(\psi_{\mathrm{out}})-f_{\mathrm{in}}(\psi_{\mathrm{in}})\\\
=\frac{1}{2}(\psi^{[>0]}_{\mathrm{out}},\psi^{[<0]}_{\mathrm{out}})-\frac{1}{2}(\psi_{\mathrm{out}}^{+},\psi_{\mathrm{out}}^{-})-G(\psi^{-}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}})-\frac{1}{2}(\psi^{[>0]}_{\mathrm{in}},\psi^{[<0]}_{\mathrm{in}})+\frac{1}{2}(\psi_{\mathrm{in}}^{+},\psi_{\mathrm{in}}^{-}).$

####  5.2.1. Splitting the fields

The goal is to find again a symplectomorphism
$\Phi\colon\mathcal{B}\times\mathcal{V}\times\mathcal{Y}^{\prime}\to\mathcal{F}$.373737Strictly
speaking, the range of $\Phi$ is not $\mathcal{F}$, but rather a certain
regularization of $\mathcal{F}$ more suitable for quantization. See the
discussion in [16, Section 9.2.3]. Here the trick is that we keep the space of
fluctuations $\mathcal{Y}$ as above in Equation (43). In ghost number zero,
the map $\Phi$ is defined as follows. For boundary values
$\psi^{-}_{\mathrm{in}}\in\mathcal{B}_{\mathrm{in}}^{[0]}$ and
$\psi^{Q}_{\mathrm{out}}\in\mathcal{B}_{\mathrm{out}}^{[0]}$ and fluctuations
$\psi^{-}_{\mathrm{fl}},\psi^{+}_{\mathrm{fl}}\in\mathcal{Y}$ (recall that
$\psi^{-}_{\mathrm{fl}}(0)=\psi^{+}_{\mathrm{fl}}(1)=0$), we let

$\psi^{-}(t)=\begin{cases}\psi^{-}_{\mathrm{fl}}(t)&\mathrm{for}\;\;t>0,\\\
\psi^{-}_{\mathrm{in}}&\mathrm{for}\;\;t=0,\end{cases}$

and

$\psi^{+}(t)=\begin{cases}\psi^{+}_{\mathrm{fl}}(t)&\mathrm{for}\;\;t<1,\\\
F(\psi^{-}_{\mathrm{fl}}(1),\psi^{Q}_{\mathrm{out}})&\mathrm{for}\;\;t=1.\end{cases}$

The map $\Phi$ is given by

(52)
$\Phi(\psi^{[>0]}_{\mathrm{in}},\psi^{-}_{\mathrm{in}},\psi^{[>0]}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}},\psi_{\mathrm{res}},\psi_{\mathrm{fl}},A)=\psi^{-}(\psi^{-}_{\mathrm{in}},\psi^{-}_{\mathrm{fl}})+\psi^{+}(\psi^{+}_{\mathrm{fl}},\psi^{-}_{\mathrm{fl}},\psi^{Q}_{\mathrm{out}})\\\
+{\psi_{\mathrm{res}}}+\widetilde{\psi^{[>0]}_{\mathrm{in}}}+\widetilde{\psi^{[>0]}_{\mathrm{out}}}+\psi_{\mathrm{fl}}^{[\neq
0]}+A.$

In nonzero ghost number, this coincides with the splitting considered in the
previous section. In what follows, we will discuss only the physical sector,
i.e., the part in ghost number 0. The analysis in the ghost sector proceeds
exactly as in Section 5.1 and results in the ghost effective action
$S_{\mathrm{gh}}$ described in Proposition 5.2.

####  5.2.2. Effective action

Again, we can use the gauge-fixing lagrangian
$\mathcal{L}\subset\mathcal{Y}^{\prime}$ given by zero forms. Restricted to
$\mathcal{B}\times\mathcal{V}\times\mathcal{L}$ and fields of ghost number 0,
we have

$S^{f}[\psi^{-}_{\mathrm{in}},\psi^{Q}_{\mathrm{out}},\psi^{-}_{\mathrm{fl}},\psi^{+}_{\mathrm{fl}},dt\cdot
A_{\mathrm{res}}]=-(\psi^{-}_{\mathrm{in}},\psi^{+}_{\mathrm{fl}}(0))-G(\psi^{-}_{\mathrm{fl}}(1),\psi^{Q}_{\mathrm{out}})\\\
+\int_{I}(\psi^{+}_{\mathrm{fl}},d_{I}\psi^{-}_{\mathrm{fl}})-dt(d^{+}_{\mathfrak{g}}A^{[1]}_{\mathrm{res}},\psi^{-}_{\mathrm{fl}})-dt(d^{-}_{\mathfrak{g}}A^{[1]}_{\mathrm{res}},\psi^{+}_{\mathrm{fl}})$

where the computation is very similar to the one in the proof of Lemma 5.1.
The BV-BFV effective action is defined by

$\displaystyle Z$
$\displaystyle=e^{\frac{i}{\hbar}S_{\mathrm{eff}}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},A_{\mathrm{res}}]}=\int\mathcal{D}\psi_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}[\psi_{\mathrm{in}},\psi_{\mathrm{out}},\psi_{\mathrm{res}},\psi_{\mathrm{fl}},A_{\mathrm{res}}]}$
$\displaystyle=e^{\frac{i}{\hbar}S_{\mathrm{back}}}\int\mathcal{D}\psi_{\mathrm{fl}}\;e^{\frac{i}{\hbar}(S_{\mathrm{source}}+S_{0}+S_{\mathrm{int}})}$

where the integral is defined in terms of Feynman diagrams.

######  Proposition 5.5.

The effective action (in ghost number 0) is

(53)
$S_{\mathrm{eff}}^{\mathrm{ph}}[\psi_{\mathrm{in}}^{-},\psi_{\mathrm{out}}^{Q},A^{[1]}_{\mathrm{res}}]=-G(\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q})-(d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{in}}^{-}+\frac{1}{2}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})$

###### Proof.

In terms of Feynman diagrams, the source term creates a vertex of arbitrary
incoming valence on the out-boundary decorated by derivatives of $G$, and a
univalent (outgoing) vertex on the in-boundary decorated by
$\psi^{-}_{\mathrm{in}}$. The interaction term creates univalent in- and
outgoing bulk vertices decorated by
$d_{\mathfrak{g}}^{\pm}A^{[1]}_{\mathrm{res}}$ as in the proof of Proposition
5.2.

$\psi_{\mathrm{out}}^{Q}$$i_{1}$$i_{2}$$i_{k}$$\leadsto\frac{\partial
G(x,y)}{\partial x^{i_{1}}\cdots\partial
x^{i_{k}}}\big{|}_{(x,y)=(0,\psi_{\mathrm{out}}^{Q})}$ Figure 4. Additional
vertex in 1D AKSZ theory with a general polarization on
$X_{\mathrm{out}}^{[0]}$.

The connected Feynman diagrams contributing to the effective action are:

  1. (1)

Diagrams involving the $G$-vertex on the out-boundary. The outgoing half-edges
can connect either to the bulk vertex involving $\psi^{+}_{\mathrm{fl}}$ or
the vertex on the in-boundary (see Figure 5). Summing over all valences, we
obtain the Taylor series in $x$ of $G(x,y)$ in the first argument at
$(0,\psi_{\mathrm{out}}^{Q})$ evaluated on
$\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}$. Hence, by
analyticity of $G$ those vertices sum up to

$-G(\psi_{\mathrm{in}}^{-}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q}).$

  2. (2)

Diagrams involving the univalent incoming bulk vertex. Here the outgoing half-
edges connect to either the vertex on the in-boundary or an outgoing bulk
vertex, giving

$-(d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},\psi^{-}_{\mathrm{in}}+\frac{1}{2}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}).$

Those diagrams are the same as in the linear case (Figure 3).

$\gamma_{\mathrm{in}}$$\gamma_{\mathrm{in}}$$\gamma_{\mathrm{in}}$$\psi_{\mathrm{out}}^{Q}$
Figure 5. Additional Feynman diagrams in 1D AKSZ with general polarization on
$X^{[0]}_{\mathrm{out}}$.

In total, we obtain the effective action (53). ∎

######  Remark 5.6.

In the main case of interest for this paper, the target
$\mathfrak{g}=\Omega^{\bullet}(M)$ is infinite-dimensional and the propagator
contains a delta form as the “inverse” of the pairing
$(\tau_{1},\tau_{2})=\int_{M}\tau_{1}\wedge\tau_{2}$ (cf. Remark 4.1).
However, our computations here are still valid. Indeed, even though Feynman
diagrams contain products of delta functions, since all these diagrams are
actually trees, no problematic terms like $\delta(0)$ arise when computing the
integrals.

######  Proposition 5.7.

The lagrangian generated by (53) is the evolution relation in
$X_{\mathrm{in}}^{[0]}\times X_{\mathrm{out}}^{[0]}$.

###### Proof.

We know that in the $\psi^{\pm}$ variables, the evolution relation is given by
$\psi^{-}_{\mathrm{out}}=\psi^{-}_{\mathrm{in}}+d^{-}_{\mathfrak{g}}A^{[1]}_{\mathrm{res}}$,
$\psi^{+}_{\mathrm{out}}=\psi^{+}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}$.
The lagrangian generated by
$S_{\mathrm{eff}}^{\mathrm{ph}}[\psi_{\mathrm{in}}^{-},\psi_{\mathrm{out}}^{Q},A^{[1]}_{\mathrm{res}}]$
is

$\displaystyle-\psi^{+}_{\mathrm{in}}=\frac{\partial
S_{\mathrm{eff}}^{\mathrm{ph}}}{\partial\psi^{-}_{\mathrm{in}}}$
$\displaystyle=-F(\psi_{\mathrm{in}}^{-}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi^{Q}_{\mathrm{out}})+d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}$
$\displaystyle=-F(\psi^{-}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}})+d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}}=-\psi^{+}_{\mathrm{out}}+d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}},$
$\displaystyle\psi^{P}_{\mathrm{out}}=\frac{\partial
S_{\mathrm{eff}}^{\mathrm{ph}}}{\partial\psi^{Q}_{\mathrm{out}}}$
$\displaystyle=-\frac{\partial
G(\psi^{-}_{\mathrm{out}},\psi_{\mathrm{out}}^{Q})}{\partial\psi^{Q}_{\mathrm{out}}},$
$\displaystyle 0=\frac{\partial S_{\mathrm{ph}}}{\partial
A^{[1]}_{\mathrm{res}}}$
$\displaystyle=d_{\mathfrak{g}}^{+}\psi_{\mathrm{in}}^{-}+d_{\mathfrak{g}}^{-}\psi^{+}_{\mathrm{out}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}.$

This lagrangian coincides with the evolution relation. ∎

####  5.2.3. Modified quantum master equation

Let us also comment on the mQME. Again, we can compute the BV bracket (we
ignore higher ghosts for simplicity)

$\frac{1}{2}\\{S_{\mathrm{eff}},S_{\mathrm{eff}}\\}_{\mathrm{res}}=\\{S_{\mathrm{eff}}^{\mathrm{gh}},S_{\mathrm{eff}}^{\mathrm{ph}}\\}=-(\psi^{[1]}_{\mathrm{out}}-\psi^{[1]}_{\mathrm{in}},d_{\mathfrak{g}}^{-}F(\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q})+d_{\mathfrak{g}}^{+}\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}).$

As before, we have
$\Omega_{\mathrm{in}}=-(d_{\mathfrak{g}}^{+}\psi_{\mathrm{in}}^{[1]},\psi^{-}_{\mathrm{in}})-i\hbar(d_{\mathfrak{g}}^{-}\psi_{\mathrm{in}}^{[1]},\frac{\delta}{\delta\psi^{-}_{\mathrm{in}}})$
and

$\displaystyle Z^{-1}\Omega_{\mathrm{in}}Z$
$\displaystyle=-(d_{\mathfrak{g}}^{+}\psi_{\mathrm{in}}^{[1]},\psi^{-}_{\mathrm{in}})Z-(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{in}},F(\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q}))+(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{in}},d_{\mathfrak{g}}^{+}A^{[1]}_{\mathrm{res}})$
$\displaystyle=-(\psi^{[1]}_{\mathrm{in}},d_{\mathfrak{g}}^{-}F(\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q})+d_{\mathfrak{g}}^{+}\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}}).$

Thus, the mQME is equivalent to

(54)
$Z^{-1}\Omega_{\mathrm{out}}Z=(\psi^{[1]}_{\mathrm{out}},d_{\mathfrak{g}}^{-}F(\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}},\psi_{\mathrm{out}}^{Q})+d_{\mathfrak{g}}^{+}\psi^{-}_{\mathrm{in}}+d_{\mathfrak{g}}^{+}d_{\mathfrak{g}}^{-}A^{[1]}_{\mathrm{res}})=(\psi^{[1]}_{\mathrm{out}},d_{\mathfrak{g}}\psi^{[0]}_{\mathrm{out}}).$

The operator $\Omega_{\mathrm{out}}$ acting on ghost number 0 fields should be
obtained as a quantization of $\Theta(\psi)$ in the
$\psi^{P}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}}$ variables,

(55)
$\Theta(\psi^{P}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}})=(d_{\mathfrak{g}}^{+}\psi^{[1]}_{\mathrm{out}},\psi^{-}(\psi^{P}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}}))+(d_{\mathfrak{g}}^{-}\psi^{[1]}_{\mathrm{out}},\psi^{+}(\psi^{P}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}})).$

The standard quantization $\Omega_{\mathrm{out}}^{\mathrm{std}}$ of (55) —
i.e., replacing all $\psi^{P}_{\mathrm{out}}$ variables with
$-i\hbar\delta/\delta\psi^{Q}_{\mathrm{out}}$ and moving all derivatives to
the right - satisfies (54) to $0$-th order in $\hbar$, but there are terms of
higher order in $\hbar$ corresponding to higher derivatives in
$\psi^{Q}_{\mathrm{out}}$ acting on $G$. To prove the mQME to all orders, one
would have to find quantum corrections to
$\Omega^{\mathrm{std}}_{\mathrm{out}}$ such that these terms are cancelled and
the deformed operator still squares to 0.

######  Remark 5.8.

A particularly simple case occurs when
$\psi(\psi^{P},\psi^{Q})=\psi^{P}+\psi^{Q}$. A rather trivial example of this
case is $\psi^{Q}=\psi^{+},\psi^{P}=\psi^{-}$. A nontrivial example will be
considered in Section 6.3. In this case, we may define

$\Omega_{\mathrm{out}}=\left(d_{\mathfrak{g}}\psi^{[1]}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}}-i\hbar\frac{\delta}{\delta\psi^{Q}_{\mathrm{out}}}\right).$

Then, from $\partial G/\partial\psi^{Q}=\psi^{P}$ we immediately get
$Z^{-1}\Omega_{\mathrm{out}}Z=(d_{\mathfrak{g}}\psi^{[1]}_{\mathrm{out}},\psi^{Q}_{\mathrm{out}}+\psi^{P}_{\mathrm{out}})=(\psi^{[1]}_{\mathrm{out}},d_{\mathfrak{g}}\psi^{[0]}_{\mathrm{out}})$,
i.e., Equation (54), and hence the mQME, are satisfied. In general, we have
the mQME whenever the constraints are linear both in the original and the new
momenta, see [16, Section 12].

######  Remark 5.9.

Let $Z_{\mathrm{nl},\perp}$ be the partition function with transversal ghost
polarization and a general polarization in ghost number 0, to be precise, we
are choosing the $(\psi^{-},\psi^{[<0]})$ on the $\mathrm{in}$-boundary and
the $(\psi^{Q},\psi^{[>0]})$ on the $\mathrm{out}$-boundary. In this case,
there are no residual fields, and following a computation similar to the
above, one finds383838There are no bulk vertices in this polarization. The two
contributing terms come from multivalent $\mathrm{out}$-boundary vertices in
ghost number 0 and univalent boundary vertices in the ghost sector.

$Z_{\mathrm{nl},\perp}=\exp\left(-\frac{i}{\hbar}G(\psi^{-}_{\mathrm{in}},\psi^{Q}_{\mathrm{out}})+\frac{i}{\hbar}\psi^{[<0]}_{\mathrm{in}}\psi^{[>0]}_{\mathrm{out}}\right).$

The mQME for this partition function is just
$(\Omega_{\mathrm{in}}+\Omega_{\mathrm{out}})Z=0$, since there are no residual
fields. We can observe that the only obstruction for the mQME to hold is the
existence of a suitable $\Omega_{\mathrm{out}}$ in a general polarization.
Then, one can obtain the partition function $Z_{\mathrm{nl},\parallel}$ -with
parallel ghost polarization and $(\psi^{-},\psi^{Q})$-representation in ghost
number 0, as given by (53) by composition of the partition function
$Z_{\mathrm{l},\parallel}$ with parallel ghost polarization and linear
polarization in ghost number 0, given by (46) with the partition function
$Z_{\mathrm{nl},\perp}$: $Z_{\mathrm{nl},\parallel}=Z_{\mathrm{nl},\perp}\circ
Z_{\mathrm{l},\parallel}$. Since we know that $Z_{\mathrm{l},\parallel}$
satisfies the mQME, $Z_{\mathrm{nl},\parallel}$ will satisfy it if
$Z_{\mathrm{nl},\perp}$ does.

###  5.3. 3D nonabelian Chern–Simons with parallel ghost polarization and
antiholomorphic-to-holomorphic polarization in ghost degree zero

Next, we return to the example of 3D Chern–Simons with parallel ghost
polarization. In this context, it is convenient to use the traditional
notation for the components of the superfield $\mathcal{A}$:

$\mathcal{A}=c+A+A^{*}+c^{*},$

where $\phi^{*}$ denotes the BV antifield of the field $\phi$.

In this section we will use some special notations for field components (as
compared to Section 4): ${a^{1,0}=\mathsf{A}_{\mathrm{fl}}^{1,0}}$,
$a^{0,1}=\mathsf{A}_{\mathrm{fl}}^{0,1}$, $c=\mathsf{A}^{0}$,
$\mathsf{A}^{*}=\mathsf{A}^{2}$, $\sigma=\mathsf{A}^{0}_{I\,\mathrm{res}}$.

####  5.3.1. Abelian case

The action with polarization terms is:

$S^{f}=\int_{I\times\Sigma}\frac{1}{2}\mathcal{A}d\mathcal{A}+\int_{\\{1\\}\times\Sigma}\frac{1}{2}\left(\mathsf{A}^{1,0}\mathsf{A}^{0,1}+c\mathsf{A}^{*}\right)-\int_{\\{0\\}\times\Sigma}\frac{1}{2}\left(\mathsf{A}^{0,1}\mathsf{A}^{1,0}+c\mathsf{A}^{*}\right).$

The space of fields is:

$\mathcal{F}=\Omega^{\bullet}(I,\Omega^{1,0}\oplus\Omega^{0,1}\oplus\Omega^{0}[1]\oplus\Omega^{2}[-1])$

— here $\Omega^{p}$ in the coefficients stands for $\Omega^{p}(\Sigma)$. It is
fibered over

$\mathcal{B}=(\Omega^{0,1}\oplus\Omega^{0}[1])\bigoplus(\Omega^{1,0}\oplus\Omega^{0}[-1])\;\;\ni((\mathsf{A}^{0,1}_{\mathrm{in}},c_{\mathrm{in}}),\;(\mathsf{A}^{1,0}_{\mathrm{out}},c_{\mathrm{out}}))$

with fiber

$\mathcal{Y}=\Omega^{\bullet}(I,\\{0\\};\Omega^{0,1})\oplus\Omega^{\bullet}(I,\\{1\\};\Omega^{1,0})\oplus\Omega^{\bullet}(I,\\{0,1\\};\Omega^{0}[1])\oplus\Omega^{\bullet}(I;\Omega^{2}[-1]).$

The space of residual fields is given by the (relative) cohomology in
$I$-direction:

$\mathcal{V}=H^{\bullet}(I,\\{0,1\\};\Omega^{0}[1])\oplus
H^{\bullet}(I;\Omega^{2}[-1])\quad\ni(dt\cdot\sigma,\mathsf{A}^{*}_{\mathrm{res}}).$

The gauge-fixing lagrangian $\mathcal{L}$ in the fiber of
$\mathcal{Y}\rightarrow\mathcal{V}$ is given by setting to zero the
(relatively) exact 1-form components of fields along $I$.

Thus, on $\mathcal{L}$ we have

$\displaystyle\mathrm{gh}=0\mathrel{\mathop{\mathchar 58\relax}}\quad$
$\displaystyle
A^{(1)}=\widetilde{\mathsf{A}}^{1,0}_{\mathrm{out}}+\widetilde{\mathsf{A}}^{0,1}_{\mathrm{in}}+a^{1,0}+a^{0,1}+dt\cdot\sigma,$
$\displaystyle\mathrm{gh}=1\mathrel{\mathop{\mathchar 58\relax}}\quad$
$\displaystyle
A^{(0)}=\widetilde{c}_{\mathrm{out}}+\widetilde{c}_{\mathrm{in}}+c_{\mathrm{fl}},$
$\displaystyle\mathrm{gh}=-1\mathrel{\mathop{\mathchar 58\relax}}\quad$
$\displaystyle
A^{(2)}=\mathsf{A}^{*}_{\mathrm{res}}+\mathsf{A}^{*}_{\mathrm{fl}},$
$\displaystyle\mathrm{gh}=-2\mathrel{\mathop{\mathchar 58\relax}}\quad$
$\displaystyle A^{(3)}=0,$

with tilde denoting the discontinuous extension by zero from $t=1$ or $t=0$,
respectively. Fluctuations are understood to satisfy

$a^{1,0}|_{t=1}=0,\quad a^{0,1}|_{t=0}=0,\quad
c_{\mathrm{fl}}|_{t=0}=c_{\mathrm{fl}}|_{t=1}=0,\quad\int_{0}^{1}dt\;\mathsf{A}^{*}_{\mathrm{fl}}=0.$

The gauge-fixed polarized action is:

$S^{f}|_{\mathcal{L}}=\\\
\int_{I\times\Sigma}a^{1,0}d_{I}a^{0,1}+\int_{I\times\Sigma}dt\,(a^{1,0}+a^{0,1})d_{\Sigma}\sigma+\int_{\Sigma}\mathsf{A}^{1,0}_{\mathrm{out}}a^{0,1}\big{|}_{t=1}-\int_{\Sigma}\mathsf{A}^{0,1}_{\mathrm{in}}a^{1,0}\big{|}_{t=0}\\\
+\int_{I\times\Sigma}\mathsf{A}^{*}_{\mathrm{fl}}d_{I}c_{\mathrm{fl}}-\int_{\Sigma}(\mathsf{A}^{*}_{\mathrm{res}}+\mathsf{A}^{*}_{\mathrm{fl}}\big{|}_{t=1})c_{\mathrm{out}}+\int_{\Sigma}(\mathsf{A}^{*}_{\mathrm{res}}+\mathsf{A}^{*}_{\mathrm{fl}}\big{|}_{t=0})c_{\mathrm{in}}.$

The propagators are given by:

(56) $\displaystyle\langle a^{0,1}(t,z)a^{1,0}(t^{\prime},z^{\prime})\rangle$
$\displaystyle=-i\hbar\,\theta(t-t^{\prime})\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}d\bar{z}\,dz^{\prime},$
(57) $\displaystyle\langle
c_{\mathrm{fl}}(t,z)\mathsf{A}^{*}_{\mathrm{fl}}(t^{\prime},z^{\prime})\rangle$
$\displaystyle=-i\hbar\,(\theta(t-t^{\prime})-t)\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}dz^{\prime}\,d\bar{z}^{\prime}.$

The corresponding effective action is:

(58)
$S^{\mathrm{eff}}=\int_{\Sigma}\mathsf{A}^{1,0}_{\mathrm{out}}\mathsf{A}^{0,1}_{\mathrm{in}}+\mathsf{A}^{1,0}_{\mathrm{out}}\bar{\partial}\sigma+\mathsf{A}^{0,1}_{\mathrm{in}}\partial\sigma-\frac{1}{2}\partial\sigma\bar{\partial}\sigma-\mathsf{A}^{*}_{\mathrm{res}}(c_{\mathrm{out}}-c_{\mathrm{in}}).$

######  Remark 5.10 (Hamilton–Jacobi property, mQME).

Notice that (58) coincides with (46) above upon specializing
$\mathfrak{g}=\Omega^{\bullet}(\Sigma)$, $X^{+}=\Omega^{1,0}(\Sigma)$,
$X^{-}=\Omega^{0,1}(\Sigma)$. Thus (58) satisfies the modified quantum master
equation, and the $\mathrm{gh}=0$ part of (58) generates the evolution
relation of abelian Chern–Simons theory.

######  Remark 5.11 (Integrating out residual fields).

As in Section 4.2.2, we can integrate out the residual fields
$\sigma,\mathsf{A}_{\mathrm{res}}^{*}$ by choosing a Riemannian metric
compatible with the complex structure and decomposing fields as
$\sigma=\sigma_{c}+\underline{\sigma},\mathsf{A}^{*}_{\mathrm{res}}=\mu\cdot\mathsf{A}^{*}_{\mathrm{res},c}+\underline{\mathsf{A}^{*}_{\mathrm{res}}}$.
As expected, the result differs from (34) only in the ghost sector:

$Z_{*}[c_{\mathrm{out}},c_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}}]=\delta(c_{\mathrm{out},c}-c_{\mathrm{in},c})\big{(}{\det}^{\prime}_{\Omega^{0}(\Sigma)}\Delta_{g}\big{)}^{-\frac{1}{2}}\cdot
e^{\frac{i}{\hbar}\mathbb{I}(\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}})}.$

Here $\mathbb{I}$ is given by (35).

####  5.3.2. Nonabelian case

In the nonabelian Chern–Simons theory with coefficients in a semisimple Lie
algebra $\mathcal{G}$ (corresponding to a compact group393939To simplify the
notations, and to be able to write expressions like $g^{-1}\partial g$ below,
we will assume that $G$ is a matrix group. Otherwise, we should use left/right
translations in $G$. $G$), the superfield is
$\mathcal{A}\in\Omega^{\bullet}(I\times\Sigma,\mathcal{G}[1])$ and all the
splittings are as before, just with components understood as
$\mathcal{G}$-valued forms, paired in the quadratic part of the action via the
Killing form $\langle,\rangle$ on $\mathcal{G}$. The interaction term of the
nonabelian theory, when restricted to the gauge-fixing lagrangian, yields

$S_{\mathrm{int}}=\frac{1}{6}\int\langle\mathcal{A},[\mathcal{A},\mathcal{A}]\rangle=-\int_{\Sigma}\int_{I}dt\langle
a^{1,0},\mathrm{ad}_{\sigma}a^{0,1}\rangle-\int_{\Sigma}\int_{I}dt\langle
c_{\mathrm{fl}},\mathrm{ad}_{\sigma}(\mathsf{A}^{*}_{\mathrm{res}}+\mathsf{A}^{*}_{\mathrm{fl}})\rangle.$

This adds two new bivalent vertices and a univalent vertex to the Feynman
rules.

Let us introduce the following notations:

(59) $\displaystyle F_{+}(x)=\frac{x}{1-e^{-x}}=\sum_{n\geq
0}(-1)^{n}\frac{B_{n}}{n!}x^{n},\quad F_{-}(x)=-\frac{x}{e^{x}-1}=-\sum_{n\geq
0}\frac{B_{n}}{n!}x^{n},$ (60) $\displaystyle
j(\sigma)=\sum_{n=2}^{\infty}\frac{B_{n}}{n\cdot
n!}\mathrm{tr}_{\mathcal{G}}(\mathrm{ad}_{\sigma})^{n}=\mathrm{tr}_{\mathcal{G}}\log\frac{\sinh\frac{\mathrm{ad}_{\sigma}}{2}}{\frac{\mathrm{ad}_{\sigma}}{2}},$

with $B_{n}$ the Bernoulli numbers,
$B_{0}=1,\;B_{1}=-\frac{1}{2},\;B_{2}=\frac{1}{6},\;B_{3}=0,\;B_{4}=-\frac{1}{30},\ldots$

In the following lemma we assume that $\sigma$ is in a sufficiently small
neighborhood of zero in $\mathcal{G}$, see Remark 5.13 below for details.

######  Lemma 5.12.

The partition function of the nonabelian Chern–Simons theory on a cylinder is
$Z=e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$ with the following effective action:

(61) $S^{\mathrm{eff}}=\\\
=\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},e^{-\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{0,1}_{\mathrm{in}}\rangle+\langle\mathsf{A}^{1,0}_{\mathrm{out}},\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}\circ\bar{\partial}\sigma\rangle+\langle\mathsf{A}^{0,1}_{\mathrm{in}},\frac{e^{\mathrm{ad}_{\sigma}}-1}{\mathrm{ad}_{\sigma}}\circ\partial\sigma\rangle\\\
-\langle\partial\sigma,\frac{e^{-\mathrm{ad}_{\sigma}}+\mathrm{ad}_{\sigma}-1}{(\mathrm{ad}_{\sigma})^{2}}\circ\bar{\partial}\sigma\rangle-\langle\mathsf{A}^{*}_{\mathrm{res}},F_{+}(\mathrm{ad}_{\sigma})\circ
c_{\mathrm{out}}+F_{-}(\mathrm{ad}_{\sigma})\circ
c_{\mathrm{in}}\rangle-i\hbar\mathbb{W}(\sigma).$

In (61), the $1$-loop correction $\mathbb{W}$ stands for the contribution of
“ghost wheels” — cycles of $n\geq 1$ ghost-antifield propagators (at the
vertices, they interact with the residual field $\sigma$). These graphs are
ill-defined in the chosen axial gauge; their formal evaluation yields the
expression

(62) $\mathbb{W}(\sigma)=\sum_{n\geq 1}\frac{B_{n}}{n\cdot
n!}\,\mathrm{tr}_{C^{\infty}(\Sigma,\mathcal{G})}(\mathrm{ad}_{\sigma})^{n}=\mathrm{tr}_{C^{\infty}(\Sigma)}j(\sigma)\cdot$

This expression heuristically stands for the “sum over points $z$ of $\Sigma$”
of $j(\sigma(z))$.

We refer the reader to [16, Section 11.3] for a one-dimensional toy model of
this statement.

###### Proof.

One has the following classes of Feynman diagrams contributing to the
effective action:

Figure 6. Feynman diagrams in nonabelian theory on a cylinder with “parallel
ghost” polarization.

Here the solid lines represent the “physical propagator” (56) and the dashed
lines represent the “ghost propagator” (57).

These diagrams are calculated easiest by introducing the propagators dressed
with $\sigma$-insertions:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\langle a^{0,1}(t,z)\otimes
a^{1,0}(t^{\prime},z^{\prime})\rangle_{\mathrm{dressed}}$
$\displaystyle=-i\hbar\,\theta(t-t^{\prime})\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}d\bar{z}\,dz^{\prime}\;\sum_{k=0}^{\infty}\int_{t^{\prime}<t_{1}<\cdots<t_{k}<t}dt_{1}\cdots
dt_{k}(-\mathrm{ad}_{\sigma})^{k}$
$\displaystyle=-i\hbar\,\theta(t-t^{\prime})\,e^{-(t-t^{\prime})\mathrm{ad}_{\sigma}}\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}d\bar{z}\,dz^{\prime},$
$\displaystyle\langle c_{\mathrm{fl}}(t,z)$
$\displaystyle\otimes\mathsf{A}^{*}_{\mathrm{fl}}(t^{\prime},z^{\prime})\rangle_{\mathrm{dressed}}=-i\hbar\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}dz^{\prime}\,d\bar{z}^{\prime}\cdot$
$\displaystyle\cdot$
$\displaystyle\sum_{k=0}^{\infty}\underbrace{\int_{t_{1},\ldots,t_{k}\in[0,1]}dt_{1}\cdots
dt_{k}\;(\theta(t-t_{1})-t)\,(\theta(t_{1}-t_{2})-t_{1})\cdots(\theta(t_{k}-t^{\prime})-t_{k})}_{\begin{cases}\frac{B_{k+1}(1-t^{\prime})-B_{k+1}(t-t^{\prime})}{(k+1)!},\;\;t>t^{\prime}\\\
(-1)^{k}\frac{B_{k+1}(t^{\prime}-t)-B_{k+1}(t^{\prime})}{(k+1)!},\;\;t<t^{\prime}\end{cases}}\;(-\mathrm{ad}_{\sigma})^{k}$
$\displaystyle=-i\hbar\,\delta^{(2)}(z-z^{\prime})\frac{i}{2}dz^{\prime}\,d\bar{z}^{\prime}\cdot\frac{e^{(t^{\prime}-t+\theta(t-t^{\prime}))\mathrm{ad}_{\sigma}}-e^{t^{\prime}\mathrm{ad}_{\sigma}}}{e^{\mathrm{ad}_{\sigma}}-1}.$

Here $B_{k}(t)$ are the Bernoulli polynomials.

Computing the tree Feynman diagrams (i)–(vi) in Figure 6, we have the
following.

  1. (i)

$\displaystyle\wick{\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\c{1}{a^{0,1}}\big{|}_{t=1}\rangle\int_{\Sigma}\langle\c{1}{a^{1,0}}\big{|}_{t=0},\mathsf{A}^{0,1}_{\mathrm{in}}\rangle}=\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},e^{-\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{0,1}_{\mathrm{in}}\rangle$.
Here the contraction is the dressed propagator.

  2. (ii)

$\displaystyle\wick{\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\c{1}{a^{0,1}}\big{|}_{t=1}\rangle\int_{I\times\Sigma}dt\,\langle\c{1}{a^{1,0}},\bar{\partial}\sigma\rangle}=\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\int_{0}^{1}dt\,e^{-(1-t)\mathrm{ad}_{\sigma}}\circ\bar{\partial}\sigma\rangle\\\
=\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}\circ\bar{\partial}\sigma\rangle$.

  3. (iii)

Similarly to (ii),
$\displaystyle\wick{-\int_{I\times\Sigma}dt\,\langle\partial\sigma,\c{1}{a^{0,1}}\rangle\int_{\Sigma}\langle\c{1}{a^{1,0}}\big{|}_{t=0},\mathsf{A}^{0,1}_{\mathrm{in}}\rangle}=-\int_{\Sigma}\langle\partial\sigma,\int_{0}^{1}dt\,e^{-t\mathrm{ad}_{\sigma}}\mathsf{A}^{0,1}_{\mathrm{in}}\rangle\\\
=-\langle\partial\sigma,\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{0,1}_{\mathrm{in}}\rangle=\langle\frac{e^{\mathrm{ad}_{\sigma}}-1}{\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{0,1}_{\mathrm{in}},\partial\sigma\rangle$.

  4. (iv)

$\displaystyle\wick{-\int_{I\times\Sigma}dt\,\langle\partial\sigma,\c{1}{a^{0,1}}\rangle\int_{I\times\Sigma}dt^{\prime}\,\langle\c{1}{a^{1,0}},\bar{\partial}\sigma\rangle}=-\int_{\Sigma}\langle\partial\sigma,\int_{0}^{1}dt\int_{0}^{1}dt^{\prime}\,e^{-(t-t^{\prime})\mathrm{ad}_{\sigma}}\circ\bar{\partial}\sigma\rangle\\\
=-\int_{\Sigma}\langle\partial\sigma,\frac{e^{-\mathrm{ad}_{\sigma}}+\mathrm{ad}_{\sigma}-1}{(\mathrm{ad}_{\sigma})^{2}}\circ\bar{\partial}\sigma\rangle$.

  5. (v)

$\displaystyle\wick{-\int_{I\times\Sigma}dt\,\langle\mathrm{ad}_{\sigma}\mathsf{A}^{*}_{\mathrm{res}},\c{1}{c_{\mathrm{fl}}}\rangle\int_{\Sigma}\langle\c{1}{\mathsf{A}_{\mathrm{fl}}^{*}}\big{|}_{t=1},c_{\mathrm{out}}\rangle}-\int_{\Sigma}\langle
A^{*}_{\mathrm{res}},c_{\mathrm{out}}\rangle\\\
=-\int_{\Sigma}\langle\mathrm{ad}_{\sigma}\mathsf{A}^{*}_{\mathrm{res}},\int_{0}^{1}dt\,\frac{e^{(1-t)\mathrm{ad}_{\sigma}}-e^{\mathrm{ad}_{\sigma}}}{e^{\mathrm{ad}_{\sigma}}-1}\circ
c_{\mathrm{out}}\rangle-\int_{\Sigma}\langle
A^{*}_{\mathrm{res}},c_{\mathrm{out}}\rangle\\\
=-\int_{\Sigma}\langle\mathsf{A}_{\mathrm{res}}^{*},\frac{\mathrm{ad}_{\sigma}}{1-e^{-\mathrm{ad}_{\sigma}}}\circ
c_{\mathrm{out}}\rangle$.

  6. (vi)

Similarly to (v),
$\displaystyle\wick{\int_{I\times\Sigma}dt\,\langle\mathrm{ad}_{\sigma}\mathsf{A}^{*}_{\mathrm{res}},\c{1}{c_{\mathrm{fl}}}\rangle\int_{\Sigma}\langle\c{1}{\mathsf{A}_{\mathrm{fl}}^{*}}\big{|}_{t=0},c_{\mathrm{in}}\rangle}+\int_{\Sigma}\langle
A^{*}_{\mathrm{res}},c_{\mathrm{in}}\rangle\\\
=\int_{\Sigma}\langle\mathrm{ad}_{\sigma}\mathsf{A}^{*}_{\mathrm{res}},\int_{0}^{1}dt\,\frac{e^{(1-t)\mathrm{ad}_{\sigma}}-1}{e^{\mathrm{ad}_{\sigma}}-1}\circ
c_{\mathrm{in}}\rangle+\int_{\Sigma}\langle
A^{*}_{\mathrm{res}},c_{\mathrm{in}}\rangle\\\
=\int_{\Sigma}\langle\mathsf{A}^{*}_{\mathrm{res}},\frac{\mathrm{ad}_{\sigma}}{e^{\mathrm{ad}_{\sigma}}-1}\circ
c_{\mathrm{in}}\rangle$.

Thus, the Feynman diagrams (i)-(vi) in Figure 6 yield the $O(\hbar^{0})$ part
of the answer (61).

Next, consider the one-loop graphs in Figure 6. The “physical wheels” —
diagrams (vii) — vanish due to the form of the propagator (56): they are
proportional to

$\int_{t_{1},\ldots,t_{k}\in[0,1]}dt_{1}\cdots
dt_{k}\,\theta(t_{1}-t_{2})\theta(t_{2}-t_{3})\cdots\theta(t_{k-1}-t_{k})\theta(t_{k}-t_{1})=0.$

Finally, consider the “ghost wheels” — diagrams (viii). The propagator (57) is
the integral kernel of an operator $K^{\mathrm{gh}}=\mathrm{id}\otimes
K^{\mathrm{gh}}_{I}$ acting on $C^{\infty}(\Sigma)\otimes\Omega^{\bullet}(I)$
with $K^{\mathrm{gh}}_{I}\mathrel{\mathop{\mathchar
58\relax}}f(t)+dt\,g(t)\mapsto\int_{0}^{1}dt^{\prime}\,(\theta(t-t^{\prime})-t)\,g(t^{\prime})$.
As a regularization, let us replace $C^{\infty}(\Sigma)$ with $C^{\infty}(X)$,
with $X$ a finite set of points — the set of vertices of some triangulation of
the surface $\Sigma$. In particular, $C^{\infty}(X)$ is a finite-dimensional
vector space. Then, the regularized value of the ghost wheel diagram (viii)
with $k$ $\sigma$-insertions is the supertrace:

$-i\hbar\;\mathrm{str}_{C^{\infty}(X)\otimes\Omega^{\bullet}(I,\mathcal{G})}(-K^{\mathrm{gh}}\,dt\,\mathrm{ad}_{\sigma})^{k}=-i\hbar\,\mathrm{tr}_{C^{\infty}(X)}\,\mathrm{str}_{\Omega^{\bullet}(I,\mathcal{G})}(-K^{\mathrm{gh}}_{I}\,dt\,\mathrm{ad}_{\sigma})^{k}.$

For the supertrace over the interval, we have (see, e.g., [35]):

$\mathrm{str}_{\Omega^{\bullet}(I,\mathcal{G})}(-K^{\mathrm{gh}}_{I}\,dt\,\mathrm{ad}_{\sigma})^{k}\\\
=\mathrm{tr}_{\mathcal{G}}\int_{t_{1},\ldots,t_{k}\in[0,1]}(\theta(t_{1}-t_{2})-t_{1})dt_{2}\mathrm{ad}_{\sigma}\cdots(\theta(t_{k-1}-t_{k})-t_{k-1})dt_{k}\mathrm{ad}_{\sigma}(\theta(t_{k}-t_{1})-t_{1})dt_{1}\mathrm{ad}_{\sigma}\\\
=\frac{B_{k}}{k!}\mathrm{tr}_{\mathcal{G}}(\mathrm{ad}_{\sigma})^{k}.$

Summing over the values of $k\geq 1$ and taking into account the symmetric
factor $1/k$ (due to the automorphisms of the wheel graph), we obtain

$\sum_{k\geq
1}\frac{1}{k}\mathrm{str}_{\Omega^{\bullet}(I,\mathcal{G})}(-K^{\mathrm{gh}}_{I}\,dt\,\mathrm{ad}_{\sigma})^{k}=j(\sigma),$

with $j$ as in (60). Thus, finally, the total contribution of graphs (viii) to
the effective action is

$-i\hbar\,\mathbb{W}=-i\hbar\,\mathrm{tr}_{C^{\infty}(X)}j(\sigma)=-i\hbar\sum_{z\in
X}j(\sigma(z)).$

Trying to pass to a limit of dense triangulation $X$ obviously leads to an
ill-defined result here.

Put another way, the regularized computation of a ghost wheel diagram is:

$-i\hbar\wick{\sum_{z_{1}\in
X}\int_{I}dt_{1}\langle\c{4}{\mathsf{A}^{*}_{\mathrm{fl}}},-\mathrm{ad}_{\sigma}\c{1}{c_{\mathrm{fl}}}\rangle\sum_{z_{2}\in
X}\int_{I}dt_{2}\langle\c{1}{\mathsf{A}^{*}_{\mathrm{fl}}},-\mathrm{ad}_{\sigma}\c{2}{c_{\mathrm{fl}}}\rangle\;\;\c{2}{}\cdots\c{3}{}\;\;\sum_{z_{k}\in
X}\int_{I}dt_{k}\langle\c{3}{\mathsf{A}^{*}_{\mathrm{fl}}},-\mathrm{ad}_{\sigma}\c{4}{c_{\mathrm{fl}}}\rangle}\\\
=i\hbar\sum_{z\in
X}\int_{I}dt_{1}\cdots\int_{I}dt_{k}(\theta(t_{1}-t_{2})-t_{1})\cdots(\theta(t_{k-1}-t_{k})-t_{k-1})(\theta(t_{k}-t_{1})-t_{k})\mathrm{tr}_{\mathcal{G}}(-\mathrm{ad}_{\sigma})^{k}\\\
=-i\hbar\,\frac{B_{k}}{k!}\sum_{z\in
X}\mathrm{tr}_{\mathcal{G}}(\mathrm{ad}_{\sigma(z)})^{k}.$

where the contractions are the nondressed propagators (57) with the delta form
in $z$ replaced with Kronecker symbol $\delta_{zz^{\prime}}$. In this
regularized setup we understand the fields
$c_{\mathrm{fl}},\mathsf{A}^{*}_{\mathrm{fl}},\sigma$ as supported at the
vertices of $X$; fields $c_{\mathrm{fl}},\mathsf{A}^{*}_{\mathrm{fl}}$ also
depend on $t\in[0,1]$. ∎

######  Remark 5.13.

In Lemma 5.12 we assumed that the residual field $\sigma$ takes values in a
sufficiently small neighborhood of zero in $\mathcal{G}$, so that the sums of
Feynman diagrams in Figure 6 converge.404040Curiously, the issue of
convergence arises only in diagrams (v), (vi), (viii) — the diagrams involving
ghosts. In fact, they converge iff $\sigma$ is valued in
$B_{0}\subset\mathcal{G}$ where $B_{0}$ is the connected component of the
origin in

$\left\\{\sigma\in\mathcal{G}\,\big{|}\,{\det}_{\mathcal{G}}\frac{\sinh\frac{\mathrm{ad}_{\sigma}}{2}}{\frac{\mathrm{ad}_{\sigma}}{2}}\neq
0\right\\}\qquad\subset\quad\mathcal{G}.$

In other words, $B_{0}$ is the subset of $\mathcal{G}$ where all eigenvalues
of $\mathrm{ad}_{\sigma}$ lie in the open interval $(-2\pi i,2\pi i)\subset
i\mathbb{R}$. Thus, we are assuming that $\sigma$ takes values in
$B_{0}\subset\mathcal{G}$ (cf. the discussion of the Gribov region in the
context of 2D Yang-Mills in [30, Section 2.4.1]). Furthermore, note that the
exponential map $\exp\mathrel{\mathop{\mathchar
58\relax}}\mathcal{G}\rightarrow G$ is a diffeomorphism from $B_{0}$ onto its
image $\exp(B_{0})$. Moreover, $\exp(B_{0})$ is an open dense subset of $G$.

####  5.3.3. Group-valued parametrization of the residual field

Let us parametrize the residual field $\sigma$ by a group-valued map
$g=e^{-\sigma}\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow G$.

######  Lemma 5.14.

The effective action (61) can be rewritten as

(63)
$S^{\mathrm{eff}}=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\,\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\,\partial
g\rangle\\\ +\langle\mathsf{A}^{*}_{\mathrm{res}},F_{-}(\mathrm{ad}_{\log
g})\circ c_{\mathrm{out}}+F_{+}(\mathrm{ad}_{\log g})\circ
c_{\mathrm{in}}\rangle\Big{)}+\mathrm{WZW}(g)-i\hbar\mathbb{W}.$

Here

(64) $\mathrm{WZW}(g)=-\frac{1}{2}\int_{\Sigma}\langle\partial g\cdot
g^{-1},\bar{\partial}g\cdot
g^{-1}\rangle-\frac{1}{12}\int_{I\times\Sigma}\langle
d\widetilde{g}\cdot\widetilde{g}^{-1},[d\widetilde{g}\cdot\widetilde{g}^{-1},d\widetilde{g}\cdot\widetilde{g}^{-1}]\rangle.$

is the Wess–Zumino–Witten action, where $\widetilde{g}=e^{(t-1)\sigma}$ is the
extension of $g$ to a mapping $I\times\Sigma\rightarrow G$, interpolating
between $\widetilde{g}=g$ at $t=0$ and $\widetilde{g}=1$ at $t=1$.414141By the
standard result on Wess–Zumino terms, $\mathrm{WZW}(g)\bmod
4\pi^{2}\mathbb{Z}$ is independent of the choice of $\widetilde{g}$
interpolating between $g$ on one end of the cylinder and $1$ on the other.

######  Remark 5.15.

Under the convergence assumption that $\sigma$ is valued in $B_{0}$ (see
Remark 5.13), or equivalently that $g$ is valued in $\exp(B_{0})$ — a
contractible open dense subset of $G$, $\mathrm{WZW}(g)$ is a single-valued
function of $g$, and hence $S^{\mathrm{eff}}$ is also a single-valued
expression. If $g$ is allowed to roam the entire group $G$, $\mathrm{WZW}(g)$
(and thus $S^{\mathrm{eff}}$) becomes multi-valued, defined only
$\bmod\,4\pi^{2}\mathbb{Z}$. In the latter case, for
$e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$ to be a single-valued expression, one
needs $\hbar=\frac{2\pi}{k}$ with $k\in\mathbb{Z}$ an integer level. The fact
that quantization of $\hbar$ is necessary in one case but not in the other can
be traced to the fact that the Cartan $3$-form (whose pullback by
$\widetilde{g}$ is the integrand in the second term in the r.h.s. of (64))
represents a nontrivial cohomology class on $G$ but is exact when restricted
to $\exp(B_{0})$.

###### Proof of Lemma 5.14.

First terms in (61) and (63) obviously match. We have

$\displaystyle g^{-1}\partial
g=e^{\sigma}\int_{0}^{1}d\tau\,e^{-\tau\sigma}(-\partial\sigma)e^{-(1-\tau)\sigma}$
$\displaystyle=\int_{0}^{1}d\tau\,e^{\tau\mathrm{ad}_{\sigma}}(-\partial\sigma)$
$\displaystyle=\frac{e^{\mathrm{ad}_{\sigma}}-1}{\mathrm{ad}_{\sigma}}(-\partial\sigma),$
$\displaystyle\bar{\partial}g\cdot
g^{-1}=\int_{0}^{1}d\tau\,e^{-\tau\sigma}(-\bar{\partial}\sigma)e^{-(1-\tau)\sigma}e^{\sigma}$
$\displaystyle=\int_{0}^{1}d\tau\,e^{-\tau\mathrm{ad}_{\sigma}}(-\bar{\partial}\sigma)$
$\displaystyle=\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}(-\bar{\partial}\sigma).$

Thus, second and third terms in (61) and (63) also match. Next, evaluating the
Wess-Zumino term on our preferred extension $\widetilde{g}=e^{(t-1)\sigma}$,
we have

(65) $-\frac{1}{12}\int_{I\times\Sigma}\langle
d\widetilde{g}\cdot\widetilde{g}^{-1},[d\widetilde{g}\cdot\widetilde{g}^{-1},d\widetilde{g}\cdot\widetilde{g}^{-1}]\rangle\\\
=-\frac{1}{4}\int_{\Sigma}\int_{0}^{1}dt\Big{\langle}\sigma,\int_{0}^{1-t}d\tau\int_{0}^{1-t}d\tau^{\prime}\Big{[}e^{-\tau\sigma}(-d\sigma)e^{-(1-t-\tau)\sigma}e^{(1-t)\sigma},e^{-\tau^{\prime}\sigma}(-d\sigma)e^{-(1-t-\tau^{\prime})\sigma}e^{(1-t)\sigma}\Big{]}\Big{\rangle}\\\
=\frac{1}{4}\int_{\Sigma}\int_{0}^{1}dt\int_{0}^{1-t}d\tau\int_{0}^{1-t}d\tau^{\prime}\Big{\langle}d\sigma,\Big{[}\sigma,e^{(\tau^{\prime}-\tau)\mathrm{ad}_{\sigma}}d\sigma\Big{]}\Big{\rangle}=\frac{1}{2}\int_{\Sigma}\Big{\langle}d\sigma,\Big{[}\sigma,\Big{(}\frac{\sinh\mathrm{ad}_{\sigma}-\mathrm{ad}_{\sigma}}{(\mathrm{ad}_{\sigma})^{3}}\Big{)}\,d\sigma\Big{]}\Big{\rangle}\\\
=\frac{1}{2}\int_{\Sigma}\Big{\langle}d\sigma,\Big{(}\frac{\sinh\mathrm{ad}_{\sigma}-\mathrm{ad}_{\sigma}}{(\mathrm{ad}_{\sigma})^{2}}\Big{)}\,d\sigma\Big{\rangle}.$

The WZW kinetic term is:

(66) $-\frac{1}{2}\int_{\Sigma}\langle\partial g\cdot
g^{-1},\bar{\partial}g\cdot g^{-1}\rangle\\\
=-\frac{1}{2}\int_{\Sigma}\int_{0}^{1}d\tau\int_{0}^{1}d\tau^{\prime}\Big{\langle}e^{-\tau\sigma}(-\partial\sigma)e^{-(1-\tau)\sigma}e^{\sigma},e^{-\tau^{\prime}\sigma}(-\bar{\partial}\sigma)e^{-(1-\tau^{\prime})\sigma}e^{\sigma}\Big{\rangle}\\\
=-\frac{1}{2}\int_{\Sigma}\int_{0}^{1}d\tau\int_{0}^{1}d\tau^{\prime}\Big{\langle}\partial\sigma,e^{(\tau-\tau^{\prime})\mathrm{ad}_{\sigma}}\bar{\partial}\sigma\Big{\rangle}=-\int_{\Sigma}\Big{\langle}\partial\sigma,\frac{\cosh\mathrm{ad}_{\sigma}-1}{(\mathrm{ad}_{\sigma})^{2}}\bar{\partial}\sigma\Big{\rangle}.$

Putting the kinetic term (66) and the Wess-Zumino term (65) together, we
obtain

$\mathrm{WZW}(g)=\int_{\Sigma}-\Big{\langle}\partial\sigma,\frac{\cosh\mathrm{ad}_{\sigma}-1}{(\mathrm{ad}_{\sigma})^{2}}\bar{\partial}\sigma\Big{\rangle}+\Big{\langle}\partial\sigma,\frac{\sinh\mathrm{ad}_{\sigma}-\mathrm{ad}_{\sigma}}{(\mathrm{ad}_{\sigma})^{2}}\,\bar{\partial}\sigma\Big{\rangle}\\\
=-\int_{\Sigma}\Big{\langle}\partial\sigma,\frac{e^{-\mathrm{ad}_{\sigma}}+\mathrm{ad}_{\sigma}-1}{(\mathrm{ad}_{\sigma})^{2}}\,\bar{\partial}\sigma\Big{\rangle}.$

Thus, finally, $\mathrm{WZW}$ term in (63) coincides with the fourth term in
(61).

Ghost terms and the 1-loop contributions in (61) and (63) are identified
directly.

∎

####  5.3.4. A comment on ghost wheels

To understand the role of the term $\mathbb{W}$ (ghost wheels) in (63), recall
that, for $\mu_{G}$ the Haar measure on the group $G$ and $\mu_{\mathcal{G}}$
the Lebesgue measure on the Lie algebra, one has
$\exp^{*}\mu_{G}=e^{j}\cdot\mu_{\mathcal{G}}$, with $j$ the function on
$\mathcal{G}$ defined by the formula (60). Therefore, the half-density on the
space of residual fields associated to the effective action (63) is,
heuristically, the following:424242 Recall, see [38], that on an
$(n|n)$-dimensional odd symplectic supermanifold $(\mathcal{M},\omega)$, half-
densities can be understood as cohomology classes of the differential
$\omega\wedge$ acting on differential forms $\Omega^{\bullet}(\mathcal{M})$.
Moreover, if $(x^{i},\xi_{i})$ are Darboux coordinates, $i=1,\ldots,n$, with
$x^{i}$ the even coordinates, then each cohomology class has a unique
representative of the form $\rho(x,\xi)dx^{1}\cdots
dx^{n}\in\Omega^{n}(\mathcal{M})$ corresponding to the half-density
$\rho(x,\xi)\prod_{i}d^{\frac{1}{2}}xD^{\frac{1}{2}}\xi$, with $\rho$ some
function. In (67), “$\sim$” refers this choice of preferred representative for
a half-density.

(67)
$e^{\frac{i}{\hbar}S^{\mathrm{eff}}}\mathcal{D}^{\frac{1}{2}}\sigma\;\mathcal{D}^{\frac{1}{2}}\mathsf{A}^{*}_{\mathrm{res}}\sim
e^{\frac{i}{\hbar}S^{\mathrm{eff}}}\mathcal{D}\sigma\\\
=e^{\frac{i}{\hbar}S^{\mathrm{eff}\,(0)}}\;\;``\prod_{z\in\Sigma}\underbrace{e^{j(\sigma(z))}\mu_{\mathcal{G}}(\sigma(z))}_{\mu_{G}(g(z))}"=e^{\frac{i}{\hbar}S^{\mathrm{eff}\,(0)}}\;\mathcal{D}g.$

Here $S^{\mathrm{eff}\,(0)}$ stands for (63) without the $\mathbb{W}$
term434343 The superscript $(0)$ means the “$0$-loop part,” corresponding to
the expansion in powers of $\hbar$:
$S^{\mathrm{eff}}=\sum_{k=0}^{\infty}(-i\hbar)^{k}S^{\mathrm{eff}\,(k)}$. In
the present case, we have only $k=0,1$ terms.  — the latter was used in
transforming the functional measure from the pointwise product of Lebesgue
measures for $\sigma$ to the product of Haar measures for $g$. The equivalence
(67) of a half-densities is an extension of a rigorous result presented in
[16] for a finite-dimensional system.

The odd symplectic form on residual fields is

(68) $\displaystyle\omega_{\mathrm{res}}$
$\displaystyle=\int_{\Sigma}\langle\delta\mathsf{A}^{*}_{\mathrm{res}},\delta\sigma\rangle=\delta\int_{\Sigma}\langle\mathsf{A}^{*}_{\mathrm{res}},\delta\sigma\rangle$
$\displaystyle=\delta\int_{\Sigma}\langle\mathsf{A}^{*}_{\mathrm{res}},-\frac{\mathrm{ad}_{\sigma}}{1-e^{-\mathrm{ad}_{\sigma}}}\circ(\delta
g\cdot
g^{-1})\rangle=\delta\int_{\Sigma}\langle-\frac{\mathrm{ad}_{\sigma}}{e^{\mathrm{ad}_{\sigma}}-1}\circ\mathsf{A}^{*}_{\mathrm{res}},\delta
g\cdot g^{-1}\rangle$ $\displaystyle=\int_{\Sigma}\langle\delta g^{*},\delta
g\rangle,$

where we introduced the notation

(69) $g^{*}=-g^{-1}\cdot\big{(}F_{+}(\mathrm{ad}_{\log
g})\circ\mathsf{A}^{*}_{\mathrm{res}}\big{)}=\big{(}F_{-}(\mathrm{ad}_{\log
g})\circ\mathsf{A}^{*}_{\mathrm{res}}\big{)}\cdot g^{-1}$

— a reparametrization of the residual field $\mathsf{A}^{*}_{\mathrm{res}}$
such that $(g,g^{*})$ form Darboux coordinates on $\mathcal{V}$.

Rewritten in terms of the parametrization $(g,g^{*})$ for residual fields, the
half-density (67) becomes

(70)
$e^{\frac{i}{\hbar}S^{\mathrm{eff}}}\mathcal{D}^{\frac{1}{2}}\sigma\;\mathcal{D}^{\frac{1}{2}}\mathsf{A}^{*}_{\mathrm{res}}=e^{\frac{i}{\hbar}S^{\mathrm{eff}\,(0)}}\mathcal{D}^{\frac{1}{2}}g\;\mathcal{D}^{\frac{1}{2}}g^{*}.$

I.e., in the $(g,g^{*})$-parametrization, the ghost loops go away and the
effective action has no quantum corrections.

######  Remark 5.16.

In the context of BV formalism, it is natural to think of $S^{\mathrm{eff}}$
as a “log-half-density” (see, e.g., [35, section 2.6]) on the space of
residual fields, rather than a function, i.e., behaving under a change of
Darboux coordinates as

$S^{\mathrm{eff}}_{[x,\xi]}(x,\xi)=S^{\mathrm{eff}}_{[x^{\prime},\xi^{\prime}]}(x^{\prime},\xi^{\prime})-i\hbar\log\mathrm{sdet}\frac{\partial(x,\xi)}{\partial(x^{\prime},\xi^{\prime})},$

so that one has
$e^{\frac{i}{\hbar}S^{\mathrm{eff}}_{[x,\xi]}(x,\xi)}d^{\frac{1}{2}}xD^{\frac{1}{2}}\xi=e^{\frac{i}{\hbar}S^{\mathrm{eff}}_{[x^{\prime},\xi^{\prime}]}(x^{\prime},\xi^{\prime})}d^{\frac{1}{2}}x^{\prime}D^{\frac{1}{2}}\xi^{\prime}$.
Here the superdeterminant (Berezinian) $\mathrm{sdet}\cdots$ is the Jacobian
of the transformation. With that in mind, the effective action (61) is
$S^{\mathrm{eff}}_{[\sigma,\mathsf{A}^{*}_{\mathrm{res}}]}$ — relative to the
coordinate system $(\sigma,\mathsf{A}^{*}_{\mathrm{res}})$ on $\mathcal{V}$.
On the other hand, $S^{\mathrm{eff}\,(0)}$ given by (63) without the
$-i\hbar\mathbb{W}$ term is $S^{\mathrm{eff}}_{[g,g^{*}]}$ — relative to the
coordinate system $(g,g^{*})$.

####  5.3.5. Effective action vs. Hamilton–Jacobi

Denote

(71)
$\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)=\\\
=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\partial
g\rangle\Big{)}+\mathrm{WZW}(g)$

— the effective action (63) restricted to fields of ghost number zero and
without the $O(\hbar)$ term.

Function (71) produces, as a generalized generating function (see [16,
Appendix A] and Section 2), with $g$ an auxiliary parameter, the following
lagrangian
$L\subset\overline{\mathcal{F}^{\partial}_{\mathrm{in}}}\times\mathcal{F}^{\partial}_{\mathrm{out}}$
in the phase space for the boundary of the cylinder:

(72) $\displaystyle L=\Big{\\{}$
$\displaystyle\mathsf{A}|_{t=1}=\mathsf{A}_{\mathrm{out}}^{1,0}+\underbrace{g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}-\bar{\partial}g\cdot
g^{-1}}_{\mathsf{A}^{0,1}\big{|}_{t=1}=\frac{\delta\mathbb{I}}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}}}\;,$
$\displaystyle\mathsf{A}|_{t=0}=\mathsf{A}_{\mathrm{in}}^{0,1}+\underbrace{g^{-1}\mathsf{A}^{1,0}_{\mathrm{out}}g+g^{-1}\cdot\partial
g}_{\mathsf{A}^{1,0}\big{|}_{t=0}=-\frac{\delta\mathbb{I}}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}}}\;\;\Big{|}\;\;Y=0\Big{\\}},$

where we denoted444444 The notation $Y=-\frac{\delta\mathbb{I}}{\delta g\cdot
g^{-1}}$ means that variation of $\mathbb{I}$ under the variation of $g$ is
$\delta_{g}\mathbb{I}=-\int_{\Sigma}\langle\delta g\cdot g^{-1},Y\rangle$.

(73) $Y=-\frac{\delta\mathbb{I}}{\delta g\cdot g^{-1}}=\\\
=[\mathsf{A}_{\mathrm{out}}^{1,0},g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}]+(\bar{\partial}-\mathrm{ad}_{\bar{\partial}g\cdot
g^{-1}})\mathsf{A}_{\mathrm{out}}^{1,0}+\partial(g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1})-\partial(\bar{\partial}g\cdot
g^{-1}).$

######  Lemma 5.17.

The lagrangian (72) generated by the functional (71) — the tree part of the
effective action, restricted to $\mathrm{gh}=0$ fields — coincides with the
evolution relation in
$\overline{\mathcal{F}^{\partial}_{\mathrm{in}}}\times\mathcal{F}^{\partial}_{\mathrm{out}}$
for Chern–Simons theory on the cylinder $I\times\Sigma$.

###### Proof.

We are restricting our attention only the to $\mathrm{gh}=0$ connection field
$\mathsf{A}+dt\cdot a$ with $\mathsf{A}$ a $t$-dependent $1$-form on $\Sigma$
and $a$ a $t$-dependent $0$-form on $\Sigma$ (both are $\mathcal{G}$-valued).
The equation of motion — zero-curvature condition — $F_{\mathsf{A}+dt\,a}=0$
splits into

(74) $\displaystyle
d_{\Sigma}\mathsf{A}+\frac{1}{2}[\mathsf{A},\mathsf{A}]=0,$ (75)
$\displaystyle\partial_{t}\mathsf{A}=(d_{\Sigma}+[\mathsf{A},-])a.$

Equation (75) says that $\mathsf{A}$ changes by a continuous gauge
transformation on $\Sigma$ as $t$ changes, with $a$ the infinitesimal
generator. Thus,

(76)
$\mathsf{A}|_{t=1}=g\,\mathsf{A}|_{t=0}\,g^{-1}+gd_{\Sigma}g^{-1},\quad\mathrm{with}\;\;g=P\overleftarrow{\exp}\left(-\int_{0}^{1}dt\,a\right).$

This implies that we can recover the $(1,0)$ component of $\mathsf{A}$ at
$t=0$ from its known value at $t=1$ and can recover the $(0,1)$ component at
$t=1$ from its known value at $t=0$. Thus,

(77) $\displaystyle\mathsf{A}|_{t=1}$
$\displaystyle=\mathsf{A}^{1,0}_{\mathrm{out}}+\mathsf{A}^{0,1}|_{t=1}=\mathsf{A}^{1,0}_{\mathrm{out}}+g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}+g\bar{\partial}g^{-1},$
(78) $\displaystyle\mathsf{A}|_{t=0}$
$\displaystyle=\mathsf{A}^{0,1}_{\mathrm{in}}+\mathsf{A}^{1,0}|_{t=0}=\mathsf{A}^{0,1}_{\mathrm{in}}+g^{-1}\mathsf{A}^{1,0}_{\mathrm{out}}g+g^{-1}\partial
g.$

Note that these two equations coincide with the first two equations in (72).
Next, equation (74) means that the curvature of $\mathsf{A}$ must vanish on
$\Sigma\times\\{t\\}$ for any $t$. In fact, it suffices to verify it just for
one value of $t$, because for all others it would follow from (75). Checking
(74) at $t=1$, we have

(79)
$F_{\mathsf{A}}\big{|}_{t=1}=\underbrace{\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}+\partial(g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}-\bar{\partial}g\cdot
g^{-1})}_{d_{\Sigma}\mathsf{A}|_{t=1}}+[\mathsf{A}^{1,0}_{\mathrm{out}},g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}-\bar{\partial}g\cdot
g^{-1}]\quad=0.$

This equation coincides with the constraint $Y=0$ in (72) coming from equating
to zero the variation of the generating function $\mathbb{I}$ in the auxiliary
parameter $g$.

Thus, we have checked that the lagrangian in the boundary phase space induced
from the equations of motion (the evolution relation) coincides with the
lagrangian generated by $\mathbb{I}$. ∎

######  Remark 5.18.

The function $\mathbb{I}$ given by (71) is also the Hamilton–Jacobi action
(see [16, Section 7.2]): it is the evaluation of the Chern–Simons action with
polarization terms, restricted to degree-zero fields,

$S^{f}_{\mathrm{ph}}=\int_{I\times\Sigma}\Big{(}\frac{1}{2}\langle
A,dA\rangle+\frac{1}{6}\langle
A,[A,A]\rangle\Big{)}+\int_{\\{1\\}\times\Sigma}\frac{1}{2}\langle\mathsf{A}^{1,0},\mathsf{A}^{0,1}\rangle-\int_{\\{0\\}\times\Sigma}\frac{1}{2}\langle\mathsf{A}^{0,1},\mathsf{A}^{1,0}\rangle,$

on any connection $1$-form $A$ solving the “evolution equation”
$\iota_{\frac{\partial}{\partial t}}F_{A}=0$ subject to boundary conditions
$(A|_{t=1})^{1,0}=\mathsf{A}^{1,0}_{\mathrm{out}}$,
$(A|_{t=0})^{0,1}=\mathsf{A}^{0,1}_{\mathrm{in}}$ and with the parallel
transport of $A$ along the interval $I\times\\{z\\}$ given by $g(z)\in G$ for
any $z\in\Sigma$. One proves this by an explicit computation similar to the
proof of Lemma 5.14, picking a convenient gauge equivalent representative for
$A=\mathsf{A}+a\,dt$ with $a$ constant along $I$ (but allowed to vary in
$\Sigma$ direction). Here we are using gauge-invariance of Chern–Simons action
$\bmod\,4\pi^{2}\mathbb{Z}$ with respect to gauge transformations trivial on
the boundary.

####  5.3.6. Quantum master equation

Quantum BFV operators on in- and out-states
$\Omega_{\mathrm{in}},\Omega_{\mathrm{out}}$ are given by canonical
quantization of the boundary BFV action

$S^{\mathrm{BFV}}_{\Sigma}=\pm\int_{\Sigma}\langle
c,F_{\mathsf{A}}\rangle+\langle\mathsf{A}^{*},\frac{1}{2}[c,c]\rangle$

with $\pm$ corresponding to out-/in-boundary. Explicitly, quantum BFV
operators are454545The BFV operator (81), its generalization to the case of
Wilson lines intersecting the boundary — see (95) below — and its cohomology
in genus zero were discussed in [1].

(80) $\displaystyle\Omega_{\mathrm{out}}$
$\displaystyle=\int_{\Sigma}\left\langle
c_{\mathrm{out}},\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}-i\hbar(\partial+[\mathsf{A}^{1,0}_{\mathrm{out}},-])\frac{\delta}{\delta\mathsf{A}^{1,0}_{\mathrm{out}}}\right\rangle-i\hbar\left\langle\frac{1}{2}[c_{\mathrm{out}},c_{\mathrm{out}}],\frac{\delta}{\delta
c_{\mathrm{out}}}\right\rangle,$ (81) $\displaystyle\Omega_{\mathrm{in}}$
$\displaystyle=\int_{\Sigma}\left\langle
c_{\mathrm{in}},-\partial\mathsf{A}^{0,1}_{\mathrm{in}}-i\hbar(\bar{\partial}+[\mathsf{A}^{0,1}_{\mathrm{in}},-])\frac{\delta}{\delta\mathsf{A}^{0,1}_{\mathrm{in}}}\right\rangle-i\hbar\left\langle\frac{1}{2}[c_{\mathrm{in}},c_{\mathrm{in}}],\frac{\delta}{\delta
c_{\mathrm{in}}}\right\rangle.$

######  Lemma 5.19.

The partition function $Z=e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$ with
$S^{\mathrm{eff}}$ given by (61), (63) satisfies the modified quantum master
equation

(82)
$(\Omega_{\mathrm{out}}+\Omega_{\mathrm{in}}-\hbar^{2}\Delta_{\mathrm{res}})Z=0$

with
$\Delta_{\mathrm{res}}=\int_{\Sigma}\langle\frac{\delta}{\delta\sigma},\frac{\delta}{\delta\mathsf{A}^{*}_{\mathrm{res}}}\rangle$
the BV Laplacian on residual fields.

###### Proof.

Given the ansatz $Z=e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$, the equation (82)
can be written as

(83)
$Z^{-1}\Omega_{\mathrm{in}}Z+Z^{-1}\Omega_{\mathrm{out}}Z+\frac{1}{2}\\{S^{\mathrm{eff}},S^{\mathrm{eff}}\\}_{\mathrm{res}}-i\hbar\Delta_{\mathrm{res}}S^{\mathrm{eff}}\stackrel{{\scriptstyle!}}{{=}}0$

with $\\{,\\}_{\mathrm{res}}$ the odd Poisson bracket on residual fields
associated with the symplectic structure (68). Moreover, using the
decomposition
$S^{\mathrm{eff}}=S^{\mathrm{eff}\,(0)}-i\hbar\,\mathbb{W}(\sigma)$, the mQME
can be further rewritten as

(84)
$Z^{-1}\Omega_{\mathrm{in}}Z+Z^{-1}\Omega_{\mathrm{out}}Z+\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}_{\mathrm{res}}-\\\
-i\hbar\Big{(}\\{S^{\mathrm{eff}\,(0)},\mathbb{W}\\}_{\mathrm{res}}+\Delta_{\mathrm{res}}S^{\mathrm{eff}\,(0)}\Big{)}\stackrel{{\scriptstyle!}}{{=}}0.$

It is easiest to compute the term
$\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}_{\mathrm{res}}$
using $(g,g^{*})$ \- parametrization of residual fields. We have

$\displaystyle S^{\mathrm{eff}\,(0)}$ $\displaystyle=$
$\displaystyle\mathbb{I}+\int_{\Sigma}\langle
g^{*},c_{\mathrm{out}}g-gc_{\mathrm{in}}\rangle,$
$\displaystyle\frac{\delta}{\delta g}S^{\mathrm{eff}\,(0)}$ $\displaystyle=$
$\displaystyle-g^{-1}Y+g^{*}c_{\mathrm{out}}+c_{\mathrm{in}}g^{*},$
$\displaystyle\frac{\delta}{\delta g^{*}}S^{\mathrm{eff}\,(0)}$
$\displaystyle=$ $\displaystyle c_{\mathrm{out}}g-gc_{\mathrm{in}},$

with $\mathbb{I}$ as in (71) and $Y$ as in (73). Thus,

$\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}_{\mathrm{res}}=S^{\mathrm{eff}\,(0)}\left(\int_{\Sigma}\langle\frac{\overleftarrow{\delta}}{\delta
g},\frac{\overrightarrow{\delta}}{\delta
g^{*}}\rangle\right)S^{\mathrm{eff}\,(0)}\\\ =\int_{\Sigma}\langle
c_{\mathrm{out}}-gc_{\mathrm{in}}g^{-1},-Y+gg^{*}c_{\mathrm{out}}+gc_{\mathrm{in}}g^{*}\rangle.$

Acting on the partition function with the boundary BFV operators yields

$\displaystyle Z^{-1}\Omega_{\mathrm{out}}Z$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\Big{\langle}c_{\mathrm{out}},\bar{\partial}\mathsf{A}^{1,0}_{\mathrm{out}}+(\partial+[\mathsf{A}^{1,0}_{\mathrm{out}},-])(g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}-\bar{\partial}g\;g^{-1})\Big{\rangle}-\frac{1}{2}\langle[c_{\mathrm{out}},c_{\mathrm{out}}],gg^{*}\rangle,$
$\displaystyle Z^{-1}\Omega_{\mathrm{in}}Z$ $\displaystyle=$
$\displaystyle\int_{\Sigma}\Big{\langle}c_{\mathrm{in}},-\partial\mathsf{A}^{0,1}_{\mathrm{in}}-(\bar{\partial}+[\mathsf{A}^{0,1}_{\mathrm{in}},-])(g^{-1}\mathsf{A}^{1,0}_{\mathrm{out}}g+g^{-1}\partial
g)\Big{\rangle}+\frac{1}{2}\langle[c_{\mathrm{in}},c_{\mathrm{in}}],g^{*}g\rangle.$

Putting together these computations, we find that

(85)
$Z^{-1}\Omega_{\mathrm{in}}Z+Z^{-1}\Omega_{\mathrm{out}}Z+\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}_{\mathrm{res}}=0$

— all the terms in this combination cancel out. This gives us the mQME in the
leading order $O(\hbar^{0})$.

For the two remaining terms,
$\\{S^{\mathrm{eff}\,(0)},\mathbb{W}\\}_{\mathrm{res}}$ and
$\Delta_{\mathrm{res}}S^{\mathrm{eff}\,(0)}$, we will use the
$(\sigma,\mathsf{A}^{*}_{\mathrm{res}})$-parametrization for residual fields.
The variation of $j(\sigma)$ (see (60)) in $\sigma$ is

$\delta_{\sigma}j(\sigma)=\mathrm{tr}_{\mathcal{G}}P(\mathrm{ad}_{\sigma})\mathrm{ad}_{\delta\sigma},\qquad\mathrm{where}\quad
P(x)=\frac{1}{2}\coth\frac{x}{2}-\frac{1}{x}.$

Therefore, using (62), we have

(86) $\displaystyle\\{S^{\mathrm{eff}\,(0)},\mathbb{W}\\}_{\mathrm{res}}$
$\displaystyle=\sum_{z\in\Sigma}\mathrm{tr}_{\mathcal{G}}P(\mathrm{ad}_{\sigma})[-F_{+}(\mathrm{ad}_{\sigma})\circ
c_{\mathrm{out}}-F_{-}(\mathrm{ad}_{\sigma})\circ c_{\mathrm{in}},\;\bullet]$
$\displaystyle=\sum_{z\in\Sigma}\mathrm{tr}_{\mathcal{G}}P(\mathrm{ad}_{\sigma})[-c_{\mathrm{out}}+c_{\mathrm{in}},\;\bullet].$

Here the last simplification relies on the identity

(87) $\mathrm{tr}_{\mathcal{G}}[(\mathrm{ad}_{x})^{a}\circ
y,(\mathrm{ad}_{x})^{b}\bullet]=0\qquad\mathrm{for}\;\;a\geq 1,\;b\geq 0,$

which follows from the cyclic property of the trace and Jacobi identity. We
also have

(88)
$\Delta_{\mathrm{res}}S^{\mathrm{eff}\,(0)}=\sum_{z\in\Sigma}-\mathrm{tr}_{\mathcal{G}}\sum_{r,s\geq
0}\frac{B_{r+s+1}}{(r+s+1)!}(\mathrm{ad}_{\sigma})^{r}\mathrm{ad}_{\bullet}\Big{(}(-1)^{r+s+1}(\mathrm{ad}_{\sigma})^{s}c_{\mathrm{out}}-(\mathrm{ad}_{\sigma})^{s}c_{\mathrm{in}}\Big{)}\\\
=\sum_{z\in\Sigma}\mathrm{tr}_{\mathcal{G}}\sum_{r,s\geq
0}\frac{B_{r+s+1}}{(r+s+1)!}\Big{[}(-1)^{r+s+1}(\mathrm{ad}_{\sigma})^{s}c_{\mathrm{out}}-(\mathrm{ad}_{\sigma})^{s}c_{\mathrm{in}},(\mathrm{ad}_{\sigma})^{r}\bullet\Big{]}\\\
\underset{(\ref{trace
identity})}{=}\sum_{z\in\Sigma}\mathrm{tr}_{\mathcal{G}}\sum_{r\geq
0}\frac{B_{r+1}}{(r+1)!}[c_{\mathrm{out}}-c_{\mathrm{in}},(\mathrm{ad}_{\sigma})^{r}\bullet]=\sum_{z\in\Sigma}\mathrm{tr}_{\mathcal{G}}\;\mathrm{ad}_{c_{\mathrm{out}}-c_{\mathrm{in}}}\circ
P(\mathrm{ad}_{\sigma}).$

Comparing (86) and (88), we see that they exactly cancel each other pointwise
on $\Sigma$. Thus,

$\\{S^{\mathrm{eff}\,(0)},\mathbb{W}\\}_{\mathrm{res}}+\Delta_{\mathrm{res}}S^{\mathrm{eff}\,(0)}=0.$

Together with (85), this finishes the proof of mQME (84). ∎

######  Remark 5.20.

The check of the mQME above clearly breaks into two parts:

  1. (a)

The classical part

(89)
$Z^{-1}\Omega_{\mathrm{in}}Z+Z^{-1}\Omega_{\mathrm{out}}Z+\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}=0,$

which is unambiguous and requires no regularization.

  2. (b)

The quantum part

$\\{S^{\mathrm{eff}\,(0)},\mathbb{W}\\}_{\mathrm{res}}+\Delta_{\mathrm{res}}S^{\mathrm{eff}\,(0)}=0,$

which makes sense with the same regularization as the one used in the proof of
Lemma 5.12: replacing $\Sigma$ with the set of vertices of a triangulation.

Aside: mQME and Polyakov-Wiegmann formula. The classical part of the mQME,
equation (89), itself splits into two parts: terms involving the antifield
$g^{*}$ and others. The terms involving $g^{*}$ cancel due to invariance of
the inner product. The cancellation of the remaining terms can be understood
in terms of the WZW model as follows. The part of the effective action
$\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)$
defined in (71) can be identified as the WZW action coupled to two external
chiral gauge fields
$\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}}$, see, e.g.,
eq. (4.5) in [23, §4.2]. This coupling is sometimes called “gauging the
$G_{L}\times G_{R}$ symmetry”, for instance in [42]. For us it is more natural
to call it the $G_{\mathrm{in}}\times G_{\mathrm{out}}$-action. Explicitly,
the action of $(h_{\mathrm{in}},h_{\mathrm{out}})\in G\times G$ on
$(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)$ is

(90)
$(h_{\mathrm{in}},h_{\mathrm{out}})\cdot(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)=\left({}^{h_{\mathrm{in}}}(\mathsf{A}^{0,1}_{\mathrm{in}}),{}^{h_{\mathrm{out}}}(\mathsf{A}^{1,0}_{\mathrm{out}});h_{\mathrm{out}}gh_{\mathrm{in}}^{-1}\right).$

It is well known that under the transformation (90) $\mathbb{I}$ is not
invariant, but transforms according to the Polyakov–Wiegmann [37] formula:

(91)
$\mathbb{I}\left({}^{h_{\mathrm{in}}}(\mathsf{A}^{0,1}_{\mathrm{in}}),{}^{h_{\mathrm{out}}}(\mathsf{A}^{1,0}_{\mathrm{out}});h_{\mathrm{out}}gh_{\mathrm{in}}^{-1}\right)=\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)-\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},0;h_{\mathrm{in}})+\mathbb{I}(0,\mathsf{A}^{1,0}_{\mathrm{out}};h_{\mathrm{out}}).$

We claim that this equation is just the finite version of the classical part
of the mQME, eq. (89) above. To see this, consider a path
$(h_{\mathrm{in}}(t),h_{\mathrm{out}}(t))$ of gauge transformations starting
at the identity and compute the derivative of (91) at $t=0$. The computation
is quite straightforward and we just sketch it: using eq. (73), we get

$\left.\frac{d}{dt}\right|_{t=0}\mathbb{I}\left(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}},h_{\mathrm{out}}(t)gh_{\mathrm{in}}(t)^{-1}\right)=\int_{\Sigma}\langle\dot{h}_{\mathrm{out}}-g\dot{h}_{\mathrm{in}}g^{-1},Y\rangle.$

Upon identifying
$c_{\mathrm{in}}=\dot{h}_{\mathrm{in}},c_{\mathrm{out}}=\dot{h}_{\mathrm{out}}$
this gives the piece of
$\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}$ of (89) not
involving $g^{*}$. Then, we find that

$\left.\frac{d}{dt}\right|_{t=0}\mathbb{I}\left({}^{h_{\mathrm{in}}(t)}(\mathsf{A}^{0,1}_{\mathrm{in}}),\mathsf{A}^{1,0}_{\mathrm{out}},g\right)+\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},0,h_{\mathrm{in}}(t))\\\
=-\bar{\partial}_{\mathsf{A}^{0,1}_{\mathrm{in}}}\dot{h}_{\mathrm{in}}(g^{-1}A^{1,0}_{\mathrm{out}}g+g^{-1}\partial
g)-\int\langle\dot{h}_{\mathrm{in}},Y\big{|}_{g=1,A^{0,1}_{\mathrm{out}}=0}\rangle=Z^{-1}\Omega_{\mathrm{in}}Z\bigg{|}_{g^{*}=0,c_{\mathrm{in}}=\dot{h}_{\mathrm{in}},c_{\mathrm{out}}=\dot{h}_{\mathrm{out}}}$

and similarly for the action of $h_{\mathrm{out}}$. Overall, we find

$\left.\frac{d}{dt}\right|_{t=0}\left[\mathbb{I}\left({}^{h_{\mathrm{in}}}(\mathsf{A}^{0,1}_{\mathrm{in}}),{}^{h_{\mathrm{out}}}(\mathsf{A}^{1,0}_{\mathrm{out}});h_{\mathrm{out}}gh_{\mathrm{in}}^{-1}\right)+\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},0,h_{\mathrm{in}}(t))-\mathbb{I}(0,\mathsf{A}^{1,0}_{\mathrm{out}},h_{\mathrm{out}}(t)\right]\\\
=\left[Z^{-1}\Omega_{\mathrm{in}}Z+Z^{-1}\Omega_{\mathrm{out}}Z+\frac{1}{2}\\{S^{\mathrm{eff}\,(0)},S^{\mathrm{eff}\,(0)}\\}\right]_{g^{*}=0,c_{\mathrm{in}}=\dot{h}_{\mathrm{in}},c_{\mathrm{out}}=\dot{h}_{\mathrm{out}}},$

which proves the claim that (a part of) the $\hbar=0$ part of the mQME is
equivalent to the infinitesimal Polyakov–Wiegmann formula. We will comment
further on the relationship between Chern–Simons theory on $\Sigma\times I$
and WZW theory on $\Sigma$ in Section 5.3.8 below.

######  Remark 5.21.

In the mQME (82) and the proof above we were using the
$(g,\mathsf{A}^{*}_{\mathrm{res}})$-parametrization of residual fields for the
BV Laplacian. The corresponding statement for the BV Laplacian in
$(g,g^{*})$-parametrization,

$\Delta_{[g,g^{*}]}=\int_{\Sigma}\langle\frac{\delta}{\delta
g},\frac{\delta}{\delta g^{*}}\rangle=\int_{\Sigma}\langle\frac{\delta}{\delta
g\;g^{-1}},\frac{\delta}{g\,\delta g^{*}}\rangle,$

is:

(92)
$(\Omega_{\mathrm{out}}+\Omega_{\mathrm{in}}-\hbar^{2}\Delta_{[g,g^{*}]})\;e^{\frac{i}{\hbar}S^{\mathrm{eff}\,(0)}}=0.$

Note that here we should not be including the $-i\hbar\mathbb{W}$ term in the
effective action, cf. Remark 5.16. The proof of (92) is exactly as before in
the order $O(\hbar^{0})$. In the order $O(\hbar^{1})$, we have

$\Delta_{[g,g^{*}]}S^{\mathrm{eff}\,(0)}=\sum_{z\in\Sigma}\frac{1}{2}\mathrm{div}_{T^{*}[-1]G}\big{\\{}\langle
g^{*},c_{\mathrm{out}}g-gc_{\mathrm{in}}\rangle,\bullet\big{\\}}.$

The hamiltonian vector field generated by the ghost term in the effective
action is the cotangent lift to $T^{*}[-1]G$ of the vector field

$X=\langle c_{\mathrm{out}},\frac{\partial}{\partial g\;g^{-1}}\rangle-\langle
c_{\mathrm{in}},\frac{\partial}{g^{-1}\partial g}\rangle.$

— This is a sum of a right-invariant and a left-invariant vector field on $G$.
Since the Haar measure is bi-invariant, $X$ has divergence zero. Therefore,

$\Delta_{[g,g^{*}]}S^{\mathrm{eff}\,(0)}=\sum_{z\in\Sigma}\mathrm{div}_{G}X=0.$

Ultimately, to avoid the ambiguity as to whether we should be including the
term $-i\hbar\mathbb{W}$ into the partition function or not, we can use the
invariant formulation where the partition function is a _half-density_ (rather
than a function) on residual fields and the BV Laplacian is the canonical BV
Laplacian on half-densities. Then the mQME is

$(\Omega_{\mathrm{out}}+\Omega_{\mathrm{in}}-\hbar^{2}\Delta_{\mathrm{res}}^{\mathrm{can}})\,Z^{\mathrm{can}}=0,$

where
$Z^{\mathrm{can}}=e^{\frac{i}{\hbar}S^{\mathrm{eff}}_{[x,\xi]}}d^{\frac{1}{2}}xD^{\frac{1}{2}}\xi$.
Here $(x,\xi)$ can be any Darboux coordinate system on $\mathcal{V}$, e.g.,
$(\sigma,\mathsf{A}^{*}_{\mathrm{res}})$ or $(g,g^{*})$.

Summary. Summarizing the main results of Section 5.3, we have the following:

  * •

The canonical partition function of the nonabelian theory on the cylinder
$[0,1]\times\Sigma$ with the “parallel ghost” polarization is:
$Z^{\mathrm{can}}=e^{\frac{i}{\hbar}S^{\mathrm{eff}}_{[g,g^{*}]}}\mathcal{D}^{\frac{1}{2}}g\,\mathcal{D}^{\frac{1}{2}}g^{*}$
where the effective action relative to the coordinate system $(g,g^{*})$ on
$\mathcal{V}$ is given explicitly by

(93) $S^{\mathrm{eff}}_{[g,g^{*}]}=\\\
=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\partial
g\rangle\Big{)}+\mathrm{WZW}(g)\\\ +\int_{\Sigma}-\langle
c_{\mathrm{out}},g\,g^{*}\rangle+\langle c_{\mathrm{in}},g^{*}g\rangle.$

In particular, there are no quantum corrections in
$S^{\mathrm{eff}}_{[g,g^{*}]}$.

  * •

$Z^{\mathrm{can}}$ satisfies the modified quantum master equation.

  * •

The restriction of $S^{\mathrm{eff}}_{[g,g^{*}]}$ to ghost number zero fields
is the Hamilton–Jacobi action, i.e., is the generalized generating function
for the evolution relation of the classical theory obtained by evaluating the
classical action on a solution of the evolution equations, see Section 2.

######  Remark 5.22.

The relation between 3D nonabelian Chern–Simons theory and the (gauged) WZW
model was studied from different angles in the literature. The closest
discussion to ours, perhaps, was in [11]: $G/G$ WZW theory was recovered from
Chern–Simons on a cylinder, using essentially the same gauge fixing and
polarization as the ones we employ. But there are crucial differences in the
two approaches. We have an explicit Feynman diagram computation of the
partition function and prove the QME and the gauge invariance property at the
quantum level. In [11], on the other hand, quantum gauge invariance was
assumed and was used to evaluate the Chern–Simons partition function.

####  5.3.7. “Vertical” Wilson lines

One can enrich Chern–Simons theory with Wilson line observables given
classicaly by the parallel transport of the connection field $A$ along a curve
$\gamma$ ending on the boundary; the parallel transport is evaluated in some
linear representation $\rho$ of $G$ on a vector space $R$.

Let us consider a very simple case: several “vertical” Wilson lines with
$\gamma_{j}=I\times\\{z_{j}\\}$ connecting the in- and out-boundaries of the
cylinder $I\times\Sigma$; here $z_{j}$ are a collection of points on $\Sigma$,
$j=1,\ldots,n$. We are fixing representations $\rho_{j}$ for the Wilson lines,
with $R_{j}$ the respective representation spaces.

Figure 7. Vertical Wilson lines.

Note that for our choice of gauge fixing, we have for the Wilson lines

$W_{j}=\rho_{j}(P\overleftarrow{\exp}(-\int_{\gamma_{j}}A))=\rho_{j}(e^{-\sigma(z_{j})})=\rho_{j}(g(z_{j}))\qquad\in\mathrm{End}(R_{j}).$

I.e., vertical Wilson lines depend only on the residual fields.

Thus, the partition function of the theory enriched with vertical Wilson lines
is:

(94)
$Z_{I\times\Sigma,\\{\gamma_{j}\\}}=Z_{I\times\Sigma}\cdot\bigotimes_{j}\rho_{j}(g(z_{j}))$

with $Z_{I\times\Sigma}=e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$ the partition
function without the Wilson lines.

The space of out-states is given by functionals of
$\mathsf{A}^{1,0}_{\mathrm{out}},c_{\mathrm{out}}$ with values in
$\bigotimes_{j}R_{j}$, while the space of in-states is given by functionals of
$\mathsf{A}^{0,1}_{\mathrm{in}},c_{\mathrm{in}}$ also with values in
$\bigotimes_{j}R_{j}^{*}$.464646Our convention is that the partition function
is not a homomorphism from in-states to out-states (depending on residual
fields), but an element of
$\mathcal{H}_{\mathrm{out}}\widehat{\otimes}\mathcal{H}_{\mathrm{in}}$, i.e.,
without dualization of the in-factor. In this description, gluing of two
cylinders involves a pairing between the out-states of the first cylinder
$\mathcal{H}_{\mathrm{out}}^{I}$ and the in-states
$\mathcal{H}_{\mathrm{in}}^{II}$ of the second.  The BFV operators are:474747
See [1] for a construction of these BFV operators from the presentation of
Wilson lines via a path integral over auxiliary fields supported on
$\gamma_{j}$ (the Alekseev–Faddeev–Shatashvili formula).

(95)
$\Omega_{\mathrm{out}}^{\Sigma,\\{z_{j}\\}}=\Omega_{\mathrm{out}}^{\Sigma}+i\hbar\sum_{j}\rho_{j}(c_{\mathrm{out}}(z_{j}))\;\;,\quad\Omega_{\mathrm{in}}^{\Sigma,\\{z_{j}\\}}=\Omega_{\mathrm{in}}^{\Sigma}+i\hbar\sum_{j}\rho_{j}^{*}(c_{\mathrm{in}}(z_{j})),$

where $\Omega_{\mathrm{out}}^{\Sigma}$, $\Omega_{\mathrm{in}}^{\Sigma}$ are
the BFV operators for the theory without the Wilson lines, given by (80),
(81); $\rho_{j}^{*}$ is the dual representation to $\rho_{j}$ with
representation space $R_{j}^{*}$.

As a direct consequence of Lemma 5.19, one has that the partition function
with Wilson lines (94) satisfies the modified quantum master equation:

$(\Omega_{\mathrm{out}}^{\Sigma,\\{z_{j}\\}}+\Omega_{\mathrm{in}}^{\Sigma,\\{z_{j}\\}}-\hbar^{2}\Delta_{\mathrm{res}})\,Z_{I\times\Sigma,\\{\gamma_{j}\\}}=0.$

Here we understand that the $\rho_{j}^{*}$ term in $\Omega_{\mathrm{in}}$ acts
on the second factor in $\rho_{j}(g(z_{j}))\in R_{j}\otimes R_{j}^{*}$, while
the $\rho_{j}$ term in $\Omega_{\mathrm{out}}$ acts on the first factor.

####  5.3.8. The CS-WZW correspondence: WZW theory as an effective theory of
Chern–Simons

Equation (93) is evidence of a strong relationship between the Chern–Simons
theory on a manifold with boundary $\Sigma$ and the WZW theory on the Riemann
surface $\Sigma$. This relationship has, of course, already been subject to a
lot of scrutiny after Witten’s seminal article [41]. In the approach of this
paper, this relationship stems from the fact that the gauged WZW action
emerges as an effective action of the Chern–Simons theory, as is clear from
eq. (93). To be precise, the following two theories are equivalent:

  1. i)

The BV-BFV effective theory of Chern–Simons on a $I\times\Sigma$, restricted
to the gauge-fixing lagrangian $\mathcal{L}=\\{g^{*}=0\\}\subset T^{*}[-1]G$.

  2. ii)

The WZW theory with gauged “$G_{\mathrm{in}}\times
G_{\mathrm{out}}$”-symmetry.

This is a very strong statement of equivalence: It means that essentially all
quantities computed from the action functional in gauged WZW theory have an
expression in Chern–Simons theory. We summarize this relationship in Table 1
below.

Object in CS on $I\times\Sigma$ | Object in gauged WZW on $\Sigma$
---|---
Effective action $S^{\mathrm{eff}}_{[g,g*]}$ | Gauged WZW action $\mathbb{I}(\mathsf{A}^{0,1}_{\mathrm{in}},\mathsf{A}^{1,0}_{\mathrm{out}};g)$
mQME $(\Omega-\Delta_{\mathrm{res}})Z=0$ |  Polyakov–Wiegmann formula (91) (group 1-cocycle property)
Expectation value $W$ of Wilson line $\gamma=I\times\\{z\\}$ in rep. $\rho$ | Field insertion $\rho(g(z))$
Table 1. The CS-WZW correspondence

######  Remark 5.23.

One might wonder why in Table 1 on the left hand side we have objects defined
in the quantization on the Chern–Simons theory, while on the right-hand side
we have entirely classical objects in the WZW model. This apparent puzzle is
resolved by the observation that on the left-hand side we are seeing only the
semiclassical limit of the quantum Chern–Simons theory (which in this case
happens to be exact, since there are no loop contributions).484848Up to the
subtleties concerning ghost loops discussed above.

######  Remark 5.24 (Nonequivalent gauge-fixing lagrangians).

Instead of $\mathcal{L}=\\{g^{*}=0\\}$, one could restrict the effective
Chern–Simons action (93) also to another lagrangian
$\mathcal{L}^{\prime}\subset T^{*}[-1]G$ given by $g=1$. In that way, one
obtains

(96)
$S^{\mathrm{eff}}_{[g,g^{*}]}\bigg{|}_{g=1}=\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}}\rangle+\int_{\Sigma}\langle
g^{*},c_{\mathrm{in}}-c_{\mathrm{out}}\rangle.$

Upon integrating $Z$ over $\mathcal{L}$ or $\mathcal{L^{\prime}}$, one obtains
two $(\Omega_{\mathrm{in}}+\Omega_{\mathrm{out}})$-cocycles $Z_{1}$, $Z_{2}$.
$Z_{1}$ is concentrated in ghost degree 0 (we will discuss it in more detail
in the next subsection) while

$Z_{2}=\exp\left(\frac{i}{\hbar}\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}}\rangle\right)\delta(c_{\mathrm{in}}-c_{\mathrm{out}})$

has nonzero ghost number (formally, it is infinite,
$\mathrm{gh}=\dim\Omega^{0}(\Sigma,\mathfrak{g})$). Therefore, $\mathcal{L}$
and $\mathcal{L}^{\prime}$ provide an example of nonequivalent gauge-fixing
lagrangians.

In the $(g,g^{*})$-coordinates we can define a particularly simple gauge-
fixing lagrangian $\mathcal{L}$ given by $g^{*}=0$ (for the general remarks in
this section, we will allow ourselves to ignore issues arising from possible
zero modes).

(97)
$S^{\mathrm{eff}}\big{|}_{\mathcal{L}}=\mathrm{WZW}(g)+\langle\mathsf{A}^{1,0}_{\mathrm{out}},g\mathsf{A}^{0,1}_{\mathrm{in}}g^{-1}\rangle-\langle\mathsf{A}^{1,0}_{\mathrm{out}},\bar{\partial}g\cdot
g^{-1}\rangle-\langle\mathsf{A}^{0,1}_{\mathrm{in}},g^{-1}\partial
g\rangle\equiv\mathbb{I}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}};g].$

Here
$\mathbb{I}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}};g]$
is the standard way of gauging the WZW action, see, e.g., eq. (4.5) in [23,
§4.2]. We can then express the Chern–Simons partition function on
$I\times\Sigma$ as

(98)
$Z_{I\times\Sigma}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}},c_{\mathrm{in}},c_{\mathrm{out}}]\equiv
Z_{I\times\Sigma}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}}]=\int_{g}\exp\frac{i}{\hbar}\mathbb{I}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}};g]\mathcal{D}g$

(notice the partition function does not depend on
$c_{\mathrm{in}},c_{\mathrm{out}}$). This is the definition of the partition
function $Z_{\mathsf{A}}^{\mathrm{WZW}}$ of gauged WZW, see, e.g., eq. (4.7)
in [23, §4.2]. Here we abbreviate
$\mathsf{A}=(\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}})$.
Similarly, we see that a correlator in the gauged WZW theory can be expressed
as the partition function of Chern–Simons theory enriched with Wilson lines:

(99)
$\langle\rho_{1}(g(z_{1}))\otimes\ldots\otimes\rho_{n}(g(z_{n}))\rangle_{\mathsf{A}}\\\
=\int_{g}\rho_{1}(g(z_{1}))\otimes\ldots\otimes\rho_{n}(g(z_{n}))e^{\frac{i}{\hbar}\mathbb{I}[g,\mathsf{A}]}\mathcal{D}g=\int_{g}Z_{I\times\Sigma,\\{\gamma_{j}\\}}\mathcal{D}g.$

For the purposes of this subsection, we will treat the path integral
expressions on the right hand side of (98) and (99) heuristically. In the
literature, these objects are typically defined via representation theory. In
this paper, we are typically interpreting path integral expressions as defined
via Feynman graphs and rules, but for WZW the absence of a natural linear
structure on the target (the group $G$) obstructs the treatment of the path
integral as a perturbed Gaussian integral. We will therefore simply assume
that the partition function exists and defines an element in
$\Omega$-cohomology. In [1] it was shown that in genus 0, the
$\Omega$-cohomology with $n$ Wilson lines ending on the boundary can be
identified with the $n$-point space of conformal blocks. We expect this to
hold for all genera, and assume it for the purpose of the next section. We
summarize the content of the CS-WZW correspondence after integrating over
$\mathcal{L}$ in Table 2 below.

Object in CS on $I\times\Sigma$ | Object in gauged WZW on $\Sigma$
---|---
CS partition function $Z_{I\times\Sigma}[\mathsf{A}^{1,0}_{\mathrm{out}},\mathsf{A}^{0,1}_{\mathrm{in}}]$ | Gauged WZW partition function $Z_{\mathsf{A}}^{\mathrm{WZW}}$
Expectation value $W$ of $n$ Wilson lines $\gamma=I\times\\{z_{i}\\}$ in rep. $\rho_{1},\ldots\rho_{n}$ |  Gauged WZW correlator $\langle\rho_{1}(g(z_{1}))\cdots\rho_{n}(g(z_{n}))\rangle_{\mathsf{A}}$
$\Omega$-cohomology with $n$ Wilson lines |  $n$-point space of conformal blocks
Table 2. The CS-WZW correspondence after integrating over the gauge-fixing
lagrangian $\mathcal{L}$.

####  5.3.9. An application: Holomorphic factorization of the WZW theory

We will now discuss an application of the correspondence observed above. The
arguments in this section will be more heuristic in nature.

Suppose we fix the boundary condition on one side, e.g., fix the
antiholomorphic boundary condition at the $\mathrm{in}$-boundary by setting
$\mathsf{A}^{0,1}_{\mathrm{in}}=0$ (remember that in our treatment of boundary
conditions, setting $\mathsf{A}^{0,1}_{\mathrm{in}}=0$ means that
$\mathsf{A}\big{|}_{\Sigma_{\mathrm{in}}}\in\Omega^{1,0}(\Sigma,\mathfrak{g})$
), and take the $\mathrm{out}$-boundary in the
$\mathsf{A}^{1,0}_{\mathrm{out}}$-representation. We will call such a cylinder
a “chiral cylinder.”

$\Sigma_{\mathrm{in}}$$\mathsf{A}^{0,1}_{\mathrm{in}}=0$$\Sigma_{\mathrm{out}}$$\mathsf{A}^{1,0}_{\mathrm{out}}$
Figure 8. A chiral cylinder: antiholomorphic boundary conditions on
$\Sigma_{\mathrm{in}}$, $\mathsf{A}^{1,0}_{\mathrm{out}}$-representation on
$\Sigma_{\mathrm{out}}$. Gray indicates that we fix a boundary condition on
this boundary, while hatching indicates we fix only the polarization.

After integrating out $g$, we obtain the partition function
$\psi(\mathsf{A}_{\mathrm{out}}^{1,0})$ of a “chiral gauged WZW theory,” i.e.,
a WZW theory coupled to a chiral gauge field, see, e.g., [42].494949Where
Witten suggests that “this in fact can be regarded as the essential relation
between the WZW model and Chern–Simons theory.” This partition function is not
a number, but rather a holomorphic gauge invariant section of a line bundle
over the space of connections on $\Sigma$.505050In our approach,
holomorphicity simply follows from the fact that it depends only on
$\mathsf{A}^{1,0}_{\mathrm{out}}$, while gauge invariance is the statement
that $\Omega_{\mathrm{out}}\psi=0$. We can glue the chiral cylinder to another
“antichiral” cylinder with opposite boundary conditions (see Figure 9).

*= Figure 9. Gluing a chiral and an antichiral cylinder into a cylinder with opposite chiral boundary conditions. Gray indicates a fixed boundary condition, hatching indicates a polarized boundary.

In this way, we obtain – as explained in [42] – the square of the norm of
$\psi$.515151A missing factor of $\int_{\Sigma}A^{1,0}A^{0,1}$ in comparison
with [42] comes from the gluing conventions in BV-BFV, see Appendix A. Here,
the “norm square” should be taken with respect to a well-defined inner product
on the $\Omega$-cohomology, i.e., the finite-dimensional moduli space of
gauge-invariant holomorphic sections.525252See, for instance, [25] for a more
detailed discussion in genus 0, or [19] for a discussion in genus 1. The
authors do not know of an explicit construction of this inner product in
higher genera. On the other hand, from the general principles of the BV-BFV
formalism, we will then obtain the partition function of Chern–Simons theory
with opposite chiral boundary conditions, which is given by specializing to
$\mathsf{A}^{1,0}_{\mathrm{out}}=\mathsf{A}^{1,0}_{\mathrm{in}}=0$ in (98):

(100) $|\psi|^{2}=Z_{I\times\Sigma}^{\mathrm{CS}}=Z_{\Sigma}^{\mathrm{WZW}}.$

Here on the left-hand side we have the norm-square of the partition function
of chiral WZW, in the middle we have Chern–Simons partition function on the
cylinder with opposite chiral boundary conditions, while on the right-hand
side we have the definition of the WZW partition function. Equation (100) is
sometimes called “holomorphic factorization of the WZW model”, because one
sees that the partition function of the full WZW model — which does not vary
holomorphically on the moduli space of conformal structures — splits into a
sum of products of holomorphic and antiholomorphic factors, which do depend
(anti)holomorphically on the complex structure. Thus, holomorphic
factorization of the WZW model follows from the self-similarity of the
Chern–Simons partition function on cylinders.
Using the results of Section 5.3.7, in particular equation (94), these results
for the partition function generalize to correlators in chiral and full WZW.
Namely, suppose we are given $n$ Wilson lines colored by representations
$R_{1},\ldots,R_{n}$, and let $V=\otimes_{j}R_{j}$. Then, the Chern–Simons
partition function with Wilson lines on a chiral cylinder
$\psi_{\\{\gamma_{j}\\}}$ is naturally a degree zero element of
$V\otimes\mathcal{H}_{\mathrm{out}}$ with $\mathcal{H}_{\mathrm{out}}$ the
space of functionals of $\mathsf{A}^{1,0}_{\mathrm{out}},c_{\mathrm{out}}$
with values in $V^{*}$. Gluing with an antichiral cylinder, we obtain the
Chern–Simons partition function with Wilson lines and opposite boundary
conditions - the correlator
$\langle\rho_{1}(g(z_{1}))\cdots\rho_{n}(g(z_{n}))\rangle\in V\otimes V^{*}$
in a full WZW model. On the other hand, explicitly computing the BV-BFV gluing
we obtain535353 The gluing is defined as a formal functional integral, but its
restriction to the finite-dimensional $\Omega$-cohomology gives rise to a
well-defined inner product. $(\psi,\overline{\psi})\in V\otimes V^{*}$. Here
$(\cdot,\cdot)$ is the inner product on the space on $n$-point conformal
blocks. Thus, we obtain the generalization of (100) to the case with Wilson
lines:

(101)
$(\psi_{\\{\gamma_{j}\\}},\overline{\psi}_{\\{\gamma_{j}\\}})=Z^{CS}_{I\times\Sigma,\\{\gamma_{j}\\}}=\langle\rho_{1}(g(z_{1}))\cdots\rho_{n}(g(z_{n}))\rangle.$

###  5.4. 3D nonabelian Chern–Simons theory in holomorphic-to-holomorphic
polarization

Next, consider the nonabelian Chern–Simons theory on $\Sigma\times[0,1]$ with
polarizations as in Section 4.1. The residual fields

$\mathsf{A}^{+}_{I\,\mathrm{res}}=\mathsf{A}^{0}_{{I\,\mathrm{res}}}+\mathsf{A}^{1,0}_{{I\,\mathrm{res}}},\qquad\mathsf{A}^{-}_{\mathrm{res}}=\mathsf{A}^{0,1}_{\mathrm{res}}+\mathsf{A}^{2}_{\mathrm{res}}$

and the gauge fixing are as in Section 4.1 (but now all forms are
$\mathcal{G}$-valued). We will use the notations
$\sigma=\mathsf{A}^{0}_{I\,\mathrm{res}}$,
$\lambda=\mathsf{A}^{0,1}_{\mathrm{res}}$ for $\mathrm{gh}=0$ residual fields,
as in Section 4.1.1.

In this subsection we will present only the results; the computations are
similar to those of Section 5.3.

The Feynman diagrams for the partition function
$Z=e^{\frac{i}{\hbar}S^{\mathrm{eff}}}$ are:

Figure 10. Feynman diagrams in nonabelian theory on a cylinder in holomorphic-
to-holomorphic polarization.

Here the “physical wheels” (viii) and the “ghost wheels” (ix) cancel each
other, due to the form of propagators in the chosen polarization.

Calculating the Feynman diagrams, one finds the following expression for the
effective action:

(102)
$S^{\mathrm{eff}}=S^{\mathrm{eff}}_{\mathrm{ph}}+S^{\mathrm{eff}}_{\mathrm{gh}},$

where the part depending only on “physical” ($\mathrm{gh}=0$) fields (the
contribution of diagrams (i), (ii), (iii)) is

(103)
$S^{\mathrm{eff}}_{\mathrm{ph}}=\int_{\Sigma}\langle\lambda,\partial\sigma\rangle+\left\langle\mathsf{A}^{1,0}_{\mathrm{out}},\frac{\mathrm{ad}_{\sigma}}{e^{\mathrm{ad}_{\sigma}}-1}\circ\lambda+\left(-\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}+\frac{1}{\mathrm{ad}_{\sigma}}\right)\circ\bar{\partial}\sigma\right\rangle\\\
+\left\langle\mathsf{A}^{1,0}_{\mathrm{in}},-\frac{\mathrm{ad}_{\sigma}}{1-e^{-\mathrm{ad}_{\sigma}}}\circ\lambda+\left(\frac{1}{1-e^{-\mathrm{ad}_{\sigma}}}-\frac{1}{\mathrm{ad}_{\sigma}}\right)\circ\bar{\partial}\sigma\right\rangle$

and the ghost-dependent part (the contribution of diagrams (iv)–(vii)) is

(104) $S^{\mathrm{eff}}_{\mathrm{gh}}=\int_{\Sigma}\left\langle
c_{\mathrm{out}},\sum_{k\geq
0}\frac{B_{k}^{-}}{k!}(\mathrm{ad}_{\sigma})^{k}\mathsf{A}^{2}_{\mathrm{res}}-\sum_{k\geq
0}\frac{B^{-}_{k+1}}{(k+1)!}(\mathrm{ad}_{\sigma})^{k}\bar{\partial}\mathsf{A}^{1,0}_{I\,\mathrm{res}}\right.\\\
\left.+\sum_{k,l\geq
0}\frac{B^{-}_{k+l+1}}{(k+l+1)!}(\mathrm{ad}_{\sigma})^{k}\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}(\mathrm{ad}_{\sigma})^{l}\lambda-\sum_{k,l\geq
0}\frac{B^{-}_{k+l+2}}{(k+l+2)!}(\mathrm{ad}_{\sigma})^{k}\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}(\mathrm{ad}_{\sigma})^{l}\bar{\partial}\sigma\right\rangle\\\
+\left\langle c_{\mathrm{in}},-\sum_{k\geq
0}\frac{B_{k}^{+}}{k!}(\mathrm{ad}_{\sigma})^{k}\mathsf{A}^{2}_{\mathrm{res}}+\sum_{k\geq
0}\frac{B^{+}_{k+1}}{(k+1)!}(\mathrm{ad}_{\sigma})^{k}\bar{\partial}\mathsf{A}^{1,0}_{I\,\mathrm{res}}\right.\\\
\left.-\sum_{k,l\geq
0}\frac{B^{+}_{k+l+1}}{(k+l+1)!}(\mathrm{ad}_{\sigma})^{k}\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}(\mathrm{ad}_{\sigma})^{l}\lambda+\sum_{k,l\geq
0}\frac{B^{+}_{k+l+2}}{(k+l+2)!}(\mathrm{ad}_{\sigma})^{k}\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}(\mathrm{ad}_{\sigma})^{l}\bar{\partial}\sigma\right\rangle.$

Here $B_{n}^{\pm}$ are the Bernoulli numbers with $B^{\pm}_{1}=\pm\frac{1}{2}$
and with $B^{+}_{n}=B^{-}_{n}$ the usual Bernoulli numbers for $n\neq 1$
(thus, $B^{-}_{n}=B_{n}$ are the standard Bernoulli numbers for
$n=0,1,2,\ldots$):

$n$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\cdots$
---|---|---|---|---|---|---|---|---
$B^{+}_{n}$ | $1$ | $+\frac{1}{2}$ | $\frac{1}{6}$ | $0$ | -$\frac{1}{30}$ | $0$ | $\frac{1}{42}$ | $\cdots$
$B^{-}_{n}$ | $1$ | $-\frac{1}{2}$ | $\frac{1}{6}$ | $0$ | -$\frac{1}{30}$ | $0$ | $\frac{1}{42}$ | $\cdots$

######  Remark 5.25.

Another form of the ghost-dependent part of the effective action (104), with
sums over $k,l$ below evaluated explicitly, is:

$S^{\mathrm{eff}}_{\mathrm{gh}}=\int_{\Sigma}\Big{\langle}c_{\mathrm{out}},\frac{\mathrm{ad}_{\sigma}}{e^{\mathrm{ad}_{\sigma}}-1}\circ\mathsf{A}^{2}_{\mathrm{res}}+\Big{(}-\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}+\frac{1}{\mathrm{ad}_{\sigma}}\Big{)}\circ\bar{\partial}\mathsf{A}^{1,0}_{I\,\mathrm{res}}\\\
+\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}\circ\lambda-\frac{\mathrm{ad}_{\sigma}}{1-e^{-\mathrm{ad}_{\sigma}}}\mathrm{ad}\left(\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{1,0}_{I\,\mathrm{res}}\right)\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}\circ\lambda\\\
-\frac{1}{\mathrm{ad}_{\sigma}}\mathrm{ad}_{\mathsf{A}^{1,0}_{I\,\mathrm{res}}}\frac{1}{\mathrm{ad}_{\sigma}}\bar{\partial}\sigma+\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}\mathrm{ad}\left(\frac{e^{\mathrm{ad}_{\sigma}}-1}{\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{1,0}_{I\,\mathrm{res}}\right)\frac{1}{1-e^{-\mathrm{ad}_{\sigma}}}\circ\bar{\partial}\sigma\Big{\rangle}\\\
+\Big{\langle}c_{\mathrm{in}},\Big{(}\sigma\rightarrow-\sigma,\mathsf{A}^{2}_{\mathrm{res}}\rightarrow-\mathsf{A}^{2}_{\mathrm{res}}\Big{)}\Big{\rangle}.$

Here the coefficient of $c_{\mathrm{in}}$ is obtained from the coefficient of
$c_{\mathrm{out}}$ by replacing $\sigma$ with $-\sigma$ and replacing
$\mathsf{A}^{2}_{\mathrm{res}}$ with $-\mathsf{A}^{2}_{\mathrm{res}}$.

Next, one can introduce a new parametrization of the space of residual fields
by a group-valued map $g\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow
G$, a $(0,1)$-form $\Lambda$, a $(1,0)$-form $\Lambda^{*}$ and a $2$-form
$g^{*}$:545454$\Lambda,\Lambda^{*}$ are $\mathcal{G}$-valued forms while
$g^{*}$ is a $\mathcal{G}$-valued form translated by $g^{-1}$. The ghost
number is $0$ for $g,\Lambda$ and is $-1$ for $g^{*},\Lambda^{*}$.

$\displaystyle g$ $\displaystyle=e^{-\sigma},$ $\displaystyle\Lambda$
$\displaystyle=\mbox{coefficient of $\mathsf{A}^{1,0}_{\mathrm{out}}$ in
(\ref{hol-hol non-ab S_eff phys part})}$
$\displaystyle=\frac{\mathrm{ad}_{\sigma}}{e^{\mathrm{ad}_{\sigma}}-1}\circ\lambda+\left(-\frac{1}{e^{\mathrm{ad}_{\sigma}}-1}+\frac{1}{\mathrm{ad}_{\sigma}}\right)\circ\bar{\partial}\sigma,$
$\displaystyle\Lambda^{*}$
$\displaystyle=\frac{1-e^{-\mathrm{ad}_{\sigma}}}{\mathrm{ad}_{\sigma}}\circ\mathsf{A}^{1,0}_{I\,\mathrm{res}},$
$\displaystyle g^{*}$ $\displaystyle=(\mbox{coefficient of $c_{\mathrm{in}}$
in (\ref{hol-hol non-ab S_eff ghost part}) })\cdot g^{-1}.$

This change of parametrization has Jacobian $1$ and changes one Darboux
coordinate system with respect to the BV symplectic form on $\mathcal{V}$ into
another one:

$\omega_{\mathrm{res}}=\int_{\Sigma}\langle\delta\sigma,\delta\mathsf{A}^{2}_{\mathrm{res}}\rangle+\langle\delta\lambda,\delta\mathsf{A}^{1,0}_{I\,\mathrm{res}}\rangle=\int_{\Sigma}\langle\delta
g,\delta g^{*}\rangle+\langle\delta\Lambda,\delta\Lambda^{*}\rangle.$

In terms of this new parametrization, the effective action (102) can be
written more concisely:

(105)
$S^{\mathrm{eff}}=-\mathrm{WZW}(g^{-1})-\int_{\Sigma}\langle\Lambda,\partial
g\cdot g^{-1}\rangle\\\
+\int_{\Sigma}\langle\mathsf{A}^{1,0}_{\mathrm{out}},\Lambda\rangle+\langle\mathsf{A}^{1,0}_{\mathrm{in}},g^{-1}\Lambda
g+g^{-1}\bar{\partial}g\rangle+\langle
c_{\mathrm{out}},-g\,g^{*}+\bar{\partial}\Lambda^{*}+[\Lambda,\Lambda^{*}]\rangle+\langle
c_{\mathrm{in}},g^{*}g\rangle,$

where $\mathrm{WZW}$ is the Wess–Zumino–Witten action defined as in (64).

The effective action (105) satisfies the following properties:

  * •

Its restriction to $\mathrm{gh}=0$ fields satisfies the Hamilton–Jacobi
property, i.e., it is the generalized generating function for the evolution
relation of nonabelian Chern–Simons theory. From this identification one can
see that, on-shell, $\Lambda$ can be interpreted as the $(0,1)$-component of
the connection field at $t=1$.555555There is, of course, a similar change of
variables where instead we take the $(0,1)$-component at $t=0$ to be a
coordinate on $\mathcal{V}$. It leads to an expression for $S^{\mathrm{eff}}$
where the symmetry between in/out boundaries is broken in the opposite way to
(105): the coefficients of $\mathsf{A}^{1,0}_{\mathrm{in}}$,
$c_{\mathrm{out}}$ are simple and the coefficients of
$\mathsf{A}^{1,0}_{\mathrm{out}}$, $c_{\mathrm{in}}$ are more complicated.

  * •

One has the modified quantum master equation

$(\Omega_{\mathrm{out}}+\Omega_{\mathrm{in}}-\hbar^{2}\Delta_{\mathrm{res}})e^{\frac{i}{\hbar}S^{\mathrm{eff}}}=0$

with the boundary BFV operators

$\Omega_{\mathrm{out}/\mathrm{in}}=\int_{\Sigma}\big{\langle}c,\pm\bar{\partial}\mathsf{A}^{1,0}-i\hbar(\partial+[\mathsf{A}^{1,0},-])\frac{\delta}{\delta\mathsf{A}^{1,0}}\big{\rangle}-i\hbar\big{\langle}\frac{1}{2}[c,c],\frac{\delta}{\delta
c}\big{\rangle}.$

Here the sign $\pm$ is $+$ for out-boundary and $-$ for in-boundary; we also
suppressed the $\mathrm{out}/\mathrm{in}$ subscript in the boundary fields
$\mathsf{A}^{1,0}$ and $c$.

##  6\. BV-BFV approach to higher-dimensional Chern–Simons theories

The observations on abelian Chern–Simons theory in Sections 3 and 4 generalize
readily to cylinders $I\times M$ of other dimensions $d$. Observe that $d$
must be odd because we want the field $\mathcal{A}$ to belong to the
superspace $\Omega^{\bullet}(I\times M)$ or $\Pi\Omega^{\bullet}(I\times M)$
and, in either case, the BV action $S=\int_{I\times
M}\frac{1}{2}\mathcal{A}\wedge d\mathcal{A}$ is even iff $d$ is odd.

In the following, we will actually focus on the graded case where the field
$\mathcal{A}$ belongs to the graded space $\Omega^{\bullet}(I\times M)[k]$ for
some integer $k$ and the BV action has degree zero. This forces $d=2k+1$. If
$k$ were even, we would have $\mathcal{A}\wedge
d\mathcal{A}=\frac{1}{2}d\mathcal{A}^{2}$, so the BV action would have no bulk
contribution. Therefore, we will have to assume that $k$ is odd. To
summarize:565656There are other ways to get at this results. For example, we
may consider the classical theory with action $S_{\text{cl}}=\int_{I\times
M}\frac{1}{2}AdA$ with $A$ a $k$-form for some $k$. This immediately forces
$d=2k+1$. For $k$ even, we have $AdA=\frac{1}{2}dA^{2}$, so the classical
action has no bulk term. We then have to assume $k$ odd. Another option is to
consider the AKSZ construction with target $\mathbb{R}[k]$, for some $k$,
endowed with a symplectic form. If we denote by $x$ the coordinate, the
general $2$-form is $\omega=f(x)dxdx$ with $f$ a function. If $k$ is even,
$dxdx=0$ and $\omega$ is degenerate, so we have to assume $k$ odd. In this
case $f(x)=a+bx$ for some real numbers $a$ and $b$. Now $\omega$ is closed iff
$b=0$ and nondegenerate iff $a\not=0$ (we may, e.g., take $a=\frac{1}{2}$). We
then have that $\omega$ has degree $2k$. Since we want to produce a BV form
(i.e., degree equal to $-1$) on $I\times M$ by the AKSZ construction, we need
$d=2k+1$.

$d=\dim(I\times M)=2k+1,\quad k=2l+1.$

The case $k=1$ has been considered in Section 4. We will briefly describe the
general case before turning to the next example of interest, $k=3$.

Next we assume that the $2k$-dimensional manifold $M$ is closed and oriented.
Again, we can construct a BV-BFV theory by the AKSZ construction as

$\mathcal{F}=\mathrm{Map}(T[1](I\times
M),\mathbb{R}[k])=\Omega^{\bullet}(I\times M)[k]$

and rewrite this space of fields in the form

$\mathcal{F}=\Omega^{\bullet}(I,\Omega^{\bullet}(M)[k]),$

exhibiting the theory as a 1-dimensional Chern–Simons theory with coefficients
in $\mathfrak{g}=\Omega^{\bullet}(M)$. The BV action is then, mimicking (18),

$S=\int_{I\times M}\frac{1}{2}\mathcal{A}\wedge
d\mathcal{A}=\int_{I}\frac{1}{2}(\mathcal{A},d_{I}\mathcal{A})+\frac{1}{2}(\mathcal{A},d_{M}\mathcal{A}),$

where $d=d_{I}+d_{M}$ and $(a,b)=\int_{M}a\wedge b$. Again, the field
$\mathcal{A}$ can be split as $\mathcal{A}=\mathsf{A}+dt\cdot\mathsf{A}_{I}$
and the boundary phase space is
$\mathcal{F}^{\partial}_{M}=\Omega^{\bullet}(M)[k]$ with Noether 1-form

$\alpha=\frac{1}{2}\int_{\\{1\\}\times
M}\mathsf{A}\wedge\delta\mathsf{A}-\frac{1}{2}\int_{\\{0\\}\times
M}\mathsf{A}\wedge\delta\mathsf{A}.$

Next, assume that $M$ carries a complex structure. Then we can split the space
of complexified $k$-forms as

$\Omega^{k}_{\mathbb{C}}(M)=\bigoplus_{j_{1}+j_{2}=k}\Omega^{j_{1},j_{2}}(M).$

Given that $k$ is odd, the splitting

(106)
$\Omega^{k}_{\mathbb{C}}(M)=\underbrace{\bigoplus_{j=0}^{l}\Omega^{k-j,j}(M)}_{\Omega^{k}_{+}(M)}\oplus\underbrace{\bigoplus_{j=l+1}^{k}\Omega^{k-j,j}(M)}_{\Omega^{k}_{-}(M)}$

provides a splitting into lagrangian subspaces of $\Omega^{k}_{\mathbb{C}}(M)$
(which is the degree 0 subspace of $\mathcal{F}^{\partial}_{M}$). The
splitting (20) generalizes to

$\underbrace{\Omega^{\bullet}_{\mathbb{C}}(M)}_{\mathfrak{g}_{\mathbb{C}}}=\underbrace{\bigoplus_{j=0}^{k-1}\Omega^{j}_{\mathbb{C}}(M)\oplus\Omega^{k}_{+}(M)}_{\mathfrak{g}^{+}_{\mathbb{C}}}\oplus\underbrace{\Omega^{k}_{-}(M)\oplus\bigoplus_{j=k+1}^{d-1}\Omega_{\mathbb{C}}^{j}(M)}_{\mathfrak{g}^{-}_{\mathbb{C}}}.$

Correspondingly, we split the fields $\mathsf{A}$ into its components
$\mathsf{A}=\mathsf{A}^{+}+\mathsf{A}^{-}$, and similarly for
$\mathsf{A}_{I}$. The de Rham differential restricted to the subcomplex
$\Omega^{k-1}(M)\to\Omega^{k}_{+}(M)\oplus\Omega^{k}_{-}(M)\to\Omega^{k+1}(M)$
splits as $d_{M}=d_{M}^{+}+d_{M}^{-}$ as in Section 5.1,575757 In more detail,
for a $(p,q)$-form $\alpha$ with $p+q=k-1$ we have
$d_{M}^{+}\alpha=\begin{cases}d_{M}\alpha,&q<l,\\\ \partial\alpha,&q=l,\\\
0,&q>l.\end{cases}$ and vice versa for $d_{M}^{-}$. see also Figure 11 below.

Figure 11. Splitting of $\Omega^{p,q}$ into $\mathfrak{g}^{+}$ (below the
thick line) and $\mathfrak{g}^{-}$ (above the thick line), in the case $k=3$.
Solid arrows are components of $d^{-}$, dashed arrows are components of
$d^{+}$. Horizontal arrows (dashed or not) are $\partial$, vertical ones are
$\bar{\partial}$.

In particular, we have
$(\mathsf{A}^{+},d_{M}\mathsf{A}^{+})=(\mathsf{A}^{+},d_{M}^{-}\mathsf{A}^{+})$.
Before turning to the particular example of $k=3$, let us briefly have a look
at the general form of the partition function in the two polarizations
considered already in the last section.

###  6.1. Partition functions in $4l+3$-dimensional CS theories

First, let us take $\mathfrak{g}^{+}$ as the base of the polarization on both
ends of the cylinder. This means that

$\mathcal{B}=\mathfrak{g}^{+}[k]\oplus\mathfrak{g}^{+}[k]\ni(\mathsf{A}^{+}_{\rm{in}},\mathsf{A}^{+}_{\rm{out}})$

with fiber

$\mathcal{Y}=\Omega^{\bullet}(I,\partial
I;\mathfrak{g}^{+}[k])\oplus\Omega^{\bullet}(I,\mathfrak{g}^{-}[k]).$

Again, we can gauge fix the polarized theory by choosing

$\mathcal{V}=dt\cdot\mathfrak{g}^{+}[k]\oplus
1\cdot\mathfrak{g}^{-}[k]\ni(dt\cdot\mathsf{A}^{+}_{I,\rm{res}},\mathsf{A}^{-}_{\rm{res}})$

and using Hodge decomposition (7) with chain contraction (8) and corresponding
propagator (9). Thus, we obtain the splitting

$\mathcal{A}=\widetilde{\mathsf{A}}_{\rm{in}}^{+}+\widetilde{\mathsf{A}}_{\rm{out}}^{+}+dt\cdot\mathsf{A}^{+}_{I,\rm{res}}+\mathsf{A}^{-}_{\rm{res}}+\mathsf{A}^{+}_{\rm{fl}}+\mathsf{A}^{-}_{\rm{fl}},$

where
$\widetilde{\mathsf{A}}_{\rm{in}}^{+},\widetilde{\mathsf{A}}_{\rm{out}}^{+}$
are the discontinuous extensions of
$\mathsf{A}^{+}_{\rm{in}},\mathsf{A}^{+}_{\rm{out}}$ into the bulk. In terms
of this splitting, we can rewrite the “perturbation” $\frac{1}{2}\int_{I\times
M}\mathcal{A}\wedge d_{M}\mathcal{A}$ as

$\frac{1}{2}\int_{I\times M}\mathcal{A}\wedge
d_{M}\mathcal{A}=\int_{I}dt\int_{M}\mathsf{A}^{+}_{I,\mathrm{res}}d_{M}(\mathsf{A}^{-}_{\rm{res}}+\mathsf{A}^{+}_{\rm{fl}}+\mathsf{A}^{-}_{\rm{fl}}),$

since $dt\cdot\mathsf{A}^{+}_{I,\mathrm{res}}$ is the only term containing a
$dt$. By definition (see (7)) the $\mathsf{A}^{-}_{\mathrm{fl}}$ fluctuations
have vanishing integral over $I$. Applying the fact that
$d_{M}=d_{M}^{+}+d_{M}^{-}$, we obtain

$\frac{1}{2}\int_{I\times M}\mathcal{A}\wedge
d_{M}\mathcal{A}=\int_{I}dt\left(\int_{M}\mathsf{A}^{+}_{I,\mathrm{res}}d_{M}\mathsf{A}^{-}_{\rm{res}}+\int_{M}\mathsf{A}^{+}_{I,\mathrm{res}}d_{M}^{-}\mathsf{A}^{+}_{\mathrm{fl}}\right).$

The BV-BFV partition function - as in (22) - is then given by

(107)
$Z(\mathsf{A}^{+}_{\rm{in}},\mathsf{A}^{-}_{\rm{out}},\mathsf{A}^{+}_{I,\rm{res}},\mathsf{A}^{-}_{\rm{res}})=\\\
=\int_{\mathcal{Y}^{\prime}_{K-ex}\subset\mathcal{Y}^{\prime}}\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}S^{f}\left(\widetilde{\mathsf{A}^{+}_{\mathrm{in}}}+\widetilde{\mathsf{A}^{+}_{\mathrm{out}}}+\mathsf{A}^{+}_{\mathrm{fl}}+\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}+dt\cdot\mathsf{A}^{+}_{I\,\mathrm{res}}\right)}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;e^{\frac{i}{\hbar}\big{(}\int_{I\times
M}\mathsf{A}^{-}_{\mathrm{fl}}d_{I}\mathsf{A}^{+}_{\mathrm{fl}}+\int_{\\{1\\}\times
M}\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}-\int_{\\{0\\}\times
M}\mathsf{A}^{+}_{\mathrm{in}}\mathsf{A}^{-}+\int_{I\times
M}\frac{1}{2}\mathcal{A}\,d_{M}\mathcal{A}\big{)}}\\\
=\int\mathcal{D}\mathsf{A}^{+}_{\mathrm{fl}}\;\mathcal{D}\mathsf{A}^{-}_{\mathrm{fl}}\;\exp\frac{i}{\hbar}\Big{(}\int_{I\times
M}\mathsf{A}^{-}_{\mathrm{fl}}\,d_{I}\mathsf{A}^{+}_{\mathrm{fl}}+\int_{M}\mathsf{A}^{+}_{\mathrm{out}}\,(\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}\big{|}_{t=1})-\\\
-\int_{M}\mathsf{A}^{+}_{\mathrm{in}}\,(\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{-}_{\mathrm{fl}}\big{|}_{t=0})+\int_{M}\mathsf{A}^{+}_{I\,\mathrm{res}}d_{M}\mathsf{A}^{-}_{\mathrm{res}}+\int_{I\times
M}dt\;\mathsf{A}^{+}_{I\,\mathrm{res}}\,d_{M}^{-}\mathsf{A}^{+}_{\mathrm{fl}}\Big{)}.$

This is structurally the same formula as in Section 4 before, with the
difference that the pairing on residual fields in slightly more complicated in
this case. The Feynman diagrams defining this functional integral are the same
as in (23) and yield

$Z=\exp\frac{i}{\hbar}\int_{M}\Big{(}(\mathsf{A}^{+}_{\mathrm{out}}-\mathsf{A}^{+}_{\mathrm{in}})\;\mathsf{A}^{-}_{\mathrm{res}}+\mathsf{A}^{+}_{I\,\mathrm{res}}\,d_{M}\mathsf{A}^{-}_{\mathrm{res}}+\frac{1}{2}(\mathsf{A}^{+}_{\mathrm{out}}+\mathsf{A}^{+}_{\mathrm{in}})\;d_{M}^{-}\mathsf{A}^{+}_{I\,\mathrm{res}}\Big{)}.$

Similarly, the partition function in the holomorphic-to-antiholomorphic
polarization, with space of boundary conditions

$\mathcal{B}=\mathfrak{g}^{-}[k]\oplus\mathfrak{g}^{+}[k]\ni(\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}})$

and space of residual fields

$\mathcal{V}=dt\cdot\mathfrak{g}_{\mathbb{C}}[k-1]\oplus(1-t)\cdot\mathfrak{g}^{+}[k]\oplus
t\cdot\mathfrak{g}^{-}[k]\;\;\ni\;\;dt\cdot\mathsf{A}_{I\,\mathrm{res}}+(1-t)\cdot\mathsf{A}^{+}_{\mathrm{res}}+t\cdot\mathsf{A}^{-}_{\mathrm{res}},$

is

$\displaystyle Z(\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{+}_{\mathrm{out}}$
$\displaystyle;\mathsf{A}_{I\,\mathrm{res}},\mathsf{A}^{+}_{\mathrm{res}},\mathsf{A}^{-}_{\mathrm{res}})$
$\displaystyle=\exp\frac{i}{\hbar}\Bigg{(}\int_{M}\
-\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}_{\mathrm{in}}+\mathsf{A}^{+}_{\mathrm{out}}\mathsf{A}^{-}_{\mathrm{res}}-\mathsf{A}^{-}_{\mathrm{in}}\mathsf{A}^{+}_{\mathrm{res}}+\frac{1}{2}\mathsf{A}^{-}_{\mathrm{res}}\mathsf{A}^{+}_{\mathrm{res}}+$
$\displaystyle+\frac{1}{2}\int_{M}\mathsf{A}_{I\,\mathrm{res}}^{+}\;(d_{M}^{-}\mathsf{A}^{+}_{\mathrm{res}}+d_{M}\mathsf{A}^{-}_{\mathrm{res}})+\frac{1}{2}\int_{M}\mathsf{A}_{I\,\mathrm{res}}^{-}d_{M}\mathsf{A}_{\mathrm{res}}^{+}\Bigg{)}.$

###  6.2. Parallel ghost polarization

We can also choose again the “parallel ghost” polarization discussed in
Section 5.1. To be more explicit, and in preparation for the next section, let
us fix $k=3$. Then we have $\mathfrak{g}=\Omega^{\bullet}(M,\mathbb{C})$ where
$M$ is a 6-dimensional manifold with a complex structure, and
$X=\mathfrak{g}[3]$.585858We use freely the notation from Section 5.1. For
later purposes, let us suppose that $M$ is endowed with a Kähler metric $g$.
We use the complex structure to define the polarization of the $\mathrm{gh}=0$
component of $X$:

$X^{[0]}=\Omega^{3}_{\mathbb{C}}(M)=\underbrace{\Omega^{3,0}(M)\oplus\Omega^{2,1}(M)}_{X^{+}}\oplus\underbrace{\Omega^{1,2}(M)\oplus\Omega^{0,3}(M)}_{X^{-}}.$

Correspondingly, we split the field $\mathsf{A}=\mathsf{A}^{\leq
2}+\mathsf{A}^{3,+}+\mathsf{A}^{3,-}+\mathsf{A}^{\geq 4}$ and similarly for
$\mathsf{A}_{I}$. We denote by $\mathcal{P}^{[<0],-}$ the polarization given
by

$\mathcal{P}^{[<0],-}=\left\\{\frac{\delta}{\delta\mathsf{A}^{\geq
4}},\frac{\delta}{\delta\mathsf{A}^{3,-}}\right\\},$

whose base is parametrized by $(\mathsf{A}^{\leq 2},\mathsf{A}^{3,+})$, and by
$\mathcal{P}^{[<0],+}$ the similar polarization with $-$ and $+$ exchanged.
The maps $d_{\mathfrak{g}}^{+},d_{\mathfrak{g}}^{-}\colon X^{[1]}\to X^{[0]}$
defined in Section 5.2 are given by projecting the de Rham differential

$d_{M}\colon\Omega^{2}_{\mathbb{C}}(M)\to\Omega^{3}_{\mathbb{C}}(M)$

to $X^{\pm}$. Explicitly, they are given by

$\displaystyle d^{+}_{\mathfrak{g}}\big{|}_{\Omega^{2,0}}$
$\displaystyle=d_{M}\big{|}_{\Omega^{2,0}},\qquad$ $\displaystyle
d^{+}_{\mathfrak{g}}\big{|}_{\Omega^{1,1}}=\partial_{M}\big{|}_{\Omega^{1,1}},\qquad$
$\displaystyle d^{+}_{\mathfrak{g}}\big{|}_{\Omega^{0,2}}=0,$ $\displaystyle
d^{-}_{\mathfrak{g}}\big{|}_{\Omega^{2,0}}$ $\displaystyle=0,\qquad$
$\displaystyle
d^{-}_{\mathfrak{g}}\big{|}_{\Omega^{1,1}}=\bar{\partial}_{M}\big{|}_{\Omega^{1,1}},\qquad$
$\displaystyle
d^{-}_{\mathfrak{g}}\big{|}_{\Omega^{0,2}}=d_{M}\big{|}_{\Omega^{0,2}},$

see Figure 11. Now, we consider the cylinder $I\times M$ with with
$\mathcal{P}^{[<0],+}$ on the in-boundary and $\mathcal{P}^{[<0],-}$ on the
out-boundary. We then have the following fields in the effective action
(referring to notation from Section 5.1):

  * •

$\psi_{\mathrm{out}}^{+}=\mathsf{A}^{3,+}_{\mathrm{out}}=\mathsf{A}^{3,0}_{\mathrm{out}}+\mathsf{A}^{2,1}_{\mathrm{out}}$
— physical boundary field on out-boundary,

  * •

$\psi^{-}_{\mathrm{in}}=\mathsf{A}^{3,-}_{\mathrm{in}}=\mathsf{A}^{0,3}_{\mathrm{in}}+\mathsf{A}^{1,2}_{\mathrm{in}}$
— physical boundary field on in-boundary,

  * •

$(\psi^{[>0]}_{\mathrm{in}},\psi^{[>0]}_{\mathrm{out}})=(\mathsf{A}_{\mathrm{in}}^{\leq
2},\mathsf{A}_{\mathrm{out}}^{\leq 2})$ — boundary fields in higher ghost
number (collected in a superfield),

  * •

$A^{[1]}_{\mathrm{res}}=\mathsf{A}^{2}_{I\,\mathrm{res}}=\mathsf{A}_{I\,\mathrm{res}}^{2,0}+\mathsf{A}_{I\,\mathrm{res}}^{1,1}+\mathsf{A}_{I\,\mathrm{res}}^{0,2}$
— 2-form, residual field in ghost number 0,

  * •

$(A^{[>1]}_{\mathrm{res}},\psi_{\mathrm{res}})=(\mathsf{A}^{<2}_{I\,\mathrm{res}},\mathsf{A}_{\mathrm{res}}^{>3})$
— residual fields of higher ghost number (form degree $<2$) or negative ghost
number (form degree $>3$).

The effective action (46) then reads

$S_{\mathrm{eff}}[\mathsf{A}_{\mathrm{out}},\mathsf{A}_{\mathrm{in}},\mathsf{A}_{I\,\mathrm{res}},\mathsf{A}_{\mathrm{res}}]=S_{\mathrm{ph}}+S_{\mathrm{gh}},$

where

$\displaystyle S_{\mathrm{ph}}$
$\displaystyle=\int_{M}\mathsf{A}^{3,0}_{\mathrm{out}}\mathsf{A}^{0,3}_{\mathrm{in}}+\mathsf{A}^{2,1}_{\mathrm{out}}\mathsf{A}^{1,2}_{\mathrm{in}}+\mathsf{A}^{3,0}_{\mathrm{out}}\bar{\partial}\mathsf{A}_{I\,\mathrm{res}}^{0,2}+\mathsf{A}^{0,3}_{\mathrm{in}}\partial\mathsf{A}_{I\,\mathrm{res}}^{2,0}$
$\displaystyle+\int_{M}\mathsf{A}^{1,2}_{\mathrm{in}}\partial\mathsf{A}_{I\,\mathrm{res}}^{1,1}+\mathsf{A}^{1,2}_{\mathrm{in}}\bar{\partial}\mathsf{A}_{I\,\mathrm{res}}^{2,0}+\mathsf{A}^{2,1}_{\mathrm{out}}\partial\mathsf{A}_{I\,\mathrm{res}}^{0,2}+\mathsf{A}^{2,1}_{\mathrm{out}}\bar{\partial}\mathsf{A}_{I\,\mathrm{res}}^{1,1}$
$\displaystyle+\frac{1}{2}\int_{M}\bar{\partial}\mathsf{A}_{I\,\mathrm{res}}^{1,1}\partial\mathsf{A}_{I\,\mathrm{res}}^{1,1},$
$\displaystyle S_{\mathrm{gh}}$ $\displaystyle=\int_{M}(\mathsf{A}^{\leq
2}_{\mathrm{out}}-\mathsf{A}^{\leq
2}_{\mathrm{in}})\mathsf{A}^{>3}_{\mathrm{res}}+\mathsf{A}_{\mathrm{res}}^{>3}d_{M}\mathsf{A}_{I\,\mathrm{res}}^{<2}.$

A similar formula holds for higher-dimensional Chern–Simons theories. Some
comments:

  1. i)

As a consequence of Propositions 5.4 and 5.3, the effective action satisfies
the modified quantum master equation, and the $\mathrm{gh}=0$ part
$S_{\mathrm{ph}}$ satisfies the generalized Hamilton–Jacobi equations. In
particular, $S_{\mathrm{ph}}$ can be identified with the HJ action.

  2. ii)

One can rewrite $S_{\mathrm{ph}}$ as

$S_{\mathrm{ph}}=\int_{M}(\mathsf{A}^{3,+}_{\mathrm{out}}-d_{\mathfrak{g}}^{+}\mathsf{A}^{2}_{I\,\mathrm{res}})(A^{3,-}_{\mathrm{in}}+d_{\mathfrak{g}}^{-}\mathsf{A}_{I\,\mathrm{res}}^{2})-\frac{1}{2}\bar{\partial}A^{1,1}_{I\,\mathrm{res}}\partial
A^{1,1}_{I\,\mathrm{res}}$

— a higher-dimensional version of an abelian gauged WZW model, see also
footnote 22.

####  6.2.1. Pushforward over residual fields

The space of residual fields is

$\mathcal{V}=\\{dt\cdot\mathsf{A}^{\leq
2}_{I\,\mathrm{res}}+\mathsf{A}_{\mathrm{res}}^{\geq 4}\\}\;\;=\Omega^{\leq
2}_{\mathbb{C}}(M)[2]\oplus\Omega_{\mathbb{C}}^{\geq 4}(M)[3].$

In particular, the components and their ghost numbers are

field | $\mathsf{A}^{0}_{I\,\mathrm{res}}$ | $\mathsf{A}^{1}_{I\,\mathrm{res}}$ | $\mathsf{A}^{2}_{I\,\mathrm{res}}$ | $\mathsf{A}^{4}_{\mathrm{res}}$ | $\mathsf{A}^{5}_{\mathrm{res}}$ | $\mathsf{A}^{6}_{\mathrm{res}}$
---|---|---|---|---|---|---
ghost number | 2 | 1 | 0 | -1 | -2 | -3

A gauge-fixing lagrangian can be constructed by using the Hodge decomposition
for $\bar{\partial}$: Namely, using the Kähler metric $g$, we decompose

$\Omega^{p,q}(M)=H^{p,q}(M)\oplus\Omega^{p,q}_{\bar{\partial}-\mathrm{ex}}\oplus\Omega^{p,q}_{\bar{\partial}-\mathrm{coex}}$

where the middle and rightmost terms denote the spaces of
$\bar{\partial}$-exact and $\bar{\partial}^{*}$-exact forms, respectively. The
gauge-fixing lagrangian $\mathcal{L}\subset\mathcal{V}$ is then defined as

(108) $\mathcal{L}=\bigoplus_{p+q\leq
2}\Omega^{p,q}_{\bar{\partial}-\mathrm{coex}}\oplus\bigoplus_{p+q\geq
4}\left(H^{p,q}(M)\oplus\Omega^{p,q}_{\bar{\partial}-\mathrm{coex}}\right).$

Restricted to this gauge-fixing lagrangian, the effective action is
nondegenerate in residual fields and the integral gives

$Z_{*}\propto\delta(\bar{\partial}\mathsf{A}^{3,0}_{\mathrm{out}}+\partial\mathsf{A}^{2,1}_{\mathrm{out},\bar{\partial}\mathrm{-ex}})\,\delta(\bar{\partial}\mathsf{A}^{1,2}_{\mathrm{in}}+\partial\mathsf{A}^{0,3}_{\mathrm{in},\bar{\partial}\mathrm{-ex}})\,\delta(\mathsf{A}^{\leq
2}_{\mathrm{out},\mathrm{harm}}-\mathsf{A}^{\leq
2}_{\mathrm{in},\mathrm{harm}})\cdot\\\
\cdot\exp\frac{i}{\hbar}\bigg{(}\int_{M}\big{(}\mathsf{A}^{3,0}_{\mathrm{out}}\mathsf{A}^{0,3}_{\mathrm{in}}+\mathsf{A}^{2,1}_{\mathrm{out}}(\mathrm{id}-P_{\bar{\partial}\mathrm{-ex}})\mathsf{A}^{1,2}_{\mathrm{in}}\big{)}-\\\
-\frac{1}{2}\int_{M\times
M}\bar{\partial}\mathsf{A}^{2,1}_{\mathrm{out}}(x)K(x,x^{\prime})\bar{\partial}\mathsf{A}^{2,1}_{\mathrm{out}}(x^{\prime})-\frac{1}{2}\int_{M\times
M}\partial\mathsf{A}^{1,2}_{\mathrm{in}}(x)K(x,x^{\prime})\partial\mathsf{A}^{1,2}_{\mathrm{in}}(x^{\prime})\bigg{)},$

where $K(x,x^{\prime})$ is the integral kernel of the inverse of the operator
$(\partial\bar{\partial})$ restricted to
$\Omega^{1,1}_{\bar{\partial}-\mathrm{coex}}$.

###  6.3. 7D Chern–Simons and Kodaira–Spencer action functional

We now turn our attention to 7-dimensional Chern–Simons theory on a cylinder
with a particular polarization on the out-boundary. This polarization was
first discovered by Hitchin [29]. It was used in [26] to argue that the semi-
classical approximation of the Chern–Simons wave function can be expressed in
terms of the Kodaira–Spencer action functional introduced in [9] whose
classical solutions are deformations of complex structures on a Kähler
manifold (see Appendix B for a brief review of the Kodaira-Spencer theory).
Here we argue that this semiclassical approximation is in fact exact in the
axial gauge. From the general arguments of the BV-BFV formalism, it follows
that a change of gauge fixing will result in an $\Omega$-exact change of the
partition function, hence its $\Omega$-cohomology class is well-defined and
given by the Kodaira–Spencer partition function. A caveat is that in this
section we do not take care of determinants arising in Gaussian path
integrals. Those might lead to anomalies similar to the discussion of Remark
4.4, and would have to be treated separately.

####  6.3.1. General polarizations in $4l+3$-dimensional Chern–Simons theory

Using the results of Section 5.2, one can consider also more general
polarizations in higher-dimensional Chern–Simons theories, in dimension
$d=2k+1=4l+3$.

Suppose that $\mathcal{P}^{[0]}$ is any polarization on
$X^{[0]}=\Omega^{k}_{\mathbb{C}}(M)$ such that we have local coordinates
$\mathsf{A}^{Q}$ on the base and $\mathsf{A}^{P}$ on the fibers, and let
$G=G(\mathsf{A}^{-},\mathsf{A}^{Q})$ be the corresponding generating function.
From Section 5.2 we know that the partition function of abelian Chern–Simons
theory with $\mathcal{P}^{[<0],+}$-polarization on the in-boundary and
$\mathcal{P}^{[<0],P}$-polarization on the out-boundary is
$Z=\exp(\frac{i}{\hbar}S_{\mathrm{eff}})$ with
$S_{\mathrm{eff}}=S_{\mathrm{ph}}+S_{\mathrm{gh}}$ and

$\displaystyle
S_{\mathrm{ph}}[\mathsf{A}^{-}_{\mathrm{in}},\mathsf{A}^{Q}_{\mathrm{out}},\mathsf{A}_{I\,\mathrm{res}}]$