BF Theory and Wilson Surfaces
Let G be a Lie group and denote by \({\mathfrak {g}}\) its Lie algebra. Moreover, consider a principal G-bundle P over some d-manifold \(\Sigma _d\) and construct the adjoint bundle of P, denoted by \({\mathrm {ad}}P\), given as the frame bundle \(P\times ^{\mathrm {Ad}}{\mathfrak {g}}\) with respect to the adjoint representation \({\mathrm {Ad}}:G\rightarrow {\mathrm {Aut}}({\mathfrak {g}})\) and let \({\mathrm {ad}}^*P\) denote its coadjoint bundle. Let \({\mathscr {A}}\) be the affine space of connection 1-forms on P and \({\mathscr {G}}\) the group of gauge transformations. For a connection \(A\in {\mathscr {A}}\), let \({\mathrm {d}}_A\) be the covariant derivative on \(\Omega ^\bullet (\Sigma _d,{\mathrm {ad}}P)\) and \(\Omega ^\bullet (\Sigma _d,{\mathrm {ad}}^*P)\). Let \(A\in {\mathscr {A}}\) and \(B\in \Omega ^{d-2}(M,{\mathrm {ad}}^*P)\) and define the BF action by
$$\begin{aligned} S(A,B):=\int _{\Sigma _d} \langle B,F_A\rangle , \end{aligned}$$
(175)
where \(\langle ,\rangle \) denotes the extension of the adjoint and coadjoint type for the canonical pairing between \({\mathfrak {g}}\) and \({\mathfrak {g}}^*\) to differential forms.
Remark 9.1
(Abelian BF theory). The abelian BF action, i.e., the action for the case where \({\mathfrak {g}}={\mathbb {R}}\), in fact arises as the unperturbed part of many different AKSZ theories such as the Poisson sigma model or Chern–Simons theory. In fact, for the abelian case we have \((A,B)\in \Omega ^\bullet (\Sigma _d)[1]\oplus \Omega ^\bullet (\Sigma _d)[d-1]\) such that \(F_A={\mathrm {d}}A\) and thus we get an action \(S=\int _{\Sigma _d}B\wedge {\mathrm {d}}A\).
The solutions to the Euler–Lagrange equations \(\delta S=0\) for S defined as in (226) are given by
$$\begin{aligned} {\mathrm {M}}_{{\mathrm {cl}}}=\left\{ (A,B)\in {\mathscr {A}}\times \Omega ^{d-2}(\Sigma _d,{\mathrm {ad}}^*P)\,\big |\, F_A=0,{\mathrm {d}}_AB=0\right\} \end{aligned}$$
(176)
Remark 9.2
One can check that the BF action is invariant under the action of
$$\begin{aligned} {{\widetilde{\mathscr {G}}}}:={\mathscr {G}}\rtimes \Omega ^{d-3}(\Sigma _d,{{\,\mathrm{ad}\,}}^* P), \end{aligned}$$
(177)
where \({\mathscr {G}}\) acts on \(\Omega ^{d-3}(\Sigma _d,{{\,\mathrm{ad}\,}}^* P)\) by the coadjoint action. For \((g,\sigma )\in {\widetilde{\mathscr {G}}}\) and \((A,B)\in {\mathscr {A}}\times \Omega ^{d-2}(\Sigma _d,{{\,\mathrm{ad}\,}}^*P)\) we have an action
$$\begin{aligned} A\mapsto A^g,\qquad B\mapsto B^{(g,\sigma )}={{\,\mathrm{Ad}\,}}^*_{g^{-1}}B+{\mathrm {d}}_{A^g}\sigma . \end{aligned}$$
(178)
It is then easy to check that \(S(A^g,B^{(g,\sigma )})=S(A,B)\).
Consider an embedded submanifold \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\) and consider the pullback bundle of P by i according to the diagram
We can now formulate an important type of classical action which is important for the study of higher-dimensional knots [27].
Definition 9.3
(Wilson surface action). The Wilson surface action is given by
$$\begin{aligned} W(\alpha ,\beta ,A,B;i):=\int _{\Sigma _{d-2}}\langle \alpha ,{\mathrm {d}}_{i^*A}\beta +i^*B\rangle , \end{aligned}$$
(179)
where \(\alpha \in \Omega ^0(\Sigma _{d-2},{{\,\mathrm{ad}\,}}i^*P)\) and \(\beta \in \Omega ^{d-3}(\Sigma _{d-2},{{\,\mathrm{ad}\,}}^*i^*P)\).
Definition 9.4
(Wilson surface observable). The Wilson surface observable is given by
$$\begin{aligned} {\mathcal {W}}_{\Sigma _{d-2}}(A,B;i):=\int {\mathscr {D}}[\alpha ]{\mathscr {D}}[\beta ]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }W(\alpha ,\beta ,A,B;i)} \end{aligned}$$
(180)
Remark 9.5
The expectation values of Wilson surface observables in fact give certain higher-dimensional knot invariants [26]. These invariants are based on the construction of invariants by Bott [14] giving the generalization to a family of isotopy invariants for long knots \({\mathbb {R}}^{n}\hookrightarrow {\mathbb {R}}^{n+2}\) for odd \(n\ge 3\), which are based on constructions involving combinations of configuration space integrals. In [57] it was proven that these invariants are of finite type for the case of long ribbon knots and that they are related to the Alexander polynomial for these type of knots. Further generalizations based on this construction, in particular for rectifiable knots, have been given in [43, 44].
BV Formulation of BF Theory
We can consider BF theory in terms of its BV extension. The BV space of fields is given by
$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}=\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\oplus \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}^*P)[d-2], \end{aligned}$$
(181)
where \({\mathscr {A}}=\Omega ^1(\Sigma _d,{{\,\mathrm{ad}\,}}P)\). We will denote the superfields in \({\mathcal {F}}_{\Sigma _d}\) by \(({\varvec{A}},{\varvec{B}})\). Note that there is an induced Lie bracket \([[,]]\) on \(\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\) which is induced by the Lie bracket on \({\mathfrak {g}}\).
Remark 9.6
If we consider local coordinates on \({\mathfrak {g}}\) with corresponding basis \((e_i)\), we have
$$\begin{aligned} {[}[a,b]]=(-1)^{{\mathrm {gh}}(a)\deg (b)}a^ib^jf_{ij}^ke_k, \end{aligned}$$
(182)
where \(f_{ij}^k\) denotes the structure constants of \({\mathfrak {g}}\).
Moreover, for \({\varvec{A}}\in \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\) we get the curvature
$$\begin{aligned} {\varvec{F}}_{{\varvec{A}}}=F_{A_0}+{\mathrm {d}}_{A_0}{\varvec{a}}+\frac{1}{2}[[{\varvec{a}},{\varvec{a}}]], \end{aligned}$$
(183)
where \(A_0\) is any reference connection and \({\varvec{a}}:={\varvec{A}}-A_0\in \Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1]\).
Definition 9.7
(BV action for BF theory). The BV action for BF theory is defined by
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}({\varvec{A}},{\varvec{B}})=\int _{\Sigma _d}\langle \langle {\varvec{B}},{\varvec{F}}_{{\varvec{A}}}\rangle \rangle , \end{aligned}$$
(184)
where \(\langle \langle ,\rangle \rangle \) is the extension to forms of the adjoint and coadjoint type of the canonical pairing between \({\mathfrak {g}}\) and \({\mathfrak {g}}^*\). For two forms a, b we have
$$\begin{aligned} \langle \langle a,b\rangle \rangle =(-1)^{{\mathrm {gh}}(a)\deg (b)}\langle a,b\rangle , \end{aligned}$$
(185)
We can see that
$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}=T^*[-1]\Omega ^\bullet (\Sigma _d,{{\,\mathrm{ad}\,}}P)[1], \end{aligned}$$
(186)
hence we have a canonical symplectic structure \(\omega _{\Sigma _d}\) on \({\mathcal {F}}_{\Sigma _d}\). Similarly as before, let us denote the odd Poisson bracket induced by \(\omega _{\Sigma _d}\) by \(\{,\}_{\omega _{\Sigma _d}}\) and note that \({\mathcal {S}}_{\Sigma _d}\) satisfies the CME
$$\begin{aligned} \left\{ {\mathcal {S}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d}\right\} _{\omega _{\Sigma _d}}=0. \end{aligned}$$
(187)
The cohomological vector field \(Q_{\Sigma _d}\) is given as the Hamiltonian vector field of \({\mathcal {S}}_{\Sigma _d}\), thus \(Q_{\Sigma _d}=\left\{ {\mathcal {S}}_{\Sigma _d},\right\} _{\omega _{\Sigma _d}}\). Note that
$$\begin{aligned} Q_{\Sigma _d}({\varvec{A}})=(-1)^d{\varvec{F}}_{{\varvec{A}}},\qquad Q_{\Sigma _d}({\varvec{B}})=(-1)^d{\mathrm {d}}_{{\varvec{A}}}{\varvec{B}}. \end{aligned}$$
(188)
If we choose a volume element \(\mu \) which is compatible with \(\omega _{\Sigma _d}\), we can define the BV Laplacian by
$$\begin{aligned} \Delta :f\mapsto \frac{1}{2}{{\,\mathrm{div}\,}}_\mu \{f,\}_{\omega _{\Sigma _d}}. \end{aligned}$$
(189)
Then we can show that the QME holds:
$$\begin{aligned} \delta _{{\mathrm {BV}}}{\mathcal {S}}_{\Sigma _d}=\left\{ {\mathcal {S}}_{\Sigma _d},{\mathcal {S}}_{\Sigma _d}\right\} _{\omega _{\Sigma _d}}-2{\mathrm {i}}\hbar \Delta {\mathcal {S}}_{\Sigma _d}=0. \end{aligned}$$
(190)
This is in fact true since \(\Delta {\mathcal {S}}_{\Sigma _d}=0\). Moreover, as expected, we have \(\delta _{{\mathrm {BV}}}^2=0\).
Formal Global BF Theory from the AKSZ Construction
Let us consider the case of abelian BF theory. Note that in this case the Wilson surface action is given by
$$\begin{aligned} W(\alpha ,\beta ,A,B;i):=\int _{\Sigma _{d-2}} \alpha ({\mathrm {d}}\beta +i^*B), \end{aligned}$$
(191)
where \({\mathrm {d}}\) is the de Rham differential on \({\mathbb {R}}\). Solving the Euler–Lagrange equations for \(\delta W=0\), we get that the ciritical points are solutions to
$$\begin{aligned} {\mathrm {d}}\alpha&=0, \end{aligned}$$
(192)
$$\begin{aligned} {\mathrm {d}}\beta +i^*B&=0. \end{aligned}$$
(193)
We want to deal with B perturbatively, that means we can consider solutions to \({\mathrm {d}}\alpha ={\mathrm {d}}\beta =0\) instead and hence we look at solutions of the form \(\alpha =const\) and \(\beta =0\). This means that the constant field \(\alpha \) is going to take the place of the background field. The Wilson surface observable is then given by
$$\begin{aligned} {\mathcal {W}}_{\Sigma _{d-2}}(A,B;i)=\int {\mathscr {D}}[\alpha ]{\mathscr {D}}[\beta ]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }\int _{\Sigma _{d-2}}\alpha {\mathrm {d}}\beta }\int _{x\in {\mathbb {R}}}\mu (x){\text {e}}^{\frac{{\mathrm {i}}}{\hbar }x\int _{\Sigma _{d-2}}i^*B}, \end{aligned}$$
(194)
where \(\mu \) is a volume element on the moduli space of classical solutions for the auxiliary theory which is given by
$$\begin{aligned} {\mathrm {M}}_{{\mathrm {cl}}}=\left\{ (\alpha ,\beta )\in \Omega ^0(\Sigma _{d-2})\oplus \Omega ^{d-3}(\Sigma _{d-2})\,\big |\, \alpha =const,\, \beta =0\right\} \cong {\mathbb {R}}. \end{aligned}$$
(195)
By abbuse of notation we will also denote the perturbation of \(\alpha \) around \(x\in {\mathbb {R}}\) by \(\alpha \). Moreover, if we assume that P is a trivial bundle, not for the abelian case, we get
$$\begin{aligned} {\mathcal {F}}_{\Sigma _d}&\cong \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}^*[d-2] \end{aligned}$$
(196)
$$\begin{aligned}&\cong {{\,\mathrm{Map}\,}}_{{\mathrm {GrMnf}}}(T[1]\Sigma _d,{\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]). \end{aligned}$$
(197)
Remark 9.8
The assumption that P is trivial is similar to a formal lift, whereas the background field is given by a constant critical point of the form (x, 0) with constant background field \(x:T[1]\Sigma _d\rightarrow {\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]\). In fact, it induces a linear split theory as in Definition 6.7.
Remark 9.9
(CE complex and \(L_\infty \)-structure). Let \({\mathfrak {g}}\) be a Lie algebra and consider the differential graded algebra
$$\begin{aligned} {\mathrm {CE}}({\mathfrak {g}}):=\left( \bigwedge ^\bullet {\mathfrak {g}}^*,{\mathrm {d}}_{{\mathrm {CE}}}\right) \cong \left( C^\infty ({\mathfrak {g}}[1]),Q\right) . \end{aligned}$$
(198)
This is called the Chevalley–Eilenberg algebra of \({\mathfrak {g}}\) [28]. The real valued Chevalley–Eilenberg complex is given by
$$\begin{aligned} 0\rightarrow {{\,\mathrm{Hom}\,}}\left( \bigwedge ^0{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}{{\,\mathrm{Hom}\,}}\left( \bigwedge ^1{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}{{\,\mathrm{Hom}\,}}\left( \bigwedge ^2{\mathfrak {g}},{\mathbb {R}}\right) \xrightarrow {{\mathrm {d}}_{{\mathrm {CE}}}}\cdots \end{aligned}$$
(199)
endowed with the Chevalley–Eilenberg differential
$$\begin{aligned} {\mathrm {d}}_{\mathrm {CE}}:{{\,\mathrm{Hom}\,}}\left( \bigwedge ^{n}{\mathfrak {g}},{\mathbb {R}}\right) \rightarrow {{\,\mathrm{Hom}\,}}\left( \bigwedge ^{n+1}{\mathfrak {g}},{\mathbb {R}}\right) \end{aligned}$$
given by
$$\begin{aligned}&({\mathrm {d}}_{\mathrm {CE}}F)(X_1,\ldots ,X_{n+1}) \nonumber \\&\quad :=\sum _{j=1}^{n+1}(-1)^{j+1}X_iF(X_1,\ldots ,{\widehat{X}}_j,\ldots ,X_{n+1}) \nonumber \\&\qquad +\sum _{1\le j<k\le n+1}(-1)^{j+k}F([X_j,X_k],X_1,\ldots ,{\widehat{X}}_j,\ldots ,{\widehat{X}}_k,\ldots ,X_{n+1}), \end{aligned}$$
(200)
where the hat means that these elements are omitted. Denote by \((\xi ^{i})\) the coordinates on \({\mathfrak {g}}[1]\) of degree \(+1\). Then Q has to be of the form
$$\begin{aligned} Q=-\frac{1}{2}f_{ij}^k\xi ^{i}\xi ^{j}\frac{\partial }{\partial \xi ^{k}}, \end{aligned}$$
where \(f_{ij}^k\) are the structure constants of \({\mathfrak {g}}\). Note that a function \(F\in {{\,\mathrm{Hom}\,}}\left( \bigwedge ^n{\mathfrak {g}},{\mathbb {R}}\right) \) corresponds to an element in \(C_n^\infty ({\mathfrak {g}}[1])\) such that the Chevalley–Eilenberg differential is indeed mapped to Q under the isomorphism
$$\begin{aligned} F(X_{j_1}\wedge \ldots \wedge X_{j_n})=:F_{j_1\ldots j_n}\longleftrightarrow \frac{1}{n!}\xi ^{j_1}\cdots \xi ^{j_n}F_{j_1\ldots j_n}. \end{aligned}$$
In fact, for a graded vector space \({\mathfrak {g}}=\bigoplus _{k\in {\mathbb {Z}}}{\mathfrak {g}}_k\), the differential graded algebra \((C^\infty ({\mathfrak {g}}),Q)\) corresponds to an \(L_\infty \)-algebra which is actually given by the Chevalley–Eilenberg algebra \({\mathrm {CE}}({\mathfrak {g}}[-1])\) of the \(L_\infty \)-algebra \({\mathfrak {g}}[-1]\). The dual of the cohomological vector field Q is given by a codifferential D of homogenous degree \(+1\) on \({\widehat{{\text {Sym}}}}({\mathfrak {g}})\cong {\widehat{{\text {Sym}}}}({\mathfrak {g}}[-1])\). The isomorphism is induced by the shift isomorphism \(s:{\mathfrak {g}}\xrightarrow {\sim } {\mathfrak {g}}[1]\). The codifferential D decomposes into a sum \(D=\sum _{j\ge 1}{\bar{D}}_j\) such that the restrictions
$$\begin{aligned} D_j:={\bar{D}}_j\big |_{{\widehat{{\text {Sym}}}}^j({\mathfrak {g}})}:{\widehat{{\text {Sym}}}}^j({\mathfrak {g}})\rightarrow {\mathfrak {g}} \end{aligned}$$
satisfy
$$\begin{aligned} \ell _j=(-1)^{\frac{1}{2}j(j-1)+1}s^{-1}\circ D_j\circ s^{\otimes j},\qquad \forall j\ge 1. \end{aligned}$$
Note that since \(Q^2=0\), we get \(D^2=0\). Such a codifferential induces a classical Grothendieck connection as in Sect. 6.1.
Remark 9.10
(\(L_\infty \)-structure on \(\Omega ^\bullet \)). If \({\mathfrak {g}}\) is endowed with a (curved) \(L_\infty \)-structure, we can view
$$\begin{aligned} \Omega ^\bullet (\Sigma _{d},{\mathfrak {g}})=\bigoplus _{\begin{array}{c} r+j=k \\ 0\le r\le d\\ j\in {\mathbb {Z}} \end{array}}\Omega ^r(\Sigma _d)\otimes {\mathfrak {g}}_{j} \end{aligned}$$
as a (curved) \(L_\infty \)-algebra. The \(L_\infty \)-structure arises as the linear extension of the higher brackets
$$\begin{aligned} {\hat{\ell }}_1(\alpha _1\otimes X_1)&:={\mathrm {d}}_{\Sigma _d}\alpha _1\otimes X_1+(-1)^{\deg (\alpha _1)}\alpha _1\otimes \ell _1(X_1) \end{aligned}$$
(201)
$$\begin{aligned} {\hat{\ell }}_n(\alpha _1\otimes X_1,\ldots ,\alpha _n\otimes X_n)&:=(-1)^{n\sum _{j=1}^{n}\deg (\alpha _j)+\sum _{j=0}^{n-2}\deg (\alpha _{n-j})\sum _{k=1}^{n-j-1}\deg (X_k)} \nonumber \\&\quad \times (\alpha _1\wedge \cdots \wedge \alpha _n)\otimes \ell _n(X_1,\ldots ,X_n) \end{aligned}$$
(202)
for \(n\ge 2\), \(\alpha _1,\ldots ,\alpha _n\in \Omega ^\bullet (\Sigma _d)\) and \(X_1,\ldots ,X_n\in {\mathfrak {g}}\). If \({\mathfrak {g}}\) is cyclic, and \(\Sigma _d\) is compact, oriented without boundary, there is a natural cyclic inner product on \(\Omega ^\bullet (\Sigma _d,{\mathfrak {g}})\) given by
$$\begin{aligned}&\langle \alpha _1\otimes X_1,\alpha _2\otimes X_2\rangle _{\Omega ^\bullet (\Sigma _d,{\mathfrak {g}})} \nonumber \\&\quad =(-1)^{\deg (\alpha _2)\deg (X_1)}\int _{\Sigma _d}\alpha _1\wedge \alpha _2\langle X_1,X_2\rangle _{{\mathfrak {g}}} \end{aligned}$$
(203)
for \(\alpha _1,\alpha _2\in \Omega ^\bullet (\Sigma _d)\) and \(X_1,X_2\in {\mathfrak {g}}\).
BV Extension of Wilson Surfaces
We will now construct the BV extended observable for the auxiliary codimension 2 theory in the case where P is a trivial bundle. Let
$$\begin{aligned} {\mathcal {F}}_{\Sigma _{d-2}}\cong \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}^*[d-2] \end{aligned}$$
(204)
endowed with the symplectic form \(\omega _{\Sigma _{d-2}}\) which induces the corresponding BV bracket \(\{,\}_{\omega _{\Sigma _{d-2}}}\). For auxiliary superfields \(({\varvec{\alpha }},{\varvec{\beta }})\in {\mathcal {F}}_{\Sigma _{d-2}}\) and ambient superfields \(({\varvec{A}},{\varvec{B}})\in {\mathcal {F}}_{\Sigma _d}\) we have the following definition:
Definition 9.11
(BV extended Wilson surface action). The BV extended Wilson surface action is given by
$$\begin{aligned} {\varvec{W}}^0_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i)=\int _{\Sigma _{d-2}}\langle \langle {\varvec{\alpha }},{\mathrm {d}}_{i^*{\varvec{A}}}{\varvec{\beta }}+i^*{\varvec{B}}\rangle \rangle . \end{aligned}$$
(205)
Remark 9.12
As it was shown in [27], we can extend \({\varvec{W}}^0_{\Sigma _{d-2}}\), regarded as a function on embeddings \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\), to a form-valued function \({\varvec{W}}_{\Sigma _{d-2}}\) on these embeddings by setting
$$\begin{aligned} {\varvec{W}}_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i):=\pi _*\langle \langle {\varvec{\alpha }},{\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\beta }}+{\mathrm {ev}}^*{\varvec{B}}\rangle \rangle , \end{aligned}$$
(206)
where \({\mathrm {ev}}\) denotes the evaluation map of embeddings \(\Sigma _{d-2}\hookrightarrow \Sigma _d\) and \(\pi _*\) denotes the integration along the fiber \(\Sigma _{d-2}\).
Proposition 9.13
([27]). The Wilson surface action satisfies a modified version of the dCME, i.e., we have
$$\begin{aligned} {\mathrm {d}}{\varvec{W}}_{\Sigma _{d-2}}-(-1)^d\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _d}}-\frac{1}{2}\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _{d-2}}}=0\nonumber \\ \end{aligned}$$
(207)
Remark 9.14
Proposition 9.13 is a consequense of the fact that
$$ {\mathrm {d}}\int _{\Sigma _{d-2}}=(-1)^d\int _{\Sigma _{d-2}}{\mathrm {d}}$$
and (188).
Denote by \(\Delta _{\Sigma _{d-2}}\) the BV Laplacian for the auxiliary theory. Then we get the following proposition.
Proposition 9.15
([27]). Define the vector field
$$\begin{aligned} {\varvec{Q}}_{\Sigma _{d-2}}=\left\{ {\varvec{W}}_{\Sigma _{d-2}},\right\} _{\omega _{\Sigma _{d-2}}}, \end{aligned}$$
(208)
which acts on generators by
$$\begin{aligned} {\varvec{Q}}_{\Sigma _{d-2}}({\varvec{\alpha }})=(-1)^d{\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\alpha }},\qquad {\varvec{Q}}_{\Sigma _{d-2}}({\varvec{\beta }})=(-1)^d({\mathrm {d}}_{{\mathrm {ev}}^*{\varvec{A}}}{\varvec{\beta }}+{\mathrm {ev}}^*{\varvec{B}}).\nonumber \\ \end{aligned}$$
(209)
Assume that the formal measure \({\mathscr {D}}[{\varvec{\alpha }}]{\mathscr {D}}[{\varvec{\beta }}]\) is invariant with respect to the vector fields (209). Then we have
$$\begin{aligned}&{\mathrm {d}}{\varvec{W}}_{\Sigma _{d-2}}-(-1)^d\left( \delta _{{\mathrm {BV}}}{\varvec{W}}_{\Sigma _{d-2}}+\frac{1}{2}\left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _d}}\right) \nonumber \\&\quad +\frac{1}{2}\left( \left\{ {\varvec{W}}_{\Sigma _{d-2}},{\varvec{W}}_{\Sigma _{d-2}}\right\} _{\omega _{\Sigma _{d-2}}}-2{\mathrm {i}}\hbar \Delta _{\Sigma _{d-2}}{\varvec{W}}_{\Sigma _{d-2}}\right) =0 \end{aligned}$$
(210)
Remark 9.16
Note that the assumption of invariance of the formal measure implies that \(\Delta _{\Sigma _{d-2}}{\varvec{W}}_{\Sigma _{d-2}}=0\).
Definition 9.17
(BV extended Wilson surface observable). We define the BV extended Wilson surface observable as
$$\begin{aligned} {\varvec{{\mathcal {W}}}}_{\Sigma _{d-2}}({\varvec{A}},{\varvec{B}};i)=\int {\mathscr {D}}[{\varvec{\alpha }}]{\mathscr {D}}[{\varvec{\beta }}]{\text {e}}^{\frac{{\mathrm {i}}}{\hbar }{\varvec{W}}_{\Sigma _{d-2}}({\varvec{\alpha }},{\varvec{\beta }},{\varvec{A}},{\varvec{B}};i)} \end{aligned}$$
(211)
Formulation by Hamiltonian Q-Bundles
Let \({\mathcal {M}}={\mathfrak {g}}[1]\oplus {\mathfrak {g}}^*[d-2]\). Denote by \(x:{\mathfrak {g}}[1]\rightarrow {\mathfrak {g}}\) the degree 1 \({\mathfrak {g}}\)-valued coordinate on \({\mathfrak {g}}[1]\) and let \(x^*:{\mathfrak {g}}^*[d-2]\rightarrow {\mathfrak {g}}^*\) be the \({\mathfrak {g}}^*\)-valued coordinate on \({\mathfrak {g}}^*[d-2]\) of degree \(d-2\).
Then we can consider a trivial Hamiltonian Q-bundle over \({\mathcal {M}}\) given by the fiber data
$$\begin{aligned} {\mathcal {N}}&={\mathfrak {g}}\oplus {\mathfrak {g}}^*[d-3], \end{aligned}$$
(212)
$$\begin{aligned} {\mathcal {V}}_{\mathcal {E}}&=\left\langle [x,y],\frac{\partial }{\partial y}\right\rangle +\left\langle {{\,\mathrm{ad}\,}}_x^*y^*,\frac{\partial }{\partial y^*}\right\rangle +(-1)^d\left\langle x^*,\frac{\partial }{\partial y^*}\right\rangle , \end{aligned}$$
(213)
$$\begin{aligned} \omega _{\mathcal {N}}&=\left\langle \delta y^*,\delta y\right\rangle , \end{aligned}$$
(214)
$$\begin{aligned} \alpha _{\mathcal {N}}&=\langle y^*,\delta y\rangle , \end{aligned}$$
(215)
$$\begin{aligned} \Theta _{{\mathcal {E}}}&=\left\langle y^*,[x,y]\right\rangle +\langle x^*,y\rangle , \end{aligned}$$
(216)
where y is the \({\mathfrak {g}}\)-valued coordinate of degree 0 on \({\mathfrak {g}}\) and \(y^*\) is the \({\mathfrak {g}}^*\)-valued coordinate of degree \(d-3\) on \({\mathfrak {g}}^*[d-3]\). For an embedding \(i:\Sigma _{d-2}\hookrightarrow \Sigma _d\) we get the auxiliary theory
$$\begin{aligned} {\mathcal {F}}^{\mathcal {N}}_{\Sigma _{d-2}}&=\Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}\oplus \Omega ^\bullet (\Sigma _{d-2})\otimes {\mathfrak {g}}^*[d-3], \end{aligned}$$
(217)
$$\begin{aligned} \omega ^{\mathcal {N}}_{\Sigma _{d-2}}&=(-1)^d\int _{\Sigma _{d-2}}\langle \delta {\varvec{y}}^*,\delta {\varvec{y}}\rangle , \end{aligned}$$
(218)
$$\begin{aligned} {\mathcal {S}}^{\mathcal {N}}_{\Sigma _{d-2}}&=\int _{\Sigma _{d-2}}\langle {\varvec{y}}^*,{\mathrm {d}}_{\Sigma _{d-2}}{\varvec{y}}\rangle +\int _{\Sigma _{d-2}}\langle {\varvec{y}}^*,[i^*{\varvec{A}},{\varvec{y}}]\rangle +\int _{\Sigma _{d-2}}\langle i^*{\varvec{B}},{\varvec{y}}\rangle . \end{aligned}$$
(219)
Note that \({\mathcal {M}}\) is a differential graded symplectic manifold with the following data:
$$\begin{aligned} Q_{\mathcal {M}}&=\left\langle \frac{1}{2}[x,x],\frac{\partial }{\partial x}\right\rangle +\left\langle {{\,\mathrm{ad}\,}}_x^*x^*,\frac{\partial }{\partial x^*}\right\rangle , \end{aligned}$$
(220)
$$\begin{aligned} \omega _{\mathcal {M}}&=\langle \delta x^*,\delta x\rangle , \end{aligned}$$
(221)
$$\begin{aligned} \alpha _{\mathcal {M}}&=\langle x^*,\delta x\rangle , \end{aligned}$$
(222)
$$\begin{aligned} \Theta _{\mathcal {M}}&=\frac{1}{2}\left\langle x^*,[x,x]\right\rangle . \end{aligned}$$
(223)
Hence the ambient theory is given by
$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}&=\Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}[1]\oplus \Omega ^\bullet (\Sigma _d)\otimes {\mathfrak {g}}^*[d-2]\ni ({\varvec{A}},{\varvec{B}}), \end{aligned}$$
(224)
$$\begin{aligned} \omega _{\Sigma _d}&=\int _{\Sigma _d}\langle \delta {\varvec{B}},\delta {\varvec{A}}\rangle , \end{aligned}$$
(225)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}&=\int _{\Sigma _d}\left\langle {\varvec{B}},{\varvec{F}}_{{\varvec{A}}}\right\rangle =\int _{\Sigma _d}\left\langle {\varvec{B}},{\mathrm {d}}_{\Sigma _d}{\varvec{A}}+\frac{1}{2}[{\varvec{A}},{\varvec{A}}]\right\rangle . \end{aligned}$$
(226)
Note that (226) is exactly the BF action as in Definition 9.7. In the case of abelian BF theory, i.e., when \({\mathfrak {g}}={\mathbb {R}}\), we get that \(Q_{\mathcal {M}}=0\), \(\Theta _{\mathcal {M}}=0\) and the ambient theory
$$\begin{aligned} {\mathcal {F}}^{\mathcal {M}}_{\Sigma _d}&=\Omega ^\bullet (\Sigma _d)[1]\oplus \Omega ^\bullet (\Sigma _d)[d-2], \end{aligned}$$
(227)
$$\begin{aligned} \omega _{\Sigma _d}&=\int _{\Sigma _d}\delta {\varvec{B}}\wedge \delta {\varvec{A}}, \end{aligned}$$
(228)
$$\begin{aligned} {\mathcal {S}}_{\Sigma _d}&=\int _{\Sigma _d}{\varvec{B}}\wedge {\mathrm {d}}_{\Sigma _d}{\varvec{A}}. \end{aligned}$$
(229)
Remark 9.18
The constructions presented in this paper are expected to extend to manifolds with boundary. Using the constructions as in [25] together with the quantum BV-BFV formalism [21, 23], one can show how the formal global observables for split AKSZ sigma models on the boundary induce a more general gauge condition as the dQME which is called modified differential Quantum Master Equation (mdQME). This condition also handles the boundary part which arises as the ordered standard quantization \(\Omega \) of the boundary action \({\mathcal {S}}^\partial \) of dgeree \(+1\), induced by the underlying BFV manifold, plus some higher degree terms. The mdQME is then given as some annihilation condition for the formal global boundary observable \({\mathcal {O}}^\partial \). In fact, it is annihilated by the quantum Grothendieck BFV operator \(\nabla _{\mathsf {G}}:={\mathrm {d}}-{\mathrm {i}}\hbar \Delta +\frac{{\mathrm {i}}}{\hbar }\Omega \) (see [24, 25]), which means that \(\nabla _{\mathsf {G}}{\mathcal {O}}^\partial =0\).