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Relational symplectic groupoid quantization for constant poisson structures

Abstract

As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results are also addressed. In particular, the paper includes an extension to space–times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a “differential” version of the classical and quantum master equations. The quantization constructed in this paper induces Kontsevich’s deformation quantization on the underlying Poisson manifold, i.e., the Moyal product, which is known in full details. This allows focussing on the BV-BFV technology and testing it. For the inexperienced reader, this is also a practical and reasonably simple way to learn it.

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Notes

  1. We write \(\partial _i\) for \(\frac{\partial }{\partial x^{i}}\).

  2. This formula seems of course a bit strange without the precise derivation of the underlying objects. To get a full understanding of these objects we refer to [22] or [15].

  3. Cyclically ordered means that if we start from 0 and move counterclockwise on the unit circle we will first meet 1 and then \(\infty \). One can also regard the unit circle here as a projective space of the real line where the point \(\infty \) actually represents the identification of \(a\rightarrow \infty \) and \(a\rightarrow -\infty \) for \(a\in \mathbb {R}\).

  4. Note that the whole moduli space \({\mathcal {M}}\) of critical points of the kinetic term modulo symmetries is a vector bundle over \({\mathscr {P}}\) with fiber at \(x\in {\mathscr {P}}\) given by \(H^1(\Sigma )\otimes T^*_x{\mathscr {P}}\). The fiber directions will be taken care of completely by the residual fields to be introduced below. Moreover, there is a canonical choice of background for \(\eta \), namely, \(\eta =0\). For these reasons we only take \({\mathcal {M}}_0\) as space of backgrounds.

  5. One should do this before passing to formal coordinates, but the result is the same, see [11].

  6. As anticipated in Footnote 4, the ghost number zero component of the second summand takes care of the \(\eta \)-direction of the moduli space of critical points. On the other hand, the ghost number zero component of the first summand in general only sees a formal neighborhood of the X-component of the moduli space.

  7. Similar computations were done in [19] for 1-dimensional gravity.

  8. Note that here \({\mathrm {d}}\) is the de Rham differential on \({\mathcal {M}}_0\).

  9. If we take smooth maps, then \(({\mathcal {G}},\omega )\) is a weak-symplectic Fréchet manifold.

  10. We thank the referee for pointing out this simple argument.

  11. Recall that \({\mathbb {X}}\) and \(\mathbb {E}\) denote the \(\widehat{\textsf {X}}\) and \(\widehat{\varvec{\eta }}\) components of boundary fields, respectively.

  12. Recall from [17] that the gluing of two propagators along a common boundary is again a propagator for the glued manifold.

References

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Acknowledgements

We thank I. Contreras and P. Mnëv for useful discussions and comments. We are especially grateful to the referee for pointing out some small errors and for very precious comments and suggestions.

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Correspondence to Alberto S. Cattaneo.

Additional information

This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology).

Appendices

Appendix 1: Computations of the mdQMEs of Sect. 7

In this appendix we report the most relevant computations from [29].

Computation for \(L_3\)

We want to show the mdQME for \(L_3\), i.e.,

$$\begin{aligned} \nabla ^{\partial L_3}_\textsf {G}\widehat{\psi }_{L_3}=\left( {\mathrm {d}}+\mathrm {i}\hbar \triangle +\frac{\mathrm {i}}{\hbar }\varOmega ^{(3)}\right) \widehat{\psi }_{L_3}=0 \end{aligned}$$
(19)

As it was already mentioned in Sect. 7, (19) reduces to the equation \(\varOmega ^{(3)}\widehat{\psi }_{L_3}=0\). Moreover, recall that \(\varOmega ^{(3)}\) is given by (see the general construction of [17]) the sum of

$$\begin{aligned} \varOmega ^{(3)}_{\text {pert}}&=-\frac{1}{2}\alpha ^{ij}\left( \hbar ^2\int _{\partial _1M}\frac{\delta }{\delta {\mathbb {X}}_i}\frac{\delta }{\delta {\mathbb {X}}_j}-\int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \end{aligned}$$
(20)
$$\begin{aligned} \varOmega ^{(3)}_0&=\mathrm {i}\hbar \left( \int _{\partial _1M} {\mathrm {d}}{\mathbb {X}}_i\frac{\delta }{\delta {\mathbb {X}}_i}+\int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta }{\delta \mathbb {E}_i}+\int _{\partial _1M}{\mathrm {d}}x^{i}\frac{\delta }{\delta {\mathbb {X}}_i}+\int _{\partial _2M}\mathbb {E}_i \wedge {\mathrm {d}}x^{i}\right) , \end{aligned}$$
(21)

since \(\widehat{\varvec{\eta }}=-\mathrm {i}\hbar \frac{\delta }{\delta \widehat{\textsf {X}}}\) is the conjugated momentum. Let us also look at some terms of \(\varOmega ^{(3)}\) with a derivative by defining

$$\begin{aligned} \varOmega ^{(3)}_{{\mathbb {X}},\text {pert}}&:=-\frac{\hbar ^2}{2}\alpha ^{ij}\int _{\partial _1M}\frac{\delta }{\delta {\mathbb {X}}_i}\frac{\delta }{\delta {\mathbb {X}}_j}, \end{aligned}$$
(22)
$$\begin{aligned} \varOmega _{\mathbb {X}}^{(3)}&:=\mathrm {i}\hbar \int _{\partial _1M}{\mathrm {d}}{\mathbb {X}}_i\frac{\delta }{\delta {\mathbb {X}}_i}, \end{aligned}$$
(23)
$$\begin{aligned} \varOmega _{\mathbb {E}}^{(3)}&:=\mathrm {i}\hbar \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta }{\delta {\mathbb {E}}_i} \end{aligned}$$
(24)

Applying \(\widetilde{\varOmega }=\varOmega ^{(3)}_{\text {pert}}+\varOmega ^{(3)}_{\mathbb {X}}+\varOmega ^{(3)}_\mathbb {E}\) to the state \(\widehat{\psi }_{L_3}\), we get

$$\begin{aligned} \widetilde{\varOmega }\widehat{\psi }_{L_3}=T_{L_3}\widetilde{\varOmega }\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}=T_{L_3}\left( \varOmega ^{(3)}_{\text {pert}}+\varOmega ^{(3)}_{\mathbb {X}}+\varOmega ^{(3)}_\mathbb {E}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$

Recall also that \(T_{L_3}=1\). We want to compute each contribution of the different parts of \({\widetilde{\varOmega }}\). Let us therefore first apply \(\varOmega ^{(3)}_{\text {pert}}\) and observe

$$\begin{aligned} \varOmega ^{(3)}_{\text {pert}}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}&=-\frac{\hbar ^2}{2}\alpha ^{ij}\int _{\partial _1M}\frac{\delta }{\delta {\mathbb {X}}_i}\frac{\delta }{\delta {\mathbb {X}}_j}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(25)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(26)
$$\begin{aligned}&=-\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) \int _{\partial _1M} \frac{\delta }{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_j}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(27)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(28)
$$\begin{aligned}&=-\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) ^2\int _{\partial _1M}\left( \frac{\delta ^2{\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}+\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_j}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}\right) \end{aligned}$$
(29)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}\end{aligned}$$
(30)
$$\begin{aligned}&=-\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) ^2\int _{\partial _1M}\left( \frac{\delta ^2{\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j}+\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_j}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(31)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(32)

Now we need to express the functional derivatives in (31) in terms of the propagator and the fields. Each term is given by a sum and the only terms which contribute to the derivatives are

$$\begin{aligned} \frac{\delta ^2{\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j}&=\frac{\delta ^2{\mathcal {S}}^{\text {eff}}_{\partial ^{(1)}M}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j}+\frac{\delta ^2{\mathcal {S}}^{\text {eff}}_{\partial ^{(2)}M}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j} \end{aligned}$$
(33)
$$\begin{aligned} \frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i}&=\frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}+\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i} \end{aligned}$$
(34)
$$\begin{aligned} \frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_j}&=\frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}+\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}, \end{aligned}$$
(35)

since the other terms of the effective action do not depend on the \({\mathbb {X}}\)-field. Now we get

$$\begin{aligned}&\frac{\delta ^2{\mathcal {S}}^{\text {eff}}_{\partial ^{(k)}M}}{\delta {\mathbb {X}}_i\delta {\mathbb {X}}_j}=0,\quad \frac{\delta {\mathcal {S}}_{\partial ^{(k)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}=\int _{\partial _1M\times \partial _{2}^{(k)}M}\zeta _{02}\wedge \pi _{2}^*\mathbb {E}_i,\quad \nonumber \\&\quad \frac{\delta {\mathcal {S}}_{\partial ^{(k)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}=\int _{\partial _1M\times \partial _{2}^{(k)}M}\zeta _{03}\wedge \pi _{2}^*\mathbb {E}_j \end{aligned}$$
(36)

and hence

$$\begin{aligned} \frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial M}^{\text {eff}}}{\delta {\mathbb {X}}_j}= & {} \frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}+\frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j} +\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial ^{(1)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}\nonumber \\&+\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}_{\partial ^{(2)} M}^{\text {eff}}}{\delta {\mathbb {X}}_j}. \end{aligned}$$
(37)

This shows that the application of \(\varOmega ^{(3)}_{\text {pert}}\) to the state \(\widehat{\psi }_{L_3}\) gives

$$\begin{aligned} \varOmega ^{(3)}_{\text {pert}}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}&=\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times C_2(\partial _2^{(1)}M)}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(38)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times [\partial _2^{(1)}M\times \partial _2^{(2)}M]}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(39)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times [\partial _2^{(2)}M\times \partial _2^{(1)}M]}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(40)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times C_2(\partial _2^{(2)}M)}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(41)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(42)

For these terms we get the diagrams as in Fig. 15. Recall that \(u_0\in \partial _1M,u_1\in M\) and \(u_2,u_3,u_\nu \in \partial _2^{(k)}M\) for some \(k\in \{1,2\}\). Now we want to compute the corresponding terms for \(\varOmega ^{(3)}_\mathbb {E}\). We get

$$\begin{aligned} \varOmega ^{(3)}_\mathbb {E}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}&=\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {eff}}_{\partial ^{(1)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(43)
$$\begin{aligned}&\quad +\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {eff}}_{\partial ^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(44)
$$\begin{aligned}&\quad +\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(1)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(45)
$$\begin{aligned}&\quad +\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(46)
$$\begin{aligned}&\quad +\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(47)
Fig. 15
figure 15

The diagrams for the terms of \(\varOmega ^{(3)}_{\text {pert}}\). a For \(\partial _2^{(1)}M\). b For \(\partial _2^{(2)}M\). c For \(\partial _{2}^{(1)}M\times \partial _2^{(2)}M\)

Let us now compute each term individually. We will start with (43) and observe

$$\begin{aligned}&\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {eff}}_{\partial ^{(1)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}\nonumber \\&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( -\int _{\partial _2^{(1)}M\times \partial _1M}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(48)

For (44) we get

$$\begin{aligned}&\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {eff}}_{\partial ^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}\nonumber \\&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( -\int _{\partial _2^{(2)}M\times \partial _1M}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(49)

For the term (45) we get

$$\begin{aligned}&\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(1)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(50)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij} \left( \int _{C_2(\partial _2^{(1)}M)}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \widehat{\zeta }^\partial _{23} \wedge \pi _2^*\mathbb {E}_j\right. \nonumber \\&\qquad \left. +\int _{C_2(\partial _2^{(1)}M)}\pi _1^*\mathbb {E}_i \wedge \widehat{\zeta }^\partial _{23}\wedge \pi _2^*{\mathrm {d}}\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(51)

Now using integration by parts we get

$$\begin{aligned}&\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{C_2(\partial _2^{(1)}M)} \pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \widehat{\zeta }^\partial _{23}\wedge \pi _2^*\mathbb {E}_j\right. \nonumber \\&\qquad \left. +\int _{C_2(\partial _2^{(1)}M)}\pi _1^*\mathbb {E}_i\wedge \widehat{\zeta }^\partial _{23}\wedge \pi _2^*{\mathrm {d}}\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(52)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{C_2(\partial _2^{(1)}M)}{\mathrm {d}}(\pi _1^*\mathbb {E}_i\wedge \pi _2^*\mathbb {E}_j)\wedge \widehat{\zeta }^\partial _{23}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(53)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2^{(1)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(54)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{C_2(\partial _2^{(1)}M)}\pi _1^*\mathbb {E}_i\wedge \pi _2^*\mathbb {E}_j\wedge {\mathrm {d}}\widehat{\zeta }^\partial _{23}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(55)

Because of the fact that we have vanishing cohomology, we can use Stokes’ theorem to compute \({\mathrm {d}}\widehat{\zeta }^\partial \). We get

$$\begin{aligned} {\mathrm {d}}\widehat{\zeta }^\partial _{23}:={\mathrm {d}}\widehat{\zeta }^\partial (u_2,u_3)=\int _{\partial _1M}\zeta _{02}\wedge \zeta _{03}. \end{aligned}$$

Hence (55) can be written as

$$\begin{aligned} \mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times C_2(\partial _2^{(1)}M)}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(56)

The same procedure holds for (46), and thus

$$\begin{aligned}&\mathrm {i}\hbar \left( \int _{\partial _2^{(1)} M\sqcup \partial _2^{(2)}M}{\mathrm {d}}\mathbb {E}_i \frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(57)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2} \alpha ^{ij}\left( \int _{\partial _2^{(2)}M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(58)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij} \left( \int _{\partial _1M\times C_2(\partial _2^{(2)}M)}\zeta _{02}\wedge \zeta _{03}\wedge \pi _{2,1}^*\mathbb {E}_i \wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(59)

The boundary propagator \(\widehat{\zeta }^\partial \) is no longer in (54) and (55) since we have to integrate over the fiber of the configuration space, which implies that by the property of \(\widehat{\zeta }^\partial \) its value is constant 1 on the fiber. Moreover, since the diagonal is a copy of the manifold itself, we get that integration over \(\partial C_2(\partial _2^{(k)}M)\) is then actually given by integration over \(\partial _2^{(k)}M\) with the remaining form \(\mathbb {E}_i\wedge \mathbb {E}_j\), i.e., evaluated at the same point for \(k\in \{1,2\}\). For the term (47) we have the same principle and thus

$$\begin{aligned}&\mathrm {i}\hbar \left( \int _{\partial _2^{(1)}M\sqcup \partial _2^{(2)}M} {\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2^{(1)} M\sqcup \partial _2^{(2)}M}}{\delta \mathbb {E}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(60)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2} \alpha ^{ij}\left( \int _{\partial [\partial _2^{(1)}M\times \partial _2^{(2)}M]}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(61)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{\partial _1M\times [\partial _2^{(1)}M\times \partial _2^{(2)}M]}\zeta _{02}\wedge \zeta _{03} \wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(62)

Moreover, we can observe that (61) vanishes, since we get integration over the double boundary and the fields vanish on the endpoints of the boundary. Now we need to compute the terms for \(\varOmega ^{(3)}_{\mathbb {X}}\). Therefore, we get

$$\begin{aligned} \varOmega _{{\mathbb {X}}}\text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}&=\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \int _{\partial _1M} {\mathrm {d}}{\mathbb {X}}_i\frac{\delta {\mathcal {S}}_{\partial ^{(1)}M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(63)
$$\begin{aligned}&\quad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \int _{\partial _1M} {\mathrm {d}}{\mathbb {X}}_i\frac{\delta {\mathcal {S}}_{\partial ^{(2)}M}^{\text {eff}}}{\delta {\mathbb {X}}_i}\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(64)

The term in (63) is then

$$\begin{aligned} \mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( -\int _{\partial _2^{(1)}M\times \partial _1M}\pi _1^*\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathrm {d}}{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(65)

The term in (64) is then

$$\begin{aligned} \mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( -\int _{\partial _2^{(2)}M\times \partial _1M}\pi _1^*\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathrm {d}}{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(66)

Now if we combine (65) with (48) and (66) with (49) and using again integration by parts we get

$$\begin{aligned}&\left( \int _{\partial _2^{(1)}M\times \partial _1M}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i +\int _{\partial _2^{(1)}M\times \partial _1M}\pi _1^*\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathrm {d}}{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(67)
$$\begin{aligned}&\quad =\left( \int _{\partial _2^{(1)}M\times \partial _1M} {\mathrm {d}}(\pi _1^*\mathbb {E}_i\wedge \pi _2^*{\mathbb {X}}_i) \wedge \zeta _{0\nu }\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(68)
$$\begin{aligned}&\quad =\left( \int _{\partial [\partial _2^{(1)}M\times \partial _1M]} \pi _1^*\mathbb {E}_i\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(69)
$$\begin{aligned}&\qquad +\left( \int _{\partial _2^{(1)}M\times \partial _1M}\pi _1^*\mathbb {E}_i\wedge {\mathrm {d}}\zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(70)

and

$$\begin{aligned}&\left( \int _{\partial _2^{(2)}M\times \partial _1M}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i +\int _{\partial _2^{(2)}M\times \partial _1M}\pi _1^*\mathbb {E}_i\wedge \zeta _{0\nu }\wedge \pi _2^*{\mathrm {d}}{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}}\end{aligned}$$
(71)
$$\begin{aligned}&\quad =\left( \int _{\partial _2^{(2)}M\times \partial _1M} {\mathrm {d}}(\pi _1^*\mathbb {E}_i\wedge \pi _2^*{\mathbb {X}}_i) \wedge \zeta _{0\nu }\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(72)
$$\begin{aligned}&\quad =\left( \int _{\partial [\partial _2^{(2)}M\times \partial _1M]} \pi _1^*\mathbb {E}_i\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(73)
$$\begin{aligned}&\qquad +\left( \int _{\partial _2^{(2)}M\times \partial _1M}\pi _1^*\mathbb {E}_i \wedge {\mathrm {d}}\zeta _{0\nu }\wedge \pi _2^*{\mathbb {X}}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar } {\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(74)

respectively. Now again we can use that there is no cohomology and which implies that \({\mathrm {d}}\zeta =0\), and thus the terms (70) and (74) vanish. Moreover, the terms (69) and (73) also vanish because of the principle we already had before. Now the term (38) cancels with (56), the term (39) cancels with (62) the term (41) cancels with (59), and finally the term in (42) cancels with the sum of the terms (54) and (58). Finally, for \(\int _{\partial _1M}{\mathrm {d}}x^{i}\frac{\delta }{\delta {\mathbb {X}}_i}\) we get a term \(-\int _{\partial _2M\times M}\mathbb {E}_i\wedge \zeta _{0\nu }\wedge {\mathrm {d}}x^{i}=-\int _{\partial _2M}\mathbb {E}_i\wedge {\mathrm {d}}x^{i}\) which cancels the multiplicative term in \(\varOmega ^{(3)}_0\). Now since \({\mathrm {d}}\widehat{\psi }_{L_3}=0\), the mdQME holds for \(\widehat{\psi }_{L_3}\), because \(\triangle =0\) without cohomology.

Computation for \(L_1\)

Now we need to do the same computations for \(M=L_1\) but with the difference that we have cohomology which means that \(\triangle \not =0\). We need to show the mdQME for \(L_1\), i.e.,

$$\begin{aligned} \nabla _\textsf {G}^{\partial L_1}\widehat{\psi }_{L_1}^{\frac{\delta }{\delta {\mathbb {X}}}} =\left( {\mathrm {d}}+\mathrm {i}\hbar \triangle +\frac{\mathrm {i}}{\hbar }\varOmega ^{(1)}_{\frac{\delta }{\delta {\mathbb {X}}}}+\right) \widehat{\psi }^{\frac{\delta }{\delta {\mathbb {X}}}}_{L_1}=0, \end{aligned}$$
(75)

where

$$\begin{aligned} \varOmega ^{(1)}_{\frac{\delta }{\delta {\mathbb {X}}}}&=\int _{\partial _2M}\left( \mathrm {i}\hbar {\mathrm {d}}\mathbb {E}_i\frac{\delta }{\delta \mathbb {E}_i}-\hbar ^2\mathbb {E}_i \wedge {\mathrm {d}}x^{i}+\frac{1}{2}\alpha ^{ij}\mathbb {E}_i\wedge \mathbb {E}_j\right) , \end{aligned}$$
(76)
$$\begin{aligned} \triangle&=\sum _{i=1}^n(-1)^{1+\deg z_i}\frac{\partial }{\partial z_i}\frac{\partial }{\partial z^{\dagger }_i} \end{aligned}$$
(77)

We can use the formula \(\deg z_i=1-\deg \chi _i\), where \(\{[\chi _i]\}\) is a basis for the cohomology for some representatives \(\chi _i\) (see [17]), and since we have the cohomology of the disk, we get that \(\deg \chi _i=0\) and hence \(\deg z_i=1\). Therefore, we have an even exponent and only the coefficients \(+1\). Now define \(\varOmega ^{(1)}_\mathbb {E}:=\mathrm {i}\hbar \int _{\partial _2M}{\mathrm {d}}\mathbb {E}_i\frac{\delta }{\delta \mathbb {E}_i}\). Then we get

$$\begin{aligned} \varOmega ^{(1)}_{\mathbb {E}}\widehat{\psi }^{\frac{\delta }{\delta {\mathbb {X}}}}_{L_1}&=\varOmega ^{(1)}_\mathbb {E}T_{L_1}\text {e}^{\frac{\mathrm {i}}{\hbar }\left( {\mathcal {S}}^{\text {eff}}_{\partial M}\right) }=T_{L_1}\varOmega ^{(1)}_{\mathbb {E}}\text {e}^{\frac{\mathrm {i}}{\hbar }\left( {\mathcal {S}}^{\text {pert,eff}}_{\partial _2 M}+{\mathcal {S}}^{\text {eff}}_{\partial _2M}+\tilde{{\mathcal {S}}}^{\text {pert,eff}}_{\partial _2M}\right) } \end{aligned}$$
(78)
$$\begin{aligned}&=T_{L_1}\left( \mathrm {i}\hbar \int _{\partial _2M}{\mathrm {d}}\mathbb {E}_i\frac{\delta }{\delta \mathbb {E}_i}\text {e}^{\frac{\mathrm {i}}{\hbar }\left( {\mathcal {S}}^{\text {pert,eff}}_{\partial _2 M}+{\mathcal {S}}^{\text {eff}}_{\partial _2M}+\tilde{{\mathcal {S}}}^{\text {pert,eff}}_{\partial _2M}\right) }\right) \end{aligned}$$
(79)
$$\begin{aligned}&=T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \int _{\partial _2M}\left( {\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2 M}}{\delta \mathbb {E}_i}+{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {eff}}_{\partial _2M}}{\delta \mathbb {E}_i}+{\mathrm {d}}\mathbb {E}_i\frac{\delta {\mathcal {S}}^{\text {pert,eff}}_{\partial _2 M}}{\delta \mathbb {E}_i} \right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(80)
$$\begin{aligned}&=T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{C_2(\partial _2 M)} \pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \widehat{\zeta }^\partial _{23}\wedge \pi _2^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(81)
$$\begin{aligned}&\quad +T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{C_2(\partial _2 M)} \pi _1^*\mathbb {E}_i\wedge \widehat{\zeta }^\partial _{23}\wedge \pi _2^*{\mathrm {d}}\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(82)
$$\begin{aligned}&\quad +T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \int _{\partial _2M}z^{i}{\mathrm {d}}\mathbb {E}_i\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(83)
$$\begin{aligned}&\quad +T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \alpha ^{ij}\left( \int _{\partial _2M}z_i^\dagger {\mathrm {d}}\mathbb {E}_j\wedge \tau \right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}} \end{aligned}$$
(84)

Again, with integration by parts, we get that (81) together with (82) gives

$$\begin{aligned}&T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij}\left( \int _{\partial _2M}\mathbb {E}_i\wedge \mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}\nonumber \\&\quad +\,T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij} \left( \int _{C_2(\partial _2M)}\pi _1^*\mathbb {E}_i\wedge {\mathrm {d}}\widehat{\zeta }^\partial _{23}\wedge \pi _2^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(85)

Now we can use (7) and we get that the second term of (85) is given by

$$\begin{aligned}&T_{L_1}\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \frac{1}{2}\alpha ^{ij} \left( \int _{C_2(\partial _2M)}\pi _1^*\mathbb {E}_i\wedge \pi _1^*\tau \wedge \pi _2^*\mathbb {E}_j\right. \nonumber \\&\quad \left. -\int _{C_2(\partial _2M)}\pi _1^*\mathbb {E}_i\wedge \pi _2^*\tau \wedge \pi _2^*\mathbb {E}_j\right) \text {e}^{\frac{\mathrm {i}}{\hbar }{\mathcal {S}}^{\text {eff}}_{\partial M}}. \end{aligned}$$
(86)

Hence the term which arises from \(\triangle \) cancels with (86). Moreover, we also get a term \(\hbar ^2\int _{\partial _2M}\mathbb {E}_i \wedge {\mathrm {d}}x^{i}\), which cancels with the first multiplicative term of \(\varOmega ^{(1)}_{\frac{\delta }{\delta {\mathbb {X}}}}\), and since (83) and (84) vanish, and clearly \({\mathrm {d}}\widehat{\psi }^{\frac{\delta }{\delta {\mathbb {X}}}}_{L_1}=0\), we get that the (75) holds. Recall that the mdQME for \(\widehat{\psi }_{L_1}^{\frac{\delta }{\delta \mathbb {E}}}\) is trivially satisfied, and \(T_{L_1}=1\).

Appendix 2: Computations for the associativity of the gluing of Sect. 8

We want to show (17). We claim that

$$\begin{aligned} \partial _t\widehat{\psi }_{{\mathcal {M}}^n}(t)&=\varOmega ^{(3)} \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi _1^*\mathbb {E}_i\wedge \kappa ^n\wedge \pi _2^*{\mathbb {X}}_i\right) \end{aligned}$$
(87)
$$\begin{aligned}&\quad +\varOmega ^{(3)}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t) \frac{1}{2}\alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta ^{n,t}_{12} \wedge \kappa ^{n}_{13}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi ^*_{2,2}\mathbb {E}_j\right) \end{aligned}$$
(88)
$$\begin{aligned}&\quad +\varOmega ^{(3)}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t) \frac{1}{2}\alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\kappa _{12}^n\wedge \zeta ^{n,t}_{13} \wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \end{aligned}$$
(89)
$$\begin{aligned}&=\varOmega ^{(3)}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(90)

with

$$\begin{aligned} \mathcal {A}= & {} \int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi ^*_1\mathbb {E}_i\wedge \kappa ^n\wedge \pi _2^*{\mathbb {X}}_i+\frac{1}{2}\alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \kappa ^n_{13} \wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\\&+\frac{1}{2}\alpha ^{ij} \int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\kappa _{12}^n\wedge \zeta _{13}^{n,t} \wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j, \end{aligned}$$

Indeed, we can first observe that

$$\begin{aligned} \partial _t\widehat{\psi }_{{\mathcal {M}}^n}(t) =\left( \frac{\mathrm {i}}{\hbar }\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) (\partial _t{\mathcal {S}}_{{\mathcal {M}}^n}^{\text {eff}}(t)), \end{aligned}$$
(91)

This shows that we only have to compute \(\partial _t{\mathcal {S}}_{{\mathcal {M}}^n}^{\text {eff}}(t)\). Let us first look at the free term \({\mathcal {S}}_{{\mathcal {M}}^n}^{\text {free,eff}}(t)\) of the action. We get that its derivative is given by

$$\begin{aligned} \partial _t{\mathcal {S}}_{{\mathcal {M}}^n}^{\text {free,eff}}(t)= & {} \partial _t\left( -\int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}} \pi _1^*\mathbb {E}_i\wedge (\zeta +t{\mathrm {d}}\kappa ^n)\wedge \pi _2^*{\mathbb {X}}_i\right) \nonumber \\= & {} -\int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi _1^*\mathbb {E}_i\wedge {\mathrm {d}}\kappa ^n\wedge \pi _2^*{\mathbb {X}}_i. \end{aligned}$$
(92)

The derivative corresponding to the perturbation term \({\mathcal {S}}_{{\mathcal {M}}^n}^{\text {pert,eff}}(t)\) is given by

$$\begin{aligned} \partial _t{\mathcal {S}}_{{\mathcal {M}}^n}^{\text {pert,eff}}(t)= & {} \alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta ^{n,t}_{12}\wedge \partial _t\zeta ^{n,t}_{13} \wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\nonumber \\= & {} \alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta ^{n,t}_{12}\wedge {\mathrm {d}}\kappa ^n_{13}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j. \end{aligned}$$
(93)

Hence we get that

$$\begin{aligned} \partial _t\widehat{\psi }_{{\mathcal {M}}^n}(t)= & {} \left( \frac{\mathrm {i}}{\hbar }\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)\left( -\int _{\partial _2{\mathcal {M}} \times \partial _1{\mathcal {M}}}\pi _1^*\mathbb {E}_i\wedge {\mathrm {d}}\kappa ^n\wedge \pi _2^*{\mathbb {X}}_i\right. \nonumber \\&\left. +\alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta ^{n,t}_{12}\wedge {\mathrm {d}}\kappa ^n_{13}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) . \end{aligned}$$
(94)

Now we want to compute \(\varOmega ^{(3)}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \). We get

$$\begin{aligned} \varOmega ^{(3)}(\widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A})&=\left( \varOmega ^{(3)}_{0} +\varOmega ^{(3)}_{\text {pert}}\right) \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(95)
$$\begin{aligned}&=\varOmega ^{(3)}_0\left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) +\varOmega ^{(3)}_{\text {pert}}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(96)
$$\begin{aligned}&=\underbrace{\left( \varOmega ^{(3)}_0\widehat{\psi }_{{\mathcal {M}}^n}(t)\right) \mathcal {A}}_{(\star )}+\underbrace{\left( \varOmega ^{(3)}_0\mathcal {A}\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)}_{(\star \star )} +\underbrace{\varOmega ^{(3)}_{\text {pert}}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t) \mathcal {A}\right) }_{(\star \star \star )}. \end{aligned}$$
(97)

Let us first compute term \((\star \star \star )\). We get

$$\begin{aligned}&\varOmega ^{(3)}_{\text {pert}}\left( \widehat{\psi }_{{\mathcal {M}}^n}(t) \cdot \mathcal {A}\right) =-\frac{\hbar ^2}{2}\alpha ^{ij}\int _{\partial _1M} \frac{\delta }{\delta {\mathbb {X}}_i}\frac{\delta }{\delta {\mathbb {X}}_j} \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \nonumber \\&\quad +\frac{1}{2} \alpha ^{ij}\int _{\partial _2M}\mathbb {E}_i\wedge \mathbb {E}_j \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(98)
$$\begin{aligned}&\quad =-\frac{\hbar ^2}{2}\alpha ^{ij}\int _{\partial _1M}\frac{\delta }{\delta {\mathbb {X}}_i} \left[ \left( \frac{\mathrm {i}}{\hbar }\right) \frac{\delta {\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}}{\delta {\mathbb {X}}_j} \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right. \nonumber \\&\qquad \left. + \frac{\delta \mathcal {A}}{\delta {\mathbb {X}}_j} \widehat{\psi }_{{\mathcal {M}}^n}(t)\right] +\frac{1}{2}\alpha ^{ij}\int _{\partial _2M}\mathbb {E}_i\wedge \mathbb {E}_j \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(99)
$$\begin{aligned}&\quad =-\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) ^2 \int _{\partial _1M}\frac{\delta {\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}}{\delta {\mathbb {X}}_i}\frac{\delta {\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}}{\delta {\mathbb {X}}_j}\widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\end{aligned}$$
(100)
$$\begin{aligned}&\quad +\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) \int _{\partial _1M}\frac{\delta \mathcal {A}}{\delta {\mathbb {X}}_i} \frac{\delta {\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}}{\delta {\mathbb {X}}_j} \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(101)
$$\begin{aligned}&\quad +\frac{\hbar ^2}{2}\alpha ^{ij}\left( \frac{\mathrm {i}}{\hbar }\right) \int _{\partial _1M}\frac{\delta \mathcal {A}}{\delta {\mathbb {X}}_j} \frac{\delta {\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}}{\delta {\mathbb {X}}_i} \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(102)
$$\begin{aligned}&\quad +\frac{1}{2}\alpha ^{ij}\int _{\partial _2M}\mathbb {E}_i\wedge \mathbb {E}_j \left( \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\right) \end{aligned}$$
(103)

Analyzing the terms, we get that term (101) is given by

$$\begin{aligned} -\frac{\hbar ^2}{2}\alpha ^{ij}\left( \int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \zeta _{13}^{n,t}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \mathcal {A}\widehat{\psi }_{{\mathcal {M}}^n}(t) , \end{aligned}$$
(104)

term (102) by

$$\begin{aligned} \frac{\hbar ^2}{2}\alpha ^{ij}\left( \int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \kappa _{13}^n\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t), \end{aligned}$$
(105)

and term (103) by

$$\begin{aligned} \frac{\hbar ^2}{2}\alpha ^{ij}\left( \int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{13}^{n,t} \wedge \kappa _{12}^n\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t), \end{aligned}$$
(106)

We can take the sum of (105) and (106) to obtain

$$\begin{aligned} {\hbar ^2}\alpha ^{ij}\left( \int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t} \wedge \kappa _{13}^n\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t), \end{aligned}$$
(107)

Now we want to compute \((\star )\). We get

$$\begin{aligned}&\left( \varOmega ^{(3)}_0\widehat{\psi }_{{\mathcal {M}}^n}(t)\right) \mathcal {A}=\left( \frac{\mathrm {i}}{\hbar }\right) \left( \varOmega ^{(3)}_0{\mathcal {S}}^{\text {eff}}_{{\mathcal {M}}^n}\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\end{aligned}$$
(108)
$$\begin{aligned}&\quad =-\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \overbrace{\int _{\partial _2{\mathcal {M}} \times \partial _1{\mathcal {M}}}\pi _{1}^*{\mathrm {d}}\mathbb {E}_i\wedge \zeta _{12}^{n,t} \wedge \pi _{2}^*{\mathbb {X}}_i+\int _{\partial _2{\mathcal {M}} \times \partial _1{\mathcal {M}}}\pi _{1}^*\mathbb {E}_i\wedge \zeta _{12}^{n,t} \wedge \pi _{2}^*{\mathrm {d}}{\mathbb {X}}_i}^{\text {integration by parts}=\int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}} \pi _1^*\mathbb {E}_i\wedge \zeta ^{n,t}_{02}\wedge \pi _2^*{\mathbb {X}}_i=0}\right) \nonumber \\&\qquad \times \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\end{aligned}$$
(109)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \frac{1}{2}\alpha ^{ij} \int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \zeta _{13}^{n,t} \wedge \pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \pi ^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\end{aligned}$$
(110)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \left( \frac{1}{2}\alpha ^{ij} \int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t} \wedge \zeta _{13}^{n,t}\wedge \pi _1^*\mathbb {E}_i\wedge \pi ^*{\mathrm {d}}\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)\mathcal {A}\end{aligned}$$
(111)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \frac{\mathrm {i}}{\hbar }\right) \Bigg (\underbrace{\int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}{\mathrm {d}}(\zeta ^{n,t}_{02}\wedge \zeta ^{n,t}_{03})\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j}_{=0} \nonumber \\&\qquad +\left( \frac{1}{2}\alpha ^{ij}\int _{\partial _2{\mathcal {M}}}\mathbb {E}_i\wedge \mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t)\Bigg )\mathcal {A}. \end{aligned}$$
(112)

The term \((\star \star )\) gives us

$$\begin{aligned}&\left( \varOmega ^{(3)}_0\mathcal {A}\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) =\mathrm {i}\hbar \left( \int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi _1^*{\mathrm {d}}\mathbb {E}_i\wedge \kappa ^n_{01}\wedge \pi _2^*{\mathbb {X}}_i\right. \nonumber \\&\qquad \left. +\int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi _1^*\mathbb {E}_i\wedge \kappa ^n_{01}\wedge \pi _2^*{\mathrm {d}}{\mathbb {X}}_i\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(113)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \kappa ^n_{13}\wedge \pi _{2,1}^*{\mathrm {d}}\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(114)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge \kappa ^n_{13}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*{\mathrm {d}}\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(115)
$$\begin{aligned}&\quad =\mathrm {i}\hbar \left( \int _{\partial _2{\mathcal {M}}\times \partial _1{\mathcal {M}}}\pi _1^*\mathbb {E}_i\wedge {\mathrm {d}}\kappa ^n_{12}\wedge \pi _2^*{\mathbb {X}}_i\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(116)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \alpha ^{ij}\int _{\partial _1{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{02}^{n,t}\wedge \kappa ^{n}_{03}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(117)
$$\begin{aligned}&\qquad +\mathrm {i}\hbar \left( \alpha ^{ij}\int _{{\mathcal {M}}\times C_2(\partial _2{\mathcal {M}})}\zeta _{12}^{n,t}\wedge {\mathrm {d}}\kappa ^{n}_{13}\wedge \pi _{2,1}^*\mathbb {E}_i\wedge \pi _{2,2}^*\mathbb {E}_j\right) \widehat{\psi }_{{\mathcal {M}}^n}(t) \end{aligned}$$
(118)

Rearranging the terms, and by the fact that \(\widehat{\psi }_{{\mathcal {M}}^n}(t)\) satisfies the mdQME, we see that (17) holds.

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Cattaneo, A.S., Moshayedi, N. & Wernli, K. Relational symplectic groupoid quantization for constant poisson structures. Lett Math Phys 107, 1649–1688 (2017). https://doi.org/10.1007/s11005-017-0959-6

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  • DOI: https://doi.org/10.1007/s11005-017-0959-6

Keywords

  • Deformation quantization
  • Poisson sigma model
  • Symplectic groupoid
  • BV-BFV formalism
  • Formal geometry

Mathematics Subject Classification

  • Primary 53D55
  • Secondary 53D17
  • 57R56
  • 81T70