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Globalization for perturbative quantization of nonlinear split AKSZ sigma models on manifolds with boundary. (English) Zbl 1430.58007

The authors construct perturbative partition functions of certain AKSZ (Alexandrov, Kontsevich, Schwartz and Zaboronsky) theories, on manifolds with and without boundary, that vary in a covariant way as the point of expansion changes. This is achieved combining the BV-BFV (Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky) [the first author, P. Mnev and N. Reshetikhin, ibid. 357, No. 2, 631–730 (2018; Zbl 1390.81381)] formalism with methods of formal geometry [F. Bonechi, the first author and P. Mnev, J. High Energy Phys. No. 1, Paper No. 099, 27 p. (2012; Zbl 1306.81290); R. Bott, “Some aspects of invariant theory in differential geometry”, in: Differential operators on manifolds. Berlin: Springer. 49–145 (2010; doi:10.1007/978-3-642-11114-3_2); the first author, G. Felder and L. Tomassini, Duke Math. J. 115, No. 2, 329–352 (2001; Zbl 1037.53063); I. M. Gel’fand and D. A. Kazhdan, Dokl. Akad. Nauk Ser. Fiz. 200, 269–272 (1971; Zbl 0238.58001)]. The modified quantum master equation is \( \left(\hbar^{2}\Delta_{\mathcal{V}_{\Sigma}}+\Omega_{\partial\Sigma}\right)\psi_{\Sigma}=0 \), i.e., the partition function \(\psi_{\Sigma}\) is closed with respect to the coboundary operator \(\hbar^{2}\Delta_{\mathcal{V}_{\Sigma}}+\Omega_{\partial\Sigma}\) when the space of fields is linear while if the space of fields is nonlinear then it should be linearized. The authors, using methods of formal geometry, define a covariant partition function \(\tilde{\psi}_{\Sigma}\), an inhomogeneous differential form with values in the vector bundle \(\hat{\mathcal{H}}_{\Sigma,tot}\) over the target with fiber over \(x\) the space of states of the BV-BFV quantization around \(x\) and they show that it satisfies the mdQME (modified differential quantum master equation) \( \left(d_{x}-ih\Delta_{\mathcal{V}_{\Sigma}}+\frac{i}{h}\Omega_{\partial \Sigma}\right) \tilde{\psi}_{\Sigma}=0 \) (as called by the authors), and they show that the quantum Grothendieck operator \(\nabla_{G}:=d_{x}-i\hbar\Delta_{\mathcal{V}_{\Sigma}}+\frac{i}{h}\Omega_{\partial\Sigma}\) squares to zero. The main results of the paper are Theorem 4.6 (mdQME for split AKSZ theories), Theorem 4.8 which states that for the quantum GBFV operator \(\nabla_{G}\) holds \(\left(\nabla_{G}\right)^{2}\equiv0\), and Theorem 5.1 (Covariant change of data). The paper concludes with two comprehensible appendices. Specifically, in Appendix A the authors recall the compactification of various configuration spaces, and in Appendix B they recall notions of formal geometry and its extension to graded manifolds. In the opinion of the reviewer, the authors have written an excellent paper with results that contribute to the further understanding and research in the area of quantization in field theory.

MSC:

58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
81T99 Quantum field theory; related classical field theories
58A50 Supermanifolds and graded manifolds
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