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Register a new userOn the Globalization of the Poisson Sigma Model in the BV-BFV Formalism
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We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum GBFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology.
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These notes give an introduction to the mathematical framework of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms. Some of the presented content was given as a mini course by the first author at the 2018 QSPACE conference in Benasque.
We show how to derive asymptotic charges for field theories on manifolds with asymptotic boundary, using the BV-BFV formalism. We also prove that the conservation of said charges follows naturally from the vanishing of the BFV boundary action, and show how this construction generalises Noethers procedure. Using the BV-BFV viewpoint, we resolve the controversy present in the literature, regarding the status of large gauge transformation as symmetries of the asymptotic structure. We show that even though the symplectic structure at the asymptotic boundary is not preserved under these transformations, the failure is governed by the corner data, in agreement with the BV-BFV philosophy. We analyse in detail the case of electrodynamics and the interacting scalar field, for which we present a new type of duality to a sourced two-form model.
We describe a globalization construction for the Rozansky-Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. After having introduced the necessary background, we define an AKSZ sigma model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky-Witten model. The relations with other relevant constructions in the literature are discussed. Moreover, we split the model as a $BF$-like theory and we construct a perturbative quantization of the model in the quantum BV-BFV framework. In this context, we are able to prove the modified differential Quantum Master Equation and the flatness of the quantum Grothendieck BFV operator. Additionally, we provide a construction of the BFV boundary operator in some cases.
We show how the BV-BFV formalism provides natural solutions to descent equations, and discuss how it relates to the emergence of holographic counterparts of given gauge theories. Furthermore, by means of an AKSZ-type construction we reproduce the Chern-Simons to Wess-Zumino-Witten correspondence from infinitesimal local data, and show an analogous correspondence for BF theory. We discuss how holographic correspondences relate to choices of polarisation relevant for quantisation, proposing a semi-classical interpretation of the quantum holographic principle.
A globalized version of a trace formula for the Poisson Sigma Model on the disk is presented by using its formal global picture in the setting of the Batalin-Vilkovisky formalism. This global construction includes the concept of zero modes. Moreover, for the symplectic case of the Poisson Sigma Model with cotangent target, the globalized trace reduces to a symplectic construction which was presented by Grady, Li and Li for 1-dimensional Chern-Simons theory (topological quantum mechanics). In addition, the connection between this formula and the Nest-Tsygan theorem and the Tamarkin-Tsygan theorem is explained.
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