Quantum Chern–Simons Theories on Cylinders: BV-BFV Partition Functions

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I × Σ , with I an interval and dim Σ = 4 l + 2 , in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Σ = 0 is considered...

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Bibliographic Details
Published in:Communications in mathematical physics 2023, Vol.398 (1), p.133-218
Main Authors: Cattaneo, Alberto S., Mnev, Pavel, Wernli, Konstantin
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Field theory
Field theory (physics)
Formalism
Mathematical and Computational Physics
Mathematical Physics
Partitions (mathematics)
Physics
Physics and Astronomy
Quantum Physics
Quantum theory
Relativity Theory
Theoretical
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Summary:We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I × Σ , with I an interval and dim Σ = 4 l + 2 , in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Σ = 0 is considered as a toy example. We show that one can identify—for certain choices of residual fields—the “physical part” (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton–Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton–Jacobi actions, and quantization, arXiv:2012.13270 ), without any quantum corrections. This Hamilton–Jacobi action is the action functional of a conformal field theory on Σ . For dim Σ = 2 , this implies a version of the CS-WZW correspondence. For dim Σ = 6 , using a particular polarization on one end of the cylinder, the Chern–Simons partition function is related to Kodaira–Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04513-8