Abstract
The goal of this note is to give a brief overview of the BV-BFV formalism developed by the first two authors and Reshetikhin in (Cattaneo et al., Commun Math Phys 332(2), 535–603, 2014) [9], (Cattaneo et al., Perturbative Quantum Gauge Theories on Manifolds with Boundary, 2015) [10] in order to perform perturbative quantisation of Lagrangian field theories on manifolds with boundary, and present a special case of Chern–Simons theory as a new example.
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Notes
- 1.
Slight abuse of language as we are actually considering Fresnel integrals, i.e. with complex exponent.
- 2.
This leads to another theory with a larger space of fiels called canonical BF theory, see [15].
- 3.
Also known as background fields, slow fields, infrared fields.
- 4.
Otherwise known as fast fields or ultraviolet fields.
- 5.
This is basically a choice of coordinates and canonically conjugate momenta, similar to the p and q variables in quantum mechanics.
- 6.
This definition differs from the one in [9] by a purely conventional sign \((-1)^n\) in front of \(\delta S\).
- 7.
There are some subtleties arising from the regularisation of higher functional derivatives that would be too much for the purpose of this note.
- 8.
Also known to physicists as 2-point function or - slightly abusing language - Green’s function.
- 9.
In the sense that we compute it formally as a Gaussian (or rather, Fresnel) integral.
- 10.
Actually, a semiclassical expansion around the classical solution given by the trivial connection.
- 11.
From now on, we will make use of Einstein summation (sums over repeated indices are implied).
- 12.
These contributions can be ignored if the Lie algebra is unimodular (i.e. the structure constants satisfy \(f^i_{ik} = 0\)) or the Euler characteristic of M is 0. We will restrict ourselves to these cases.
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Acknowledgements
A.S.C. and K.W. acknowledge partial support of SNF Grants No. 200020-149150/1 and PDFMP2_137103. 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). P. M. acknowledges partial support of RFBR Grant No. 13-01-12405-ofi-m.
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Cattaneo, A.S., Mnev, P., Wernli, K. (2017). Split Chern–Simons Theory in the BV-BFV Formalism. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes Lega, A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-65427-0_9
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