Chern–Simons invariants on hyperbolic manifolds and topological quantum field theories
- Cite this article as:
- Bonora, L., Bytsenko, A.A. & Gonçalves, A.E. Eur. Phys. J. C (2016) 76: 625. doi:10.1140/epjc/s10052-016-4468-z
Abstract
We derive formulas for the classical Chern–Simons invariant of irreducible SU(n)-flat connections on negatively curved locally symmetric three-manifolds. We determine the condition for which the theory remains consistent (with basic physical principles). We show that a connection between holomorphic values of Selberg-type functions at point zero, associated with R-torsion of the flat bundle, and twisted Dirac operators acting on negatively curved manifolds, can be interpreted by means of the Chern–Simons invariant. On the basis of the Labastida–Mariño–Ooguri–Vafa conjecture we analyze a representation of the Chern–Simons quantum partition function (as a generating series of quantum group invariants) in the form of an infinite product weighted by S-functions and Selberg-type functions. We consider the case of links and a knot and use the Rogers approach to discover certain symmetry and modular form identities.
Partition functions for supersymmetric gauge theories on spheres
A. A. Bytsenko,; M. Chaichiany; and A. E. Goncalves;
Departamento de Fsica, Universidade Estadual de Londrina,
Caixa Postal 6001, Londrina-Parana, Brazil
Department of Physics, University of Helsinki,
P.O. Box 64, FI-00014 Helsinki, Finland
aabyts@gmail.com
masud.chaichian@helsinki.
aedsongoncalves@gmail.com
Received 15 October 2018
Accepted 19 October 2018
Published 30 October 2018
In this paper we brie
y review the main idea of the localization technique and its exten-
sion suitable in supersymmetric gauge eld theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit the so-called Faa di Bruno's formula and show that multipartite partition functions can be written in the form of expansion series of
the Bell polynomials. Applying the restricted specialization argument we show that q-in nite-product representation of partition functions admits presentation in terms of thePatterson{Selberg (or the Ruelle-type) spectral functions.