Abstract
Zero modes are an essential part of topological field theories, but they are frequently also an obstacle to the explicit evaluation of the associated path integrals. In order to address this issue in the case of Ray-Singer Torsion, which appears in various topological gauge theories, we introduce a massive variant of the Ray-Singer Torsion which involves determinants of the twisted Laplacian with mass but without zero modes. This has the advantage of allowing one to explicitly keep track of the zero mode dependence of the theory. We establish a number of general properties of this massive Ray-Singer Torsion. For product manifolds M = N × S1 and mapping tori one is able to interpret the mass term as a flat ℝ+ connection and one can represent the massive Ray-Singer Torsion as the path integral of a Schwarz type topological gauge theory. Using path integral techniques, with a judicious choice of an algebraic gauge fixing condition and a change of variables which leaves one with a free action, we can evaluate the torsion in closed form. We discuss a number of applications, including an explicit calculation of the Ray-Singer Torsion on S1 for G = PSL(2, R) and a path integral derivation of a generalisation of a formula of Fried for the torsion of finite order mapping tori.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Blau and G. Thompson, A New Class of Topological Field Theories and the Ray-Singer Torsion, Phys. Lett. B 228 (1989) 64 [INSPIRE].
G. Horowitz, Exactly Soluble Diffeomorphism Invariant Theories, Commun. Math. Phys. 125 (1989) 417.
E. Witten, Topology Changing Amplitudes in (2 + 1)-Dimensional Gravity, Nucl. Phys. B 323 (1989) 113 [INSPIRE].
M. Blau and G. Thompson, Topological Gauge Theories of Antiymmetric Tensor Fields, Annals Phys. 205 (1991) 130.
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].
E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].
B. Rusakov, Loop Averages and Partition Functions in U(N) Gauge Theory on Two-Dimensional Manifolds, Mod. Phys. Lett. A 05 (1990) 693.
M. Blau and G. Thompson, Quantum Yang-Mills Theory on Arbitrary Surfaces, Int. J. Mod. Phys. A 7 (1992) 3781.
M. Blau and G. Thompson, Lectures on 2-D gauge theories: Topological aspects and path integral techniques, in Summer School in High-energy Physics and Cosmology, Trieste Italy, June 14–July 30 1993, pp. 0175–244 [hep-th/9310144] [INSPIRE].
D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Topological field theory, Phys. Rept. 209 (1991) 129 [INSPIRE].
M. Blau and G. Thompson, Do Metric Independent Classical Actions lead to Topological Field Theories?, Phys. Lett. B 22 (1991) 535.
D. Johnson, A Geometric Form of Casson’s Invariant and its Connection to Reidemeister Torsion, unpublished lecture notes (1988).
M. Celada, D. González and M. Montesinos, BF gravity, Class. Quant. Grav. 33 (2016) 213001 [arXiv:1610.02020] [INSPIRE].
L. Freidel and S. Speziale, On the relations between gravity and BF theories, SIGMA 8 (2012) 032 [arXiv:1201.4247] [INSPIRE].
A.S. Schwarz, The Partition Function of Degenerate Quadratic Functional and Ray-Singer Invariants, Lett. Math. Phys. 2 (1978) 247 [INSPIRE].
A. Schwarz, The Partition Function of a Degenerate Functional, Commun. Math. Phys. 67 (1979) 1.
A.S. Schwarz and Y.S. Tyupkin, Quantization of antisymmetric tensors and Ray-Singer torsion, Nucl. Phys. B 242 (1984) 436 [INSPIRE].
D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math 7 (1971) 145.
D. Ray, Reidemeister Torsion and the Laplacian on Lens Spaces, Adv. Math. 4 (1970) 109.
J. Cheeger, Analytic Torsion and Reidemeister Torsion, Proc. Natl. Acad. Sci. U.S.A. 74 (1977) 2651.
J. Cheeger, Analytic torsion and the heat equation, Annals Math. 109 (1979) 259.
W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math 28 (1978) 233.
D. Freed, Reidemeister Torsion, Spectral Sequences, and Brieskorn Spheres, J. Reine Angew. Math. 429 (1992) 75.
M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nucl. Phys. B 408 (1993) 345 [hep-th/9305010] [INSPIRE].
D. Ray and I. Singer, Analytic Torsion, Partial Differential Equations, Proc. Sympos. Pure Math. 23 (1971) 167.
M. Blau and G. Thompson, Chern-Simons theory on S1-bundles: Abelianisation and q-deformed Yang-Mills theory, JHEP 05 (2006) 003 [hep-th/0601068] [INSPIRE].
M. Blau and G. Thompson, Chern-Simons Theory on Seifert 3-Manifolds, JHEP 09 (2013) 033 [arXiv:1306.3381] [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24 (2020) 6 1475.
M. Blau, K.M. Keita, K.S. Narain and G. Thompson, Chern-Simons theory on a general Seifert 3-manifold, Adv. Theor. Math. Phys. 24 (2020) 279 [arXiv:1812.10966] [INSPIRE].
D. Fried, Lefschetz Formulas for Flows, in The Lefschetz Centennial Conference. Part III, Contemp. Math. 58 (1987) 19.
M. Blau, I. Jermyn and G. Thompson, Solving topological field theories on mapping tori, Phys. Lett. B 383 (1996) 169 [hep-th/9605095] [INSPIRE].
M. Blau and G. Thompson, Diagonalization in Map(M,G), Commun. Math. Phys. 172 (1995) 639 [hep-th/9402097].
U. Bunke, Lectures on Analytic Torsion, (2015) [https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/sixtorsion.pdf].
P. Mnev, Lecture notes on torsions, arXiv:1406.3705 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev. D 28 (1983) 2567 [Erratum ibid. 30 (1984) 508] [INSPIRE].
L. Borsten, M.J. Duff and S. Nagy, Odd dimensional analogue of the Euler characteristic, JHEP 12 (2021) 178 [arXiv:2105.13268] [INSPIRE].
M. Blau and G. Thompson, Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds, arXiv:1603.01149 [INSPIRE].
L.P. Singh and F. Steiner, Fermionic Path Integrals, the Nicolai Map and the Witten Index, Phys. Lett. B 166 (1986) 155 [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
H.P. McKean and I.M. Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967) 43 [INSPIRE].
V. Patodi, Curvature and the Eigenforms of the Laplace Operator, J. Diff. Geom. 5 (1971) 233.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2206.12268
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Blau, M., Kakona, M. & Thompson, G. Massive Ray-Singer torsion and path integrals. J. High Energ. Phys. 2022, 230 (2022). https://doi.org/10.1007/JHEP08(2022)230
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2022)230