Abelian analytic torsion and symplectic volume
Archivum mathematicum, Tome 51 (2015) no. 3, pp. 175-187 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifert data.
This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifert data.
DOI : 10.5817/AM2015-3-175
Classification : 53C25, 53D10, 58J28
Keywords: analytic torsion; contact torsion; Chern-Simons theory; Sasakian three-manifold; quantum field theory
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McLellan, B.D.K. Abelian analytic torsion and symplectic volume. Archivum mathematicum, Tome 51 (2015) no. 3, pp. 175-187. doi: 10.5817/AM2015-3-175

[1] Atiyah, M.F., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry, I. Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. | DOI | MR | Zbl

[2] Atiyah, M.F., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry, II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432. | DOI | MR | Zbl

[3] Beasley, C., Witten, E.: Non-abelian localization for Chern-Simons theory. J. Differential Geom. 70 (2005), 183–323. | MR | Zbl

[4] Belov, D. M., Moore, G.W.: Classification of abelian spin Chern-Simons theories. hep-th/0505235, (2005).

[5] Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford University Press, 2008. | MR | Zbl

[6] Braverman, M.: New proof of the Cheeger-Müller theorem. Ann. Global Anal. Geom. 23 (1) (2002), 77–92. | DOI | MR

[7] Cheeger, J.: Analytic torsion and the heat equation. Ann. of Math. 109 (1979), 259–300. | DOI | MR | Zbl

[8] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology. Comm. Math. Phys. 129 (1990), 486–522. | DOI | MR | Zbl

[9] Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds. Progr. Math., vol. 246, Birkhäuser, Basel, 2006. | MR | Zbl

[10] Furata, M., Steer, B.: Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points. Adv. Math. 96 (1) (1992), 38–102. | DOI | MR

[11] Jeffrey, L.C., McLellan, B.D.K.: Non-abelian localization for $U(1)$ Chern-Simons theory. Geometric Aspects of Analysis and Mechanics, Conf. Proc. (in honour of the $65^{th}$ birthday of Hans Duistermaat), Birkhäuser, 2009. | MR

[12] Kawasaki, T.: The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math. 16 (1) (1979), 151–159. | MR | Zbl

[13] Kirk, P., Klassen, E.: Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of $T^{2}$. Comm. Math. Phys. 153 (3) (1993), 521–557. | DOI | MR

[14] McLellan, B.D.K.: Localization in abelian Chern-Simons theory. J. Math. Phys. 54 (2013), arXiv:1208.1724[math-ph]. | MR | Zbl

[15] Müller, W.: Analytic torsion and R-torsion on Reimannian manifolds. Adv. Math. 28 (1978), 233–305. | DOI

[16] Neumann, W.D., Raymond, F.: Seifert manifolds, plumbing, $\mu $-invariant and orientation reversing maps, Algebraic and geometric topology . Proc. Sympos., Univ. California, Santa Barbara, California, Lecture Notes in Math., 664, Springer, Berlin, (1978), 163–196, 1977. | DOI | MR

[17] Nicolaescu, L.I.: Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations. Comm. Anal. Geom. 8 (5) (2000), 1027–1096. | DOI | MR | Zbl

[18] Orlik, P.: Seifert Manifolds. Lecture Notes in Math., Springer-Verlag, Berlin, 1972. | MR | Zbl

[19] Orlik, P., Vogt, E., Zieschang, H.: Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten. Topology 6 (1967), 49–64. | DOI | MR | Zbl

[20] Prasolov, V.V., Sossinsky, A.B.: Knots, links, braids and 3-manifolds: An introduction to the new invariants in low-dimensional topology. Amer. Math. Soc. (1996), Vol. 154 of Translations of Mathematical Monographs. | MR

[21] Ray, D., Singer, I.: Analytic torsion. Proc. Sympos. Pure Math. 23 (1973), 167–182. | MR | Zbl

[22] Reidemeister, K.: Überdeckungen von Komplexen. J. Reine Angew. Math. 173 (1935), 164–173. | Zbl

[23] Rumin, M.: Formes différentielles sur les variétés de contact. J. Differential Geom. 39 (2) (1994), 281–330. | MR | Zbl

[24] Rumin, M., Seshadri, N.: Analytic torsions on contact manifolds. Ann. Inst. Fourier 62 (2012), 727–782. | DOI | MR | Zbl

[25] Schwarz, A.S.: The partition function of degenerate functional. Comm. Math. Phys. 67 (1979), 1–16. | DOI | MR

[26] Williams, F.L.: Lectures on zeta functions, L-functions and modular forms with some physical applications. A window into zeta and modular physics, MSRI Publications, 2010. | MR | Zbl

[27] Witten, E.: Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), 661–692. | MR | Zbl

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