| Title:
|
Abelian analytic torsion and symplectic volume (English) |
| Author:
|
McLellan, B.D.K. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
175-187 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
analytic torsion |
| Keyword:
|
contact torsion |
| Keyword:
|
Chern-Simons theory |
| Keyword:
|
Sasakian three-manifold |
| Keyword:
|
quantum field theory |
| MSC:
|
53C25 |
| MSC:
|
53D10 |
| MSC:
|
58J28 |
| idZBL:
|
Zbl 06487029 |
| idMR:
|
MR3397270 |
| DOI:
|
10.5817/AM2015-3-175 |
| . |
| Date available:
|
2015-09-09T09:49:02Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144428 |
| . |
| Reference:
|
[1] Atiyah, M.F., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry, I.Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. Zbl 0297.58008, MR 0397797, 10.1017/S0305004100049410 |
| Reference:
|
[2] Atiyah, M.F., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry, II.Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432. Zbl 0314.58016, MR 0397798, 10.1017/S0305004100051872 |
| Reference:
|
[3] Beasley, C., Witten, E.: Non-abelian localization for Chern-Simons theory.J. Differential Geom. 70 (2005), 183–323. Zbl 1097.58012, MR 2192257 |
| Reference:
|
[4] Belov, D. M., Moore, G.W.: Classification of abelian spin Chern-Simons theories.hep-th/0505235, (2005). |
| Reference:
|
[5] Boyer, C.P., Galicki, K.: Sasakian Geometry.Oxford University Press, 2008. Zbl 1155.53002, MR 2382957 |
| Reference:
|
[6] Braverman, M.: New proof of the Cheeger-Müller theorem.Ann. Global Anal. Geom. 23 (1) (2002), 77–92. MR 1952860, 10.1023/A:1021227705930 |
| Reference:
|
[7] Cheeger, J.: Analytic torsion and the heat equation.Ann. of Math. 109 (1979), 259–300. Zbl 0412.58026, MR 0528965, 10.2307/1971113 |
| Reference:
|
[8] Dijkgraaf, R., Witten, E.: Topological gauge theories and group cohomology.Comm. Math. Phys. 129 (1990), 486–522. Zbl 0703.58011, MR 1048699, 10.1007/BF02096988 |
| Reference:
|
[9] Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds.Progr. Math., vol. 246, Birkhäuser, Basel, 2006. Zbl 1099.32008, MR 2214654 |
| Reference:
|
[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. MR 1185787, 10.1016/0001-8708(92)90051-L |
| Reference:
|
[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 2809473 |
| Reference:
|
[12] Kawasaki, T.: The Riemann-Roch theorem for complex V-manifolds.Osaka J. Math. 16 (1) (1979), 151–159. Zbl 0405.32010, MR 0527023 |
| Reference:
|
[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. MR 1218931, 10.1007/BF02096952 |
| Reference:
|
[14] McLellan, B.D.K.: Localization in abelian Chern-Simons theory.J. Math. Phys. 54 (2013), arXiv:1208.1724[math-ph]. Zbl 1280.81102, MR 3076394 |
| Reference:
|
[15] Müller, W.: Analytic torsion and R-torsion on Reimannian manifolds.Adv. Math. 28 (1978), 233–305. 10.1016/0001-8708(78)90116-0 |
| Reference:
|
[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. MR 0518415, 10.1007/BFb0061699 |
| Reference:
|
[17] Nicolaescu, L.I.: Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations.Comm. Anal. Geom. 8 (5) (2000), 1027–1096. Zbl 1001.58004, MR 1846125, 10.4310/CAG.2000.v8.n5.a3 |
| Reference:
|
[18] Orlik, P.: Seifert Manifolds.Lecture Notes in Math., Springer-Verlag, Berlin, 1972. Zbl 0263.57001, MR 0426001 |
| Reference:
|
[19] Orlik, P., Vogt, E., Zieschang, H.: Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten.Topology 6 (1967), 49–64. Zbl 0147.23503, MR 0212831, 10.1016/0040-9383(67)90013-4 |
| Reference:
|
[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 1414898 |
| Reference:
|
[21] Ray, D., Singer, I.: Analytic torsion.Proc. Sympos. Pure Math. 23 (1973), 167–182. Zbl 0273.58014, MR 0339293 |
| Reference:
|
[22] Reidemeister, K.: Überdeckungen von Komplexen.J. Reine Angew. Math. 173 (1935), 164–173. Zbl 0012.12604 |
| Reference:
|
[23] Rumin, M.: Formes différentielles sur les variétés de contact.J. Differential Geom. 39 (2) (1994), 281–330. Zbl 0973.53524, MR 1267892 |
| Reference:
|
[24] Rumin, M., Seshadri, N.: Analytic torsions on contact manifolds.Ann. Inst. Fourier 62 (2012), 727–782. Zbl 1264.58027, MR 2985515, 10.5802/aif.2693 |
| Reference:
|
[25] Schwarz, A.S.: The partition function of degenerate functional.Comm. Math. Phys. 67 (1979), 1–16. MR 0535228, 10.1007/BF01223197 |
| Reference:
|
[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. Zbl 1225.11105, MR 2648361 |
| Reference:
|
[27] Witten, E.: Supersymmetry and Morse theory.J. Differential Geom. 17 (1982), 661–692. Zbl 0499.53056, MR 0683171 |
| . |
