×

Topological conformal field theories and gauge theories. (English) Zbl 1139.32006

Author’s abstract: This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, that is, a kind of open string theory. If the integral of these forms converges, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on a four manifold arises as one of these Chern-Simons type gauge theories.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
81T13 Yang-Mills and other gauge theories in quantum field theory
81T45 Topological field theories in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M Alexandrov, M Kontsevich, A Schwarz, O Zabronovsky, Geometry of the master equation and topological field theory, · Zbl 1073.81655 · doi:10.1142/S0217751X97001031
[2] S Axelrod, I M Singer, Chern-Simons perturbation theory, World Sci. Publ., River Edge, NJ (1992) 3 · Zbl 0813.53051
[3] A Cattaneo, P Cotta-Ramusino, F Fucito, M Martellini, M Rinaldi, A Tanzini, M Zeni, Four-dimensional Yang-Mills theory as a deformation of topological BF theory, · Zbl 0927.58023
[4] K Costello, A dual point of view on the ribbon graph decomposition of moduli space, · Zbl 1131.32008 · doi:10.2140/gt.2007.11.1637
[5] K Costello, Topological conformal field theories and Calabi-Yau categories, · Zbl 1171.14038 · doi:10.1016/j.aim.2006.06.004
[6] S K Donaldson, R P Thomas, Gauge theory in higher dimensions, Oxford Univ. Press (1998) 31 · Zbl 0926.58003
[7] P B Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Studies in Advanced Mathematics, CRC Press (1995) · Zbl 0856.58001
[8] H Kajiura, Noncommutative homotopy algebras associated with open strings, · Zbl 1136.81399 · doi:10.1142/S0129055X07002912
[9] R M Kauffmann, Moduli space actions on the Hochschild cochains of a Frobenius algebra, · Zbl 1169.55005 · doi:10.4171/JNCG/22
[10] M Kontsevich, Feynman diagrams and low-dimensional topology, Progr. Math. 120, Birkhäuser (1994) 97 · Zbl 0872.57001
[11] M Kontsevich, Talk at the Hodge Centennial conference, Edinburgh (2003)
[12] M Kontsevich, Talks at University of Miami (2004)
[13] M Kontsevich, Y Soibelman, Homological mirror symmetry and torus fibrations, World Sci. Publ., River Edge, NJ (2001) 203 · Zbl 1072.14046
[14] M Kontsevich, Y Soibelman, Notes on A-infinity algebras, A-infinity categories and noncommutative geometry I, (2006) · Zbl 1114.14027
[15] C C M Liu, Moduli of \(J\)-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an \(S^1\)-equivariant pair,
[16] E Looijenga, Intersection theory on Deligne-Mumford compactifications (after Witten and Kontsevich), Astérisque (1993) 4, 187 · Zbl 0821.14005
[17] M Markl, Transferring \(A_{\infty}\) (strongly associative) structures, · Zbl 1112.18007
[18] M Martellini, M Zeni, The BF formalism for Yang-Mills theory and the t’Hooft algebra, · Zbl 0939.81042
[19] S A Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices (1999) 153 · Zbl 0995.32013 · doi:10.1155/S1073792899000070
[20] A Schwarz, Geometry of Batalin-Vilkovisky quantisation, · Zbl 0786.58017
[21] A Schwarz, A model and generalised Chern-Simons,
[22] A Sen, B Zwiebach, Background independent algebraic structures in closed string field theory, Comm. Math. Phys. 177 (1996) 305 · Zbl 0848.17037 · doi:10.1007/BF02101895
[23] R P Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on \(K3\) fibrations, J. Differential Geom. 54 (2000) 367 · Zbl 1034.14015
[24] T Tradler, M Zeinalian, Algebraic string operations, · Zbl 1144.55012 · doi:10.1007/s10977-007-9005-2
[25] E Witten, Chern-Simons gauge theory as a string theory, Progr. Math. 133, Birkhäuser (1995) 637 · Zbl 0844.58018
[26] E Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004) 189 · Zbl 1105.81061 · doi:10.1007/s00220-004-1187-3
[27] B Zwiebach, Oriented open-closed string theory revisited, Ann. Physics 267 (1998) 193 · Zbl 0914.53046 · doi:10.1006/aphy.1998.5803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.