Topological conformal field theories and gauge theories

Costello, Kevin¹

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago IL 60637, USA

Geometry & topology, Tome 11 (2007) no. 3, pp. 1539-1579.

Résumé

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 converged, 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.

Détail

DOI : 10.2140/gt.2007.11.1539

Keywords: moduli spaces, heat kernels, gauge theory

Bibliographie

[1] M Alexandrov, M Kontsevich, A Schwarz, O Zabronovsky, Geometry of the master equation and topological field theory,

[2] S Axelrod, I M Singer, Chern–Simons perturbation theory, from: "Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol 1, 2 (New York, 1991)", World Sci. Publ., River Edge, NJ (1992) 3

[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,

[4] K Costello, A dual point of view on the ribbon graph decomposition of moduli space,

[5] K Costello, Topological conformal field theories and Calabi–Yau categories,

[6] S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe (Oxford, 1996)", Oxford Univ. Press (1998) 31

[7] P B Gilkey, Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Studies in Advanced Mathematics, CRC Press (1995)

[8] H Kajiura, Noncommutative homotopy algebras associated with open strings,

[9] R M Kauffmann, Moduli space actions on the Hochschild cochains of a Frobenius algebra,

[10] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97

[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, from: "Symplectic geometry and mirror symmetry (Seoul, 2000)", World Sci. Publ., River Edge, NJ (2001) 203

[14] M Kontsevich, Y Soibelman, Notes on A–infinity algebras, A–infinity categories and noncommutative geometry I, (2006)

[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

[17] M Markl, Transferring $A_{\infty}$ (strongly associative) structures,

[18] M Martellini, M Zeni, The BF formalism for Yang–Mills theory and the t'Hooft algebra,

[19] S A Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices (1999) 153

[20] A Schwarz, Geometry of Batalin–Vilkovisky quantisation,

[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

[23] R P Thomas, A holomorphic Casson invariant for Calabi–Yau 3–folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367

[24] T Tradler, M Zeinalian, Algebraic string operations,

[25] E Witten, Chern–Simons gauge theory as a string theory, from: "The Floer memorial volume", Progr. Math. 133, Birkhäuser (1995) 637

[26] E Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004) 189

[27] B Zwiebach, Oriented open-closed string theory revisited, Ann. Physics 267 (1998) 193