Volume 11, issue 3 (2007)

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Topological conformal field theories and gauge theories

Kevin Costello

Geometry & Topology 11 (2007) 1539–1579

arXiv: math/0605647


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.

moduli spaces, heat kernels, gauge theory
Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 81T13
Forward citations
Received: 9 June 2006
Accepted: 7 May 2007
Published: 23 July 2007
Proposed: Ralph Cohen
Seconded: Jim Bryan, Lothar Goettsche
Kevin Costello
Department of Mathematics
University of Chicago
5734 S. University Avenue
Chicago IL 60637