Geometry & Topology

Topological conformal field theories and gauge theories

Kevin Costello

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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.

Article information

Geom. Topol., Volume 11, Number 3 (2007), 1539-1579.

Received: 9 June 2006
Accepted: 7 May 2007
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 81T13: Yang-Mills and other gauge theories [See also 53C07, 58E15]

moduli spaces heat kernels gauge theory


Costello, Kevin. Topological conformal field theories and gauge theories. Geom. Topol. 11 (2007), no. 3, 1539--1579. doi:10.2140/gt.2007.11.1539.

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