Topological Field Theory Interpretation of String Topology

Abstract

The string bracket introduced by Chas and Sullivan [math.GT/9911159] is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the $S^1$-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of $G{L}_n$ with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.

27 pages, 2 figures

References in corpus (3)

• String Topology
• Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops
• Loop observables for BF theories in any dimension and the cohomology of knots

Cited by in corpus (10)

• Wilson surfaces and higher dimensional knot invariants
• Configuration spaces and Vassiliev classes in any dimension
• String topology for stacks
• A-model and generalized Chern-Simons theory
• String Bracket and Flat Connections
• Quantization of the Lie bialgebra of string topology
• Link Invariants for Flows in Higher Dimensions
• Deformation of string topology into homotopy skein modules
• Homological Quantum Field Theory
• Invariants from classical field theory