Topological sigma models

Edward Witten

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663 Scopus citations

Abstract

A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surface Σ to an arbitrary almost complex manifold M. It possesses a fermionic BRST-like symmetry, conserved for arbitrary Σ, and obeying Q2=0. In a suitable version, the quantum ground states are the 1+1 dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure of M) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.

Original languageEnglish (US)
Pages (from-to)411-449
Number of pages39
JournalCommunications In Mathematical Physics
Volume118
Issue number3
DOIs
StatePublished - Sep 1988
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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