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Chern-Simons Gauge Theory As A String Theory

Edward Witten

TL;DR

By embedding three-dimensional Chern-Simons theory into a topological string framework, the paper shows that a topological sigma model with target $T^*M$ reproduces CS perturbation theory, with instanton effects mapped to Wilson lines and a space-time interpretation via open string field theory. The A-model and B-model provide dual realizations: the A-model yields CS on a boundary 3-manifold $M$ through flat boundary connections, while the B-model captures holomorphic bundle data on Calabi–Yau targets; in the large $t$ limit these reduce to familiar cohomological data on $M$ or $X$. Boundary conditions, boundary terms, and large-$t$ localization are developed, and the disc-level open-string amplitudes reproduce the Chern–Simons action, including Massey products that encode deformations of flat connections. Kontsevich’s observations about observables and descent are connected to the world-sheet formulation, and instanton or boundary contributions are organized via open-string field theory, revealing a background-independent space-time interpretation for the A-model CS sector, with the B-model offering a holomorphic counterpart. Closed-string contributions are discussed as finite but largely decoupled from the open-string sector, suggesting a consistent, finite framework where open-string dynamics capture the essential CS physics and its generalizations.

Abstract

Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory.

Chern-Simons Gauge Theory As A String Theory

TL;DR

By embedding three-dimensional Chern-Simons theory into a topological string framework, the paper shows that a topological sigma model with target reproduces CS perturbation theory, with instanton effects mapped to Wilson lines and a space-time interpretation via open string field theory. The A-model and B-model provide dual realizations: the A-model yields CS on a boundary 3-manifold through flat boundary connections, while the B-model captures holomorphic bundle data on Calabi–Yau targets; in the large limit these reduce to familiar cohomological data on or . Boundary conditions, boundary terms, and large- localization are developed, and the disc-level open-string amplitudes reproduce the Chern–Simons action, including Massey products that encode deformations of flat connections. Kontsevich’s observations about observables and descent are connected to the world-sheet formulation, and instanton or boundary contributions are organized via open-string field theory, revealing a background-independent space-time interpretation for the A-model CS sector, with the B-model offering a holomorphic counterpart. Closed-string contributions are discussed as finite but largely decoupled from the open-string sector, suggesting a consistent, finite framework where open-string dynamics capture the essential CS physics and its generalizations.

Abstract

Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory.

Paper Structure

This paper contains 30 sections, 103 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Riemann Surfaces And Gauge Theories
  3. Axioms
  4. Vertex Operators And Ghosts
  5. A Conundrum
  6. Realizations Via Topological Field Theories
  7. The ${\bf A}$ Model
  8. The ${\bf B}$ Model
  9. Boundary Conditions
  10. Boundary Terms
  11. The ${\bf A}$ Model
  12. The ${\bf B}$ Model
  13. Large $t$ Limit
  14. The ${\bf A}$ Model
  15. The ${\bf B}$ Model
  16. ...and 15 more sections

Figures (9)

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  • ...and 4 more figures