[Paper Review] Chern-Simons Gauge Theory As A String Theory

Edward Witten|arXiv (Cornell University)|Jul 28, 1992

Black Holes and Theoretical PhysicsPhysics and Astronomy24 references93 citations

TL;DR

This paper proposes that three-dimensional Chern-Simons gauge theory can emerge as a string theory through a topological sigma model with target space $T^*M$, where instanton contributions in the worldsheet theory generate Wilson line insertions in the spacetime Chern-Simons action. The key result is that the perturbative expansion of this string model reproduces Chern-Simons perturbation theory, establishing a background-independent, gauge-invariant spacetime interpretation of the theory.

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.

Motivation & Objective

• To explore whether Chern-Simons gauge theory in three dimensions can be interpreted as a string theory, despite the breakdown of the usual ghost-matter decoupling in topological string models.
• To construct a worldsheet model using a topological sigma model on $T^*M$ that reproduces the perturbative structure of Chern-Simons theory.
• To determine the spacetime interpretation of the open-string version of this model, showing it is equivalent to Chern-Simons theory in perturbation theory.
• To investigate the role of closed strings in the model, particularly their potential to cancel anomalies without affecting open string amplitudes.
• To clarify the connection between topological string theory, Chern-Simons theory, and mathematical invariants such as those studied by Kontsevich.

Proposed method

• Construct a topological sigma model with target space $T^*M$, where $M$ is a three-manifold, to serve as the worldsheet theory.
• Use the fact that there are no finite-action instantons with boundary in $M$, but virtual instantons at infinity contribute to the path integral.
• Show that the counting of these virtual instantons at infinity reproduces the perturbative expansion of Chern-Simons theory as formulated by Axelrod and Singer.
• Apply open-string field theory to derive the spacetime effective action, identifying it as Chern-Simons theory with Wilson line insertions.
• Analyze the closed string sector by attaching closed strings to finite-length open string propagators, with moduli corresponding to the length and attachment point.
• Demonstrate that the closed string propagator annihilates physical states (due to $b_0$ acting on harmonic forms), leading to decoupling of closed strings from open string amplitudes.

Experimental results

Research questions

• RQ1 Can Chern-Simons gauge theory in three dimensions be realized as a string theory without assuming decoupling of matter and ghosts?
• RQ2 How do instanton contributions in a topological sigma model on $T^*M$ give rise to Wilson line insertions in the spacetime Chern-Simons action?
• RQ3 What is the spacetime interpretation of the open-string version of this topological string model?
• RQ4 What role do closed strings play in the model, especially in relation to anomalies and finiteness?
• RQ5 How does this construction relate to Kontsevich’s observations on Chern-Simons theory and topological invariants?

Key findings

• The perturbative expansion of the topological sigma model on $T^*M$ reproduces the perturbative Chern-Simons theory as defined by Axelrod and Singer.
• Wilson line insertions in the spacetime Chern-Simons theory arise from virtual instantons at infinity in the worldsheet sigma model.
• The open-string field theory of the model yields a spacetime action equivalent to three-dimensional Chern-Simons gauge theory in perturbation theory.
• Closed strings decouple from open string amplitudes because the closed string propagator annihilates physical states via the $b_0$ operator, preserving finiteness.
• The model provides the first complete determination of a background-independent, gauge-invariant spacetime interpretation of a string theory.
• The construction explains Kontsevich’s observations on Chern-Simons theory through the lens of topological string theory and virtual instantons.