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A String Theory for Two-Dimensional Yang-Mills Theory II

Ofer Aharony, Suman Kundu, Tal Sheaffer

TL;DR

We develop a string-theoretic dual for zero-coupling 2D Yang–Mills theory by extending the Horava–Polyakov worldsheet action to include boundaries that model Wilson loops. The construction uses two gauges (conformal and induced) to handle smooth mappings and degenerations such as branch points, folds, and orientation-reversing tubes, and it includes a boundary-compatible localization mechanism that yields an Euler-number weighted sum over worldsheet mappings with $N^{\chi(\text{worldsheet})}$. The main result is a detailed, cross-checked mapping between worldsheet configurations (holomorphic, mixed, twists, ORTs) and Gross–Taylor’s large-$N$ Wilson-loop results at zero coupling on the sphere, plane, and torus, supporting the conjectured equality of the two descriptions in this limit. The work lays groundwork for finite-coupling extensions and generalizations to other gauge groups, matter content, and non-orientable sectors, thereby advancing a concrete string dual for 2D YM and informing broader holographic approaches to large-$N$ gauge theories.

Abstract

In earlier work we proposed a string theory dual to two dimensional Yang-Mills theory at zero coupling (which can also be thought of as a $BF$ theory), given by a Polyakov-like generalization of Ho\v rava's topological rigid string theory, and we showed that it correctly reproduces (in the $1/N$ expansion) several partition functions of $SU(N)$ Yang-Mills theory. In the present paper, we generalise this to Wilson loop expectation values by adding boundaries with one Dirichlet and one Neumann boundary condition to our string worldsheets. We discuss in detail several examples, including examples where the worldsheet has branch points or orientation-reversing tubes, or where the Wilson loop has one or more self-intersections, and we show that in all of them the string theory reproduces the known Yang-Mills expectation values. We argue that examples with orientation-reversing tubes or self-intersecting Wilson loops cannot be brought to the conformal gauge, so we analyse them in a different gauge.

A String Theory for Two-Dimensional Yang-Mills Theory II

TL;DR

We develop a string-theoretic dual for zero-coupling 2D Yang–Mills theory by extending the Horava–Polyakov worldsheet action to include boundaries that model Wilson loops. The construction uses two gauges (conformal and induced) to handle smooth mappings and degenerations such as branch points, folds, and orientation-reversing tubes, and it includes a boundary-compatible localization mechanism that yields an Euler-number weighted sum over worldsheet mappings with . The main result is a detailed, cross-checked mapping between worldsheet configurations (holomorphic, mixed, twists, ORTs) and Gross–Taylor’s large- Wilson-loop results at zero coupling on the sphere, plane, and torus, supporting the conjectured equality of the two descriptions in this limit. The work lays groundwork for finite-coupling extensions and generalizations to other gauge groups, matter content, and non-orientable sectors, thereby advancing a concrete string dual for 2D YM and informing broader holographic approaches to large- gauge theories.

Abstract

In earlier work we proposed a string theory dual to two dimensional Yang-Mills theory at zero coupling (which can also be thought of as a theory), given by a Polyakov-like generalization of Ho\v rava's topological rigid string theory, and we showed that it correctly reproduces (in the expansion) several partition functions of Yang-Mills theory. In the present paper, we generalise this to Wilson loop expectation values by adding boundaries with one Dirichlet and one Neumann boundary condition to our string worldsheets. We discuss in detail several examples, including examples where the worldsheet has branch points or orientation-reversing tubes, or where the Wilson loop has one or more self-intersections, and we show that in all of them the string theory reproduces the known Yang-Mills expectation values. We argue that examples with orientation-reversing tubes or self-intersecting Wilson loops cannot be brought to the conformal gauge, so we analyse them in a different gauge.
Paper Structure (37 sections, 102 equations, 10 figures)

This paper contains 37 sections, 102 equations, 10 figures.

Table of Contents

  1. Introduction and summary
  2. Review of results from the 1990s
  3. Gross Taylor: Partition function and Wilson loop
  4. Hořava's Harmonic Sigma model
  5. Hořava-Polyakov formalism with boundaries
  6. Branch points, folds and Wilson loop boundaries
  7. Conformal gauge
  8. Induced gauge
  9. The worldsheet path integral
  10. Wilson loop expectation values in 2D YM
  11. $G=0$, $\gamma_1$, $k=1$ : Disk
  12. $n_1=1$, $n_2=0$
  13. $n_1=2$, $n_2=1$
  14. $n_1 = \tilde{n}_1 = 1$, $\tilde{n}_2 = 1$ or $n_2 = \tilde{n}_2 = 1$, $n_1 = 1$
  15. $G=0$, $\gamma_8$, $k=1$ : Twist
  16. ...and 22 more sections

Figures (10)

  • Figure 1: Three Wilson loops on the sphere, with numbers denoting the different regions.
  • Figure 2: A different drawing of the "figure 8" Wilson loop, which is more suitable when the area of the second region goes to infinity.
  • Figure 3: A different drawing of the two-circle Wilson loop from figure \ref{['fig:All WL']}, which is more suitable when the second area goes to infinity.
  • Figure 4: A Wilson loop with 3 self-intersections, and four different contributions to its expectation value.
  • Figure 5: Two Wilson loops on the torus, depicting a specific worldsheet contributing to their expectation value.
  • ...and 5 more figures