Symmetric Group Gauge Theories and Simple Gauge/String Dualities

Abstract

We study two-dimensional topological gauge theories with gauge group equal to the symmetric group $S_n$ and their string theory duals. The simplest such theory is the topological quantum field theory of principal $S_n$ fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of $n$ are coded in a partial permutation algebra. We propose a generalization of the partial permutation algebra to any symmetric orbifold topological quantum field theory and show that the latter theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov-Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

arXiv comment36 pages, 7 figures, Journal version. Augmented introduction

References in corpus (31)

• Matrix String Theory
• Two Dimensional QCD is a String Theory
• Elliptic Genera of Symmetric Products and Second Quantized Strings
• Emergent Spacetime and Holographic CFTs
• Correlation functions for M^N/S_N orbifolds
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Two Dimensional QCD as a String Theory
• Three-point functions for M^N/S^N orbifolds with N=4 supersymmetry
• Shifted Schur Functions
• Twists and Wilson Loops in the String Theory of Two Dimensional QCD
• Deriving the ${\text{AdS}}_3/{\text{CFT}}_2$ Correspondence
• Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
• Quantum Cohomology and Virasoro Algebra
• Diagrams for Symmetric Product Orbifolds
• Gravity from CFT on $S^N(X)$ CFT: Symmetries and Interactions
• D-branes and K-theory in 2D topological field theory
• A perturbative CFT dual for pure NS-NS AdS${}_3$ strings
• Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds
• Lattice Topological Field Theory in Two Dimensions
• Partition functions of the tensionless string
• The equivariant Gromov-Witten theory of P^1
• Spin Hurwitz numbers and topological quantum field theory
• What is the Simplest Gauge-String Duality?
• Localization of virtual classes
• On double Hurwitz numbers with completed cycles
• Lecture notes on 2-dimensional defect TQFT
• The Topological Symmetric Orbifold
• Second quantized Frobenius algebras
• 2d TQFT and baby universes
• Virasoro constraints for target curves
• The Chiral Ring of a Symmetric Orbifold and its Large N Limit