Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A & B model approaches to surface operators and Toda theories

Can Kozcaz, Sara Pasquetti, Niclas Wyllard

TL;DR

The paper develops a cohesive, multi-formalism framework for surface operators in 4d N=2 SU(2) theories and their Toda/Liouville duals, unifying B-model topological recursion, A-model toric-brane methods, and AGT-inspired instanton counting. It demonstrates that multiple surface operators correspond to multiple toric branes and to degenerate CFT insertions, with explicit checks in T_2 and SU(2) with N_f=4, including open amplitudes, mirror maps, and 4d/5d limits. The results provide a calculationally robust bridge between CFT degenerate states and gauge-theory observables across duality frames, and they open new avenues for studying 2d Toda theories and nonperturbative dualities. This work thus extends the AGT correspondence to surface operators, offering a powerful toolkit for exploring strong-coupling regimes and higher-rank generalizations.

Abstract

It has recently been argued by Alday et al that the inclusion of surface operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d N=2 gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.

A & B model approaches to surface operators and Toda theories

TL;DR

The paper develops a cohesive, multi-formalism framework for surface operators in 4d N=2 SU(2) theories and their Toda/Liouville duals, unifying B-model topological recursion, A-model toric-brane methods, and AGT-inspired instanton counting. It demonstrates that multiple surface operators correspond to multiple toric branes and to degenerate CFT insertions, with explicit checks in T_2 and SU(2) with N_f=4, including open amplitudes, mirror maps, and 4d/5d limits. The results provide a calculationally robust bridge between CFT degenerate states and gauge-theory observables across duality frames, and they open new avenues for studying 2d Toda theories and nonperturbative dualities. This work thus extends the AGT correspondence to surface operators, offering a powerful toolkit for exploring strong-coupling regimes and higher-rank generalizations.

Abstract

It has recently been argued by Alday et al that the inclusion of surface operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d N=2 gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.

Paper Structure

This paper contains 19 sections, 134 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of surface operators and summary of conjectures
  3. CFT approach: Degenerate operators in Toda field theory
  4. The $\mathcal{T}_{3,0}(A_1)$ theory
  5. The $\mathcal{T}_{4,0}(A_1)$ theory
  6. $\mathcal{W}$-algebra degenerate states and $\mathrm{SU}(N)$ surface operators
  7. Number of parameters in the general Toda three-point function
  8. B-model approach: The topological recursion
  9. Surface operators in the $T_2$ theory
  10. Surface operators in the $\mathrm{SU}(2)$ theory with $N_f=4$
  11. A-model approach: Toric branes and the topological vertex
  12. Toda theory (chiral) three-point functions from toric geometry?
  13. Some facts about the geometric engineering of $\mathrm{SU}(N)$ theories
  14. Surface operators as toric branes
  15. AGT approach: Gauge theory and instanton counting
  16. ...and 4 more sections

Figures (8)

  • Figure 1: The toric diagram for the $T_2$ geometry.
  • Figure 2: Closed topological vertex with a choice of preferred direction.
  • Figure 3: The toric diagram for the $T_3$ geometry.
  • Figure 4: The toric version of $\mathrm{SU}(N)$ with $N_f=2N_c$.
  • Figure 5: The $\widetilde{T}_2$ strip.
  • ...and 3 more figures