M2-brane surface operators and gauge theory dualities in Toda

TL;DR

The work links M2-brane surface operators in class S theories to 2d ${\cal N}=(2,2)$ quivers and to degenerate operators in ${A}_{N_f-1}$ Toda CFT, showing that sphere partition functions with surface insertions equal Toda correlators with a degenerate puncture. It provides a detailed dictionary connecting Fayet-Iliopoulos parameters to puncture positions and uses this to realize 2d Seiberg- and Kutasov-- Schwimmer-type dualities as geometric symmetries (braiding, crossing, and momentum conjugation) of Toda blocks. The paper further derives new Toda CFT data: explicit conformal blocks, braiding matrices, and fusion rules, and extends the duality web to arbitrary Toda degenerates via quivers and collisions yielding irregular punctures. This geometric perspective highlights deep connections between 4d/2d coupled systems, surface operator dynamics, and the algebraic structure of Toda CFT with potential implications for exact results in class S theories.

Abstract

We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.

Figures (3)

• Figure 1: Example of how multiple Toda CFT degenerate punctures map to a quiver gauge theory.
• Figure 2: A 4d/2d quiver, its corresponding brane diagram, and Toda CFT correlator.
• Figure 3: Sequence of Seiberg dualities on the quiver with all $\epsilon_k=-1$.