On Crossing Symmmetry and Modular Invariance in Conformal Field Theory and S Duality in Gauge Theory

TL;DR

The paper investigates how S-duality in 4d gauge theories can be derived from 2d CFT structures, specifically crossing symmetry and modular invariance in Liouville theory, within the AGT framework. Using the AGT correspondence, it relates the partition functions of $N=2$ $SU(2)$ with four fundamentals and $N=4$ $SU(2)$ to Liouville correlation functions on a punctured sphere and on a torus, with the coupling data encoded in the surface's complex structure. The main results show explicit duality transformations: the four-point Liouville crossing symmetry reproduces the $q' = 1/q$ map for the $N=2$ theory, while the Liouville torus partition function yields the full $N=4$ partition function (including $U(1)$ factors) under modular transformations, consistent with the AGT dictionary and parameter identifications $\epsilon_1=b$, $\epsilon_2=1/b$, $Q=b+1/b$. The work strengthens the 6d $(0,2)$ class S viewpoint, generalizes to Toda theories for higher rank, and provides a concrete computational bridge between gauge theory dualities and CFT data, with implications for extracting spectral and Seiberg–Witten information from Liouville/Toda theories.

Abstract

In this note, we explore the relation between crossing symmetry and modular invariance in conformal field theory and S-duality in gauge theory. It is shown that partition functions of different S dual theories of N=2 SU(2) gauge theory with four fundamentals can be derived from the crossing symmetry of the Liouville four point function. We also show that the partition function of N=4 SU(2) gauge theory can be derived from the Liouville partition function on torus.

Figures (4)

• Figure 1: An elastic scattering with incoming particles of momentum $p_1$, $p_2$ and outgoing particles of momentum $p_3$, $p_4$. We indicate the contribution from $s$ channel and $t$ channel. The field theory amplitude is constructed from the sum of those contributions.
• Figure 2: Crossing symmetry of four point function in conformal field theory.
• Figure 3: Different degeneration limits of the punctured sphere, this corresponds to different weakly coupled S-dual theory of $N=2$$SU(2)$ gauge theory with four fundamentals.
• Figure 4: On the left hand side is the Riemann surface associated with the $SU(2)$ gauge theory in one particular S dual frame, the Coulomb branch parameter is $a$. One the right hand side, we draw the $s$ channel contribution to the four point function of Liouville theory.