Elliptic hypergeometric integrals and 't Hooft anomaly matching conditions

TL;DR

This work shows that all 't Hooft anomaly matching conditions for Seiberg dual pairs can be derived from the $SL(3,\mathbb{Z})$-modular transformation properties of the kernels of dual superconformal indices, realized via modified elliptic gamma functions. By constructing modified indices $I_E^{mod}$ and $I_M^{mod}$ and proving their modular equivalence, the authors connect modular invariance to anomaly coefficients through Diophantine relations that reproduce known anomalies such as $SU(N_f)^3$ and $U(1)_R^3$. They demonstrate that the previously proposed total ellipticity condition is not sufficient for all anomalies, highlighting the central role of modularity in encoding anomaly matching. The results offer a universal, mathematically grounded framework for dualities in 4d $\mathcal{N}=1$ theories and point to deep connections with potential physical interpretations on spaces like $\mathbb{T}^3\times\mathbb{R}$ and dimensional reductions to 3d theories.

Abstract

Elliptic hypergeometric integrals describe superconformal indices of 4d supersymmetric field theories. We show that all 't Hooft anomaly matching conditions for Seiberg dual theories can be derived from $SL(3,\mathbb{Z})$-modular transformation properties of the kernels of dual indices.

Figures (1)

• Figure 1: A one-loop triangle diagram describing anomalies.

Theorems & Definitions (3)

• Theorem 1: Rains, Spiridonov, 2004
• Theorem 2
• Theorem 3