On a modular property of N=2 superconformal theories in four dimensions

TL;DR

This note analyzes how the Schur index of 4d ${\cal N}=2$ SCFTs transforms under modular transformations of the torus parameter $\tau$, linking ${\it S^3\times S^1}$ partition functions at different radii. For Lagrangian theories of type $A_n$, a modified index ${\cal I}^T$ satisfies a simple modular relation that factors in the conformal central charge $c$ and the number of vector multiplets $n_V$, reflecting an ${\it SL(2,\mathbb{Z})}$ structure. The analysis is extended to the non-Lagrangian ${\cal N}=2$ SCFT ${\cal E}_6$ via Argyres-Seiberg duality, yielding a matching modular formula with $c_{\cal E_6}$ in the exponent. Overall, the results indicate a broad modular framework for Schur indices across class ${\cal S}$ theories and motivate a duality-invariant, geometric interpretation of these partition functions.

Abstract

In this note we discuss several properties of the Schur index of N=2 superconformal theories in four dimensions. In particular, we study modular properties of this index under SL(2,Z) transformations of its parameters.

Figures (2)

• Figure 1: The torus on the (complexified) flavor fugacity plane $z=e^{2\pi i \, \zeta}$. The two sides of the annulus are to be identified, $z\sim q\,z$. The dashed line is the unit circle: this is the ${\cal A}$-cycle of the torus around which the flavor symmetry is integrated over when it is being gauged.
• Figure 2: The modular transformation interchanges the integrations over the two cycles: before performing modular transformation the integration is along the ${\cal A}$-cycle of the torus in the gauge chemical potential complex plane and after the transformation the integration is along the ${\cal B}$-cycle of the dual torus. The dashed lines represent integration contours.