Siegel Paramodular Forms and Sparseness in AdS$_3$/CFT$_2$

TL;DR

This paper explores the use of Siegel paramodular forms to count polar states in symmetric product orbifold CFTs, providing new formulas and insights into the sparsity of spectra relevant for AdS$_3$/CFT$_2$ dualities.

Contribution

It introduces five specific examples of Siegel paramodular forms with exact counting formulas, including new cases that suggest dual supergravity theories.

Findings

Reproduces known supergravity counting results.

Provides four new counting formulas for polar states.

Shows the low energy spectrum is very sparse in these models.

Abstract

We discuss the application of Siegel paramodular forms to the counting of polar states in symmetric product orbifold CFTs. We present five special examples and provide exact analytic counting formulas for their polar states. The first example reproduces the known result for type IIB supergravity on AdS$_3\times S^3\times K3$, whereas the other four examples give new counting formulas. Their crucial feature is that the low energy spectrum is very sparse, which suggests the existence of a suitable dual supergravity theory. These examples open a path to novel realizations of AdS$_3$/CFT$_2$.