A Twist in the Dyon Partition Function

TL;DR

This paper introduces twisted indices in supersymmetric string theories, revealing their modular properties, wall-crossing behavior, and a novel connection to black hole entropy via orbifold geometries.

Contribution

It computes twisted indices in type IIB string theories on K3 x T^2 and T^6, uncovering their modular structure and explaining their growth with black hole entropy.

Findings

Twisted indices share properties with helicity trace indices.

Partition functions are modular forms of a subgroup of Sp(2,Z).

Large charge behavior relates to black hole entropy via orbifold geometries.

Abstract

In four dimensional string theories with N=4 and N=8 supersymmetries one can often define twisted index in a subspace of the moduli space which captures additional information on the partition function than the ones contained in the usual helicity trace index. We compute several such indices in type IIB string theory on K3 x T^2 and T^6, and find that they share many properties with the usual helicity trace index that captures the spectrum of quarter BPS states in N=4 supersymmetric string theories. In particular the partition function is a modular form of a subgroup of Sp(2,Z) and the jumps across the walls of marginal stability are controlled by the residues at the poles of the partition function. However for large charges the logarithm of this index grows as 1/n times the entropy of a black hole carrying the same charges where n is the order of the symmetry generator that is used to define the twisted index. We provide a macroscopic explanation of this phenomenon using quantum entropy function formalism. The leading saddle point corresponding to the attractor geometry fails to contribute to the twisted index, but a Z_n orbifold of the attractor geometry produces the desired contribution.