Seiberg-Witten prepotential for E-string theory and random partitions

TL;DR

This paper derives a Nekrasov-type formula for the Seiberg-Witten prepotential in six-dimensional E_8 string theory compactified to four dimensions, revealing its modular properties and anomaly behavior.

Contribution

It provides the first explicit Nekrasov-type expression for the E-string theory prepotential, connecting BPS states, modularity, and winding/momentum contributions.

Findings

Derived a Nekrasov-type formula for the E-string prepotential

Confirmed the modular properties of the prepotential

Proved the prepotential satisfies the modular anomaly equation

Abstract

We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E_8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E_8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.