Counting BPS Blackholes in Toroidal Type II String Theory

TL;DR

This paper derives a U-duality invariant formula for counting BPS black hole states in toroidal type II string theory, using a novel topological partition function despite the vanishing elliptic genus.

Contribution

It introduces a new counting method for BPS states in a D1-D5 system with toroidal compactification, incorporating extra symmetries and a topological partition function.

Findings

Derived a U-duality invariant degeneracy formula.

Found agreement with supergravity counting results.

Utilized a topological partition function with fermion number insertions.

Abstract

We derive a $U$-duality invariant formula for the degeneracies of BPS multiplets in a D1-D5 system for toroidal compactification of the type II string. The elliptic genus for this system vanishes, but it is found that BPS states can nevertheless be counted using a certain topological partition function involving two insertions of the fermion number operator. This is possible due to four extra toroidal U(1) symmetries arising from a Wigner contraction of a large $\mathcal{N}=4$ algebra $\mathcal{A}_{\kappa,\kappa'}$ for $\kappa' \to \infty$. We also compare the answer with a counting formula derived from supergravity on $AdS_3\times S^3 \times T^4$ and find agreement within the expected range of validity.