Modular Symmetries of N=2 Black Holes

TL;DR

The paper addresses how classical extremal $N=2$ black holes in $S$-$T$-$U$ models transform under dualities, using the triality group to make horizon moduli and entropy manifestly duality-invariant. It derives explicit horizon values for $S$, $T$, and $U$ from extremizing the central charge, and shows the entropy obeys a triality-invariant formula ${\cal S}/\pi = \sqrt{ \langle M,M\rangle \langle N,N\rangle - (M\cdot N)^2 } = \langle N,N\rangle \Re S$. The work classifies entropy-vanishing configurations, connects to $N=4$ heterotic compactifications, and discusses how one-loop perturbative corrections break triality, proposing a conjectured one-loop entropy ${\cal S}_{1-\text{loop}} = \pi \langle N,N\rangle \Re S_{invar}$ with an invariant dilaton $S_{invar}$; it also outlines non-perturbative implications via dualities to $IIA$ on Calabi–Yau manifolds and associated monodromies. Overall, it clarifies the symmetry structure of the classical BPS spectrum and lays groundwork for incorporating quantum and non-perturbative corrections in a duality-consistent way.

Abstract

We discuss the transformation properties of classical extremal N=2 black hole solutions in S-T-U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest. We also comment on quantum corrections and we make a conjecture for the one-loop corrected entropy.