STU Black Holes and String Triality

TL;DR

The paper addresses how non-perturbative dualities organize the horizon entropy of supersymmetric black holes in an $\mathcal{N}=2$ STU model and its duals. It develops a symplectic-covariant framework for stabilization equations and derives a moduli-independent area formula with $[SL(2,\mathbb{Z})]^3$ symmetry, linking STU black holes to stringy $(S|TU)$ partners via a specific $Sp(8,\mathbb{Z})$ map. The authors provide explicit double-extreme STU black-hole solutions, show how moduli are fixed at the horizon, and establish an exact duality between the STU theory and its dual without a prepotential, including the relation $A^{\text{STU}}(p,q)=A^{\text{S|TU}}(\hat p,\hat q)$. This work broadens the understanding of black-hole entropy in Calabi–Yau moduli spaces, reveals a fully non-perturbative realization of $S,T,U$ dualities in the democratic STU framework, and suggests avenues for connecting 4D STU results to 5D area formulas and potential quantum corrections.

Abstract

We find double-extreme black holes associated with the special geometry of the Calabi-Yau moduli space with the prepotential F=STU. The area formula is STU-moduli independent and has ${[SL(2,Z)]}^3$ symmetry in space of charges. The dual version of this theory without prepotential treats the dilaton S asymmetric versus T,U-moduli. We display the dual relation between new (STU) black holes and stringy (S|TU) black holes using particular Sp(8, Z) transformation. The area formula of one theory equals that of the dual theory when expressed in terms of dual charges. We analyse the relation between (STU) black holes to string triality of black holes: (S|TU), (T|US), (U|ST) solutions. In the democratic STU-symmetric version we find that all three S and T and U duality symmetries are non-perturbative and mix electric and magnetic charges.

Figures (2)

• Figure 1: Duality transformations in the STU-model. The fundamental field strengths are not located on one side.
• Figure 2: Duality transformations in the S,T and U strings. The fundamental field strengths are located on one side and two duality symmetries are perturbative. The field strengths have different indices from Figure one, because the ones here are fundamental $S$-string fields.