Freezing of Moduli by N=2 Dyons

TL;DR

In d=4, N=2 supergravity, the authors define double-extreme black holes as extremal, supersymmetric solutions with coincident horizons and moduli frozen to charge-dependent values. They derive stabilization equations linking electric/magnetic charges (p^Λ,q_Λ) to the vector-multiplet moduli through the central charge Z and covariantly holomorphic sections, showing the ADM mass M equals the extremal value |Z| when the moduli are fixed. They obtain explicit fixed moduli for two models: (i) the prepotential F = -i X^0 X^1, yielding z = (q_0 - i p^1)/(q_1 - i p^0) and M^2 = |q_0 q_1 + p^0 p^1|; (ii) the ST[2,n] family (including STU) with S = (p·q)/p^2 - i sqrt(p^2 q^2 - (p·q)^2)/p^2 and analogous T,U expressions, with M^2 = sqrt(p^2 q^2 - (p·q)^2). The results demonstrate concrete, charge-driven moduli stabilization in N=2 vacua and highlight how hypermultiplet moduli decouple from this stabilization, with implications for moduli stabilization and potential SUSY-breaking scenarios.

Abstract

In N=2 ungauged supergravity we have found the most general double-extreme dyonic black holes with arbitrary number n_v of constant vector multiplets and n_h of constant hypermultiplets. They are double-extreme: 1) supersymmetric with coinciding horizons, 2) the mass for a given set of quantized charges is extremal. The spacetime is of the Reissner-Nordstrom form and the vector multiplet moduli depend on dyon charges. As an example we display n_v complex moduli as functions of 2(n_v+1) electric and magnetic charges in a model related to a classical Calabi-Yau moduli space. A specific case includes the complex S, T, U moduli depending on 4 electric and 4 magnetic charges of 4 U(1) gauge groups.