Flux vacua as supersymmetric attractors

TL;DR

The paper establishes that a class of Type IIB flux vacua on Calabi–Yau orientifolds with O3/O7 can be described by algebraic attractor equations analogous to those for supersymmetric black holes, recasting the DW=0, W ≠ 0 conditions into flux–moduli relations governed by the covariant central charge $Z = e^{K/2} W$. By making the $SL(2,\mathbb{Z})$ symmetry manifest and using a symplectic structure from special geometry, the authors derive explicit attractor relations $p^\Lambda = i \bar{Z} L^\Lambda - i Z \bar{L}^\Lambda$, $q_\Lambda = i \bar{Z} M_\Lambda - i Z \bar{M}_\Lambda$ and show a compact form for the 4-form flux: $F_4 = [e^{K} (\bar{W} \Omega + W \bar{\Omega})]_{fix} = 2 \mathrm{Re}(Z \hat{\bar{\Omega}})|_{fix}$. They illustrate the framework with an M-theory on K3×K3 example, where $|Z|_{fix}^2$ matches a black-hole entropy-like quantity $\sqrt{\det Q}$, highlighting a deep connection between flux vacua and attractor physics. The results offer a practical algebraic route to fix moduli from quantized fluxes and underpin KKLT-type constructions by linking flux-induced minima to black-hole attractor structure.

Abstract

We derive algebraic attractor equations describing supersymmetric flux vacua of type IIB string theory in terms of the doublet of the 3-form fluxes, F and H. These equations are similar to the attractor equations for moduli fixed by the charges near the horizon of the supersymmetric black holes.