Complex-structure independence of large black brane cycle volumes

Let $X$ be a Calabi–Yau threefold. An even-dimensional homology class $[\Sigma]$ is called large when, for example, $[\Sigma]=N[\Sigma_0]$ with $N\gg1$. If the associated black brane equations of motion have a solution strictly in the interior of the Kähler cone and this solution is an attractor, the associated connected locally volume-minimizing representative $\Sigma$ is a large black brane cycle (LBBC). Let $t_0$ denote the corresponding attractor values of the moduli. Complex-structure independence conjecture. For an LBBC $\Sigma$ in $X$, at the attractor values $t_0$, the volume of $\Sigma$ is asymptotically independent of the complex-structure moduli; more precisely, $\lim_{N\to\infty}\operatorname{vol}([\Sigma])/N$ is independent of the complex-structure moduli and of $N$. This claim is presented as a physics proof motivated by the decoupling of complex-structure moduli from the non-holomorphic volume calculation, and concerns the large-charge asymptotic regime.