BPS Microstates and the Open Topological String Wave Function

TL;DR

The paper extends the OSV framework to open topological strings by conjecturing $Z_{\mathrm{BPS}}^{\mathrm{open}}=|\psi_{\mathrm{top}}^{\mathrm{open}}|^2$ for BPS states bound to open D-branes. It develops a microscopic testing ground using Type IIA on a local Calabi–Yau with a Riemann surface, where degeneracies are computable via $q$-deformed Yang–Mills, and shows a consistent factorization of open and closed sectors in a solvable example. It further interprets open string amplitudes through an open–closed duality with ghost branes and discusses the wave-function nature and background dependence of open topological strings, including the attractor-like moduli shifts. The work provides a concrete bridge between open string wave functions and BPS state counting, with implications for how non-perturbative completions of the open topological string encode black hole microphysics.

Abstract

It has recently been conjectured that the closed topological string wave function computes a grand canonical partition function of BPS black hole states in 4 dimensions: Z_BH=|psi_top|^2. We conjecture that the open topological string wave function also computes a grand canonical partition function, which sums over black holes bound to BPS excitations on D-branes wrapping cycles of the internal Calabi-Yau: Z^open_BPS=|psi^open_top|^2. This conjecture is verified in the case of Type IIA on a local Calabi-Yau threefold involving a Riemann surface, where the degeneracies of BPS states can be computed in q-deformed 2-dimensional Yang-Mills theory.

Figures (10)

• Figure 1: The disc $C$ in the fiber of ${\mathcal{L}}_{p+2g-2}$ over a point $P$ on the Riemann surface $\Sigma$; $C$ meets ${\mathcal{D}}$ only at $P$, and the boundary of $C$ lies on the Lagrangian submanifold representing this asymptotic infinity.
• Figure 2: A rough toric representation of the behavior of $X$ in a neighborhood of a singularity of the vector field $v$ described in the text. Two of the three $U(1)$ actions making up the toric fiber are the rotations of the line bundles ${\mathcal{L}}_{-p} \oplus {\mathcal{L}}_{p+2g-2}$ and the third is the action of $v$. The toric base of the divisor ${\mathcal{D}}$ on which the D4-branes are wrapped is indicated, as is the base of the Lagrangian submanifold representing the asymptotic infinity. The disc $C$ ends on this Lagrangian submanifold, meeting ${\mathcal{D}}$ at the single point $P$.
• Figure 3: The 1-cycle $\gamma$ and its dual cycle $D$ inside $L$.
• Figure 4: The operator $\delta_M\left(e^{i \oint_{\gamma} A}, e^{i \phi}\right)$ cuts $\Sigma$ into two pieces.
• Figure 5: The vertex representation of $X = {\mathcal{O}}(-p) \oplus {\mathcal{O}}(p-2) \to \mathbb C\mathbb P^1$, with a stack of $M$ branes with complexified holonomy $U = e^u$, a stack of infinitely many ghost branes with complexified holonomy $U'_1 = e^{u'_1}$, and a stack of infinitely many ghost antibranes with complexified holonomy $U'_2 = e^{u'_2}$.