Non-perturbative effects and wall-crossing from topological strings

TL;DR

This work demonstrates that the Gopakumar–Vafa invariants governing topological strings can compute and resum specific non-perturbative D-brane instanton effects, notably D1/D(-1) contributions to the hypermultiplet moduli space in 4d ${\cal N}=2$ IIB compactifications, and extends the framework to ${\cal N}=1$ via fluxes, orientifolds, and gauge branes. By employing the c-map, M-theory one-loop analyses, and T-duality, the authors connect D-brane instantons to topological-string amplitudes, revealing a mechanism by which non-perturbative terms remain continuous across walls of BPS stability, including threshold and marginal cases. They also offer a new physical interpretation for matrix-model instantons, relate DT and GV invariants in various chambers, and discuss how D6/D2/D0 and D5/D1/D(-1) sectors fit into this unified picture through twistor-space formalisms and linear approximations. These insights provide a computationally powerful, largely chamber-independent handle on non-perturbative effects with potential implications for moduli stabilization and phenomenology in string compactifications. Overall, the paper strengthens the link between topological-string theory and physical non-perturbative dynamics in four-dimensional effective theories, especially in how wall-crossing is encoded and ameliorated by topological data.

Abstract

We argue that the Gopakumar-Vafa interpretation of the topological string partition function can be used to compute and resum certain non-perturbative brane instanton effects of type II CY compactifications. In particular the topological string A-model encodes the non-perturbative corrections to the hypermultiplet moduli space metric from general D1/D(-1)-brane instantons in 4d N=2 IIB models. We also discuss the reduction to 4d N=1 by fluxes and/or orientifolds and/or D-branes, and the prospects to resum brane instanton contributions to non-perturbative superpotentials. We argue that the connection between non-perturbative effects and the topological string underlies the continuity of non-perturbative effects across lines of BPS stability. We also confirm this statement in mirror B-model matrix model examples, relating matrix model instantons to non-perturbative D-brane instantons. The computation of non-perturbative effects from the topological string requires a 3d circle compactification and T-duality, relating effects from particles and instantons, reminiscent of that involved in the physical derivation of the Kontsevich-Soibelmann wall-crossing formula.

Figures (5)

• Figure 1: Typical quiver theories describing D-branes at walls of threshold stability in 4d ${\cal N}=2$ (a) and ${\cal N}=1$ (b) theories.
• Figure 2: a) Toric diagram and web diagram for the complex cone over $dP_1$. For clarity, in the web diagram we show the collapsed cycles with some finite but small size. b) The quiver and dimer diagrams for the theory..
• Figure 3: a) Toric diagram and web diagram for the partial blow-up to a conifold singularity. For clarity, in the web diagram we show the collapsed cycles with some finite but small size, while the blown-up cycles are shown with finite and large size. b) The quiver and dimer diagram for the resulting conifold singularity. They are obtained from the $dP_1$ ones by recombining the nodes 1 with 3, and 2 with 4.
• Figure 4: Phases of the cycles $C_{12}$ (blue line), $C_{13}$ (purple line) and $C_{13}$ (grey line) in the generic large $N$ model with widened cuts. The phases are plotted against the distance $\delta$ of cut 2 to the real axis. There are three possible candidates for wall crossing. They correpond to the points where the three phases coincide (including $\delta=0$).
• Figure 5: Same as figure \ref{['cuts']}, but for the case where there is just one eigenvalue distributed among the extrema of the potential $W$, and the cuts are collapsed to points. Of the three points where the phases align, only those with $\delta\neq 0$ represent a real line of stability Townsend:1990.