Instanton-Monopole Correspondence from M-Branes on $\mathbb{S}^1$ and Little String Theory

TL;DR

The paper investigates M5–M2 brane configurations with a compact transverse circle, revealing a rich SL(2,Z)×SL(2,Z) modular structure and two dual gauge theories connected by fiber-base duality. Using refined topological string theory on the dual Calabi–Yau X_N, it constructs non-perturbative partition functions that can be expanded along either modular parameter, and proves that compact BPS counts are linear combinations of non-compact counts, uncovering a novel instanton–monopole correspondence and a T-duality between IIa and IIb little string theories. It further demonstrates how the NS-limit of the compact free energies encodes elliptic genera of affine A_{N-1} monopole moduli spaces and derives explicit relations between compact and non-compact counting functions for various configurations. The work provides new insights into tensionless string dynamics in six dimensions and proposes concrete conjectures for the elliptic and χ_y genera of monopole moduli spaces, with several nontrivial consistency checks via series expansions and modular properties.

Abstract

We study BPS excitations in M5-M2-brane configurations with a compact transverse direction, which are also relevant for type IIa and IIb little string theories. These configurations are dual to a class of toric elliptically fibered Calabi-Yau manifolds $X_N$ with manifest $SL(2, \mathbb{Z}) \times SL(2,\mathbb{Z})$ modular symmetry. They admit two dual gauge theory descriptions. For both, the non-perturbative partition function can be written as an expansion of the topological string partition function of $X_N$ with respect to either of the two modular parameters. We analyze the resulting BPS counting functions in detail and find that they can be fully constructed as linear combinations of the BPS counting functions of M5-M2-brane configurations with non-compact transverse directions. For certain M2-brane configurations, we also find that the free energies in the two dual theories agree with each other, which points to a new correspondence between instanton and monopole configurations. These results are also a manifestation of T-duality between type IIa and IIb little string theories.

Figures (3)

• Figure 1: Brane configuration: The M5-branes are all located at the origin in $\mathbb{R}^4_\perp$, wrapped around $\mathbb{T}^2$ and stretched along the $(6)$-direction.
• Figure 2: The toric web diagram of the Calabi-Yau threefold $X_N$ dual to the brane configuration. Both the horizontal and vertical directions are compactified on $\mathbb{S}^1$s, defining the diagram on $\mathbb{T}^2$.
• Figure 3: The building block of partition function of configuration of M5-branes wrapping a circle.