M String, Monopole String and Modular Forms

TL;DR

The paper establishes a concrete link between six-dimensional M-string BPS spectra and five-dimensional monopole-string (m-string) moduli space elliptic genera, via a NS-limit regularization that restores necessary ISO(2) symmetry. It shows that M-string free energies $ ilde F^{(k_1, dots,k_{N-1})}$ encode the elliptic genera of relative m-string moduli spaces for gcd$(k_1, dots,k_{N-1})=1$, with Taub-NUT and Atiyah-Hitchin spaces serving as key checks. A central technical advance is the construction of holomorphic Jacobi forms $T^{(K)}$ by summing over all partitions of a fixed total $K$, which transform covariantly under congruence subgroups and are conjectured to be expressible via Hecke transforms of $T^{(1)}$. The results illuminate a rich modular structure underlying BPS bound-state counting and suggest a unified framework connecting M-strings, m-strings, and noncompact hyperkähler geometry through refined topological strings and holomorphic Jacobi forms. These insights advance the program of extracting elliptic genera for monopole moduli spaces from higher-dimensional string constructions and hint at broader dualities across M-/F-/Type II frames.

Abstract

We study relations between M-strings (one-dimensional intersections of M2-branes and M5-branes) in six dimensions and m-strings (magnetically charged monopole strings) in five dimensions. For specific configurations, we propose that the counting functions of BPS bound-states of M-strings capture the elliptic genus of the moduli space of m-strings. We check this proposal for the known cases, the Taub-NUT and Atiyah-Hitchin spaces for which we find complete agreement. Furthermore, we analyze the modular properties of the M-string free energies, which do not transform covariantly under SL(2,Z). However, for a given number of M-strings, we find that there exists a unique combination of unrefined genus-zero free energies that transforms as a Jacobi form under a congruence subgroup of SL(2,Z). These combinations correspond to summing over different numbers of M5-branes and make sense only if the distances between them are all equal. We explain that this is a necessary condition for the m-string moduli space to be factorizable into relative and center-of-mass parts.

Figures (4)

• 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: One cycle of the M2-brane wraps around the two-torus formed by $(x_1, x_5)$ by $(w_1, w_5) = (3, 2)$ times. The resulting M-string tension is given by $T = T_{M_2} (R_2 w_1 + R_5 w_5)$. Likewise, the M-wave propagates along the same cycle of the M2-brane. Note that the cycle lies within the M5-worldvolume.
• Figure 3: Topic diagram of the CY3fold which gives five-dimensional ${\cal N}=1^*$ theory with two choices for preferred direction. Here, $m$ is the Kähler parameter of the $\mathbb{P}^1$ which corresponds to the $(1,-1)$ line and $(\tau-m)$ is the Kähler parameter of the horizontal (red) line in (a).
• Figure 4: An M5-brane is contractible whenever on both sides of it a single M2-brane ends. The contracted M5-brane contributes $W(\tau, m, \epsilon_1, \epsilon_2)$ to the free energy.