M-Strings

TL;DR

The paper investigates M-strings arising from M2-branes suspended between parallel M5-branes, computing their elliptic genus under twists preserving 2d $(2,0)$ supersymmetry. It unveils a deep link to a codimension-one parameter subspace where the elliptic genus matches that of a $(4,4)$ $A_{n-1}$ quiver theory, while for $N>1$ the result aligns with a $(4,0)$ sigma-model on the Hilbert scheme ${\rm Hilb}^N({\mathbb{R}}^4)$ with right-moving fermions coupled to a bundle $V=E\oplus E^*$. The analysis employs refined topological strings and Nekrasov instanton calculus to compute the topological partition functions and their modular properties, revealing domain-wall partition functions $D_{\nu_L\nu_R}$ that describe boundaries between varying numbers of M2-branes. Beyond the M-string sector, the work connects to M5-brane partition functions on ${S^4\times T^2}$ and ${S^5\times S^1}$, and checks consistency with elliptic genera of quiver theories in special limits. The results illuminate nontrivial bound-state structure induced by circle compactification and provide a unified framework for BPS degeneracies across dual descriptions, with potential extensions to ADE/D/E theories and domain-wall dynamics.

Abstract

M2 branes suspended between adjacent parallel M5 branes lead to light strings, the `M-strings'. In this paper we compute the elliptic genus of M-strings, twisted by maximally allowed symmetries that preserve 2d (2,0) supersymmetry. In a codimension one subspace of parameters this reduces to the elliptic genus of the (4,4) supersymmetric A_{n-1} quiver theory in 2d. We contrast the elliptic genus of N M-strings with the (4,4) sigma model on the N-fold symmetric product of R^4. For N=1 they are the same, but for N>1 they are close, but not identical. Instead the elliptic genus of (4,4) N M-strings is the same as the elliptic genus of (4,0) sigma models on the N-fold symmetric product of R^4, but where the right-moving fermions couple to a modification of the tangent bundle. This construction arises from a dual A_{n-1} quiver 6d gauge theory with U(1) gauge groups. Moreover we compute the elliptic genus of domain walls which separate different numbers of M2 branes on the two sides of the wall.

Figures (17)

• Figure 1: The brane and toric geometry. The red line marks mean to identify the toric legs or branes with each other and therefore describe a compactified direction which is associated to the gauge coupling $\tau$ in the gauge theory. The length of the $(1,1)$ branes is associated to the mass of the $\mathcal{N}=2^*$ theory. Last but not least the separation of the branes maps to the Coulomb branch parameter $t_f$ of the gauge theory.
• Figure 2: The system of M2 and M5 branes. The M5 branes are depicted in yellow whereas the M2 brane is blue. They intersect at the torus $T^2$ which is depicted in green.
• Figure 3: The torus $T^2$ and its cycles. In (a), the rectangular torus is depicted. While going around the circle with radius $R_1$ one twists with the mass rotation, and along the circle $R_0$ one introduces the $\epsilon_i$ rotations. In (b), the same geometry is depicted where we use the holomorphy of the result to move the twisting along the A-cycle by $m$ to twisting around the B-cycle by $t_m$.
• Figure 4: The toric diagram of elliptic Calabi-Yau threefold $X_{N}$.
• Figure 5: Toric diagram of the geometry giving rise to $SU(2)$${\cal N}=2^{*}$ theory. The preferred direction is taken to be vertical.