Superconformal Partition Functions and Non-perturbative Topological Strings

TL;DR

The paper introduces a non-perturbative definition of refined topological strings, enabling exact computations of five-dimensional superconformal partition functions on $S^5$ and six-dimensional superconformal indices via a product of three topological-string amplitudes with $SL(3,\mathbb{Z})$-type coupling inversions. By expressing the $S^5$ partition function as a Coulomb-branch integral that sums over all BPS states, the authors connect to a non-perturbative completion of topological strings, implemented through a three-factor structure of $Z^{top}$ with modular transforms. They provide concrete constructions, including the massless vector contributions, a detailed SU(2) example, and analytic properties of the resulting $Z$, and they outline a possible M-theory derivation from twisted Taub-NUT setups. Extending to 6d, they derive superconformal indices for ${\cal N}=(1,0)$ and ${\cal N}=(2,0)$ theories, including multiple M5-branes, and propose intriguing modular identities that may simplify the 6d/5d/4d web of indices via elliptic Calabi–Yau geometries. The work suggests a unifying framework in which BPS data from topological strings fully capture supersymmetric partition functions across dimensions, with potential practical implications for exactly solvable sectors of M-theory compactifications.

Abstract

We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S^5 and the superconformal index of a large number of 6 dimensional (2,0) and (1,0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3,Z) modular transformations acting by inverting the coupling constants of the refined topological string play a key role.

Figures (5)

• Figure 1: $\mathbb{P}^1\times \mathbb{P}^1$ geometry corresponding to SU(2) theory on the squashed five-sphere. The non-perturbative topological string computed from this geometry is to be integrated over $a$.
• Figure 2: Squashed $S^3$ viewed as a torus fibered over the interval. At the ends of the interval one of the two circles degenerates. On the left half of the geometry, as one goes around the red (dashed) circle, the second circle is twisted by $2\pi i\tau$. In gluing the left and right halves, one must interchange the two circles of $T^2$. On the right half, in going around the blue circle the red circle gets twisted by $-2\pi i/\tau$.
• Figure 3: Squashed $S^5$ as a $T^3$ fibration over a triangle: the cube, whose opposite faces are identified, represents the torus. At the edges of the triangles the torus collapses to a $T^2$; at the vertices it collapses to $S^1$. At each vertex we also display the correctly normalized equivariant parameters corresponding to the three circles.
• Figure 4: Toric diagram for the geometry that engineers the $\mathcal{N}=2^*$$U(1)$ theory in five dimensions. The toric plane is compactified to a cylinder, and the horizontal edges are identified with each other.
• Figure 5: The periodic toric geometry for the $A_1$ case. For $A_{n-1}$ case we get $n$ horizontal lines. The $t_i$ are the Coulomb branch parameters and $m$ corresponds to the mass of the adjoint in the $N=2^*$ theory.