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 S5 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.
Highlights
Compute relevant amplitudes for supersymmetric partition functions of superconformal theories
The perturbative parts of the superconformal partition functions were computed for certain gauge theories on S5 [3,4,5,6,7], and using this ingredient and the condition that the BPS content captured by topological strings behaves as the fundamental degrees of freedom of the theory, an idea advanced in [2], we propose a way to compute the full answer for superconformal partition functions on S5, and a non-perturbative definition for topological strings
Upon further compactification on S5, we can use the resulting nonperturbative topological string on elliptic Calabi-Yau threefold to compute the partition function on S5. This leads to the partition function of the 6d theory on S1 × S5, i.e. it leads to the computation of the 6d superconformal index, where mj correspond to fugacities for flavor symmetry and τ1,2 correspond to parameters of supersymmetric rotations on S5
Summary
One of the common themes that have emerged in the study of superconformal theories in various dimensions is the important role played by the BPS states that arise when one moves away from the superconformal fixed point (see [2] and references therein). We are interested in computing the partition functions of these theories on S3 and S5, respectively To this end, it is instructive to review the case of N = 2 superconformal theories on the squashed three-sphere Sb3. We recall the partition function for superconformal gauge theories on the squashed three-sphere, whose gauge and matter content are provided respectively by vector and chiral multiplets. We would like to clarify the relation with BPS states and open topological string theory For this purpose, it is convenient to strip away the prefactors from the double sine function and define πi S2(z|ω1, ω2) = exp − 2 B2,2(z|ω1, ω2) S2(z|ω1, ω2). Using the building block of the double sine function we can write down the contribution of particles of charges ni, nj under U(1) gauge factors and flavor factors respectively with central terms (xi, mj) (before gauging) and spins s:. We discuss how this can be presented in the context of 3d theories living on M5 branes wrapped on special Lagrangian 3-cycles, using open topological string amplitudes
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