Five-dimensional gauge theories and the local B-model

TL;DR

This paper develops a new algebraic framework to compute the prepotential of the topological B-model on local Calabi--Yau geometries related to five-dimensional super Yang--Mills theories, with tests confirming its accuracy.

Contribution

It introduces a novel algebraic approach to construct Picard--Fuchs equations for these geometries, applicable to both simply-laced and non-simply-laced gauge groups.

Findings

Accurate non-perturbative comparisons with gauge theory prepotentials.

Construction of Picard--Fuchs operators from Frobenius manifolds.

Ruling out certain integrable system proposals for non-simply-laced groups.

Abstract

We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi--Yau geometries related to the circle compactification of five-dimensional $\mathcal{N}=1$ super Yang--Mills theory with simple gauge group. In the simply-laced case, we construct Picard--Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated to the extended affine Weyl groups of type $\mathrm{ADE}$. In general, we propose a purely algebraic construction of Picard--Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg--Witten curve, encompassing non-simply-laced cases as well. We offer several precision tests of our proposal. Whenever a candidate spectral curve is known from string theory/brane engineering, we perform non-perturbative comparisons with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. We also employ our formalism to rule out some proposals from the theory of integrable systems of Seiberg--Witten geometries for non-simply laced gauge groups.