Five-dimensional gauge theories and the local B-model

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 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 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.

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