Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
Robbert Dijkgraaf, Lotte Hollands, Piotr Sulkowski, Cumrun Vafa
TL;DR
This paper provides a unifying framework that connects four-dimensional supersymmetric gauge theories to two-dimensional free fermion systems realized on I-branes formed by intersecting D4- and D6-branes along a spectral curve $\Sigma$. By exploiting a chain of dualities, it shows how gauge-theory data maps to topological-string amplitudes on non-compact Calabi–Yau manifolds, with the string coupling realized as a background $B$-field that renders the geometry non-commutative. Across the full quantum theory, higher-genus corrections are naturally encoded in holonomic $D$-modules, which serve as quantum deformations of the classical spectral curve. The work further derives the McKay–Nakajima correspondence and level-rank duality from the I-brane/affine-algebra perspective and extends these ideas to curved I-branes relevant for ${\cal N}=2$ theories, while highlighting the role of Donaldson–Thomas invariants and the topological vertex in counting BPS states. Overall, the paper provides a comprehensive string-theoretic and algebraic framework for understanding exact holomorphic quantities in SUSY gauge theories through intersecting branes, free fermions, and noncommutative geometry.
Abstract
We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-commutative. Including all string loop corrections the free fermion theory is elegantly formulated in terms of holonomic D-modules that replace the classical holomorphic curve in the quantum case.