Supersymmetric gauge theories, intersecting branes and free fermions

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 twodimensional 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 noncommutative. 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.

The omega deformation from string and M-theory

“…A similar non-commutativity also appears in the related limit of the Ω-deformation corresponding to the topological string [18,19]. In our case the geometric interpretation of the quantization is more direct, with the symplectic form on the curve realized as the pullback of M-theory four-form flux to an M5-brane.…”

“…[19,33,39]). By realizing the four-dimensional gauge theory as the dynamics of a geometrically engineered Calabi-Yau singularity in type IIA string theory, the gauge theory can be lifted to a five-dimensional theory living on the same singularity in M-theory, and then re-compactified with Melvin boundary conditions to yield an Ω-deformed gauge theory in four dimensions, which is ultraviolet-completed to the topological string on the Calabi-Yau singularity.…”

“…The second factor x 6,8,9,10 = C 2 has a complex structure with respect to which the Riemann surface is embedded holomorphically. With respect to that complex structure, the flux ω C 2 in [19] has Hodge numbers (2, 0) and (0, 2) only, and consequently has vanishing integral on the Riemann surface. In the solutions presented here, the flux ω C 2 has a component with Hodge numbers (1, 1) and generically has a nonzero integral along the Riemann surface (see eq.…”

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

“…Conjecturally it should be related to a higher rank version of local Donaldson-Thomas theory of curves provided the later were defined. This construction provides a mathematical framework for some of the results obtained in the string theory literature [25,20].…”

Nonsupersymmetric flux vacua and perturbed 𝒩 = 2 systems

“…In particular, the SU(N) Seiberg-Witten curve of gauge theory [48,49] is geometrically identified with the curve (3.14) underlying the Calabi-Yau. A T -duality along the compact circle in the uv-fiber, followed by a lift to M-theory, translates [50] this geometry into a system of an M5-brane which wraps the Riemann surface Σ SW and fills R 3,1 . In the IIA limit, this system is related to a HananyWitten type brane configuration in type IIA, where one has two NS5-branes with N D4-branes stretching between them [51,52].…”