Five-dimensional gauge theories and local mirror symmetry

Abstract

We study the dynamics of 5-dimensional gauge theory on M 4 × S 1 by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and shall show that the prepotential of the 5-dimensional SU (2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local F 2 , i.e. Hirzebruch surface F 2 lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit R → 0 (R denotes the radius of S 1 ) and also the result of the uncompactified 5-dimensional theory at R → ∞.We also discuss SU (2) gauge theory with 1 ≤ N f ≤ 4 matter in vector representations and show that they are described by the geometry of the local F 2 blown up at N f points.

Parabolic Refined Invariants and Macdonald Polynomials

“…In conclusion, one obtains a series of asymptotic refined invariants counted by equivariant K-theoretic invariants as in [53]. Similar results have been obtained in [37,43,19,53,33,34,20,32,41,44,35], where the resulting…”

“…Geometric engineering is used in Section 5 to relate the stable pair generating function (1.4) to a D-brane quiver quantum mechanical partition function. Analogous results in the physics literature were obtained in [37,43,19,53,33,34,20,32,41,44,35] while a general mathematical theory of geometric engineering is currently being developed by Nekrasov and Okounkov in [52]. The treatment in Section 5 follows the usual approach in the physics where χ T y (V(γ)) is the equivariant Hirzebruch genus of a vector bundle V(γ) on the nested Hilbert scheme N (γ).…”

5d E n Seiberg-Witten curve via toric-like diagram

“…When the geometry is toric, the corresponding SW curve can be computed also from the toric diagram which can be reinterpreted as the dual graph of the (p, q) 5-brane web [18]. It followed that the SW curve for five-dimensional Sp(1) gauge theory with N f ≤ 4 flavors was reproduced as mirror curve of the corresponding Calabi-Yau geometry [19]. While it can be toric for N f ≤ 5, the corresponding local del Pezzo surfaces for N f = 6, 7 are non-toric, and thus there have been difficulties finding corresponding (p, q) 5-brane web diagrams.…”

Wallcrossing and cohomology of the moduli space of Hitchin pairs

“…The integer p = deg(M 1 ) corresponds to the level of the five dimensional Chern-Simons term [57]. Therefore by analogy with [42,18,52,30,31,19,29,39,43,32], the refined topological string partition function of Z should be related with the equivariant instanton partition function Z (p) inst (Q, ǫ 1 , ǫ 2 , a 1 , a 2 , y), which has been constructed in [52]. As explained in detail in the next subsection, Z (p) inst (Q, ǫ 1 , ǫ 2 , a 1 , a 2 , y) is the generating function for the χ y -genus of a certain holomorphic bundle on a partial compactification of the instanton moduli space.…”