In this paper we propose a systematic construction of mirrors of nonabelian two dimensional (2,2) supersymmetric gauge theories. Specifically, we propose a construction of B-twisted Landau-Ginzburg orbifolds whose correlation functions match those of A-twisted supersymmetric gauge theories, and whose critical loci reproduce quantum cohomology and Coulomb branch relations in A-twisted gauge theories, generalizing the Hori-Vafa mirror construction. We check this proposal in a wide variety of examples. For instance, we construct mirrors corresponding to Grassmannians and two-step flag manifolds, as well as complete intersections therein, and explicitly check predictions for correlation functions and quantum cohomology rings, as well as other properties. We also consider mirrors to examples of gauge theories with U(k), U(k 1 ) × U(k 2 ), SU(k), SO(2k), SO(2k + 1), and Sp(2k) gauge groups and a variety of matter representations, and compare to results in the literature for the original two dimensional gauge theories. Finally, we perform consistency checks of conjectures of Aharony et al that a two dimensional (2,2) supersymmetric pure SU(k) gauge theory flows to a theory of k − 1 free twisted chiral multiplets, and also consider the analogous question in pure SO(3) theories. For one discrete theta angle, the SO(3) theory behaves the same as the SU(2) theory; for the other, supersymmetry is broken. We also perform consistency checks of analogous statements in pure supersymmetric SO and Sp gauge theories in two dimensions.

A graded quiver with superpotential is a quiver whose arrows are assigned degrees c ∈ {0, 1, · · · , m}, for some integer m ≥ 0, with relations generated by a superpotential of degree m − 1. Ordinary quivers (m = 1) often describe the open string sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d N = 1 supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with m = 2 and m = 3 similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2d N = (0, 2) and 0d N = 1 gauge theories, respectively. In this work, we further explore the correspondence between m-graded quivers with superpotential, Q (m) , and CY (m + 2)-fold singularities, X m+2 . For any m, the open string sector of the topological B-model on X m+2 can be described in terms of a graded quiver. We illustrate this correspondence explicitly with a few infinite families of toric singularities indexed by m ∈ N, for which we derive "toric" graded quivers associated to the geometry, using several complementary perspectives. Many interesting aspects of supersymmetric quiver gauge theories can be formally extended to any m; for instance, for one family of singularities, dubbed C(Y 1,0 (P m )), that generalizes the conifold singularity to m > 1, we point out the existence of a formal "duality cascade" for the corresponding graded quivers. arXiv:1811.07016v1 [hep-th]

In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study (0,2) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring.December 2015

We study 2d N = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY 4 ) singularities. On general grounds, the holomorphic sector of these theories-matter content and (classical) superpotential interactions-should be fully captured by the topological B-model on the CY 4 . By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A ∞ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY 4 geometry. We also suggest a relation between triality of N = (0, 2) gauge theories and certain mutations of exceptional collections of sheaves. 0d N = 1 supersymmetric quivers, corresponding to D-instantons probing CY 5 singularities, can be discussed similarly. Contents 1. Introduction 2 2. D1-brane quivers and 2d N = (0, 2) theories 7 2.1 N = (0, 2) quiver gauge theory from B-branes at a CY 4 singularity 8 2.2 D1-brane on C 4 15 2.3 Orbifolds C 4 /Γ 20 2.4 Fractional branes on a local P 3 24 2.5 Fractional branes on a local P 1 × P 1 38 3. Triality and mutations of exceptional collections 44 3.1 Triality acting on N = (0, 2) supersymmetric quivers 45 3.2 Triality and the C 4 /Z 4 quiver 47 3.3 Triality from mutation-a conjecture 51 4. D-instanton quivers and gauged matrix models 55 4.1 N = 1 gauged matrix model from B-branes at a CY 5 singularity 56 4.2 D(−1)-brane on C 5 58 4.3 Orbifolds C 5 /Γ 60 A. Dimensional reductions 63 B. Fractional D3-branes on a local P 2 65 B.1 Fractional branes and supersymmetric quivers 65 B.2 Seiberg duality as mutation 69 C. A ∞ structure and N = (0, 2) quiver 70 C.1 An algebraic preliminary 70 C.2 Ext algebra and N = (0, 2) quiver 71 C.3 General procedure to compute the higher products 73 D. Higher products on a local P 1 × P 1 74-1 -1 That is, fields X IJ in the fundamental of U (N I ) and in the antifundamental of U (N J ). 2 Very recently, these 2d and 0d quivers were also related to cluster algebras [22].10 This expression is only formal. The N = (0, 2) superpotential that appears in the gauge theory Lagrangian is the usual Tr Λ I J I (X)), since superspace treats Λ andΛ asymmetrically. This formal W first appeared in [22]. It elegantly encodes the algebraic structure of the N = (0, 2) quiver relations. This point is further discussed in Appendix C.11 That is, a family of maps satisfying certain consistency conditions [59].The J-terms of the remaining 8 fields are given explicitly by:One can check that:Tr(E Γ (s) J Γ (s) ) = Tr C i M jk A k D ij , (2.81)

In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2) mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian manifolds.