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.

We study two-dimensional N =(0, 2) supersymmetric gauged linear sigma models (GLSMs) using supersymmetric localization. We consider N =(0, 2) theories with an R-symmetry, which can always be defined on curved space by a pseudotopological twist while preserving one of the two supercharges of flat space. For GLSMs which are deformations of N =(2, 2) GLSMs and retain a Coulomb branch, we consider the A/2-twist and compute the genus-zero correlation functions of certain pseudo-chiral operators, which generalize the simplest twisted chiral ring operators away from the N =(2, 2) locus. These correlation functions can be written in terms of a certain residue operation on the Coulomb branch, generalizing the Jeffrey-Kirwan residue prescription relevant for the N =(2, 2) locus. For abelian GLSMs, we reproduce existing results with new formulas that render the quantum sheaf cohomology relations and other properties manifest. For non-abelian GLSMs, our methods lead to new results. As an example, we briefly discuss the quantum sheaf cohomology of the Grassmannian manifold. Keywords: Supersymmetry, Topological Field Theory. A. Conventions and review of N =(0, 2) supersymmetry 40 A.1 Curved space conventions 40 A.2 N =(0, 2) supersymmetry in flat space 40 B. Elementary properties of the Grothendieck residue 43 C. One-loop determinants 44 C.1 Matter determinant for A/2-twisted GLSM with (2, 2) locus 44 C.2 Matter determinant for the B/2-twisted model 46 D.Čech-cohomology-based results for the correlation functions 46 D.1 P 1 × P 1 46 D.2Čech-cohomology-based results for F 1 49 R[E I ] = r I + 1 , R[J I ] = 1 − r I , (2.49) and such that Tr( Λ I E I ) and Tr(Λ I J I ) are gauge invariant.Anomaly cancelation imposes further constraints on the matter content and on the R-charge assignment. Let us decompose the gauge algebra g into semi-simple factors g α

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We propose a new holographic program of gravity in which we introduce a surface stress tensor. Our proposal differs from Verlinde's in several aspects. First, we use an open or a closed screen. Second, the temperature is not necessary, but a surface energy density and pressure are introduced. The surface stress tensor is proportional to the extrinsic curvature. Third, the energy we use is Brown-York energy and the equipartition theorem is violated by a non-vanishing surface pressure. We discuss holographic thermodynamics of a gas of weak gravity and find a chemical potential, and then show that Verlinde's program does not lead to reasonable thermodynamics. The holographic entropy is similar to the Bekenstein entropy bound. entropic gravity, entropy bound, the holographic principle

In this paper we extend the nonabelian mirror proposal of two of the authors from twodimensional gauge theories with connected gauge groups to the case of O(k) gauge groups with discrete theta angles. We check our proposed extension by counting and comparing vacua in mirrors to known dual two-dimensional (S)O(k) gauge theories. The mirrors in question are Landau-Ginzburg orbifolds, and for mirrors to O(k) gauge theories, the critical loci of the mirror superpotential often intersect fixed-point loci, so that to count vacua, one must take into account twisted sector contributions. This is a technical novelty relative to mirrors of gauge theories with connected gauge groups, for which critical loci do not intersect fixed-point loci and so no orbifold twisted sector contributions are pertinent. The vacuum computations turn out to be a rather intricate test of the proposed mirrors, in particular as untwisted sector states in the mirror to one theory are often exchanged with twisted sector states in the mirror to the dual. In cases with nontrivial IR limits, we also check that central charges computed from the Landau-Ginzburg mirrors match those expected for the IR SCFTs.