Kentaro Hori

Mirror symmetry in three-dimensional gauge theories, quivers and D-branes

We construct and analyze dual N = 4 supersymmetric gauge theories in three dimensions with unitary and symplectic gauge groups. The gauge groups and the field content of the theories are encoded in quiver diagrams. The duality exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters. We analyze the classical and the quantum moduli spaces of the theories and construct an explicit mirror map between the mass parameters and the Fayet-Iliopoulos parameters of the dual. The results generalize the relation between ALE spaces and moduli spaces of SU(n) and SO(2n) instantons. We interpret some of these results from the string theory viewpoint, for SU(n) by analyzing T-duality and extremal transitions in type II string compactifications, for SO(2n) by using D-branes as probes. Finally, we make a proposal for the moduli space of vacua of these theories in the absence of matter.

BRANES AND N = 2 THEORIES IN TWO DIMENSIONS

Type IIA brane configurations are used to construct N = 2 supersymmetric gauge theories in two dimensions. Using localization of chiral multiplets in ten-dimensional space-time, supersymmetric non-linear sigma models with target space such as CPn−1 and the Grassmann manifolds are studied in detail. The quantum properties of these models are realized in M-theory by taking the strong Type IIA coupling limit. The brane picture implies an equivalence between the parameter space of N = 2 supersymmetric theories in two dimensions and the moduli space of vacua of N = 2 supersymmetric gauge theories in four dimensions. Effects like level-rank duality are interpreted in the brane picture as continuation past infinite coupling. The BPS solitons of the CPn−1 model are identified as topological excitations of a membrane and their masses are computed. This provides the brane realization of higher rank tensor representations of the flavor group.

Dynamics of N = 2 Supersymmetric Gauge Theories in Three Dimensions

We study the structure of the moduli spaces of vacua and superpotentials of N = 2 supersymmetric gauge theories in three dimensions. By analyzing the instanton corrections, we compute the exact superpotentials and determine the quantum Coulomb and Higgs branches of the theories in the weak coupling regions. We find candidates for non-trivial N = 2 superconformal field theories at the singularities of the moduli spaces. The analysis is carried out explicitly for gauge groups U(Nc) and SU(Nc) with N f flavors. We show that the field theory results are in complete agreement with the intersecting branes picture. We also compute the exact superpotentials for arbitrary gauge groups and arbitrary matter content.

Mirror Symmetry in Three-Dimensional Gauge Theories, SL(2, Z) and D-Brane Moduli Spaces

We construct intersecting D-brane configurations that encode the gauge groups and field content of dual N = 4 supersymmetric gauge theories in three dimensions. The duality which exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters is derived from the SL (2, Z) symmetry of the type IIB string. Using the D-brane configurations we construct explicitly this mirror map between the dual theories and study the instanton corrections in the D-brane world-volume theory via open string instantons. A general procedure to obtain mirror pairs is presented and illustrated. We encounter transitions among different field theories that correspond to smooth movements in the D-brane moduli space. We discuss the relation between the duality of the gauge theories and the level-rank duality of affine Lie algebras. Examples of other dual theories are presented and explained via T-duality and extremal transitions in type II string compactifications. Finally we discuss a second way to study instanton corrections in the gauge theory, by wrapping 5-branes around six-cycles in M-theory compactified on a Calabi-Yau 4-fold.

Branes and mirror symmetry in N = 2 supersymmetric gauge theories in three dimensions

We use brane configurations and SL(2, Z) symmetry of the type IIB string to construct mirror N = 2 supersymmetric gauge theories in three dimensions. The mirror map exchanges Higgs and Coulomb branches, Fayet-Iliopoulos and mass parameters and U(1)R symmetries. Some quantities that are determined at the quantum level in one theory are determined at the classical level of the mirror. One such example is the complex structure of the Coulomb branch of one theory, which is determined quantum mechanically. It is mapped to the complex structure of the Higgs branch of the mirror theory, which is determined classically. We study the generation of N = 2 superpotentials by open D-string instantons in the brane configurations.

Quantum Cohomology and Virasoro Algebra

Abstract We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds M ( c 1 ( M ) > 0) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler ect. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to χ(M), the Euler characteristic of the manifold M .

Kähler potential and higher derivative terms from M-theory fivebrane

The construction of four-dimensional supersymmetric gauge theories via the fivebrane of M-theory wrapped around a Riemann surface has been successfully applied to the computation of holomorphic quantities of field theory. In this paper we compute non-holomorphic quantities in the eleven-dimensional supergravity limit of M-theory. While the Kahler potential on the Coulomb of N = 2 theories is correctly reproduced, higher derivative terms in the N = 2 effective action differ from what is expected for the four-dimensional gauge theory. For the Kahler potential of N = 1 theories at an abelian Coulomb phase, the result again differs from what is expected for the four-dimensional gauge theory. Using a gravitational back-reaction method for the fivebrane we compute the metric on the Higgs branch of N = 2 gauge theories. Here we find an agreement with the results expected for the gauge theories. A similar computation of the metric on N = 1 Higgs branches yields information on the complex structure associated with the flavor rotation in one case and the classical metric in another. We discuss what four-dimensional field theory quantities can be computed via the fivebrane in the supergravity limit of M-theory.

Branes and dynamical supersymmetry breaking

We study dynamical supersymmetry breaking in four dimensions using the fivebrane of M theory, in particular for the Izawa-Yanagida-Intriligator-Thomas (IYIT) model, which we realize as the worldvolume theory of a certain M theory fivebrane configuration. From the brane point of view, supersymmetry is broken when a holomorphic configuration with the proper boundary conditions does not exist. We discuss the difference between explicit and spontaneous supersymmetry breaking and between runaway behavior and having a stable vacuum. As a preparation for the study of the IYIT model, we examine a realization of the orientifold four-plane in M theory. We derive known as well as new results on the moduli spaces of N = 2 and N = 1 theories with symplectic gauge groups. These results are based on a hypothesis that a certain intersection of the fivebrane and the Z_2 fixed plane breaks supersymmetry. In the IYIT model, we show that the brane exhibits runaway behavior when the flavor group is gauged. On the other hand, if the flavor group is not gauged, we find that the brane does not run away. We suggest that a stable supersymmetry-breaking vacuum is realized in the region beyond the reach of the supergravity approximation.

Non-Abelian conifold transitions and N = 4 dualities in three dimensions

We show how the Higgs mechanism for non-Abelian N = 2 gauge theories in four dimensions is geometrically realized in the context of type II strings as transitions among compactifications of Calabi-Yau 3-folds. We use this result and T-duality of a further compactification on a circle to derive N = 4, d = 3 dual field theories. This reduces the dualities for N = 4 gauge systems in three dimensions to perturbative symmetries of string theory. Moreover, we find that the dual of a gauge system always exists but may or may not correspond to a Lagrangian system. In particular, we verify a conjecture of Intriligator and Seiberg that an ordinary gauge system is dual to compactification of exceptional tensionless string theory down to three dimensions.

Consistency condition for fivebrane in M Theory on R5/Z2Orbifold

We derive some consistency conditions for fivebrane in M theory on R^5/Z_2 orbifold from the quantization law for the antisymmetric tensor field. We construct consistent fivebrane configurations in R^5/Z_2 type orbifold that exhibit the correct low energy dynamics of N=2 SQCD in four dimensions with symplectic and orthogonal gauge groups. This leads us to propose the M theory realization of orientifold four-planes of various types, and we study their properties by applying the consistency conditions.