Narrow Results

Starting from an supersymmetric electric gauge theory with the gauge group Sp(N_c) × SO(2N′_c) with fundamentals for the first gauge group factor and a bifundamental, we apply Seiberg dual to the symplectic gauge group only and arrive at the supersymmetric dual magnetic gauge theory with dual matters including the gauge singlets and superpotential. By analyzing the F-term equations of the dual magnetic superpotential, we describe the intersecting brane configuration of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory.

We construct the type IIA nonsupersymmetric meta-stable brane configuration consisting of (2k+1) NS5-branes and D4-branes where the electric gauge theory superpotential has an order (2k+2) polynomial for the bifundamentals. We find a rich pattern of nonsupersymmetric meta-stable states as well as the supersymmetric stable ones. By adding the orientifold 4-plane to this brane configuration, we also describe the intersecting brane configuration of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of corresponding gauge theory.

A summary is reported on our previous publications about four-dimensional supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of a-maximization, we studied decoupling operators in the infrared and the renormalization flow of the theory. We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory.

Three-branes at a given toric Calabi–Yau singularity lead to different phases of the conformal field theory related by toric (Seiberg) duality. Using the dimer model/brane tiling description in terms of bipartite graphs on a torus, we find a new invariant under Seiberg duality, namely the Klein j-invariant of the complex structure parameter in the distinguished isoradial embedding of the dimer, determined by the physical R-charges. Additional number theoretic invariants are described in terms of the algebraic number field of the R-charges. We also give a new compact description of the a-maximization procedure by introducing a generalized incidence matrix.

With a bird's-eye view, we survey the landscape of Calabi–Yau threefolds, compact and noncompact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its principles on powerful computers and experimenting with the vast mathematical data, new physics can be learnt. It is hoped that this interdisciplinary glimpse will be of some use to the beginning student.

We study two M5-branes on $A_1$ ALE space. We introduce some M2-branes suspended between the M5-branes. Then, the boundaries of M2-branes look like strings. We call them “M-strings.” The M-strings have ${𝒩}=(4,0)$ supersymmetry by considering the brane configuration on $A_1$ ALE space. We calculate the partition function of M-strings by using the refined topological vertex formalism. We find that the supersymmetry of M-strings gets enhanced to ${𝒩}=(4,4)$ by tuning some Kähler parameters. Furthermore, we discuss another possibility of the enhancement of supersymmetry which is different from the above one.

We consider the partition function of super-Yang–Mills theories defined on ${𝕋}^2\times {\mathrm{\Sigma }}_g$. This path integral can be computed by the localization. The one-loop determinant is evaluated by the elliptic genus. This elliptic genus gives trivial result in our calculation. As a result, we obtain a theory defined on the Riemann surface.

In this paper we extend work on exotic two-dimensional (2,2) supersymmetric gauged linear sigma models (GLSMs) in which, for example, geometries arise via nonperturbative effects, to (0,2) theories, and in so doing find some novel (0,2) GLSM phenomena. For one example, we describe examples in which bundles are constructed physically as cohomologies of short complexes involving torsion sheaves, a novel effect not previously seen in (0,2) GLSMs. We also describe examples related by RG flow in which the physical realizations of the bundles are related by quasi-isomorphism, analogous to the physical realization of quasi-isomorphisms in D-branes and derived categories, but novel in (0,2) GLSMs. Finally, we also discuss (0,2) deformations in various duality frames of other examples.

We compute the partition functions of ${𝒩}=1$ gauge theories on $S^2\times {ℝ}_{𝜀}^2$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of ${ℝ}_{𝜀}^2$. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the ${𝒩}=1$ partition functions on the $\mathrm{\Omega }$-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

We discuss intriguing relations between 5d supersymmetric gauge theories of eight supercharges based on 5-brane web and cubic prepotential and propose new relations between SU gauge theories and SO/Sp gauge theories when the parameters of SU gauge theories are specially tuned.

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S ~ TrW_αW^α, with the relation S^h = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.

First, we give a brief review of recent development of lattice formulations for supersymmetric Yang-Mills (SYM) theories with extended supersymmetry, which preserves a part of supersymmetry on lattice. For cases of two dimensions, we can see that lattice models in such formulations lead to the target continuum theories with no fine-tuning. Namely, supersymmetries or some other symmetries not realized on the lattice are automatically restored in the continuum limit.

Next, we consider a mass deformation to and present its lattice formulation with keeping two supercharges. It provides a nonperturbative framework to investigate IIA matrix string theory. Moreover, since it has fuzzy sphere solutions around which four-dimensional theory is deconstructed, it will serve a nonperturbative formulation of four-dimensional which requires no fine-tuning. The rank of the gauge group is not restricted to large N. It opens a quite interesting possibility to test AdS/CFT correspondence in a stringy regime where string loop effects cannot be neglected. Also, for two-dimensional , a similar argument is possible to obtain four-dimensional on noncommutative space.

We consider two supersymmetric gauge theories connected by an interface and the gravity dual of this system. This interface is expressed by a fuzzy funnel solution of Nahmfs equation in the gauge theory side. The gravity dual is a probe D5-brane in AdS₅ × S⁵. The potential energy between this interface and a test particle is calculated in both the gauge theory side and the gravity side by the expectation value of a Wilson loop. In the gauge theory it is evaluated by just substituting the classical solution to the Wilson loop. On the other hand it is done by the on-shell action of the fundamental string stretched between the AdS boundary and the D5-brane in the gravity. We show the gauge theory result and the gravity one agree with each other.

Bipartite graphs, especially drawn on Riemann surfaces, have of late assumed an active rôle in theoretical physics, ranging from MHV scattering amplitudes to brane tilings, from dimer models and topological strings to toric AdS/CFT, from matrix models to dessins d’enfants in gauge theory. Here, we take a brief and casual promenade in the realm of brane tilings, quiver SUSY gauge theories and dessins, serving as a rapid introduction to the reader.