Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family of measures on a space of maps from to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of -valued three-forms on M. We show that these measures are positive, and that the measures are Borel probability measures. As an application we show that formulas arising from expectations in the measures reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures , where M is a homology three-sphere, will yield the Casson invariant of M. Dedicated to the memory of Raoul Bott Supported in part by NSF grant DMS 04/05670.     

Non-Abelian Multiple Vortices in Supersymmetric Field Theory

In this paper, we consider a system of non-Abelian multiple vortex equations governing coupled SU(N) and U(1) gauge and Higgs fields which may be embedded in a supersymmetric field theory framework. When the underlying domain is doubly periodic, we prove the existence and uniqueness of an n-vortex solution under a necessary and sufficient condition explicitly relating the domain size to the vortex number n and the Higgs boson masses. When the underlying domain is the full plane, we use a constructive approach to establish the existence and uniqueness of an n-vortex solution.     

Supersymmetric U(1) Gauge Field Theory with Massive Gauge Field

A new mechanism to introduce the mass of U(1) gauge field in supcrsymmctric U(1) gauge theory is discussed.The modelhas the strict local U(1) gauge symmetry and supersymmetry.Because we introduce two vector superfields simultaneously,the model contains a massive U(1) gauge field as well as a massless U(1) gauge field.     

Non-Abelian Vortices in Supersymmetric Gauge Field Theory via Direct Methods

Vortices in supersymmetric gauge field theory are important constructs in a basic conceptual phenomenon commonly referred to as the dual Meissner effect which is responsible for color confinement. Based on a direct minimization approach, we present a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations governing color-charged multiply distributed flux tubes, which provide an essential mechanism for linear confinement. Over a doubly periodic domain, existence results are obtained under explicitly stated necessary and sufficient conditions that relate the size of the domain, the vortex numbers, and the underlying physical coupling parameters of the models. Over the full plane, existence results are valid for arbitrary vortex numbers and coupling parameters. In all cases, solutions are unique.     

Integrable Lattice Spin Models from Supersymmetric Dualities

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang–Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories and make clear notes on the special functions used by several authors.     

Generalized Supersymmetric Perturbation Theory

Using the basic ingredient of supersymmetry, a simple alternative approach is developed to perturbation theory in one-dimensional non-relativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve tedious calculations which appear in the available perturbation theories. The model applicable in the same form to both the ground state and excited bound states, unlike the recently introduced supersymmetric perturbation technique which, together with other approaches based on logarithmic perturbation theory, are involved within the more general framework of the present formalism.     

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4{\mathcal N=4} supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2{\mathcal N=2} and the N=2^*{\mathcal N=2^*} supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2{\mathcal N=2} superconformal gauge theory is treated similarly.     

Supersymmetric Quantum Theory and Non-Commutative Geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-) K?hler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail. Received: 1 April 1997 / Accepted: 24 November 1998     

Ergodicity Breaking and Localization of the Nicolai Supersymmetric Fermion Lattice Model

We investigate dynamics of a supersymmetric fermion lattice model introduced by Nicolai (J Phys A 9:1497–1505, 1976). We show that the Nicolai model has infinitely many local constants of motion for its Heisenberg time evolution, and therefore ergodicity (with respect to thermal equilibrium states) breaks. It has infinitely many degenerated classical ground states. This phenomena is considered as localization at zero temperature. From a viewpoint of perturbation theory, we explain why delocalization is suppressed at zero temperature despite its disorder-free translation-invariant quantum interaction.     

Generalized Self-Duality for the Supersymmetric Yang-Mills Theory with a Scalar Multiplet

Abstract

Generalized Self-Duality Equations for the Supersymmetric Yang-Mills Theory with a Scalar Multiplet are Presented in Terms of Component Fields and Superfields as Well.     

Decorated Ising Square Lattice in a Magnetic Field

Physics of the Solid State - The Ising model on a decorated square lattice with various parameters of exchange interactions in an external magnetic field is studied. A comparison is made with the...     

Entanglement Entropy Scaling Law in the Ground State of Supersymmetric Fermion Lattice Model

Journal of Experimental and Theoretical Physics - We investigate the entanglement entropy in the ground state of the supersymmetric fermion lattice model. The Hamiltonian of this model is block...     

格点Z_4 Higgs场－规范场体系的相结构

用Ｍｉｇｄａｌ－Ｋａｄａｎｏｆｆ重整化群方法对Ｚ４Ｈｉｇｇｓ场─—规范场体系进行了研究，给出了四维时空的相互作用耦合常数递推关系和体系的相图，发现Ｈｉｇｇｓ相与禁闭相是连通的．     

格点Z3规范场–Higgs场耦合体系的相结构

用Migdal–Kadanoff重整化群方法研究了四维时空中的格点Z₃规范场与Z₃-Higgs标量场耦合体系,得到了该体系的耦合常数递推关系和相图,结果表明Z3耦合体系的耦合常数递推关系中的项与Z₂、Z₄耦合体系的耦合常数递推关系中的项不存在一一对应关系.