Higher bundle gerbes and cohomology classes in gauge theories

The notion of a higher bundle gerbe is introduced to give a geometric realisation of the highder degree integral cohomology of certain manifolds. We consider examples using the infinite-dimensional spaces arising in gauge theories.     

Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH ⁴(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH ³(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH ⁴(BG,Z) toH ³(G,Z). We generalize this correspondence to topological spin theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ ₂ graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.     

Comparison of lattice gauge theories with gauge groups Z2 and SU(2)

We study a model of a pure Yang Mills theory with gauge group SU(2) on a lattice in Euclidean space. We compare it with the model obtained by restricting variables to $\text{Z}$₂. An inequality relating expectation values of the Wilson loop integral in the two theories is established. It shows that confinement of static quarks is true in our SU(2) model whenever it holds for the corresponding $\text{Z}$₂-model. The SU(2) model is shown to have high and low temperature phases that are distinguished by a qualitatively different behavior of the 't Hooft disorder parameter.     

Symmetry groups and non-abelian cohomology

We consider the implementation of symmetry groups of automorphisms of an algebra of observables in a reducible representation whose multipliers in general are non-commuting operators in the commutant of the representation. The multipliers obey a non-abelian cocycle relation which generalizes the 2-cohomology of the group. Examples are given from the theory of spin algebras and continuous tensor products. For typeI representations we show that the multiplier can be chosen to lie in the centre, giving an isomorphism with abelian theory.Dedicated to Res Jost and Arthur Wightman     

Improved QCD vacuum for gauge groups SU(3) and SU(4)

The effective energy for a covariantly constant background field in a pure Yang-Mills theory is calculated to loop order one. For gauge group SU(3) [SU(4)] it is found that the “vacuum” configuration consists of two [three] constant color magnetic fields of equal non-zero magnitude orthogonal to each other in color as well as physical space. We conclude that the structure of the Copenhagen vacuum is more complex than previously expected.     

Formal Group Laws for Affine Kac-Moody groups and group quantization

We describe a method for obtaining Formal Group Laws from the structure constants of Affine Kac-Moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group. As an intermediate step we get an explicit expression for two-cocycles on Loop Groups. The programme is applied to the AffineSU(2) group.Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de Formacion del Personal Investigador, and the Comision Asesora de Investigacion Cientifica y Tecnica (CAICYT)On leave of absence from the IFIC, Centro Mixto Universidad de Valencia — C.S.I.C. and the Departamento de Fisica Teorica de la Universidad de Valencia     

Gromov-Witten classes,quantum cohomology,and enumerative geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given.     

Indefinite harmonic forms and gauge theory

Indecomposable representations have been extensively used in the construction of conformal and de Sitter gauge theories. It is thus noteworthy that certain unitary highest weight representations have been given a geometric realization as the unitary quotient of an indecomposable representation using indefinite harmonic forms [RSW]. We apply this construction toSU (2,2) and the de Sitter group. The relation is established between these representations and the massless, positive energy representations ofSU (2,2) obtained in the physics literature. We investigate the extent to which this construction allows twistors to be viewed as a gauge theory ofSU (2,2). For the de Sitter group, on which the gauge theory of singletons is based, we find that this construction is not directly applicable.     

Kac-Moody current algebras of D = 2 massless gauge theories,their representations and applications

We give a classification of the Kac-Moody current algebras of all the possible massless fermion-gauge theories in two dimensions. It is shown that only Kac-Moody algebras based on A_N, B_N, C_N, and D_N in the Cartan classification with all possible central charge occur. The representation of local fermion fields and simply laced Kac-Moody algebras with minimal central charge in terms of free boson fields on a compactified space is discussed in detail, where stress is laid on the role played by the boundary conditions on the various collective modes. Fractional solitons and the possible soliton representation of certain nonsimply laced algebras is also analysed. We briefly discuss the relationship between the massless bound state sector of these two-dimensioned gauge theories and the critically coupled two-dimensional nonlinear sigma model, which share the same current algebra. Finally we briefly discuss the relevance of Sp(n) Kac-Moody algebras to the physics of monopole-fermion systems.     

SU(3)L × SU(3)R as gauge groups of weak and associated charges

Consequences of an SU(3)_L × SU(3)_R gauge group underlying weak-electromagnetic and associated interactions are studied. A specific pattern of symmetry breaking relates the lepton mass spectrum and mixing pattern, the effective (anti) neutrino-induced neutral current and (approximate) parity conservation in the neutral-current sector.     

Kac-Moody groups,topology of the Dirac determinant bundle,and fermionization

The relation between Kac-Moody groups and algebras and the determinant line bundle of the massless Dirac operator in two dimensions is clarified. Analogous objects are studied in four space-time dimensions and a generalization of Witten's fermionization mechanism is presented in terms of the topology of the Dirac determinant bundle.     

Unified gauge theories and bilinear invariant forms

Gauge theories based on a simple Lie algebra orU(1) have only one coupling constant. It is shown that there are no other Lie algebras with this property.     

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle at level 2. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations being finite is determined by the cohomology of the manifold.     

Infinite-dimensional gauge groups and special nonlinear gravitons

A gauge theory in flat space—time, in which the gauge algebra is the (infinite-dimensional) algebra of vector fields on a surface, determines a curved space—time metric. This note deals with some completely integrable examples, concentrating on the N → ∞ limit of the Euler—Arnol'd equations [geodesics on SO(N)]. In this case, the metric turns out to be flat, which points the way to a coordinate transformation that solves the original equations.