http://inspirehep.net/record/499969/files/arXiv:hep-th_9905111.pdf

Large N Field Theories, String Theory and Gravity

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

/pdf/quantum-field-theory-and-the-jones-polynomial-5897tp4ud9.pdf

Quantum field theory and the Jones polynomial

https://cds.cern.ch/record/152341/files/198407016.pdf

Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory

Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7. Regular rings 8. Flatness revisited 9. Derivations 10. I-smoothness 11. Applications of complete local rings Appendices References Index.

Commutative Ring Theory

We review the holographic correspondence between "eld theories and string/M theory, focusing on the relation between compacti"cations of string/M theory on Anti-de Sitter spaces and conformal "eld theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the "eld theory is approximated by classical or semiclassical gravity. We focus on the case of the N"4 supersymmetric gauge theory in four dimensions, but we discuss also "eld theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 04.65.#e; 11.15.!q; 11.25.!w; 11.30.Pb; 12.38.!t

/pdf/large-n-field-theories-string-theory-and-gravity-1kug5wi7zu.pdf

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this construction shows, in particular, that in the simplest case of the sl (2|1) level 1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras.

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

Conformal algebra is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those ``based'' on the punctured complex plane, like the Virasoro algebra and loop algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standard properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for most of the important examples.

Cohomology of Conformal Algebras

We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms \(\omega_j, j \in {\mathbb {Z}}_+\), of the variational complex Ω. If these forms are exact, i.e., ωj are variational derivatives of some local functionals ∫ hj, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is \(\fancyscript {V}\) is “normal”; in particular, for arbitrary \(\fancyscript {V}\), any closed form in Ω becomes exact if we add to \(\fancyscript {V}\) a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

/pdf/poisson-vertex-algebras-in-the-theory-of-hamiltonian-ums5k968wt.pdf

Poisson vertex algebras in the theory of Hamiltonian equations

This volume, dedicated to Kostant on his 65th birthday, is a collection of 22 papers by international leaders in the fields of Lie theory, geometry, algebra and math physics. The papers broadly reflect the range of Kostant's work.

Lie theory and geometry : in honor of Bertram Kostant

Conformal algebra is an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in a conformal field theory. This is a review of recent developments in the subject.

Formal distribution algebras and conformal algebras

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