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

Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g=sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.

A q -difference analogue of U(g) and the Yang-Baxter equation

Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper

Infinite Dimensional Lie Algebras

Preface. 1: Wightman axioms and vertex algebras. 1.1: Wightman axioms of a QFT. 1.2: d = 2 QFT and chiral algebras. 1.3: Definition of a vertex algebra. 1.4: Holomorphic vertex algebras. 2: Calculus of formal distributions. 2.1: Formal delta-function. 2.2: An expansion of a formal distribution a(z,w). 2.3: Locality. 2.4: Taylor's formula. 2.5: Current algebras. 2.6: Conformal weight and the Virasoro algebra. 2.7: Lie superalgebras of formal distributions and conformal superalgebras. 3: Local fields. 3.1: Normally ordered product. 3.2: Dong's lemma. 3.3: Wick's theorem and a "non-commutative" generalization. 3.4: Restricted and field representations of Lie superalgebras of formal distributions. 3.5: Free (super)bosoms. 3.5: Free (super)fermions. 4: Structure theory of vertex algebras. 4.1: Consequences of translation covariance. 4.2: Quasisymmetry. 4.3: Superalgebras, ideals, and tensor products. 4.4: Uniqueness theorem. 4.5: Existence theorem. 4.6: Borcherds OPE formula. 4.7: Vertex algebras associated to Lie superalgebras of formal distributions. 4.8: Borcherds identity. 4.9: Graded and Mobius conformal vertex algebras. 4.10: Conformal vertex algebras. 4.11: Field algebras. 5: Examples of vertex algebras and their applications. 5.1: Charged free fermions. 5.2: Boson-fermion correspondence and KP hierarchy. 5.3: gl and W. 5.4: Lattice vertex algebras. 5.5: Simple lattice vertex algebras. 5.6: Root lattice vertex algebras and affine vertex algebras. 5.7: Conformal structure for affine vertex algebras. 5.8: Superconformal vertex algebras. 5.9: On classification of conformal superalgebras. Bibliography. Index

Vertex algebras for beginners

Basic representations of affine Lie algebras and dual resonance models

Infinite-dimensional Lie algebras, theta functions and modular forms

This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations.The first is the canonical commutation relations of the infinite-dimensional Heisenberg Algebra (= oscillator algebra). The second is the highest weight representations of the Lie algebra gl∞ of infinite matrices, along with their applications to the theory of soliton equations, discovered by Sato and Date, Jimbo, Kashiwara and Miwa. The third is the unitary highest weight representations of the current (= affine Kac-Moody) algebras. These algebras appear in the lectures twice, in the reduction theory of soliton equations (KP → KdV) and in the Sugawara construction as the main tool in the study of the fourth incarnation of the main idea, the theory of the highest weight representations of the Virasoro algebra.This book should be very useful for both mathematicians and physicists. To mathematicians, it illustrates the interaction of the key ideas of the representation theory of infinite-dimensional Lie algebras; and to physicists, this theory is turning into an important component of such domains of theoretical physics as soliton theory, theory of two-dimensional statistical models, and string theory.

Victor G. Kac

Papers

Infinite Dimensional Lie Algebras

Vertex algebras for beginners

Basic representations of affine Lie algebras and dual resonance models

Infinite-dimensional Lie algebras, theta functions and modular forms

Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras