Physics and Mathematics of Strings

The following sections are included:

• FOREWORD A.M. Polyakov

• There are some things that never die… Lars Brink

• FOREWORD Daniel Friedan

• CONTENTS

The following sections are included:

• INTRODUCTION

• DIFFEOMORPHISMS OUT OF GAUGE TRANSFORMATIONS

• CONSEQUENCES FOR WZNW ACTIONS

• THE CASE OF SL(3)

• ACKNOWLEDGEMENTS

• REFERENCES

The following sections are included:

• Introduction

• General Properties of a Conformal Field Theory
  • Conformal Invariance and Conformal Ward Identities
  • Conformal blocks and duality

• Degenerate Conformal Families. Coulomb Gas Repretation
  • Generalities
  • Coulomb gas regresentation. Fusion rules
  • Virasoro characters and null vectors
  • Sample computations: conformal blocks and braiding matrices

• Current Algebras and Wess-Zumino-Witten Models
  • Generalities
  • Representation and fusion rules
  • Partition functions. modular invariants

• Fusion Algebras. The Verlinde Conjecture
  • The fusion algebra and its properties
  • Proof of the Verlinde conjecture
  • Constraints from rationality

• Duality and Quantum Groups
  • Motivation
  • Quantam groups and their representations for q a root of unity

• Explicit Examples and Modular Properties
  • WZW models
  • Rational gaussian models
  • Modular properties

• Solvable Lattice Models and Conformal Field Theories
  • Vertez Models
    • Regular lattice
    • Nonregular lattice: Z-invariance
  • RSOS-models⁶⁵
  • Conformal field theory

• Knots Conformal Field Theories and Statiscal Mechanics
  • Knot theory
  • State models: Turaev and Kauffman invariants
  • Rational conformal field theories and knots invariants
  • Knots on arbitrary 3–manifolds

• References

Let V be the n dimensional irreducible U_q(gl(n,C))-module and let ν_¼ (¼ = 1,&, n) be its natural basis. There is an irreducible decomposition where T is a standard tableau with N nodes. It is shown that in the q^±l → 0 limit V(T) is spanned by decomposable vectors and that the correspondence is given by the Robinson-Shensted correspondence.

We describe in the case of free bosonic and fermionic theories the sewing procedure, that is a very convenient way for constructing correlation functions of these theories on an arbitrary Riemann surface from their knowledge on the sphere. The fundamental object that results from this construction is the N-point g-loop vertex. It summarizes the information of all correlation functions of the theory on an arbitrary Riemann surface. We then check explicitly the bosonization rules and derive some useful formulas.

Recently infinite-dimensional grassmannians and their supergeneralizations were used to study conformal two-dimensional fields and strings.¹ In particular the super Mumford form (holomorphic square root from the superstring measure on moduli space) was expressed in Ref. 2 through super analog of Sato τ-function, In this paper we present results about supergrassmannians and super τ-functions. Some of these assertions generalize the results of Ref. 2; in these cases we omit the proofs. Let us denote by H_N (or simply by H) the space of square integrable functions f(z, θ₁, … ,θ_N) where z is a complex number, |z| = 1 and θ₁, … , θN are elements of some Grassmann algebra Λ_N. We introduce in H a bilinear scalar product:

Particular perturbations of a 2D Conformal Field Theory leading to Integrable massive Quantum Field Theories are examined. The mass spectra and S-matricee for some models, including the field theory of the Ising Model with magnetic field and “thermal” deformations of the tricritical Ising and 3-state Potts models, are proposed. The hidden Lie-algebraic structures of these spectra and their relation to the Toda systems are discussed.

We study a new class of representations of affine Kac-Moody algebras, so-called Wakimoto modules, introduced in our previous work⁹. These modules have boson realization, endowed with special objects-chains and intertwining operators. We give explicit constructions of these objects, using composition vertex operators and point out the connection with Virasoro algebra and Walgebras. Our results yield explicit description of primary fields in WZW-models and allow to give integral representation of correlation functions in these models. We'also use Wakimoto modules for investigation of the structure of highest weight modules on the singular hyperplane .

Aspects of the conformal group in two-dimensional Minkowski space-time are reviewed. Representations of the algebra and of the group, i. e. of infinitesimal generators and of finite group elements, are given within a field theoretic Schrödinger representation for bosons and fermions. The role of self-dual fields in representing conformal transformations is stressed, their kinematics and dynamics are shown to obey the Hill-Feynman boson-fermion duality.

In the previous works [1-3], started in 1986, the grounds of the operator theory of the interacting closed bosonic strings were constructed. Our theory is the direct development of the classical algebraic operator theory of non-interacting bosonic string of Virasoro. Mandelstam et. a1. [4-5]. This theory corresponds to the case of the Riemann surface of the genus g = 0. The approach which we proposed unifies the ideas of the operator formalism with the ideas of the geometrical approach of Polyakov and others [6-8].…

The following sections are included:

• Motivations

• N=2 effective Lagrangians[6][14][15]

• Some moonshine: singularities at low c

• Singularity Theory

• Various varieties

• Discussion

• Acknowledgements

• References

A technique for performing string perturbation theory in flat space time is reviewed with particular emphasis on the features arising as a consequence of the Ramond string complications. Details on results obtained so far are presented.

The Goddard-Kent-Olive construction is used to derive various conformal and superconformal theories, including a large class of N = 2 models recently discovered by Kazama and Suzuki. Various approaches to compactification based on exactly soluble conformal field theories, including Gepner’s proposal for using the N = 2 minimal models, are sketched. Recent progress in understanding N = 2 models and Calabi-Yau spaces using mathematical techniques of singularity theory is described.

The following sections are included:

• INTRODUCTION

• ORIGIN OF THE STRING SCALE PARAMETER

• THE PLANCK CONSTANT AS ULTRAVIOLET CUT-OFF

• PHYSICS MUCH ABOVE THE FUNDAMENTAL LENGTH

• PHYSICS BELOW THE FUNDAMENTAL LENGTH

• ENLARGED UNCERTAINTY AND EQUIVALENCE PRINCIPLES?

• A SCALE-FREE (TOPOLOGICAL) PHASE?

Many of the most elusive problems in string theory involve the search for the proper interpretation of ideas such as conformal invariance and duality that date back to the early days of the subject. In this paper, some modest new insights are offered about some of these old ideas.

The following sections are included:

• Introduction

• The Minimal Covariant Action

• Covariantized Light-Cone Approach

• Harmonic Superspace

• Actions with only first-class constraints

• Twistors

• Conclusions

• References

A formal (and speculative) construction is given of the space of all conformal field theories of given conformal central charge - and also of the space of all classical ground states of string - using only intrinsic structure in the space of all closed Riemann surfaces.