On The Characters of Parafermionic Field Theories

We study cosets of the type $H_l/U(1)^r$, where $H$ is any Lie algebra at level $l$ and rank $r$. These theories are parafermionic and their characters are related to the string functions, which are generating functions for the multiplicities of weights in the affine representations. An identity for the characters is described, which apply to all the algebras and all the levels. The expression is of the Rogers Ramanujan type. We verify this conjecture, for many algebras and levels, using Freudenthal Kac formula, which calculates the multiplicities in the affine representations, recursively, up to some grade. Our conjecture encapsulates all the known results about these string functions, along with giving a vast wealth of new ones.Comment: 13 pages. The fortran program ALGEBRA.for is available from the source file

Galois groups in rational conformal field theory II. The discriminant

We express the discriminant of the polynomial relations of the fusion ring, in any conformal field theory, as the product of the rows of the modular matrix to the power -2. The discriminant is shown to be an integer, always, which is a product of primes which divide the level. Detailed formulas for the discriminant are given for all WZW conformal field theories.Comment: 19 pages, one table. Minor typos correcte

On New Conformal Field Theories with Affine Fusion Rules

Some time ago, conformal data with affine fusion rules were found. Our purpose here is to realize some of these conformal data, using systems of free bosons and parafermions. The so constructed theories have an extended $W$ algebras which are close analogues of affine algebras. Exact character formulae is given, and the realizations are shown to be full fledged unitary conformal field theories.Comment: Minor correction in an example and some typo

The SO(2r)2\boldsymbol{SO(2r)_2} string functions as q\boldsymbol{q}-diagrams

We discuss our conjecture for simply laced Lie algebras level two string functions of mark one fundamental weights and prove it for the $SO(2r)$ algebra. To prove our conjecture we introduce $q$-diagrams and examine the diagrammatic interpretations of known identities by Euler, Cauchy, Heine, Jacobi and Ramanujan. Interestingly, the diagrammatic approach implies these identities are related in the sense that they represent the first few terms in an infinite series of diagrammatic identities. Furthermore, these diagrammatic identities entail all the identities needed to prove our conjecture as well as generalise it to all $SO(2r)$ level two string functions. As such, our main objective is proving these series of diagrammatic identities thus extending the works mentioned and establishing our conjecture for the $SO(2r)$ level two string functions.Comment: 39 page

New String Theories And Their Generation Number

New heterotic string theories in four dimensions are constructed by tensoring a nonstandard SCFT along with some minimal SCFT's. All such theories are identified and their particle generation number is found. We prove that from the infinite number of new heterotic string theories only the {6} theory predicts three generations as seen in nature which makes it an interesting candidate for further study.Comment: 18 pages, 1 table and no figure