Doron Gepner

String Theory on Group Manifolds

A number of issues concerning affine Lie algebras and string propagation on group manifolds are addressed. We show that a 1 + 1 dimensional quantum field theory which gives a realization of current algebra (for any non-abelian Lie group G) will always give rise to an “integrable” representation. It is known that string propagation on the group manifold can give rise to a realization of current algebra for any G and any k, but precisely which representations occur for given k has not been determined previously. We do this here by studying modular invariance and by making a semiclassical study for large k. These results permit a complete description of the operator product algebra. Some examples based on SO(3) and SU(3)/Z3 are worked out in detail.

Space-Time Supersymmetry in Compactified String Theory and Superconformal Models

Abstract Space-time supersymmetric compactified string theories which obey spin and statistics, tachyon free, and modular invariant, are constructed by compactifying on arbitrary sums of N = 2 minimal superconformal models. Heterotic-type compactifications with realistic phemomenology are described. The theories offer a considerably simpler structure than standard supersymmetric compactifications, gauge groups which contain E6 with chiral fermions in realistic grand unification representations, and no free parameters. Also given are the partition functions and the classification of modular invariants for the N = 2 minimal superconformal models.

Exactly Solvable String Compactifications on Manifolds of Su(N) Holonomy

A variety of heterotic string compactifications on the K3 surface, manifolds of SU(3) holomony, and higher holomony manifolds, are solved exactly. An example of the quintic hypersurface in CP 4 is worked out in detail. It is conjectured, and demonstrated in part, that any supersymmetric compactification of the heterotic string with an N =2 superconformal theory is equivalent to a compactification on a manifold of SU( N ) holonomy, and in particular an arbitrary gluing of the discrete models with c=9 gives a solvable heterotic string compactification on some Calabi-Yau manifold. Calabi-Yau compactifications are seen to be exact vacua of string theory, retaining their topological and geometrical characteristics. Previously unknown enhanced gauge symmetries are found to arise for certain backgrounds.

Modular Invariant Partition Functions for Parafermionic Field Theories

The partition functions of the parafermionic theories of Zamolodchikov and Fateev are computed in terms of the string functions of affine algebra. Conditions for modular invariance are given and a complete set of solutions is found. The partition functions in the presence of twisted boundary conditions are also given. All the solutions are shown to be orbifolds of some principal ones.

New Conformal Field Theories Associated with Lie Algebras and their Partition Functions

Abstract New conformal field theories are defined as a generalization of the vertex operator construction to arbitrary level, k, and arbitrary simple Lie algebra. Exact scaling dimensions and partition functions are given in terms of the string functions of affine algebra. The theories have a discrete symmetry group which is the root lattice modulo k times the long roots lattice. Twisted partition functions are given, and the general solutions to modular invariance are described as orbifolds of principal ones.

On the spectrum of 2D conformal field theories

Abstract Possible unitary statistical models and SU(2) current algebra theories are classified up to certain “levels” of the Virasoro and Kac-Moody algebras. A connection that is found between the Virasoro and SU(2) Kac-Moody characters is used to generate unitary statistical models from the SU(2) theories. Using the “fusion rules” of the operator product algebra of these theories, we are able to check the consistency of the solutions, and to write down their operator product algebra. The connection between the two algebras extends also to the fusion rules.

Field identification in coset conformal field theories

Abstract The fields appearing in G/H coset conformal field theories are shown to obey certain relations. These relations, in conjunction with the modular transformations imply a certain identification of fields. The use of these relations and identifications leads to the correct modular invariant spectrum of the theories. The consistency of this spectrum with the operator product algebra is discussed.

Yukawa couplings for Calabi-Yau string compactification

Abstract Recently, we have described string theories based on N = 2 superconformal theories. It was argued that all such string theories correspond to string propagation on Calabi-Yau manifolds. We compute here the Yukawa couplings for massless particles in the representation 27 of E 6 (generations), in some examples, and show that the quasi-topological result of the field-theory approximation holds exactly. This is a non-trivial quantitative agreement which further supports the geometric interpretation of these string theories, as well as giving an explicit demonstration of the quasi-topological nature of these couplings.

On the connection between WZW and free field theories

Abstract A large class of primary fields which appear at any level of the WZW theories (of types A N , B N , C N , D N , E 6 , and E 7 ) are shown to posses simple power-like four-point functions. As a consequence, these fields, which are in 1-1 correspondence with the center of the covering group, may be written as symmetrized products of level one fields. The latter are known to be related to free fermions (A N , B N , D N ) or free bosons (A N , D N , E 6 , E 7 ). Our results indicate that a relation to free field theory exists also for the case of C N .

Scalar field theory and string compactification

Abstract The new N = 2 conformal field theories based on SU( n + 1)/SU( n ) × U(1) recently constructed as G/H theories are shown to be equivalent to certain supersymmetric scalar field theories. The chiral fields and their structure constants are described for any scalar field theories. This allows the calculation of the Yukawa couplings 27 3 in any supersymmetric heterotic string compactification based on some scalar field theory. The generation structure and Yukawa couplings are then shown to be given by a geometrical formula which follows from the manifold which is described by the potential.