The Orbifold-String Theories of Permutation-Type: III. Lorentzian and Euclidean Space-Times in a Large Example

To illustrate the general results of the previous paper, we discuss here a large concrete example of the orbifold-string theories of permutation-type. For each of the many subexamples, we focus on evaluation of the \emph{target space-time dimension} $\hat{D}_j(\sigma)$, the \emph{target space-time signature} and the \emph{target space-time symmetry} of each cycle $j$ in each twisted sector $\sigma$. We find in particular a gratifying \emph{space-time symmetry enhancement} which naturally matches the space-time symmetry of each cycle to its space-time dimension. Although the orbifolds of $\Z_{2}$-permutation-type are naturally Lorentzian, we find that the target space-times associated to larger permutation groups can be Lorentzian, Euclidean and even null (\hat{D}_{j}(\sigma)=0), with varying space-time dimensions, signature and symmetry in a single orbifold.Comment: 36 page

Two Large Examples in Orbifold Theory: Abelian Orbifolds and the Charge Conjugation Orbifold on su(n)

Recently the operator algebra and twisted vertex operator equations were given for each sector of all WZW orbifolds, and a set of twisted KZ equations for the WZW permutation orbifolds were worked out as a large example. In this companion paper we report two further large examples of this development. In the first example we solve the twisted vertex operator equations in an abelian limit to obtain the twisted vertex operators and correlators of a large class of abelian orbifolds. In the second example, the twisted vertex operator equations are applied to obtain a set of twisted KZ equations for the (outer-automorphic) charge conjugation orbifold on su(n \geq 3).Comment: 58 pages, v2: three minor typo

The orbifold-string theories of permutation-type: II. Cycle dynamics and target space-time dimensions

We continue our discussion of the general bosonic prototype of the new orbifold-string theories of permutation type. Supplementing the extended physical-state conditions of the previous paper, we construct here the extended Virasoro generators with cycle central charge $\hat{c}_j(\sigma)=26f_j(\sigma)$, where $f_j(\sigma)$ is the length of cycle $j$ in twisted sector $\sigma$. We also find an equivalent, reduced formulation of each physical-state problem at reduced cycle central charge $c_j(\sigma)=26$. These tools are used to begin the study of the target space-time dimension $\hat{D}_j(\sigma)$ of cycle $j$ in sector $\sigma$, which is naturally defined as the number of zero modes (momenta) of each cycle. The general model-dependent formulae derived here will be used extensively in succeeding papers, but are evaluated in this paper only for the simplest case of the "pure" permutation orbifolds.Comment: 32 page

Flat Connections for Characters in Irrational Conformal Field Theory

Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find that the affine-Virasoro characters of ICFT satisfy heat-like differential equations with flat connections. As a first example, we solve the system for the general $g/h$ coset construction, obtaining an integral representation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple $g$, noting a simplification for the subspace of theories which possess a non-trivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally-extended loop group.Comment: harvmac (answer b to question) 40 pages. LBL-35718, UCB-PTH-94/1

Ward Identities for Affine-Virasoro Correlators

Generalizing the Knizhnik-Zamolodchikov equations, we derive a hierarchy of non-linear Ward identities for affine-Virasoro correlators. The hierarchy follows from null states of the Knizhnik-Zamolodchikov type and the assumption of factorization, whose consistency we verify at an abstract level. Solution of the equations requires concrete factorization ans\"atze, which may vary over affine-Virasoro space. As a first example, we solve the non-linear equations for the coset constructions, using a matrix factorization. The resulting coset correlators satisfy first-order linear partial differential equations whose solutions are the coset blocks defined by Douglas.Comment: 53 pages, Latex, LBL-32619, UCB-PTH-92/24, BONN-HE-92/2

Unification of the General Non-Linear Sigma Model and the Virasoro Master Equation

The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affine Lie algebra) of the WZW model, while the Einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form $L_{ij} \partial x^i \partial x^j$ in the background of a general sigma model. The requirement that these operators satisfy the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field $L_{ij}$ couples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed.Comment: 18 pages, Latex. Talk presented by MBH at the NATO Workshop `New Developments in Quantum Field Theory', June 14-20, 1997, Zakopane, Polan

Infinite Dimensional Free Algebra and the Forms of the Master Field

We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic derivation of the Gopakumar-Gross form. For action theories, these forms of the master field immediately give a number of new free-algebraic packagings of the planar Schwinger-Dyson equations.Comment: 39 pages. Expanded historical remark