arXiv subjects

M. B. Halpern

Publications and source records attributed to M. B. Halpern.

49 records · Page 3Linked to original sources

arXiv1996

Unified Einstein-Virasoro Master Equation in the General Non-Linear Sigma Model

The Virasoro master equation (VME) describes the general affine-Virasoro construction $T=L^{ab}J_aJ_b+iD^a \dif J_a$ in the operator algebra of the WZW model, where $L^{ab}$ is the inverse inertia tensor and $D^a $ is the improvement vector. In this paper, we generalize this construction to find the general (one-loop) Virasoro construction in the operator algebra of the general non-linear sigma model. The result is a unified Einstein-Virasoro master equation which couples the spacetime spin-two field $L^{ab}$ to the background fields of the sigma model. For a particular solution $L_G^{ab}$, the unified system reduces to the canonical stress tensors and conventional Einstein equations of the sigma model, and the system reduces to the general affine-Virasoro construction and the VME when the sigma model is taken to be the WZW action. More generally, the unified system describes a space of conformal field theories which is presumably much larger than the sum of the general affine-Virasoro construction and the sigma model with its canonical stress tensors. We also discuss a number of algebraic and geometrical properties of the system, including its relation to an unsolved problem in the theory of $G$-structures on manifolds with torsion.

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arXiv1996

Semi-Classical Blocks and Correlators in Rational and Irrational Conformal Field Theory

The generalized Knizhnik-Zamolodchikov equations of irrational conformal field theory provide a uniform description of rational and irrational conformal field theory. Starting from the known high-level solution of these equations, we first construct the high-level conformal blocks and correlators of all the affine-Sugawara and coset constructions on simple g. Using intuition gained from these cases, we then identify a simple class of irrational processes whose high-level blocks and correlators we are also able to construct.

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arXiv1995

The WZW Model as a Dynamical System on Affine Lie Groups

Working directly on affine Lie groups, we construct several new formulations of the WZW model. In one formulation WZW is expressed as a one-dimensional mechanical system whose variables are coordinates on the affine Lie group. When written in terms of the affine group element, this formulation exhibits a two-dimensional WZW term. In another formulation WZW is written as a two-dimensional field theory, with a three-dimensional WZW term, whose fields are coordinates on the affine group. On the basis of these equivalent formulations, we develop a translation dictionary in which the new formulations on the affine Lie group are understood as mode formulations of the conventional WZW formulation on the Lie group. Using this dictionary, we also express WZW as a three-dimensional field theory on the Lie group with a four-dimensional WZW term.

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arXiv1995

The Generic World-Sheet Action of Irrational Conformal Field Theory

We review developments in the world-sheet action formulation of the generic irrational conformal field theory, including the non-linear and the linearized forms of the action. These systems form a large class of spin-two gauged WZW actions which exhibit exotic gravitational couplings. Integrating out the gravitational field, we also speculate on a connection with sigma models.

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arXiv1995

Irrational Conformal Field Theory

This is a review of irrational conformal field theory, which includes rational conformal field theory as a small subspace. Central topics of the review include the Virasoro master equation, its solutions and the dynamics of irrational conformal field theory. Discussion of the dynamics includes the generalized Knizhnik-Zamolodchikov equations on the sphere, the corresponding heat-like systems on the torus and the generic world- sheet action of irrational conformal field theory.

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arXiv1994

Irrational Conformal Field Theory on the Sphere and the Torus

I review the foundations of irrational conformal field theory (ICFT), which includes rational conformal field theory as a small subspace. Highlights of the review include the Virasoro master equation and the generalized Knizhnik-Zamolodchikov equations for the correlators of ICFT on the sphere and the torus.

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arXiv1994

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.

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arXiv1993

Linearized Form of the Generic Affine-Virasoro Action

Halpern and Yamron have given a Lorentz, conformal, and Diff S$_2$-invariant world-sheet action for the generic irrational conformal field theory, but the action is highly non-linear. In this paper, we introduce auxiliary fields to find an equivalent linearized form of the action, which shows in a very clear way that the generic affine-Virasoro action is a Diff S$_2$-gauged WZW model. In particular, the auxiliary fields transform under Diff S$_2$ as local Lie $g$ $\times$ Lie $g$ connections, so that the linearized affine-Virasoro action bears an intriguing resemblance to the usual (Lie algebra) gauged WZW model.

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arXiv1993

Flat Connections and Non-Local Conserved Quantities in Irrational Conformal Field Theory

Irrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine-Virasoro Ward identities describe the biconformal correlators of ICFT. We reformulate the Ward identities as an equivalent linear partial differential system with flat connections and new non-local conserved quantities. As examples of the formulation, we solve the system of flat connections for the coset correlators, the correlators of the affine-Sugawara nests and the high-level $n$-point correlators of ICFT.

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arXiv1993

Recent Progress in Irrational Conformal Field Theory

In this talk, I will review the foundations of irrational conformal field theory (ICFT), which includes rational conformal field theory as a small subspace. Highlights of the review include the Virasoro master equation, the Ward identities for the correlators of ICFT and solutions of the Ward identities. In particular, I will discuss the solutions for the correlators of the $g/h$ coset constructions and the correlators of the affine-Sugawara nests on $g\supset h_1 \supset \ldots \supset h_n$. Finally, I will discuss the recent global solution for the correlators of all the ICFT's in the master equation.

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arXiv1993

Solving the Ward Identities of Irrational Conformal Field Theory

The affine-Virasoro Ward identities are a system of non-linear differential equations which describe the correlators of all affine-Virasoro constructions, including rational and irrational conformal field theory. We study the Ward identities in some detail, with several central results. First, we solve for the correlators of the affine-Sugawara nests, which are associated to the nested subgroups $g\supset h_1 \supset \ldots \supset h_n$. We also find an equivalent algebraic formulation which allows us to find global solutions across the set of all affine-Virasoro constructions. A particular global solution is discussed which gives the correct nest correlators, exhibits braiding for all affine-Virasoro correlators, and shows good physical behavior, at least for four-point correlators at high level on simple $g$. In rational and irrational conformal field theory, the high-level fusion rules of the broken affine modules follow the Clebsch-Gordan coefficients of the representations.

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arXiv1992

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 ansätze, 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.

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arXiv1991

The Lie h-Invariant Conformal Field Theories and the Lie h-Invariant Graphs

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory.

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