A Large Distance Expansion for Quantum Field Theory

Using analyticity of the vacuum wave-functional under complex scalings, the vacuum of a quantum field theory may be reconstructed from a derivative expansion valid for slowly varying fields. This enables the eigenvalue problem for the Hamiltonian to be reduced to algebraic equations. Applied to Yang-Mills theory this expansion leads to a confining force between quarks.Comment: 5 pages, latex, invited talk at the Second International Sakharov Conference on Physics, May 199

Solving Virasoro Constraints on Integrable Hierarchies via the Kontsevich-Miwa Transform

We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge $Q$. Then the solutions to the constraints are given by the theory which has the same field content as the David-Distler-Kawai theory: it consists of a minimal matter scalar with background charge $Q$, dressed with an extra `Liouville' scalar. The construction is based on a generalization of the Kontsevich parametrization of the KP times achieved by introducing into it Miwa parameters which depend on the value of $Q$. Under the thus defined Kontsevich-Miwa transformation, the Virasoro constraints are proven to be equivalent to a master equation depending on the parameter $Q$. The master equation is further identified with a null-vector decoupling equation. We conjecture that $W^{(n)}$ constraints on the KP hierarchy are similarly related to a level-$n$ decoupling equation. We also consider the master equation for the $N$-reduced KP hierarchies. Several comments are made on a possible relation of the generalized master equation to {\it scaled} Kontsevich-type matrix integrals and on the form the equation takes in higher genera.Comment: 23pp (REVISED VERSION, 10 April 1992

On the Equivalence of Affine sl(2) and N=2 Superconformal Representation Theories

There exist two different languages, the ^sl(2) and N=2 ones, to describe similar structures; a dictionary is given translating the key representation-theoretic terms related to the two algebras. The main tool to describe the structure of ^sl(2) and N=2 modules is provided by diagrams of extremal vectors. The ^sl(2) and N=2 representation theories of certain highest-weight types turn out to be equivalent modulo the respective spectral flows.Comment: 14 pages, LaTeX209, needs bezier.sty. Contribution to the proceedings of the 30th Int. Symposium Ahrenshoop on the theory of elementary particles, Buckow, Germany, August 27--31, 199

Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra

We evaluate characters of irreducible representations of the N=2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG-resolution of the admissible N=2 representations and also a new 3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every $\hat{sl}(2)$ character as a linear combination of spectral-flow transformed N=2 characters; this identity involves a finite number of N=2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N=2 characters as contour integrals of admissible $\hat{sl}(2)$ characters.Comment: LaTeX2e: amsart, 34pp. An overall sign error corrected in (4.33) and several consequent formulas, and the presentation streamlined in Sec.4.2.3. References added. To appear in Nucl. Phys.

Singular Vectors and Topological Theories from Virasoro Constraints via the Kontsevich-Miwa Transform

We use the Kontsevich-Miwa transform to relate the different pictures describing matter coupled to topological gravity in two dimensions: topological theories, Virasoro constraints on integrable hierarchies, and a DDK-type formalism. With the help of the Kontsevich-Miwa transform, we solve the Virasoro constraints on the KP hierarchy in terms of minimal models dressed with a (free) Liouville-like scalar. The dressing prescription originates in a topological (twisted N=2) theory. The Virasoro constraints are thus related to essentially the N=2 null state decoupling equations. The N=2 generators are constructed out of matter, the `Liouville' scalar, and $c=-2$ ghosts. By a `dual' construction involving the reparametrization $c=-26$ ghosts, the DDK dressing prescription is reproduced from the N=2 symmetry. As a by-product we thus observe that there are two ways to dress arbitrary $d\leq1$ or $d\geq25$ matter theory, that allow its embedding into a topological theory. By th e Kontsevich-Miwa transform, which introduces an infinite set of `time' variables $t_r$, the equations ensuring the vanishing of correlators that involve BRST-exact primary states, factorize through the Virasoro generators expressed in terms of the $t_r$. The background charge of these Virasoro generators is determined by the topological central charge.Comment: 62p. LaTeX, CERN-TH.6752, IMAFF-92/8, revised (minor corrections, typos) easy-fontversio

Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models

We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with the fusion of irreducible and logarithmic modules in the (1,p) Virasoro logarithmic minimal models.Comment: 19 page

Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model

We generalize the string functions C_{n,r}(tau) associated with the coset ^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended ^sl(2)_k conformal field model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered ``logarithmic parafermionic characters,'' are given by A_{n,r}(tau), B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra characters of the (p=k+2,1) logarithmic model. We study the properties of A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string functions C_{n,r}, and evaluate the modular group representation generated from A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some nonsense in B.3.3. correcte

Logarithmic Conformal Field Theories via Logarithmic Deformations

We construct logarithmic conformal field theories starting from an ordinary conformal field theory -- with a chiral algebra C and the corresponding space of states V -- via a two-step construction: i) deforming the chiral algebra representation on V\tensor End K[[z,1/z]], where K is an auxiliary finite-dimensional vector space, and ii) extending C by operators corresponding to the endomorphisms End K. For K=C^2, with End K being the two-dimensional Clifford algebra, our construction results in extending C by an operator that can be thought of as \partial^{-1}E, where \oint E is a fermionic screening. This covers the (2,p) Virasoro minimal models as well as the sl(2) WZW theory.Comment: LaTeX, 35 pages, 4 eps figures. v2: references adde

On A Superfield Extension of The ADHM Construction and N=1 Super Instantons

We give a superfield extension of the ADHM construction for the Euclidean theory obtained by Wick rotation from the Lorentzian four dimensional N=1 super Yang-Mills theory. In particular, we investigate the procedure to guarantee the Wess-Zumino gauge for the superfields obtained by the extended ADHM construction, and show that the known super instanton configurations are correctly obtained.Comment: 22 pages, LaTeX, v2: typos corrected, references adde

Triplectic Quantization: A Geometrically Covariant Description of the Sp(2)-symmetric Lagrangian Formalism

A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of coordinates (`fields') have two superpartners (`antifields'). The quantization on such a triplectic manifold requires introducing several specific differential-geometric objects, whose properties we study. These objects are then used to impose a set of generalized master-equations that ensure gauge-independence of the path integral. The theory thus quantized is shown to extend to a level-1 theory formulated on a manifold that includes antifields to the Lagrange multipliers. We also observe intriguing relations between triplectic and ordinary symplectic geometry.Comment: Revised version -- our treatment in Section 5 has been extended and several pedagogical notes inserted in Sections 2--4; more references added