Non-Perturbative String Equations for Type 0A

Well-defined non-perturbative formulations of the physics of string theories, sometimes with D-branes present, were identified over a decade ago, from a careful study of double scaled matrix models. Following recent work which recasts some of those old results in the context of type 0 string theory, a study is made of a much larger family of models, which are proposed as type 0A models of the entire superconformal minimal series coupled to gravity. This gives many further examples of important physical phenomena, including non-perturbative descriptions of transitions between D-branes and fluxes, tachyon condensation, and holography. In particular, features of a large family of non-perturbatively stable string equations are studied, and results are extracted which pertain to type 0A string theory, with D-branes and fluxes, in this large class of backgrounds. For the entire construction to work, large parts of the spectrum of the supergravitationally dressed superconformal minimal models and that of the gravitationally dressed bosonic conformal minimal models must coincide, and it is shown how this happens. The example of the super-dressed tricritical Ising model is studied in some detail.Comment: 29 pages LaTe

Tachyon Condensation, Open-Closed Duality, Resolvents, and Minimal Bosonic and Type 0 Strings

Type 0A string theory in the (2,4k) superconformal minimal model backgrounds and the bosonic string in the (2,2k-1) conformal minimal models, while perturbatively identical in some regimes, may be distinguished non-perturbatively using double scaled matrix models. The resolvent of an associated Schrodinger operator plays three very important interconnected roles, which we explore perturbatively and non-perturbatively. On one hand, it acts as a source for placing D-branes and fluxes into the background, while on the other, it acts as a probe of the background, its first integral yielding the effective force on a scaled eigenvalue. We study this probe at disc, torus and annulus order in perturbation theory, in order to characterize the effects of D-branes and fluxes on the matrix eigenvalues. On a third hand, the integrated resolvent forms a representation of a twisted boson in an associated conformal field theory. The entire content of the closed string theory can be expressed in terms of Virasoro constraints on the partition function, which is realized as wavefunction in a coherent state of the boson. Remarkably, the D-brane or flux background is simply prepared by acting with a vertex operator of the twisted boson. This generates a number of sharp examples of open-closed duality, both old and new. We discuss whether the twisted boson conformal field theory can usefully be thought of as another holographic dual of the non-critical string theory.Comment: 37 pages, some figures, LaTe

Non-Local Matrix Generalizations of W-Algebras

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras.Comment: 43 pages, a reference and a remark on the conformal properties for $U_1\ne 0$ adde

On the support of the Ashtekar-Lewandowski measure

We show that the Ashtekar-Isham extension of the classical configuration space of Yang-Mills theories (i.e. the moduli space of connections) is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices. These results are then used to prove that the classical configuration space is contained in a zero measure subset of this extension with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure. Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the ``extended" configuration space.Comment: 22 pages, Tex, Preprint CGPG-94/3-

Higher Dimensional Classical W-Algebras

Classical $W$-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These $W$-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of Diff$S(1)$ is taken by the algebra of generators of local diffeomorphisms in two dimensions.Comment: 22 pages, Plain TeX, KUL-TF-92/19, US-FT/6-9

Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.Comment: 61 pg

A One-Parameter Family of Hamiltonian Structures for the KP Hierarchy and a Continuous Deformation of the Nonlinear \W_{\rm KP} Algebra

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting \W-algebra is a one-parameter deformation of \W_{\rm KP} admitting a central extension for generic values of the parameter, reducing naturally to \W_n for special values of the parameter, and contracting to the centrally extended \W_{1+\infty}, \W_\infty and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to \w_{\rm KP}. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of \widehat{\W}_\infty which contracts to a new nonlinear algebra of the \W_\infty-type.Comment: 31 pages, compressed uuencoded .dvi file, BONN-HE-92/20, US-FT-7/92, KUL-TF-92/20. [version just replaced was truncated by some mailer

Noncompact SL(2,R) spin chain

We consider the integrable spin chain model - the noncompact SL(2,R) spin magnet. The spin operators are realized as the generators of the unitary principal series representation of the SL(2,R) group. In an explicit form, we construct R-matrix, the Baxter Q-operator and the transition kernel to the representation of the Separated Variables (SoV). The expressions for the energy and quasimomentum of the eigenstates in terms of the Baxter Q-operator are derived. The analytic properties of the eigenvalues of the Baxter operator as a function of the spectral parameter are established. Applying the diagrammatic approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into a product of certain operators each depending on a single separated variable.Comment: 29 pages, 12 figure

Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct the $W_n^l$ algebras, first discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky.Comment: 41 page

The Robinson-Trautman Type III Prolongation Structure Contains K2_2

The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K$_2$, which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of B\"acklund transformations to evolve new solutions.Comment: 20 pages, plain TeX, no figures, submitted to Commun. Math. Phy