Virasoro Constraints Research Articles

Virasoro constraints and flows in hermitian matrix models

The Virasoro constraints are written in terms of the solution to the hermitian matrix model string equation. An explicit solution is found to the first two linear constraints. This solution is used in an asymptotic limit to study the renormalization group flow between multicritical points. We are led to the conclusion that there is no flow between odd and even multicritical points due to a nonperturbative instability, while flows between odd multicritical points are not prevented.

Matrix models of two-dimensional gravity and Toda theory

Recurrent relations for orthogonal polynomials, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equivalent to the equations of the Toda-chain hierarchy supplemented by additional Virasoro constraints. This is the case even before the continuum limit is taken. When the odd times are suppressed, the Volterra hierarchy arises, its continuum limit being the KdV hierarchy. The unitary-matrix model gives rise to a sort of quantum deformation of the Volterra equation. We call it the modified Volterra hierarchy, since in the analogous continuum limit it may turn into the mKdV. For multimatrix models the Toda lattice hierarchy of the type A ∞ appears. The time-dependent partition functions are given by τ-functions — the sections of the determinant bundle over the infinite-dimensional grassmannian, associated with Riemann surfaces of spectral parameters. However, because of the Virasoro constraints matrix models correspond to a very restricted subset of the grassmannian, intimately related to the W ∞ Lie subalgebra of UGL(∞). The continuum limit, leading to multicritical behavior is adequately described in terms of peculiar operators, defined as hypergeometric polynomials of the original matrix fields and resembling a sort of handle-gluing operator.

Conformal fields. From Riemann surfaces to integrable hierarchies

I discuss the idea of translating ingredients of conformal field theory into the language of hierarchies of integrable differential equations. Primary conformal fields are mapped into (differential or matrix) operators living on the phase space of the hierarchy, whereas operator insertions of, e.g., a current or the energy-momentum tensor, become certain vector fields on the phase space and thus acquire a meaning independent of a given Riemann surface. A number of similarities are observed between the structures arising on the hierarchy and those of the theory on the world-sheet. In particular, there is an analogue of the operator product algebra with the Cauchy kernel replaced by its “off-shell” hierarchy version. Also, hierarchy analogues of certain operator insertions admit two (equivalent, but distinct) forms, resembling the “bosonized” and “fermionized” versions respectively. As an application, I obtain a useful reformulation of the Virasoro constraints of the type that arise in matrix models, as a system of equations on dressing (or Lax) operators (rather than correlation functions, i.e., residues or traces). This also suggests an interpretation in terms of a 2D topological field theory, which might be extended to a correspondence between Virasoro-constrained hierarchies and topological theories.

Multi-matrix model and 2D Toda multi-component hierarchy

We study the integrability of the hermitian matrix-chain model at finite N. The integrable system, constructed from the matrix integrals using orthogonal polynomials is identified with the two-dimensional Toda system with multi-component hierarchy. We derive the Lax equations, the zero curvature conditions and an infinite number of conserved quantities for this 2D Toda hierarchy. The partition function of the matrix model is proved to be the “tau-function” of this Toda system. Also, using our formalism, we derive the Virasoro constraints on the partition function of the multi-matrix model for the first time.

Continuum versus discrete Virasoro in one-matrix models

Virasoro constraints are imposed on the partition function of the one-matrix models at the discrete level before the continuum limit is taken. After a preliminary discussion of the role of the Virasoro constraints and the associated Virasoro group orbit structure in the proof of universality, the interrelation between discrete and continuum Virasoro constraints is considered. As an intermediate step, the model of complex matrices is discussed. An appropriate change of time variables for the continuum limit (Kazakov's or “admissible”) as well as time-dependent rescales of the partition function are introduced in the approaches of orthogonal polynomials and loop equations. Even after these modifications, Virasoro constraints do not possess a nice continuum limit in the model of complex matrices because of additional constant terms in the Virasoro generators. These terms are eliminated in the case of the reduced hermitean matrix model so that the results of Fukuma et al. and Dijkgraaf et al. are rigorously derived.

W 1+∞-type constraints in matrix models at finite N

A systematic method is presented to obtain a large class of constraint equations for matrix models at finite N. These constraints are associated with the higher order differential operators with respect to the eigenvalues of the matrices, {{ λ n ∂ λ m }} with n, m⩾0. In the case of the one-matrix model, we find that the constraints for lower m are reducible to the Virasoro constraints. We derive a class of new constraint equations for the matrix chains.

Loops and unoriented strings

We derive and analyze Schwinger-Dyson equations for Dyson's Coulomb gas at arbitrary temperature. This system interpolates between theories of oriented and unoriented strings. The Schwinger-Dyson equations can be formulated as Virasoro constraints constructed from a boson with a temperature-dependent background charge. We describe a geometric interpretation of these equations, calculate correlation functions perturbatively in 1/ N and discuss a high-low temperature duality exhibited by the Coulomb gas. We also interpret the double scaling continuum limit of the loop equations.

A DEFORMATION OPERATOR OF THE VIRASORO ALGEBRAIC CONSTRAINTS IN MATRIX MODELS

Following the method of Dijkgraaf et al. a deformation operator P(q) is constructed from the KP hierarchy theory. It is explicitly shown that the deformation operator P(q) can be used to define a recursion relation of the Virasoro constraints on the two- and three-matrix models theory, respectively.

Recursion relations in topological gravity with minimal matter

A complete set of recursion relations is obtained for correlation functions in two-dimensional topological gravity coupled with the ADE series of topological minimal models, by demanding the consistency of the contact algebra of physical operators. When formulated as linear constraints on the partition function, they coincide with the Virasoro constraints and the W-constraints that have been conjectured and partially verified in the multi-matrix models of two-dimensional quantum gravity. These recursion relations give a complete solution of the theory.

Reduced unitary matrix models and the hierarchy of τ-functions

We study reductions of unitary one-matrix models. The unitary model admits an especially rich class of reductions of which the widely known symmetric model is only one. The partition function of a certain class of reduced models is shown to be a product of distinct Toda chain τ-functions. Virasoro constraints are also derived for the case of unitary models. It is claimed, in analogy with the reduced hermitian model, that in the continuous limit the Virasoro constraints must be imposed on a fractional power of the original partition function.

Topological gravity with minimal matter

Topological minimal matter, obtained by twisting the minimal N = 2 superconformal field theory, is coupled to two-dimensional topological gravity. The free field formulation of the coupled system allows explicit representations of BRST charge, physical operators and their correlation functions. The contact terms of the physical operators may be evaluated by extending the argument used in a recent solution of topological gravity without matter. The consitency of the contact terms in correlation functions implies recursion relations which coincide with the Virasoro constraints derived from the multi-matrix models. Topological gravity with minimal matter thus provides the field theoretic description for the multi-matrix models of two-dimensional quantum gravity.

SYMMETRIES IN NON-PERTURBATIVE 2-d QUANTUM GRAVITY

Based on the Korteweg-de Vries (KdV) hierarchy formulation of non-perturbative 2-d quantum gravity, we investigate the symmetries of the system. Particularly, it is shown that the recently found Virasoro constraints are due to the non-isospectral symmetries of KdV hierarchy, which can be interpreted as SL(2, C) invariance and the independence of the moduli of the auxiliary infinite genus hyperelliptic Riemann surfaces which appear in the Krichever construction of soliton solutions.

Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity

We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

Recursion relation for the string S-matrix

We derive a recursion relation of the S-matrix for the open bosonic string by using the Witten string field theory. This relation is derived from a constraint equation analogous to the Virasoro constraints of the τ-function in the matrix model or topological gravity. This constraint equation may give the non-perturbative string S-matrix in non-trivial background fields.

Virasoro constraints for D2n + 1−, E6−, E7−, E8−type minimal models coupled to 2D gravity

We find Virasoro constraints for D_(2n + 1−), E_6−, E_7−, E_8−type models analogous to the recently discovered Virasoro constraints for An-type models by Fukuma et al., and Dijkgraaf et al. We verify that the proposed Virasoro constraints give operator scaling dimensions identical to those found by Kostov. We check that these Virasoro constraints and, more generally, W-algebra constraints can be used to express correlation functions with non-primary operator in terms of correlation functions of primary operators only.

On the origin of Virasoro constraints in matrix model: lagrangian approach

A simple way of derivation of Virasoro and W-constraints from explicit on-shell invariance of the action of matrix models is described. Non-trivial corrections to naive Virasoro and W-generators are related to “quantum effects”, namely the change of integration measure. This lagrangian approach provides a clear and unified treatment of constraints in all matrix models.

Interacting physical states of the bosonic string. II

A preliminary discussion on the importance of the contact terms in relation with the existence of many string vacua is presented. The problem of the BRST invariance of the string states transition amplitudes is considered in the sigma model approach to the string. It is shown that the BRST invariance is controlled by a gravitational (Diff) anomaly, and its connections with the standard Virasoro constraints are illustrated. The tachyon amplitudes are analysed and the BRST invariant dilaton vertex operator for non flat world-sheet is constructed.

Algebraic quantization on a group and nonabelian constraints

A generalization of a previous group manifold quantization formalism is proposed. In the new version the differential structure is circumvented, so that discrete transformations in the group are allowed, and a nonabelian group replaces the ordinary (central)U(1) subgroup of the Heisenberg-Weyl-like quantum group. As an example of the former we obtain the wave functions associated with the system of two identical particles, and the latter modification is used to account for the Virasoro constraints in string theory.

Free bosonic string field theory without supplementary fields

A covariant local action for free bosonic string fields is constructed without the use of supplementary fields. The open string case is treated in detail. Up to a mathematical conjecture that is likely to hold it is shown that the Virasoro constraints arise as a special choice of gauge. The kinetic operator turns out to be extremely simple, the gauge transformation law arising rather implicitly. The case of closed strings is briefly discussed.

The string field embedded in a loop space geometry

We study the equations of motion for a loop geometry that we introduced in an earlier paper. These geometrical constraints are necessary in order to satisfy Q 2 = 0, where Q is a BRST charge that encodes the “local” reparametrization invariance related to the loop differential geometry. In the linearized version of these equations, the zeroth order is satisfied by d = 26 and α 0 = = 1, while the first order is satisfied by a string field that essentially obeys the Virasoro constraints, thus reproducing the equations satisfied by free string fields. We are therefore encouraged to speculate that the higher order terms, which are not studied in this paper, correspond to string field interactions.