Virasoro Action Research Articles

Matrix model partition function by a single constraint

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of {hat{w}}-operators. In this letter, we demonstrate that even more is true: a singlew-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of two requirements: either a string equation imposed on the KP/Toda tau -function or a pair of Virasoro generators. This mysterious single-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to super integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with potential of higher degree, when the constraint algebra is a larger W-algebra, and neither W-representation, nor superintegrability are understood well enough.

Loop-corrected Virasoro symmetry of 4D quantum gravity

Recently a boundary energy-momentum tensor Tzz has been constructed from the soft graviton operator for any 4D quantum theory of gravity in asymptotically flat space. Up to an “anomaly” which is one-loop exact, Tzz generates a Virasoro action on the 2D celestial sphere at null infinity. Here we show by explicit construction that the effects of the IR divergent part of the anomaly can be eliminated by a one-loop renormalization that shifts Tzz.

Tau Function and Virasoro Action for the n × n KdV Hierarchy

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group L as positive and negative subgroups \({L_\pm}\) and a commuting sequence in the Lie algebra \({\mathcal{L}_+}\) of \({L_+}\). Given \({f\in L_-}\), there is a formal inverse scattering solution u f of the hierarchy. When there is a 2 co-cycle on \({\mathcal{L}}\) that vanishes on both \({\mathcal{L}_+ {\rm and} \mathcal{L}_-}\), Wilson constructed for each \({f\in L_-}\) a tau function \({\tau_f}\) for the hierarchy. In this third paper, we prove the following results for the n × n KdV hierarchy: (1) The second partials of \({\ln\tau_f}\) are differential polynomials of the formal inverse scattering solution u f . Moreover, \({u_f}\) can be recovered from the second partials of \({\ln\tau_f}\). (2) The natural Virasoro action on \({\ln\tau_f}\) constructed in the second paper is given by partial differential operators in \({\ln\tau_f}\). (3) There is a bijection between phase spaces of n × n KdV hierarchy and Gelfand-Dickey (GD n ) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the n × n KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD n hierarchy under the bijection.

Virasoro Action on Imaginary Verma Modules and the Operator Form of the KZ-Equation

We define the Virasoro algebra action on imaginary Verma modules for affine $${\mathfrak{sl}(2)}$$ and construct an analogue of the Knizhnik–Zamolodchikov equation in the operator form. Both these results are based on a realization of imaginary Verma modules in terms of sums of partial differential operators.

The higher flows of harmonic maps and an application to the Virasoro action

One characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of λ which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a −1-flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Using Grozman’s formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S 1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H 1-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S 1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S 1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.

On conformal field theory and stochastic Loewner evolution

We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.

Symmetries, sato theory, and tau functions

We relate Virasoro (conformal) symmetries with a subset of generators of the algebra and show their connection to the symmetries of [8] arising from Virasoro action on gauge operat tors . We a1so show explicitly their relation to the classical mastersymmetries of degree one (providing a new method of generating such mastersymmetries) and we establish connections with the impor tant operator . This leads to a Hamiltonian form for the Virasoro flows in a single potential context.

Some remarks on symmetries, dressing, and virasoro action

With background reference, to work of Bullough, Case, Chen, Cheng, Fuchssteiner, Grinevic, Konopelcenko, Lee, Li, Lin, Matsukidaira, G. and W. Oevel, Orlov, Satsuma, Semikhatov, Shulman, and Strampp, general dressing formulas for KP situations are developed, based on wave and adjoint wave functions of the bare operators, and applications to conformal symmetries are indicated. First order mastersymmetries are exhibited in a coherent framework, with some new observations, and precise connections to Virasoro operators are spelled out, leading to the standard hereditary algebras, etc.

Virasoro algebra action on integrable hierarchies and Virasoro constraints in matrix models: Matrix models from the viewpoint of integrable hierarchies

The action of the Virasoro algebra on integrable hierarchies of non-linear equations and on related objects (“Schrödinger” differential operators) is investigated. The method consists in pushing forward the Virasoro action to the wave function of a hierarchy, and then reconstructing its action on the dressing and Lax operators. This formulation allows one to observe a number of suggestive similarities between the structures involved in the description of the Virasoro algebra on the hierarchies and the structure of conformal field theory on the world-sheet. This includes, in particular, an “off-shell” hierarchy version of operator products and of the Cauchy kernel. In relation to matrix models, which have been observed to be effectively described by integrable hierarchies subjected to Virasoro constraints, I propose to define general Virasoro-constrained hierarchies also in terms of dressing operators, by certain equations which carry the information of the hierarchy and the Virasoro algebra simultaneously and which suggest an interpretation as operator versions of recursion/loop equations in topological theories. These same equations provide a relation with integrable hierarchies with quantized spectral parameter introduced recently. The formulation in terms of dressing operators allows a scaling (continuum limit) of discrete (i.e. lattice) hierarchies with the Virasoro constraints into “continuous” Virasoro-constrained hierarchies. In particular, the KP hierarchy subjected to the Virasoro constraints is recovered as a scaling limit of the Virasoro-constrained Toda hierarchy. The dressing operator method also makes it straightforward to identify the full symmetry algebra of Virasoro-constrained hierarchies, which is related to the family of W ∞( J) algebras introduced recently. For the KP hierarchy subjected to the Virasoro constraints depending on a parameter which may be interpreted as a conformal weight of an abstract bc system, we find the Borel subalgebra of W ∞( J), which we describe as an extension by the L 2 Virasoro generator of the “wedge”, or higher-spin, algebra B λ=J 2−J . Reductions to the generalized N-KdV hierarchies and the analysis of the corresponding constraints are also carried out. In particular, the W N algebra in the N-reduced case is related to W ∞ through a hamiltonian reduction and the sl( N) Kac-Moody algebra is observed.