Bott Virasoro Group Research Articles

Berezin Quantization, Conformal Welding and the Bott–Virasoro Group

Following Nag–Sullivan, we study the representation of the group Diff+(S1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Diff}^+(S^1)$$\\end{document} of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott–Virasoso cocycle class, and leads to a surprising identity for the Takhtajan–Teo energy functional on Diff+(S1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Diff}^+(S^1)$$\\end{document}.

Edge Wave and Boundary Layer of Vortex Matter.

We show that vortex matter, that is, a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with signlike eddies, exhibits (i)a boundary layer of vorticity (vorticity layer) and (ii)a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation, exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to the odd viscosity of vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect.

New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map.

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $$M$$ , the metric is defined only on a sub-bundle $${{\mathrm{\mathcal {H}}}}$$ of the tangent bundle $$TM$$ , called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on $${{\mathrm{\mathcal {H}}}}$$ , and normal geodesics that depend both on $${{\mathrm{\mathcal {H}}}}$$ itself and on the metric on $${{\mathrm{\mathcal {H}}}}$$ . In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups with invariant sub-Riemannian structure. As examples, we consider the group of sense-preserving diffeomorphisms $${{\mathrm{Diff}}}\, S^1$$ of the unit circle and the Virasoro–Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant Kahlerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa–Holm, Hunter–Saxton, KdV, and other known non-linear PDE.

The energy functional on the Virasoro–Bott group with the L 2-metric has no local minima

The geodesic equation for the right invariant L 2-metric (which is a weak Riemannian metric) on each Virasoro–Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular, solutions of KdV do not define locally length-minimizing paths.

Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

The Virasoro-Bott group endowed with the right-invariant L 2-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS

Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler–Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero–Bogoyavlenskii–Schiff equation and various other (2 + 1)-dimensional Korteweg–deVries (KdV) type systems follow from this construction. Using the right invariant H1inner product on the Lie algebra of loop Bott–Virasoro group, we formulate the Euler–Poincaré framework of the (2 + 1)-dimensional of the Camassa–Holm equation. This equation appears to be the Camassa–Holm analogue of the Calogero–Bogoyavlenskii–Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter–Saxton equation. Finally, we give an Euler–Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler–Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.

EXTENDED BOTT–VIRASORO ALGEBRA, SEMIDIRECT PRODUCT, ⋆-LIE ALGEBRA OF DIFFEOMORPHISM AND NONCOMMUTATIVE INTEGRABLE SYSTEMS

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S1) ⋉ C∞(S1) algebra using the action of ⋆-deformed matrix Hill's operators Δ⋆ on the space of ⋆-deformed tensor densities. The centrally extended semidirect product algebra [Formula: see text] is a Lie algebra of extended semidirect product of the Bott–Virasoro group [Formula: see text]. The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ⋆ on ⋆ deformed tensor-densities on S1 allow us to construct noncommutative two component Korteweg–de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of ⋆-deformed quantization of the centrally extended semidirect product algebra [Formula: see text] and two component noncommutative KdV equations.

On the classification of the coadjoint orbits of the Sobolev Bott–Virasoro group

The purpose of this paper is to give the classification of the Bott–Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott–Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott–Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied.

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.

Energy characteristics of subordination chains

We consider subordination chains of simply connected domains with smooth boundaries in the complex plane. Such chains admit Hamiltonian and Lagrangian interpretations through the Löwner–Kufarev evolution equations. The action functional is constructed and its time variation is obtained. It represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.

Euler–Poincaré flows and leibniz structure of nonlinear reaction–diffusion type systems

In this paper we present Euler–Poincaré formulation of the Fisher, Fitzhugh–Nagumo, Burgers–Huxley and extended Fitzhugh–Nagumo and extended Burgers–Huxley type nonlinear reaction–diffusion systems. All these flows are related to infinite dimensional almost Poisson manifolds and the corresponding Lie–Poisson structures yield Leibniz brackets, a bracket endowed with both symmetric and skewsymmetric parts. The symmetric part contributes the diffusion part of the ssystem. The properties exhibited by the reaction–diffusion systems defined in this way are in general very different from the standard Hamiltonian mechanics since the dynamics are controlled by the standard Poisson brackets. Moreover, all the nonlinear reaction–diffusion systems under consideration are Euler–Poincaré flows on the dual of Kirillov’s superalgebra associated to the Bott–Virasoro group.

Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems

We show that all the Antonowicz–Fordy type coupled KdV equations have the same symmetry group and similar bi-Hamiltonian structures. It turns out that their configuration space is Diff ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ , where Diff ( S 1 ) ˆ is the Bott–Virasoro group of orientation preserving diffeomorphisms of the circle, and all these systems can be interpreted as equations of a geodesic flow with respect to L 2 metric on the semidirect product space Diff ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ .

Euler-Poincaré Formalism of Coupled KdV Type Systems and Diffeomorphism Group on S1

This paper describes a wide class of coupled KdV equations. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2-component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 metric on the semidirect product group , where Diff( S 1 ) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg-Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on .

Geodesic flow on the Bott–Virasoro group and deformed Hunter–Saxton equation

We study the deformed Hunter–Saxton equation proposed by Manna and Neveu (2003 Preprint physics/0303085). It is known that the Manna–Neveu system is integrable admitting a Lax pair. In this paper, we show that this equation can be explained as an equation of the geodesic flow associated with a right-invariant metric on the Bott–Virasoro group. We also discuss the bi-Hamiltonian structure of the deformed Hunter–Saxton equation.

Geodesic flow on (super-) Bott–Virasoro group and Harry Dym family

We show various Harry Dym type equations and Super Harry Dym equations, introduced by Brunelli, Das, and Popowicz [2003 Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys. 44, 4756–4767 (2003)], follow from the geodesic flows on the Bott–Virasoro group and its supersymmetric generalization, superconformal group. In fact, their bi-Hamiltonian structures can be derived from the Lie Poisson structures on the (super-) Bott–Virasoro orbit. We also show that that m2KdV or Calogero–Degasperis equation is also connected to the Bott–Virasoro group.

Geometry of the Kaup-Newell equation

It is known that the KdV equation appears naturally in the geometry of the orientation preserving diffeomorphic group Diff( S 1). It is a geodesic flow of a L 2 metric on the Bott-Virasoro group. The nonlinear Schrödinger equation (NLSE) is an evolution equation analogous to the KdV equation which describes an isospectral deformation of the first order 2 × 2 matrix differential operator, yet the family of NSLE has not been studied via diffeomorphic groups. In this paper we derive the Kaup-Newell equation and its generalizations are the Euler-Poincaré flows on the space of first-order scalar (or matrix) differential operators. We show that the operators involved in the flow generated by the action of Vect( S 1) are neither Poisson nor skew symmetric. We also discuss the relation between the KdV flow and the Kaup-Newell flow.

Projective and affine connections on S1 and integrable systems

It is known that the Korteweg–de Vries (KdV) equation is a geodesic flow of an L 2 metric on the Bott–Virasoro group. This can also be interpreted as a flow on the space of projective connections on S 1. The space of differential operators Δ ( n) = ∂ n + u 2 ∂ n−2 +⋯+ u n form the space of extended or generalized projective connections. If a projective connection is factorizable Δ ( n) =( ∂−(( n+1)/2−1) p 1)⋯( ∂+( n−1)/2 p n ) with respect to quasi primary fields p i ’s, then these fields satisfy ∑ i=1 n (( n+1)/2− i) p i =0. In this paper we discuss the factorization of projective connection in terms of affine connections. It is shown that the Burgers equation and derivative non-linear Schrödinger (DNLS) equation or the Kaup–Newell equation is the Euler–Arnold flow on the space of affine connections.

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys.40 (1997), 31–39). The Ito equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L2 metric on the semidirect product space Diff s \(({\text{S}}^{\text{1}} \widehat{) \odot }\)C∞(S1), where Diff s (S1) is the group of orientation-preserving Sobolev Hs diffeomorphisms of the circle. We also study the geodesic flows with respect to H1 metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.

Diffeomorphism, periodic KdV and C. Neumann system

After reviewing few known facts regarding KdV equation as an Euler equation in the coadjoint orbit of the Virasoro–Bott group we are able to establish an interesting connection between the projective vector field, a vector field leaving fixed a given projective connection, and the C. Neumann system using the idea of Knörrer and Moser. We show that certain quadratic function of a projective field satisfies C. Neumann system.