Equivalence of the Drinfeld-Sokolov reduction to a bi-Hamiltonian reduction

We show that the Drinfeld-Sokolov reduction is equivalent to a bi-Hamiltonian reduction, in the sense that these two reductions, although different, lead to the same reduced Poisson (more correctly, bi-Hamiltonian) structure. In order to do this, we heavily use the fact that they are both particular cases of a Marsden-Ratiu reduction.This work has been supported by the Italian MURST and by the GNFM of the Italian CNR.     

Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view

We show that the Drinfeld-Sokolov reduction can be framed in the general theory of bihamiltonian manifolds, with the help of a specialized version of a reduction theorem for Poisson manifolds by Marsden and Ratiu.This work has been supported by the Italian MURST and by the GNFM of the Italian C.N.R.     

Integrability of linear and nonlinear evolution equations and the associated nonlinear fourier transforms

The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.     

Invariants of linear combinations of transpositions

It is shown that for a one-parameter set H _N of linear combinations of N(N–1)/2 elementary transpositions {P _jk } (1j at arbitrary natural N3, one can construct a variety {I _m } (3mN) of operators which commute with H _N . Being applied to SU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction.     

AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories

We give a detailed exposition of the Alexandrov–Kontsevich–Schwarz– Zaboronsky superfield formalism using the language of graded manifolds. As a main illustrating example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin–Vilkovisky master action for the model.      

Multiplication of solutions for linear overdetermined systems of partial differential equations

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE’s contains the Cauchy–Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.     

Exact symplectic structures and a classical model for the Dirac electron

We show how the classical model for the Dirac electron of Barut and coworkers can be obtained as a Hamiltonian theory by constructing an exact symplectic form on the total space of the spin bundle over spacetime.     

The analytic torsion of a cone over an odd dimensional manifold

We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W$W$. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Müller theorem  and  for a manifold with boundary, according to Brüning and Ma (2006) [5]. We also prove Poincaré duality for the analytic torsion of a cone.     

Solution of the Cauchy problem for a continuous limit of the Toda lattice and its superextension

A supersymmetric equation associated with a continuum limit of the classical superalgebra sl(n/n+1) is constructed. This equation can be considered as a superextension of a continuous limit of t the Toda lattice with fixed end-points or, in other words, as a supersymmetric version of the heavenly equation. A solution of the Cauchy problem for the continuous limit of the Toda lattice and for its superextension is given using some formal reasonings.     

On the interaction of solitons for a class of integrable systems in the spacetime R n+1

A class of integrable systems of nonlinear partial differential equations in the spacetime R ⁿ⁺¹ is introduced. Single and multi-soliton solutions are constructed by using the Darboux matrix method. It is proved that as t±, a k multi-soluton solution splits asymptotically into k single solitons. Moreover, the interaction between solitons is elastic if we consider their magnitudes only.The work is C. H. Gu supported by the Chinese National Program for fundamental research nonlinear science.     

The lie bialgebroid of a Poisson-Nijenhuis manifold

We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures.     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

Twisted Diff S 1-action on loop groups and representations of the virasoro algebra

A modified Hamiltonian action of Diff S ¹on the phase space LG ^C /G^C, where LG is a loop group, is defined by twisting the usual action by a left translation in LG. This twisted action is shown to be generated by a nonequivariant moment map, thereby defining a classical Poisson bracket realization of a central extension of the Lie algebra diff_C S ¹. The resulting formula expresses the Diff S ¹generators in terms of the left LG translation generators, giving a shifted modification of both the classical and quantum versions of the Sugawara formula.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation.     

A supersymmetric integrable system for arbitrary N

On the dual space to the loop algebra of the Lie algebra of vector fields on a super N-circle, there lives an integrable dynamical system.     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

Symplectic manifolds with Lagrangian fibration

A structure theorem is presented for certain kinds of symplectic manifold with a Lagrangian fibration. As a corollary, the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of symplectic manifold considered in the theorem for which in addition, a certain cohomology class vanishes. These results and techniques are then applied to two situations in classical mechanics where symplectic manifolds foliated by Lagrangian submanifolds arise, namely, the Legendre transformation and Hamilton-Jacobi theory.     

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M ₀, respectively. The nonuniqueness of the local injections t: M ₀M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, C ^k fields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.Research work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR) and by the Italian Ministry of Public Education through the research project Geometria e Fisica.     

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.