2003/01/13 by A. Levin, Levin, A., M. A. Olshanetsky +2
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #hep-th #math.DG
paper · pdf · doi:10.48550/arxiv.hep-th/0301078
Latex, 34 pages
arxiv created 2003/01/13 · openalex publication_date 2003/01/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting with a Lie algebroid \cal A over a space M we lift its action to the canonical transformations on the affine bundle \cal R over the cotangent bundle T^*M. Such lifts are classified by the first cohomology H1(\cal A). The resulting object is a Hamiltonian algebroid \cal AH over \cal R with the anchor map from \G(\cal AH) to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize Lie algebras of canonical transformations. We prove that the BRST operator for \cal AH is cubic in the ghost fields as in the Lie algebra case. The Poisson sigma model is a natural example of this construction. Canonical transformations of its phase space define a Hamiltonian algebroid with the Lie brackets related to the Poisson structure on the target space. We apply this scheme to analyze the symmetries of generalized deformations of complex structures on Riemann curves \Sig,n of genus g with n marked points .We endow the space of local \GL-opers with the Adler-Gelfand-Dikii (AGD) Poisson brackets. It allows us to define a Hamiltonian algebroid over the phase space of WN-gravity on \Sig,n. The sections of the algebroid are Volterra operators on \Sig,n with the Lie brackets coming from the AGD bivector. The symplectic reduction defines the finite-dimensional moduli space of WN-gravity and in particular the moduli space of the complex structures \bp on \Sig,n deformed by the Volterra operators.