2007/12/26 by A. Levin, Levin, A., M. A. Olshanetsky +2
Mathematics · #14E20 #54C40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.QA #msc:14E20 #msc:54C40
paper · pdf · doi:10.48550/arxiv.0712.3828
36 pages,AMS-LaTeX 1.2, Essentially revised and elaborated version of hep-th/0010043
arxiv created 2007/12/26 · openalex publication_date 2007/12/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The space of generalized projective structures on a Riemann surface Σ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of WN-gravity on Σ×ℝ. This space is a generalization of the space of projective structures on the Riemann surface. We define the moduli space of WN-gravity as a symplectic quotient with respect to the canonical action of a special class of Lie algebroids. This moduli space describes in particular the moduli space of deformations of complex structures on the Riemann surface by differential operators of finite order, or equivalently, by a quotient space of Volterra operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids, because they are constructed by means of AGD bivector on the space of opers restricted on a circle. The AGD-algebroids are particular case of Lie algebroids related to a Poisson sigma-model. The moduli space of the generalized projective structure can be described by cohomology of a BRST-complex.