2000/10/06 by A. Levin, Levin, A., Olshanetsky +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.hep-th/0010043
openalex publication_date 2000/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting from a Lie algebroid \cal A over a space V we lift its action to the canonical transformations on the principle affine bundle \cal R over the cotangent bundle T^*V. Such lifts are classified by the first cohomology H1(\cal A). The resulting object is the Hamiltonian algebroid \cal AH over \cal R with the anchor map from \G(\cal AH) to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize the 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. To illustrate this construction we analyze two topological field theories. First, we define a Lie algebroid over the space V3 of \SL-opers on a Riemann curve \Sig,n of genus g with n marked points. The sections of this algebroid are the second order differential operators on \Sig,n. The algebroid is lifted to the Hamiltonian algebroid over the phase space of W3-gravity. We describe the BRST operator leading to the moduli space of W3-gravity. In accordance with the general construction the BRST operator is cubic in the ghost fields. We present the Chern-Simons explanation of our results. The second example is the Hamiltonian algebroid structure in the Poisson sigma-model invoked by Cattaneo and Felder to describe the Kontsevich deformation quantization formula. The hamiltonian description of the Poisson sigma-model leads to the Lie algebraic form of the BRST operator.