Hamiltonian Algebroid Symmetries in W-gravity and Poisson sigma-model

Abstract

Starting from a Lie algebroid $\mathcal{A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle $\mathcal{R}$ over the cotangent bundle $T^{\ast }V$. Such lifts are classified by the first cohomology $H^1(\mathcal{A})$. The resulting object is the Hamiltonian algebroid ${\mathcal{A}}^H$ over $\mathcal{R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize the Lie algebras of canonical transformations. We prove that the BRST operator for ${\mathcal{A}}^H$ 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 $V_3$ of $\SL$-opers on a Riemann curve $\Si_{g,n}$ of genus g with n marked points. The sections of this algebroid are the second order differential operators on $\Si_{g,n}$. The algebroid is lifted to the Hamiltonian algebroid over the phase space of $W_3$-gravity. We describe the BRST operator leading to the moduli space of $W_3$-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.

Several changes, the definition of algebroid representation is modified, typos corrected, LateX, 28 pages