Lie Algebroids as Gauge Symmetries in Topological Field Theories

Abstract

The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The ``coordinate part'' of the Hamiltonian phase space is the Poisson manifold $M$ and the Lie algebroid brackets are defined by means of the Poisson bivector. The Lie algebroid action defined on $M$ can be lifted to the phase space. The main observation is that the classical BRST operator has the same form as in the case of the Lie groups action. Two examples are analyzed. In the first, $M$ is the space of $SL(3,C)$-opers on Riemann curves with the Adler-Gelfand-Dikii brackets. The corresponding Hamiltonian system is the $W_3$-gravity. Its phase space is the base of the algebroid bundle. The sections of the bundle are the second order differential operators on Riemann curves. They are the gauge symmetries of the theory. The moduli space of $W_3$ geometry of Riemann curves is the symplectic quotient with respect to their action. It is demonstrated that the nonlinear brackets and the second order differential operators arise from the canonical brackets and the standard gauge transformations in the Chern-Simons field theory, as a result of the partial gauge fixing. The second example is $M=C^4$ endowed with the Sklyanin brackets. The symplectic reduction with respect to the algebroid action leads to a generalization of the rational Calogero-Moser model. As in the previous example the Sklyanin brackets can be derived from a ``free theory.'' In this case it is a ``relativistic deformation'' of the $SL(2,C)$ Higgs bundle over an elliptic curve.