标题： 作为拓扑场论中的规范对称性的李代数胚 显示英文标题

标题： Lie Algebroids as Gauge Symmetries in Topological Field Theories

摘要： 李代数胚是李代数的推广。 它们特别作为研究具有第一类约束的动力系统的一种数学工具而出现。 在这里我们考虑由一类特殊的李代数胚生成的哈密顿系统的规范对称性。 哈密顿相空间的“坐标部分”是泊松流形$M$，而李代数胚括号是通过泊松二向量定义的。 定义在$M$上的李代数胚作用可以提升到相空间。 主要观察结果是，经典BRST算子的形式与李群作用的情况相同。 分析了两个例子。 在第一个例子中，$M$是黎曼曲线上的$SL(3,C)$-操作子的空间，具有Adler-Gelfand-Dikii括号。 相应的哈密顿系统是$W_3$-引力。 其相空间是代数胚丛的基空间。 该丛的截面是黎曼曲线上的二阶微分算子。 它们是该理论的规范对称性。 黎曼曲线的$W_3$几何的模空间是相对于它们的作用的辛商。 证明了非线性括号和二阶微分算子是从 Chern-Simons 场论中的规范对称性通过部分规范固定的结果。 第二个例子是带有 Sklyanin 括号的$M=C^4$。 相对于代数胚作用的辛约化导致了有理 Calogero-Moser 模型的推广。 与前一个例子一样，Sklyanin 括号可以从一个“自由理论”中导出。 在这种情况下，它是椭圆曲线上的$SL(2,C)$ Higgs丛的“相对论性变形”。 显示英文摘要

摘要： 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.