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Target space equivariant cohomological structure of the Poisson sigma model. (English) Zbl 1038.53079

A formulation of the Poisson sigma model in which the target space Poisson manifold \((M,\overline{\omega})\) carries a Hamilton action of some finite-dimensional Lie algebra \(\mathcal H\) is discussed. Such a Poisson sigma model has a hidden equivariant cohomological structure that makes it akin to the cohomological theories and determines the structure of the action and properties of the gauge invariant observables. The analysis relies on the abstract algebraic framework, called operation. One can define an \(\mathcal H\)-operation over the space of functions on the superbundle \(\Pi T\Pi T^*M\) and its \(\mathcal H\)-equivariant extension \(\text{Fun}(\Pi T\Pi T^*M)\hat\otimes W(\mathcal H)\), where \(W(\mathcal H)\) is the Weil operation of the Lie algebra \(\mathcal H\). One can construct a de Rham superfield realization of this \(\mathcal H\)-equivariant operation. This leads to an \(\mathcal H\)-operation over a formal graded associative algebra of superfields, referred to as \(\mathcal H\)-Hamilton de Rham superfield operation, whose \(\mathcal H\)-basic cohomology is intimately related to the \(\mathcal H\)-equivariant cohomology of \(\Pi T\Pi T^*M\). This de Rham superfield formalism is used to explore efficiently the implications of the Batalin-Vilkoviski master equation. The action of the Poisson sigma model is \[ \mathcal S_\pi=\int_\Sigma\mu\left(y_i{\text d}x^i+\frac{1}{2}\pi^{ij}(x) y_iy_j-{\text d}\gamma^ah_a(x)\right), \] where \(x^i,y_i,\gamma^a\) are generators of the \(\mathcal H\)-Hamilton de Rham superfield operation, \(\pi^{ij}\) is a bivector on \(M\), \(\mu\) is the integration supermeasure of \(\Sigma\) and \(h_a\) are the functions of \(M\) corresponding via the action to generators of \(\mathcal H\). The action satisfies the classical master equation \(\{\mathcal S_\pi,\mathcal S_\pi\}=0\) if and only if \(\pi^{ij}\) is a Poisson bivector for which \(h_a\) are Casimirs. If \(\pi^{ij}\) Schouten commutes with \(\overline{\omega}^{ij}\) then \(\mathcal S_\pi\) is a representative of a degree 0 \(\mathcal H\)-basic cohomology class of Hamilton de Rham superfield operation. Thus, the Batalin-Vilkoviski cohomology and the \(\mathcal H\)-basic Hamilton de Rham superfield cohomology are compatible.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B56 Cohomology of Lie (super)algebras
58A50 Supermanifolds and graded manifolds
81T45 Topological field theories in quantum mechanics
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