Instantons, Poisson structures and generalized Kähler geometry. (English) Zbl 1110.53056

The notion of generalized Kähler manifold is defined by a pair \(J_1\), \(J_2\) of commuting generalized complex structures, and according to M. Gualtieri [Generalized complex geometry. http://arxiv.org./list/math.DG/0401221,2004] the manifold has an equivalent interpretation, e.g., that it is equipped with two complex structures \(I_+\) and \(I_-\), a metric \(g\), Hermitian with respect to both and connections \(\nabla^+\) and \(\nabla^-\) compatible with these structures but with skew torsion \(db\) and \(-db\) for a 2-form \(b\).
This paper begins with the stuy of generalized Kähler manifold with \(J_1\) and \(J_2\) such that each one is the \(B\)-field transform of a symplectic structure determined by a closed form \(\exp(B+iw)\), and by the use of the corresponding \(I_+\) and \(I_-\), it is proven that \(g([I_+,I_-]X,Y)\) defines a holomorphic Poisson structure. The following sections show how to introduce a bi-Hermitian structure on the moduli space of gauge-equivalent classes of solutions to the anti-self-dual Yang-Mills equations. Finally, the paper gives a quotient construction which demonstrates the problem of making a generalized Kähler structure descend to the quotient, although a quotient construction for the instanton moduli space has not yet been found in this study.


53C55 Global differential geometry of Hermitian and Kählerian manifolds
53D17 Poisson manifolds; Poisson groupoids and algebroids
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