## 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.

### MSC:

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

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