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Asian Journal of Mathematics
Volume 26 (2022)
Number 5
Dirac structures on the space of connections
Pages: 617 – 634
DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n5.a2
Authors
Abstract
We shall investigate the Dirac structures on the space of connections over threemanifolds and over four-manifolds. We show that the space of irreducible connections on the trivial $\mathrm{SU}(n)$-bundle over a three-manifold is canonically endowed with a Dirac structure twisted by a $3$-form. We also give a family of Dirac structures twisted by the $3$-form on the space of irreducible connections on the trivial $\mathrm{SU}(n)$-bundle over a four-manifold. These twisted Dirac structures induce (non-twisted) Dirac structures on the space of flat connections. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of the corresponding Dirac structure over the four-manifold. We also discuss the action of the group of gauge transformations over these Dirac structures.
Keywords
Dirac structures, symplectic structures, space of connections
2010 Mathematics Subject Classification
53D05, 53D30, 53Z05, 58B20, 81S10, 81T30
Received 17 June 2022
Accepted 6 July 2022
Published 13 April 2023