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

Yuji Hirota (Laboratory of Basic Education (Mathematics), Azabu University, Chuo-ku, Sagamihara City, Kanagawa, Japan)

Tosiaki Kori (Department of Mathematics, Graduate School of Science and Engineering, Waseda University, Tokyo, Japan)

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