A new 2-form for connections on surfaces with boundary
Let \((P, G, \Sigma, \pi)\) be a principal bundle, where \(G\) is a compact connected Lie group (with Lie algebra \(\mathfrak g\)) and \(\Sigma\) a compact oriented surface with one boundary and genus \(g\geq 1\). We denote by \(\mathcal A\) the infinite-dimensional space of all connections on \(P\), by \(\mathcal G\) the group of all gauge transformations of \(P\) and by \({\mathcal G}_P\) the subgroup of \(\mathcal G\) consisting of gauge transformations fixing pointwise a certain fiber. Under the above assumptions, \(P\) is a trivial bundle, \(\mathcal A\) is reduced (by means of a fixed section) to the space of all \({\mathfrak g}\)-valued 1- forms on \(\Sigma\) and \(\mathcal G\) is reduced to the space of all smooth \(G\)- valued maps on \(\Sigma\). Hence, the action of \(\mathcal G\) on \(\mathcal A\) (by means of pull-backs) reduces to gauge transformations on the base \(\Sigma\). In this setting the authors introduce a new 2-form \(\Omega_{\text{new}}\) on \(\mathcal A\), which is gauge invariant and descends to a 2-form \(\overline\Omega_{\text{new}}\) on the moduli space \({\mathcal A}^{fl}/{\mathcal G}_p\), if \({\mathcal A}^{fl}\) denotes the space of flat connections. \(\Omega_{\text{new}}\) extends, in a sense, the natural symplectic form \(\Omega\) of \(\mathcal A\) which, however, does not descend to a 2-form on \({\mathcal A}^{fl}/{\mathcal G}_p\). An essential part of the paper is devoted to an explicit expression of \(\overline\Omega_{\text{new}}\) and its exterior derivative in terms of holonomy variations. \(\overline\Omega_{\text{new}}\) is also related to the symplectic structure of the ``extended moduli space'' due to \textit{L. C. Jeffrey} [Math. Ann. 298, No. 4, 667-692 (1994; Zbl 0794.53017)].
scientific article; zbMATH DE number 811272