Prequantization of the moduli space of flat \(\text{PU}(p)\)-bundles with prescribed boundary holonomies (Q2339228)

Let \(\Sigma\) be a compact oriented surface with \(s\) boundary components and \(G\) a compact connected Lie group. The moduli space \(\mathbb M\) of flat \(G\)-bundles over \(\Sigma\), with prescribed boundary holonomies on the conjugacy classes \(\mathcal{C_1},\dots,\mathcal{C}_s\) is well known [\textit{M. F. Atiyah} and \textit{R. Bott}, Philos. Trans. R. Soc. Lond., Ser. A 308, 523--615 (1983; Zbl 0509.14014); \textit{W. M. Goldman}, Adv. Math. 54, 200--225 (1984; Zbl 0574.32032)] to be a symplectic manifold, where the symplectic form is defined by a choice of invariant inner product on the Lie algebra of \(G\). The article under review is concerned with the geometric prequantization of \(\mathbb M\). When \(G\) is simply connected, then \(\mathbb M\) is connected and the prequantization is ensured once a certain integrality condition is satisfied [\textit{A. Alekseev} et al., J. Symplectic Geom. 1, No. 1, 1--46 (2001; Zbl 1033.53074); \textit{J.-M. Bismut} and \textit{F. Labourie}, in: Surveys in differential geometry. Boston, MA: International Press. 97--311 (1999; Zbl 0997.53066); \textit{E. Meinrenken}, Enseign. Math. (2) 49, No. 3--4, 307--333 (2003; Zbl 1061.53034)]. When \(G\) is not simply connected though, difficulties arise: First, \(\mathbb{M}\) may have multiple connected components; second, the integrality condition is no longer sufficient. Only partial results exist for non-simply connected Lie groups \(G\), in particular only the case \(G=\mathrm{SO}(3)=\mathrm{PU}(2)\) has been resolved, assuming the surface \(\Sigma\) has prescribed holonomies on the boundary [\textit{D. Krepski} and \textit{E. Meinrenken}, Q. J. Math. 64, No. 1, 235--252 (2013; Zbl 1272.53077)]. The author considers a non-simply connected Lie group \(G\) and provides a full description of the connected components of the moduli space \(\mathbb M\) associated with the quotient of \(G\) by its center \(Z\). Then the author considers the group \(\mathrm{PU}(p)\) for \(p>2\) prime and calculates the associated prequantization obstruction. This obstruction is a certain integrality condition of cohomological nature. The author mentions in the introduction that the approach described in this article does not apply to other cases of non-simply connected structure groups.