Verlinde formulas for nonsimply connected groups (Q2417743)

If $G$ is a compact, simple, simply connected Lie group, $\Sigma_g$ is a compact oriented surface of genus $g$ without boundary, and $M_G(\Sigma_g)=\text{Hom}(\pi_1(\Sigma_g),G)/G$ the representation variety, then by choosing standard generators of the fundamental group, $M_G(\Sigma_g)=\Phi^{-1}(e)/G$, where $\Phi:G^{2g}\to G$, $(a_1,b_1,\dots,a_g,b_g)\mapsto\prod\limits_{i=1}^ga_ib_ia_i^{-1}b_i^{-1}$. By interpreting $M_G(\Sigma_g)$ as a moduli space of flat $G$-bundles over $\Sigma_g$ modulo gauge transformations, \textit{M. F. Atiyah} and \textit{R. Bott} in [Philos. Trans. R. Soc. Lond., A 308, 523--615 (1983; Zbl 0509.14014)] defined a symplectic structure on the smooth part of $M_G(\Sigma_g)$, with 2-form depending on the choice of an invariant inner product $B$ on the Lie algebra $\mathfrak{g}$. If $B$ is a $k$-th multiple of the basic inner product on $\mathfrak{g}$ for some integral $k\in\mathbb N$, then $M_G(\Sigma_g)$ acquires a prequantum line bundle $L$, and after suitable desingularization, one can define a quantization ${\mathcal{Q}}(M_G(\Sigma_g))\in\mathbb Z$ as the index of a spin-c Dirac operator with coefficients in $L$. This index is given by the symplectic Verlinde formula $(*)\,{\mathcal{Q}}(M_G(\Sigma_g))=\sum\limits_{\lambda\in P_k}S_{0,\lambda}^{2-2g}$, where $P_k$ is the finite set of level $k$ weights of $G$ and $S_{0,\lambda}$ are components of the $S$-matrix. In [Commun. Math. Phys. 206, No. 3, 691--736 (1999; Zbl 0962.17017)], \textit{J. Fuchs} and \textit{C. Schweigert} proposed a generalization of the Verlinde formula $(*)$ to the space $M_{G'}(\Sigma_g)=\text{Hom}(\pi_1(\Sigma_g),G')/G'$, where $G'=G/Z$ is a nonsimply connected simple Lie group with a finite subgroup $Z\subset Z(G)$ of the center $Z(G)$. In general, this space has several connected components $M_{G'}(\Sigma_g)_{(c)}$ indexed by the elements $c\in Z$. They suggested that the resulting formula would involve the natural action $\bullet_k$ of the center $Z(G)$ on $P_k$ and suitable phase factors $\varepsilon(c_1,\dots,c_{2g})\in\text{U}(1)$. In this paper, the authors prove the symplectic version of the Fuchs-Schweigert conjectures for arbitrary compact, simple $G'$, for surfaces with at most one boundary component. For the entire collection see [Zbl 1412.22001].