Symplectic geometry and the Verlinde formulas (Q2739538)

Let \(\Sigma\) be a Riemann surface with marked points and let \(G\) be a connected simply connected compact Lie group, \(G_\mathbb{C}\) denoting the complexification of \(G\). Let \(M\) be the moduli space of semistable \(G_\mathbb{C}\)-bundles on \(\Sigma\). The Verlinde formula computes the dimension of the space of holomorphic sections of the canonical line bundle over \(M\). The authors give a proof of the Verlinde formula for every simple Lie group \(G\) by methods of symplectic geometry.NEWLINENEWLINENEWLINEThe proof is very involved in the sense that it uses a number of results in several branches of mathematics. The first step is the description of the smoothness of the orbifold moduli space \(M/G\). The second step is to give the proof of Witten's formula for the symplectic volume of the moduli spaces. The third step is the construction of the orbifold line bundle \(\lambda^p\) over \(M/G\). The authors use also a formula of \textit{L. C. Jeffrey} and \textit{F. Kirwan} [Topology 36, 647-693 (1977; Zbl 0876.55007)] and \textit{K. Liu} [Math. Res. Lett. 3, 743-762 (1996; Zbl 0871.58014)] by using the formalism of the moment maps. Finally the authors give a residue formula for the index of the Dirac operator on \(M/G\), and give, using the Riemann-Roch-Kawasaki theorem, the proof of the Verlinde formula by comparing the index of Dirac operator and the expression \(V_{g,q} (\theta_1, \dots,\theta_s)\) of Verlinde for \(\theta_1, \dots, \theta_s\in \Lambda_+ \cap\overline R\) and \(g\in\mathbb{N}\), \(q\in \mathbb{Z}\), where \(R\) is the root system of \(G\), \(\overline R\) the lattice generated by \(R\) and \(\Lambda_+\) denotes the set of positive weights of \(G\).NEWLINENEWLINENEWLINEReviewer's note: The result in this paper was announced in the authors' previous paper [C.R. Acad. Sci. Paris 325, 1009-1014 (1997; Zbl 0889.14013)] in which the authors note that C. Sorger announced a proof of the Verlinde formula. However, the reference of Sorger is deleted from the references of this paper.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00007].