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The Verlinde formulas as fixed point formulas. (English) Zbl 1033.53074

Let \(M\) be a compact finite-dimensional Hamiltonian \(G\)-manifold with \(G\)-equivariant pre-quantized line bundle \(L\). There is a Spin\(^c\)-Dirac operator \(\partial^L\). The \(G\)-equivariant index \(\text{Ind}_G \partial^L(M) \in R(G)\) can be computed from (1) the Atiyah-Segal-Singer theorem in terms of fixed points data, or (2) from the fact that quantization commutes with reduction principle in terms of indices of symplectic quotients. Let \(M_{G, a} = \mu^{-1}(a)/G\) be the reduced space (symplectic quotient, provided \(a\) is a regular value of the moment map \(\mu: M \to {\mathfrak g}^*\) and \(G\) acts freely on \(\mu^{-1}(a)\)) and \(L_{G, a}\) be the induced pre-quantized line bundle. The principle \[ \text{Ind}_G \partial^L(M) = \text{Ind}\, \partial^{L_{G, a}}(M_{G, a}) \] has been proved by many authors with various methods. To mention just a few, E. Meinrenken and R. Sjamaar [Topology 38, 699–763 (1999; Zbl 0928.37013)] proved it by the symplectic cut, Y. Tian and W. Zhang [Invent. Math. 132, 229–259 (1998; Zbl 0944.53047)] by an analytic method as well as C. Teleman [Ann. Math. (2) 152, 1–43 (2000; Zbl 0980.53102)] for more general coefficients by algebraic methods. Note that there is a refined version of the principle \(H^*(M, L)^G = H^*(M_{G, a}, L_{G, a})\) by constructing a quasi-homomorphism of the Dolbeault complexes from an analytic approach by W. Zhang [Commun. Contemp. Math. 1, 281–293 (1999; Zbl 0970.53044)]; and \(H^*(M, L)^G = H^*(M//G, L)\) for the geometric quotient by Teleman (a kind of Morse (in)equality).
The paper under review is to extend this principle or calculation from finite dimensional theory to infinite dimensional theory. Let \(\hat{M}\) be an infinite-dimensional Hamiltonian \(LG\)-manifold with reduced space \(\hat{\mu}^{-1}(a)/LG_a = M_{LG, a}\) a compact and finite-dimensional symplectic orbifold. Assumptions in the paper imply that \(M_{LG, a}\) is finite-dimensional and compact. Therefore, the results known can be applied. The main principle in A. Pressley and G. B. Segal [Loop groups, Oxford University Press (1998; Zbl 0638.22009)] is that one can extend the finite-dimensional compact Lie group results to the loop group results. See also [W. Li, Contemp. Math. 322, 195–215 (2003; Zbl 1033.57014)] for extension of symplectic Floer cohomology to the infinite symplectic manifold case. The main result of the paper under review is the same analogue for the \(LG\)-fixed points contributions (Theorem 4.3). The identifications of fixed point contributions come from the symplectic cross-section theorem for Hamiltonian loop groups (analog of Guillemin-Sternberg cross section theorem in finite-dimensional and compact groups), and from a further symplectic action of the lattice.
The application for \(\widehat{M} = {\mathcal M}(\Sigma) = {\mathcal A}_{\text{flat}}(\Sigma)/{\mathcal G}_{\partial}(\Sigma)\) and \(LG = {\mathcal G}(\partial \Sigma)\) gives the reduced space \(\widehat{M}/LG = M(\Sigma)\) of dimension \(2(g+h-1)\dim G\), where \(\Sigma\) is a compact connected oriented genus \(g\) surface with \(h\) boundary components. Theorem 5.4 (Verlinde formula) follows easily from the previous result in section 4. The last section 6 provides proofs of the fixed points contributions in the equivariant index formula.
It would be nice to know if the refined quantization conjecture in Teleman and Zhang holds for the \(LG\)-Hamiltonian manifolds. The infinite dimensional version of the Morse (in)equality must have some significant roles in mathematical physics.

MSC:

53D30 Symplectic structures of moduli spaces
58J20 Index theory and related fixed-point theorems on manifolds
53D20 Momentum maps; symplectic reduction
53C27 Spin and Spin\({}^c\) geometry
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