A proof for the Verlinde formula. (English) Zbl 0809.14009

Faltings, Gerd

J. Algebr. Geom. 3, No. 2, 347-374 (1994).

Let \(k\) be an algebraically closed field of characteristic zero and write \(K = k((t))\) for the field of fractions of \(V = k[[t]]\). Also, let \(C\) be a semistable curve over \(V\) with smooth projective generic fibre \(C_ K\) of genus \(\geq 2\), and with special fibre \(C_ s\) the union of smooth projective lines \(\mathbb{P}^ 1\) meeting transversely in simple double points. Let \(G\) denote a semisimple simply connected algebraic group. Finally, write \({\mathcal M} (G) = {\mathcal M}_ K (G)\) for the moduli-stack of \(G\)-torsors on \(C_ K\). In a previous paper [J. Algebr. Geom. 2, No. 3, 507-568 (1993; Zbl 0790.14019)] the author showed that the inverse of the cohomology determinant of any \(G\)-module defines a line-bundle \({\mathcal L} = {\mathcal L}_ c\) (for a certain level \(c \in \mathbb{Z})\) on \({\mathcal M}_ K (G)\). The Verlinde formula gives an expression for \(\dim \Gamma ({\mathcal M}_ K(G), {\mathcal L})\). There are several approaches to it, one being the method of conformal field theory where one calculates the dimension of the ‘space of vacua’. Here it is shown by algebro-geometric methods that this dimension is equal to that of \(\Gamma ({\mathcal M}_ K (G), {\mathcal L})\). An important ingredient for the construction of \(G\)-torsors over \(C_ K\) is the loop group \(LG\) translated into the following one: Writing \(T \subset G\) for the maximal torus, the above mentioned functions for bundles correspond to functions on \(T\). Do they extend to \(\overline T\), where \(T \times \mathbb{G}_ m \subset \overline T\) is a suitable form embedding such that \(T\subset \overline{T}\) is defined over \(V\)? The answer is shown to be affirmative (in a somewhat more general setting). To obtain an explicit formula for \(\dim \Gamma ({\mathcal M} (G), {\mathcal L}_ c)\) one invokes the structure of the so-called ‘fusion algebra’ \({\mathcal F}_ c\). This is a \(\mathbb{C}\)-vector space with basis the irreducible \(G\)-modules which give rise to nontrivial integrable \(LG\)- representations and equipped with a suitable (associative) product. One is led to state a conjecture on the characters of \({\mathcal F}_ c\), and assuming the validity of this conjecture, a formula for \(\dim \Gamma ({\mathcal M} (G), {\mathcal L}_ c)\) ensures: \[ \dim \Gamma ({\mathcal M} (G), {\mathcal L}_ c) = \sum_{\gamma \in \Gamma^{\text{reg}}/W} \left | {\bigl | D (\gamma) \bigr |^ 2 \over \# \Gamma_ c} \right |^{g-1}, \] where \(\Gamma_ c \subset T\) is a well-defined subgroup (depending on \(c)\), \(W\) is the Weyl-group acting on \(\Gamma_ c\), \(\Gamma^{\text{reg}}\) is the set of elements where the action is free, and \(D(\gamma)\) is the Weyl-denominator. In an appendix (‘Some Lie- theory’) the conjecture is proved for the classical groups and for \(G_ 2\).

Reviewer: W.W.J.Hulsbergen (Haarlem)

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14H10	Families, moduli of curves (algebraic)
14H60	Vector bundles on curves and their moduli

Keywords:

space of vacua; torsors; Verlinde formula; conformal field theory; loop group

Citations:

Zbl 0790.14019