Given a simple, simply connected complex Lie group \(G\), the Verlinde formula is a combinatorial function \(V^ G: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\) associated with \(G\). The expressions \(V^ G (k,g)\) were first introduced by E. Verlinde in the context of conformal quantum field theory [cf. E. Verlinde, Nucl. Physics B, Field Theory and Statistical Systems 300, No. 3, 360-376 (1988)]. Their significance in algebraic geometry stems from the (originally conjectural) fact that they are related to the Hilbert functions of moduli spaces of semi-stable vector bundles over compact Riemann surfaces of genus \(g\). The corresponding relation between those Hilbert functions of moduli spaces and the “Verlinde numbers” \(V^ G (k,g)\) used to be called the “Verlinde conjecture” for the respective moduli spaces, and its verification, mainly in the cases of \(G= \text{SU} (n)\) and \(G= \text{SL} (n)\), has been a central topic of research in the last five years.
More precisely, the Verlinde conjecture, in most general setting, can be roughly stated as follows. Let \(C\) be a compact Riemann surface of genus \(g\), and let \({\mathcal M}_ C^ G\) be the (factually existing) moduli space of principal G-bundles over \(C\). Then there is an ample line bundle \({\mathcal L}\) over \({\mathcal M}^ G_ C\), a so-called generalized theta bundle, such that \(\dim_ \mathbb{C} H^ 0 ({\mathcal M}_ C^ G, {\mathcal L}^{\otimes k})= V^ G (k_ h, g)\) for any \(k\in \mathbb{Z}\), where \(h\) denotes the dual Coxeter number of the group \(G\).
In this form, and for \(G= \text{SL} (n)\), the Verlinde conjecture has recently been verified by G. Faltings [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]; A. Beauville and Y. Laszlo [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)]; S. Kumar, M. S. Narasimhan and A. Ramanathan [Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)]; A. Bertram and the author [Topology 32, No. 3, 599-609 (1993; Zbl 0798.14004)], and others in less general cases. – In the present paper under review, the author discusses the origin and properties of the Verlinde formulas \(V^ G (k,g)\) and, in addition, their connection with the famous Witten conjectures in the intersection theory of moduli spaces of algebraic curves. After a brief overview of the structure of topological field theories, fusion algebras and Verlinde’s formal calculus, the explicit behavior of \(V^ G (k,g)\) as a function of \(k\) is studied, again in the special case of \(\text{SL} (n)\). The main result is a residue formula for \(V^{\text{SL} (n)} (k,g)\) yielding the deep fact that these numbers are integer-valued polynomials in \(k\). The author then shows how this residue formula for \(V^{\text{SL} (n)} (k,g)\) can be related, via the Grothendieck- Hirzebruch-Riemann-Roch theorem, to Witten’s conjectures on the intersection numbers of moduli spaces of curves [cf. E. Witten, J. Geom. Phys. 9, No. 4, 303-368 (1992; Zbl 0768.53042)]. Assuming the validity of Witten’s formulas, a quick proof of the Verlinde conjecture (as stated above) is given.
This very instructive approach puts the Verlinde formulas into a wider context, and provides some more evidence of Witten’s conjectures from this now well-established complex of results.
For the entire collection see [Zbl 0810.00011].