Reciprocity Laws in the Verlinde Formulae for the Classical Groups

Abstract

Abstract. The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of G-bundles on curves are conjectured but not proved.

Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

“…On the geometric side, since SO(m) is not simply connected the moduli spaces for orthogonal groups will be disconnected, and any reasonable approach to strange duality must take into account all components. It was this observation that led to the conjectural Verlinde type formula of Oxbury-Wilson [44], which is proved below.…”

Heisenberg invariant quartics and  C (2) for a curve of genus four

“…The Chern character of Q can be computed by Grothendieck-Riemann-Roch (see [1,chapter VIII]) and is g + 1 − 4θ, where θ denotes the fundamental class of Θ in the Jacobian. We therefore conclude from (21) and (23) that ch(Q 1 ) = (g + 1 − 4θ)e 2θ .…”

“…where t −p0 : J 1 → J denotes translation by p 0 . Applying π * to both sides of (22) we obtain (21). The Chern character of Q can be computed by Grothendieck-Riemann-Roch (see [1,chapter VIII]) and is g + 1 − 4θ, where θ denotes the fundamental class of Θ in the Jacobian.…”

“…Using the Verlinde formula (see [10] or [21]) one observes that the respective dimensions of these spaces are:…”

“…where t −p 0 : J 1 → J denotes translation by p 0 . Applying π * to both sides of (22) we obtain (21).…”