1. Verlinde bundles 1.1. Flatness constraint. Let Mg be the moduli space of nonsingular curves of genus g ≥ 2. Let

be the moduli space of rank r degree d semistable bundles on nonsingular genus g curves. The space Ug(r, r(g − 1)) carries a canonical theta divisor Θr = {(C,E → C) with h 0(C,E) 6 = 0}. For levels k ≥ 1, the divisors Θkr are known to have no higher cohomology on the fibers of µ. The µ-pushforwards of the powers of the associated line bundle give the Verlinde vector bundles on Mg, Vr,k = µ⋆Θ k r. The rank of Vr,k is given by the well-known Verlinde formula [9]. For all ranks r and levels k, the Verlinde bundle Vr,k carries a projectively flat connec-tion defined by Hitchin [1, 3, 5]. As a basic consequence, the Verlinde bundle satisifes the topological constraint (1) chVr,k = rankVr,k · exp c1(Vr,k) rankVr,k ∈ H∗(Mg), where chi Vr,k is the