Commun. Math. Phys. 257, 1–28 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1341-6 Communications in Mathematical Physics A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories Christian Blanchet L.M.A.M., Universit´e de Bretagne-Sud, BP 573, 56017 Vannes, France. E-mail:
[email protected] Received: 20 March 2003 / Accepted: 30 December 2004 Published online: 15 April 2005 – © Springer-Verlag 2005 Abstract: A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU (N ). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context. 0. Introduction Given a simple, simply connected complex Lie group G, the Verlinde formula [35] is a combinatorial function VG : (K, g) → VG (K, g) associated with G (here the integers K and g are respectively the level and the genus). In conformal field theory this formula gives the dimension of the so called conformal blocks. Its combinatorics was intensively studied since this formula has a deep interpretation as the rank of a space of generalized theta functions (sections of some bundle over the moduli space of G-bundles over a Riemann surface) [6, 5, 15, 28]. See [8, 9], for a development using methods of symplectic geometry. We will consider here a purely topological approach to Verlinde formulas related with SU (N ). The genus g Verlinde formula associated with a modular category [30] is the dimension of the TQFT-module of a genus g surface; the general formula is given in [30, IV,12.1.2]. Various constructions of modular categories are known, either from quantum groups [2, 4, 29] or from skein theory [34, 11, 7]. The geometric Verlinde formula for the group SU (N ) at level K is recovered from the so called SU (N, K) modular category. This modular category can be obtained either from the quantum group Uq sl(N ) when 2 C. Blanchet q = s 2 is a primitive (N + K)th root of unity or from Homfly skein theory. Its simple objects correspond to the weights in the fundamental alcove. One may also consider a modular category with less simple objects. This was done for gcd(N, K) = 1 by restricting to representations whose heighest weight is in the root lattice, and was called the projective or P SU (N) theory [17, 36, 22, 18, 19]. Using an appropriate choice of the framing parameter in Homfly skein theory, we have obtained in [11] a variant which is defined for all N, K. We are not aware of a quantum group approach to these reduced modular categories for gcd(N, K) > 1. Nevertheless we find it convenient to call them P U (N, K) modular categories. In our construction the simple object corresponding to the deformation of the determinant of the vector representation of sl(N ) may be non-trivial; we think that a version of the quantum group Uq (gl(N )) could be used here. As is well known, the Verlinde formula for the SU (N, K) modular category coincides with the formula in conformal field theory for the group SU (N ); dN,K (g) = VSU (N) (K, g). We show that for the P U (N, K) modular category the Verlinde formula is dN,K (g) , d˜N,K (g) = N g where N = N gcd(N,K) . These integral numbers satisfy the level-ra