Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V (n) ‫؍‬ 0 for n < 0, V0 ‫؍‬ ‫,1ރ‬ and the contragredient module V is isomorphic to V as a V-module; (ii) every ‫-ގ‬gradable weak V-module is completely reducible; (iii) V is C 2-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation ‫ۋ‬ ؊1͞ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of ‫ۋ‬ ؊1͞ , and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category. I n 1987, by comparing fusion algebras with certain algebras obtained in the study of conformal field theories on genus-one Riemann surfaces, Verlinde (1) conjectured that the matrices formed by the fusion rules are diagonalized by the matrix given by the action of the modular transformation ‫ۋ‬ Ϫ1͞ on the space of characters of a rational conformal field theory. From this conjecture, Verlinde obtained the famous Verlinde formulas for the fusion rules and, more generally, for the dimensions of the spaces of conformal blocks on Riemann surfaces of arbitrary genera. In the particular case of the conformal field theories associated to affine Lie algebras (the Wess-Zumino-NovikovWitten models), the Verlinde formulas give a surprising formula for the dimensions of the spaces of sections of the ''generalized theta divisors,'' which has given rise to a great deal of excitement and new mathematics. See the works by Tsuchiya et al. In 1988, Moore and Seiberg (6) showed on a physical level of rigor that the Verlinde conjecture is a consequence of the axioms for rational conformal field theories. This result of Moore and Seiberg is based on certain polynomial equations that they derived from the axioms for rational conformal field theories (6, 7). Moore and Seiberg further demonstrated that these polynomial equations are actually conformal-field-theoretic analogues of the tensor category theory for group representations. Their work greatly advanced our understanding of the structure of conformal field theories. In particular, the notion of modular tensor category was later introduced to summarize the properties of the Moore-Seiberg polynomial equations and has played a central role in the development of conformal field theories and 3D topological field theories. See, for example, refs. 8 and 9 for the theory of modular tensor categories, their applications, and references to many important works done by mathematician...

Uniqueness of open/closed rational CFT with given algebra of open states

“…The 3-d TFT should then encode the behaviour of conformal blocks under sewing as well as the action of the mapping class group. (This is known to be true for genus zero and genus one if V obeys the conditions of theorem 2.1 in [46], but for higher genus it still remains open.) Note that in the second step, i.e.…”

“…For a rational CFT, the representation category Rep(V) is ribbon [45] and even modular [46]. Motivated by the path-integral formulation in the case of Chern-Simons theories [47,48] one identifies the spaces of states that the 3-d TFT associated to the modular tensor category Rep(V) assigns to surfaces with fibers of the bundles of conformal blocks.…”

Logarithmic torus amplitudes

“…Under the assumption that the modular differential equation is in fact of order 3p − 1 = 8, we can determine it uniquely by requiring it to be solved by this vacuum character. Explicitly we find − 5897 124416 E 4 (q)E 6 (q)cod (6) cod (4) cod (2) + 10889 55296 (E 4 (q)) 2 cod (8) cod (6) cod (4) cod (2) + 157 432 E 6 (q)cod (10) cod (8) cod (6) cod (4) cod (2) − 21 16 E 4 (q)cod (12) cod (10) cod (8) cod (6) cod (4) cod (2) + cod (16) cod (14) cod (12) cod (10) cod (8) cod (6) cod (4) cod (2) T (q) .…”

“…The structure of these theories is well understood: in particular, the characters of the irreducible representations transform into one another under modular transformations [1] (see also [2]), and the modular S-matrix determines the fusion rules via the Verlinde formula [3]. (A general proof for this has only recently been given in [4].) On the other hand, it is clear that rational conformal field theories are rather special, and it is therefore important to understand the structure of more general classes of conformal field theories.…”

Structure and Classification of Superconformal Nets

“…The SU (N ) k nets and the Virasoro nets Vir c with c < 1 are both modular. We expect all local conformal completely rational nets to be modular (see [31] for results of similar kind). Furthermore, if B is a modular local conformal net and C an irreducible extension of B, then C is also modular [37], that allows to check the modularity property in several cases.…”