Communications in Contemporary Mathematics

We continue our study in [16] of the Gamma limit of the Abelian Chern–Simons–Higgs energy

We study the Dirichlet boundary value problem on a bounded domain Ω ⊂ ℝ^N. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ* in the class of all solutions with finite Morse index, λ* being the extremal value associated to the nonlinear eigenvalue problem.

We study the principal subspaces, introduced by Feigin and Stoyanovsky, of the level 1 standard modules for with l ≥ 2. In this paper, we construct exact sequences which give us a complete set of recursions that characterize the graded dimensions of the principal subspaces of these representations. This problem can be viewed as a continuation of a new program to obtain Rogers–Ramanujan-type recursions, which was initiated by Capparelli, Lepowsky and Milas. In order to prove the exactness of the sequences, we use intertwining vertex operators and we supply a proof of the completeness of a list of relations for the principal subspaces. By solving these recursions we recover the graded dimensions of the principal subspaces, previously obtained by Georgiev using a different method.

We consider the problem

We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) V_(n) = 0 for n < 0, V₍₀₎ = ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C₂-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Then the matrices formed by the fusion rules among the irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ ↦ -1/τ on the space of characters of irreducible V-modules. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation τ ↦ -1/τ is symmetric.