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Whittaker modules and fusion rules for the Weyl vertex algebra, affine vertex algebras and their orbifolds

The topic of this thesis are two problems in the vertex operator algebra theory: determination of fusion rules and the orbifold problem. For the fusion rules problem we study the example of Weyl vertex algebra, also known as the β γ ghost system. This is a nonrational vertex algebra, hence we give a first proof of a Verlinde formula for non-rational VOAs and confirm the Verlinde type conjecture given by D. Ridout and S. Wood in [70]. For the orbifold problem, we extend a theorem given in the Dong-Mason quantum Galois theory paper [41], from the category of ordinary modules to the whole category of weak modules. The proof given by Dong and Mason necessarily involves Zhu’s theory, and therefore can not be extended to the category of weak modules. In particular, we study the example of Whittaker modules for the Weyl vertex algebra and Heisenberg VOA. In the first part of this thesis, we calculate fusion rules in the category of weight modules for the Weyl vertex algebra. Our proof is entirely vertex-algebraic and it uses the theory of intertwining operators for vertex algebras and the fusion rules for the affine vertex superalgebra L1(gl(1|1)). Moreover, we explicitly construct the intertwining operators involved. We also prove a general irreducibility result which relates irreducible weight modules for the Weyl vertex algebra M to irreducible weight modules for L1(gl(1|1)). In the second part of this thesis we prove a theorem on irreducible weak V-module W and an automorphism g of finite order. Here either W ◦ g^i is non- isomorphic to W for all i, in which case W is an irreducible V^g–module, or W ∼= W ◦ g in which case W is a direct sum of p irreducible V hgi–modules. The key idea of our proof was to consider a “big” module for the vertex algebra, constructed as a direct sum of modules W^i ◦ g. Moreover, we present a counterexample for the expansion of our theorem to the case of infinitedimensional group G for the irreducible Weyl algebra modules of Whittaker type.

vertex algebra, vertex operator algebra, ordinary modules, weak modules, Whittaker modules, Weyl vertex algebra, Heisenberg vertex operator algebra, quantum Galois theory, orbifold problem, fusion rules, intertwining operators, Verlinde formula, fusion algebra, simple current, vertex algebra automorphism, lattice vertex superalgebra, Verma module, Clifford vertex algebra

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