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Fusion of perturbed defects in conformal field theory

The infinite-dimensional symmetry algebra of a conformal field theory (CFT), the Virasoro algebra, is generated by the holomorphic and anti-holomorphic part of the stress tensor. Besides such ‘chiral symmetries’ the CFT also has an integrable symmetry, that is, infinite families of commuting conserved charges. In this thesis a step towards combining these two symmetries into a single formalism is taken, by identifying integrable structures of a CFT through studying the representation category of the underlying chiral algebra. Then by introducing defects in the system, conserved charges can be constructed by perturbing certain conformal defects. Starting from an abelian rigid braided monoidal category C one defines an abelian rigid monoidal category CF which captures some aspects of perturbed conformal defects in two-dimensional CFT. Namely, for V a rational vertex operator algebra one considers the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in CF corresponds to a conformal defect condition together with a direction of perturbation. To each object in CF one assigns a perturbed defect operator on the space of states of the CFT and then shows that the assignment factors through the Grothendieck ring of CF . This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.