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x, 81 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St). / Committee in charge: Victor Ostrik, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Jonathan Brundan, Member, Mathematics; Alexander Kleshchev, Member, Mathematics; Michael Kellman, Outside Member, Chemistry

<p>Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part.</p><p>The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis.</p><p>The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.</p>

Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part. The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis. The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.

The main goal of this thesis is to construct a tensor product on the 2-category BFC-A of braided fusion categories containing a symmetric fusion category A. We achieve this by introducing the new notion of Z(A)-crossed braided categories. These are categories enriched over the Drinfeld centre Z(A) of the symmetric fusion category. We show that Z(A) admits an additional symmetric tensor structure, which makes it into a 2-fold monoidal category. ByTannaka duality, A= Rep(G) (or Rep(G; w)) for a finite group G (or finite super-group (G,w)). Under this identication Z(A) = VectG[G], the category of G-equivariant vector bundles over G, and we show that the symmetric tensor product corresponds to (a super version of) to the brewise tensor product. We use the additional symmetric tensor product on Z(A) to define the composition in Z(A)-crossed braided categories, whereas the usual tensor product is used for the monoidal structure. We further require this monoidal structure to be braided for the switch map that uses the braiding in Z(A). We show that the 2-category Z(A)-XBF is equivalent to both BFC=A and the 2-category of (super)-G-crossed braided categories. Using the former equivalence, the reduced tensor product on BFC-A is dened in terms of the enriched Cartesian product of Z(A)-enriched categories on Z(A)-XBF. The reduced tensor product obtained in this way has as unit Z(A). It induces a pairing between minimal modular extensions of categories having A as their Mueger centre.

Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study a subalgebra J_C of J corresponding to the subregular cell C of W . We prove a factorization theorem that allows us to compute products in J_C without inputs from Kazhdan-Lusztig theory, then discuss two applications of this result. First, we describe J_C in terms of the Coxeter diagram of (W, S) in the case (W, S) is simply- laced, and deduce more connections between the diagram and J_C in some other cases. Second, we prove that for certain specific Coxeter systems, some subalgebras of J_C are free fusion rings, thereby connecting the algebras to compact quantum groups arising in operator algebra theory.

In this thesis, we consider q-deformations of multiplicative Hypertoric varieties, where q&isin;&Kopf;^x for &Kopf; an algebraically closed field of characteristic 0. We construct an algebra D_q of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use D_q to construct an Azumaya algebra on an l-twist of the multiplicative Hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumaya algebras.