Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 148-150). Also available in electronic version. Access restricted to campus users.

The Noether charge algebra of brane actions is typically modified by a topological anomalous term. The underlying cohomology of this anomalous term is investigated, and it is shown that the anomalous term possesses a gauge freedom. The result is that the anomalous term generates a parameterized family of topological charge algebras. When fermionic charges are taken to be nonvanishing, the known algebras underlying extended superspace formulations of the action appear in these families. This phenomena is investigated for minimal p-branes, Dp-branes and (p, q)-strings. The algebras resulting from the D-brane actions are shown to allow the construction of extended superspace actions without worldvolume gauge fields. It is shown that the actions are !-symmetric, and that the symmetry is generated by a right action. The global and local symmetry transformations of the Born-Infeld gauge field are thus shown to be described geometrically by left/right actions of the underlying extended supertranslation group. An equivalence class construction is proposed for the description of compact fermionic dimensions. In this construction, open strings in extended superspace translate to closed strings in compact superspace, and fermionic topological charges may be realized by closed strings. The differential underlying the descent construction for Noether charge algebras is shown to be naturally described as a dual of the de Rham differential. The ghost fields used in the construction are shown to be described geometrically as a vielbein with respect to this differential.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 91-92). Also available for download via the World Wide Web; free to University of Oregon users.

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Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1995. / Includes bibliographical references (p. 80-85).

Thesis (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.

In this thesis, we consider the question of describing the eigenvalues of a distinguished family of invariant differential operators associated to a Lie superalgebra g and a g-module W, called the "Capelli basis", via evaluation of certain classes of supersymmetric functions, called the interpolation super Jack polynomials. Finding the eigenvalues of the Capelli basis is referred to the Capelli Eigenvalue Problem. The eigenvalue formula depends on the chosen parametrization of the highest weight vectors in the decomposition of the superpolynomial algebra P(W), and consequently on the choice of a Borel subalgebra. In this thesis, we give a solution for each conjugacy class of Borel subalgebras, which we call a refined solution to the Capelli Eigenvalue Problem. Given the pair (g, W), we investigate the formulae for the eigenvalues of the Capelli operators associated to the completely reducible and multiplicity-free modules for two cases: diagonal and symmetric cases. In the former case, we show that we can express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of the b-highest weight of the irreducible component for any Borel subalgebra b. In the latter case, we show with a concrete counterexample that we cannot expect the results to be as strong as in the first case for all Borel subalgebras. We then express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of a piecewise affine map on the span of b-highest weights of the irreducible submodules of P(W), with respect to different decreasing Borel subalgebras b.

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N &gt = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N &gt = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N &gt = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

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