Aspects of the Affine Superalgebra sl(2|l) at Fractional Level Ph.D. Thesis by Gavin Balfour Johnstone, April 2001 In this thesis we study the affine superalgebra sl(2|l; C) at fractional levels of the form k = l/u-l,uєN\{l}. It is for these levels that admissible representations exist, which transform into each other under modular transformations. In the second chapter we review background material on conformal field theory, particularly the Wess-Zumino-Witten model and the connection with modular transformations. The superalgebra sl(2|l;C) is introduced, as is its affine version. The next chapter studies the modular transformation properties of sl(2|l;C) characters. We derive formulae for these transformations for all levels of the form K = 1/u-1,uєN\{1}. We also investigate some modular invariant combinations of characters and find two series of modular invariants, analogous to the A- and D-series of the classification of sl{2) modular invariants. In chapter 4 we turn to the study of fusion rules. We concentrate on the case k = -1/2. By considering the decoupling of singular vectors, we are able to find consistent fusion rules for this particular level. These fusion rules correspond to a modular invariant found in chapter 3. This study suggests that one may consistently define a conformal field theory based on sl(2|l;C) at fractional level.

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Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [85]-89).

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thèse--Genève. / Bibliographical footnotes.

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: S. G. Guralnik. Includes bibliographical references (leaves 117-118). Also available online.

This thesis reports a study of the quantization of generally covariant and nonlinear field theories. It begins by reviewing some existing general theories in Chapter 2 and Chapter 3, Chapter 2 deals with general classical theories while Chapter 3 examines various quantization schemes. The model field derived from the Lagrangian density ℓ = 1/4 ℇ[super]lkμλ (A [sub] k,t – A [sub] k,t)(A, {sub{ A[sub]μ,λ – A[sub]λ,μ). is proposed in Chapter 4 especially for the study of general covariance. It is demonstrated that for this field general covariance alone does not appear to bring in anything physically new, a discussion is given on the differences between general covariance and Lorentz covariance. In subsequent chapters a generally covariant and nonlinear model field, a 4-surface of stationary 4-volume embedded in a 5-dimensional Pseudo-Euclidean space, is investigated. Firstly a manifestly covariant quantization programme is carried out. The model field is then examined in a special coordinate frame for the study of its nonlinearity. Various treatments of the intrinsic nonlinearity are examined starting with conventional perturbation theory in Chapter 6. The usual divergence problem in quantum field theory appears, in particular in the self-energy calculation of a one-particle state. A new variational method is proposed in Chapter 8 which is able to lead to finite results for one-particle states. The thesis is concluded with a chapter discussing some general problems involved and a chapter containing suggestions for further work.

Field theory is used to describe the material content of the universe throughout its entire history, and an oscillating cosmological model without a singularity is presented. In our theory, the “cosmological fluid” is described by a classical scalar field that undergoes a series of phase transitions over the lifetime of the universe. Each transition corresponds to a discontinuous change in the equation of state of the field. In general, for an FRW universe and a given equation of state, we show that the field potential V(Φ) may be derived from the solution of Piccati’s equation. The resulting expression for V(Φ) includes parameters whose values are determined from the boundary conditions. In our theory, we employ the standard cosmological model and the fundamental Planck quantities to provide these boundary conditions. We thereby determine the scalar field Lagrangian for the entire history of the universe. The resulting cosmological model is free of any singularities, and includes an early inflationary epoch. Inflation arises in our theory as a consequence of the initial conditions. The theory describes a universe that is very cold at its minimum radius, although it heats rapidly during the initial inflationary era. This increase in the temperature of the scalar field during inflation is a direct consequence of applying classical thermodynamics under the assumed conditions for the early universe, and does not depend on the fine-tuning of free parameters. Inflation continues until a maximum possible physical temperature (the Planck temperature) is attained, at which point a phase transition occurs and the standard model era begins. By relating the temperature of the scalar field in our theory to the radiation temperature in the standard model universe, it is possible to establish a thermodynamic constraint on a more complete theory of matter for the early universe. Although, in principle, inflation occurs for any equation of state where p < -(1/3)p, we find that the initial equation of state must be p ≈ -p if the later epochs of the universe are to resemble the standard model. In particular, we find that Ho = 33 - 44 km sec-1 Mpc-1 is the value of the Hubble parameter a t the current epoch that is least sensitive to the initial equation of state. / Graduate