Date of Degree

9-2016

Document Type

Dissertation

Degree Name

Ph.D.

Program

Physics

Advisor(s)

V. Parameswaran Nair

Scott O. Wilson

Committee Members

Alexios Polychronakos

Thomas Tradler

S. G. Rajeev

Subject Categories

Algebra | Elementary Particles and Fields and String Theory | Geometry and Topology | Quantum Physics

Keywords

principal bundle, holonomy, gauge theory, entropy, states on C*-algebras, GNS construction

Abstract

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description of the Gelfand-Naimark-Segal construction as a canonical procedure from states on C*-algebras to representations of C*-algebras equipped with pure state.

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