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Full-Text Articles in Physical Sciences and Mathematics
Representing The Derivative Of Trace Of Holonomy, Jeffrey Peter Kroll
Representing The Derivative Of Trace Of Holonomy, Jeffrey Peter Kroll
Dissertations, Theses, and Capstone Projects
Trace of holonomy around a fixed loop defines a function on the space of unitary connections on a hermitian vector bundle over a Riemannian manifold. Using the derivative of trace of holonomy, the loop, and a flat unitary connection, a functional is defined on the vector space of twisted degree 1 cohomology classes with coefficients in skew-hermitian bundle endomorphisms. It is shown that this functional is obtained by pairing elements of cohomology with a degree 1 homology class built directly from the loop and equipped with a flat section obtained from the variation of holonomy around the loop. When the …
Some 2-Categorical Aspects In Physics, Arthur Parzygnat
Some 2-Categorical Aspects In Physics, Arthur Parzygnat
Dissertations, Theses, and Capstone Projects
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 …
On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller
On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller
Dissertations, Theses, and Capstone Projects
The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form …