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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Smooth Loops And Loop Bundles, Sergey Grigorian
Smooth Loops And Loop Bundles, Sergey Grigorian
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Maurer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and …
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 …
Ambient Prime Geodesic Theorems On Hyperbolic 3-Manifolds, Lindsey A. Dever, Djordje Milićević
Ambient Prime Geodesic Theorems On Hyperbolic 3-Manifolds, Lindsey A. Dever, Djordje Milićević
Mathematics Faculty Research and Scholarship
We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
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 …
Conformally Parallel Spin(7) Structures On Solvmanifolds, Selman Uğuz
Conformally Parallel Spin(7) Structures On Solvmanifolds, Selman Uğuz
Turkish Journal of Mathematics
In this paper we review the Spin(7) geometry in relation to solvmanifolds. Starting from a 7-dimensional nilpotent Lie group N endowed with an invariant G_2 structure, we present an example of a homogeneous conformally parallel Spin(7) metric on an associated solvmanifold. It is thought that this paper could lead to very interesting and exciting areas of research and new results in the direction of (locally conformally) parallel Spin(7) structures.
There Is No Triangulation Of The Torus With Vertex Degrees 5, 6, . . ., 6, 7 And Related Results: Geometric Proofs For Combinatorial Theorems, Ivan Izmestiev, Robert B. Kusner, Günter Rote, Boris Springborn, John M. Sullivan
There Is No Triangulation Of The Torus With Vertex Degrees 5, 6, . . ., 6, 7 And Related Results: Geometric Proofs For Combinatorial Theorems, Ivan Izmestiev, Robert B. Kusner, Günter Rote, Boris Springborn, John M. Sullivan
Robert Kusner
There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies …