Saikat Chatterjee

Parallel Transport over Path Spaces

We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework.

NEGATIVE FORMS AND PATH SPACE FORMS

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces.

Generalized forms and vector fields

The generalized vector is defined on an n-dimensional manifold. The interior product and Lie derivative acting on generalized p-forms, −1 ≤ p ≤ n are introduced. The generalized commutator of two generalized vectors is defined. Adding a correction term to Cartans formula, the generalized Lie derivatives action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.

TWISTED-PRODUCT CATEGORICAL BUNDLES

Abstract Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categorical sense, developing the relationship between functorial sections of such bundles and trivializations. We construct functorial cocycles with values in categorical groups using a suitable family of locally defined functors on the object space of the base category. Categorical product bundles being too rigid to give a widely applicable model for local triviality, we introduce the notion of a twisted-product categorical bundle. We relate such bundles to decorated categorical bundles that contain more information, specifically parallel transport data.

GEOMETRIC STRUCTURES ON PATH SPACES

Let M be a C∞ manifold, and let

Double Category Related to Path Space Parallel Transport and Representations of Lie 2 Groups

{\mathcal P}M

Path Space Forms and Surface Holonomy

be the space of all smooth maps from [0, 1] to M. We investigate geometric structures on

Geodesics on path spaces and double category

{\mathcal P}M

Geometric prequantization on the path space of a prequantized manifold

constructed from the geometric structures on M. In particular, we show that a generalized (almost) complex structure on M produce a generalized (almost) complex structure on

A Morphism Double Category and Monoidal Structure

{\mathcal P}M