A Global Geometric Approach to Parallel Transport of Strings in Gauge Theory

Abstract

This paper is motivated by recent developments of higher gauge theory. Different from its style of using higher category theory, we try to describe the concept of higher parallel transport within setting of classical principal bundle theory. From this perspective, we obtain a global geometric proof on a generalized 3-dimensional nonabelian Stokes' theorem related to parallel transport on surfaces. It can be naturally extended to four dimension when the underlying crossed module is replaced by crossed 2-module. This 4-dimensional Stokes' theorem yields a global formula for volume holonomy, and also guarantees its gauge invariance. In the process, we also find composition formulas for parallel transport on volumes which have been seen in the case of surfaces.

Topological invariant of 4-manifolds based on a 3-group

“…It is straightforward to show that the corresponding equations of motion reduce to vacuum Einstein field equations. Thus the action (17) is classically equivalent to general relativity. The construction of the action ( 17) is analogous to the Plebanski model, where general relativity is constructed by adding a simplicity constraint to the BF theory based on the Lorentz group.…”

“…Here, the first row is the topological sector and the familiar simplicity constraint for gravity from (17), while the second row contains the appropriate simplicity constraints for Yang Mills field, featuring the Lagrange multipliers λ I and ζ abI . The action (22) provides two dynamical equations -the equation for A I ,…”

“…The first row is the topological sector (27) and the simplicity constraint for gravity from the action (17), the second row contains two new simplicity constraints featuring the Lagrange multiplier 1-forms λ and Λ ab and the 0-form H abc , and the third row features the mass term for the scalar field. The action (28) has two dynamical equations of motion -the equation for the scalar field φ is the covariant Klein-Gordon equation,…”

“…In order to overcome this problem, a new approach has been developed within the framework of higher gauge theory (for a review of higher gauge theory, see [16,17], and for its applications in physics see [18][19][20][21][22][23][24][25][26][27][28][29]). Within higher gauge theory formalism, one generalizes the BF action, which is based on some Lie group, to a 2BF action based on the 2-group structure.…”

Topological invariant of 4-manifolds based on a 3-group

“…In order to overcome this problem, a new approach has been developed within the framework of higher gauge theory (for a review of higher gauge theory, see [16,17], and for its applications in physics see [18][19][20][21][22][23][24][25][26][27][28][29]). Within higher gauge theory formalism, one generalizes the BF action, based on some Lie group, to an 2BF action based on the 2-group structure.…”