Higher gauge theory and the $B$-field
Gauge theory as a theory on curves
There are many ways to describe gauge theories, and the connections they generate, but to describe where higher gauge theory stems from, perhaps the best way to do so is to describe them in terms of how they impact the motion of curves.
Let's consider a principal bundle $\pi : P \to M$. For concreteness, let's pick electromagnetism, with a principal bundle locally equivalent to $\mathrm{U}(1)$. As a manifold itself, $P$ admits a tangent bundle $TP$, which we can split into a vertical and horizontal part :
\begin{equation} TP = VP \oplus HP \end{equation}The vertical subbundle is uniquely determined by the property that it is the tangent of the fibers, in other words
Connection on a string
In a similar manner to the case of a point particle, we can wonder how such a connection would affect a string, or $1$-brane, or, in a more global sense, a worldsheet, rather than a point (or $0$-brane, or a curve). In this case, it may be useful to define an object similar to a connection, but this time for a surface. These objects are called $2$-connections.
Last updated : 2021-10-27 10:49:51