Higher gauge theory and M-theory
Abstract
In this thesis, the emerging field of higher gauge theory will be discussed, particularly
in relation to problems arising in M-theory, such as selfdual strings and the
so-called (2,0) theory. This thesis will begin with a Nahm-like construction for selfdual
strings using loop space, the space of loops on spacetime. This construction maps
solutions of the Basu-Harvey equation, the BPS equation arising in the description
of multiple M2-branes, to solutions of a selfdual string equation on loop space. Furthermore,
all ingredients of the construction reduce to those of the ordinary Nahm
construction when compactified on a circle with all loops restricted to those wrapping
the circle. The rest of this thesis, however, will not involve loop space. We will
see a Nahm-like construction for the case of infinitely many selfdual strings, suspended
between two M5-branes. This is possible since the limit taken renders the
fields describing the M5-branes abelian. This avoids the problem which the rest of
this thesis focuses on: What fields describe multiple M5-branes? The answer is likely
to involve higher gauge theory, a categorification of gauge theory which describes the
parallel transport of extended objects. Any theories which involves 3-algebras, including
current M2-brane models and the Lambert-Papageorgakis M5-brane model,
are examples of higher gauge theories. Recently, a class of models with N = (1, 0)
supersymmetry have been found, with significant overlap with algebraic structures
in higher gauge theory. This overlap suggests that the full N = (2, 0) theory could
involve semistrict L∞-algebras. Finally, we will see some explicit selfdual string
solutions, which may fit into these frameworks.