Towards an M5‐Brane Model II: Metric String Structures

Abstract

In this paper, we develop the mathematical formulation of metric string structures. These play a crucial role in the formulation of certain six-dimensional superconformal field theories and we believe that they also underlie potential future formulations of the (2,0)-theory. We show that the connections on non-abelian gerbes usually introduced in the literature are problematic in that they are locally gauge equivalent to connections on abelian gerbes. Connections on string structures form an exception and we introduce the general concept of an adjusted Weil algebra leading to potentially interacting connections on higher principal bundles. Considering a special case, we derive the metric extension of string structures and the corresponding adjusted Weil algebra. The latter lead to connections that were previously constructed by hand in the context of gauged supergravities. We also explain how the Leibniz algebras induced by an embedding tensor in gauged supergravities fit into our picture.

Towards an M5-brane model. Part III. Self-duality from additional trivial fields

“…This paper is the third in a series of papers [1,2] attempting to systematically overcome the obstacles for constructing a classical M5-brane model, see also [3]. There are good reasons to be skeptical about the existence of such a model, cf.…”

“…also [7]. A problem arising here is that consistency of this parallel transport requires a part of the curvature to vanish, which locally renders the connection gauge equivalent to the connection on an abelian gerbe [2,8]. One can circumvent this issue by using special higher gauge algebras which allow for an adjustment, i.e.…”

“…essentially an alternative definition of the curvature expressions [2]. An important example that we will use in our model in this paper are adjusted string structures [9][10][11][12], or rather their metric extension [2], which are higher analogues of spin bundles. String structures allow for the definition of an adjusted higher parallel transport [13].…”

“…The mathematical interpretation of this structure was unclear; it is actually an EL ∞ -algebra [27]. Using the adjusted metric string structures of [2,3] makes it evident that this theory is mathematically well-defined and consistent: there are obvious notions of connections on higher principal bundles and of a higher parallel transport that underlie the theory.…”

Global Double Field Theory is Higher Kaluza‐Klein Theory

“…Hence principal n ‐bundles are a natural framework to deal with higher gauge theories (see [10]), which are a global formalization of n ‐form fields in physics. For instance higher gauge theory has been used by [111], [103], [112] to formulate a 6 d superconformal field theory which must be regarded as a very promising step in the direction of developing the desired M5‐brane worldvolume theory.…”