Higher Yang–Mills Theory

Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of ‘higher-dimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2-group’, which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a ‘Lie crossed module’: a pair of Lie groups G, H with a homomorphism t:H → G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing’s ideas on the geometry of nonabelian gerbes, one can define ‘principal 2-bundles’ for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a g-valued 1-form together with an h-valued 2-form, and its curvature consists of a g-valued 2-form together with a h-valued 3-form. We generalize the Yang–Mills action for this sort of connection, and use this to derive ‘higher Yang– Mills equations’. Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions.