Gauge theory, loop groups, and string topology
Seminar Room 1, Newton Institute
In this talk I will describe a joint project with John Jones, in which we relate the gauge theory of a principal bundle G --> P --> M to the string topology spectrum of the principal bundle. This spectrum has, as its homology the homology of the corresponding adjoint bundle. In this study G can be any topological group. In the universal case when P is contractible, the adjoint bundle is equivalent to the free loop space, LM, and this spectrum realizes the original Chas-Sullivan string topology homological structure. One of our main results is to identify the group of units the string topology spectrum, and to relate it to the gauge group of the original bundle. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections, to the setting of principal fiberwise spectra over a manifold, and show how it allows us to do explicit calculations. We also show how, in the universal case, an action of a Lie group on a manifold yields a representation of the loop group on the string topology spectrum. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum of a principal bundle is the "linearization" of the gauge group.