Bundle gerbes applied to quantum field theory

This paper reviews recent work on a new geometric object called a bundle ge rbe and discusses some new examples arising in quantum field theory. One ap plication is to an Atiyah-Patodi-Singer index theory construction of the bu ndle of fermionic Fock spaces parameterized by vector potentials in odd spa ce dimensions and a proof that this leads in a simple manner to the known S chwinger terms (Mickelsson-Faddeev cocycle) for the gauge group action. Thi s gives an explicit computation of the Dixmier-Douady class of the associat ed bundle gerbe. The method also works in other cases of fermions in extern al fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in th e case of vector potentials. Another example, in which the bundle gerbe cur vature plays a role, arises from the WZW model on Riemann surfaces. A furth er example is the "existence of string structures" question. We conclude by showing how global Hamiltonian anomalies fit within this framework.