Matthew Lapa (Illinois Physics) Gauged WessZumino actions, equivariant cohomology, and the electromagnetic response of symmetryprotected topological phases Abstract: I will introduce the notion of a symmetryprotected topological (SPT) phase protected by the symmetry of a group G, and then present a calculation of the electromagnetic response of some bosonic SPT phases with G=U(1) in all dimensions. Remarkably, we find that the magnitude of the response of these bosonic SPT phases in spacetime dimensions 2m1 or 2m differs from that of their more familiar fermionic counterparts by a numerical factor of m!, in agreement with previous results in low dimensions. The calculation uses a description of an SPT phase in terms of a nonlinear sigma model (NLSM) with theta term for the bulk and WessZumino term for the boundary. The target space of the NLSM is a sphere of a particular dimension, and a crucial part of the NLSM description is an action of the group G=U(1) on the target space. I will show that the bulk response of the SPT phase can be deduced from the form of the gauged WessZumino action describing the boundary coupled to the electromagnetic field. The construction of the gauged WessZumino action is related to the U(1)equivariant cohomology of the sphere, and I will explain this connection in detail. In particular, for evendimensional spheres our result is equivalent to an equivariant extension of the volume form on the sphere with respect to the U(1) symmetry. On the other hand, for odddimensional spheres our result gives a physical interpretation for why such an extension fails. This talk is based on the paper arXiv:1611.03504 written together with ChaoMing Jian, Peng Ye, and Taylor L. Hughes. 
