Mathematical and Theoretical Physics Seminar: "Gauged Wess-Zumino actions, equivariant cohomology, and the electromagnetic response of symmetry-protected topological phases"
(sign-up)Matthew Lapa, University of Illinois at Urbana Champaign
Physics and Mathematics
I will introduce the notion of a symmetry-protected 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 2m-1 or 2m differs from that of their more familiar fermionic counterparts by a numerical factor of m! (factorial), 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 Wess-Zumino 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 Wess-Zumino action describing the boundary coupled to the electromagnetic field. The construction of the gauged Wess-Zumino action is related to the U(1)-equivariant cohomology of the sphere, and I will explain this connection in detail. In particular, for even-dimensional 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 odd-dimensional 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 Chao-Ming Jian, Peng Ye, and Taylor L. Hughes.
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