Talks

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.

In crystals, quantum electrons can be spatially distributed in a way that the bulk solid supports macroscopic electric multipole moments, which are deeply 

related with emergence of topology insulators in condensed matter systems. However, unlike the classical electric multipoles in open space, 

defining electric multipoles in crystals is a non-trivial task. So far, only the dipolar moment, namely polarization, has been successfully defined and served as a classic example of topological insulators. 

In this talk, we propose the general definition, i.e., many-body invariants, for electric multipoles in crystals, which are related with recently-discovererd higher-order topological insulators. 

Our invariants are designed to measure the distribution of electron charge in unit cells and thus can detect multipole moments purely from the bulk ground state wavefunctions. 

We provide analytic as well as numerical proofs for our invariants. Application of our invariants to spin systems as well as various other aspects of the many-body invariants will be briefly discussed.

I argue that we do not understand gauge theory as well as we think when boundaries are present. I will briefly explain the conceptual and technical issues that arise at the boundary.  I will then propose a tentative resolution, which requires us to think of theories not in spacetime, but in field-space.