Conference

I will discuss several examples of novel continuous phase transitions, primarily in 3+1-D, that are beyond the standard Landau paradigm of order parameter fluctuations. These provide non-trivial examples of deconfined quantum critical points.

In the study of three-dimensional gapped models, two-dimensional gapped states can be considered as a free resource. This is the basic idea underlying our proposal of the notion of `foliated fracton order'. Using this idea, we have found that many of the known type-I fracton models, like the X-cube model and the checkerboard model, have the same foliated fracton order. In this talk, I will present three-dimensional fracton models with a different kind of foliated fracton order. The previously known foliated fracton order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry. The new order corresponds to a twisted version of the gauge theory where the system before gauging has nontrivial order protected by the subsystem planar symmetries. I will discuss a way to identify the nontrivial order by compactifying the system in the z direction and analyzing the resulting two dimensional order.

The classical Hall algebra of the category of representations of one-loop quiver is isomorphic to the ring of symmetric functions, and Hall-Littlewood polynomials arise naturally as the images of objects. I will talk about a second "fusion" product on this algebra, whose structure constants are given by counting of bundles with nilpotent endomorphisms on P^1 with restrictions at 0, 1 and infinity. The two products together make up a structure closely related to the elliptic Hall algebra. In the situations when bundles can be explicitly enumerated, I will explain how this leads to q,t-identities conjectured by combinatorists, such as the shuffle conjecture and its generalizations. This is a joint project with Erik Carlsson.