Scientific Series

In the past few years we have witnessed a flurry of activity in the field of topological phases of matter. An outstanding theme in the field is the study of the interplay between geometry and topology of many-body wave functions, which has attracted the attention of condensed matter and high-energy physicists. In this talk, I will present the quantum field-theoretic descriptions of the fascinating novel phenomena emergent from intertwined geometry and topology, which are vividly exemplified by the geometric responses in fractional quantum Hall systems. For the strict topological limit, where only the global topology of space matters, the fractional quantum Hall systems are characterized by their universal properties such as fractional quantum Hall conductivity and a degeneracy on surfaces with the topology of a torus. Quite surprisingly, these topological fluids also couple to the geometry and have universal responses to the adiabatic deformations of the background geometry. These responses are given by a Wen-Zee term, Hall viscosity term, and gravitational Chern-Simons term. Through the field-theoretic approaches of the topological fluids, I will for the first time show how to derive all the universal geometric responses. To account for the coupling to the background geometry, I show that the concept of “flux attachment” must be modified in the curved space and use it to derive the responses from Chern-Simons theories for all the known fractional quantum Hall states. I also apply these results to the theory of the anisotropic quantum Hall systems, where the geometric responses play a central role in understanding their universal physics.

We derive in the framework of soft collinear effective field theory (SCET) a Lagrangian describing the t-channel exchange of Glauber quarks, which are incorporated through fermionic potential operators in the effective theory. The Wilson line structure of the operators, which is derived from matching calculations and the symmetries of the effective theory, describe additional soft and collinear emissions from a fermionic t-channel exchange in the forward scattering limit to all orders. Lightcone singularities in the fermionic potential are regulated using a rapidity regulator, whose corresponding renormalization group evolution gives rise to the quark reggeization at amplitude level.

Many-body entanglement can lead to exotic phases of matter beyond conventional symmetry breaking paradigm. Those exotic phases may contain fractionalized quasiparticles and emergent gauge fields. In this talk,  I will focus on a wide class of long-range entangled phases—quantum spin liquid. In quantum spin liquids, the spins are entangled in some intricate fashion giving rise to interesting physics such as emergent topological field theory and QED3 theory. I will show in detail how such exotic physics can emerge in simple spin systems.

The notion of Positive Representations is a new research program devoted to the representation theory of split real quantum groups, initiated in a joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of compact quantum groups, but at the same time also serves some new properties that are not available in the compact case. In this talk, I will survey the recent developments and describe some of its relations to other areas of mathematics. 

I will discuss some results on double loop groups that point to geometric phenomena about double affine flag varieties and double affine Grassmannian. One result of this study is a definition of double affine Kazhdan-Lusztig polynomials.

It is commonly believed that quantum information is not lost in a black hole. Instead, it is encoded into non-local degrees of freedom in some clever way; like a quantum error-correcting code. In this talk, I will discuss how one may resolve some paradoxes in quantum gravity by using the theory of quantum error-correction. First, I will introduce a simple toy model of the AdS/CFT correspondence based on tensor networks and demonstrate that the correspondence between the AdS gravity and CFT is indeed a realization of quantum codes. I will then show that the butterfly effect/scrambling in black holes can be interpreted as non-local encoding of quantum information and can be quantitatively measured by out-of-time ordered correlations. Finally I will describe a simple decoding protocol for reconstructing a quantum state from the Hawking radiation and suggest a physical interpretation as a traversable wormhole in an AdS black hole. The decoding protocol also provides an attractive platform for laboratory experiments for measuring out-of-time ordered correlation functions as it clearly distinguishes unitary scrambling from non-unitary decoherence. 

For a family of finite rate stabilizer codes, one can define two distinct error correction thresholds: the usual "block" threshold for the entire code, and the single-qubit threshold, where we only care about the stability of a single encoded qubit corresponding to a randomly chosen conjugate pair of logical X and Z operators.  Our main result is that in the case of erasures, for hyperbolic surface codes related to a {p,q} tiling of the hyperbolic plane, it is the latter threshold that coincides exactly with the infinite-graph edge percolation transition.  I will also discuss likely generalizations to more general codes and other error models. This is joint work with Nicolas Delfosse.

General relativity taught us that space time is dynamical and quantum theory posits that dynamical objects are quantum. Hence the Newtonian notion of space time as a passive stage where physics takes place needs to be replaced by a notion of quantum space time which is interacting with all other quantum matter fields.

I will present recent developments that aim at constructing quantum space time. In particular I will explain how  topological quantum field theories give rise to new quantum geometry realizations and how these serve as starting points  for the construction of a  dynamics  of quantum gravity, which is consistent over all scales.  Such a dynamics will then determine the properties of quantum space time.  

I discuss a string theoretic approach to integrable lattice models. This approach provides a unified perspective on various important notions in lattice models, and relates these notions to four-dimensional N = 1 supersymmetric field theories and their surface operators.  I explain how my construction connects to Costello's work and the Nekrasov-Shatashvili correspondence.

The properties of physical processes reflect themselves in the structure of the relevant observables. This idea has been largely exploited for the flat space  S-matrix, whose analytic structure is determined by locality and unitary, the two pillars which our current understanding of nature is based on. In this context, it  has been possible to find new mathematical structures whose properties turn out to be the ones we ascribe to scattering processes in flat-space, so that both  unitarity and locality can be viewed as emergent from some more fundamental structure. However, the S-matrix does certainly not exhaust all the physical information contained in a theory and thus an interesting question to ask is whether and how we can get a similar and deeper understanding of other observables.

In this talk I will report on some recent progress towards such a formulation for quantities such as the universe wave-function and late-time correlators. In particular, I will focus on the perturbative representations of such quantities, their analytic properties and their geometrical interpretation as volumes of certain new polytopes. I  will consider a specific toy model which allows to extend most of the discussion to other set-ups than the cosmological one, such as flat and AdS space-times.

Topological phases of matter serve as one of the most striking examples of the richness and novelty of quantum systems with many degrees of freedom.  In contrast to conventional matter, they are characterized by both non-local properties and non-classical notions such as entanglement.  I will discuss two broad categories of topological phases, distinguished by whether or not they possess fractionalized “anyon” excitations that are neither bosons nor fermions.  I will demonstrate that entanglement not only provides an understanding of such phases but also enables the transmutation between these two categories of topological phases. 

This talk concerns a family of special functions common to the study of quantum conformal blocks and hypergeometric solutions to q-KZB type equations.  In the first half, I will explain two methods for their construction -- as traces of intertwining operators between representations of quantum affine algebras and as certain theta hypergeometric integrals we term Felder-Varchenko functions.  I will then explain our proof by bosonization the first case of Etingof-Varchenko's conjecture that these constructions are related by a simple renormalization.

The second half of the talk will concern applications to affine Macdonald theory.  I will present refinements of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr.  I will then explain how to prove the first non-trivial cases of these conjectures by combining the methods of the first half and well-chosen applications of the elliptic beta integral.   The second half of this talk is joint work with E. Rains and A. Varchenko.