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The Quantum Geometric Langlands Topological Field Theory

Periodic Reporting for period 4 - QuantGeomLangTFT (The Quantum Geometric Langlands Topological Field Theory)

Reporting period: 2019-12-01 to 2021-05-31

This ERC-funded five-year project seeks to understand the fundamental relationship between the nature of space in dimensions 2, 3, and 4, and a number of important algebraic structures introduced in the past two decades, and intimately tied to developments in mathematical physics, algebraic geometry, and most of all geometric representation theory. Geometric representation theory is the study of symmetries of a space, and how those symmetries impact the types of information the space can encode. This is an important paradigm in mathematical physics, where the typical spaces one encounters describe configurations of interacting particles in some system, and the information of interest is various physical quantities such as charge, spin, magnetic field, etc. In algebraic geometry, one encodes both these spaces of interest, and the information that they carry, using systems of polynomial equations, in many variables, and the introduction of algebraic techniques.

The proposal seeks to address an important question at the interface of all of these fields: over the past two decades, experts in the field of topological field theory -- the study of how topological properties of a space or space-time impact the kinds of information the space can carry, and how that information glues as we glue together the spacetimes coherently -- have understood that there should be a 4-dimensional topological field theory associated to the data of a quantum group. Here a quantum group is a fundamental deformation of an algebraic group of symmetries. Algebraically these quantum groups by now are quite well understood, and indeed many connections to topological field theories in dimension 3 have been rigourously developed. However, the constructions so far have had strong limitations on the nature of the quantum groups which are considered -- typically they are required to "modular" categories, which have found a number of applications for instance to statistical mechanics and quantum computing, but are ultimately too compact to yield examples related to algebraic geometry and representation theory.
In the first stages of the project, we have developed a rich machinery for dealing with these topological field theories in the non-modular setting, i.e. the completely general setting of rigid braided tensor categories. These constructions build on cutting-edge techniques of several teams in algebraic topology and homotopy theory -- Lurie, Ayala-Francis-Tannaka, Haugseng, Scheimbauer, Freed-Teleman, and many others -- to define the theory. They furthermore apply a powerful set of tools from category theory and geometric representation theory to transform the abstract formalism given to us by the homotopy theorists into very concrete sets of equations, which in turn capture a number of important quantum symmetries studied in the quantum groups literature for decades. This immediately provides a conduit between the two a priori quite distant fields, which we may exploit to prove many new results. Our current state of the art is as follows: we have constructed and thoroughly analyzed the resulting algebraic structures coming from surfaces -- that is, spaces in dimension 2. We have now turned our attention to a rigorous construction of the long-predicted invariants in dimension three. These have been foreshadowed in works of Walker, Freed-Teleman, Lurie, Crane-Yetter-Kauffman, and many others, but only in recent years, and only with the tools we have been building can these constructions be brought to their full strength.

In the final stages of the product we have obtained a number of important applications of our earlier foundational work. We have proved a long-standing and well-known conjecture of the physicist Edward Witten, stating that certain very natural Hilbert spaces of quantum states on a 3-dimensional space time, comprised of so-called Wilson loop expectations, is actually a finite-dimensional vector space. In related work we proved that the boundary operators for this theory -- coming from the value of the theory on surfaces -- have a strong finiteness and invertibility property relative to their classical degeneration. This established a refined version of the Unicity Conjecture of Bonahon and Wong. Finally, we established a fundamental conjecture of Freed, Teleman and Walker, which describes the general algebraic structure one needs to capture four-dimensional symmetries -- anomalies, in physical parlance -- acting on 3-dimensional topological field theories. In establishing their conjecture, we in fact found new and important examples extending the original formulation. Finally, we have taken the first step towards understanding the algebraic structures involved in spaces with defects, by computing categories of surface operators in a 4-dimensional theory in the framework of cluster algebras and canonical quantization.
This work has advanced the state of the art in contemporary mathematical understanding of topological quantum field theory in dimensions 2, 3, and 4. Quantum field theory is the language of physics -- specifically high energy particle physics, and topological quantum field theory is a rigorous mathematical approach to this all-important field. By formulating and then proving a myriad of important mathematical statements underpinning topological quantum field theory, we further progress the century-long quest to develop a fully working framework for quantum field theory.
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