Representation theory seeks to classify and describe the possible realizations of symmetries, and to exploit symmetry by providing a tool to decompose symmetric structures into elementary constituents. Representation theory has been an essential tool in quantum physics almost from its inception, providing for example the structure of atomic orbitals. Gauge theories, quantum theories built directly out of the structure of local symmetry, are the language of much of high energy physics, in particular the Standard Model, which describes all the fundamental forces besides gravity. Gauge theory in turn has had a tremendous impact on low dimensional topology and geometry. This project is concerned with the reversal of the relationship between representation theory and gauge theory: applying the structure of gauge theory as a powerful organizing framework for representation theory in the abstract. In this paradigm, different representation theories are encoded by different models of gauge theory, and inherit a radically new and uniform structure from the behavior of observables and defects in gauge theory. Thus the fundamental symmetries of nature become powerful tools to understand the most abstract questions in algebra and analysis -- in particular some of the deepest structures we know in algebra (the Langlands program, responsible for the resolution of Fermat's Last Theorem) derive from the symmetry between electricity and magnetism. These connections and synergies will be developed under this project, both in the PI's research and in his extensive expository work, including writing a graduate text introducing this paradigm to a broader audience for the first time.
Gauge theories are quantum field theories built directly out of local Lie group symmetry. Conversely, one can view many aspects of representation theory of Lie groups through the lens of gauge theory, which provides a powerful organizing principle for representation theory through the medium of low-dimensional topology. The object of this project is to develop and disseminate the perspective of representation theory as gauge theory. The project details two primary research projects inspired by developments in gauge theory. The first is the exploitation of a new source of commutative symmetry algebras in geometric representation theory inspired by Seiberg-Witten geometry of gauge theory and uncovered in the PI's recent work. The PI will apply spectral decomposition with respect to these symmetries in a variety of contexts, including Lusztig's theory of character sheaves, the homology of character varieties of surfaces, and the geometric Langlands correspondence. The second is the development of the new "Betti" (or topological) form of the Geometric Langlands Correspondence introduced by the PI, which he intends to reduce to elementary building blocks by proving an "automorphic Verlinde formula." The case of genus one appears accessible, and has implications for much-studied topics in representation theory. In addition the PI intends to engage in extensive expository writing, including a graduate text and interdisciplinary expository work with theoretical high energy physicists.
