Periodic Reporting for period 2 - NCST (Non-compact Chern-Simons Theory, Positive Representations, and Cluster Varieties)
Période du rapport: 2023-01-01 au 2024-06-30
Broadly speaking, the project addresses questions on the interface of Poisson geometry, representation theory, functional analysis, and quantum topology. The first field, Poisson geometry, can be thought of as a mathematical language for classical mechanics. The next two, representation theory and functional analysis, constitute the language for quantum mechanics. The last one, quantum topology, provides mathematical tools for topological quantum field theories (TQFT. Finally, the combination of all 4 of them, gives a broad machinery applicable, for example, to the study of gauge theories. More specifically, my research in this rather broad collection of mathematical areas revolves around the so-called, cluster algebras and cluster varieties. The latter may be understood as a yet another mathematical tool that provides an infinite number of extremely convenient coordinate systems (exponentiated Darboux coordinates), in which a large number of non-trivial calculations get significantly simplified.
The types of mathematical problems this project addresses are essentially of the following 3 kinds. The first one concerns constructions when certain explicitly computable algebraic data (such as a representation of an algebra) is attached to a geometric object (such as a surface), in a way that distinguishes these objects up to their topology. In this way we can speak of topological invariants of surfaces. The second kind are problems in representation theory and Poisson geometry, which either relate representations of different algebras, thus building connections between the corresponding physical systems, or relate constructions in quantum algebra to their quasi-classical analogues, thus bridging the quantum and classical systems. Finally, the third type of problems I work on is the search of cluster structures on classical or quantum objects, in order to apply the powerful machinery of cluster coordinates to questions above.
My proposal has two main objectives. The first one is to establish a proof of the celebrated Modular Functor Conjecture in higher Teichmüller theory. In broad terms, the conjecture seeks to construct a partial data of a 4-dimensional TQFT, providing topological invariants of low-dimensional manifolds. However, the construction utilizes functional analytic rather than algebraic representation theory, thus bringing us one step closer to models of actual interest to broader physics community. The second major objective is to establish cluster structure on the so-called Coulomb branches of the so-called 4d N=2 quiver gauge theories. One may think that the theory takes a group and a surface as an input, and produces a Poisson manifold as one of its outputs, the so-called Coulumb branch. Those have been of great interest for physics community for several decades and has been finally given a mathematical definition by Braverman, Finkelberg, and Nakajima less than a decade ago. It was conjectured by Gaiotto that their Coulomb branches possess cluster structure and the second objective is to settle that conjecture.
The types of mathematical problems this project addresses are essentially of the following 3 kinds. The first one concerns constructions when certain explicitly computable algebraic data (such as a representation of an algebra) is attached to a geometric object (such as a surface), in a way that distinguishes these objects up to their topology. In this way we can speak of topological invariants of surfaces. The second kind are problems in representation theory and Poisson geometry, which either relate representations of different algebras, thus building connections between the corresponding physical systems, or relate constructions in quantum algebra to their quasi-classical analogues, thus bridging the quantum and classical systems. Finally, the third type of problems I work on is the search of cluster structures on classical or quantum objects, in order to apply the powerful machinery of cluster coordinates to questions above.
My proposal has two main objectives. The first one is to establish a proof of the celebrated Modular Functor Conjecture in higher Teichmüller theory. In broad terms, the conjecture seeks to construct a partial data of a 4-dimensional TQFT, providing topological invariants of low-dimensional manifolds. However, the construction utilizes functional analytic rather than algebraic representation theory, thus bringing us one step closer to models of actual interest to broader physics community. The second major objective is to establish cluster structure on the so-called Coulomb branches of the so-called 4d N=2 quiver gauge theories. One may think that the theory takes a group and a surface as an input, and produces a Poisson manifold as one of its outputs, the so-called Coulumb branch. Those have been of great interest for physics community for several decades and has been finally given a mathematical definition by Braverman, Finkelberg, and Nakajima less than a decade ago. It was conjectured by Gaiotto that their Coulomb branches possess cluster structure and the second objective is to settle that conjecture.
My work to date has been mostly devoted to proving the two flagship conjectures of the proposal. I believe that at the moment we have all details of both proofs, and expect to have finished both preprints by the conclusion of the project.
The PhD students of mine supported by the ERC grant have been working on the following projects. David Cueto has been searching for the cluster structure on multiplicative Nakajima quiver varieties, the mirror side of Gaiotto's conjecture. By now David obtained some important partial results, and is now preparing a preprint. Matthew Tyson has been exploring applications of quantum cluster algebra to the representation theory of quantum groups. He, as well is preparing his first preprint and is working on generalizing obtained results.
Finally, let me briefly describe contributions made by my postdocs. Léa Bittmann wrote and co-aothored 3 papers, which focused on connections between representation theory of quantum groups, quantum affine algebras, and cluster algebra. Nitin Chidambaram has written 3 papers on various connections between topological recursion, Hurwitz numbers, and Whittaker vectors of W-algebras. Finally, Mikhail Bershtein who joined my team last year has co-authored a preprint studying conformal blocks and blow-up relations on Nekrasov partition functions. Works of Chidambaram and Bershtein concern one of the most interesting potential applications of this project, namely a connection between representation theory of quantum groups and that of W-algebras or in physics language, between certain topological quantum field theories and conformal field theories.
The PhD students of mine supported by the ERC grant have been working on the following projects. David Cueto has been searching for the cluster structure on multiplicative Nakajima quiver varieties, the mirror side of Gaiotto's conjecture. By now David obtained some important partial results, and is now preparing a preprint. Matthew Tyson has been exploring applications of quantum cluster algebra to the representation theory of quantum groups. He, as well is preparing his first preprint and is working on generalizing obtained results.
Finally, let me briefly describe contributions made by my postdocs. Léa Bittmann wrote and co-aothored 3 papers, which focused on connections between representation theory of quantum groups, quantum affine algebras, and cluster algebra. Nitin Chidambaram has written 3 papers on various connections between topological recursion, Hurwitz numbers, and Whittaker vectors of W-algebras. Finally, Mikhail Bershtein who joined my team last year has co-authored a preprint studying conformal blocks and blow-up relations on Nekrasov partition functions. Works of Chidambaram and Bershtein concern one of the most interesting potential applications of this project, namely a connection between representation theory of quantum groups and that of W-algebras or in physics language, between certain topological quantum field theories and conformal field theories.
While the proofs of the two major conjectures remain to be written, these results are beyond the state of the art. I expect to have finished the corresponding preprints by the end of the project, and plan to submit them to top mathematics journals. Other expected results include, but are not limited to, applications to representation theory, classical and quantum integrable systems, functional analytic versions of quantum groups
In my opinion, the most exciting and unexpected outcome of our work are the three new research directions, which definitely go beyond the state of the art. The first one concerns the study of cluster Poisson varieties under Hamiltonian reduction. Roughly, there is a natural way to obtain interesting physical systems from simpler ones, via reducing dimension of the latter in a very controlled way. This approach, however, has never been explored in cluster setting, and my group is pioneering this research. We have obtained exciting partial results, which are directly related to the main objectives of the proposal, and I fully expect the techniques we develop to find a very wide range of applications.
The second research direction is the study of certain loop versions of cluster varieties. This has been a very sought after open question, which remained open for the last 2 decades. The ingenious construction suggested by Mikhail Bershtein seem to have all the desired properties, and have already been successfully applied to several open questions. This shall bridge the field of cluster algebras with that of W-algebras and conformal field theory, opening a new chapter in both.
Finally, the part 2 of my proposal, which concerns representation theory of quantum groups, got a completely unexpected development, arguably more interesting than most of the initially specified objectives. Namely, we were able to apply cluster algebraic techniques towards constructing locally compact quantum groups. The latter is a successful framework developed for the study of quantum integrable systems, which involves both the representation theory and functional analysis. However, so far it lacked any non-trivial Lie theoretic examples, except one found 3 decades ago. Our work finally allows to carry over all classical Lie groups into this framework on the one hand, and provides a powerful language to study cluster representation theory on the other.
In my opinion, the most exciting and unexpected outcome of our work are the three new research directions, which definitely go beyond the state of the art. The first one concerns the study of cluster Poisson varieties under Hamiltonian reduction. Roughly, there is a natural way to obtain interesting physical systems from simpler ones, via reducing dimension of the latter in a very controlled way. This approach, however, has never been explored in cluster setting, and my group is pioneering this research. We have obtained exciting partial results, which are directly related to the main objectives of the proposal, and I fully expect the techniques we develop to find a very wide range of applications.
The second research direction is the study of certain loop versions of cluster varieties. This has been a very sought after open question, which remained open for the last 2 decades. The ingenious construction suggested by Mikhail Bershtein seem to have all the desired properties, and have already been successfully applied to several open questions. This shall bridge the field of cluster algebras with that of W-algebras and conformal field theory, opening a new chapter in both.
Finally, the part 2 of my proposal, which concerns representation theory of quantum groups, got a completely unexpected development, arguably more interesting than most of the initially specified objectives. Namely, we were able to apply cluster algebraic techniques towards constructing locally compact quantum groups. The latter is a successful framework developed for the study of quantum integrable systems, which involves both the representation theory and functional analysis. However, so far it lacked any non-trivial Lie theoretic examples, except one found 3 decades ago. Our work finally allows to carry over all classical Lie groups into this framework on the one hand, and provides a powerful language to study cluster representation theory on the other.