Periodic Reporting for period 4 - CatDT (Categorified Donaldson-Thomas Theory)
Période du rapport: 2021-05-01 au 2023-04-30
This project concerned the foundations and applications of categorified Donaldson-Thomas (DT) theory. DT theory started its life as a mathematically rigorous way to recover invariants from certain types of six-dimensional manifolds called Calabi-Yau 3-folds. These are mysterious objects which are central to string theory, which tells us that the fundamental invariants of subatomic particles are determined by the geometry of six extra "curled up" dimensions to our universe, alongside the four that we see, and that these extra dimensions form a Calabi-Yau 3-fold. Making sense of physical predictions, and understanding the enumerative theory of Calabi-Yau 3-folds within mathematics, has led to decades of intensive study and beautiful work, along with unforeseen and deep connections with many other areas of mathematics. One of the main goals of this project was to deepen these connections, by exploiting a categorical refinement of the numbers in DT theory (it is an enumerative theory) to vector spaces, and mixed Hodge structures.
One major application of DT theory, applied in a more general context of 3-Calabi-Yau categories than just those coming from geometry, is in quantum cluster algebras. Quantum cluster algebras are certain algebraic structures, admitting a relatively quick definition, but with a certain infinite recursion built into this definition that makes them very hard to study or understand in general. In line with the simplicity of the definition, they turn out to be a controlling structure in many different areas of mathematics. During the project I strengthened the connections between DT theory and quantum cluster algebras, specifically through the use of scattering diagrams in joint work with Travis Mandel, proving one of the main open conjectures regarding quantum cluster algebras.
Another major application is to nonabelian Hodge theory, which is the study of the topology and geometry of spaces of two different types of objects attached to Riemann surfaces. During the project I and other team members made significant advances in understanding the cohomology of spaces of representations of preprojective algebras via cohomological DT theory. This in turn produced deep links with Yangians and other quantum groups, as well as the study of Nakajima quiver varieties in geometric representation theory.
One major application of DT theory, applied in a more general context of 3-Calabi-Yau categories than just those coming from geometry, is in quantum cluster algebras. Quantum cluster algebras are certain algebraic structures, admitting a relatively quick definition, but with a certain infinite recursion built into this definition that makes them very hard to study or understand in general. In line with the simplicity of the definition, they turn out to be a controlling structure in many different areas of mathematics. During the project I strengthened the connections between DT theory and quantum cluster algebras, specifically through the use of scattering diagrams in joint work with Travis Mandel, proving one of the main open conjectures regarding quantum cluster algebras.
Another major application is to nonabelian Hodge theory, which is the study of the topology and geometry of spaces of two different types of objects attached to Riemann surfaces. During the project I and other team members made significant advances in understanding the cohomology of spaces of representations of preprojective algebras via cohomological DT theory. This in turn produced deep links with Yangians and other quantum groups, as well as the study of Nakajima quiver varieties in geometric representation theory.
During the project I and the other members of the project research group worked on applications of cohomological Donaldson-Thomas theory in cluster algebras, geometric representation theory and algebraic geometry.
Cluster algebras:
The idea of using scattering diagram techniques to prove positivity results in cluster algebras had already had spectacular success in the work of Gross, Hacking Keel and Kontsevich, who managed to prove both the positivity and strong positivity conjectures in cluster algebras using these techniques. By combining my earlier work with Sven Meinhardt on the integrality conjecture in cohomological Donaldson-Thomas theory with the quantum scattering diagram theory that Travis is an expert in, we were able to prove the quantum strong positivity conjecture in the preprint "Strong positivity for quantum theta bases of quantum cluster algebras", published in Inventiones Mathematicae in 2021
Nonabelian Hodge theory:
One of my main goals here was to prove a version of nonabelian Hodge theory for moduli stacks of objects, where we no longer identify non-isomorphic objects, and in addition we remember the data of automorphism groups of objects. This was finally achieved in joint work with team members Lucien Hennecart and Sebastian Schlegel Mejia, resulting in the preprint "BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks", uploaded to the Arxiv in 2022.
2-Calabi-Yau categories:
In the preprint "Purity and 2-Calabi-Yau categories", uploaded to the Arxiv in 2021, my main result is that the celebrated decomposition of Beilinson, Bernstein, Deligne and Gabber is true for the morphism from the moduli stack of objects in a 2-Calabi-Yau category to its coarse moduli space. This has deep implications for the study of the Borel-Moore homology of moduli stacks of objects in these categories, which I have exploited in a subsequent series of papers dedicated to geometric representation theory with team members Lucien Hennecart and Sebastian Schlegel Mejia.
Algebraic geometry:
With Francesca Carocci we have been working on proving the strong rationality conjecture via the approach outlined in my preprint "Refined invariants of finite-dimensional Jacobi algebras", published on the Arxiv in March 2019. So far we have succeeded in producing numerous constructions relating the vanishing cycle cohomology of Pandharipande-Thomas moduli spaces to the geometric representation theory of CoHAs, and in future work we intend to use these constructions to complete this project.
Somehow, even though there are no compact curves in complex three dimensional space A^3, the approach outlined above makes a strong prediction for the (affinized) BPS Lie algebra associated to the category of compactly supported sheaves on A^3, which controls the entire cohomological DT theory of this threefold. Defining what all this means, as well as verifying this prediction, was the result obtained in my paper Affine BPS algebras, W algebras, and the cohomological Hall algebra of A^2.
Cluster algebras:
The idea of using scattering diagram techniques to prove positivity results in cluster algebras had already had spectacular success in the work of Gross, Hacking Keel and Kontsevich, who managed to prove both the positivity and strong positivity conjectures in cluster algebras using these techniques. By combining my earlier work with Sven Meinhardt on the integrality conjecture in cohomological Donaldson-Thomas theory with the quantum scattering diagram theory that Travis is an expert in, we were able to prove the quantum strong positivity conjecture in the preprint "Strong positivity for quantum theta bases of quantum cluster algebras", published in Inventiones Mathematicae in 2021
Nonabelian Hodge theory:
One of my main goals here was to prove a version of nonabelian Hodge theory for moduli stacks of objects, where we no longer identify non-isomorphic objects, and in addition we remember the data of automorphism groups of objects. This was finally achieved in joint work with team members Lucien Hennecart and Sebastian Schlegel Mejia, resulting in the preprint "BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks", uploaded to the Arxiv in 2022.
2-Calabi-Yau categories:
In the preprint "Purity and 2-Calabi-Yau categories", uploaded to the Arxiv in 2021, my main result is that the celebrated decomposition of Beilinson, Bernstein, Deligne and Gabber is true for the morphism from the moduli stack of objects in a 2-Calabi-Yau category to its coarse moduli space. This has deep implications for the study of the Borel-Moore homology of moduli stacks of objects in these categories, which I have exploited in a subsequent series of papers dedicated to geometric representation theory with team members Lucien Hennecart and Sebastian Schlegel Mejia.
Algebraic geometry:
With Francesca Carocci we have been working on proving the strong rationality conjecture via the approach outlined in my preprint "Refined invariants of finite-dimensional Jacobi algebras", published on the Arxiv in March 2019. So far we have succeeded in producing numerous constructions relating the vanishing cycle cohomology of Pandharipande-Thomas moduli spaces to the geometric representation theory of CoHAs, and in future work we intend to use these constructions to complete this project.
Somehow, even though there are no compact curves in complex three dimensional space A^3, the approach outlined above makes a strong prediction for the (affinized) BPS Lie algebra associated to the category of compactly supported sheaves on A^3, which controls the entire cohomological DT theory of this threefold. Defining what all this means, as well as verifying this prediction, was the result obtained in my paper Affine BPS algebras, W algebras, and the cohomological Hall algebra of A^2.
The project has led to the settling of a number of conjectures in areas connected to cohomological DT theory, where those connections themselves have been established as part of the project. In the first half of the project, the main achievement was the settling of the strong quantum cluster positivity conjecture via a combination of cohomological wall-crossing and scattering diagram techniques, in line with the Description of the Action of the original project proposal. This resulted in the above-mentioned paper "Strong positivity for quantum theta bases of quantum cluster algebras" with Travis Mandel (hired as part of the project team) which settled this conjecture, as well as significantly developing the theory of quantum theta functions in homological mirror symmetry.
More recently, the main progress beyond the state of the art has come via the connections between cohomological DT theory and the moduli theory of objects in 2-Calabi-Yau categories, via dimensional reduction. This resulted in the foundational paper "Purity and 2-Calabi-Yau categories", which is still under review. This paper established a version of the famous decomposition theorem for moduli stacks of objects in categories across geometric representation theory, nonabelian Hodge theory and algebraic geometry, thereby laying the foundations for much subsequent work, undertaken with Lucien Hennecart and Sebastian Schlegel Mejia (both hired as part of the project). Highlights of this project so far include the nonabelian Hodge isomorphism for stacks, and a proof of the Bozec Schiffmann positivity conjecture, both of which are beyond the state of the art as it stood at the beginning of the project.
More recently, the main progress beyond the state of the art has come via the connections between cohomological DT theory and the moduli theory of objects in 2-Calabi-Yau categories, via dimensional reduction. This resulted in the foundational paper "Purity and 2-Calabi-Yau categories", which is still under review. This paper established a version of the famous decomposition theorem for moduli stacks of objects in categories across geometric representation theory, nonabelian Hodge theory and algebraic geometry, thereby laying the foundations for much subsequent work, undertaken with Lucien Hennecart and Sebastian Schlegel Mejia (both hired as part of the project). Highlights of this project so far include the nonabelian Hodge isomorphism for stacks, and a proof of the Bozec Schiffmann positivity conjecture, both of which are beyond the state of the art as it stood at the beginning of the project.