Periodic Reporting for period 2 - SCHEMES (Schobers, Mutations and Stability)
Reporting period: 2022-07-01 to 2023-12-31
Mirror symmetry is a manifestation of string theory that predicts a certain symmetry between complex geometry and symplectic geometry. Mirror symmetry is justified on physical grounds but makes nonetheless strong and testable predictions about purely mathematical concepts. A celebrated example is the prediction by physicists of the number of rational curves of a given degree in a generic quintic threefold which went far beyond classical enumerative geometry.
One of the main actors in mirror symmetry is the "Stringy Kähler Moduli Space" which is the moduli space of complex structures on the mirror partner of a Calabi-Yau manifold. The SKMS is not rigorously defined as mirror symmetry itself is not rigorous, but in many cases there are precise heuristics available to characterize it.
Mirror symmetry predicts the existence of an action of the fundamental group of the SKMS on the derived category of coherent sheaves of a Calabi-Yau manifold. This prediction has only been verified in a limited number of cases.
One of the main actors in mirror symmetry is the "Stringy Kähler Moduli Space" which is the moduli space of complex structures on the mirror partner of a Calabi-Yau manifold. The SKMS is not rigorously defined as mirror symmetry itself is not rigorous, but in many cases there are precise heuristics available to characterize it.
Mirror symmetry predicts the existence of an action of the fundamental group of the SKMS on the derived category of coherent sheaves of a Calabi-Yau manifold. This prediction has only been verified in a limited number of cases.
1. With Anya Nordskova we have understood the work of Bridgeland and Stern on the relation between mutations and cluster mutations via the work Kuleshov and Orlov. This considerably shortens the original approach.
2. With Špela Špenko we have understood the decategorification of the perverse schober associated to a quasi-symmetric representation. We have shown that the corresponding representation of the fundamental group is given by the GKZ system.
3. With Špela Špenko we have used homological mirror symmetry in the toric, not necessarily quasi-symmetric, case to prove that the fundamental group of the SKMS acts on the derived category of the toric boundary. We have again established a strong connection with the GKZ system.
2. With Špela Špenko we have understood the decategorification of the perverse schober associated to a quasi-symmetric representation. We have shown that the corresponding representation of the fundamental group is given by the GKZ system.
3. With Špela Špenko we have used homological mirror symmetry in the toric, not necessarily quasi-symmetric, case to prove that the fundamental group of the SKMS acts on the derived category of the toric boundary. We have again established a strong connection with the GKZ system.
All results listed in the previous progress represent progress beyond the state of the art with, in our opinion 3., being the most significant one.
With Špela Špenko we are currently trying extend our results on the toric boundary to the full toric variety. A substantial roadblock is given by the fact that there is no truly convenient definition of the Fukaya category of an LG-model. With Daniel Kaplan we are circumventing this problem by considering the equivariant version of the conjecture. The non-equivariant version would follow by "quotienting out the group action".
We are currently trying to extend 1. to certain weak Del Pezzo varieties and stacks. Somewhat surprisingly this is a substantial challenge, the reason being that the knowledge in these cases (represented by work of Ishii, Okawa and Uehara) is much less complete than in the Del Pezzo case. We expect to be able to at least understand some weighted projective spaces.
With Špela Špenko we are currently trying extend our results on the toric boundary to the full toric variety. A substantial roadblock is given by the fact that there is no truly convenient definition of the Fukaya category of an LG-model. With Daniel Kaplan we are circumventing this problem by considering the equivariant version of the conjecture. The non-equivariant version would follow by "quotienting out the group action".
We are currently trying to extend 1. to certain weak Del Pezzo varieties and stacks. Somewhat surprisingly this is a substantial challenge, the reason being that the knowledge in these cases (represented by work of Ishii, Okawa and Uehara) is much less complete than in the Del Pezzo case. We expect to be able to at least understand some weighted projective spaces.