Lectures by Todor Milanov

mathematical physics algebraic geometry

Audience: Researchers in the topic
Seminar series times: Monday 06:30-08:00, Tuesday 06:30-08:00, Wednesday 06:30-08:00, Thursday 06:30-08:00, Friday 06:30-08:00 in your time zone, UTC
Organizer: Satoshi Nawata*
*contact for this listing

Zoom ID: 9383671691 Password: 123456

Videos: www.youtube.com/playlist?list=PLAEGnJOLOreqHmnMaxXUP83C4OJNOZCkI

If the quantum cohomology of a smooth projective variety X, or more generally a projective variety with an orbifold structure, is semisimple, then Dubrovin and Zhang have constructed an integrable hierarchy that governs the Gromov--Witten (GW) invariants of X. The main motivation for the work that I would like to present is to construct an infinite dimensional Grassmannian that parametrizes the solutions of the Dubrovin--Zhang hierarchies. Similar description was proposed first by Mikio Sato for the KP hierarchy and it is very useful, because it allows us to study integrable hierarchies via the methods of representation theory of infinite dimensional Lie algebras. Based on ideas of Givental, I developed a method for proving that the generating function of GW invariants satisfies a given system of Hirota Bilinear Equations (HBEs). This is exactly what I am planning to talk about in my lectures. Note however, that the existence of the HBEs is still not clear. Based on joint work with B. Bakalov, I can only speculate that the HBEs, if they exist at all, should come from the representation theory of a certain class of lattice vertex algebras, i.e., in principal, we have reduced the problem of constructing HBEs in GW theory to a problem about lattice vertex algebras. Our ideas with Bakalov should be tested in an interesting example, such as, X=CP^2 or X= orbifold structure on P^1 with negative orbifold Euler characteristics. At this point I am not ready to lecture about the relation to lattice VOA, but if time permits, I will try to say something. Here is a list of topics that I am planning to cover

1. Frobenius manifolds and quantum cohomology

2. Givental's higher genus reconstruction

3. Quantization formalism

4. Vertex operators and phase factors

5. Hirota Bilinear Equations: an example.

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