Seminar “Modern Mathematical Physics”

Date and Time: Tuesday, 20 April 2021, at 1:00 PM

Venue: Online conference on Zoom, Bogoliubov Laboratory of Theoretical Physics

Link to join on Zoom

Seminar topic: «Special Bohr – Sommerfeld geometry: variation»

Speaker: Nikolay Tyurin

Abstract:

My talk will be devoted to a big programme which was presented for the following reason. In Geometric Quantization one can distinguish two approaches. The first one, which was started in the constructions of Sourio and Konstant, deals with prequantization pair (𝐿, a) where 𝐿 is a bundle and a is a hermitian connection on the bundle such that its curvature form 𝐹_𝑎 = 2πiω. Another approach was called Lagrangian since it deals with lagrangian submanifolds which satisfy so-called Bohr – Sommerfeld condition. This approach was started in the works of Maslov and his collaborators.

Some times ago we found a construction that can help in the unification of these two approaches. For a given symplectic manifold (𝑀, ω) understood as the phase space of a classical mechanical system, we take the direct product ℙ 𝛤(𝑀, 𝐿) × 𝐵_𝑆 where the first component corresponds to smooth sections of the prequantization bundle and the second component is the moduli space of Bohr – Sommerfeld lagrangian submanifold, and in this direct product we define certain universal object which joins the Bundle approach and Lagrangian approach to GQ.

In the talk, we describe this universal space 𝑈_𝑆𝐵𝑆 and present its properties, in particular, we show how certain finite dimensional objects can be derived from the picture if our symplectic manifold (𝑀, ω) admits the second ingredients – compatible integrable complex structure 𝐼.