MS Seminar (Mathematics - String Theory)

Speaker: Alexander Voronov (U. Minnesota)
Title: Introduction to Quantum Deformation Theory
Date: Tue, Jul 13, 2010, 15:30 - 17:00
Place: Balcony A
Abstract: This is the first meeting of a subseminar on Quantum Deformation Theory, which is a new topic in mathematics emerging under the influence of physics.

Quantum deformation theory is based on the Quantum Master Equation (QME), also known as the Batalin-Vilkovisky (BV) Master Equation: d S + h \Delta S + 1/2 {S,S} = 0, inasmuch as classical deformation theory is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan Equation: d S + 1/2 {S,S} = 0.

In classical deformation theory, there are two sides of the story, abstract deformation theory, coming from the works of Deligne,
Schlessinger, Stasheff, Goldman, Millson, Kontsevich, and Soibelman, and concrete deformation theories, such as deformations of complex structures (Kodaira-Spencer), associative algebras (Gerstenhaber), and many others. Abstract deformation theory takes a dg Lie algebra as a primary object and studies the CME, the associated deformation functor and its moduli space. Concrete deformation theory presents a dg Lie algebra governing the deformation problem and uses the specifics of the concrete situation to understand the local structure of the moduli space, such as smoothness, formality, obstructions, virtual dimension, etc.

In quantum deformation theory, there is just a tip of the iceberg beginning to appear. There are a few papers which may be viewed as making first steps in abstract quantum deformation theory: Quantum Backgrounds and QFT by Jae-Suk Park, Terilla, and Tradler; Modular Operads and Batalin-Vilkovisky Geometry by Barannikov; and Smoothness Theorem for Differential BV Algebras by Terilla.

There is no general theory of quantum deformations, and it is not understood what quantum deformations are in concrete examples. The goal of the seminar will be to set up the problems, study what is known, and hopefully make further steps in quantum deformation theory. During the first meeting, I plan to give a review of classical deformation theory and set up the problem of quantum deformation theory.