Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 1

Special Issue in Honor of Yuri Manin: Part 3 of 3

Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau

Quantizing Deformation Theory II

Pages: 125 – 152

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n1.a3


Alexander A. Voronov (School of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.; and Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba, Japan)


A quantization of classical deformation theory, based on the Maurer–Cartan Equation $dS + \frac{1}{2} [S, S] = 0$ in dg‑Lie algebras, a theory based on the Quantum Master Equation $dS + \hslash \Delta S + \frac{1}{2} \lbrace S, S \rbrace = 0$ in dg‑BV‑algebras, is proposed. Representability theorems for solutions of the Quantum Master Equation are proven. Examples of “quantum” deformations are presented.


deformation theory, Maurer–Cartan equation, quantum master equation, differential graded manifold, BV-algebra

2010 Mathematics Subject Classification

Primary 14D15, 16E45. Secondary 81T70.

The author is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan, and a Collaboration grant from the Simons Foundation (#282349).

Received 21 June 2018

Published 6 February 2020