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# 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

#### Author

#### Abstract

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.

#### Keywords

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