Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

On the $\operatorname{BV}$ structure on the cohomology of moduli space

Pages: 755 – 773

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a13


Sümeyra Sakallı (Max Planck Institute for Mathematics, Bonn, Germany)

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


The question of vanishing of the $\operatorname{BV}$ operator on the cohomology of the moduli space of Riemann surfaces is investigated. The $\operatorname{BV}$ structure, which comprises a $\operatorname{BV}$ operator and an antibracket, is identified, vanishing theorems are proven, and a counter-example is provided.


moduli space of Riemann surfaces, Deligne–Mumford compactification, spectral sequence, mixed Hodge structure, stable cohomology, tautological class, $\operatorname{BV}$-algebra, Maurer–Cartan equation, quantum master equation

2010 Mathematics Subject Classification

Primary 14F17, 14H15. Secondary 32G15.

A.V. was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan, and a Collaboration grant from the Simons Foundation (#585720).

Received 30 December 2019

Accepted 8 June 2020

Published 11 November 2020