Letters in Mathematical Physics

, Volume 107, Issue 8, pp 1515–1543 | Cite as

The MV formalism for \({\mathrm{IBL}}_\infty \)- and \({\mathrm{BV}}_\infty \)-algebras

  • Martin MarklEmail author
  • Alexander A. Voronov


We develop a new formalism for the quantum master equation \(\Delta e^{S/\hbar } = 0\) and the category of \(\mathrm{IBL}_\infty \)-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of \(\mathrm{IBL}_\infty \)-algebras and \(\mathtt{L}_\infty \)-algebras and, on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.


\(\mathrm{MV}\)-algebra \(\mathrm{IBL}_\infty \)-algebra Master equation Transfer 

Mathematics Subject Classification

08C05 18G55 (Primary) 16E45 58A50 (Secondary) 



The authors are indebted to D. Bashkirov and the anonymous referees for useful remarks, and to J. Latschev for sharing a preliminary version of his paper with K. Cieliebak and K. Fukaya. The first author also wishes to express his thanks to M. Doubek, B. Jurčo and I. Sachs for introducing him to the jungle of \(\mathrm{IBL}_\infty \)-algebras and acknowledge the hospitality of the University of Minnesota where he held the position of a Distinguished Ordway Visitor during the last stages of the work on this article. The second author gratefully acknowledges support from the Graduate School of Mathematical Sciences, the University of Tokyo, and the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Mathematical Institute of the AcademyPrague 1Czech Republic
  2. 2.Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  4. 4.Kavli IPMU (WPI), UTIASThe University of TokyoKashiwaJapan

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