Communications in Mathematical Physics

, Volume 367, Issue 1, pp 215–240 | Cite as

Quantum \({L_\infty}\) Algebras and the Homological Perturbation Lemma

  • Martin Doubek
  • Branislav Jurčo
  • Ján PulmannEmail author


Quantum \({L_\infty}\) algebras are a generalization of \({L_\infty}\) algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum \({L_\infty}\) algebra via the homological perturbation lemma and show that it’s given by a Feynman diagram expansion, computing the effective action in the finite-dimensional Batalin–Vilkovisky formalism. We also construct a homotopy between the original and this effective quantum \({L_\infty}\) algebra.


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The research of M.D. and B.J. was supported by grant GAČR P201/12/G028. B.J. wants to thankMPIMin Bonn for hospitality. J.P. was supported by NCCR SwissMAP of the Swiss National Science Foundation and had also benefited from support by the project SVV-260089 of the Charles University. We would like to thank the anonymous reviewer, for suggestions that streamlined the paper considerably, and also for explaining to us what is now Remark 7. B.J. thanks Martin Markl and Owen Gwilliam for discussions. J.P would like to thank Florian Naef and Pavol Ševera for numerous discussions and Pavel Mnev for useful pointers. Finally, we would like to thank Lada Peksová for her useful comments on an earlier draft of this paper.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsMathematical Institute, Charles UniversityPragueCzech Republic
  2. 2.Section of MathematicsUniversity of GenevaGenevaSwitzerland

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