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

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Abstract

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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References

  1. Albert, C.: Batalin–Vilkovisky Gauge-Fixing via Homological Perturbation Theory. http://www-math.unice.fr/~patras/CargeseConference/ACQFT09_CarloALBERT.pdf. Accessed 22 Feb 2019

  2. Barannikov S.: Solving the noncommutative Batalin–Vilkovisky equation. Lett. Math. Phys. 103(6), 605–628 (2013) arXiv:1004.2253

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Batalin I., Vilkovisky G.: Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981) http://www.sciencedirect.com/science/article/pii/0370269381902057. Accessed 22 Feb 2019

    Article  ADS  MathSciNet  Google Scholar 

  4. Blanes S., Casas F., Oteo J.A., Ros J.: The Magnus expansion and some of its applications. Phys. Rep. 470(5–6), 151–238 (2008) arXiv:0810.5488

    ADS  MathSciNet  Google Scholar 

  5. Braun, C., Maunder, J.: Minimal models of quantum homotopy Lie algebras via the BV-formalism. arXiv:1703.00082

  6. Braun C., Lazarev A.: Unimodular homotopy algebras and Chern–Simons theory. J. Pure Appl. Algebra 219(11), 5158–5194 (2015) arXiv:1309.3219

    Article  MathSciNet  MATH  Google Scholar 

  7. Brown, R.: The twisted Eilenberg–Zilber theorem. Simposio di Topologia (Messina, 1964)’, Edizioni Oderisi, Gubbio 33–37 (1965) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.8277. Accessed 22 Feb 2019

  8. Cattaneo, A.S., Mnev, P., Reshetikhin, N.: A cellular topological field theory. arXiv:1701.05874

  9. Chuang J., Lazarev A.: Feynman diagrams and minimal models for operadic algebras. J. Lond. Math. Soc. 81(2), 317–337 (2010) arXiv:0802.3507

    Article  MathSciNet  MATH  Google Scholar 

  10. Chuang J., Lazarev A.: Abstract Hodge decomposition and minimal models for cyclic algebras. Lett. Math. Phys. 89(1), 33–49 (2009) arXiv:0810.2393

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Costello, K.: Renormalisation and the Batalin–Vilkovisky formalism. arXiv:0706.1533

  12. Costello, K.: Renormalization and Effective Field Theory. Mathematical surveys and monographs. American Mathematical Society, (2011)

  13. Crainic, M.: On the perturbation lemma, and deformations. arXiv:math/0403266

  14. Eilenberg S., Lane S.M.: On the groups H(\({\Pi}\), n), I. Ann. Math. 58(1), 55–106 (1953) http://www.jstor.org/stable/1969820. Accessed 22 Feb 2019

    Article  MathSciNet  Google Scholar 

  15. Granåker, J.: Unimodular L-infinity algebras arXiv:0803.1763

  16. Gugenheim V.K.A.M.: On the chain-complex of a fibration. Illinois J. Math. 16(3), 398–414 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gwilliam, O.: Factorization algebras and free field theories. (2013). http://people.math.umass.edu/~gwilliam/thesis.pdf. Accessed 22 Feb 2019

  18. Huebschmann, J.: The homotopy type of \({F\Psi^q}\). The complex and symplectic cases. In: Applications of Algebraic \({K}\)-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), vol. 55 of Contemp. Math., pp. 487–518. Amer. Math. Soc., Providence, RI, (1986)

  19. Johnson-Freyd T.: Homological perturbation theory for nonperturbative integrals. Lett. Math. Phys. 105(11), 1605–1632 (2012) arXiv:1206.5319

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Kajiura H.: Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19(01), 1–99 (2003) arXiv:math/0306332

    Article  MathSciNet  MATH  Google Scholar 

  21. Khudaverdian H.M.: Semidensities on odd symplectic supermanifolds. Commun. Math. Phys. 247(2), 353–390 (2004) arXiv:math/0012256

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Lambe L., Stasheff J.: Applications of perturbation theory to iterated fibrations. Manuscripta Math. 58(3), 363–376 (1987) http://link.springer.com/10.1007/BF01165893. Accessed 22 Feb 2019

    Article  MathSciNet  MATH  Google Scholar 

  23. Markl M.: Loop homotopy algebras in closed string field theory. Commun. Math. Phys. 221(2), 367–384 (2001) arXiv:hep-th/9711045

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. McDuff D., Salamon D.: Introduction to Symplectic Topology. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  25. Mnev, P.: Discrete BF theory. arXiv:0809.1160

  26. Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Muenster K., Sachs I.: Quantum open-closed homotopy algebra and string field theory. Commun. Math. Phys. 321(3), 769–801 (2011) arXiv:1109.4101

    Article  ADS  MathSciNet  Google Scholar 

  28. Muenster K., Sachs I.: Homotopy classification of Bosonic string field theory. Commun. Math. Phys. 330(3), 1227–1262 (2012) arXiv:1208.5626

    Article  ADS  MathSciNet  Google Scholar 

  29. Schlessinger M., Stasheff, J.: Deformation theory and rational homotopy type. arXiv:1211.1647

  30. Pulmann, J.: S-matrix and homological perturbation lemma. Diploma thesis, Charles University in Prague, (2016). https://is.cuni.cz/webapps/zzp/detail/161931/?lang=en. Accessed 22 Feb 2019

  31. Schwarz A.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155(2), 249–260 (1993) arXiv:hep-th/9205088

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Shih W.: Homologie des espaces fibrés. Publications Mathématiques de l’IHES 13, 5–87 (1962)

    Article  MATH  Google Scholar 

  33. Zwiebach B.: Closed string field theory: quantum action and the Batalin–Vilkovisky master equation. Nucl. Phys. B 390(1), 33–152 (1993) arXiv:hep-th/9206084

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Prague, 186 75, Czech Republic

    Martin Doubek & Branislav Jurčo

  2. Section of Mathematics, University of Geneva, Geneva, Switzerland

    Ján Pulmann

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  1. Martin Doubek
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  2. Branislav Jurčo
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  3. Ján Pulmann
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Correspondence to Ján Pulmann.

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Communicated by C. Schweigert

Dedicated to the memory of Martin Doubek

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Doubek, M., Jurčo, B. & Pulmann, J. Quantum \({L_\infty}\) Algebras and the Homological Perturbation Lemma. Commun. Math. Phys. 367, 215–240 (2019). https://doi.org/10.1007/s00220-019-03375-x

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