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Homotopy Cartan calculus and inner deformations of \(A_\infty \)-algebras

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Abstract

We consider inner deformations of families of \(A_\infty \)-algebras. With the help of noncommutative Cartan’s calculus, we prove the invariance of Hochschild (co)homology under inner deformations. The invariance also holds for cyclic cohomology classes that satisfy some additional conditions. Applications to dg-algebras and QFT problems are briefly discussed.

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Notes

  1. Therefore, it is not just a minimal model of A.

  2. Notice that the graded Leibniz rules (hG1) and (hG7) are compatible with Koszul’s sign convention if one thinks of \(\cup \) as an object of degree 1.

  3. By definition, A[m] is a graded vector space with \(A[m]_n=A_{n+m}\).

  4. We provide the proof because that in [11] contains an unfortunate misprint.

  5. Recall that \(c^{j_1\ldots j_l}\) are symmetric in upper indices.

  6. For \(\Psi _0\), the deformation boils down to the trivial shift \(t\mapsto t+s\).

References

  1. Stasheff, J.: The (secret?) homological algebra of the Batalin-Vilkovisky approach. Contemp. Math. 219, 195–210 (1998)

    Article  MathSciNet  Google Scholar 

  2. Jurčo, B., Raspollini, L., Sämann, C., Wolf, M.: \(L_\infty \)-algebras of classical field theories and the Batalin-Vilkovisky formalism. Fortschr. Phys. 67(7), 1900025 (2019)

    Article  MathSciNet  Google Scholar 

  3. Zwiebach, B.: Closed string field theory: Quantum action and the B-V master equation. Nucl. Phys. B 390, 33–152 (1993), arXiv:hep-th/9206084 [hep-th]

  4. Kajiura, H.: Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007), arXiv:math/0306332 [math-qa]

  5. Kajiura, H., Stasheff, J.: Homotopy algebras inspired by classical open-closed string field theory. Commun. Math. Phys. 263(3), 553–581 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  6. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003), arXiv:q-alg/9709040 [q-alg]

  7. Sharapov, A., Skvortsov, E.: Formal higher spin gravities. Nucl. Phys. B 941, 838–860 (2019). arXiv:1901.01426 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  8. Sharapov, A., Skvortsov, E.: \(A_\infty \) algebras from slightly broken higher spin symmetries. JHEP 09, 024 (2019). arXiv:1809.10027 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  9. Maldacena, J., Zhiboedov, A.: Constraining conformal field theories with a slightly broken higher spin symmetry. Class. Quant. Gr. 30(10), 104003 (2013)

  10. Gerasimenko, P., Sharapov, A.A., Skvortsov, E.D.: Slightly broken higher spin symmetry: general structure of correlators. JHEP 2022, 1–31 (2022)

    Article  Google Scholar 

  11. Sharapov, A., Skvortsov, E.: Cup product on \(A_\infty \)-cohomology and deformations. J. Noncommut. Geom. 15(1), 223–240 (2021)

    Article  MathSciNet  Google Scholar 

  12. Gel’fand, I.M., Daletskii, Y.L., Tsygan, B.L.: On a variant of noncommutative differential geometry. Dokl. Akad. Nauk SSSR 308(6), 1293–1297 (1989)

    Google Scholar 

  13. Getzler, E.: Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology. In: Quantum deformations of algebras and their representations, vol. 7, pp. 65–78. Bar-Ilan University, Ramat-Gan (1993)

  14. Nest, R., Tsygan, B.: On the cohomology ring of an algebra. In: Advances in geometry, pp. 337–370. Springer (1999)

  15. Tamarkin, D., Tsygan, B.: The ring of differential operators on forms in noncommutative calculus. In: Lyubich, M., and Takhtajan, L. (eds.) Graphs and Patterns in Mathematics and Theoretical Physics, Proceedings of Symposia in Pure Mathematics, pp. 105–138. American Mathematical Society (2005)

  16. Dolgushev, V., Tamarkin, D., Tsygan, B.L.: Noncommutative calculus and the Gauss–Manin connection. In: Higher structures in geometry and physics, pp. 139–158. Springer (2011)

  17. Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78, 59–73 (1963)

    Article  MathSciNet  Google Scholar 

  18. Kadeishvili, T.: Cohomology \(C_{\infty }\)-algebra and Rational Homotopy Type, arXiv:0811.1655 [math.AT]

  19. Markl, M.: A cohomology theory for A(m)-algebras and applications. J. Pure Appl. Algebra 83(2), 141–175 (1992)

    Article  MathSciNet  Google Scholar 

  20. Lada, T., Markl, M.: Strongly homotopy Lie algebras. Commun. Algebra 23, 2147–2161 (1994)

    Article  MathSciNet  Google Scholar 

  21. Lada, T.: Commutators of \(A_\infty \) structures. Contemp. Math. 227, 227–233 (1999)

    Article  MathSciNet  Google Scholar 

  22. Gerstenhaber, M., Voronov, A.: Higher operations on Hochschild complex. Funct. Anal. Appl. 29, 1–6 (1995)

    Article  MathSciNet  Google Scholar 

  23. Cuntz, J., Skandalis, G., Tsygan, B.: Cyclic Homology in Non-Commutative Geometry. Springer, Berlin (2004)

    Book  Google Scholar 

  24. Fiorenza, D., Kowalzig, N.: Highe brackets on cyclic and negative cyclic (Co)homology. Int. Math. Res. Not. 2020(23), 9148–9209 (2018)

    Article  Google Scholar 

  25. Kadeishvili, T.: The structure of the A(\(\infty \))-algebra, and the Hochschild and Harrison cohomologies., Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR (1998)

  26. Stasheff, J., Lada, T.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993)

    Article  MathSciNet  Google Scholar 

  27. Kadeishvili, T.: On the homology theory of fibre spaces. Russ. Math. Surv. 35, 231–238 (1980)

    Article  Google Scholar 

  28. Getzler, E., Jones, J.D.S.: \(A_\infty \)-algebras and the cyclic bar complex. Ill. J. Math. 34, 256–283 (1990)

    MathSciNet  MATH  Google Scholar 

  29. Loday, J.-L.: Cyclic Homology. Springer, Cham (1998)

    Book  Google Scholar 

  30. Reinhold, B.: \(L_\infty \)-agebras and their cohomology. Emerg. Sci. 3, 4 (2019)

    Google Scholar 

  31. Tsujishita, T.: Homological method of computing invariants of systems of differential equations. Differ. Geom. Appl. 1, 3–34 (1991)

    Article  MathSciNet  Google Scholar 

  32. Bryant, R.L., Griffiths, P.A.: Characteristic cohomology of differential systems (I): general theory. J. Am. Math. Soc. 8, 507–596 (1995)

    MathSciNet  MATH  Google Scholar 

  33. Barnich, G., Grigoriev, M.: A Poincare lemma for sigma models of AKSZ type. J. Geom. Phys. 61, 663–674 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  34. Anderson, I.M., Torre, C.G.: Asymptotic conservation laws in classical field theory. Phys. Rev. Lett. 77, 4109–4113 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  35. Sharapov, A.A.: Variational Tricomplex. Global Symmetries Conser. Gauge Syst. SIGMA 12, 098 (2016)

    MATH  Google Scholar 

  36. Hood, C., Jones, J.: Some algebraic properties of cyclic homology groups, K-Theory 1 no. 4, (1987)

  37. Cieliebak, K., Volkov, E.: Eight flavors of cyclic homology. Kyoto J. Math. 61(2), 495–541 (2021)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work of A. Sh. was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘BASIS’. The work of E.S. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101002551). The results of Sect. 7 were obtained under exclusive support of the Ministry of Science and Higher Education of the Russian Federation (project No. FSWM-2020-0033).

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Authors and Affiliations

  1. Physics Faculty, Tomsk State University, Lenin Ave. 36, Tomsk, Russia, 634050

    Alexey A. Sharapov

  2. Service de Physique de l’Univers, Champs et Gravitation, Université de Mons, 20 place du Parc, 7000, Mons, Belgium

    Evgeny D. Skvortsov

  3. Lebedev Institute of Physics, Leninsky Ave. 53, Moscow, Russia, 119991

    Evgeny D. Skvortsov

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Correspondence to Alexey A. Sharapov.

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Sharapov, A.A., Skvortsov, E.D. Homotopy Cartan calculus and inner deformations of \(A_\infty \)-algebras. Lett Math Phys 112, 61 (2022). https://doi.org/10.1007/s11005-022-01557-8

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