Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 110, 34 pages      arXiv:2307.07497      https://doi.org/10.3842/SIGMA.2026.110

The Geometric Deformation of Curved $L_\infty$ Algebras and Lie Algebroids

Xiaoyi Cui
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, P.R. China

Received November 26, 2025, in final form October 10, 2026; Published online November 09, 2026

Abstract
While $L_\infty$ algebras are fundamental structures in differential geometry and mathematical physics, the geometric information encoded in such structures is often implicit. We address the following question: What constitutes a geometrically meaningful deformation of an $L_\infty$ algebra arising from vector bundles, and how can such deformations classify new geometric invariants? Inspired by nonabelian extension theory of Lie algebras, we define geometric deformations of curved $L_\infty$ algebras constructed from a vector bundle $V\to M$, and demonstrate that such deformations uniquely correspond to Lie algebroid structures on $V$. Explicit computations reveal that a representative of the first Atiyah-Chern class is exact in the total Weil complex, with its primitive representing the modular class in Lie algebroid cohomology. In the case of action Lie algebroids, the leading-order Atiyah-Chern classes correspond to the equivariant Chern characters. Applications to BV theories show that the geometric deformations naturally generate Poisson sigma models. These results provide a coherent framework for deriving field theories from geometric deformations of $L_\infty$ algebras.

Key words: curved $L_\infty$ algebras; Lie algebroids; geometric deformation; Atiyah-Chern classes; modular class; Batalin-Vilkovisky formalism.

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References

  1. Agore A.L., Militaru G., Extending structures for Lie algebras, Monatsh. Math. 174 (2014), 169-193, arXiv:1301.5442.
  2. Alexandrov M., Schwarz A., Zaboronsky O., Kontsevich M., The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12 (1997), 1405-1429, arXiv:hep-th/9502010.
  3. Arias Abad C., Crainic M., The Weil algebra and the Van Est isomorphism, Ann. Inst. Fourier (Grenoble) 61 (2011), 927-970, arXiv:0901.0322.
  4. Arias Abad C., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. 663 (2012), 91-126, arXiv:0901.0319.
  5. Bai C., Guo L., Ni X., Generalizations of the classical Yang-Baxter equation and $\mathcal O$-operators, J. Math. Phys. 52 (2011), 063515, 17 pages, arXiv:1302.0408.
  6. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Math. Wiss., Vol. 298, Springer, Berlin, 1992.
  7. Blaom A.D., Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc. 358 (2006), 3651-3671, arXiv:math/0404313.
  8. Brahic O., Zambon M., $L_\infty$-actions of Lie algebroids, Commun. Contemp. Math. 24 (2022), 2150013, 62 pages, arXiv:1708.06415.
  9. Calaque D., Campos R., Nuiten J., Lie algebroids are curved Lie algebras, Selecta Math. (N.S.) 32 (2026), 63, 90 pages, arXiv:2103.10728.
  10. Calaque D., Van den Bergh M., Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010), 1839-1889, arXiv:0708.2725.
  11. Cattaneo A.S., Felder G., A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611, arXiv:math/9902090.
  12. Cattaneo A.S., Felder G., Tomassini L., From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002), 329-352, arXiv:math/0012228.
  13. Cattaneo A.S., Mnev P., Reshetikhin N., Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 (2014), 535-603, arXiv:1201.0290.
  14. Cattaneo A.S., Mnev P., Reshetikhin N., Perturbative quantum gauge theories on manifolds with boundary, Comm. Math. Phys. 357 (2018), 631-730, arXiv:1507.01221.
  15. Chen Z., Stiénon M., Xu P., From Atiyah classes to homotopy Leibniz algebras, Comm. Math. Phys. 341 (2016), 309-349, arXiv:1204.1075.
  16. Chen Z., Xiang M., Xu P., Atiyah and Todd classes arising from integrable distributions, J. Geom. Phys. 136 (2019), 52-67, arXiv:1711.11253.
  17. Costello K., A geometric construction of the Witten genus, II, arXiv:1112.0816.
  18. Costello K., Gwilliam O., Factorization algebras in quantum field theory. Vol. 2, New Math. Monogr., Vol. 41, Cambridge University Press, Cambridge, 2021.
  19. Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681-721, arXiv:math/0008064.
  20. Crainic M., On the perturbation lemma, and deformations, arXiv:math/0403266.
  21. Crainic M., Fernandes R.L., Secondary characteristic classes of Lie algebroids, in Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys., Vol. 662, Springer, Berlin, 2005, 157-176.
  22. Dolgushev V., Covariant and equivariant formality theorems, Adv. Math. 191 (2005), 147-177, arXiv:math/0307212.
  23. Evens S., Lu J.H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. 50 (1999), 417-436, arXiv:dg-ga/9610008.
  24. Fedosov B.V., A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213-238.
  25. Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179, arXiv:math/0007132.
  26. Grady R., Gwilliam O., Lie algebroids as $L_\infty$ spaces, J. Inst. Math. Jussieu 19 (2020), 487-535, arXiv:1604.00711.
  27. Grady R., Li Q., Li S., Batalin-Vilkovisky quantization and the algebraic index, Adv. Math. 317 (2017), 575-639, arXiv:1507.01812.
  28. Gwilliam O., Grady R., One-dimensional Chern-Simons theory and the $\hat A$ genus, Algebr. Geom. Topol. 14 (2014), 2299-2377, arXiv:1110.3533.
  29. Kalkman J., BRST model for equivariant cohomology and representatives for the equivariant Thom class, Comm. Math. Phys. 153 (1993), 447-463.
  30. Kapranov M., Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), 71-113, arXiv:alg-geom/9704009.
  31. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, arXiv:q-alg/9709040.
  32. Kosmann-Schwarzbach Y., Magri F., Poisson-Lie groups and complete integrability. I. Drinfeld bialgebras, dual extensions and their canonical representations, Ann. Inst. H. Poincaré Phys. Théor. 49 (1988), 433-460.
  33. Kotov A., Strobl T., Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries, J. Geom. Phys. 135 (2019), 1-6, arXiv:1904.05809.
  34. Mehta R.A., Stiénon M., Xu P., The Atiyah class of a dg-vector bundle, C. R. Math. Acad. Sci. Paris 353 (2015), 357-362, arXiv:1502.03119.
  35. Seol S., Stiénon M., Xu P., Dg manifolds, formal exponential maps and homotopy Lie algebras, Comm. Math. Phys. 391 (2022), 33-76, arXiv:2106.00812.
  36. Vaintrob A.Y., Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), 428-429.
  37. Xiang M., Atiyah and Todd classes of regular Lie algebroids, Sci. China Math. 66 (2023), 1569-1592, arXiv:2110.04720.

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