A review of Lie superalgebra cohomology for pseudoforms

Cremonini, Carlo Alberto

Archivum mathematicum, Tome 58 (2022) no. 5, pp. 269-286

Abstract (VO)

This note is based on a short talk presented at the “42nd Winter School Geometry and Physics” held in Srni, Czech Republic, January 15th–22nd 2022. We review the notion of Lie superalgebra cohomology and extend it to different form complexes, typical of the superalgebraic setting. In particular, we introduce pseudoforms as infinite-dimensional modules related to sub-superalgebras. We then show how to extend the Koszul-Hochschild-Serre spectral sequence for pseudoforms as a computational method to determine the cohomology groups induced by sub-superalgebras. In particular, we show as an example the case of $\mathfrak{osp}(1\mid 4)$ and choose $\mathfrak{osp}(1\mid 2) \times \mathfrak{sp} (2)$ as sub-algebra. We finally comment on some physical applications of such new cohomology classes related to super-branes. The note is a compact version of [10].

Détail

DOI : 10.5817/AM2022-5-269

Classification : 17B56, 17B81
Keywords: Lie superalgebras; cohomology; pseudoforms; integral forms; infinite-dimensional representations

Bibliographie

[1] Achúcarro, A., Evans, J.M., Townsend, P.K., Wiltshire, D.L.: Super p-branes. Phys. Lett. B 198 (4) (1987), 441–446. | DOI | MR

[2] Baranov, M.A., Schwarz, A.S.: Multiloop contribution to string theory. JETP Lett. 42 (1985), 419–421. | MR

[3] Belopolsky, A.: New geometrical approach to superstrings. [arXiv:hep-th/9703183 [hep-th]].

[4] Bernstein, I.N., Leites, D.A.: Integral forms and the Stokes formula on supermanifolds. Funkt. Anal. Pril. 11 (1977), 55. | MR

[5] Cacciatori, S.L., Noja, S., Re, R.: The unifying double complex on supermanifolds. Doc. Math. (2022), 489–518. | DOI

[6] Castellani, L., D’Auria, R., Fré, P.: Supergravity and superstrings: A Geometric perspective. Singapore: World Scientific 1, 2, 3 (1991), 1375–2162. | MR

[7] Catenacci, R., Cremonini, C.A., Grassi, P.A., Noja, S.: Cohomology of Lie superalgebras: Forms, integral forms and coset superspaces. [arXiv:2012.05246 [hep-th]]. | MR

[8] Catenacci, R., Grassi, P.A., Noja, S.: Superstring field theory, superforms and supergeometry. J. Geom. Phys. 148 (2020), 103559. | DOI | MR

[9] Chevalley, C., Eilenberg, S.: Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63 (1948), 85. | DOI | MR | Zbl

[10] Cremonini, C.A., Grassi, P.A.: Generalised cocycles and super p-branes. [arXiv:2206.03394 [hep-th]].

[11] Cremonini, C.A., Grassi, P.A.: Pictures from super Chern-Simons theory. JHEP 03 (2020), 043. | DOI | MR

[12] Cremonini, C.A., Grassi, P.A.: Super Chern-Simons theory: Batalin-Vilkovisky formalism and $A_\infty $ algebras. Phys. Rev. D 102 (2) (2020), 025009. | DOI | MR

[13] Cremonini, C.A., Grassi, P.A.: Self-dual forms in supergeometry I: The chiral boson. Nuclear Phys. B 973 (2021), 115596. | MR

[14] Cremonini, C.A., Grassi, P.A., et alii, : In preparation.

[15] Duff, M.J.: The conformal brane-scan: an update. [arXiv:2112.13784 [hep-th]]. | MR

[16] Erler, T., Konopka, S., Sachs, I.: Resolving Witten’s superstring field theory. JHEP 12 (2014), 1550018. | MR

[17] Fiorenza, D., Sati, H., Schreiber, U.: Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields. Int. J. Geom. Methods Mod. Phys. 12 (2014), 1550018. | DOI | MR

[18] Frappat, L., Sorba, P., Sciarrino, A.: Dictionary on Lie Algebras and Superalgebras. Academic Press, 2000. | MR

[19] Friedan, D., Martinec, E.J., Shenker, S.H.: Conformal invariance, supersymmetry and string theory. Nuclear Phys. B 271 (1986), 93–165. | MR

[20] Fuks, D.: Cohomology of infinite-dimensional Lie algebras. Springer, New York, 1986. | MR | Zbl

[21] Hochschild, G., Serre, J.P.: Cohomology of Lie Aalgebras. Ann. of Math. 57 (2) (1953), 591–603. | DOI | MR

[22] Kac, V.G.: Lie superalgebras. Adv. Math. 26 (1977), 8–96. | DOI | MR | Zbl

[23] Koszul, J.L.: Homologie et cohomologie des algebres de Lie. Bull. Soc. Math. France 78 (1950), 65–127. | DOI | MR

[24] Lebedev, A., Leites, D.A., Shereshevskii, I.: Lie superalgebra structures in cohomology spaces of Lie algebras with coefficients in the adjoint representation. Amer. Math. Soc. Transl. Ser. 2 213 (2005), 157–172. | MR

[25] Lehrer, G.I., Zhang, R.B.: The first fundamental theorem of invariant theory for the orthosymplectic supergroup. Commun. Math. Phys. 349 (2) (2017), 661–702. | DOI | MR

[26] Lehrer, G.I., Zhang, R.B.: The second fundamental theorem of invariant theory for the orthosymplectic supergroup. Nagoya Math. J. 242 (2021), 52–76. | DOI | MR

[27] Leites, D.A.: Representations of Lie superalgebras. Theor. Math. Phys. 52 (1982), 764–766. | DOI | MR

[28] Manin, Y.I.: Gauge field theory and complex geometry. Springer, Berlin, 1988, Translated from the Russian by N. Koblitz and J. R. King. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 289. | MR

[29] Noja, S.: On the geometry of forms on supermanifolds. [arXiv:2111.12841 [math.AG]].

[30] Noja, S., Re, R.: A note on super Koszul complex and the Berezinian. Ann. Mat. Pura Appl. (4) 201 (2022), 403–421. | MR

[31] Ogievetskii, O.V., Penkov, I.B.: Serre duality for projective supermanifolds. Funct. Anal. Appl. 18 (1984), 68–70. | DOI | MR

[32] Penkov, I.B.: D-modules on supermanifolds. Invent. Math. 71 (1983), 501–512. | DOI | MR

[33] Scheunert, M., Zhang, R.B.: Cohomology of Lie superalgebras and of their generalizations. J. Math. Phys. 39 (1998), 5024–5061. | DOI | MR

[34] Su, Y., Zhang, R.B.: Cohomology of Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$. Proc. London Math. Soc. 94 (2007), 91–136. | MR

[35] Su, Y., Zhang, R.B.: Mixed cohomology of Lie superalgebras. J. Algebra 549 (2020), 1–29. | DOI | MR

[36] Sullivan, D.: Infinitesimal computations in topology. Publications Mathématiques de l'IHÉS 47 (1977), 269–331. | DOI | MR

[37] Verlinde, E.P., Verlinde, H.L.: Multiloop calculations in covariant superstring theory. Phys. Lett. B 192 (1987), 95–102. | DOI | MR

[38] Witten, E.: Notes on supermanifolds and integration. Pure Appl. Math. Quart. 15 (1) (2019), 3–56. | DOI | MR