Previous |  Up |  Next

Article

Keywords:
F-manifolds; Frobenius manifolds; Lie algebroids
Summary:
In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.
References:
[1] Audin, M.: Symplectic geometry in Frobenius manifolds and quantum cohomology. J. Geom. Phys. 25 (1–2) (1998), 183–204. DOI 10.1016/S0393-0440(97)00026-0 | MR 1611969
[2] Crainic, M., Fernandes, R.L.: Lectures integrability Lie brackets. Geom. Topol. Monogr. 17 (2011), 1–107. MR 2795150
[3] David, L., Strachan, I.A.B.: Dubrovin’s duality for F-manifolds with eventual identities. Adv. Math. 226 (4) (2011), 4031–4060. DOI 10.1016/j.aim.2010.11.006 | MR 2770440
[4] Dotsenko, V.: Algebraic structures of F-manifolds via pre-Lie algebras. Ann. Mat. Pura Appl. (4) 198 (2019), 517–527. DOI 10.1007/s10231-018-0787-z | MR 3927168
[5] Dubrovin, B.: Geometry of 2D topological field theories. Lecture Notes in Math., vol. 1620, Springer, 1996. MR 1397274 | Zbl 0841.58065
[6] Dubrovin, B.: On almost duality for Frobenius manifolds. Amer. Math. Soc. Transl. 212 (2004), 75–132. MR 2070050
[7] Dubrovin, B.: WDVV Equations and Frobenius Manifolds. Encyclopedia of Mathematical Physics, vol. 1, Elsevier, 2006, pp. 438–447.
[8] Dufour, J.P., Zung, N.T.: Poisson Structures and Their Normal Forms. Birkhauser, 2000. MR 2178041
[9] Fernandes, R.L.: Lie algebroids, holonomy and characteristic classes. Adv. Math. 170 (1) (2002), 119–179. DOI 10.1006/aima.2001.2070 | MR 1929305
[10] Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge University Press, 2004. MR 1924259
[11] Hertling, C., Manin, Y.: Weak Frobenius manifolds. Internat. Math. Res. Notices 6 (1999), 277–286. DOI 10.1155/S1073792899000148 | MR 1680372 | Zbl 0960.58003
[12] Hitchin, N.: Frobenius manifolds. Gauge Theory and Symplectic Geometry, Springer, 1997. MR 1461570
[13] Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Springer, 2005. MR 2109686
[14] Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press, 2005. MR 2157566 | Zbl 1078.58011
[15] Manetti, M.: Lectures on deformation of complex manifolds. Rendiconti di Matematica 24 (2004), 1–183. MR 2130146
[16] Manin, Y.: Mirrors, functoriality, and derived geometry. arXiv:1708.02849.
[17] Manin, Y.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. Amer. Math. Soc. Colloq. Publ. 47 (1999), xiv+303 pp. MR 1702284 | Zbl 0952.14032
[18] Manin, Y.: F-manifolds with flat structure and Dubrovin’s duality. Adv. Math. 198 (1) (2005), 5–26. DOI 10.1016/j.aim.2004.12.003 | MR 2183247
[19] Manin, Y.: Grothendieck-Verdier duality patterns in quantum algebra. Izv. Ross. Akad. Nauk Ser. Mat. 81 (4) (2017), 158–166. MR 3682786
[20] Weinstein, A.: Linearization problems Lie algebroids and Lie groupoids. Lett. Math. Phys. 52 (2000), 93–102. DOI 10.1023/A:1007657920231 | MR 1800493
Partner of
EuDML logo