nLab Frobenius manifold

Idea

A Frobenius manifold [Dubrovin 1996] is a flat Riemannian manifold with compatible Frobenius algebra-structure on the fibers of its tangent bundle.

Related concepts

• quantum cohomology ring

References

The original articles:

• Boris Dubrovin, Geometry of 2d topological field theories, in: Integrable Systems and Quantum Groups, Lecture Notes in Mathematics 1620 (1996) [arXiv:hep-th/9407018, doi:10.1007/BFb0094793]

• Boris Dubrovin, Geometry and Analytic Theory of Frobenius Manifolds, Extra Volume ICM II (1998) 315–326 [arXiv:math/9807034]

Monograph with relation to quantum cohomology:

• Yuri Manin: Frobenius manifolds, quantum cohomology, and moduli spaces, Colloqium Publications 47, Amer. Math. Soc. (1999) [ams:]

See also:

• Wikipedia, Frobenius manifold

Generalization to supergeometry (Frobenius supermanifolds) and relation to quantum cohomology of complex projective space:

• Yuri I. Manin, Sergei A. Merkulov: Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$, Topol. Methods in Nonlinear Analysis 9 (1997) 107-161 [arXiv:alg-geom/9702014, doi:10.12775/TMNA.1997.006]

Frobenius manifolds in singularity theory, along with a generalization (F-manifolds):

• Claus Hertling: Frobenius manifolds and moduli spaces for singularities, Cambridge University Press (2002) [doi:10.1017/CBO9780511543104]

Obtaining Frobenius manifolds from W-algebras:

• Yassir I. Dinar: Frobenius manifolds from regular classical $W$-algebras, Advances in Mathematics 226 Issue 6 (2011) 5018-5040 [arXiv:1001.0611, doi:10.1016/j.aim.2010.12.024]

• Yassir I. Dinar: Frobenius Manifolds from Subregular Classical $W$-algebras, International Mathematics Research Notices 2013 12 (2013) [doi:10.1093/imrn/rns121]

• Yassir I. Dinar: Algebraic classical $W$-algebras and Frobenius manifolds, Lett Math Phys 111 115 (2021) [arXiv:1911.00271, doi:10.1007/s11005-021-01458-2]

Survey:

• Yassir Dinar: Dubrovin-Frobenius Manifolds from Classical $W$-Algebras, talk at CQTS @ NYU Abu Dhabi (Sep 2024) [slides: pdf]