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Semisimple Frobenius (super)manifolds and quantum cohomology of Pr
Topological Methods in Nonlinear Analysis. 1997. Vol. 9. No. 1. P. 107–161.
Merkulov S., Manin Y. I.
Keywords: Frobenius manifolds
Merkulov S., International Mathematics Research Notices 1998 No. 14 P. 727–733
Added: October 1, 2026
Basalaev A., Takahashi A., International Mathematics Research Notices 2022 Vol. 2022 No. 19 P. 14865–14922
For any triple of positive integers A′=(a′1,a′2,a′3) and c∈C∗, cusp polynomial fA′=xa′11+xa′22+xa′33−c−1x1x2x3 is known to be mirror to Geigle–Lenzing orbifold projective line P1a′1,a′2,a′3. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial fA′ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ...
Added: September 9, 2022
Basalaev A., Dunin-Barkowski P., Natanzon S., Journal of Physics A: Mathematical and Theoretical 2021 Vol. 54 No. 11 P. 1–25
We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to AN, DN and BN singularities. In the case of AN Frobenius manifolds our hierarchy turns out to coincide with the KP hierarchy; for BN ...
Added: March 19, 2021
Buryak A., Shadrin S., Letters in Mathematical Physics 2010 Vol. 93 No. 3 P. 243–252
In this note we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on ...
Added: October 5, 2020
Buryak A., Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 475–493
By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In ...
Added: May 22, 2020
Basalaev A., Buryak A., International Mathematics Research Notices 2021 Vol. 2021 No. 7 P. 5460–5491
A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types A and D that gives a solution of the open WDVV equations. For the A-singularity, the resulting solution describes ...
Added: April 21, 2020
Ionov A., Journal of Geometry and Physics 2019 Vol. 140 P. 125–130
We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F_{k,n} = ∑^n_{i=1} x^k_i invariant under the natural S_n-action. ...
Added: November 8, 2019
Basalaev A., Journal of Geometry and Physics 2018 Vol. 2014 No. 84 P. 73–86
We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an arbitrary field K⊂C via the theory of modular forms. By an arithmetic property of an elliptic curve Eτ defined ...
Added: February 26, 2019
Belavin A. A., Gepner D., Kononov Y., Theoretical and Mathematical Physics 2016 Vol. 189 No. 3 P. 1775–1789
We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the ...
Added: February 19, 2017
Dunin-Barkowski P., Norbury P., Orantin N. et al., Journal of the Institute of Mathematics of Jussieu 2019 Vol. 18 No. 3 P. 449–497
We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold. ...
Added: December 22, 2016
Ionov A., / Series arXiv "math". 2016. No. 1611.03962.
We apply the technique of the paper "The abelian/nonabelian correspondence and Frobenius manifolds" by I. Ciocan-Fontanine, B. Kim, C. Sabbah to construct Saito primitive forms for Gepner singularities. ...
Added: November 16, 2016