Frobenius Manifolds and a New Class of Extended Affine Weyl Groups of A-type (II)

Abstract

With the help of the local isomorphism between the Hurwitz space $M_{0;k-1,l-k-r+1,r-1}$ and the orbit space $\mathcal {M}_{k,k+r}(A_l)$, we will show the existence of a Frobenius manifold structure on the orbit space $\mathcal {M}_{k,k+r}(A_l)\setminus \Sigma _r$ of the extended affine Weyl group $\widetilde{W}^{(k,k+r)}(A_l)$ for $1\le k<k+r\le l$.

Keywords

Frobenius manifolds / Extended affine / Weyl groups

References

[1] Brini A, Carlet G, Romano S, Rossi P. Rational reductions of the 2D-Toda hierarchy and mirror symmetry. J. Eur. Math. Soc., 2017, 19(3): 835-880

[2] Chang X-K, Hu X-B, Li S-H. Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type. Nonlinearity, 2018, 31: 4746-4775

[3] Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups, pp. 120–348. Springer, Berlin, Heidelberg (1996)

[4] Dubrovin B, Zhang Y. Extended affine Weyl groups and Frobenius manifolds. Compos. Math., 1998, 111: 167-219

[5] Dubrovin B, Strachan IAB, Zhang Y, Zuo D. Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau-Ginzburg superpotentials. Adv. Math., 2019, 351: 897-946

[6] Li, S.-H., Yu, G.-F.: Rank shift conditions and reductions of 2-Toda theory arXiv:1908.08725v2

[7] Ma, S.L., Zuo, D.: Frobenius manifolds and a new class of extended affine Weyl groups of BCD-type. J. Geom. Phys. 180 (2022), Paper No. 104622, 26pp

[8] Milanov T, Shen Y, Tseng H-H. Gromov-Witten theory of Fano orbifold curves, gamma integral structures and ADE-Toda hierarchies. Geom. Topol., 2016, 20: 2135-2218

[9] Zuo, D.: Frobenius manifolds and a new class of extended affine Weyl groups of A-type. Lett. Math. Phys. 110, 1903–1940 (2020)

[10] Zuo, D.: Frobenius manifolds and orbit spaces of reflection groups and their extensions. Nonlinear systems and their remarkable mathematical structures. Vol. 3—contributions from China, 227–246, CRC Press, Boca Raton, FL, 2022

Funding

national natural science foundation of china(11671371)