Frobenius manifold for the nodal quiver
Atsuki Nakago, Yuuki Shiraishi, and Atsushi Takahashi
Journal of Singularities
volume 30 (2026), 255-299
Received: 5 September 2025. In revised form: 24 December 2025
Abstract:
Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. Given a deformation of the Weierstrass elliptic function, we construct a primitive form normalized to be compatible with the period map for integral cycles, and obtain a Frobenius structure whose Frobenius potential is defined over the rational numbers. We also construct a Frobenius structure using elliptic Weyl group invariants (as opposed to Jacobi group invariants), and establish an isomorphism between these two Frobenius structures. We further examine the relationship between the degree of the Lyashko--Looijenga map modulo the modular group and the number of full exceptional collections up to the braid group action and translations, as well as the associated Gamma-integral structure.
Author(s) information:
Atsuki Nakago
Department of Mathematics
Graduate School of Science
The University of Osaka,
Toyonaka
Osaka, 560-0043, Japan
email: u400778f@ecs.osaka-u.ac.jp
Yuuki Shiraishi
School of Economics and Management
University of Hyogo
Nishiku Kobe
Hyogo, 651-2197, Japan
email: s912y025@guh.u-hyogo.ac.jp
Atsushi Takahashi
Department of Mathematics
Graduate School of Science
The University of Osaka,
Toyonaka
Osaka, 560-0043, Japan
email: takahashi@math.sci.osaka-u.ac.jp