arXiv subjects
Publications and source records attributed to Atsuki Nakago.
Recently, Chang--Haiden--Schroll shows that the braid group action on full exceptional collections in a triangulated category is not transitive but has infinitely many orbits in general. Their proof is based on a geometric model and the theory of branched coverings such as Birman--Hilden theory. This paper provides a simple algebraic proof of their theorem.
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.