The number of full exceptional collections modulo spherical twists for extended Dynkin quivers

TL;DR

This paper determines the count of full exceptional collections modulo spherical twists for extended Dynkin quivers, revealing connections to Frobenius manifolds and mirror symmetry, and generalizing known formulas from the Dynkin case.

Contribution

It introduces a recursive formula for counting exceptional collections in extended Dynkin cases, extending Deligne's Dynkin formula with categorical and geometric interpretations.

Findings

Number of collections matches the degree of the Lyashko--Looijenga map.

Recursion generalizes Deligne's Dynkin case formula.

Results suggest links to stability conditions and mirror symmetry.

Abstract

This paper calculates the number of full exceptional collections modulo an action of a free abelian group of rank one for an abelian category of coherent sheaves on an orbifold projective line with a positive orbifold Euler characteristic, which is equivalent to the one of finite dimensional modules over an extended Dynkin quiver of ADE type by taking their derived categories. This is done by a recursive formula naturally generalizing the one for the Dynkin case by Deligne whose categorical interpretation is due to Obaid--Nauman--Shammakh--Fakieh--Ringel. Moreover, the number coincides with the degree of the Lyashko--Looijenga map of the Frobenius manifold for the orbifold projective line, which hints a consistency in some problems in Bridgeland's stability conditions and mirror symmetry.