Seiberg Duality conjecture for star-shaped quivers and finiteness of Gromov-Witten thoery for D-type quivers

TL;DR

This paper proves the Seiberg duality conjecture for star-shaped quivers, showing the equivalence of Gromov-Witten theories for mutation-related varieties and establishing finiteness properties for D-type quivers.

Contribution

It demonstrates the validity of Seiberg duality for star-shaped quivers and proves the finiteness of Gromov-Witten theory for D-type quivers after finite mutations.

Findings

Gromov-Witten theories for mutation-related star-shaped quivers are equivalent.

D-type quivers return to their original Gromov-Witten data after finite mutations.

Finiteness of Gromov-Witten theory for D3-type quiver varieties.

Abstract

This is the second work on Seiberg Duality. This work proves that the Seiberg duality conjecture holds for star-shaped quivers: the Gromov-Witten theories for two mutation-related varieties are equivalent. In particular, it is known that a $D$-type quiver goes back to itself after finite times quiver mutations, and we further prove that Gromov-Witten theory together with k\"ahler variables of a $D_3$-type quiver variety return to the original ones after finite times quiver mutations.