Weyl Groups and Cluster Structures of Families of Log Calabi-Yau Surfaces
Time: 2019-10-07
Published By: He Liu
Speaker(s): Yan Zhou (Peking University)
Time: 14:00-16:00 October 10, 2019
Venue: Room 29, Quan Zhai, BICMR
Cluster algebras were first introduced by Fomin and Zelevinsky as an algebraic framework to study total positivity and Lusztig'sdual canonical bases in semisimple Lie groups. Later, cluster structures are discovered in many other areas in mathematics - Teichmüller theory, quiver representations, quantum field theory... In the first half this talk, we will introduce cluster varieties from the perspective of the mirror symmetry program for log Calabi-Yau varieties as proposed by Gross-Hacking-Keel - also suggested by Abouzaid, Kontsevich-Soibelman, Siebert... In the second half of the talk, we will discuss the interplay of Weyl groups and cluster structures on families of positive log CY surfaces. On universal families, the action of Weyl groups gives us a simple, concept understanding of the Donaldson-Thomas transformation. Restricted to degenerate subfamilies, Weyl groups permute disjoint subfans in the cluster scattering diagrams and help us to construct a large class of log CY varieties with non-equivalent atlases of cluster torus charts, generalizing the well-known case of the moduli of PGL2-local systems on 1-punctured genus 1 Riemann surface.
