Speaker: Sergey Galkin
Title: Tropical del Pezzo surfaces and quantum cluster surfaces
Date: June 14, 2013, 9:30 - 10:30
Place: IRMA, Strasbourg
Given a tropical del Pezzo surface one may consider a tropical version of Fukaya-Oh-Ohta-Ono obstruction, the generating function for Maslov index 2 tropical discs. This potential is not invariant with respect to tropical modifications, and it depends on the choice of generic point: two such potentials are related by a birational change of coordinates prescribed by Auroux. Conjecturally the similar story happens with non necessary tropical varieties of any dimension, however in case of tropical surfaces one can assign q-coefficients to tropical discs by Block-Goettsche-Itenberg-Mikhalkin's recipe to deform the Laurent polynomials into functions on two-dimensional quantum torus. Thus mirror partners of del Pezzo surface have structure of (quantum) cluster varieties. For del Pezzo surfaces of degrees 1 and 2 not a single FOOO's potential have been computed so far, but the method above predicts what should they be equal to.