Holomorphic disk potential for exotic monotone Lagrangian tori in CP2
Symplectic geometry and topology
05 November 15:30 - 16:30
Grigory Mikhalkin - University of Geneva
In their 2010 IPMU preprint Galkin and Usnich have suggested a construction of an infinite family of mutations (birational coordinate changes) of the potential x + y + 1/xy into some other Laurent polynomials. They associated such mutations to the so-called Markov triples (integer a,b,c with the property a^2+b^2+c^2=3abc). In their turn, these Markov triples correspond to the weighted projective planes P(a,b,c) that admit smoothing to the ordinary plane CP2=P(1,1,1). In the same preprint Galkin and Usnich have conjectured that the resulting Laurent polynomials correspond to the holomorphic disk potentials of an (infinite) family of exotic monotone Lagrangian tori in CP2. In its weaker version (correspondence on the level of Newton polygons of the potentials) this conjecture was proved in the 2014 preprint of Renato Vianna. This result already implies existence of infinitely many of distinct monotone Lagrangian tori. The talk will present a joint work with Sergey Galkin establishing the original version of the conjecture, i.e. also taking into account correspondence on the level of the coefficients of the potentials.
University of Cambridge