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In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng.

via Zoom: https://harvard.zoom.us/j/94717938264

Kitaev’s quantum double models in 2D provide some of the most commonly studied examples of topological quantum order.  In particular, the ground space is thought to yield a quantum error-correcting code.  We offer an explicit proof that this is the case for arbitrary finite groups. Actually, a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region.  Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. This implies that Kitaev’s model satisfies both TQO1 and TQO2 conditions of Bravyi-Hastings-Michalakis, and so it is a topological order in the sense of B-H-M. We note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of this result. We also note that more recently Q. Yang and Z. Wang proved the same result for the more general class of Levin-Wen models, but their method of proof is very different.

via Zoom: https://harvard.zoom.us/j/779283357