Ising model and gauge

I understood cool things about gauge freedom today, during a beautiful blackboard talk by Himanshu Khanchandani (NYU), who was talking about the 2-d Ising model and how it relates to the continuum limit (which is a field theory, interestingly!). He showed that if you introduce certain kinds of linear defects into the lattice, the change to the Hamiltonian depends only on the locations of the endpoints of the line of linear defects. This is because there is a gauge freedom, which is that you can change the signs of the spin-spin interactions at a point, and also change the labeling of what constitutes the positive and negative local state. This leads to topological properties of defects. It's gorgeous! And maybe related to the problems we want to solve in machine learning with images and geometry.