Noncommutative Geometry Seminar

 It is generally believed that any phase of matter with an energy gap is described at low energies by a Topological Quantum Field Theory. It is also well-known that the classification of gapped phases of matter depends on whether the fundamental degrees of freedom are bosonic or fermionic. In theories with Lorenz invariance, statistics is correlated with spin, therefore it is natural to expect that gapped fermionic phases of matter are described by spin-TQFTs, i.e. by TQFTs which depend on a spin structure. But most models of condensed matter physics are defined on a lattice which breaks Lorenz invariance, and it is not clear when the spin-statistics relation continues to hold or even makes sense in this situation. I will discuss in some detail topological lattice models with fermions in low dimensions and argue that spin structure dependence is forced on us by locality and topological invariance. From the mathematical viewpoint, I explain a combinatorial construction of equivariant spin-cobordism invariants for low-dimensional manifolds.