Anomalies on a lattice and homotopy theory

Speaker: Anton Kapustin (California Institute of Technology)
Series: Isaac Newton Institute Seminar Series
Note: MMBW03 - Structure of entanglement

When a group G acts on a quantum spin system by circuits, one may ask it this action can be conjugated to an on-site action. Else and Nayak showed that in 1d there may be an obstruction to doing so described by a class in degree-3 group cohomology of G. More recently, similar obstructions have been constructed for group actions on 2d quantum spin systems. In this talk I will describe an interpretation of these invariants as obstructions to gauging a global symmetry. To accomplish this, I will define what “gauging a symmetry” means in mathematical terms, in a way which applies both to quantum lattice systems and QFT. A meta observation is that localizable symmetries are described by a group-theoretic gadget: a crossed precosheaf of groups. A non-abelian analog of Cech homology can then be used to define homotopy-theoretic obstructions to localizing a symmetry to regions. In one and two dimensions, this reproduces the known obstructions to onsite-ability.