The study of topological phases of matter and the invariants that define them has become a central pursuit of condensed matter physics. In particular, crystalline systems are known to host a large set of topological invariants, but the physical response properties associated to them are still not fully understood. In this talk we describe how to construct a topological response theory that makes detailed predictions about a set of crystalline topological invariants; we focus on two of them, the 'discrete shift', and a quantized charge polarization. We show that these invariants can be extracted from a lattice model by measuring the fractional charge at lattice defects such as disclinations and dislocations. To illustrate our method, we study the Hofstadter model of spinless electrons in a background magnetic field, and show that these invariants lead to new 'Hofstadter butterflies' which significantly expand the known phase diagram of the model.

(Pizza and refreshments will be served after the talk.)