Nonlinear optical response is well studied in the context of semiconductors and has gained a renaissance in studies of topological materials in the recent decade. So far it mainly deals with non-magnetic materials and it is believed to root in the Be
rry curvature of the material band structure. In this work, we revisit the general formalism for the second-order optical response and focus on the consequences of the time-reversal-symmetry ($mathcal{T}$) breaking, by a diagrammatic approach. We have identified three physical mechanisms to generate a dc photocurrent, i.e. the Berry curvature, the quantum metric, and the diabatic motion. All three effects can be understood intuitively from the anomalous acceleration. The first two terms are respectively the antisymmetric and symmetric parts of the quantum geometric tensor. The last term is due to the dynamical antilocalization that appears from the phase accumulation between time-reversed fermion loops. Additionally, we derive the semiclassical conductivity that includes both intra- and interband effects. We find that $mathcal{T}$-breaking can lead to a greatly enhanced non-linear anomalous Hall effect that is beyond the contribution by the Berry curvature dipole.
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and henc
e the Calabi-Yau structure is unique. If the quiver is a cycle, we show that the algebra is a non-commutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the
