Topological Terms and Phases of Sigma Models

TL;DR

The work develops a framework to analyze boundary conditions and anomalies for topological sigma models, generalizing ’t Hooft anomaly matching to sigma-model boundaries. It shows that invertible topological sigma models are classified by cobordism invariants $\Omega^m_{str}(X)$ and that boundary variations under homotopies are captured by a boundary term $\omega_1(h)$ with a non-flat connection whose curvature equals the bulk density, constraining RG flows of boundaries. Through explicit bosonic and fermionic examples with targets $S^2$ and $S^1$, it demonstrates how boundary degrees of freedom (e.g., Hopf-fibration couplings, chiral scalars) are necessary to preserve gauge invariance and realize anomalous boundaries. It further analyzes topological transitions in dynamical sigma models, showing how fermion parity and zero modes at critical points can drive changes in the bulk-boundary phase structure and suggesting a path to map phase diagrams using topological data.

Abstract

We study boundary conditions of topological sigma models with the goal of generalizing the concepts of anomalous symmetry and symmetry protected topological order. We find a version of 't Hooft's anomaly matching conditions on the renormalization group flow of boundaries of invertible topological sigma models and discuss several examples of anomalous boundary theories. We also comment on bulk topological transitions in dynamical sigma models and argue that one can, with care, use topological data to draw sigma model phase diagrams.