Topological Terms and Invertible Responses

A local density can suggest a topological action without defining a single-valued quantum phase on every allowed configuration. This chapter tests global definition, coefficient quantization, boundary dependence, and the distinction between a background response and a dynamical topological theory.

Enter this chapter

Begin with When Is a Topological Term Well Defined? and its comparison of the different claims.

Helpful background. Characteristic Classes and Chern–Weil Theory and Anomaly Polynomials and Inflow supply characteristic classes and the bulk–boundary relation.

Topics

1. When Is a Topological Term Well Defined? — Tests whether the exponentiated action is independent of auxiliary choices.
2. Theta Terms, Periodicity, and Vacuum Sectors — Derives periodicity from the admitted topological-charge lattice.
3. Chern–Simons Actions and Level Quantization — Checks levels against large transformations and global structure.
4. Wess–Zumino and WZW Terms — Uses extension independence to constrain Wess–Zumino phases.
5. BF Couplings and Discrete Topological Data — Relates compact higher-form couplings to discrete global information.
6. Background Responses and Invertible Phases — States the additional locality, gluing, and inverse conditions of an invertible response.

How the ideas fit together

The quantum quantity is the exponentiated action. If two allowed extensions change an action by $2\pi$ times an integer, their phases agree; otherwise the proposed term is not independent of the extension. The five-term comparison separates global definition, quantization, response, and dynamical-theory claims.

Keeping a connection fixed produces a background response. Summing over that connection can instead create nontrivial state spaces and line operators. Metric independence of the local density cannot decide invertibility or the absence of topological order. See Freed 2014, §§ 2–3.

Review the chapter

If the allowed charge values are all integers, why is $e^{i\theta Q}$ invariant under $\theta\to\theta+2\pi$? What can change that conclusion?

Solution

The added factor is $e^{2\pi iQ}=1$. Enlarging the allowed global sectors to include fractional charges can enlarge the period, unless other discrete data transform simultaneously. The actual charge lattice and the full theory specification determine periodicity.

Where to go next

Continue to Topological Gauge Theories and Symmetry TFT to see what changes when the topological fields are dynamical.

References

• Daniel S. Freed, “Anomalies and Invertible Field Theories,” Proceedings of Symposia in Pure Mathematics 88 (2014), 25–45. DOI. Open PDF.