# $\theta$ term of anomaly related with topological insulators

I am reading Ludwig's paper "Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors", and in this paper, although I am clear how they get the descent equation which introduced the relationship between anomaly and the existence of topological insultor, I am confused about the theta-term they mentioned on section VA5, they said that the integral of anomaly polynomial $\Omega_{2n+2}$ corresponds to the $\theta$ term. Could anyone help and explain to me what is this theta-term referring to and where can I find information about it?

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I think you will enjoy this talk pirsa.org/10050088 . – Heidar Apr 12 '11 at 22:36

The $\theta$-term is also known as the axion term and it's simply the $F\wedge F$ term known to particle physicists. In a more condensed-matter-friendly language, $$\Delta {\mathcal L} = \theta\left( \frac{e^2}{2\pi h} \right) \vec B \cdot \vec E$$ I don't know the optimum starting point but you may begin with

http://arxiv.org/abs/1001.3179

and its followups and references. More generally, the $\theta$-term means the integral of the anomaly polynomial. Note that the anomaly polynomial is a nice gauge-invariant expression - but in higher dimensions. The actual anomaly in the original spacetime is related to it by several operations.

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Could you explain a little more how this is related to the Chern-Simon term? – huyichen Apr 12 '11 at 17:01
@huyichen: (under some assumptions) they are related by $\text d\omega_{CS}=F\wedge F$, which essentially says that such a term is most important on the boundary of the topological insulator. Here $\omega_{CS} = F\wedge A = \text dA\wedge A$ is the Abelian Chern-Simons term, which indicates that the electromagnetic response on the boundary is similar to Quantum Hall effect. – Heidar Apr 12 '11 at 22:53