I've seen two distinct definitions for "topological" terms in the context of quantum field theory.

  1. Topological terms don't depend on the metric $g_{\mu\nu}$. This makes sense since topology is 'geometry minus length and angle'. (One consequence of this is that they don't contribute to the stress-energy tensor.)
  2. Topological terms are total derivatives, i.e. boundary terms. (One consequence of this is that they don't contribute at the classical level.)

Does the first statement imply the second? I can't see any clear reason for it to, but I've only seen examples where both are true.

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    $\begingroup$ Any total derivative term when integrated gives us a boundary term which depends only on the metric of the boundary and not of the metric of the bulk spacetime. In this sense a total derivative term is topological since it does not depend on the local structure of the bulk spacetime. $\endgroup$ – Prahar Mitra Jun 15 '16 at 7:56
  • $\begingroup$ Chern Simons theory is an counter example. It is independent of the metric, so belongs to the first class, while it is not a total derivative hence it does not belong to the second class. So the first class does not imply the second class. $\endgroup$ – user34104 Apr 18 '20 at 4:11

Topological terms of all types are always required not to depend on the metric, so their integrals will correspond to topological invariants, which serve as topological charges in quantum field theory.

However, it is important to distinguish between two the types of topological terms mentioned in the question, because they lead to different physical consequences. Please see the Deligne-Freed lectures on classical field theories.

The first type ($\theta$-terms) occurs when one takes a closed form on the target space of rank equal to the dimension of the base space $\mathcal{M}$:

$$ \omega(y) = \omega_{\alpha_1 …\alpha_n} dy^{\alpha_1}\wedge… dy^{\alpha_n}$$

pull it back to the base space and integrate:

$$\int_{\mathcal{M}}\omega_{\alpha_1 …\alpha_n} \frac{\partial y^{\alpha_1}}{ \partial x^{\beta_1}}… \frac{\partial y^{\alpha_n}}{ \partial x^{\beta_n}} dx^{\beta_1}\wedge… dx^{\beta_n}$$

The integration of this form does not require a metric.

An important subclass of this type of terms $\omega$ is a representative of a characteristic class (please see Nash and Sen section 7.22) of a fiber bundle over the target space. In this case, the topological term can be added to the Lagrangian on an even dimensional base space. $\theta$-terms are topological charges of instantons, and their inclusion in the Lagrangian is equivalent choosing a $\theta$-vacuum. Prototypes of this type of topological terms are the $\theta$- term of QCD and the winding number in the $\mathbb{C}P^1$ model.

The second type of topological terms constitute of pullbacks to the base manifold of secondary characteristic classes (please see Nash page 223). These classes live in odd dimensions. They are closed only when the gauge connection is a pure gauge. In this case they constitute of holonomies (Berry's phases) of gauge connections and higher versions of which in higher dimensions.

In contrast to characteristic classes which classify fiber bundles over manifolds, secondary characteristic classes classify flat fiber bundles. The prototypes of topological terms associated with secondary characteristic classes are the electromagnetic interaction term of a charged particle (in 1D) and the Chern-Simons term (in 3D). The pure gauge case corresponds to an Aharonov-Bohm potential in 1D and a Wess-Zumino-Witten term in 3-D.


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