• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Two definitions of topological terms in field theory

+ 6 like - 0 dislike

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.

This post imported from StackExchange Physics at 2016-07-02 16:05 (UTC), posted by SE-user knzhou

asked Jun 15, 2016 in Theoretical Physics by knzhou (115 points) [ revision history ]
edited Jul 2, 2016 by Dilaton
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.

This post imported from StackExchange Physics at 2016-07-02 16:05 (UTC), posted by SE-user Prahar

1 Answer

+ 4 like - 0 dislike

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.

This post imported from StackExchange Physics at 2016-07-02 16:05 (UTC), posted by SE-user David Bar Moshe
answered Jun 16, 2016 by David Bar Moshe (4,355 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights