Criteria to Define a (Classical) Topological Field Lagrangian? + Conjecture

Question

I have a question concerning topological field theories. I'd rather keep the discussion at the classical level, so as to concentrate on the feature of topological evolution, which is what interests me here. Actually, I have more questions, but I want to make sure I'm making sense.

Are these statements correct/equivalent to each other?:

For a classical-field theory to be topological, if the theory admits a Lagrangian formulation:

1. The Lagrangian density must be a $0$-form / $n$-form;$^*$ $n$ being the dimension of the background manifold, $\mathcal{M}$.
2. The theory has as many degrees of freedom as constraints --however hidden from view upon 1st inspection.
3. The theory has no propagation.
4. The manifold doesn't have a metric, or the metric is not involved in the "dynamics".$^{**}$

(The above are more or less implied in the literature/lectures, etc.)

It seems clear to me that topological field theories are so highly constrained that fields are allowed to evolve only "minimally". But what does that mean in general? Solutions don't have to be "frozen", except for the particular case of static solutions. Could it be something like this?,

Conjecture:

A classical field has topological evolution $\Rightarrow$ the evolution can be solved into(*) a finite set of quasi-rigid motions; meaning an expansion in terms of a finite set of self-consistent moving "entities", like, $$ \varphi_{a}\left(x\right)=\varphi_{a}\left(t,\boldsymbol{x}\right)=\sum_{\alpha}\gamma_{\alpha}\left(t\right)f_{a}\left(\boldsymbol{x}-\boldsymbol{q}_{\alpha}\left(t\right)\right) $$ where the index $\alpha$ represents topology (number of "lobes", "nodes", "poles", and the like), and the sum being finite because otherwise you could represent a propagating solution by an infinite sum of this kind. Also, I've left aside the Poincaré-transformation properties for the time being.

(*) I'm not suggesting linearity; the dependence could be more complicated, e.g., $\sum_{\alpha,\beta}\gamma_{\alpha\beta ab}\left(t\right)f_{a}\left(\boldsymbol{x}-\boldsymbol{q}_{\alpha}\left(t\right)\right)f_{b}\left(\boldsymbol{x}-\boldsymbol{q}_{\beta}\left(t\right)\right)$, the $\gamma$'s appearing as powers, etc.

$^*$ Edit: Not involving differential operators --pending revision by others.

$^{**}$ Edit: Added point. Many thanks to @ChiralAnomaly. I hope I'm making better sense and I'm not too late.

Note: If you wish to consider a theory in which there is no metric, or sensible way to define time foliations, I suppose you may consider any other parameter $s$, instead of $t$, to describe curves on the manifold, and the corresponding diffeomorphism group. Again, I hope I'm making sense. Thank you @ChiralAnomaly.

Answer 1

The name "topological field theory" isn't always used the same way. The question asks about classical topological field theory, but this answer helps make the point by listing some of the many different ways people have used the name "topological quantum field theory."

For that reason, we're probably not going to find any one-size-fits-all definition of classical topological field theory, but maybe we can still find a one-size-fits-most definition. I'll suggest idea 4 in the question as a candidate for such a definition, and then I'll comment on the other ideas 1,2,3 that were listed in the question.

Here's one candidate, which is option 4 in the question: A (classical) topological field theory is one that doesn't use a metric structure.

How does this relate to the the other ideas 1,2,3 that were listed in the question? Without trying to be mathematically precise, here are some thoughts:

1. As it stands, statement 1 is always true, not just for topological theories. Only an $n$-form can be integrated over an $n$-dimensional manifold. If the manifold can be covered by a single coordinate patch, then we can write the $n$-form as $d^nx\,L$ where the $0$-form $L$ is the usual lagrangian density. As an example, the action for the free electromagnetic field $F_{ab}\equiv\partial_a A_b-\partial_b A_a$ with a background metric $g^{ab}$ in $n$-dimensional spacetime is $\int d^nx\ L$ with $L=\sqrt{|g|} g^{ab}g^{cd}F_{ac}F_{bd}$. Nobody calls this a topological field theory, because it depends on the prescribed metric $g^{ab}$, even though $L$ and $d^nx\ L$ can be regarded as a $0$-form and an $n$-form, respectively.

2. The right way to compare statements 2 and 4 is not clear to me, because without a metric structure, there is no distinction between time and space, and without that, the distinction between "equations of motion" and "constraints" is lost. (But see the example below...)

3. The right way to compare statement 3 and 4 is also not clear to me, because without a metric structure, there is no distinction between time and space, and without that, I don't know what words like "propagation/evolution/frozen/static" should mean. (But see the example below...)

Even though I'm not sure how to compare statements 2 and 3 to statement 4, they might turn out to be closely related (after we make them unambiguous, which I won't try to do). To see why, consider the example of general relativity in empty $2+1$-dimensional spacetime. GR involves a metric structure, so the no-metric criterion (statement 4) seems to say that it's not topological. However, in contrast to the situation in four- or more-dimensional spacetime, GR in three-dimensional spacetime does not have any propagating degrees of freedom, so it does qualify as topological according to option 3, and I think we can make a similar statement about option 2. Does this mean options 2,3 are not equivalent to option 4? Not necessarily, because GR in three-dimensional spacetime can be rewritten without using a metric structure. After rewriting it that way, it qualifies as topological according to the no-metric criterion (option 4).

Another idea, but I don't know how to make it work

For quantum field theory (QFT), a mathematically elegant definition is available, which I'll call the functorial definition. (I don't know if it has an established name.) We might try to adopt a similar definition for the classical case, but we immediately run into an obstacle that I don't know how to overcome. For inspiration, I'll record the thought here anyway.

The functorial definition was originally restricted to topological QFT (TQFT), but now it is applied also to other QFTs. I cited a few references in another question. That's why I'm mentioning it here: it's versatile enough to cover all of the different flavors of "topological," all the way up to thoroughly non-topological flavors.

The functorial definition is expressed in terms of category theory. Category theory is a foundational idea in mathematics that emphasizes relationships between similar mathematical structures more than the structures themselves. The idea is that instead of defining a QFT on a single spacetime background, we can define it simultaneously on all spacetime backgrounds in a way that emphasizes how the different backgrounds (and the corresponding QFTs) are related to each other. In jargon: a QFT is a functor from a category of spacetimes-with-boundaries (related to each other by bordisms) to a category of Hilbert spaces. A functor is a map from one category to another that plays nicely with their respective structures. The category of spacetimes-with-boundaries can carry various types of structure, such as a smooth structure, a spin structure, and even a metric structure, so we can use it for all kinds of QFT, including the various flavors of TQFT described in this answer, and presumably including "ordinary" QFTs like QED and QCD.

To adapt the functorial definition to classical field theory, we need to make a couple of adjustments. We can think of classical field theory as a special case of quantum field theory, namely the case where all observables commute with each other. To make that work, we need to allow the Hilbert space to be non-separable, which is usually disallowed in quantum theory for the reasons explained in this paper (also see this paper for a less-technical review). Adjusting the functorial definition to allow non-separable Hilbert spaces seems straightforward, but I don't know how to modify it to require that all observables commute with each other -- that is, to be exclusively about classical field theory. To me, this seems like a big obstacle, because the functorial definition doesn't distinguish between self-adjoint operators and observables. It's mathematically elegant, and I love it for that reason, but in applications of QFT to physics we need to specify which operators represent observables. Specifying "all self-adjoint operators" is not always appropriate even in QFT, and it's blatantly inappropriate in classical field theory, where all observables must commute with each other. For this reason, I don't know how to adapt the functorial definition of TQFT to the classical case, but maybe it's worth thinking about.