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Maybe not a very sensible question, but I would like to know, whether there exist topological field theories (TQFT) with propagating degrees of freedom, or, conversely, theories without propagating degrees of freedom, which are not topological?

By definition, the correlation functions and observables in TQFT do not depend on the smooth deformations of spacetime, and as far as I understand, there two kinds of topological field theories -

  1. Schwarz-type TQFT, when the action functional doesn't explicitly refer to metric (BF-theory, Chern-Simons theory)
  2. Witten-type TQFT, where there is a metric $g_{\mu \nu}$ in the action, but there is BRST-like operator $Q$, and the stress-energy tensor is exact $T_{\mu \nu} = \delta G_{\alpha \beta}$ for some tensor $G_{\mu \nu}$, from which one can also infer, that observables do not depend on metric.

As far as I understand, in topological field theory there is no notion of particle in the usual sense, and all observables under consideration are integrals over manifolds, such as Wilson loops, surface Integrals, and etc. We cannot think about plane waves, point particles, but can be there motion of some non-local object?

As for the converse, I do not see an apparent candidate for theory, build from fields in some 'ordinary' sense, which correlation functions do not depend on spacetime metric, for any common theory - with scalar fields, fermions, Maxwell or Yang-Mills, the change of metric would definetely affect the behaviour of correlation functions. But maybe there is some subtle or weird theory, which has the aforementioned properties?

Finally, some concrete example. There is Self-Dual Yang-Mills theory, introduced by Chalmers and Siegel https://arxiv.org/abs/hep-th/9606061, where the action in non-supersymmetric case is: $$ \int d^{4} x \ G \wedge F $$ Where $F$ is usual Yang-Mills field tensor, and $G$ is a anti self-dual Lagrange multiplier, which transforms in the adjoint of Lie group and enforces the self-duality condition on equation of motion. This theory looks like the $BF$-theory, which is apparantly topological, however, the equation $F = \star{F}$ refers to metric explicitly, so, for this theory its topologicaliness is not manifest, not Schwarz-type TQFT, and I have not seen any references, which treat it as Witten-type TQFT. On the other hand, there are no propagating degrees of freedom. I would be very grateful if someone clarified, whether this theory is topological or not?

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  • $\begingroup$ Example of a "topological" (in a sense of independence of the correlators wrt smooth transformations of spacetime, like you wanted) theory with propagating degrees of freedom: General Relativity. $\endgroup$ May 10, 2020 at 14:33

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I think in general the division of topological field theories into Witten-type and Schwarz-type is misleading. Better to say that these are two classes of examples. But really, there's a interesting variety of different possible sensitivities to spacetime structure.

At one extreme, you have 'normal' QFTs like Yang-Mills, which depend explicitly and delicately on the spacetime metric.

A little less complicated are the conformal QFTs, in which the observables are invariant under conformal transformations. These don't have particles in the usual sense, but they do have non-trivial metric dependence.

Then there's the Donaldson theory (a twisted N=2 SUSY gauge theory in 4d). The observables in this theory don't depend on the metric, but they do depend on the smooth structure of the manifold. (Aside: Donaldson theory has point-evaluation observables, but these observables can be moved through the spacetime without changing their expectations. So point-like observables aren't necessarily incompatible with a lack of propagating degrees of freedom.)

Another intermediate example is 2d Yang-Mills theory, in which the observables (Wilson loop expectations) depend upon the overall volume of the spacetime, but not on any more detailed metric structure.

There are also QFTs that depend on spin structures, on orientations, on complex structures, pretty much anything you can think of. (One famous example: Chern-Simons calculations may be metric independent, but they do depend on a choice of framing.) There's a whole field of mathematics devoted to the study of these things.

Then finally there's the 'classic' TQFTs like BF theory, that only see the spacetime as a topological manifold.

Focusing on just one manifold doesn't really do justice to topological field theories either. There's a much richer set of behaviors that arise when you ask how the observables depend on families of metrics/spin structures/decorated bordisms. It can happen that small changes of metric leave the observables invariant, but that doesn't guarantee that big changes leave them fixed. One can discover dependence on the topology of the space of metrics. This is what happens in Gromov-Witten and Donaldson theory. This richness is one of the virtues of Witten's twisting trick.


Regarding self-dual YM theory: There's no reason to expect that it's topological in the most extreme sense. You need a Hodge star operator to say that $G$ is anti-self-dual, so the definition of the theory depends on this choice. You can make this more explicitly by writing the integral as $\int G_- \wedge F_-$. I don't know off the top of my head how much of the metric one can recover from a Hodge star.

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  • $\begingroup$ Thanks a lot! Very informative and detailed answer! $\endgroup$ May 9, 2020 at 16:24

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