As far as I know, all known quantum field theories have the same very broad structure: one gives some finite list of data in order to specify a particular QFT, then one uses some formalism to algorithmically extract various physical observables from that specifying data. (I suppose a philosopher might say that all physical theories follow this same basic pattern.)

But how this general framework works out looks very different for different types of QFTs:

  • A "standard," Lagrangian-based QFT is specified by a particular choice of Lagrangian density containing only renormalizable interaction terms. More concretely, the "input data" that determines a theory is a list of fields and a finite set of coupling constants between those fields. The observables that come out of this framework are primarily $n$-point correlation functions of the various fields. (Although not necessarily all possible correlation functions - for example, for gauge theories, only correlation functions of gauge-invariant quantities are physically observable. These correlation functions are not always the "final answer" - e.g. we might plug them into the LSZ formula to get scattering amplitudes instead. And sometimes we might want to answer questions that are not directly answered by correlation functions - e.g. the sign of the beta function, or whether a particular theory experiences a phase transition. But in principle, the correlation functions (directly or indirectly) determine all observable quantities.)
  • A conformal field theory superficially looks very different. We often do not write down any Lagrangian for a CFT, especially when working within the framework of the conformal bootstrap. "The full set of data and consistency conditions associated with a CFT is not known in general," as discussed here. But we believe that, at least for scalar fields on flat spacetime, the primary-field conformal weights $\Delta_i$ and the operator product expansion coefficients $f_{ijk}$ together form sufficient CFT data. (This CFT data must respect certain consistency constraints, like the crossing symmetry equation (which ensures the associativity of the OPE), and modular invariance in two dimensions. There may be other consistency constraints as well.) But just like in the Lagrangian case, the observables that come out of the framework are typically $n$-point correlation functions.
  • A topological field theory again looks very different. There are several different ways to formulate TQFTs: in terms of symmetric monoidal functors, or braided fusion categories, or modular tensor categories (that's seven different links). There are also Schwarz-type TQFTs, which tend to come up in condensed-matter theory, and Witten-type TQFTs, which tend to come up in high-energy theory. The basic idea is that a TQFT is a map from a spacetime topology to some topologically invariant complex number. Depending on the specific formulation, the initial data that specifies a particular choice of TQFT can be a set of $f$ symbols, or $S$ and $T$ matrices, etc., and the "observables" that come out are various topological invariants. There is usually no useful notion of $n$-point correlation functions.

Obviously, it is not at all clear how to combine these three very distinct notions into a single unifying conceptual framework. I have two related questions about the relationships between them:

  1. When used without qualification, the term "quantum field theory" usually refers to the Lagrangian-based formalism. We often intuitively think of the other two as special cases of this one. For example, we often think of a CFT as an RG fixed point of some non-conformal QFT, but we rarely write down the CFT Lagrangian explicitly. In fact, many CFTs have no known Lagrangian description at all. It is even suspected that some CFTs, like $(2, 0)$ superconformal field theory in six dimensions, have no possible Lagrangian description, although there is apparently no proven no-go theorem. Similarly, Schwarz-type TQFTs are often thought of as Lagrangian-based theories whose Lagrangians do not depend on the spacetime metric. In fact, one could perhaps think of all Schwarz-type TQFTs as also being CFTs, since any Schwarz-type TQFT's Lagrangian density trivially transforms conformally (i.e. $g_{\mu \nu}(x) \to \Lambda(x) g_{\mu \nu}(x)$) under arbitrary diffeomorphisms, since the metric does not appear at all! (Although again, in practice CFTs and TQFTs look very different.) Have either of the reasonable-seeming inclusions $\text{TQFT} \subset \text{CFT}$ or $\text{CFT} \subset \text{(Lagrangian QFT)}$ been proven or disproven? (This question is related to this one.)

  2. Do attempts to mathematically formalize QFT (e.g. the Wightman axioms, etc.) try to cover all these types of QFT's simultaneously? If not, is there any mathematical framework that unifies them? It is my understanding that certain topological QFT's have been made completely mathematically rigorous, but little progress has been made for Lagrangian-based QFTs. I'm not sure what the status of CFT is from the standpoint of mathematical rigor.

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    $\begingroup$ +1. A first: I learnt more from the question than I expect from answers. It's a summary of the subject, put far better than I ever could. Thank you for asking it. $\endgroup$ – user163104 Aug 5 '17 at 1:05
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    $\begingroup$ I don't feel like this is a real answer, but here are my thoughts on this. I believe the idea that these approaches are ultimately related can be misleading. For starters, Lagrangian QFT is not well-defined in most of the cases, so strictly speaking there isn't a theory. Bootstrap and TQFT are two different ways to provide a definition for the theory. Some special cases can be related to Lagrangian theories, or be approximately described by perturbation theory. But in general, there isn't a clear relation (why should there be?). P.S. Wightman axioms aren't satisfied by most CFT or TQFT. $\endgroup$ – Prof. Legolasov Aug 5 '17 at 11:22
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    $\begingroup$ @GennaroTedesco The whole point of the question is that in CFT, you usually don't start by defining a Lagrangian. As I mentioned, some CFTs are conjectured to not even have a Lagrangian description. And in TQFT, you usually don't calculate correlation functions (which naturally depend on the non-topologically-invariant spacetime interval), but rather topological invariants of the spacetime manifold. $\endgroup$ – tparker Aug 5 '17 at 16:36
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    $\begingroup$ @GennaroTedesco I don't think that giving one single example is a good argument that something is "usually" true. But if you don't like my use of the word "usually", you can mentally replace it with the word "sometimes". $\endgroup$ – tparker Aug 6 '17 at 1:28
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    $\begingroup$ @GennaroTedesco The "standard" QFT (KG, Dirac) with LSZ formulae (that you assumed being derived from Lagrangian formalisms) can as well be obtained without Lagrangian formalisms (see AQFT). This statement is incorrect. As a counterexample, consider 4-dimensional Yang-Mills theory. It has well-defined amplitudes at all loop orders, but an algebraic approach for it still hasn't been found. Just like that, there's a 6d (2,0) superconformal theory (mentioned in the question) that has been defined using the bootstrap, but for which no Lagrangian description has been found yet. $\endgroup$ – Prof. Legolasov Aug 6 '17 at 21:23

The three classes of QFTs you are referring to are distinguished by different symmetry assumptions (Poincare invariance, conformal invariance, and volume-preserving diffeomorphism invariance) and different background spacetimes (Minkowski, Riemann curve (or families of them), and arbitrary manifolds). Moreover, Wightman axioms only characterize the vacuum sector of a Poincare-invariant QFT.

Each set of assumptions leads to very different natural questions and constructions, hence different mathematical approaches. This explains why there is this diversity of approaches. In the light of this diversity, a uniform theory would be conceptually very shallow - too general to be restrictive and hence useful - and would break immediately into chapters distinguished by specific assumptions.

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    $\begingroup$ A few questions: (a) why must the diffeomorphisms preserve volume in order for a TQFT to remain invariant? (b) Diffeomorphism invariance naturally makes me think of general relativity (which is not just topological) rather than topological QFT. It seems to me that TQFT's are invariant under arbitrary homeomorphisms, which is a much stronger symmetry. Do you agree? (c) By "Riemann curve", do you mean "Riemann surface"? I believe that those are only the natural background spacetimes for the special case of 2D CFTs. $\endgroup$ – tparker Aug 7 '17 at 18:38
  • $\begingroup$ (d) Since the conformal and homeomorphism groups contain the Poincare group, the ground states of CFTs and TQFTs are Poincare-invariant, so shouldn't the Wightman axioms characterize them as well? $\endgroup$ – tparker Aug 7 '17 at 18:44
  • $\begingroup$ (e) Since the conformal group contains the Poincare group, under your definition any CFT should be a special case of a "standard" QFT. If indeed there are CFTs with no Lagrangian description, then by perturbing them to explicitly break the conformal symmetry while preserving the Poincare symmetry, we could presumably come up with a non-corformal, Poincare-invariant QFT with no Lagrangian description. How would we describe such a theory? $\endgroup$ – tparker Aug 7 '17 at 18:44
  • $\begingroup$ Following up on (c): it seems to me that in higher than 2D, the spacetime backgrounds for both "standard" QFTs and CFTs are the same (Minkowski spacetime). $\endgroup$ – tparker Aug 7 '17 at 18:46
  • $\begingroup$ @tparker: (a) I required volume preserving diffeomorphisms just to be cautious. It is not clear to me whether scale invariance is preserved or broken in TQFTs; I didn't study it closely enough. (b) In 2D general relativity is topological, I believe. In my view, topological field theories are just those wihout dynamical fields. (c) complex curve = real surface. (d) Possibly, but all questions look very different for different groups. Note that there are also many Galilei invariant nonrelativistic QFTs, for which Wightman's axioms are violated. Also for field theoies on curved backgrounds. $\endgroup$ – Arnold Neumaier Aug 8 '17 at 5:31

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