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It is conjectured that 6d (2,0) SCFT has no known description in terms of the action or the Lagrangian. However, it has many interesting compactifications for example 3d-3d correspondence which relates 3d Chern-Simons gauge theory with $\mathcal{N}=2$ 3D super Yang-Mills and many others.

  1. If we don't know the Lagrangian description, do we know for example the field content?

  2. Doesn't supersymmetry somehow constrain the possible interaction terms? And of course it is a CFT, so we know in principle how correlation functions look like, don't we? Do we know primary fields?

  3. How does one compactify such theory on for example 3D manifold like $S^3$?

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    $\begingroup$ You may also be interested that it has been conjectured that all lower dimensional CFTs are believed to arise from a set of 6D CFTs, the one you listed being one among them. $\endgroup$ – JamalS Jan 23 '17 at 20:04
  • $\begingroup$ That's right, I heard about that. But I'm mostly interested in the compactifications for now. $\endgroup$ – Caims Jan 23 '17 at 20:06
  • $\begingroup$ I don't understand your first sentence "It is conjectured that 6d (2,0) SCFT has no known description in terms of the action or the Lagrangian." Did you mean to include the word "known"? $\endgroup$ – tparker Aug 4 '17 at 22:24
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As you have mentioned, it is widely believed that the 6d (2,0) theories do not admit a conventional description in terms of fields and an action. Therefore, it doesn't really make sense to ask about their "field content" or their "interaction terms." Rather, like any abstract CFT, the (local) data defining such a theory consists of a list of local operators organized in unitary representations of the conformal algebra $\mathfrak{so}(6,2)$ and their OPE coefficients. Since the (2,0) theories are in fact superconformal (the result of combining supersymmetry and conformal symmetry), their local operators must actually organize into unitary representations of the larger superconformal algebra $\mathfrak{osp}(6,2|4)$.

It is believed that the (2,0) theories are (locally) uniquely labeled by a real Lie algebra $\mathfrak{g}$ (either $\mathfrak{u}(1)$ or a simple, compact ADE Lie algebra). $\mathfrak{g}=\mathfrak{u}(1)$ is a free theory of an abelian tensor multiplet, which does admit a usual Lagrangian description in terms of fields and an action (modulo some subtleties about the action of a self-dual tensor). For other $\mathfrak{g}$, the theories are isolated, strongly interacting SCFTs with no Lagrangian description. Little is known about their spectra of local operators (i.e. which $\mathfrak{osp}(6,2|4)$ representations actually define the theories), beyond the fact that they must include a conserved current multiplet including the stress-tensor.

Most of what is known about the (2,0) theories on QFT grounds is based on a) studying the low-energy theory on its moduli space, and b) compactifying to lower dimensions. Every known (2,0) theory has a moduli space of vacua which you may think of as a cone, with the conformal vacuum at the tip, and other points on the cone labeling vacua in which conformal symmetry is spontaneously broken (while supersymmetry is preserved). Moving onto the moduli space initiates an RG flow between the CFT at the origin and an IR free theory on the moduli space. At a generic point on the moduli space, the quantum field theory associated to this flow is described at low energies by an effective action of abelian tensor multiplets. When someone writes down a 6d action for a (2,0) theory, they typically mean such a moduli space effective action. Here it does make sense to ask about the allowed interactions of the low-energy fields. They are constrained by supersymmetry, conformal symmetry, R-symmetry, and anomaly cancellation. The constraints due to supersymmetry are called non-renormalization theorems.

The (2,0) theories may also be studied using QFT tools by compactifying to lower dimensions. In particular, when the (2,0) theory with Lie algebra $\mathfrak{g}$ is compactified on a circle, one obtains at low energies (far below the Kaluza-Klein scale) 5d super Yang-Mills theory with gauge algebra $\mathfrak{g}$ and gauge coupling proportional to the radius of the circle. By further compactifying one obtains a myriad of lower dimensional theories which have been extensively studied in recent years. Your question "how to compactify to 3 dimensions" would require a whole other discussion in its own right.

Everything I have said here is discussed in detail in the paper Higher Derivative Terms, Toroidal Compactification, and Weyl Anomalies in Six-Dimensional (2,0) Theories by Cordova, Dumitrescu, and Yin. See also The (2,0) superconformal bootstrap by Beem, Lemos, Rastelli, and van Rees.

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Let me add a few comments to the answer of user81003.

First of all, $6D$ ${\cal N}=(2,0)$ is not believed not to have a Lagrangian description, but it definitely doesn't have it, which follows from, say, its large $N$ behavior. Namely, the entropy scales as $N^3$, and it excludes the Lagrangian description.

Second, despite the fact that there is no Lagrangian and thus, yes, no interaction term, the notion of field content of the theory is well-defined, and the correlation functions of these fields (of course, heavily restricted by superconformal symmetry) are basically an output of the theory, along with the Wilson surfaces. To be precise, it is the $A_{N-1}$ gauge theory with a single non-Abelian supermultiplet consisting of a 2-form field with a self-dual field strength, five real scalars and fermions.

Finally, this theory is an effective theory living on the worldvolume of a stack of M5 branes, and it has a gravity dual (11D SUGRA on $AdS_7 \times S^4$) which is one of the most powerful instruments to study the theory, which is very important due to its non-Lagrangian nature. In particular, this understanding is very useful for studying compactifications of the theory.

A very extensive list of references (with a lean towards mathematical aspects of the subject) could be found in this article on ncatlab. A short review of related holography is given here. A stringy-motivated discussion of the topic is presented in "D-branes" by Johnson.

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  • $\begingroup$ If I may ask, how do you perform large N expansion of this theories? Since you don't have a Lagrangian you can't for example rescale couplings and fields or you don't even have a $\beta$ function (or do you? if yes, what's the use?). This may seem like a stupid question but I really have no clue. $\endgroup$ – Caims Jan 24 '17 at 19:40
  • $\begingroup$ @Caims I'm not sure if it can be done on the gauge theory side, but it is quite an elementary computation on the gravity side using AdS/CFT correspondence. You can find the derivation of the result after formula (6.1) in this review I already mentioned. $\endgroup$ – Andrey Feldman Jan 24 '17 at 20:35

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