(2,0) Superconformal field theory in 6 dimensions

Question

What is known about the (2,0) theory in 6 dimensions. I've heard it a rather elusive theory which is at the heart of the Electric magnetic Duality. I heard that no one can write down the action for this theory. What is the history of this theory, how was it first found? And how does one study this theory if one cannot write the action principle. Any comments, pedagogical references will be very helpful.

Answer 1

It seems important that the actual definition of the 6d (2,0)-CFT is via AdS7/CFT6 holography. 

This deserve a reminder from time to time. In his String2014 "vision talk" Physical Mathematics and the Future Greg Moore in section 5.1 lists some ideas of how to capture the 6d (2,0)-SCFT, only to conclude (on p. 28) that

As stressed to me by Edward Witten, thanks to the AdS/CFT correspondence, we do have an understanding of the complete solution of the theory in the large rank limit for su(N) theories

In the limit in which the Chern-Simons terms in the dual 7d theory dominate this even becomes essentially a mathematical precise definition, where the partition function of the 6d theory is obtained from the geometric quantization of the 7d Chern-Simons theory in direct higher analogy to how the 2d WZW model arises from 3d Chern-Simons theory (CS3/WZW2 correspondence). And this is the way that in the abelian case and for just the 2-form sector Witten has studied the 6d theory since 

• Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)

• Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012,1998 (arXiv:hep-th/9812012)

Now, it is well known that the relevant 7d Chern-Simons term (obtained by KK-compactifying the 11d SUGRA Chern-Simons term) does not just contain the abelian piece used in these articles, but contains nonabelian correction terms and is, in addition, subject to the "flux quantization" constraint. Putting this together I think one deduces a nonabelian 7d Chern-Simons theory on nonabelian 2-form gauge fields whose geometric quantization ought to give the nonabelian generalization of the above articles. 

Comments

@UrsSchreiber The (2,0) in what exactly does it reffers? Is like 2 left movers? How many susy is there preserved?

Answer 2

You are right. These (2,0) theories are theories that do not admit a Lagrangian description in the UV (generally theories not admitting a Lagrangian description must be conformal field theories) and can be ADE classified. These theories are the highest dimension non-trivial CFTs we know. You can study such theories by making various types of compactifications string dualities and so on. 

The basic idea begins by considering strings whose tension varies. This variation should take place in a space $Z$ with two extra dimensions and then embed this space into a four-dimensional (more familiar) ambient space $X$. Let $\Sigma$ be a two-sheet cover of $Z$ (remember how $\Sigma$ is the Seiberg-Witten curve). Now, imagine that $Z$ is covered by these two sheets of $\Sigma$, separated in the additional directions of $X$. Furthermore let us consider a membrane that is extended along the 2 extra dimensions of $Z$ and 1time-direction (of $X$). We can denote these coordinates as $(z,x)$ obviously with $z\in Z, \, z \in X$. Of course, $x$ is a complex coordinate and the two sheets define $x_1(z)$ and $x_2(z)$. We can consider what the tension of the membrane that is hanging between the two sheets is. It turns out that the constant tension is given by $|dx|\wedge |d\log z|$ and this membrane can be thought of as a string whose tension is a function of $z$ and satisfies $$T_{\text{string}}(z)\geq \left| \lambda_1 - \lambda_2 \right|,$$ where $\lambda_i = \left| x_i \frac{dz}{z} \right|$. Now, it turns out (to be fair I do not know a lot about it) that in M-theory appear various kinf of higher-dimensional objects with similar properties. The typical example is to consider M-theory and of course M5-branes and M-2 branes (the only type of branes of M-theory) in the product space4 $\mathbb{R}^{3,1} \times X \times \mathbb{R}^3$. Put a brane on the subspace $\mathbb{R}^{3,1} \times \Sigma \times \{0\}$. Here, $\Sigma  \subset X$ while 0 is the origin of $\mathbb{R}^3$. The resulting theory is a 4d theory. Now put one M2-brane with endpoints on the M5-brane on the space $\mathbb{R}^{0,1} \times D \times \{0\}$. Here $\mathbb{R}^{0,1} \subset \mathbb{R}^{3,1}$ (i.e. the worldline of a particle in 4d space) while $D\subset X$ with boundary on $\Sigma$. If we consider the above system as a 6d one then we do get the 6d $\mathcal{N}=(2,0)$ theory, e.g. on $\mathbb{R}^{3,1} \times Z$. By going further, we find out that this theory has a spectrum with both electrically and magnetically charged objects originating from the M-brane. It turns out that htis theory has a self-duality of the form $G = \star_6 G$ where $(\star G)_{\alpha \beta \gamma}= \epsilon_{\alpha \beta \gamma \mu \nu \sigma} G^{\mu \nu \sigma}$ (analog of the electric field in 6d). By placing the theory on a torus with coordinates $x_5,x_6$ you can consider strings wrapping these directions. Then, the field strength of the electric and magnetic fields are given by $F_{E \, \mu \nu} =  G_{6\mu \nu}$ and $F_{D \, \mu \nu } = G_{5\mu \nu}$. The duality arises from the previous duality between the $G$'s. Namely in 4d it is $F_{D} = \star_4 F_E$. Reminds you of something, right??

Ok, I 've written this in my break of this horrible and annoying calculation I am trying to do so sorry for not being more thorough. I am sure that Urs will have much more to say. The best place ot start ooking for references is the nLab article and the references therein. This reference might be interesting too. Also, I think Vafa (and maybe some other people) had nice talks on String 14 on the subject. Also try to look info for the very related AGT conjecture.