Why is the symmetry group of the $A_{N-1}$ theory the $Osp(2,6|4)$?

+ 6 like - 0 dislike
212 views

Why is the symmetry group of the $A_{N-1}$ theory the $Osp(2,6|4)$? And how do we extract the fact that the bosonic subgroup is $Spin(2,6)\times Spin(5)$?

The $A_{N-1}$ theory is the resulting 6d SCFT made out of a stack of $N$ coincident $M5$-branes. The $Spin(5)$ is supposed to be the rotation group in the transverse space of the $M-5$-branes and gives the $R$-symmetry but this is the double cover of $SO(5)$ and I do not understand why the $R$-symmetry is not given by the $SO(5)$. A reference is this one.

+ 5 like - 0 dislike

The $A_{N-1}$ theory is a $N=(2,0)$ 6d superconformal theory. So its symmetry group is the $N=(2,0)$ 6d superconformal group. Let $G$ be this group and let us try to determine $G$. Let us first work at the level of Lie superalgebras. Let $\mathfrak{g}$ be the Lie superalgebra of $G$. As in any superconformal algebra, the bosonic part of $\mathfrak{g}$ is made of the ordinary conformal algebra and of the $R$-symmetry algebra.

It is well known that the conformal algebra of the $d$-dimensional Minkowski spacetime is $so(2,d)$ (this result plays for example a key role in $AdS/CFT$ because $so(2,d)$ is naturally the Lie algebra of the isometry group of $AdS_{d+1}$). In particular, in $6$ dimensions, it is $so(2,6)$. Remark that the 6d conformal algebra $so(2,6)$ naturally contains the 6d Lorentz algebra $so(1,5)$.

To go further, one has to understand the fermionic part. The 6d Lorentz algebra has Weyl spinors with $8$ real components. The $N=(2,0)$ supersymmetric algebra contains 16 real supercharges transforming in two copies of the same chirality of this Weyl spinor. Remark that there is an exceptional isomorphism of Lie algebras $so(1,5)=sl(2, \mathbb{H})$, where $\mathbb{H}$ is the field of quaternions, and the 8 real spin representation of $so(1,5)$ is simply the fundamental representation $\mathbb{H}^2$ of $sl(2, \mathbb{H})$.

The ferminonic part of $\mathfrak{g}$ contains not only these 16 real supercharges of the standard supersymmetry algebra, which square to a spacetime translation, but also conformal supercharges which square to special conformal transformations. Conformal supercharges form another copy of two Weyl spinors with respect to $so(1,5)$. In conclusion, the full fermionic part of $\mathfrak{g}$ has $16.2=32$ real components and is the copy of two Weyl spinors of the conformal algebra $so(2,4)$.

$R$-symmetry rotates the $N=1$ superalgebra inside the $N=2$ superalgebra. As the supercharges are in quaternionic representations, the $R$-symmetry algebra is $usp(4)$ i.e. the algebra of automorphisms of $\mathbb{H}^2$. There is an exceptional isomorphism of Lie algebras $usp(4)=so(5)$ (this in clear at the level of Dynkin diagrams: $B_2$ and $C_2$ are the same).

Finally we have found that the bosonic part of $\mathfrak{g}$ is $so(2,6)\oplus usp(4)=so(2,6) \oplus so(5)$. This superalgebra is called $osp(2,6|4)$: the "$o (2,6)$" notation signals the presence of $so(2,6)$ in the bosonic part and the "$sp( |4)$" notation signals that the $R$-symmetry is $usp(4)$.

To go from Lie algebras to Lie groups, one has to see what are the representations used. The superconformal algebra $so(2,6)$ acts on the supercharges through the spinor representation so the relevant group is the full $Spin(2,6)$. Similarly, the $R$-symmetry acts on the supercharges through the fundamental representation $\mathbb{H}^2$ of $usp(4)$ which is a spinor representation of $so(5)$ so the relevant group is the full $Spin(5)$.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.