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A 6-dimensional (2,0) superconformal quantum field theory comes in Lie algebra A, D, E types. These theories do not have classical Lagrangian and are purely quantum.These theories on a torus compactification without twisting leads to the 4-dimensional ${\cal N}=4$ supersymmetric Yang-Mills theory of A, D, E, gauge group. The 6-dim theories are known to carry degrees of freedom proportional to the product of the dual Coxeter number of the group $h^\vee_G$ and the dimension of the group $d_G$.

In our recent study arXiv:2103.06044 of the black hole entropy in these 6-dim theories with a twisted circle compactification by outer-automorphism, we have encountered a very simple relation between the products $h^\vee_Gd_G$ of Lie groups.

For a group $G$ with outer-automorphism, one can associate the Langlands dual group $H^\vee$ of the subgroup $H$ obtained by outer-automorphism twisting of $G$. The relation we found is $$ h^\vee_{H^\vee} d_{H^\vee} = \frac{h^\vee_G d_G}{n_G} , $$ where $n_G=2,3,4$. $n_G$ is not exactly the order of outer-automorphism, which is 2 or 3.
The following table shows the above relation holds explicitly.

\begin{array}{||c|c|c|c|c|c||} G & h^\vee_G & d_G & n_G& H & H^\vee & h^\vee_{H^\vee} & d_{H^\vee}\\ \hline A_{2r-1} & 2r & 4r^2-1 & 2 & C_r & B_r & 2r-1 & r(2r+1)\\ A_{2r} & 2r+1 & 4r(r+1) & 4& C'_r & C'_r& r+1 & r(2r+1)\\ D_r & 2(r-1) & r(2r-1) & 2 & B_{r-1}& C_{r-1} & r & (r-1)(2r-1) \\ D_4 & 6 & 28 & 3 & G_2 & G_2 & 4 & 14 \\ E_6 & 12 & 78 & 2 & F_4 & F_4 & 9 & 52 \end{array}

For the 4-dimensional ${\cal N} =4$ supersymmetric Yang-Mills theory with the gauge group $C_r$ comes in two varieties depending on the range of the $\theta$ parameters. The S-duality leads to either gauge theory of its Langlands dual, $B_r$ or $C_r$.

One question is whether this relation has been noticed before. If so, in what context. Another is its possible implications in the representation theory.

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  • $\begingroup$ In the row of your table with $G=A_{2r}$, if $H=C_r$ shouldn't $H^\vee$ be $B_r$? EDIT: But anyways I didn't think you were really "allowed" to fold an automorphism that fixed an edge but reversed its vertices. E.g. in his note on folding Stembridge disallows this: math.lsa.umich.edu/~jrs/papers/folding.pdf. $\endgroup$
    – Sam Hopkins
    Commented Oct 7, 2022 at 14:30
  • $\begingroup$ For the folding of $A_{2r}$ is done physically in our work. I suspect that Stembridge's frame could be extended just enough to accommodate our result. $\endgroup$
    – Kimyeong Lee
    Commented Oct 9, 2022 at 13:47
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    $\begingroup$ The Langlands dual in the sense of 4d S-duality is a bit subtle for the $C_r$ case, depending on the range of the $\theta$ parameter. The 4d $C_r'$ theory is self-dual under the S-duality. $\endgroup$
    – Kimyeong Lee
    Commented Oct 9, 2022 at 13:49
  • $\begingroup$ $n_G$ appears as $c_\sigma$ in (3) of arxiv.org/abs/1404.4098v2 $\endgroup$
    – Tamas Hausel
    Commented Nov 4, 2023 at 11:07

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