I've heard that there is a belief that interacting conformal field theories do not exist in dimensions greater than 6, and in 6D the only known nontrivial CFTs are superconformal field theories. What is the argument that these theories can't exist in higher dimensions? I have a sense that 10D is rather special, and I wonder if there could exist a superconformal field theory in 10 spacetime dimensions.

An explanation of the general argument for/against higher dimensional (super)conformal field theories, and/or links to references discussing this would be very helpful.

  • $\begingroup$ There are no superconformal field theories in $D>6$ as $D=6$ is the maximal dimension in which the superconformal algebra exists. There is even a further conjecture that there are no unitary conformal field theory in $D>6$, but this has yet to be proven. $\endgroup$
    – Prahar
    Apr 23, 2017 at 21:47

1 Answer 1


As said in the comments, there are no superconformal field theories in $D>6$ dimensions. The references for this result are

  • Werner Nahm, Supersymmetries and their Representations, Nucl.Phys. B135 (1978) 149.
  • Shiraz Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2, 781 (1998) (arXiv:hep-th/9712074).
  • $\begingroup$ So even classically there is no superconformal algebra in $d>6$? Is there a simple argument why? Would it be possible to summarize the core of the proof? $\endgroup$
    – Kvothe
    Mar 8, 2022 at 10:56

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