Is it possible to have CFT in non-integer dimension?

At first sight, Wilson-Fisher fixed point in $4-\varepsilon$ gives such example.

What is known about such theories?

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    $\begingroup$ For a recent work on quantum field theory using spaces whose dimensions are not integer, including the case of non-integer spacetime dimension, see arxiv.org/abs/1911.07895 . $\endgroup$ Apr 26, 2020 at 10:48

1 Answer 1


CFTs at non integer values of the dimension $d$ can be defined by the axioms of crossing as all other CFTs. This is possible because the blocks are known analytically in $d$ $[1]$. However when $d$ is not an integer, these theories are necessarily non unitary, in that they must have states with negative norm. See $[2]$ and also $[3]$ for further details. In $[2]$, as you predicted correctly, they analyze the WF fixed point at $d = 4-\epsilon$.

The idea is that in non integer dimensions there exist additional operators called "evanescent" which vanish when $d$ hits an integer value. One example of such evanescent operators are the bilinears of fermions $$ \bar\Psi\, \Gamma_{\mu_1\cdots\mu_n}\Psi\,,\qquad \Gamma_{\mu_1\cdots\mu_n} = \gamma_{[\mu_1}\cdots\gamma_{\mu_n]}\,, $$ which, when $d \in \mathbb{N}$ vanish for $n>d$, but exist for any $n$ if $d \notin \mathbb{N}$. Those are precisely the operators that lead to a loss of unitarity.

$[1]$ Recursion Relations for Conformal Blocks J. Penedones, E. Trevisani, M. Yamazaki, 1509.00428

$[2]$ Unitarity violation at the Wilson-Fisher fixed point in 4-$\epsilon$ dimensions M. Hogervorst, S. Rychkov, B. C. van Rees, 1512.00013

$[3]$ Operator mixing in $\epsilon$-expansion: scheme and evanescent (in)dependence L. Di Pietro, E. Stamou, 1509.00428


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