# CFT in non-integer dimension

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?

• 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 . Apr 26, 2020 at 10:48

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