# Spin-statistics theorem in axiomatic (Wightman) CFT

+ 2 like - 0 dislike
124 views

In Wightman QFT one can prove spin-statistics theorem rigorously.  I was wondering, whether a similar result holds in 2D CFT, namely, that $h,\bar{h}\in \mathbb{Z}$ for bosons and $h, \bar{h}\in\frac{1}{2}\mathbb{Z}$ for fermions?

Or put it differently, since we know that for spin $s$ it holds $s\in\mathbb{Z}$ or $s\in\frac{1}{2}\mathbb{Z}$ by spin-statistics theorem, do we also now that the scaling dimension $d$ is also either $d\in\mathbb{Z}$ or $d\in\frac{1}{2}\mathbb{Z}$? Then the result would follow from $d=h+\bar{h}$ and $s= h-\bar{h}$.

asked Mar 26, 2016

In 2D there is no spin-statistics theorem. This is because of the Boson-Fermion correspondence. In 2D one often has also nonstandard statistics described in terms of braid groups rather than permutation groups. scaling dimensions need not be half-integral.

@ArnoldNeumaier Thank you for reminding me of this. I have actually forgotten of this, and now I feel dumb. But still, if I take a book on CFT, e.g. Blumenhagen, Plauschinn "Intro to CFT", their example of a free boson on a cylinder on p. 46 shows that the associated fields have dimensions $(h,\bar{h}) = (1,0)$ and $(h,\bar{h})= (0,1)$. Moreover, on p. 58 they show that a free fermion has dimensions $(h,\bar{h}) = (1/2, 0)$ and $(h,\bar{h})= (0,1/2)$. The central charges of the theories are $c=1$ and $c=1/2$, respectively, which is also suggestive. Furthermore, a mathematical text like Kac "Vertex algebras for beginners" also introduces a supercommutator on p. 6 "to include fermions".

More precisely, I am mostly interested in such statement: if we use only commutators, does this imply that $h$ and $\bar{h}$ (or $d$ and $s$ as given above) are integers?

## Your answer

 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$\hbar$ysic$\varnothing$OverflowThen 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.