2
$\begingroup$

Currently I am reading the Di Francesco-Mathieu-Sénéchal textbook on conformal field theory. Above the equation 5.52, the author argues that the EM tensor should have scaling dimension 2 and spin 2. But why did the author then conclude that both the holomorphic and anti-holomorphic dimension of the tensor are 2? Isn't it that the holomorphic dimension is 2 while the anti-holomorphic one is 0?

Another question: why does the EM tensor transform like a primary field?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

I think it is just not very well written.

For the holomorphic part $T(z)$, we should have weights $ (h,\bar h) =(2,0)$, and for the anti-holomorphic part $\tilde T(\bar z)$, we should have weights $ (h,\bar h) =(0,2)$, so for each part, we should have a scaling dimension $2$, and a spin $\pm 2$

However, the holomorphic and anti -holomorphic parts of the Energy-momentum tensor are not generally primary fields (see $5.124$ p $136$, $5.121$ p $135$) because of a possible central charge.

Improve this answer
$\endgroup$
7
  • $\begingroup$ Hi, Trimok. I've just found an errata of the DF-M-S book. It is infact h=2 while h-=0. The original text is wrong. $\endgroup$
    – Li Xinghe
    Sep 14, 2013 at 11:11
  • $\begingroup$ A further question: If the holomorphic and anti-holomorphic part of the EM tensor are not generally primary fields, then why in the equation 5.52 does the author transform the EM tensor as it were a primary field? $\endgroup$
    – Li Xinghe
    Sep 14, 2013 at 11:14
  • $\begingroup$ @LiXinghe : In fact, the discussion in 5.52 implies the transformation $z \to w=\frac{1}{z}$. Now, looking at $5.125, 5.124$, you can calculate that the Schwartzian derivative {$w;z$} is zero, so, in fact, for this particular transformation, the energy-momentum behaves as a primary field, but this is an exception, it does not work with general transformations $w(z)$. $\endgroup$
    – Trimok
    Sep 14, 2013 at 12:03
  • $\begingroup$ Thanks, Trimok. I believe you would agree my opinion that this book is not well written. Here it makes me confused. $\endgroup$
    – Li Xinghe
    Sep 15, 2013 at 0:45
  • $\begingroup$ @LiXinghe : You opinion is very ... hard!. When you are a teacher or a scientific writer, it happens very often when you want to explain a particular point, that you cannot give all details, because this will bring at a too superior level for the reader or the listener, and waste a lot of time. So, you have to make choices, and it is not always easy. Maybe, one day, you will write a physics book, or prepare a physics lesson, and you will have to face to this kind of problem. By the way, if you are satisfied with my answer, feel free to check it. $\endgroup$
    – Trimok
    Sep 15, 2013 at 13:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.