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?


1 Answer 1


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.

  • $\begingroup$ 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

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