In Volume 1 of Polchinski's String Theory book, the author works with energy--momentum tensor of CFTs till chapter 3 in a normal ordered mode, i.e. $T(z)= : \text{something}: (z)$. However, in chapter3, specifically in section $3.7$ where he treats string in curved space-time, Polchinski seems to abandon the normal ordering thing, because in equation $(3.7.12)$ he works with

$$-2\alpha' T^a_a = \beta^G_{\mu \nu} g^{ab} \partial_a X^\mu \partial_b X^\nu + i\beta^B_{\mu \nu} \epsilon^{ab} \partial_a X^\mu \partial_b X^\nu + \alpha' \beta^\Phi R,\tag1$$

But shouldn't be this be

$$-2\alpha' T^a_a = \beta^G_{\mu \nu} g^{ab} :\partial_a X^\mu \partial_b X^\nu: + i\beta^B_{\mu \nu} \epsilon^{ab} :\partial_a X^\mu \partial_b X^\nu: + \alpha' \beta^\Phi R\tag2$$ where Ricci scale terms are all also normal ordered? Why he simply drop out the normal ordering symbols?


1 Answer 1


He dropped the normal ordering symbols because he has switched from conformal normal ordering to dimensional regularisation (corresponding to $\gamma=0$) which is more usual in this context. He also explains (in eqn 3.6.24) how the operators appearing in the two schemes are related.


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