# Question on Section 9.1.3 in “Conformal Field Theory” by Philippe Di Francesco et. al

Question on Section 9.1.3 in "Conformal Field Theory" by Philippe Di Francesco et. al.

The basic idea of the Coulomb-gas formalism is to place a background charge in the system, making the $U(1)$ symmetry anomalous. This has the effect of modifying the conformal dimensions of the vertex operators and the central charge[...]

Could anyone tell me what does he mean by $U(1)$ symmetry? I didn't see any $U(1)$ symmetry here in the context...

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The non-compact $U(1)\cong \mathbb{R}$ symmetry, which the book Conformal Field Theory by Philippe Di Francesco et. al. is referring to, is the translation symmetry

$$\varphi\to\varphi+a, \qquad a\in \mathbb{R},$$

of the main boson field $\varphi$ of the Coulomb-gas formalism, reflecting the zero-mode of the $\varphi$-field.

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Every time you write an equation identifying a compact group with a non-compact group, a fairy dies. :( –  user1504 Nov 18 '12 at 16:10
thanks. Can you give some reference for the term "non-compact U(1)"? I hope to read a clear demonstration of the physical content of this "symmetry". Now I am trying to obtain the conformal Ward's identity for this symmetry... –  Yunlong Lian Nov 18 '12 at 16:35
@YunlongLian: "Noncompact $U(1)$" just means $\mathbb{R}$. Same Lie algebra, and physicists usually deal with Lie groups by constructing Lie algebra generators. –  user1504 Nov 18 '12 at 17:21
@user1504 I agree but here I can't see any advantages for introducing the "U(1) symmetry". Now I consider it as an analogy between complex scalar field correlators and vertex operator correlators since they have very similar Ward's identity. Anyway I can proceed. Thanks for you two guys :D –  Yunlong Lian Nov 18 '12 at 19:12
@YunlongLian: You should read the $U(1)$ with a lower case, for the Lie algebra. Then the abuse of notation is corrected. We just have a single infinitesimal generator, this defines $u(1)$. There are two kinds of irreducible representations, the circles (compact) and the real line (noncompact). –  Arnold Neumaier Nov 18 '12 at 19:57
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