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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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up vote 2 down vote accepted

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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