For some reason I am suddenly confused over something which should be quit elementary.

In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb C)/\mathbb Z_2$ invariance to have the form

$$\langle \phi_i(z)\phi_j(w)\rangle = \frac{d_{ij}}{(z-w)^{2h_i}}\delta_{h_i,h_j}.$$ So a necessary requirement for a non-vanishing two-point function is $h_i = h_j$. Now consider the Ghost System which contains the two primary fields $b(z)$ and $c(z)$ with the OPE's

$$T(z)b(w)\sim \frac{\lambda}{(z-w)^2}b(w) + \frac 1{z-w}\partial b(w),$$ $$T(z)c(w)\sim \frac{1-\lambda}{(z-w)^2}c(w) + \frac 1{z-w}\partial c(w).$$ These primary fields clearly don't have the same conformal weight for generic $\lambda$, $h_b\neq h_c$. However their two-point function is

$$\langle c(z)b(w)\rangle = \frac 1{z-w}.$$

Why isn't this forced to be zero? Am I missing something very trivial, or are there any subtleties here?

  • $\begingroup$ Comment to the question(v1): OP's last formula is indeed eqs. (5.107) and (5.109) in the textbook Di Franscesco et. al., CFT (up to an inessential normalization factor). $\endgroup$ – Qmechanic Nov 4 '12 at 19:31

1) Everything OP writes(v1) above his last equation is correct. The $bc$ OPE reads

$$ {\cal R}c(z)b(w) ~\sim~ \frac 1{z-w} ,$$

where ${\cal R}$ denotes radial ordering.

2) To calculate the two-point function

$$\langle c(z)b(w)\rangle $$

(which as OP writes must vanish if the conformal dimensions for $b$ and $c$ are different) is more subtle due to the presence of the ghost number anomaly, i.e. the vacuum should be prepared with certain modes of the $bc$ system, see e.g. Polchinski, String Theory, Vol. 1, Sections 2.5-2.7.

  • $\begingroup$ I am very sorry for this late response. For some reason I completely missed this answer, thought nobody had replied. I am not sure I can follow, the discussion in Polchinski is based on applying the ghost system in superstring theory and this is why the vacuum needs to be prepared with some $bc$ modes right? I am more concerned with this CFT by itself (using it for something different than string theory). In di Francesco section 5.3.3 $\langle c(z)b(w)\rangle$ is derived directly from the action, without preparing the vacuum with any modes. $\endgroup$ – Heidar Nov 4 '12 at 1:12
  • $\begingroup$ Let me give another example which (might be) related to the concern I have, but here there is for sure no ghost number anomaly. Take the usual free fermion CFT (section 5.3.2 di Francesco) where $\psi(z)$ is a (chiral) primary with $h=\frac 12$. The two-point function is given by $\langle\psi(z)\psi(w)\rangle\propto \frac 1{z-w}$. The field $\partial\psi(z)$ is a decedent field with conformal weight $h=\frac 32$. From the basic theorem cited in the question one would expect $\langle\partial\psi(z)\psi(w)\rangle$ to vanish since they don't have equal conformal weights. (continued) $\endgroup$ – Heidar Nov 4 '12 at 1:19
  • $\begingroup$ But as is typically written in CFT books (for example di Francesco section 5.3.2), this is just given by differentiation and is equation to $\langle\partial\psi(z)\psi(w)\rangle \propto \frac 1{(z-w)^2}$. This two-point function cannot respect special conformal transformations since they usually demand that $h_1 = h_2$. $\endgroup$ – Heidar Nov 4 '12 at 1:23
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    $\begingroup$ @Heidar: The resolution to your example seems to be that the decedent field $\partial\psi(z)$ is not a quasi-primary field. $\endgroup$ – Qmechanic Nov 4 '12 at 17:19
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    $\begingroup$ @Heidar: Try to work directly from the definition of a quasi-primary field, see e.g. Di Franscesco et. al. eqs. (5.22-23). I believe that the decedent field $\partial\psi(z)$ fails to behave as a quasi-primary field under special conformal transformations. $\endgroup$ – Qmechanic Nov 4 '12 at 22:12

The answer seems to be that, technically at least, the two point function $\langle b(z) c(w) \rangle$ does vanish on the sphere. In the context of the standard $bc$ ghost system that shows up in bosonic string theory, the simplest nonzero correlation function on the sphere that involves both $b$ and $c$ is $$ \langle c(z_1) c(z_2) c(z_3) c(z_4) b(w) \rangle. $$ This correlation function is a special case of (6.3.5) in Polchinski volume 1 and evaluates to $$ \frac{(z_1-z_2)(z_1-z_3)(z_2-z_3)(z_1-z_4)(z_2-z_4)(z_3-z_4)}{(z_1-w)(z_2-w)(z_3-w)(z_4-w)}. $$

The Kronecker delta function $\delta_{h_i h_j}$ that shows up in the result for $\langle \phi_i(z) \phi_j(w) \rangle$ described in OP's first formula can be argued for based on the transformations of the fields under inversion $z \to 1/z$. While $\langle b(z) c(w) \rangle$ cannot transform properly, as OP has noted, it would be reassuring to see that this five-point function does transform in the right way. Indeed, the conformal factor $$ \frac{z_1^2 z_2^2 z_3^2 z_4^2}{w^4} $$ takes precisely the right form to convert this correlation function into one involving only $1/z_i$ and $1/w$.

(Note I am using the $SL(2, R)$ invariant vacuum state to compute these correlation functions. There is also the option of using so-called Q-vacua, some of which are obtained by acting with $c$ operators at the poles of the sphere. In this case, translation invariance is lost, and the correlation function $\langle \phi_i(z) \phi_j(w) \rangle$ can depend on a further dimensionless ratio $z/w$.)


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