Notation of ghost fields $b$, $\tilde{b}$, $c$, and $\tilde{c}$ in Polchinski

Question

I am terrifically confused by the notation in Polchinski's string theory book from chapter 3 to chapter 4. The ghost action of the bosonic string in conformal gauge is (3.3.24) $$S = \frac{1}{2 \pi} \int d^2 z (b_{zz} \partial_{\overline{z}}c^z + b_{\overline{z}\overline{z}}\partial_zc^{\overline{z}}).$$ In chapter 4, he drops the indices and starts to refer to $b$, $\tilde{b}$, $c$, and $\tilde{c}$. Am I supposed to identify: \begin{align} b(z) &= b_{zz} \\ \tilde{b}(\overline{z})&=b_{\overline{z}\overline{z}}\\ c(z) &= c^{\overline{z}} \\ \tilde{c}(\overline{z}) &= c^z? \end{align} Then, with this, do the transformations such as (4.3.1c) become \begin{align} \delta_B c &= i \epsilon(c \partial + \tilde{c}\overline{\partial})c &= i \epsilon c \partial c \\ \delta_B \tilde{c} &= i \epsilon(c \partial + \tilde{c}\overline{\partial})\tilde{c} &= i \epsilon \tilde{c} \overline{\partial} \tilde{c}? \end{align} If this is not right, what is the precise functional dependence of $b$, $\tilde{b}$, $c$, and $\tilde{c}$, and how do they relate to the fields $b_{zz}, \, c^z \, b_{\overline{z}\overline{z}}, \, c^{\overline{z}}$?

Answer 1

You got your $c$'s mixed up. \begin{align} b(z,\bar{z})&:=b_{zz} \qquad &\bar{b}(z,\bar{z}) :=b_{\bar{z}\bar{z}}\\ c(z,\bar{z})&:=c^{z\phantom{z}} \qquad &\bar{c}(z,\bar{z}) :=c^{\bar{z}\phantom{\bar{z}}}\\ \partial &:=\partial_z \qquad &\bar{\partial}:=\partial_{\bar{z}}. \end{align} The action is $$S = \frac{1}{2\pi}\int\mathrm{d}^2 z\ \big(b\,\bar{\partial} c + \bar{b}\,\partial\,\bar{c}\big).$$ This is not trivially zero (in response to your comment in the original question), since $\bar{\partial}c=0$ is just the equation of motion for $c$; i.e. it is zero on-shell. This is common to all homogeneous actions (see e.g. this phys.SE answer).