Bosonic Bc Ghost System

Use a separate Euclidean worldsheet for the radial ghost formulation. Its local coordinates are dimensionless $(t,\sigma )$, its complex coordinate is $w=t+i\sigma$, and its metric is $g=e^{2{\varphi }_E}(d{t}^2+d{\sigma }^2)$ for a specified smooth real function ${\varphi }_E$. The closed cylinder has $\sigma \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$; the open strip has $0\le \sigma \le \pi$. The coordinate convention relating this presentation to the Lorentzian conformal metric $h=e^{2\varphi }(-d{\tau }^2+d{\sigma }^2)$ is the formal substitution $\tau =-it$. This convention does not assume that arbitrary smooth Lorentzian fields admit analytic continuation: the Euclidean fields and ${\varphi }_E$ are independent formulation data. Where analytic continuation exists and the continued conformal factor is real and positive, this substitution gives the indicated Euclidean signature. Locally $z=e^w$ identifies the Euclidean cylinder with the punctured plane and the strip interior with the upper half-plane.

Fix a finite complex exterior algebra $\mathrm{\Lambda }$, whose odd elements anticommute. Off shell take smooth $\mathrm{\Lambda }$-odd sections $c\in \mathrm{\Gamma }(T{\mathrm{\Sigma }}_E\otimes \mathbb{C})\otimes {\mathrm{\Lambda }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$ and $b\in \mathrm{\Gamma }({\mathrm{Sym}}^2{T}^{\ast }{\mathrm{\Sigma }}_E\otimes \mathbb{C})\otimes {\mathrm{\Lambda }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$ with $g^{ab}{b}_{ab}=0$. With $\mathrm{\nabla }$ the Levi-Civita covariant derivative of $g$, define the local Faddeev-Popov action

On a noncompact domain the action is defined for compactly supported off-shell fields, or as a local action density. In a conformal coordinate $z=x+iy$, put $b=b_{zz}$, $c=c^z$, $\bar{b}=b_{\bar{z}\bar{z}}$, $\bar{c}=c^{\bar{z}}$, $\mathrm{\partial }=({\mathrm{\partial }}_x-i{\mathrm{\partial }}_y)\mathrm{/}2$, $\bar{\mathrm{\partial }}=({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)\mathrm{/}2$, and $d^{2}z=2dxdy$. The action is $(2\pi {)}^{-1}\int d^{2}z(b\bar{\mathrm{\partial }}c+\bar{b}\mathrm{\partial }\bar{c})$. Away from insertions its equations give $\bar{\mathrm{\partial }}b=\bar{\mathrm{\partial }}c=0$ and their antiholomorphic counterparts. Thus $b(z)d{z}^2$ is a quadratic differential and $c(z){\mathrm{\partial }}_z$ a vector field: under $z^{\mathrm{\prime }}=f(z)$ their components transform as $b^{\mathrm{\prime }}(z^{\mathrm{\prime }})=(f^{\mathrm{\prime }}(z){)}^{-2}b(z)$ and $c^{\mathrm{\prime }}(z^{\mathrm{\prime }})=f^{\mathrm{\prime }}(z)c(z)$, which defines the weights $2$ and $-1$.

Quantization here means adopting odd operator-valued formal Laurent series $b(z)={\sum }_{n\in \mathbb{Z}}{b}_n{z}^{-n-2}$, $c(z)={\sum }_{n\in \mathbb{Z}}{c}_n{z}^{-n+1}$, with $\{b_m,c_n\}={\delta }_{m+n,0}$ and $\{b_m,b_n\}=\{c_m,c_n\}=0$. Their radially ordered singular contraction is $b(z)c(w)\sim (z-w{)}^{-1}$, where $\sim$ denotes equality of singular parts. Normal ordering moves annihilators to the right with the signs from odd interchanges. Adopt the chiral stress tensor $T_{gh}=-2:b\mathrm{\partial }c:-:\mathrm{\partial }bc:$ with this ordering. These are algebraic local data, not a construction of a functional measure or an analytic continuation of every classical configuration.

The algebraic radial Fock domain uses finite mode products from the SL(2)-invariant vacuum $\mathrm{\mid }0⟩$, with $b_n\mathrm{\mid }0⟩=0$ for $n\ge -1$, $c_n\mathrm{\mid }0⟩=0$ for $n\ge 2$, $gh(\mathrm{\mid }0⟩)=0$, $gh(c_n)=+1$, $gh(b_n)=-1$, and normal ordering relative to that vacuum. Closed strings have independent left/right copies; open strings on the conformal strip $0\le \sigma \le \pi$ use boundary-preserving reparametrization ghosts. In strip light-cone components at each endpoint $\sigma =0,\pi$ impose $c^{+}(\tau ,\sigma )=c^{-}(\tau ,\sigma )$ and $b_{(++)}(\tau ,\sigma )=b_{(--)}(\tau ,\sigma )$; equivalently the normal ghost component $c^{\sigma }$ and the mixed antighost component $b_{(\tau \sigma )}$ vanish. In an upper-half-plane coordinate with the boundary on the real axis these conditions are $c(z)=\bar{c}(\bar{z})$ and $b(z)=\bar{b}(\bar{z})$ on $z=\bar{z}$. Doubling reflects the antiholomorphic field to the lower half-plane using these equalities. On the doubled strip the common chiral fields are 2pi-periodic, so there is one integer-moded bc system, rather than two independent endpoint systems.

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§5.1.2–5.2 (PDF file pp.119–124; printed pp.112–117); Faddeev–Popov determinant, ghost action, OPE, stress tensor, and weights. The Laurent modes, CAR, vacuum, and ghost-number conventions are adopted locally, not attributed to these sections.

Bosonic Polyakov Model is the adopted flat tensile Polyakov model. Conformal Gauge and Residual Symmetry fixes the local conformal gauge and its boundary-compatible residual transformations. The standard local Faddeev–Popov gauge-fixing calculation in Tong, §§5.1.2–5.2, motivates the displayed action and weights; the action, mode expansions, CAR, vacuum, and grading are the formulation data adopted here. This item does not assert a global determinant or the ghost central charge.