= Solution
Use a mostly-plus <Minkowski metric> and <string tension> $T=(2\pi\alpha')^{-1}$. The <Polyakov action> is
$$
S_P=-\frac1{4\pi\alpha'}\int_\Sigma d^2\sigma\,\sqrt{-h}\,h^{ab}\partial_aX^\mu\partial_bX^\nu\eta_{\mu\nu}.
$$
Varying the auxiliary metric sets its <worldsheet stress-energy tensor> to zero and makes $h$ conformal to the induced metric. Substitution gives the equivalent <Nambu–Goto action> $-T\int\sqrt{-\det(\partial_aX\cdot\partial_bX)}$. The <string coupling> is separate from $T$. A constant <dilaton> contributes $S_\Phi=\Phi_0\chi(\Sigma)$ to the Euclidean <action>, by the <Gauss-Bonnet theorem>, including its boundary term when necessary. With $g_s=e^{\Phi_0}$, a connected <worldsheet> has weight $g_s^{-\chi}$. For an oriented closed surface of <genus> $g$, this is \b[$g_s^{2g-2}$]; with $b$ boundaries it is $g_s^{2g+b-2}$. Normalizing each external <closed string> vertex with a factor $g_s$ gives $g_s^{2g+M-2}$ for $M$ external <closed strings>. This is the <string genus expansion>.
The local <gauge symmetries> are <worldsheet diffeomorphisms>, under which $X$ is a <scalar field>, and <Weyl transformations> $h_{ab}\mapsto e^{2\omega(\sigma)}h_{ab}$. Two coordinate functions and one Weyl function locally fix the three components of the two-dimensional metric, giving <conformal gauge> $h_{ab}=e^{2\omega}\eta_{ab}$ and then $\omega=0$. Globally one instead chooses a fiducial metric and integrates over the <worldsheet moduli>; conformal Killing transformations can remain. Target-space Poincaré transformations are global symmetries, not additional local gauge freedoms.
For an infinitesimal coordinate change $v$, the trace-free metric variation is
$$
(P_1v)_{ab}=\nabla_av_b+\nabla_bv_a-h_{ab}\nabla_cv^c.
$$
After the Weyl trace is removed, gauge fixing produces the <Faddeev-Popov determinant> of this operator. An anticommuting <vector> $c^a$ and symmetric traceless <tensor> $b_{ab}$ represent that determinant through a <Grassmann Gaussian integral>. The resulting <worldsheet ghost action> is, up to field-normalization conventions,
$$
S_{bc}=\frac1{2\pi}\int d^2\sigma\sqrt{|\widehat h|}\,b^{ab}(P_1c)_{ab}.
$$
In complex coordinates this is a sum of chiral $b\bar\partial c$ and antichiral $\widetilde b\partial\widetilde c$ terms. The <worldsheet ghost fields> have <conformal weights> two and minus one. Their <central charge> is $-26$, while the $d$ free embedding <bosons> contribute $d$. In the critical dimension the total anomaly cancels, consistently with the stated nilpotence of the <BRST charge>.
Physical states are <BRST cohomology> classes at the appropriate <ghost number>: \b[$\ker Q/\operatorname{im}Q$], with $Q|\phi\rangle=0$ and $|\phi\rangle\sim|\phi\rangle+Q|\chi\rangle$. Nilpotence ensures that an exact state is closed and that the equivalence is well-defined. For the usual <open string> representatives the physical <ghost number> is one. <BRST-exact> excitations describe gauge or null directions rather than additional physical particles.
To calculate the <ghost oscillator Virasoro generators>, use the elementary fermionic identities
$$
[AB,C]=A\{B,C\}-\{A,C\}B
$$
when $A,B,C$ are odd. In the first commutator only $\{c_{-k},b_n\}=\delta_{k,n}$ survives, giving
$$
[L_m^{(b,c)},b_n]=\sum_k(m-k)b_{m+k}\delta_{k,n}=\boxed{(m-n)b_{m+n}}.
$$
In the second only $\{b_{m+k},c_n\}=\delta_{m+k+n,0}$ survives, giving
$$
[L_m^{(b,c)},c_n]=-\sum_k(m-k)\delta_{m+k+n,0}c_{-k}=\boxed{-(2m+n)c_{m+n}}.
$$
Normal-ordering constants commute with the <string oscillators> and do not change these results.
For the last request choose the <ghost oscillator vacuum> $|\downarrow\rangle$ with $b_{m>0}|\downarrow\rangle=c_{m>0}|\downarrow\rangle=b_0|\downarrow\rangle=0$. The given conditions restrict a general state to $|\phi\rangle=|\Psi\rangle\otimes|\downarrow\rangle$ with no nonzero ghost excitations. Indeed, the positive modes annihilate both members of each creation–annihilation pair, and $b_0$ selects one member of the ghost zero-mode doublet. Use the <ghost Virasoro zero-mode convention> compatible with the displayed explicit $-c_0$ term. Reordering the charge gives
$$
Q=\sum_m c_{-m}L_m^\alpha-\frac12\sum_{m,n}(m-n):c_{-m}c_{-n}b_{m+n}:-c_0.
$$
The normal-ordered cubic term annihilates $|\downarrow\rangle$. If both $c$ modes are creators, $m,n\geq0$, the $b_{m+n}$ mode is an annihilator, including $b_0$. If either $c$ is an annihilator, <normal ordering> puts an annihilator on the right. Consequently
$$
Q|\phi\rangle=c_0(L_0^\alpha-1)|\phi\rangle+\sum_{m>0}c_{-m}L_m^\alpha|\phi\rangle.
$$
These ghost states are independent. Acting with $b_0$ isolates $(L_0^\alpha-1)|\Psi\rangle$, while $b_m$, $m>0$, isolates $L_m^\alpha|\Psi\rangle$. Thus closure is equivalent to
$$
\boxed{(L_0^\alpha-1)|\Psi\rangle=0,\qquad L_m^\alpha|\Psi\rangle=0\quad(m>0).}
$$
Conversely these conditions make every term in $Q|\phi\rangle$ vanish. They are exactly the <physical-state Virasoro conditions for an open string>; the cohomology quotient additionally identifies physically equivalent representatives.
Back to article page
