= Solution
Let $h_{\mu\nu}=\partial_\mu X^a\partial_\nu X_a$ be the <induced worldvolume metric> and write $H=\gamma^{\mu\nu}h_{\mu\nu}$. Varying the embedding in the <auxiliary-metric brane action> and integrating by parts gives
$$
\boxed{\partial_\mu\!\left(\sqrt{-\gamma}\,\gamma^{\mu\nu}\partial_\nu X^a\right)=0.}
$$
The boundary contribution, including its relative factor, is
$$
\delta I\big|_{\partial\Sigma}
=2\int_{\partial\Sigma}\sqrt{|\gamma_\partial|}\,
n_\mu\gamma^{\mu\nu}\partial_\nu X_a\,\delta X^a.
$$
At initial and final times one fixes the endpoint configurations or uses variations of compact support. At a spatial boundary, free target directions require <Neumann boundary conditions> $n_\mu\gamma^{\mu\nu}\partial_\nu X^a=0$; fixed directions require <Dirichlet boundary conditions> $\delta X^a=0$. Mixed conditions must make this boundary pairing vanish. A closed <brane> has no spatial boundary. Since the <auxiliary worldvolume metric> enters without <derivatives>, its variation produces no extra boundary term.
Using $\delta\sqrt{-\gamma}=-(1/2)\sqrt{-\gamma}\gamma_{\mu\nu}\delta\gamma^{\mu\nu}$, variation of the inverse <worldvolume metric> gives
$$
\boxed{h_{\mu\nu}-\frac12\gamma_{\mu\nu}\{H-(p-1)\}=0.}
$$
Taking the trace yields $(p-1)(H-p-1)=0$. For $p\ne1$, this forces $H=p+1$, and substitution gives $\gamma_{\mu\nu}=h_{\mu\nu}$. For a nondegenerate timelike embedding, eliminating the <auxiliary worldvolume metric> therefore gives twice the <worldvolume> area, with the overall physical <brane tension> supplied by the action normalization.
For $p=1$, the traced metric equation is an identity, and the remaining equation only says $h_{\mu\nu}=(H/2)\gamma_{\mu\nu}$. It fixes the <worldsheet metric> up to a <Weyl transformation>, rather than determining it uniquely. Indeed $\sqrt{-\gamma}\gamma^{\mu\nu}$ is Weyl invariant precisely in two <worldvolume> dimensions, and the constant term vanishes precisely at $p=1$. This is the <Weyl-invariance exception for the string among branes>. In a nondegenerate interior it gives the familiar <classical equivalence of Polyakov and Nambu–Goto actions>; degeneracies at a free string endpoint must be treated through the original equations.
For the <open string>, choose <conformal gauge> $\gamma_{\mu\nu}=\operatorname{diag}(-1,1)$ on $0\leq\sigma\leq\pi$. The embedding equation is the <wave equation>, and NN means $\partial_\sigma X^a=0$ at both spatial endpoints. For these operator formulas restore the conventional overall normalization $S=-TI/2$, with $T=(2\pi\alpha\prime)^{-1}$; the overall factor does not change the preceding classical equations. Its <open-string mode expansion> is
$$
X^a=x^a+2\alpha'p^a\tau+
i\sqrt{2\alpha'}\sum_{n\ne0}\frac{\alpha_n^a}{n}e^{-in\tau}\cos(n\sigma),
\qquad \alpha_0^a=\sqrt{2\alpha'}p^a.
$$
Reality requires $(\alpha_n^a)^\dagger=\alpha_{-n}^a$. <Canonical quantization> gives
$$
[x^a,p^b]=i\eta^{ab},\qquad
[\alpha_m^a,\alpha_n^b]=m\eta^{ab}\delta_{m+n,0}.
$$
The metric equation is the vanishing of the <worldsheet stress tensor>: $(\partial_\tau X+\partial_\sigma X)^2=(\partial_\tau X-\partial_\sigma X)^2=0$. The two endpoint-compatible mode expansions contain the same <string oscillator> family. Their quadratic coefficients are the classical <Virasoro constraints>.
For the operators, use <normal ordering> with positive-index <string oscillators> as annihilators:
$$
L_m=\frac12\sum_{n\in\mathbb Z}:\alpha_{m-n}\cdot\alpha_n:,
\qquad
L_0=\alpha'p^2+\sum_{n>0}\alpha_{-n}\cdot\alpha_n.
$$
Thus $(\partial_\tau X\pm\partial_\sigma X)^2$ has modes $4\alpha'L_m$ before the quantum ordering correction. The <physical-state Virasoro conditions for an open string> are
$$
\boxed{L_m|\Phi\rangle=0\quad(m>0),\qquad
(L_0-a)|\Phi\rangle=0.}
$$
One imposes only the positive modes on kets, with the adjoint conditions on bras, as in <Gupta-Bleuler quantization>. Requiring every positive and negative mode to annihilate the same state would conflict with the <Virasoro central extension>. The intercept $a$ is the zero-mode ordering constant. The standard critical <bosonic string theory> has $d=26$ and $a=1$; in <light-cone gauge in string theory> the transverse <zero-point energy> gives $a=(d-2)/24$, while full anomaly-free Lorentz or <BRST quantization> fixes the critical values. The mass constraint is then $\alpha'M^2=N-a$, with $N=\sum_{n>0}\alpha_{-n}\cdot\alpha_n$.
To compute the <Virasoro algebra>, commute a quadratic generator with one <string oscillator>:
$$
[L_m,\alpha_n^a]=-n\alpha_{m+n}^a,\qquad [L_m,x^a]=-i\sqrt{2\alpha'}\alpha_m^a.
$$
These identities and the <Jacobi identity> imply that $[L_m,L_n]-(m-n)L_{m+n}$ commutes with every <string oscillator> and with the center-of-mass coordinates and momenta. Mode number permits a scalar term only for $m+n=0$. Its coefficient follows from the formal zero-momentum <Fock vacuum>, on which $L_0=0$. For $m>0$,
$$
L_{-m}|0\rangle=\frac12\sum_{r=1}^{m-1}
\alpha_{-r}\cdot\alpha_{-(m-r)}|0\rangle.
$$
The two possible <string oscillator> contractions give
$$
\langle0|L_mL_{-m}|0\rangle
=\frac d2\sum_{r=1}^{m-1}r(m-r)
=\frac d{12}(m^3-m).
$$
The timelike target coordinate still contributes one to this <central charge>: its two metric signs cancel in $\eta_{ab}\eta^{ab}=d$. Consequently
$$
\boxed{[L_m,L_n]=(m-n)L_{m+n}
+\frac d{12}m(m^2-1)\delta_{m+n,0}.}
$$
This is the <free-boson Virasoro central term>. It vanishes for the three global conformal modes $m=0,\pm1$.
To keep the intercept convention separate, define $\mathcal L_m=L_m-a\delta_{m,0}$. The <Virasoro zero-mode shift> changes the displayed central term to
$$
[\mathcal L_m,\mathcal L_n]=(m-n)\mathcal L_{m+n}
+\left\{\frac d{12}m(m^2-1)+2am\right\}\delta_{m+n,0}.
$$
The matter <central charge> here is $d$, not $d-2$. Covariant <worldsheet ghost fields> contribute $-26$; their inclusion cancels the anomaly at $d=26$, while the intercept is handled by the appropriate zero-mode and physical-state convention.
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