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A bc system is the anticommuting counterpart of a beta-gamma system. For weights $h$ and $1-h$, its central charge is $-2(6h^2-6h+1)$.

A beta-gamma system is a pair of commuting chiral fields with first-order action $\int \beta \bar{\partial }\gamma$. If $\gamma$ has weight $h$ and $\beta$ has weight $1-h$, its central charge is $2(6h^2-6h+1)$.

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.

To first order in ${\alpha }^{\prime }$, the conventional Euclidean dilaton coupling gives ${\overline{\beta }}_{\mu \nu }^G={\alpha }^{\prime }[R_{\mu \nu }-H_{\mu \rho \sigma }{H}_{\nu }{}^{\rho \sigma }/4+2{\nabla }_{\mu }{\nabla }_{\nu }\Phi ]$ and ${\overline{\beta }}_{\mu \nu }^B={\alpha }^{\prime }[-{\nabla }^{\rho }{H}_{\rho \mu \nu }/2+({\nabla }^{\rho }\Phi )H_{\rho \mu \nu }]$. The scalar coefficient is $(d-26)/6+{\alpha }^{\prime }[(\nabla \Phi {)}^2-{\nabla }^2\Phi /2-H^2/24]$. Here $H$ is the Kalb-Ramond field strength. Vanishing of these coefficients, including higher-order corrections, is the background-field condition for quantum Weyl transformations to remain a gauge symmetry. The scalar coefficient includes the central charge deficit and is not implied by the metric equation alone.

Bosonic string theory quantizes a relativistic string with bosonic worldsheet matter fields. In the ordinary flat critical theory, the worldsheet Weyl anomaly cancels in 26-dimensional target spacetime: the embedding fields contribute central charge $26$ and the reparameterization ghosts contribute $-26$.

Its string oscillators generate a tower of particle states. The closed-string massless sector includes a graviton, a Kalb–Ramond field and a dilaton, making the theory a useful model of how gravity emerges from a string spectrum. Its ground state is a tachyon, indicating an instability of the chosen vacuum rather than ordinary faster-than-light signalling. The absence of spacetime fermions and this instability limit its direct phenomenological use, but its quantization exposes many mechanisms shared by more elaborate string theories.^[ref]

The Brown--Henneaux central charge is the central charge of the two-dimensional conformal field theory dual to Einstein gravity on ${\mathrm{AdS}}_3$:

The BRST current is the conserved worldsheet current whose closed contour integral gives the BRST operator,For the bosonic string, one conventional holomorphic representative iswhere $b,c$ are the worldsheet ghost fields, $T_m$ is the matter stress-energy tensor, and colons denote normal ordering. The last term improves the conformal transformation law without changing the closed contour integral.

Quantum consistency requires $Q_B^2=0$. With matter central charge $c_m$ and ghost central charge $-26$, the obstructing contour residue vanishes at $c_m=26$. This does not require a nonsingular operator product expansion of the currents. For the displayed representative, even at $c_m=26$ the singular terms areThere is no simple pole, so the inner contour integral vanishes. The BRST operator can therefore be nilpotent while its local current has higher-order poles.^[ref]

A BRST charge is nilpotent when $Q^2=0$, making BRST cohomology well-defined. For the critical bosonic string theory, the matter and ghost central charges cancel, $D-26=0$, and the intercept is one. In a ghost convention with a zero-mode linear shift, the shifted total constraint generators, rather than the unshifted ones, obey the centerless Witt algebra.

For a two-dimensional conformal field theory on a spatial circle of circumference $L$, with propagation speed $v$ and left/right central charges $c_L,c_R$, the plane-vacuum sector has Casimir energyWhen $c_L=c_R=c$, the unit-radius convention $L=2\pi$, $\hslash =v=1$ gives $E_0=-c/12$. A circle of unit circumference instead gives $E_0=-\pi c/6$ in these units. Specifying radius versus circumference prevents a factor-of-$2\pi$ ambiguity.

The exponential map from a cylinder to the complex plane shifts the stress-energy tensor by its Schwarzian derivative term. Equivalently, the cylinder Hamiltonian is $H=(2\pi \hslash v/L)[L_0+{\bar{L}}_0-(c_L+c_R)/24]$. Setting both plane-vacuum weights to zero yields the formula. Other sectors can have different lowest weights. In AdS/CFT correspondence, the unit-radius result matches the renormalized vacuum energy of global three-dimensional Anti-de Sitter spacetime.^[ref]

The central charge is the coefficient of the fourth-order pole in the stress-tensor operator-product expansion, $T(z)T(w)\sim c/[2(z-w{)}^4]+\cdots$. A free real boson contributes one and a free real chiral fermion contributes one half.

A central charge in four-dimensional extended supersymmetry is a Lorentz scalar even generator appearing in $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. It commutes with the displayed translation, Lorentz algebra, and supercharge generators. An added R-symmetry automorphism can act on the central charges.

The worldsheet ghost fields form a bc system with conformal weights $2$ and $-1$. Setting $J=2$ in $c=-2(6J^2-6J+1)$ gives central charge $-26$ in each chirality. The worldsheet Weyl anomaly of $D$ free embedding bosons cancels at $D=26$.

The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through $e^{2\pi ic/8}={\mathcal{D}}^{-1}{\sum }_a{d}_a^2{\theta }_a$. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

For a current subalgebra $\mathfrak{h}\subset \mathfrak{g}$, its currents have conformal weight one under both Sugawara construction stress tensors. Their difference therefore commutes with all $\mathfrak{h}$ currents and with the $\mathfrak{h}$ stress tensor. It obeys the Virasoro algebra with central charge $c_{\mathfrak{g}}-c_{\mathfrak{h}}$. For a positive-level compact unitary vacuum representation, $\Vert L_{-2}^{\mathfrak{g}/\mathfrak{h}}|0⟩{\Vert }^2=(c_{\mathfrak{g}}-c_{\mathfrak{h}})/2\ge 0$. A Cartan subalgebra has $c_{\mathfrak{h}}=\mathrm{rank}\mathfrak{g}$, giving the lower bound on $c_{\mathfrak{g}}$ under these hypotheses.

Each of $D$ embedding bosons contributes one to the chiral central charge, and each real worldsheet Majorana fermion contributes one half. The reparameterization bc system contributes $-26$, and the superconformal ghosts contribute $+11$. Vanishing total worldsheet Weyl anomaly therefore requires $3D/2-26+11=0$, or $D=10$. This differs from the critical dimension 26 of the bosonic string, which has no worldsheet fermions or superconformal ghosts.

A supersymmetric domain wall contributes a boundary or tensorial charge to the extended anticommutator of supercharges. It is not the Lorentz-scalar central charge of an extended particle algebra. The wall's energy per unit area can cancel this charge in half the real charge combinations, giving the BPS tension bound and half-supersymmetry preservation. This does not contradict the rank classification for the ordinary algebra without wall charges.

The free boson conformal field theory has action quadratic in a scalar $X$ and central charge one. Its oscillator partition function is expressed through the Dedekind eta function.

For $d$ free target coordinates, normal ordering of $L_m={\sum }_n:{\alpha }_{m-n}\cdot {\alpha }_n:/2$ gives a matter central charge $d$. On the formal zero-momentum vacuum, the double contractions yield $⟨L_m{L}_{-m}⟩=(d/2){\sum }_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. Two target-metric contractions give ${\eta }_{ab}{\eta }^{ab}=d$, so timelike target coordinates do not subtract from this central charge. Transverse-only and covariant generators must not be confused.

A real chiral Majorana fermion with contraction $\psi (z)\psi (w)\sim 1/(z-w)$ has stress tensor $T=-:\psi \partial \psi :/2$. Fermionic Wick contractions give $T(z)\psi (w)\sim \psi (w)/(2(z-w{)}^2)+\partial \psi (w)/(z-w)$. Double contractions in $TT$ give $1/(4(z-w{)}^4)$, so the central charge is $1/2$. Fermion bilinears of different species are weight-one currents; equal-species undifferentiated bilinears vanish by antisymmetry.

The Haag–Łopuszański–Sohnius theorem allows a Z2-graded Lie superalgebra and shows that supersymmetry, including possible central charges and R-symmetries, is the nontrivial fermionic extension compatible with an interacting relativistic S-matrix.