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A bosonic string is a relativistic one-dimensional object whose embedding coordinates are bosonic fields on a two-dimensional worldsheet. Its classical motion can be described by the Nambu–Goto action, proportional to the worldsheet area, or by the equivalent Polyakov action with an auxiliary worldsheet metric.

An open string has endpoints; a closed string forms a loop and has independent left- and right-moving oscillations. Quantization produces a tower of particle states. In the ordinary flat critical bosonic string theory, anomaly cancellation selects 26 spacetime dimensions, and the closed-string spectrum includes a massless graviton. The tachyonic ground state indicates instability of the chosen vacuum.^[ref]

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]

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]

A D0-brane is a point-like D-brane in space, with a one-dimensional worldvolume consisting only of time. It belongs to type IIA superstring theory and couples electrically to its one-form Ramond–Ramond potential. “Zero” counts spatial dimensions, rather than asserting zero mass or absence of dynamics.

Its mass scales as $1/(g_s\sqrt{{\alpha }^{\prime }})$, so it is heavy at weak string coupling. Unlike an ordinary perturbative string excitation, its open-string endpoints are fixed in every spatial direction for a static brane, while the brane's own position is a dynamical collective coordinate. The low-energy description of several nearby D0-branes is quantum mechanics of their worldvolume fields; it is not a field theory with an additional extended spatial direction.^[ref]

The little group of a fixed four-momentum $p$ is its stabilizer subgroup in the Lorentz group: $G_p=\{\Lambda :\Lambda p=p\}$. It changes internal states while leaving that momentum unchanged. In four spacetime dimensions, a massive rest momentum has spatial rotations as its little group; the corresponding Spin group is SU(2), permitting half-integer spin. A nonzero future-directed massless four-momentum instead has the planar Euclidean little-group structure. Its finite-helicity quantum states have trivial translation action. These group representations, combined with the momentum orbit, underlie Wigner's classification.^[ref]

A particle polarization is a physical internal spin state at fixed four-momentum. In $D$ spacetime dimensions, massive states use the rotation little group $SO(D-1)$ or its appropriate Spin group; half-integer spin requires the cover. Finite-spin massless states use the transverse rotation part $SO(D-2)$, with trivial little-group translations. Thus a massive vector has three polarizations in four dimensions, while a photon has two.^[ref] A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ states, versus $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$ for a two-form gauge field, after imposing the field equations and identifying states related by gauge symmetry.

A worldsheet Green function inverts the quadratic kinetic operator of a scalar field on a string worldsheet, after removing any constant worldsheet zero mode. For one Euclidean target coordinate, fix $z=x+iy$, area measure $dA=dxdy$, andThe Polyakov action isThe local planar Green function isIndeed, $\partial \bar{\partial }\log |z-w{|}^2=\pi {\delta }_A(z-w)$ for the stated area measure. The arbitrary scale $\ell$ changes only an additive constant.

A common complex-coordinate convention uses $d^{2}z=2dA$ and ${\delta }_{d^{2}z}={\delta }_A/2$. With that convention the same kernel is inverted using $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }$ and the correspondingly normalized delta function. Keeping the area and delta conventions together prevents a factor-of-two mismatch. ^[ref]

On a closed compact worldsheet, the constant worldsheet zero mode obstructs an unrestricted inverse. Its removal replaces the delta source by the delta source minus the normalized zero mode projection. Boundary conditions and global geometry determine the remaining harmonic part.