Bosonic string theory

Bosonic string theory quantizes a relativistic string using only bosonic worldsheet fields. Lorentz-invariant quantization requires 26-dimensional target spacetime and contains a tachyonic ground state.

String tension is energy per unit length. The Regge-slope parameter ${\alpha }^{\prime }$ is conventionally defined by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$.

The Polyakov action introduces an independent worldsheet metric $g_{\alpha \beta }$:Its metric equation makes $g_{\alpha \beta }$ conformal to the induced worldsheet metric, reducing it classically to the Nambu–Goto action.

A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which $X^{\mu }$ is a scalar and $g_{\alpha \beta }$ is a rank-two tensor.

Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as $X^0=\tau$ for a string.

A Weyl transformation rescales the worldsheet metric locally, $g_{\alpha \beta }\mapsto e^{2\omega (\sigma )}{g}_{\alpha \beta }$, without changing $X^{\mu }$. The classical Polyakov action is Weyl invariant.

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha \beta }=e^{2\varphi }{\eta }_{\alpha \beta }$ locally.

After conformal gauge fixing, independent reparameterizations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$ remain, accompanied by a compensating Weyl transformation.

Light-cone gauge uses residual conformal transformations to make the target-space coordinate $X^{+}$ linear in worldsheet time. The Virasoro constraints then determine $X^{-}$ from the transverse fields.

The worldsheet metric equation in conformal gauge sets the stress tensor to zero, $T_{++}=T_{--}=0$. In oscillator language these are the Virasoro constraints.

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

For the traceless diffeomorphism operatorthe ghost action is

A string worldsheet $\Sigma$ is the two-dimensional surface swept out by a string in spacetime. Local coordinates are commonly denoted ${\sigma }^{\alpha }=(\tau ,\sigma )$.

The string embedding map $X:\Sigma \to \mathcal{M}$ assigns each worldsheet point its position in target spacetime.

The induced worldsheet metric is the pullback of the target-space metric:

The Nambu–Goto actionis minus the string tension times the Lorentzian area of the worldsheet.

A two-dimensional conformal field theory is invariant under local holomorphic and antiholomorphic conformal transformations.

A primary operator of weights $(h,\tilde{h})$ transforms covariantly under local conformal maps and has a universal operator-product expansion with the stress-energy tensor.

The conformal weights determine the response of a primary operator to holomorphic and antiholomorphic coordinate rescaling. Its scaling dimension is $\Delta =h+\tilde{h}$ and its spin is $h-\tilde{h}$.

An operator product expansion expresses the short-distance product of local operators as a sum of local operators multiplied by singular coefficient functions.

The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.

Normal ordering removes self-contractions from a composite operator. For free fields it places annihilation modes to the right of creation modes.

Wick theorem expresses a product of free fields as its normal-ordered product plus all possible contractions.

The Fourier modes ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.

Closed-string periodicity and the residual spatial constraint require equal left- and right-moving excitation levels, $N=\tilde{N}$.

The normal-ordering constant is the zero-point shift in the quantum Virasoro generator. Closure of the target-space Lorentz algebra fixes $a=1$ and the spacetime dimension $D=26$ for the bosonic string.

For a closed bosonic string,

The critical dimension is the target-spacetime dimension in which quantum anomalies obstructing Lorentz or Weyl symmetry cancel. It is 26 for the bosonic string.