String worldsheet

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.