String sigma model in a spacetime metric ( $S = - \frac{T}{2} \int - h h^{ab} g_{μν} (X) \partial_{a} X^{μ} \partial_{b} X^{ν}$ )

Replacing the flat target metric in the Polyakov action by a spacetime metric

g_{μν} (X)

gives the displayed string sigma model. The embedding fields

X^{μ}

map the worldsheet into target spacetime, while

h_{ab}

is an independent worldsheet metric. Classical worldsheet diffeomorphisms and Weyl transformations permit conformal gauge; quantum consistency additionally requires vanishing sigma-model beta functions.

Table of contents
- String sigma model in an isotropic plane wave String sigma model in a spacetime metric
  - Plane-wave string mode Hamiltonian String sigma model in an isotropic plane wave
  - Plane-wave string mode frequency String sigma model in an isotropic plane wave

String sigma model in an isotropic plane wave ( $d s^{2} = - 2, d X^{+} d X^{-} - m^{2} ∣ X ∣^{2} (d X^{+})^{2} + d X^{I} d X^{I}$ )

In conformal gauge and light-cone gauge in string theory

X^{+} = κ τ

, the transverse part of the string sigma model in a spacetime metric reduces to

S_{⊥} = \frac{T}{2} \int d τ d σ (\dot{X}^{I} \dot{X}^{I} - X^{' I} X^{' I} - m^{2} κ^{2} X^{I} X^{I}) .

The omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive wave equation. If the spatial period is

L

, the longitudinal momentum is

p^{+} = T κ L

in this normalization. Keeping the gauge slope explicit prevents missing a factor of

p^{+}

when changing conventions.

Plane-wave string mode Hamiltonian ( $H_{⊥} = \sum_{I, a} (\frac{p _{I a}^{2}}{2 T} + \frac{T}{2} ω_{a}^{2} q_{I a}^{2})$ )

Expand every transverse closed string coordinate in a real orthonormal Fourier basis of the spatial circle. The canonical momenta are

p_{I a} = T \overset{q}{˙}_{I a}

, and spatial orthogonality reduces the field Hamiltonian to the displayed independent harmonic oscillators. Canonical quantization gives

ω_{a} (a_{I a}^{†} a_{I a} + 1/2)

for each nonzero-frequency mode; a zero-frequency mode retains the free particle term

p_{I a}^{2} / (2 T)

. The formal zero-point energy requires a regulator; it is not automatically the massless flat-background intercept. Opposite travelling-wave oscillator combinations diagonalize the translation generator and satisfy closed-string level matching.

Plane-wave string mode frequency ( $ω_{n} = (2 πn / L)^{2} + m^{2} κ^{2}$ )

For a closed string in a string sigma model in an isotropic plane wave, insert

e^{- i ω_{n} τ + 2 πinσ / L}

into the transverse wave equation. Periodicity requires integer

n

, giving the displayed dispersion relation. The zero spatial mode is an oscillator of frequency

∣ mκ ∣

; when this mass vanishes it becomes a free particle mode and needs the position-plus-linear-time expansion.

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