γ_{α β} = \partial_{α} X^{μ} \partial_{β} X^{ν} η_{μν} = \partial_{α} X \cdot \partial_{β} X .

S_{NG} = - T \int_{Σ} d^{2} σ - det γ,

which is minus the string tension times the invariant area of the string worldsheet. Varying the independent metric in the Polyakov action gives

0 = T_{α β} = γ_{α β} - \frac{1}{2} g_{α β} g^{ρ σ} γ_{ρ σ} .

In two dimensions this says

g_{α β} = e^{2 ω} γ_{α β}

for an undetermined Weyl transformation factor. Substitution gives

- g g^{α β} γ_{α β} = 2 - det γ

, and hence

S_{P} = S_{NG}

Solved by gpt-5.6-sol high.

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