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How the Graviton Emerges from String Theory

In perturbative string theory, the graviton is not added by hand; it appears automatically as one of the vibration modes of a closed string.

1️⃣ Emergence of the graviton

  • Closed strings possess an infinite set of oscillation modes.
  • Quantization yields a tower of states labeled by excitation numbers \N_L, N_R\.
  • The lowest non-trivial excitation (level \N_L = N_R = 1\) of a closed bosonic string is
    a rank-2 tensor state:

    \

        \[ \\alpha_{-1}^{\\mu} \\, \\tilde{\\alpha}_{-1}^{\\nu} \\, \\lvert 0; k \\rangle \\]

    which decomposes into:

    • a symmetric traceless tensor \h_{\\mu\\nu}\,
    • an antisymmetric 2-form \B_{\\mu\\nu}\,
    • a scalar (the dilaton \\\phi\).

    The symmetric traceless piece \h_{\\mu\\nu}\ is massless and carries helicity \\\pm 2\: this is the graviton.

Key idea: Gravity arises because the closed string necessarily contains a massless spin-2 excitation.

2️⃣ Getting the spin and force correct

Several consistency conditions and projections are needed so the would-be graviton is truly massless, has the correct helicities, and couples as in general relativity:

Ingredient Why it mattered
Critical dimension Maintaining worldsheet reparametrization/Weyl invariance at the quantum level requires the critical dimension (\D=26\ for bosonic strings, \D=10\ for superstrings). Off-critical, anomalies spoil consistency and the would-be graviton need not stay massless.
Normal-ordering constant (“intercept”) Choosing the intercept so the level-matching and mass formula yield
\

    \[ m^2 \\,=\\, \\frac{4}{\\alpha'} (N_L - a) \\,=\\, \\frac{4}{\\alpha'} (N_R - a) \\,, \\]

gives the level \N_L = N_R = 1\ state massless (\m^2 = 0\) rather than tachyonic or massive. For closed strings, this corresponds to \a = 1\ per side.
Worldsheet supersymmetry Adding fermions on the worldsheet (superstrings) removes the tachyon and yields a stable spectrum including a massless spin-2 state.
GSO projection The Gliozzi–Scherk–Olive projection selects states with the correct worldsheet fermion number, eliminating unphysical states and ensuring the correct helicity content (only \\\pm 2\ for the graviton).
Gauge/BRST constraints Imposing the Virasoro (and, in superstrings, super-Virasoro) constraints in BRST language projects out unphysical polarizations, leaving the two physical helicities of a massless spin-2 particle.

3️⃣ Interaction and Newton’s law

The low-energy effective action for the massless closed-string modes (in the superstring) contains the Einstein–Hilbert term:

\

    \[ S \\;\\sim\\; \\frac{1}{2\\kappa^2} \\int d^{10}x \\, \\sqrt{-g} \\, R \\; + \\; \\cdots \\]

Tree-level scattering amplitudes of the \h_{\\mu\\nu}\ state reproduce, in the infrared limit, the long-range Newtonian potential. Thus, once the spectrum includes a massless spin-2 field with the right gauge constraints, its interactions automatically match general relativity at low energies.

4️⃣ Summary

  • Quantize a closed string → the level-1 state is a symmetric tensor.
  • Fix the intercept and stay at the critical dimension → the state is massless (\m^2 = 0\).
  • Apply GSO & BRST constraints → only helicities \\\pm 2\ survive (a true spin-2 graviton).
  • Compute the low-energy effective action → the Einstein–Hilbert term emerges and gravity’s force law is reproduced.