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 \.
• The lowest non-trivial excitation (level \) of a closed bosonic string is
  a rank-2 tensor state:
  \

     

  which decomposes into:

  • a symmetric traceless tensor \,
  • an antisymmetric 2-form \,
  • a scalar (the dilaton \).

  The symmetric traceless piece \ is massless and carries helicity \: 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 (\ for bosonic strings, \ 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 \     gives the level \ state massless (\) rather than tachyonic or massive. For closed strings, this corresponds to \ 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 \ 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:

Tree-level scattering amplitudes of the \ 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 (\).
• Apply GSO & BRST constraints → only helicities \ survive (a true spin-2 graviton).
• Compute the low-energy effective action → the Einstein–Hilbert term emerges and gravity’s force law is reproduced.

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