String Spectra, Scales, and Low-Energy Limits

The low-energy content of a string compactification is determined by several independent towers: string oscillators, Kaluza–Klein momenta, winding states, branes, and interaction corrections. A field-theory or supergravity limit exists only when the energies of interest lie below every tower that has been removed and when both $\alpha'$ and $g_s$ expansions are controlled.

Required background. Worldsheet sigma models and spacetime consistency supplies the perturbative background and the meaning of the $\alpha'$ expansion.

Helpful background. Gravitational EFT degrees of freedom and power counting supplies the spacetime derivative expansion. EFT truncation errors and breakdown diagnostics supplies tests for omitted thresholds.

Momentum, winding, and oscillator towers

For a closed string on a circle of radius $R$, the mass formula contains

$$M^2=\left(\frac{n}{R}\right)^2+\left(\frac{wR}{\alpha'}\right)^2 +\frac{2}{\alpha'}(N_L+N_R-a_L-a_R), \qquad N_L-N_R=nw.$$

Thus $m_{\mathrm{KK}}\sim R^{-1}$, $m_{\mathrm{wind}}\sim R/\alpha'$, and $m_s\sim\alpha'^{-1/2}$. T-duality exchanges $n\leftrightarrow w$ and $R\leftrightarrow\alpha'/R$, preventing a uniformly valid claim that small radius simply removes all compact physics. The precise intercepts and projections depend on the string theory and sector, but the scale comparison does not Polchinski 1998, Vol. 1, Chs. 7–8.

A $d$-dimensional point-particle EFT restricted to zero modes requires the same threshold separation that underlies heavy-field decoupling Appelquist and Carazzone 1975:

$$E\ll \min(m_{\mathrm{KK}},m_{\mathrm{wind}},m_s,m_{\mathrm{brane}}),$$

plus weak enough coupling to suppress virtual corrections. Supergravity additionally retains only massless string states and expands in curvature: $\alpha'\mathcal R\ll1$. A large compact volume makes winding heavy but Kaluza–Klein modes light; a small volume does the opposite. Scale separation is a property to prove, not an automatic consequence of compactification.

First application: choosing a ten- or five-dimensional description

Consider type-IIB theory on a compact factor of radius $R_K$ and an AdS factor of radius $L$. For a boundary process with bulk energy $E\sim L^{-1}$, massive string modes decouple when $L\gg\sqrt{\alpha'}$. A five-dimensional zero-mode action also requires $R_K^{-1}\gg L^{-1}$, or $R_K\ll L$. In AdS$_5\times S^5$, however, $R_K=L$, so the Kaluza–Klein gap is itself $O(L^{-1})$. Ten-dimensional supergravity can still be reliable, and selected five-dimensional fields can form a nonlinear consistent truncation, but there is no generic Wilsonian separation between the AdS and sphere towers.

The coupling hierarchy is independent. At fixed geometry, genus $h$ is weighted by $g_s^{2h-2}$, while D-brane tensions scale as $1/g_s$. Nonperturbative brane effects can be exponentially small at weak coupling even though they are invisible to every finite genus order Polchinski 1998, Vol. 2, Chs. 13–14.

Adversarial control: a light omitted tower

Choose $E=0.2m_s$ but $R=10\sqrt{\alpha'}$. Oscillator production is suppressed, yet $m_{\mathrm{KK}}=0.1m_s<E$. A truncation that retains the graviton zero mode while discarding the KK tower predicts spurious locality and incorrect thresholds. Reversing $R$ makes winding states the corresponding obstruction. This test isolates why “below the string scale” is insufficient.

The evidence ceiling is an energy- and background-dependent EFT statement. A finite hierarchy justifies a quantified truncation error; it does not define the ultraviolet theory, and an exact consistent truncation does not imply that discarded modes are heavy. The D3 decoupling limit now applies these distinctions to the original AdS/CFT proposal.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Appelquist, T., and Carazzone, J. (1975), “Infrared Singularities and Massive Fields,” Physical Review D 11, 2856–2861. doi:10.1103/PhysRevD.11.2856.
• Polchinski, J. (1998), String Theory, Vol. 1: An Introduction to the Bosonic String, Cambridge University Press. doi:10.1017/CBO9780511816079.
• Polchinski, J. (1998), String Theory, Vol. 2: Superstring Theory and Beyond, Cambridge University Press. doi:10.1017/CBO9780511618123.