Modular Invariance, Finiteness, and Misaligned Supersymmetry: New Constraints on the Numbers of Physical String States

TL;DR

This paper addresses how modular invariance constrains the distribution of bosonic and fermionic string states in non-supersymmetric, tachyon-free theories and reveals a universal mechanism called misaligned supersymmetry. It introduces the sector-averaged degeneracy $\langle a_{nn}\rangle$ to capture cross-sector cancellations and proves a main theorem that the effective growth rate $C_{\rm eff}$ of these sector averages is strictly less than the total growth rate $C_{\rm tot}$, with strong oscillatory boson/fermion patterns across levels. Two explicit examples illustrate how leading exponential terms cancel in the sector-averaged spectrum, reducing $C_{\rm eff$ and providing a concrete link to finiteness via the one-loop cosmological constant. The work connects modular-function theory with string finiteness, showing that even without full spacetime SUSY, modular invariance enforces a delicate, level-by-level cancellation that preserves ultraviolet finiteness and constrains SUSY-breaking patterns, with potential implications for hadronic string models and QCD-like spectra.

Abstract

We investigate the generic distribution of bosonic and fermionic states at all mass levels in non-supersymmetric string theories, and find that a hidden ``misaligned supersymmetry'' must always appear in the string spectrum. We show that this misaligned supersymmetry is ultimately responsible for the finiteness of string amplitudes in the absence of full spacetime supersymmetry, and therefore the existence of misaligned supersymmetry provides a natural constraint on the degree to which spacetime supersymmetry can be broken in string theory without destroying the finiteness of string amplitudes. Misaligned supersymmetry also explains how the requirements of modular invariance and absence of physical tachyons generically affect the distribution of states throughout the string spectrum, and implicitly furnishes a two-variable generalization of some well-known results in the theory of modular functions.