From a Semiclassical Dirac Field to Effective Closed Strings

We show that, if the spacetime metric is a functional of a Dirac matter field Ψ fixed by the semiclassical Einstein–Dirac equations and if that field admits a stable finite-rest-energy closed line-defect sector, then endpoint-free defect histories acquire a Nambu–Goto action at leading order in the derivative expansion in the empty-occupation sector. The construction takes Ψ as primitive, the metric as the response functional gμν = Gμν[Ψ], and the closed string as the collective-coordinate history of a stable closed Ψ-cycle, reversing the ordering used in perturbative closed-string theory in which a fundamental worldsheet CFT is primitive and the graviton is a spectrum mode. The radiative spin-2 channel is derived as the transverse-traceless projection of the metric's response to matter perturbations, δG[δΨ], rather than as an independent worldsheet graviton. Linearising the semiclassical Einstein–Dirac fixed point around a background gives the response operator Lsc[h] = 8πGδΨT[δΨ] with Lsc = δG − 8πGδgT, and the response kernel is the gauge-fixed retarded inverse of this operator on matter perturbations satisfying the linearised Dirac equation. In the Dirac-sector version of Witten's superconducting-string mechanism, a composite condensate Φ ~ Ψ̄RP†ΨL generated by Hubbard–Stratonovich bosonisation of a four-fermion channel of Ψ supplies the vortex profile, and the Jackiw–Rossi index theorem for the winding scalar-Yukawa background fixes the signed count of normalisable chiral zero modes at N+ − N− = Nchn. Occupying the positive-energy chiral branch of these zero modes below the bulk gap adds a specific positive rest-energy per unit length UDirac = Nch|n|kF2 / (4π) to the neutral Nielsen–Olesen value, produces a current-carrying effective string with Telastic(n) ≠ Ueff(n) under the leading chiral massless-carrier equation of state, and reduces to the neutral closed defect in the empty-occupation limit. The paper is deliberately scoped to the effective-string sector and does not compute the numerical value of UΨ(n), does not quantise the effective worldsheet, and does not identify the induced TT response with a quantum graviton state. The construction is a conditional derivation of an effective closed-string sector inside a semiclassical matter-response framework, not a derivation of critical superstring theory.