Worldline Localization

Abstract: We show that two elementary worldline path integrals-the thermal partition function of the harmonic oscillator and the one-loop effective action of scalar QED in a constant field strength-exhibit a natural form of supersymmetric localization. The mechanism relies on hidden fermionic symmetries of the worldline BRST formulation, rather than on standard BRST structure or physical supersymmetry. These symmetries localize the target-space trajectory. For the harmonic oscillator this yields an alternative localization derivation of the Jacobi-Poisson formula. Moreover, after the trajectory is localized, the remaining proper-time integral exhibits an emergent zero-dimensional supersymmetry generated by modular invariance, allowing the modulus T itself to be localized. For scalar QED the same structure provides a controlled computation of both the real and imaginary parts of the Euler-Heisenberg effective action. In particular, the imaginary part arises from a moduli space of circular worldline instantons, offering a localization perspective on the semiclassical exactness of the Schwinger pair-production result observed by Affleck-Alvarez-Manton.