Physical meaning of complex-coupling integrating-out in topological strings

Determine the physical interpretation of the M-theory integrating-out procedure for BPS M2-brane states, formulated via Schwinger proper-time integrals and contour deformations, when the topological string coupling λ is complex (λ = λr + iλi), and ascertain whether the associated pole-crossing Stokes jumps and non-perturbative contributions represent physically correct features of the topological string free energy.

Background

The paper proposes an exact integrating-out formula for the holomorphic-limit, non-perturbative topological string free energy based on M2-brane BPS states and evaluates it through contour integrals in complexified Schwinger proper time. For real coupling λ, the analysis identifies perturbative and non-perturbative poles and matches known non-perturbative completions in specific cases.

When λ is extended to complex values, the non-perturbative poles rotate in the complex plane and may cross integration rays, producing additional contour contributions interpreted as Stokes jumps. Although mathematically well-defined within the proposed framework, the authors state that the physical meaning of the integrating-out calculation for complex λ, and the physical validity of the resulting properties (including the Stokes jumps), is not clear and requires further investigation.