Resolving degeneracies in complex ℝ × S3 and θ-KSW

A bstract Lorentzian gravitational path integral for the Gauss-Bonnet gravity in 4 D is studied in the mini-superspace ansatz for metric. The gauge-fixed path-integral for Robin boundary choice is computed exactly using Airy -functions, where the dominant contribution comes from No-boundary geometries. The lapse integral is further analysed using saddle-point methods to compare with exact results. Picard-Lefschetz methods are utilized to find the relevant complex saddles and deformed contour of integration, thereby using WKB method to compute the integral along the deformed contour in the saddle-point approximation. However, their successful application is possible only when system is devoid of degeneracies, which in present case appear in two types: type-1 where the flow-lines starting from neighbouring saddles overlap leading to ambiguities in deciding the relevance of saddles, type-2 where saddles merge for specific choices of boundary parameters leading to failure of WKB. Overcoming degeneracies using artificial defects introduces ambiguities due to the choice of defects involved. Corrections from quantum fluctuations of scale-factor overcome degeneracies only partially (lifts type-2 completely with partial resolution of type-1 ), with the residual lifted fluently by complex deformation of ( G ℏ). Anti-linear symmetry present in various forms in the lapse action is the reason behind all the type-1 degeneracies. Any form of defect or deformation breaking anti-linearity resolves type-1 degeneracies, indicating complex deformation of ( G ℏ) as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of ( G ℏ) modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed.