Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $θ$-KSW
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2025
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| author | Ailiga, Manishankar Mallik, Shubhashis Narain, Gaurav |
| author_facet | Ailiga, Manishankar Mallik, Shubhashis Narain, Gaurav |
| contents | Lorentzian gravitational path integral for the Gauss-Bonnet gravity in $4D$ is studied in the mini-superspace ansatz for metric. The gauge-fixed path-integral for Robin boundary choice is computed exactly using {\it 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 {\it relevant} complex saddles and deformed contour of integration, thereby using WKB methods 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: {\bf type-1} where the flow-lines starting from neighbouring saddles overlap leading to ambiguities in deciding the {\it relevance} of saddles, {\bf type-2} where saddles merge for specific choices of boundary parameters leading to failure of WKB. Overcoming degeneracies using artificial {\it defects} introduces ambiguities due to the choice of {\it defects} involved. Corrections from quantum fluctuations of scale-factor overcome degeneracies only partially (lifts {\bf type-2} completely with partial resolution of {\bf type-1}), with the residual lifted fluently by complex deformation of $(G\hbar)$. {\it Anti-linear} symmetry present in various forms in the lapse action is the reason behind all the {\bf type-1} degeneracies. Any form of {\it defect} or {\it deformation} breaking anti-linearity resolves {\bf type-1} degeneracies, indicating complex deformation of $(G\hbar)$ as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of $(G\hbar)$ modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $θ$-KSW Ailiga, Manishankar Mallik, Shubhashis Narain, Gaurav High Energy Physics - Theory General Relativity and Quantum Cosmology Lorentzian gravitational path integral for the Gauss-Bonnet gravity in $4D$ is studied in the mini-superspace ansatz for metric. The gauge-fixed path-integral for Robin boundary choice is computed exactly using {\it 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 {\it relevant} complex saddles and deformed contour of integration, thereby using WKB methods 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: {\bf type-1} where the flow-lines starting from neighbouring saddles overlap leading to ambiguities in deciding the {\it relevance} of saddles, {\bf type-2} where saddles merge for specific choices of boundary parameters leading to failure of WKB. Overcoming degeneracies using artificial {\it defects} introduces ambiguities due to the choice of {\it defects} involved. Corrections from quantum fluctuations of scale-factor overcome degeneracies only partially (lifts {\bf type-2} completely with partial resolution of {\bf type-1}), with the residual lifted fluently by complex deformation of $(G\hbar)$. {\it Anti-linear} symmetry present in various forms in the lapse action is the reason behind all the {\bf type-1} degeneracies. Any form of {\it defect} or {\it deformation} breaking anti-linearity resolves {\bf type-1} degeneracies, indicating complex deformation of $(G\hbar)$ as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of $(G\hbar)$ modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed. |
| title | Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $θ$-KSW |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2507.10537 |