Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $\theta$-KSW

Abstract

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 $(G\hslash )$. {\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 $(G\hslash )$ as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of $(G\hslash )$ modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed.

arXiv commentV2: 1+64 pages, 13 figures, 10 tables, 3 appendix. text modified and references added. Published in JHEP

References in corpus (33)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Lorentzian Quantum Cosmology
• An Introduction to Resurgence, Trans-Series and Alien Calculus
• No smooth beginning for spacetime
• Testing the black-hole area law with GW150914
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• No Rescue for the No Boundary Proposal
• Hamiltonian formulation of f(Riemann) theories of gravity
• Complex actions in two-dimensional topology change
• Structure of Lefschetz thimbles in simple fermionic systems
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Review of the No-Boundary Wave Function
• The No-Boundary Proposal as a Path Integral with Robin Boundary Conditions
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• No-boundary prescriptions in Lorentzian quantum cosmology
• Complex Saddle Points and Disorder Lines in QCD at finite temperature and density
• Short Distance Freedom of Quantum Gravity
• Complex saddle points in QCD at finite temperature and density
• Uses of Complex Metrics in Cosmology
• Lessons for quantum cosmology from anti-de Sitter black holes
• The Wave Function of Simple Universes, Analytically Continued From Negative to Positive Potentials
• Robin Gravity
• Unstable no-boundary fluctuations from sums over regular metrics
• The Kontsevich-Segal Criterion in the No-Boundary State Constrains Inflation
• On Gauss-bonnet gravity and boundary conditions in Lorentzian path-integral quantization
• Exorcising Ghosts in Induced Gravity
• Surprises in Lorentzian path-integral of Gauss-Bonnet gravity
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Lorentzian Robin Universe of Gauss-Bonnet Gravity
• Constraining the topological Gauss-Bonnet coupling from GW150914
• Generalized boundary conditions in closed cosmologies
• Boundary choices and one-loop complex gravitational path integral
• Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology