The path integral of 4D Einstein-Hilbert gravity for the de Sitter–like Universe with fluctuations is investigated, and the transition amplitude from one boundary configuration to another is computed. The gravitational system is described by lapse, scale factor, and metric-fluctuation field. Variational consistency demands augmenting the bulk theory with suitable boundary action. A given boundary choice on scale factor is seen to be achievable via an infinite family of covariant boundary actions, each restricting the boundary choices for the fluctuation field. General covariance intimately ties the two boundary choices, which no longer can be chosen independently. For vanishing metric fluctuations at the boundaries, the gauge-fixed gravitational path integral disintegrates into path integral over scale factor and metric-fluctuation field, connected via only lapse integration. While the former is exactly doable, the latter is computed up to one loop, leading to one-loop corrected lapse action. Ultraviolet (UV) divergences are systematically extracted and removed by the addition of suitable counterterms, leading to finite effective action for the lapse. The lapse effective action is then utilized for computing finite transition amplitude. Contributions from virtual gravitons are seen to be secularly growing with Universe size, leading to an infrared divergent transition amplitude. The presence of nonvanishing metric fluctuation at the boundaries implies that the “no-boundary” saddles of the theory without metric fluctuations are no longer the saddles of the one-loop corrected action. The corrected saddles have the Universe starting from a nonzero size.

Elliptic division fields, obtained by adjoining p-torsion points of an elliptic curve are natural objects of study since they are analogous to cyclotomic fields. We establish a conductor discriminant bound for such family of number fields. We also formulate and conditionally prove Ihara's conjectures for Euler-Kronecker constants in this setting. This is ongoing work with Anup Dixit. https://www.imsc.res.in/~anupdixit/IMSc-CMI-NT-seminar.html

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