Monodromies and functional determinants in the CFT driven quantum cosmology

TL;DR

This paper develops a monodromy method to compute the functional determinant of a special differential operator relevant in quantum cosmology, generalizing previous single-node results to multiple oscillations of the cosmological scale factor.

Contribution

It introduces a generalized calculation of the functional determinant for operators with multiple zero mode nodes, extending prior single-node analyses in quantum cosmology.

Findings

Derived a formula for the functional determinant involving monodromy for multiple nodes

Generalized previous single-node results to multi-node cases

Applicable to the calculation of one-loop sums in cosmological models

Abstract

We apply the monodromy method for the calculation of the functional determinant of a special second order differential operator $F=-d^2/d\tau^2+{\ddot g}/g$, $\ddot g= d^2g/d\tau^2$, subject to periodic boundary conditions with a periodic zero mode $g=g(\tau)$. This operator arises in applications of the early Universe theory and, in particular, determines the one-loop statistical sum for the microcanonical ensemble in cosmology generated by a conformal field theory (CFT). This ensemble realizes the concept of cosmological initial conditions by generalizing the notion of the no-boundary wavefunction of the Universe to the level of a special quasi-thermal state which is dominated by instantons with an oscillating scale factor of their Euclidean Friedmann-Robertson-Walker metric. These oscillations result in the multi-node nature of the zero mode $g(\tau)$ of $F$, which is gauged out from its reduced functional determinant by the method of the Faddeev-Popov gauge fixing procedure. The calculation is done for a general case of multiple nodes (roots) within the period range of the Euclidean time $\tau$, thus generalizing the previously known result for the single-node case of one oscillation of the cosmological scale factor. The functional determinant of $F$ expresses in terms of the monodromy of its basis function, which is obtained in quadratures as a sum of contributions of time segments connecting neighboring pairs of the zero mode roots within the period range.