Monodromies and functional determinants in the CFT driven quantum cosmology

Abstract

We apply the monodromy method for the calculation of the functional determinant of a special second order differential operator $F=-d^2/d{\tau }^2+\ddot{g}/g$, $\ddot{g}=d^{2}g/d{\tau }^2$, 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.

11 pages, final version published in Phys. Rev. DD85, 064006 (2012)

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Cited by in corpus (1)

• Microcanonical quantum cosmology: gauge fixing and functional determinants on a circle