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We apply the monodromy method for the calculation of the functionaldeterminant 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 periodicboundary conditions with a periodic zero mode $g=g(\tau)$. This operator arisesin applications of the early Universe theory and, in particular, determines theone-loop statistical sum for the microcanonical ensemble in cosmology generatedby a conformal field theory (CFT). This ensemble realizes the concept ofcosmological initial conditions by generalizing the notion of the no-boundarywavefunction of the Universe to the level of a special quasi-thermal statewhich is dominated by instantons with an oscillating scale factor of theirEuclidean Friedmann-Robertson-Walker metric. These oscillations result in themulti-node nature of the zero mode $g(\tau)$ of $F$, which is gauged out fromits reduced functional determinant by the method of the Faddeev-Popov gaugefixing procedure. The calculation is done for a general case of multiple nodes(roots) within the period range of the Euclidean time $\tau$, thus generalizingthe previously known result for the single-node case of one oscillation of thecosmological scale factor. The functional determinant of $F$ expresses in termsof the monodromy of its basis function, which is obtained in quadratures as asum of contributions of time segments connecting neighboring pairs of the zeromode roots within the period range.

Monodromies and functional determinants in the CFT driven quantum cosmology