Vilkovisky–DeWitt effective action in multidimensional quantum gravity and antiperiodic boundary conditions

The Vilkovisky–De Witt effective action (EA) in the $d=5$ quantum gravity at zero and non-zero temperature on the background $R_4\times S_1$ where $R_4$ is the 4-dimensional space and $S_1$ is the one-dimensional sphere, is obtained. The problem of choosing periodic or antiperiodic boundary conditions for gravitational field on the sphere is considered. In all cases considered the spontaneous compactification is not physical (the conditions of spontaneous compactification are fulfilled but the 4-dimensional gravitational constant with the wrong sign is induced). The Vilkovisky–De Witt EA in the $d=5$ quantum supergravity on the background $R_4\times S_1$ at non-zero temperature is calculated.