Geodesic
Vilkovisky--DeWitt effective action in~multidimensional quantum gravity and antiperiodic boundary conditions
Teoretičeskaâ i matematičeskaâ fizika, Tome 80 (1989) no. 1, pp. 150-159

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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. 
@article{TMF_1989_80_1_a12,
     author = {I. L. Buchbinder and V. P. Dergalev and S. D. Odyntsov},
     title = {Vilkovisky--DeWitt effective action in~multidimensional quantum gravity and antiperiodic boundary conditions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {150--159},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1989_80_1_a12/}
}
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I. L. Buchbinder; V. P. Dergalev; S. D. Odyntsov. Vilkovisky--DeWitt effective action in~multidimensional quantum gravity and antiperiodic boundary conditions. Teoretičeskaâ i matematičeskaâ fizika, Tome 80 (1989) no. 1, pp. 150-159. http://geodesic.mathdoc.fr/item/TMF_1989_80_1_a12/