Gravity and global symmetries

There exists a widely held notion that gravitational effects can strongly violate global symmetries. If this is correct, it may lead to many important consequences. We argue, in particular, that nonperturbative gravitational effects in the axion theory lead to a strong violation of CP invariance unless they are suppressed by an extremely small factor g10^-82. One could hope that this problem disappears if one represents the global symmetry of a pseudoscalar axion field as a gauge symmetry of the Ogievetsky-Polubarinov-Kalb-Ramond antisymmetric tensor field. We show, however, that this gauge symmetry does not protect the axion mass from quantum corrections. The amplitude of gravitational effects violating global symmetries could be strongly suppressed by e^-S, where S is the action of a wormhole which may absorb the global charge. Unfortunately, in a wide variety of theories based on the Einstein theory of gravity the action appears to be fairly small, S10. However, we find that the existence of wormholes and the value of their action are extremely sensitive to the structure of space on the nearly Planckian scale. We consider several examples (Kaluza-Klein theory, conformal anomaly, R^2 terms) which show that modifications of the Einstein theory on the length scale l10M₏^-1 may strongly suppress violation of global symmetries. We find also that in string theory there exists an additional suppression of topology change by the factor e^-82/g^2. This effect is strong enough to save the axion theory for the natural values of the stringy gauge coupling constant.