Counterterms in extended supergravities

This chapter examines ultraviolet divergences in supergravity and presents the quantization of gauge theories with open algebras in the representation with the third ghost. Topics considered include counterterms in extended supergravities, the third ghost, and open algebra. After the first success of supergravity (2-loop finiteness of N=1 theory) there was a hope that for higher N the situation would improve. The 2-loop order and the 3-loop order are investigated. The correlation between superspace geometry, conformal anomalies, and the background formulation of quantum supergravity. The role which the third ghost plays in the quantization of theories with open algebra is clarified. It is concluded that the presented quantization of gauge theories is the most convenient way of quantization of gauge theories with open algebra containing the third ghost