New gravitational index theorems and super theorems

Abstract

Gravitational index theorems relate the number of zero eigenvalue modes of various differential operators to the gravitational topological invariants: the Euler number and the Pontryagin number. In order to apply these theorems to quantum theories such as supergravity, which involve operators of spin-0, 1 2 , 1, 3 2 and 2 , we derive index theorems for fields of arbitary spin. In this way, we discover new higher-spin theorems in addition to recovering the well-known low-spin theorems as special cases (i.e., the spin- 1 2 Atiyah-Singer theorem, the Hirzebruch signature theorem and the Gauss-Bonnet theorem). All these theorems relate operators differring by integral spin, i.e., boson to boson and fermion to fermion. We calculate en route, for fields of arbitary spin in an arbitary gravitational field, the coefficients B 4 which appear in the asymptotic expansion of the heat kernel of the corresponding differential operators. The B 4 coefficients are of interest in their own right, since they determine the one-loop counterterms, the stress tensor conformal anomalies, and the axial current anomalies. Next, we look for “super theorems” which relate the number of boson zero-modes to the number of fermion zero-modes. Such theorems may indeed be derived in the case that the gravitational field is self-dual. This in turn reveals an interesting property of supergravity: beyond one loop the theory is finite when the background fields satisfy certain self-duality conditions.