SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

P. van Nieuwenhuizen, Supergravizy 271We now discuss the reason that in (3) one has k > 1. This is because in two-components notationthe Weyl tensor has four dotted or undotted indices and is completely symmetric, so that one can onlycontract self-dual and anti-self-dual Weyl tensor among themselves. Thus, since the Weyl tensor istraceless, one needs at least two of them to obtain a nonvanishing invariant.From these superspace arguments it follows that there are in N = 1 supergravity possible dangerouscounter terms at 3, 4, 5, etc. loops. All these counter terms vanish when the superfield Wabc is self-dualor anti-self-dual (except the one-loop topological counter term which starts with ~ — 4R~,,. + R 2which is nonvanishing in topologically nontrivial spaces). The result seems to be connected with thephenomenon of helicity conservation in supergravity [178].These dangerous counterterms alsofollow fromthe tensor calculus [189].2.17. Super index theoremsIndex theorems relate the number of zero eigenvalue modes of certain elliptic differential operators,such as the Euclidean Dirac operator and the Laplace operator to the topological invariantsx = ~—~Jd4x \/~*R*RMVPU~(1)where x is the Euler number and F the Pontrjagin number (while the Hirzebruch signature is equal to= ~P in four-dimensional compact space without boundary).Let A, be the nonzero eigenvalue of the operator ~, then i(s)= ~ A ~‘ is finite at s —* 0~and countsthe number m of nonzero eigenvalues. On the other handTr e~’= ~ e~” ~ BktU~_4~/2 (2)k =()wheredenotes an asymptotic expansion for small t. Thus, the t-independent terms yield B4 = n + mwhere n is the number of zero eigenvalues. The Bk are integral invariants formed from the Riemanntensor and its covariant derivatives, involving k derivatives. Thus2 + *RMVI,,, RM’~. (3)B4 = Jd4x ~g[a*R~,.,,,,.*RMVIJ~~] +$R~,,.+ yRIn order to obtain relations between the number of zero eigenvalues of two differential operatorswith the same nonzero eigenvalues, one subtracts the two corresponding B4 coefficients so thatB4—B4’=n —n’ [174].One considers fields 4’(A, B) which are irreducible (A, B) representations of the Lorentz group, andfields 4’[A, B] which have 2A dotted and 2B undotted indices but are reducible. Next one introduces aset ofM, operators ~ 0) = ~i(A,B) —DMDM + acting ~R while on 4’(A,B). ~(0,~)differs For example, because the ~i(0,0) connection is the Klein—Gordon DM is different. operator Furthermore,~ ~)= —D,,D~g,,,,.+ RM,.. It is more convenient to consider zl[A, B] acting on Ø[A, B],—DMDbecause one can show that the number of nonzero eigenvalues in [A,B] depends-~mlyon S = A + B but