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

268 P. Lan Nieuwenhui:en. SupergraLitvAxial anomaly — The action of a gravitino in a background gravitational field= ~ (1)is gauge invariant under 64,,, = D,,(w(e)) , if the gravitational field satisfies the Einstein equationsGMV(e) = 0. Thus one must fix the gauge. Two obvious choices are~2?’(fix) = ~ yjy• 4’, 2?”(fix) = ~ y,ø(w(e))y. 4,. (2)In the second choice, the quantum action is manifestly general coordinate invariant, but one must addNielsen—Kallosh ghosts (see subsection 2.4). In this gauge one can apply the well-known Adler—Rosenberg method for obtaining the axial anomaly. Denoting the matrix element for the axial currentgoing to two gravitons (a, /3, p’) and (y, 6, p2) by MM~~, 5(p, P2)’ general coordinate invariance= 0 relates divergent to convergent form factors, and the axial anomaly A =(Pl.M +p2,M)MM,,,,.,,.,,s = 0 can be expressed in terms of finite integrals. Clearly, the contributions of thecomplex and real spin 1/2 ghosts (both commuting) are known from the spin 1/2 axial anomaly, and oneonly has to apply the Adler—Rosenberg method to the gravitino loop. The answer is —21 times theanomaly for a real anticommuting spin 1/2 field, and was first obtained by Duff and Christensen by atopological method [172]. Here we will follow the Adler—Rosenberg method. For comparison betweenthe various existing methods, see ref. [530].We must begin by writing down the axial currents. The classical axial current is the Noether currentfor the global symmetry 64’, == —! ~M~P~74jy4j (3)and transforms under gauge transformations 64,,, = D,(w(e)) into= _ia,,E M~‘~(4,,.y,,e) — iiy 5R M (4)so that the classical axial charge is gauge invariant on-shell. At the quantum level the gravitino equationacquires an extra term from the gauge fixing term [344]— 3J~U/~q, = M~”y5y,.D~i/i,, — ~ yMØy. 4,. (5)At the quantum level we also use BRST transformations rather than gauge transformations, and with.SC = A (— ~4, y,~)one finds that the axial current due to the Faddeev—Popov ghost action (whichcorresponds to F,, = —y~4’) transforms under BRST transformations as followsj~= ieC’y My 5C, 6j~p= — ~ eçti y,Ø7M75CA (6)and the extra term in (5) is produced by (6) so that again j7~+j~gives on-shell a BRST-invariant