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

370 P. van Nieuwenhuizen, SupergravityThe superdeterminant is defined bysdet M = det Mbb det M~,,= det A det’(D —(3)= det 1 M1bb det~1M 1C) deFt D.1, = det(A — BDIt is sometimes called the Berezinian. In other words, the superdeterminant is the determinant of theBose—Bose part of M times the determinant of the Fermi—Fermi part of the inverse of M. Thisdefinition should of course be shown to be the one needed in the applications. We shall show below thatin path integrals the Faddeev—Popov determinant is indeed this superdeterminant. One usually replacesthis definition by generalizing the well-known formula det A = exp(tr ln A)sdetM = exp(str ln M)str N = ~(—)“N”~ (4)ln M = ln(1 + M — 1) = ~(M — 1)k ()k_I k’where a = 0 for A = bosonic and a = 1 for A = fermionic. (These minus signs for the trace overfermionic elements are reminiscent of the minus signs for closed loops of fermions in Feynmandiagrams.) With this definition of the supertrace one has strM1M2= strM2M1. If M1,2= expN,,2, onecan use the Baker—Hausdorff rule to obtain M,M2 = exp(N, + N2 + multiple commutators) and itfollows from the definitions of the supertrace thatsdet M1 sdet M2 = sdetM,M2. (5)Using the decomposition M = ST and computing str ln S = tr in A, str in T = —tr ln(D —finds the result for sdet M From the definition of sdetM in terms of str in M, it follows thatCA ‘B), one0(sdetM) = (sdet M) str(M’OM) (6)as long as M is a supermatrix. (ln M = in M(ord) + ln(1 + M~’(ord)M(nil)) exists if M ‘(ord) exists.)Another nice way of deriving these results is to start fromM=(’~ ,g)[(~ ~)+K], K=(D21c A~B) (7)so that sdet M = det A(det D)’ sdet(1 + K). Since K is nilpotent, ln(1 + K) is well-defined, and againwith sdet(1 + K) = exp str ln(1 + K) (easy to check directly in this case) one finds1C). (8)sdet(1 + K) = det(1 — A’BDThus one finds again the result for sdet MFor supermatrices onemay define a transition rule satisfying (M1M2)T = M2TM1T. This relation holds ifone defines