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

372 P. van Nieuwenhuizen. SupergravityIn what follows we will show that(I) fixing the gauge in the path integral one finds as a generalization of the Faddeev—Popovdeterminant a superdeterminant;(ii) exponentiating this superdeterminant by means of a super Gaussian integral one finds thegeneralized ghost actions.Consider a path-integral with bosonic and fermionic symmetries. We want to multiply by unitywriting as usual1 = Jd~”d~ 1,0(Fa foe) o(B 1, — b1,)J (5)where ~(~)are the bosonic (fermionic) gauge parameters, B(F) the gauge choices and b(f) ordinary(Grassmann) variables. We must first define the Dirac delta function for fermionic arguments. SincefdOO = 1, it follows that 0(O)= 0. Thus 0(F,. —f,.) is equal to (F,. —f,.) for fermionic F,. —f,,. Making achange of integration variables (~,~)-÷ (B, F), it follows that J is the Jacobian for the change ofintegration variables (B, F)—~~ (~,~j).Theorem: The Jacobian for x” —~ x’(x’, 0’),9”’ —* 0”(x’, 0’) is given by the superdeterminant1,/3x’)] (6)—det[~x”/3x’— (19f/o0”) det(t90”’/t991,) (30’/39)~”1,(89’where it does not matter whether one uses left or right derivatives except that lJx”/90” is a rightderivative. (In all other cases we use left derivatives.)We will begin by considering linear transformation(x’\(A B\fx 7~o’i ~C DROand derive that J = sdet M In this case it is clear that 3x”/30” is a right derivative. Consider first thecase of linear changes of anticommuting variables 0’ (i = 1, N). FromJ d9~ .dONOSJ 61=1=JdOhI...dOlNO7~’~=Jdo1...deNJo~N...o~1(8)with 0” = D’,O’ it follows that J = det D’, just the inverse oL what one has for bosonic variables. Forgeneral linear transformations involving commuting x’ and anticommuting 90 one has/x’\ fA B\ fx\ /1 BD1\ fA—BD’C 0\ / I O\ fx 9~o’) = ~C D) ~e) = ~o i I ~ 0 D) ~D1C i) ~o ()The first and last transformations are shifts of integration variables 0” and x’ respectively, and since theintegrals are translationally invariant these Jacobians equal unity. The second change of integration