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

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374 P. van Nieuwenhuizen, SupergravityHermitian, it is not true that the extra terms C 1MC~+ C2MC2 vanish (their kinetic parts vanish). Notethat we did not have to assume any special properties for MSummarizing, we have the following results1 =Jfldz’~o(F~(z)—a~)sdet(8F~/3z), ~A = (x”, 0”)sdet(3FA/OzB) = fri dC~dC~exp(iC~(DFA/3z”)C~’).(17)H. The super Jacobian is the superdeterminantIn the covariant quantization of supergravity using path integrals, an essential ingredient is the superJacobian. We show here that for a general change of integration variables (x, 0) to 1(x, 0), O(x, 0) thesuper Jacobian is given by the superdeterminant:m(x, 0)/ar). (1)J = det(Ox’(i, ë)/91’) det(80Defining the supermatrices M and M’ byM — (Dx/81 8x/30\ M’ — (31/Ox 31/30 2— \a0/81 89/30)’ — ~09/8x 80/80one need specify only for Ox/0O and 31/00 whether one has left or right directives. One needs rightderivatives, in order that (Ox/31)(31/Ox) + (Ox/00)(00/Ox) be the unit matrix. Hence the super Jacobian isthe determinant of the Bose—Bose part of M (the usual result) times the determinant of theFermi—Fermi part of the inverse of M For infinitesimal changes one has J = ~ Ox’/81’ — Oem/Oem, and aproof that J equals the superdeterminant has been given by DeWitt, based on the principle thatexponentiation of this linearized result gives the superdeterminant. Here we give a direct proof valid forfinite transformations and due to Fung. -We consider the change of variables(x, 0) —* (1, 0) in two steps(i) we change 9~,... ~ to em, ... , 0~,keeping all xfixed;(ii) we change x’ x”~to 1’,. . . , I”, keeping all 0 fixed.The second change of variables has an ordinary Jacobiandet(Ox’ (1, ë)/D1’).Notice that it is essential that this change of variables is performed when the Fermi variables are alreadybarred — otherwise one would find the mixed determinant det(0x’(x, 0)101’). We now consider (i).Consider an integral of the formI=Jdxl...dxkdom...de1P(x,o)— — (3)= J dI’ . . . dl” dotm .. d~1JP(x(1, 9), 0(1, 9))
P. van Nieuwenhuizen. Supergravity 375where P is a polynomial in 0 and an arbitrary function of x, and J the super Jacobian to be determined.One can for any given i decompose P(x, 0) = A .+ O’B, where A and B, are independent of 0’. Thus,with i = 1whereI dxdO(A,+01B,)=fdxd0mf- - (4)dx dotm ••~dO2do1(A,+O1B,)~_fdx dOtm •~•d02(~9Ü’/~9ö’)B,0’ = ~‘(x’ X~z,~ . .. , 02, ~1) (5)is obtained by inverting 0’ = O’(x1, ... , x~,em, ... , 0’) for fixed x and em, ..., 0’. Thus, in going fromdo’ to di’, the Jacobian equals (Oö’/0O’)’.Next we change 02 to ë2 keeping x, 0~ o~and O~fixed. Again one can write (0O’/801)P asA2 + 02B2 where A2 and B2 do not depend on g2 and defining2(x’, ... , x”, ~m , 0~ë2 ~•‘) (6)02 = ~one finds a Jacobian (002/002)1. The total Jacobian for (i) is thus(aö’/a~)]1. (7)It is now a general theorem that for any change of variables y’ = y’(9) with i = 1, m one can writethe Jacobian as followsdet(0y’/89~)= (09m/09m)... (091/891) (8)where 9m(7) = ym(9) and 9m_1, . . , 9’ are obtained by successively inverting y = y(9) as followsyl=9l(9l,y2,.~~,ym)y2=92(7l92ySytm)yrn=9m(9l9m)(9)The proof is simple. Define F’(y1, ... , ym, 91, ... 5?fl) = — y’. Thendet(Oy’/89’) = (_)m det(OF’189’) deF’(OF’/Oy’)097091 0 ~ —1 891/0y2~ o9~/a9’ 892/a92 0... ÷ 0 —1 092/0y3.= (091/091). .. (09m/05m) (10)
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In memoriam Joel ScherkJoel Scherk
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SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

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