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

More documents

Recommendations

Info

208 P. van Nieuwenhuizen. SupergrarirvS2’( 2) = ~1. ~‘ mncd (D,,e~)e ~ 800cdD ~ tm~1~ mn (3,.e,. — 0,,e,. w,, er,..Note that the derivative D,, always contains only the spin connection, but not a connection F~,..As aresult, D,.e’ is non-zero.Next we vary the spin connection in the Rarita—Schwinger action.82’312 = ‘ ~“~~(tl/yyut/i) (80)cd) (4)Since l/i,.y5y~u~dt//,, can be written as vector terms and axial vector terms by decomposing Y~O~cd(seeappendix A)l/i,.757,ArcdtIJ,, = ~tlJ,.yS(e~~yd — ed~yC)llI,,+ ~et~,, bcdml/i,.ymt/,,, (5)and since l/’,.ylydt/i,, is symmetric in p~and o- while ~,.y’~i/i,, is antisymmetric (see appendix C), we find82’3/2 = — 8 1. ~“ ncdm (~ym~, )en ~ cd (6)Comparison of eqs. (6) and (3) thus yieldstm,, — D~etm,.= ~- (çj~ytmç/i) (7)D,.efor the field equation of the spin connection. Due to our rewriting of 2’2 as in eq. (1), one can easilyread off this result, since in eq. (6) and in eq. (1) the two -symbols are in common.With the first tetrad postulate (which is really a definition of w(e)) that 0,.em~+w,.m~(e)—= 0, one can proceed to solve from eq. (7) the spin connection ~ itself. To this purpose itis useful to introduce the contorsion tensor K,.””’ byma — mtm( ~+ ma(0,. —00,. ~, K,.w,.rn~(e) = ~e~”(8,.e~~ — 8~e~,.) — ~ (8,.emv — Opem,.) — ~em”en~(8~e~,, — 8,,e,~,)e’,.. (8)With (7) and the first tetrad postulate, one finds0,.etm~+ wtm(e) (,~ v) 0K2- 9— Kr,,,,. = ~ tli,,y~tl/~. ( )This equation is solved in the same way as one solves in general relativity for F”,.~(g)in terms of ~that is to say, one considers the identity— K~~,.)+~ — iç,.m)+ ~ — Içptm) = 2K,.,,,~. (10)
P. van Nieuwenhuizen, Supergravitv 209Substituting (9), one arrives at~ ~/i)= ~ — ~,.ynl/im + ~rny,.ll’n). (ii)Using the second tetrad postulate, 8,.e~+ ,.a~ — F~,.e”~ = 0 one can obtain an expression for F”,.~from eq. (ii). In general one defines torsion as the antisymmetric part of F”,.~= ~(F~— F~,.). (12)Clearly, F”,.~— F~,.= —K,.”~+ K~a,.,so that the torsion is given byS”,.~= —~-(t/i,.y”çfi,,). (13)In a similar manner as in eq. (10), one can solve from D,.(w(e))e’ —~s~-* ii = 0 the dependence ofw~,b(e)on the tetrad field. (Another way is to solve directly the tetrad postulate 8,.etm,, + w,.m”(e)e~~ —= 0.) Putting this result together with eq. (ii), one finds~ i/i) = ~ — R,.mn + Rmn,.)R,.p,m = —8,.e~~+ 8~e,,,,.+ ~- çl’,.y~i/i~. (14)The symbol R,.,b.m is defined to be ~ similarly for the other two R-functions. Clearly, the spinconnection w~”’(e,t/i) is supercovariant, by which is meant that if one transforms it under localsupersymmetry transformations, then it contains no 8 terms. Indeed, the terms (—~8,.ë)y”i/i,.comingfrom 8(—8,.em~)cancel against the 8 terms from Oil’,.. The symbols R,.~.rnare supercovariant bythemselves. That the solution of SI/Ow = 0 is supercovariant is an accident. For example, in theextended supergravities this equality does not hold, see subsection 6.2.1.5. Flat supergravity with torsionIn Einstein—Cartan theory, the Hubert action has a well-known symmetry f d~xeR(e, w(e)+ r) =f d~xe[R(e, 00(e)) — T + 1. (rAA,. )2] under w,.”~’1.V~T’~~0)ab + r,.~thwith T,.”~’= — ~ba but further arbitrary.The proof is trivial if one writes the terms linear in r in R,.,~flb(w(e)+r) as D,.(w(e))rpab — p.~*-~ 71 sinceone may add a Christoffel connection to D,.r~~5(because it cancels anyhow in the curl) and having thusobtained the full covariant deriv,ative, one can partially integrate to obtain a total derivative.In supergravity a similar identity holds. It is a matter of a small computation to show that [207]2’~ 2~(e, w(e, ~/~)+r)+2’312(e, t/’, w(e, t/i)+ r)= 2’~2~(e, w(e, I/i)) + 2’312(e, i/i, w(e, t/J)) + ~-~--~ (r,,~~r”’~ — (TAA,.)2) + total derivative. (1)Since w(e, I/i) is an extremum of the action, terms linear in r cancel.
Page 3 and 4: In memoriam Joel ScherkJoel Scherk
Page 6 and 7: 194 P. van Nieuwenhuizen, Supergrav
Page 12 and 13: 200 P. van Nieuwenhuizen. Supergrav
Page 14 and 15: 202 P. tan Nieuwenhuizen, Supergrav
Page 32 and 33: 220 P. ran Nieuwenhuizen. Supergrar
Page 62 and 63: 250 P. ran Nieuwenhuizen, Supergrav
Page 70 and 71:
258 P. van Nieuwenhuizen, Supergrav
Page 72 and 73:
Page 74 and 75:
262 P. van Nieuwenhuizen. Supergrav
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
268 P. Lan Nieuwenhui:en. SupergraL
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
278 P. t’an Nieuwenhuizen, Superg
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
(A V,.”)00 = ~ s*a = (D,. + ~ A,.
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
328 P. van Nicuwenhuizen, Supergrav
Page 142 and 143:
330 P. van Nieuwenhuizen Supergravi
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
354 P. van Nieuwenhuizen Supergravi
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
392 P. van Nieuwenhui’zen, Superg
Page 206 and 207:
Page 208 and 209:
Page 210:
show all

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

Create successful ePaper yourself

Delete template?

Save as template?