{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Century Schoolbook" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Tex t Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Lucida Sans Typewriter" 1 10 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "KAHLER 2-form MAtti Pitkan en" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 61 "This program uses information from Matti Pitkanen' s appendix." }}{PARA 0 "" 0 "" {TEXT -1 79 "The objective is to get so me handle on what is meant by the Kahler metric, etc." }}{PARA 0 "" 0 "" {TEXT -1 88 "It is apparent that the Kahler 2-form, has a potential equivalent to a 1-form of Action." }}{PARA 0 "" 0 "" {TEXT -1 98 "The following program computes the Field 2-form form the potential, the T opological Torsion 3-form" }}{PARA 0 "" 0 "" {TEXT -1 22 "and the Pari ty 4 form." }}{PARA 0 "" 0 "" {TEXT -1 89 "There are some slight algeb raic differences from the MAple results and what is written in" }} {PARA 0 "" 0 "" {TEXT -1 105 "MP's Appendix. There are also typo inco nsistencies in the the other sections of TDG that I have perused." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " with(liesymm):with(linalg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for close" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "setup(r,thet a,Phi,Psi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%\"rG%&thetaG%$PhiG%$ PsiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "defform(r=0,theta=0 ,Phi=0,Psi=0,a=const,b=const,c=const,k=const,mu=const,m=const,epsilon= const);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(defformG6-/%\"rG\"\"!/%& thetaGF(/%$PhiGF(/%$PsiGF(/%\"aG%&constG/%\"bGF1/%\"cGF1/%\"kGF1/%#muG F1/%\"mGF1/%(epsilonGF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " dR:=[d(r),d(theta),d(Phi),d(Psi)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#dRG7&-%\"dG6#%\"rG-F'6#%&thetaG-F'6#%$PhiG-F'6#%$PsiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "F:=(1+r^2);e0:=d(r)/F;e1:=(r*(sin( theta)*cos(Psi)*d(Phi)+sin(Psi)*d(theta)))/(2*F^(1/2));e2:=(r*(sin(the ta)*sin(Psi)*d(Phi)-cos(Psi)*d(theta)))/(2*F^(1/2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "e3:=r*(d(Psi)+cos(theta)*d(Phi))/(2*F);VOL:=e0&^ e1&^e2&^e3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&\"\"\"F&*$)%\"r G\"\"#\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e0G*&-%\"dG6#%\"r G\"\"\",&\"\"\"F,*$)F)\"\"#F*F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#e1G,$*&*&%\"rG\"\"\",&*(-%$sinG6#%&thetaGF)-%$cosG6#%$PsiGF)-%\"d G6#%$PhiGF)F)*&-F-F2F)-F5F.F)F)F)\"\"\"*$-%%sqrtG6#,&F)F)*$)F(\"\"#F;F )F;!\"\"#F)FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G,$*&*&%\"rG\"\" \",&*(-%$sinG6#%&thetaGF)-F-6#%$PsiGF)-%\"dG6#%$PhiGF)F)*&-%$cosGF1F)- F4F.F)!\"\"F)\"\"\"*$-%%sqrtG6#,&F)F)*$)F(\"\"#F%#e3G,$*&*&%\"rG\"\"\",&-%\"dG6#%$Psi GF)*&-%$cosG6#%&thetaGF)-F,6#%$PhiGF)F)F)\"\"\",&F)F)*$)F(\"\"#F7F)!\" \"#F)F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$VOLG,$*&**)%\"rG\"\"$\" \"\"-%$sinG6#%&thetaG\"\"\",&*$)-%$cosG6#%$PsiG\"\"#F+F0*$)-F-F6F8F+F0 F0-%#&^G6&-%\"dG6#F)-F@F.-F@6#%$PhiG-F@F6F0F+*$),&F0F0*$)F)F8F+F0\"\"$ F+!\"\"#F0\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "Action: =2*r*e3;Field:=wcollect(simplify(d(A)));DField:=d(Field);TOPTORS:=Acti on&^Field;PARITY:=d(TOPTORS);Ratio:=factor(getcoeff(PARITY)/getcoeff(V OL));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ActionG*&*&)%\"rG\"\"#\"\" \",&-%\"dG6#%$PsiG\"\"\"*&-%$cosG6#%&thetaGF0-F-6#%$PhiGF0F0F0F*,&F0F0 *$F'F*F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&FieldG,(*&*(%\"rG \"\"\"-%$cosG6#%&thetaGF)-%#&^G6$-%\"dG6#F(-F26#%$PhiGF)\"\"\",(F)F)*$ )F(\"\"#F7F;*$)F(\"\"%F7F)!\"\"F;*&*(F(F7,&*&F(F7-%$sinGF,F)F)*&)F(\" \"$F7FDF7F)F)-F/6$-F2F,F4F)F7F8F?!\"\"*&*&F(F7-F/6$F1-F26#%$PsiGF)F7F8 F?F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DFieldG\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(TOPTORSG,$*&*(-%$sinG6#%&thetaG\"\"\")%\"rG\" \"%\"\"\"-%#&^G6%-%\"dGF*-F56#%$PhiG-F56#%$PsiGF,F0,(F,F,*$)F.\"\"#F0F ?*$F-F0F,!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'PARITYG,$*&* ()%\"rG\"\"$\"\"\"-%$sinG6#%&thetaG\"\"\"-%#&^G6&-%\"dG6#F)-F5F.-F56#% $PhiG-F56#%$PsiGF0F+*&,&F0F0*$)F)\"\"#F+F0\"\"\",(F0F0F@FB*$)F)\"\"%F+ F0\"\"\"!\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RatioG,$*&\"\" \"F',&*$)-%$cosG6#%$PsiG\"\"#F'\"\"\"*$)-%$sinGF-F/F'F0!\"\"!#K" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Kahler Potential B = Action 1-form ." }}{PARA 0 "" 0 "" {TEXT -1 31 "Kahler 2-form J = Field 2-form." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "KAHLER_POTENTIAL:=A;KAHLER_2FORM:=wcollect(simplify(F));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%1KAHLER_POTENTIALG*&*&)%\"rG\"\"#\" \"\",&-%\"dG6#%$PsiG\"\"\"*&-%$cosG6#%&thetaGF0-F-6#%$PhiGF0F0F0F*,&F0 F0*$F'F*F0!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-KAHLER_2FORMG,(* &*(%\"rG\"\"\"-%$cosG6#%&thetaGF)-%#&^G6$-%\"dG6#F(-F26#%$PhiGF)\"\"\" ,(F)F)*$)F(\"\"#F7F;*$)F(\"\"%F7F)!\"\"F;*&*(F(F7,&*&F(F7-%$sinGF,F)F) *&)F(\"\"$F7FDF7F)F)-F/6$-F2F,F4F)F7F8F?!\"\"*&*&F(F7-F/6$F1-F26#%$Psi GF)F7F8F?F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK " 11" 0 }{VIEWOPTS 0 1 0 1 1 1803 }