Ancillary Computations for "Flux Quantization on 11d Superspace"
Author
Urs Schreiber, Hisham Sati, Grigorios Giotopoulos
Title
Ancillary Computations for "Flux Quantization on 11d Superspace"
Description
Clifford algebra checks for establishing 11d supergravity on superspace
Category
Academic Articles & Supplements
Keywords
supergravity, superspace, Bianchi identities, Clifford algebra
URL
http://www.notebookarchive.org/2024-06-1uize3c/
DOI
https://notebookarchive.org/2024-06-1uize3c
Date Added
2024-06-04
Date Last Modified
2024-06-04
File Size
300.63 kilobytes
Supplements
Rights
CC0 1.0
Download
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(*###############################################################*)(*###Ancillarycomputationsfor:####*)(*###"Flux Quantization on 11d Superspace"####*)(*##-componentofthegravitinoBianchisatisfied,theremaining-and-componentsofthegravitinoBianchifollow.Thecheckofthe-componenttakestwoordersofmagnitudemorecomputingtimethanallothercheckscombined.*)
ncatlab.org/schreiber/show/Flux+Quantization+on+11d+Superspace
##*)(*###############################################################*)(*Here:Verificationthat,assumingflux-andtorsion-Bianchis,thegravitinoBianchiisequivalenttotheRarita-Schwingerequation,whichinturnimpliestheEinsteinequation.*)(*Withattentiontotheconversedirectionthathasbeenrarelyorneverbeendiscussedintheliterature:Namelyproofthatwiththeflux-andtorsion-Bianchisandthe()
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Urs Schreiber, Hisham Sati and Grigorios Giotopoulos
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GAMMA Version 2.0β
9 December 2014
© Ulf Gran, 2001
Remark.Thecommandbyassumingtheirreducibilitycondition⋯0.Inthefollowing,firstthe-componentofthegravitinoBianchiisseentoimplythisconditionforthecomponentofthegravitinofieldstrength;whichisthenused,viaGammaExtract,toshowthatthissame-componentimpliesnofurtherconditiononbeyonditsRarita-Schwingerequationofmotion.Afterwards,theirreducibilityconditionsatisfiedbytheirrepsisused,viaGammaExtract,toshowthatthe-and-componentofthegravitinoBianchiidentityisidenticallysatisfied.(Allthisassumingthattheflux-andthetorsionBianchiidentitieshavealreadybeensolved.)
GammaExtract
providedbyGamma.msimplifiestensor-spinorexpressionsoftheformΓ⋯Γ Ξ
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(*--equation(131),forlateruse--*)H[a_]:=((1/6)*(1/3!)*Tensor[G4,{a,z1,z2,z3}]**GammaProd[{z1,z2,z3}]-(1/12)*(1/4!)*Tensor[G4,{z1,z2,z3,z4}]**GammaProd[{a,z1,z2,z3,z4}])
(*---ChecksforLemma3.2---*)(*--equation(141)--*)
In[]:=
Simplify[-(2/6)*(1/5!)*(1/3!)*3!*Binomial[5,3]*Binomial[3,3]+(2/12)*(1/5!)*(1/4!)*4!*Binomial[5,4]*Binomial[4,4]+(1/2)*(1/4!)]
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Simplify[-(2/6)*(1/5!)*(1/3!)*1*Binomial[5,1]*Binomial[3,1]+(2/12)*(1/5!)*(1/4!)*2*Binomial[5,2]*Binomial[4,2]]
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(*---ChecksforLemma3.5---*)
(*--checking⋯⋯ 84 ⋯inequation(144)--*)Simplify[GammaExtract[GammaContract[GammaExpand[GammaProd[{a1,a2,a3,a4,a5,a6,a7,b1,b2,b3,b4},{b1,b2}]**TensorSpinor[rho,{b3,b4}]-84*ASym[GammaProd[{a1,a2,a3,a4,a5}]**TensorSpinor[rho,{a6,a7}],{a1,a2,a3,a4,a5,a6,a7}]]]]]
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(*------------ChecksforLemma3.6---------------*)
(*---(151)--checking⋯ 845 ---*)Simplify[RenameDummy[GammaExtract[GammaContract[GammaExpand[GammaProd[{a1,a2,a3,a4}]**ASym[GammaProd[{a1,a2}]**Tensor[D,{a3}]**TensorSpinor[rho,{a4,a5}],{a1,a2,a3,a4,a5}]]]]]]-(*---(152)--checking+3+(1/5)- 0---*)Simplify[RenameDummy[GammaExtract[GammaContract[GammaExpand[GammaContract[GammaExpand[GammaProd[{b1,b2}]**ASym[GammaProd[{b1,b2}]**Tensor[D,{a1}]*TensorSpinor[rho,{a2,a3}],{b1,b2,a1,a2,a3}]+3*ASym[Tensor[D,{a1}]*TensorSpinor[rho,{a2,a3}],{a1,a2,a3}]+(1/5)*ASym[GammaProd[{a1,a2}]**Tensor[D,{b}]*TensorSpinor[rho,{b,a3}],{a1,a2,a3}]-ASym[GammaProd[{b,a1}]**Tensor[D,{b}]*TensorSpinor[rho,{a2,a3}],{a1,a2,a3}]]]]]]]]0
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(*------------ChecksforLemma3.7--------------*)(*+++The-componentofthegravitinoBianchi+++++*)(*---------Thecoefficientof------------*)SimplifyRenameDummyGammaContractGammaExpandGammaContractGammaExpandGammaContractGammaExpand(**)32*TensorSpinor[rho,{c,a}]+(**)(-1/3)*GammaProd[{b1,b2,b3},{c}]**ASym[GammaProd[{a,b1}]**TensorSpinor[rho,{b2,b3}],{a,b1,b2,b3}]+(**)(+1/24)*GammaProd[{a,b1,b2,b3,b4},{c},{b1,b2}]**TensorSpinor[rho,{b3,b4}]+(**)(-1/4)*GammaProd[{b1,b2},{c},{b1}]**TensorSpinor[rho,{b2,a}]+(+1/4)*GammaProd[{b1,b2},{c},{a}]**TensorSpinor[rho,{b1,b2}]+(-1/4)*GammaProd[{b1,b2},{c},{b2}]**TensorSpinor[rho,{a,b1}]
Out[]=
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(*---(163)--itscontractionsimplythegravitinoequationandirreducibility:---*)Simplify[RenameDummy[GammaContract[GammaExpand[GammaContract[GammaExpand[GammaProd[{c}]**%]]]]]]Simplify[RenameDummy[GammaContract[GammaExpand[GammaContract[GammaExpand[GammaProd[{a}]**%%]]]]]]
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(*--(165)--conversely,irreducibilityimpliesthiscomponentoftheBianchi:---*)GammaExtract[%%%]
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(*-------Thecoefficientof-------*)SimplifyRenameDummyGammaContractGammaExpandGammaContractGammaExpandGammaContractGammaExpand(**)(-1/3)*GammaProd[{b1,b2,b3},{c1,c2}]**ASym[GammaProd[{a,b1}]**TensorSpinor[rho,{b2,b3}],{a,b1,b2,b3}]+(**)(+1/24)*GammaProd[{a,b1,b2,b3,b4},{c1,c2},{b1,b2}]**TensorSpinor[rho,{b3,b4}]+(**)(-1/4)*GammaProd[{b1,b2},{c1,c2},{b1}]**TensorSpinor[rho,{b2,a}]+(+1/4)*GammaProd[{b1,b2},{c1,c2},{a}]**TensorSpinor[rho,{b1,b2}]+(-1/4)*GammaProd[{b1,b2},{c1,c2},{b2}]**TensorSpinor[rho,{a,b1}]
Out[]=
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(*--(165)--irreducibilityimpliesthiscomponentoftheBianchi:---*)GammaExtract[%]
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(*---Thecoefficientof---*)SimplifyRenameDummyGammaContractGammaExpandGammaContractGammaExpandGammaContractGammaExpand(**)(-1/3)*GammaProd[{b1,b2,b3},{c1,c2,c3,c4,c5}]**ASym[GammaProd[{a,b1}]**TensorSpinor[rho,{b2,b3}],{a,b1,b2,b3}]+(**)(+1/24)*GammaProd[{a,b1,b2,b3,b4},{c1,c2,c3,c4,c5},{b1,b2}]**TensorSpinor[rho,{b3,b4}]+(**)(-1/4)*GammaProd[{b1,b2},{c1,c2,c3,c4,c5},{b1}]**TensorSpinor[rho,{b2,a}]+(+1/4)*GammaProd[{b1,b2},{c1,c2,c3,c4,c5},{a}]**TensorSpinor[rho,{b1,b2}]+(-1/4)*GammaProd[{b1,b2},{c1,c2,c3,c4,c5},{b2}]**TensorSpinor[rho,{a,b1}]
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(*--nudgeMathematicatodiscardthetermsproportionalto---*)ASym[%,{c1,c2,c3,c4,c5}]
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(*--(165)---irreducibilityimpliesthiscomponentoftheBianchi---*)GammaExtract[%]
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(*+++Checkingthe-componentoftheGravitinoBianchi+++*)(*--p.49--*)
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(*---thetermproportionalto---*)GammaContract[GammaExpand[(-1/6)*(1/3!)*(1/11)*ASym[GammaProd[{a1,a2,a3},{a4}],{a1,a2,a3,a4}]**TensorSpinor[Xi,{}]+(-1/12)*(1/4!)*(1/11)*GammaProd[{b,a1,a2,a3,a4},{b}]**TensorSpinor[Xi,{}]+(1/(4*6))*(1/11)ASym[GammaProd[{a1,a2},{a3,a4}],{a1,a2,a3,a4}]**TensorSpinor[Xi,{}]+(1/(4*6))*(1/24)*(-1/77)*(1/5!)*GammaProd[{b1,b2},{b1,b2,a1,a2,a3,a4,c1,c2,c3,c4,c5},(*theLCsymbol*){c1,c2,c3,c4,c5}]**TensorSpinor[Xi,{}]]]
Out[]=
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(*--makethesystemrecalltheskew-symmetry--*)ASym[%,{a1,a2,a3,a4}]
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(*---thetermproportionalto---*)GammaExtract[GammaContract[GammaExpand[(-1/6)*(1/3!)*ASym[GammaProd[{a1,a2,a3}]**TensorSpinor[Xi,{a4}],{a1,a2,a3,a4}]+(-1/12)*(1/4!)*GammaProd[{b,a1,a2,a3,a4}]**TensorSpinor[Xi,{b}]+(1/(4*6))*(-2/9)*ASym[GammaProd[{a1,a2}]**ASym[GammaProd[{a3}]**TensorSpinor[Xi,{a4}],{a3,a4}],{a1,a2,a3,a4}]+(1/(4*6))*(1/24)*(5/9)*(1/5!)*GammaProd[{b1,b2},{b1,b2,a1,a2,a3,a4,c1,c2,c3,c4,c5}(*LCsymbol*)]**ASym[GammaProd[{c1,c2,c3,c4}]**TensorSpinor[Xi,{c5}],{c1,c2,c3,c4,c5}]]]]
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(*---thetermproportionalto---*)GammaExtract[GammaExpand[GammaExtract[GammaExpand[(1/(4*6))*ASym[GammaProd[{a1,a2}]**TensorSpinor[Xi,{a3,a4}],{a1,a2,a3,a4}]+(1/(4*6))*(1/24)*2*(1/5!)*GammaProd[{b1,b2},{b1,b2,a1,a2,a3,a4,c1,c2,c3,c4,c5},(*theLCsymbol*){c1,c2,c3}]**TensorSpinor[Xi,{c4,c5}]]]]]
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(*---thetermproportionalto---*)GammaExtract[GammaExpand[(1/(4*6))*(1/24)*(1/5!)GammaProd[{b1,b2},{b1,b2,a1,a2,a3,a4,c1,c2,c3,c4,c5}(*theLCsymbol*)]**TensorSpinor[Xi,{c1,c2,c3,c4,c5}]]]
Out[]=
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(*---Checkingthe-componentoftheGravitinoBianchi---*)(*--firsttostatethecovariantderivativeofinstages--*)(*--usingLemma3.6--*)(*--(132)--*)H[a_]:=((1/6)*(1/3!)*Tensor[G4,{a,z1,z2,z3}]**GammaProd[{z1,z2,z3}]-(1/12)*(1/4!)*Tensor[G4,{z1,z2,z3,z4}]**GammaProd[{a,z1,z2,z3,z4}])(*--(153)--*)barH[a_]:=((1/6)*(1/3!)*Tensor[G4,{a,z1,z2,z3}]**GammaProd[{z1,z2,z3}]+(1/12)*(1/4!)*Tensor[G4,{z1,z2,z3,z4}]**GammaProd[{a,z1,z2,z3,z4}])(*--(148)--*)FiveIndexDerivative[a1_,a2_,a3_,a4_,a5_]:=(Simplify[GammaExtract[GammaContract[GammaExpand[ASym[barH[a1]**GammaProd[{a2,a3}]**TensorSpinor[rho,{a4,a5}],{a1,a2,a3,a4,a5}]-(1/3)*ASym[Tensor[G4,{y,a1,a2,a3}]*GammaProd[{y}]**TensorSpinor[rho,{a4,a5}],{a1,a2,a3,a4,a5}]]]]])(*--(149)--*)DivergenceOfRho[a_]:=(Simplify[GammaExtract[GammaContract[GammaExpand[(5/84)*GammaProd[{x1,x2,x3,x4}]**FiveIndexDerivative[x1,x2,x3,x4,a]]]]])(*--(150)--*)ThreeIndexDerivative[c1_,c2_,c3_]:=(Simplify[GammaExtract[GammaContract[GammaExpand[GammaContract[GammaExpand[-ASym[GammaProd[{c1,c2}]**DivergenceOfRho[c3],{c1,c2,c3}]+2*barH[w]**ASym[GammaProd[{w,c1}]**TensorSpinor[rho,{c2,c3}],{w,c1,c2,c3}]+(2*5!*84/(7!*4!))*GammaProd[{c1,c2,c3,w1,w2,w3,w4,w5,w6,w7,w8}]**((12/(4!*4!))*Tensor[G4,{w1,w2,w3,w4}]*GammaProd[{w5,w6}]**TensorSpinor[rho,{w7,w8}]-(1/6!)*(1/4!)*Tensor[G4,{v1,v2,v3,v4}]*GammaProd[{u,w1,w2,w3,w4,w5,w6,v1,v2,v3,v4}]**GammaProd[{u}]**TensorSpinor[rho,{w7,w8}])]]]]]])(*--(147)--*)ExteriorDerivativeOfRho[a1_,a2_,a3_]:=((1/3)*ThreeIndexDerivative[a1,a2,a3]-(1/15)*ASym[GammaProd[{a1,a2}]**DivergenceOfRho[a3],{a1,a2,a3}]-(1/3)*GammaProd[{s1,s2}]**FiveIndexDerivative[s1,s2,a1,a2,a3])(*---nowwiththesedefinitions,theactualcheckmentionedonp.49--*)SimplifyRenameDummyGammaExtractGammaContractGammaExpandGammaContractGammaExpand(*--The-componentofthegravitinoBianchi--*)ExteriorDerivativeOfRho[a1,a2,a3]+ASym[H[a1]**TensorSpinor[rho,{a2,a3}],{a1,a2,a3}](*----*)
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(*---ChecksforLemma3.8---*)(*---DerivingtheEinsteinequationfromthegravitinoequation---*)
In[]:=
(*--expandingthe-term(B)--*)Simplify[RenameDummy[GammaContract[GammaExpand[GammaContract[GammaExpand[GammaProd[{a,b1,b2}]**((1/6)*(1/3!)*Tensor[G4,{b1,c1,c2,c3}]**GammaProd[{c1,c2,c3}]-(1/12)*(1/4!)*Tensor[G4,{c1,c2,c3,c4}]**GammaProd[{b1,c1,c2,c3,c4}])**((1/6)*(1/3!)*Tensor[G4,{b2,d1,d2,d3}]**GammaProd[{d1,d2,d3}]-(1/12)*(1/4!)*Tensor[G4,{d1,d2,d3,d4}]**GammaProd[{b2,d1,d2,d3,d4}])]]]]]-(-1/24)(GammaProd[{d0}]*Tensor[G4,{a,d1,d2,d3}]*Tensor[G4,{d0,d1,d2,d3}]-(1/8)GammaProd[{a}]*Tensor[G4,{d0,d1,d2,d3}]^2)]
Out[]=
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Γ
ad0d1
G4
d0d2d3d4
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ad0d1d2d3d4d5
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d0d1d2d6
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ad0d1d2d3d4d5d6d7
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d4d5d6d7
(*--no1-indexGamma-termleft:--*)
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%/.{GammaProd[{a1_,a2_,a3_}]->0,GammaProd[{a1_,a2_,a3_,a4_,a5_,a6_,a7_}]->0,GammaProd[{a1_,a2_,a3_,a4_,a5_,a6_,a7_,a8_,a9_}]->0}
Out[]=
0
Cite this as: Urs Schreiber, Hisham Sati, Grigorios Giotopoulos, "Ancillary Computations for "Flux Quantization on 11d Superspace"" from the Notebook Archive (2024), https://notebookarchive.org/2024-06-1uize3c
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