Quit

This worksheet computes the coefficients a, b, c, d and e, that appear in the equation (2.15) of https://arxiv.org/pdf/1610.08519.pdf

Set format output: OutputForm

<<Ricci.m

DefineBundle[tangent,4,g,{i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,i17,i18,i19,i20,i21,i222,i23,i24,i25,i22,i23,i24,i25,i26,i27,i28,i29,i30,i31,i32,i33,i34,i35,i36,i37,i38,i39,i40,i41,i42,i43,i44,i45,i46,i47,i48,i49,i50,i51,i52,i53,i54,i55,i56,i57,i58,i59,i60,i61,i62,i63},MetricType->Riemannian]

DefineTensor[k,2,Symmetries->Symmetric]

DefineTensor[χ,0]

DefineRelation[k[L[i1],U[i1]],χ]

DefineTensor[Rt,{2,2}]

DefineRelation[Rt[U[i1],U[i2],L[i3],L[i4]],Λ(g[U[i1],L[i3]]g[L[i4],U[i2]]-g[U[i1],L[i4]]g[L[i3],U[i2]])+α(k[U[i1],L[i3]]k[L[i4],U[i2]]-k[U[i1],L[i4]]k[L[i3],U[i2]])]

Simplification rules are defined below

DefineRule[kcuak1,k[L[i1],L[i3]] k[U[i3],U[i2]],k[L[i1],U[i2]]]

DefineRule[kcuak2,k[L[i1],U[i3]] k[L[i3],U[i2]],k[L[i1],U[i2]]]

DefineRule[kcuak3,k[L[i1],L[i3]] k[L[i2],U[i3]],k[L[i1],L[i2]]]

DefineRule[kcuak4,k[L[i1],U[i3]] k[L[i2],L[i3]],k[L[i1],L[i2]]]

DefineRule[kcuak5,k[L[i1],U[i3]] k[L[i3],L[i2]],k[L[i1],L[i2]]]

DefineRule[kcuak6,k[L[i1],U[i3]] k[U[i1],L[i2]],k[L[i2],U[i3]]]

DefineRule[Cuak,k[L[i1],L[i2]] k[U[i1],U[i2]],χ]

DefineRule[kcub1, k[L[i1], L[i2]]k[U[i1],L[i3]]k[U[i2],U[i3]],χ]

A general combination of the eight-independent R^3 terms (all the eight of them are independent in dimension greater than six. In dimension bwlo seven, some relations appear)

TensorSimplify[A1*Rt[L[i1],U[i3],L[i2],U[i4]]*Rt[L[i3],U[i5],L[i4],U[i6]]*Rt[L[i5],U[i1],L[i6],U[i2]]+A2*Rt[L[i1],L[i2],U[i3],U[i4]]*Rt[L[i3],L[i4],U[i5],U[i6]]*Rt[L[i5],L[i6],U[i1],U[i2]]+A3*Rt[L[i1],L[i2],L[i3],L[i4]]*Rt[U[i1],U[i2],U[i3],L[i5]]*Rt[U[i4],U[i6],U[i5],L[i6]]+A4*Rt[L[i1],L[i2],L[i3],L[i4]]*Rt[U[i1],U[i2],U[i3],U[i4]]*Rt[U[i5],U[i6],L[i5],L[i6]]+A5*Rt[L[i1],L[i2],L[i3],L[i4]]*Rt[U[i1],U[i5],U[i3],L[i5]]*Rt[U[i2],U[i6],U[i4],L[i6]]+A6*Rt[L[i1],U[i5],U[i2],L[i5]]*Rt[L[i2],U[i6],U[i3],L[i6]]*Rt[L[i3],U[i4],U[i1],L[i4]]+A7*Rt[L[i1],U[i5],L[i2],L[i5]]*Rt[U[i1],U[i6],U[i2],L[i6]]*Rt[U[i3],U[i4],L[i3],L[i4]]+A8*Rt[U[i1],U[i2],L[i1],L[i2]]*Rt[U[i3],U[i4],L[i3],L[i4]]*Rt[U[i5],U[i6],L[i5],L[i6]]]/.kcuak1/.kcuak2/.kcuak3/.kcuak4/.Cuak/.Cuak/.kcuak2/.kcuak5/.kcuak6/.kcuak2/.kcuak6

The Lagrangian evaluated on the shifted curvature Rtilde reads

f[α_]=%

And the coefficients are

e=Simplify[f'[0]/(2χ(χ-1))]

T2= Simplify[f''[0]/(4χ(χ-1))]

a=Limit[T2, χ-> 1]

T3= Simplify[(T2 -a)/(χ-1)]

c=Limit[T3, χ-> 0]

b= Simplify[(T2 - a - c*(χ-1))/(χ(χ-1))]

Final Check

ff1=2a+c//Simplify

ff2=a+2b+c//Simplify