Using xAct to calculate contractions of the torsion tensor in terms of its irreducible components

[Giorgos]

For clarity purposes, I have formulated my problem/question into the attached PDF file. I'd appreciate any and all help!

[Jose]

Hi,

There are two problems in your code:

1. You have TorsionCD (which is zero, because it is the torsion of the Levi-Civita connection) instead of TorsionCDT in a couple of places.

2. Indexed assignments need IndexSet and patterns.

So the code would be something like this:

<< xAct`xTensor`
$PrePrint = ScreenDollarIndices;
DefManifold[M, 4, {a, b, c, d, e, f, h, i, j}];
DefMetric[-1, g[-a, -b], CD];

DefCovD[CDT[-a], SymbolOfCovD -> {"#", "D"}, Torsion -> True, FromMetric -> g];

DefTensor[T[-a], M];
IndexSet[T[a_], 1/3 TorsionCDT[b, a, -b]]

DefTensor[S[-a], M]
IndexSet[S[a_], 1/6 epsilong[b, c, d, a] TorsionCDT[-b, -c, -d]]

DefTensor[q[-a, -b, -c], M]
IndexSet[q[a_, b_, c_], TorsionCDT[a, b, c] - (T[b] g[a, c] - T[c] g[a, b]) + epsilong[a, b, c, d] S[-d]]

Delta = -TorsionCDT[a, -a, -b] TorsionCDT[c, -c, b] - 1/2 TorsionCDT[a, b, c] TorsionCDT[-b, -c, -a] + 1/4 TorsionCDT[c, a, b] TorsionCDT[-c, -a, -b]

Then we can check your relation:

-Delta + (-6 T[-a] T[a] + 3/2 S[-a] S[a] + 1/2 q[a, b, c] q[-a, -b, -c]) // ContractMetric // ToCanonical

(* 0 *)

Cheers,

Jose.

[Giorgos]

Thank you! I totally missed those CDT's, and I wasn't familiar with IndexSet. Is there any way to get Mathematica to output  Delta = (-6 T[-a] T[a] + 3/2 S[-a] S[a] + 1/2 q[a, b, c] q[-a, -b, -c])? The command

Delta = -TorsionCDT[a, -a, -b] TorsionCDT[c, -c, b] - 1/2 TorsionCDT[a, b, c] TorsionCDT[-b, -c, -a] + 1/4 TorsionCDT[c, a, b] TorsionCDT[-c, -a, -b]

outputs

-(1/2)  TorsionCDT[a, b, c]   TorsionCDT[-b, -c, -a] + 1/4  TorsionCDT[-c, -a, -b]   TorsionCDT[c, a, b] - TorsionCDT[a, -a, -b]   TorsionCDT[c, -c, b]

which is basically the relation we put in. That's why you then manually check that Delta is the wanted quantity (-6 T[-a] T[a] + 3/2 S[-a] S[a] + 1/2 q[a, b, c] q[-a, -b, -c]). However, in other cases, we may not have an expression for Delta readily available.