Replicating ToCanonical see-saw spinors for SU(4)

[IN]

Hi there,

I am working in 6D and would like to use SU(4) spinors. I am trying to recreate the behaviour of the spinor package for four dimensional spinors. I am stuck on trying to reproduce ToCanonical, which uses the seesaw behaviour of spinors to simplify expressions. For example, with the following set up:

DefManifold[M4, 4, {a, b, c, d}]
DefMetric[{1, 3, 0}, g[-a, -b], CD]
DefSpinStructure[g, Spin, {A, B, C, D}, ε, σ, CDe, SpinorPrefix -> SP]

DefTensor[X[-A], M4, Dagger -> Complex]

DefTensor[Y[-A], M4, Dagger -> Complex]

both of the following expressions vanish:

ToCanonical@(X[-A] X[A])

ToCanonical@(Y[-A] X[A]+Y[A] X[-A])

In my set up, I have defined a 4D manifold without a metric.

DefManifold[Spin4, 4, {a, b, c, d}]

Now if I define two tensors:

DefTensor[X[a],Spin4]

DefTensor[Y[a],Spin4]

I would like to be able recreate the see-saw rule so that under some kind of command "SpinorSimplify" I can find that

X[a]Y[-a]+X[-a]Y[a]//SpinorSimplify

is zero.

I have tried a number of different assumptions and pattern matching to reproduce behaviour like this, but am coming up totally blank. Does anyone have any ideas?

Thank you!

[Jose]

Hi,

Spinors implements the see-saw rule by declaring that the epsilon metric is antisymmetric. Unfortunately DefMetric is not prepared to handle that automatically, so the construction of the metric must be done manually. See the code of the DefSpinStructure command in Spinors.m, in particular the code that follows the comment (* 5. Define eps and its conjugate *). Note how eps is defined using

DefTensor[eps[-A, -B], base, Antisymmetric[{1, 2}], Dagger -> Complex, ...]

and then there are things like

MetricQ[eps] ^= True;

etc. You would have to replicate that.

Cheers,

Jose.