The epsilon spinor is a rank 2 spinor which is fully skew-symmetric and whose component values are 1 or -1.

The command EpsilonSpinor(indexType, spinorType) returns the epsilon symbol of the type specified by indexType and spinorType in the current frame unless the frame is explicitly specified.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form EpsilonSpinor(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-EpsilonSpinor.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{P1}≔\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\right)$

$\mathrm{P1}:=\mathrm{dz1}\mathrm{dz2}-\mathrm{dz2}\mathrm{dz1}$

$\mathrm{P2}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P2}:=\mathrm{D\_z1}\mathrm{D\_z2}-\mathrm{D\_z2}\mathrm{D\_z1}$

$\mathrm{P3}≔\mathrm{EpsilonSpinor}\left(''cov''\,''barspinor''\right)$

$\mathrm{P3}:=\mathrm{dw1}\mathrm{dw2}-\mathrm{dw2}\mathrm{dw1}$

$\mathrm{P4}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P4}:=\mathrm{D\_z1}\mathrm{D\_z2}-\mathrm{D\_z2}\mathrm{D\_z1}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\,M\right)$

$\mathrm{dz1}\mathrm{dz2}-\mathrm{dz2}\mathrm{dz1}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,t\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{P1}≔\mathrm{EpsilonSpinor}\left(''cov''\,''spinor''\right)$

$\mathrm{P1}:=\mathrm{dz1}\mathrm{dz2}-\mathrm{dz2}\mathrm{dz1}$

$\mathrm{P2}≔\mathrm{EpsilonSpinor}\left(''con''\,''spinor''\right)$

$\mathrm{P2}:=\mathrm{D\_z1}\mathrm{D\_z2}-\mathrm{D\_z2}\mathrm{D\_z1}$

$\mathrm{P5}≔\mathrm{ContractIndices}\left(\mathrm{P2}\,\mathrm{P1}\,\left[\left[1\,1\right]\right]\right)$

$\mathrm{P5}:=\mathrm{D\_z1}\mathrm{dz1}+\mathrm{D\_z2}\mathrm{dz2}$

$\mathrm{P5}\&minus\mathrm{KroneckerDeltaSpinor}\left(''spinor''\right)$

$0\mathrm{D\_z1}\mathrm{dz1}$