Let $▿$denote covariant differentiation with respect to the given solder form $\mathrm{\σ}\.$ A type $\left(p\, q\right)$ symmetric spinor $S$with components $S_{B\cdot \cdot \cdot C}^{B\'\cdot \cdot \cdot C\'}$ ( $p$ lower unprimed indices and $q$upper primed indices) is a Killing spinor if$$${▿}_{\(A}^{\(A\'}{S}_{B\cdot \cdot \cdot C\)}^{B\'\cdot \cdot \cdot C\'\)}$ = 0.

The command KillingSpinor generates the defining system of 1st order PDE for a Killing spinor and uses pdsolve to find the solutions to these PDE.

The keyword argument coefficientvariables $\= \left[x^1 \, x^2\, ..\. \, x^k\right]$ allows the user to specify the coefficient functions in the Killing spinor $S$$$as functions of the variables $\left[x^1 \, x^2\, ..\. \, x^k\right]$ .

The exact form of the spinor $S$can be specified with the keyword argument ansatz $\= S\.$ For example, if the coordinates on the underlying manifold are $\left[x\, y\, z\right]$and $S_1\, S_2$are defined type $\left(p\, q\right)$spinors, then one may solve for Killing spinors tensors of the form $T \= f\left(y\,z\right)S_1 \+ g\left(y\,z\right)S_2$. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns $\= \left[f\left(y\,z\right)\, g\left(y\,z\right)\right]\.$

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations $\= \mathrm{EqList}\.$Here $\mathrm{EqList}$is a list of the auxiliary equations to be added to the Killing spinor equations.

If the solder form $\mathrm{\σ}$depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters$\= \mathrm{ParList}\,$where $\mathrm{ParList}$is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing spinors changes, are calculated. Additional algebraic or differential conditions may be imposed upon the parameters using the keyword argument auxiliaryequations $\= \mathrm{EqList}\.$

With keyword argument output = $''pde''\,$the defining partial differential equations for the Killing spinor are returned. The option output = $''general''$returns the general solution in terms of a number of arbitrary constants ${\mathrm{\_C}}_1$, ${\mathrm{\_C}}_2$, ... while the option output = $''list''$returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output = $''list''\.$

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

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

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

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

$g≔\mathrm{evalDG}\left(2\exp \left(2\mathrm{\rho }x\right)\mathrm{du}\&t\mathrm{du}-\mathrm{du}\&t\mathrm{dv}-\mathrm{dv}\&t\mathrm{du}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}\right)$

$g:=2{\ⅇ}^{2\mathrm{\rho }x}\mathrm{du}\mathrm{du}-\mathrm{du}\mathrm{dv}-\mathrm{dv}\mathrm{du}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}$

$\mathrm{OT}≔\mathrm{evalDG}\left(\left[-\frac{\frac{1}{2}\exp \left(-\mathrm{\rho }x\right)\mathrm{D\_u}}{\mathrm{\rho }}+\frac{\frac{1}{2}\exp \left(\mathrm{\rho }x\right)\left(2{\mathrm{\rho }}^2-1\right)\mathrm{D\_v}}{\mathrm{\rho }}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\frac{\frac{1}{2}\exp \left(-\mathrm{\rho }x\right)\mathrm{D\_u}}{\mathrm{\rho }}+\frac{\frac{1}{2}\exp \left(\mathrm{\rho }x\right)\left(1+2{\mathrm{\rho }}^2\right)\mathrm{D\_v}}{\mathrm{\rho }}\right]\right)$

$\mathrm{OT}:=\left[-\frac{{\ⅇ}^{-\mathrm{\rho }x}}{2\mathrm{\rho }}\mathrm{D\_u}+\frac{{\ⅇ}^{\mathrm{\rho }x}\left(2{\mathrm{\rho }}^2-1\right)}{2\mathrm{\rho }}\mathrm{D\_v}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\frac{{\ⅇ}^{-\mathrm{\rho }x}}{2\mathrm{\rho }}\mathrm{D\_u}+\frac{{\ⅇ}^{\mathrm{\rho }x}\left(2{\mathrm{\rho }}^2+1\right)}{2\mathrm{\rho }}\mathrm{D\_v}\right]$

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(\mathrm{OT}\right)$

$\mathrm{\σ}:=-\frac{{\ⅇ}^{\mathrm{\rho }x}\sqrt{2}}{2\mathrm{\rho }}\mathrm{du}\mathrm{D\_z1}\mathrm{D\_w1}-{\ⅇ}^{\mathrm{\rho }x}\sqrt{2}\mathrm{\rho }\mathrm{du}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{{\ⅇ}^{-\mathrm{\rho }x}\sqrt{2}}{2\mathrm{\rho }}\mathrm{dv}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}}{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}$

$\mathrm{SpinorInnerProduct}\left(\mathrm{\sigma }\,\mathrm{\sigma }\right)$

$2{\ⅇ}^{2\mathrm{\ρ}x}\mathrm{du}\mathrm{du}-\mathrm{du}\mathrm{dv}-\mathrm{dv}\mathrm{du}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}$

$\mathrm{KS1}≔\mathrm{KillingSpinors}\left(\mathrm{\sigma }\,1\,0\right)$

$\mathrm{KS1}:=\left[{\ⅇ}^{-\frac{1}{2}\mathrm{\ρ}x}\mathrm{D\_z1}\right]$

$\mathrm{KS2}≔\mathrm{KillingSpinors}\left(\mathrm{\sigma }\,0\,1\right)$

$\mathrm{KS2}:=\left[{\ⅇ}^{-\frac{1}{2}\mathrm{\ρ}x}\mathrm{dw2}\right]$

$\mathrm{KS3}≔\mathrm{KillingSpinors}\left(\mathrm{\sigma }\,1\,1\right)$

$\mathrm{KS3}:=\left[u\mathrm{D\_z1}\mathrm{dw1}\+\frac{\mathrm{I}{\ⅇ}^{-\mathrm{\ρ}x}y\mathrm{D\_z1}\mathrm{dw2}}{\mathrm{\ρ}}\+u\mathrm{D\_z2}\mathrm{dw2}\,\mathrm{D\_z1}\mathrm{dw1}\+\mathrm{D\_z2}\mathrm{dw2}\,\frac{\mathrm{D\_z1}\mathrm{dw1}}{{\mathrm{\ρ}}^2}-\frac{1}{2}\frac{\left({\ⅇ}^{\mathrm{\ρ}x}u\+{\ⅇ}^{-\mathrm{\ρ}x}v\right)\mathrm{D\_z1}\mathrm{dw2}}{{\mathrm{\ρ}}^2}\+{\ⅇ}^{\mathrm{\ρ}x}u\mathrm{D\_z2}\mathrm{dw1}\,-\frac{1}{2}\frac{{\ⅇ}^{\mathrm{\ρ}x}\mathrm{D\_z1}\mathrm{dw2}}{{\mathrm{\ρ}}^2}\+{\ⅇ}^{\mathrm{\ρ}x}\mathrm{D\_z2}\mathrm{dw1}\,{\ⅇ}^{-\mathrm{\ρ}x}\mathrm{D\_z1}\mathrm{dw2}\right]$