- #1
physicus
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Homework Statement
Given [itex]N=1 SYM[/itex] in 10 dimensions (all fields in the adjoint representation):
[itex]\int d^{10}x\, Tr\,\left( F_{MN}F^{MN}+\Psi\Gamma^M D_M\Psi\right)[/itex]
[itex]D_M\Psi=\partial_M \Psi+i[A_M,\Psi][/itex] is the gauge covariant derivative.
Reduce to 4 dimensions [itex]A_M=(A_\mu,\phi_i), \mu=0,\ldots,3, i=4,\ldots 9, \partial_i=0 \,\forall \,i[/itex] and show:
[itex]F_{MN}F^{MN}=F_{\mu\nu}F^{\mu\nu}+D_\mu \phi_i D^\mu \phi^i + [\phi_i,\phi_j][\phi^i,\phi^j][/itex]
Homework Equations
The Attempt at a Solution
My ansatz:
[itex]F_{MN}F^{MN}=F_{\mu\nu}F^{\mu\nu}+F_{\mu i}F^{\mu i}+F_{i\nu}F^{i\nu}+F_{ij}F^{ij}[/itex]
I have some trouble writing down what [itex]F_{\mu i}[/itex] and [itex]F^{\mu i}[/itex] are. I think my problem is that I only have the definition of the gauge covariant derivative acting on a field in the adjoint.
[itex]F_{\mu i}=\frac{1}{i}[D_\mu,D_i]=\frac{1}{i}(D_\mu D_i - D_i D_\mu) = ?[/itex]
Is [itex]D_i = i\phi_i[/itex]?
Thanks for any help!
physicus