# representing spinors in four dimensions as complexified forms

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The spinors appearing in the Killing spinor equation (2.2)

$$D_µ\epsilon = ∇_µ\epsilon + 1/2ℓ γ_µ\epsilon + i/4 F_{ν1ν2}γ^{ν1ν2}γ_µ\epsilon − i/ℓ A_µ\epsilon = 0$$

are Dirac spinors....

Following [9, 10, 11] these spinors can be written as complexified forms on $\mathbb{R}^2$ ; if $\Delta$ denotes the space of Dirac spinors then $\Delta = Λ∗ (\mathbb{R}^ 2 ) ⊗ \mathbb{C}$. A generic spinor $η$ can therefore be written as $η = λ1 + µ^i e^i + σe^{12}$  where $e^1$ , $e^2$ are 1-forms on $\mathbb{R^2}$ , and $i = 1, 2; e^{12} = e^1∧e^2$ .$λ$, $µ^i$ and $σ$are complex functions.

This is quoted from http://arxiv.org/pdf/hep-th/0610128.pdf .. My question is why is this space denoted by

$\Delta = Λ∗ (\mathbb{R}^ 2 ) ⊗ \mathbb{C}$

How can I understand this notation from a physics perspective knowing that the title of that section was spinors in four dimensions?

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