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I have been thinking about the definition of the notation $\cal N$ and its relation to the number of supercharges in SUSY, but still feel a little confused. In dimension 2, we usually denote, for example, $\cal N = (2,2)$ supersymmetry, where we have 2 chiral supercharge and 2 anti-chiral supercharge; but in higher dimensions we just refer to $\cal N = 1$, etc. What is the difference and why we make such different notation?

Also, I would appreciate of one can explain the exact meaning of $\cal N$, for example in 4 dimensions, and how they are related to number of supercharge $Q$ and independent spinors.

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In principle, $\mathcal{N}$ gives you the number of supercharges in your theory. There are, however, cases with more than one irreducible (pseudo-)real spinor representations. If you have $N$ charges in one and $N'$ charges in the other representation, you can denote the total number of charges as $\mathcal{N}=(N,N')$ in order to emphasize the difference. Examples would be $\mathcal{N}=(1,1)$ type IIA supergravity in ten dimensions or $\mathcal{N}=(2,2)$ supergravity in six dimensions. There also exists notation in which the latter is referred to as $\mathcal{N}=4$.

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For the relation between $\mathcal{N}$ and the number of supercharges, it depends on how many real components a spinor has (at least) in the given space-time dimension.

For instance, it is $4 \mathcal{N}$ in 4 dimensions, $8 \mathcal{N}$ in 6 dimensions, $16 \mathcal{N}$ in 10 dimensions and $32 \mathcal{N}$ in 11 dimensions.

If $\mathcal{N}=(2,2)$ in 4 dimensions, you have 8 supercharges with one chirality and 8 supercharges with the other chirality.

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