What is meant with the fact that supersymmetry with $\mathcal{N}=4$ in three (2+1) dimensions is equivalent to supersymmetry with $\mathcal{N}=2$ in ordinary four (3+1) dimensions?

Which way does one go from one formulation to the other?


In 4 dimensions the reducible, Dirac spinor representation is 4-dimensional. So N=2 SUSY in 4 dimensions means you have 2 Dirac spinors Q1 & Q2. Each has 4 components, so the total number of supercharges is 2*4=8.

In 3 dimensions theres only one irreducible spinor representation and its 2-dimensional. So you have 8 supercharges, two for each spinor. So you must have 4 of these spinors, call them Qa,Qb,Qc,Qd.

Its all about keeping track of the dimensions of the spinor representations. The general formula is 2^(d/2) if d, the spacetime dimension, is even and 2^([d-1]/2) if d is odd. So the pattern is

2 & 3 dimensions: 2 component spinors 4 & 5 dimensions: 4 component spinors 6 & 7 dimensions: 8 component spinors.

So when you reduce dimensions, you always have the same number of supercharges, its just the difference in how you organize them into spinors. Its kind of decomposition problem, how do my spinor representations break up when I reduce my dimension?


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