In my understanding, mirror symmetry in physics originates from representation of the $N=2$ superconformal algebra. Why do we need precisely 2 supersymmetries (why not 1 or 4)?

Moreover, a chiral (anti-chiral) field is defined as a state that is annihilated by $G^+_{-1/2}$ ($G^-_{-1/2}$), where $G^+_{-1/2}$ and $G^-_{-1/2}$ are coefficients of the Fourier mode expansion of some anti-commuting current $G^+(z)$ and $G^-(z)$ of conformal weight $3/2$. How should I understand this chiral (anti-chiral) field?

In $N=(2,2)$ superconformal algebra, there are four rings: $(c,c),(a,a),(a,c),(c,a)$. It is known that the first two are charge conjugate, but what does theism mean? Right and left-moving, and chiral and anti-chiral rings... these all confuse me.

This post imported from StackExchange Physics at 2014-04-02 13:01 (UCT), posted by SE-user Mathematician