• For a general ${\cal N}$ the R-symmetry group is $U({\cal N})$ but for the ${\cal N}=2$ case why is it $SU(2)$ ? I guess it is again different for ${\cal N}=4$. How does one understand this?

One denotes the generators of the R-symmetry group by $B^i$ and let $\alpha$ be the spinor index and let $a,b$ be the R-charge indices in a ${\cal N}=2$ theory. Then one writes the defining equations as,

$[Q_{\alpha a}, B^i] = -\frac{1}{2} (\tau ^i)_a ^b Q_{\alpha b}$ and $[\bar{Q}_{\dot{\alpha}}^a, B^i] = \frac{1}{2} \bar{Q}_{\dot{\alpha}}^b (\tau ^i)_b ^a$

  • But it is not clear to me as to why over and above the aforementioned equations there should be yet another charge $R$ (called the $U(1)_R$ charge) with the defining equations,

$[Q_{\alpha a},R] = Q_{\alpha a}$ and $[\bar{Q}_{\dot{\alpha} a}, R] = - \bar{Q}_{\dot{\alpha} a}$

I would like to know why the above charge should exist in ${\cal N}=2$ supersymetry separate from the R-symmetry.

Is there an analogue for this $R$ for other values of ${\cal N}$ ?

  • Typically this R-charge as defined above is anomalous. How does one see that? Also what is the analogous statement for R-symmetry?

(1) Classical $\mathcal{N}=2$ theory has R symmetry $SU(2)_R \times U(1)_R$.

(2) $[Q,B]\sim \tau Q$, for $SU(2)$ part, $\tau$ is pauli matrices. While for $[Q,R]\sim Q$, the right hand side can be seen as identity matrix. This is $U(1)_R$ part of R symmetry.

(3) The anomalous can be seen from correlation function from index theorem which says that in one instanton background, there is $2N_C$ zero mode for each left handed adjoint fermion and one zero mode for each for each fermion in fundemental and antifundamental representation. Generally the $U(1)_R$ will breaks down to $Z_{4N_c-2N_F}$ in $\mathcal{N}=2$ theory.


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