Isometry group from information about the center of the group

Question

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question.

Then he says that the zero modes are SENSITIVE to the center of SU(2), and hence the isometry group is SU(2)? What does sensitive to the center mean? Is center the subset which commutates with all elements of SU(2)? How can it tell me about the isometry group?

Answer 1

The statement "zero modes are sensitive to the center of $SU(2)$" means that the zero modes, overall parameters parameterizing the space of solutions, do change when one acts on the solution by $g\in SU(2)$ which is $g\in Z(SU(2))$, an element of the center of $SU(2)$.

"Be sensitive" is the same thing as "[for the solutions labeled by the zero modes] not to be invariant".

Note that the center of $SU(2)$ is the subgroup of all $g\in SU(2)$ that commute with all $h\in SU(2)$, $gh=hg$. For $SU(2)$, this subgroup (center) is a $Z_2$ consisting of the unit matrix and the minus unit matrix. So what they say is that when you act with the minus unit matrix in $SU(2)$ symmetry group on a solution, you get a physically inequivalent solution.

Every element of $SU(2)$ maps a solution to another solution in such a way that the distances between the solutions are preserved, so the $SU(2)$ is a group of isometries of the moduli space (the space of solutions = the space parameterized by the zero modes). The sensitivity is that each element of $SU(2)$ does a different transformation of the moduli space, so the action of the minus unit matrix isn't equivalent to the action of the unit matrix (identity element of the group).