First, recall a general $\mathcal{\mathcal{N}}=1$ theory containing only scalars and fermions. Such a theory can be described by the Lagrangian

where

$${g}_{i\stackrel{\u0304}{j}}=\frac{{\partial}^{2}K}{\partial {\varphi}_{i}\partial {\stackrel{\u0304}{\varphi}}_{\stackrel{\u0304}{j}}}.$$ | (7.1.2) |

This deﬁnes a Kähler manifold. In particular, the manifold is naturally a complex manifold. This fact is almost implicit in our formalism, since the chiral multiplets are by deﬁnition complex valued. It is instructive to recall why this was so: we have the basic supersymmetry transformation

A convention independent fact is that ${\delta}_{\alpha}{\delta}_{\stackrel{\u0307}{\alpha}}$ acting on a complex scalar involves a multiplication by $i$. In terms of the real and imaginary parts of $\varphi $, we can schematically write this fact as

where the matrix

$$I=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)$$ | (7.1.5) |

has the property ${I}^{2}=-1$. This is the crucial matrix deﬁning the complex structure of the scalar manifold of an $\mathcal{\mathcal{N}}=1$ theory.

Now, let us consider an $\mathcal{\mathcal{N}}=2$ theory consisting of scalars and fermions only. Note that this means that there are no $\mathcal{\mathcal{N}}=2$ vector multiplets. This theory has two sets of $\mathcal{\mathcal{N}}=1$ supersymmetries ${\delta}_{\alpha}^{i=1,2}$. In addition,

$${\delta}_{\alpha}^{\left(c\right)}:={c}_{1}{\delta}_{\alpha}^{1}+{c}_{2}{\delta}_{\alpha}^{2}$$ | (7.1.6) |

also generates an $\mathcal{\mathcal{N}}=1$ sub-supersymmetry when $|{c}_{1}{|}^{2}+|{c}_{2}{|}^{2}=1$. Applying the argument in the last paragraph for this $\mathcal{\mathcal{N}}=1$ subalgebra, we ﬁnd that there are matrices

which always satisfy

$$\left({I}^{\left(c\right)}\right){\phantom{\rule{0.0pt}{0ex}}}^{2}=-1.$$ | (7.1.8) |

Note that ${n}^{a}$ are real and $|{n}^{1}{|}^{2}+|{n}^{2}{|}^{2}+|{n}^{3}{|}^{2}=1$, i.e. they are on ${S}^{2}$. Denoting $\left(I,J,K\right):=\left({I}_{1},{I}_{2},{I}_{3}\right)$ for simplicity and expanding (7.1.8), one ﬁnds the relations

$${I}^{2}={J}^{2}={K}^{2}=-1,\phantom{\rule{2em}{0ex}}IJ=K=-JI,\phantom{\rule{1em}{0ex}}JK=I=-KJ,\phantom{\rule{1em}{0ex}}KI=J=-IK.$$ | (7.1.9) |

This commutation relation of $I$, $J$ and $K$ is that of a quaternion. A manifold with an action of quaternion algebra on its tangent space is called a hyperkähler manifold. Therefore we found that the scalar manifold of an $\mathcal{\mathcal{N}}=2$ theory without massless vector multiplets is hyperkähler.

Note that the $SU{\left(2\right)}_{R}$ symmetry acts on the doublet $\left({c}_{1},{c}_{2}\right)$, which is restricted to live on the three-sphere $|{c}_{1}{|}^{2}+|{c}_{2}{|}^{2}=1$. The map (7.1.7) from this $\left({c}_{1},{c}_{2}\right)$ to ${n}^{a}$ is the standard Hopf ﬁbration ${S}^{3}\to {S}^{2}$, and the index $a$ transforms as the triplet of $SU{\left(2\right)}_{R}$.

Combining with the analysis in Sec. 2.4, we see that general low-energy $\mathcal{\mathcal{N}}=2$ theory has an action of the form

such that ${K}_{h}\left({\stackrel{\u0304}{q}}_{\stackrel{\u0304}{t}},{q}_{s}\right)$ gives a hyperkähler manifold and that there is a prepotential $F\left({a}_{i}\right)$ giving ${\tau}^{ij}$ and ${K}_{v}$ via the standard formulas (2.4.7), (2.4.8) and (2.4.9).

Note that the hypermultiplet side and the vector multiplet side are completely decoupled. The dependence on the UV gauge coupling is implicitly there in the vector multiplet side. This means that the hypermultiplet side cannot receive quantum corrections depending on the gauge coupling.