What is the anti-linear involution used to construct the ADHM data and the hyper-Kähler quotient?

Question

I am trying to understand the ADHM construction. One reference I am looking at is Tachikawa's master thesis and in page 7, he defines a mapping called "anti-linear involution" and he concludes that the space is naturally a flat hyperkahler space. Why is this the case? Why those data along with this involution map make $\mathbb{X}$ a flat hyperkahler space?

Answer 1

The map $J$ is antilinear, i.e. $J(\lambda .u)=\overline{\lambda}.J(u)$ for every complex scalar $\lambda$. I don't think it should be called an involution. An involution is a map whose composite with itself is the identity. But it is easy to check that $J \circ J = -1$. An antilinear map whose square (for the composition) is $-1$ on a complex vector space $V$ is exactly what is needed to define a quaternionic structure on $V$, i.e. a structure of vector space over the quaternions whose underlying complex space is $V$. Indeed, in quaternions, we have $i$,$j$,$k$ such  that $i^2=j^2=k^2=-1$, $ij=-ji$, $k=ij$... When you have a complex vector space, you already have an action of the $i$ of complex numbers. To define a quaternionic structure, you have to define the action of the $j$. The relation $ij=-ji$ precisely means that it has to be antilinear for the complex structure and $j^2=-1$ means that it has to square to $-1$.

Remark: a true antilinear involution, i.e. whose square is $+1$, and not $-1$, defines a notion of complex conjugation and so a real structure on $V$.