Hyperkähler structure on $TS^2$

Question

What are explicit expressions for the operators $I,J$ and $K$ on the hyperkähler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x_1,x_2, \alpha_1, \alpha_2)$ where a line $m\in TS^2$ is given parametrically by $(x_1+\alpha_1t,x_2+\alpha_2t,t)$? Are they known? I did not find them in the book "The geometry and dynamics of magnetic monopoles" by Atiyah and Hitchin, and could not extract them from it.

Answer 1

I will explain how you can get the formulae for the closed $2$-forms that define the hyperKähler structure without introducing coordinates. Once you know the $2$-forms, you can find the $I$, $J$, and $K$ by well-known formulae.

Start with the fact that we can think of $TS^2$ as the space of pairs $(a,b)$ where $a$ and $b$ are vectors in $\mathbb{R}^3$ satisfying $a\cdot a = 1$ and $a\cdot b = 0$. (This embeds $TS^2$ as a submanifold of $\mathbb{R}^6 = \mathbb{R^3}\times\mathbb{R}^3$.) Let $u,v:TS^2\to\mathbb{R}^3$ be the $\mathbb{R}^3$-valued functions that satisfy $u(a,b) = a$ and $v(a,b) = b$. Evidently, they are smooth functions on $TS^2$ and satisfy $u\cdot u = 1 $ and $u\cdot v = 0$.

I'm going to be expressing things in terms of $1$-forms and $2$-forms, so it will be useful to write down some explicit $1$-forms in terms of $u$ and $v$ and their differentials. Define $\mathbb{C}^3$-valued $1$-forms on $TS^2$ by $$ \alpha = \mathrm{d}u - i\,(u\times \mathrm{d}u) $$ and $$ \beta = u\times\mathrm{d}v + i\,\bigl(u\times(u\times\mathrm{d}v)\bigr). $$ When you introduce your favorite coordinates on $TS^2$, you'll have $u$ and $v$ expressed in terms of those coordinates, so the above formulae will give you $\alpha$ and $\beta$ in terms of those coordinates as well. Since everything I write below will be expressed in terms of these forms, you should have no trouble expanding these quantities in terms of your favorite coordinates, though I admit that the answers might be tedious and unenlightening.

One can now compute that $\Upsilon = \mathrm{d}v\,\,{\hat\cdot}\,\,\alpha$ is a nowhere-vanishing $\mathbb{C}$-valued $2$-form on $TS^2$ that is closed and satisfies $\Upsilon\wedge\Upsilon = 0$. Thus, it is the holomorphic volume form for a unique complex structure on $TS^2$. [Notation: If $\phi$ and $\psi$ are $\mathbb{C}^3$-valued $1$-forms on a manifold $M$, then, the $2$-form $\phi\,\,{\hat\cdot}\,\,\psi$ is defined by the equation $$ (\phi\,\,{\hat\cdot}\,\,\psi)(x,y) = \phi(x)\cdot \psi(y) - \phi(y)\cdot\psi(x). $$ for all $x,y\in T_pM$.]

Now, let $r = v\cdot v\ge 0$, and consider the (real-valued) $2$-form $$ \omega = \frac{i}{\sqrt{1+4r}}\,\,\beta\,\,{\hat\cdot}\,\,\bar\beta + \frac{i\sqrt{1+4r}}{4}\,\,\alpha\,\,{\hat\cdot}\,\,\bar\alpha\,. $$

It is now not difficult to show that $\omega$ is closed, of type $(1,1)$, and positive with respect to the complex structure defined by $\Upsilon$. Moreover, $\Upsilon$ has (constant) unit volume with respect to the Kähler metric $g$ defined by $\omega$. In fact, $\omega^2 = 2\,\Upsilon\wedge\overline{\Upsilon}$. Thus, $\Upsilon$ is parallel with respect to $g$, which implies that the metric $g$ is hyperKähler.