References on quaternionic geometry

Question

Is there any analog, in the quaternionic setting, of Kahler potentials?

In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the quaternionic projective line $\mathbb{P}^1 (\mathbb{H}).$ In the complex setting it is well known that $\omega_{\textrm{FS}}=\frac{i}{2} \partial \overline{\partial} \log (|z|^2).$ Is there a natural extension of this form and of this potential over $\mathbb{P}^1 (\mathbb{H})$?

That is, can the standard 4-form $\Omega=\omega_i \wedge \omega_i + \omega_j \wedge \omega_j + \omega_k \wedge \omega_k$ on $\mathbb{P}^1(\mathbb{H})$ be written as some differential operator applied on some function (i.e. the potential)?

Answers, comments and references are welcome.

Answer 1

This is only a partial answer. On hypercomplex manifolds there is a notion of HyperKahler with Torsion (HKT) metrics. Such metrics are in a sense quaternionic analogues of Kahler metrics. Locally they can be presented as an application of quaternionic Hessian to a smooth function (potential), as essentially was shown in http://arxiv.org/abs/math/0402366 (see also http://arxiv.org/abs/math/0510140 for some further developments, in particular Proposition 8.2 there).

However the quaternionic projective spaces are not hypercomplex manifolds and are not covered by the above mentioned papers.

Answer 2

Actually, a more direct answer to the original question explains why there is nothing like Kähler potential theory in the hypercomplex and quaternionic cases.

Recall what a Kähler potential does: You start with a complex $n$-manifold, i.e., a $2n$-manifold $M^{2n}$ endowed with a torsion-free $\mathrm{GL}(n,\mathbb{C})$-structure $B$, represented by an integrable complex structure tensor $J:TM\to TM$. Now, $\mathrm{U}(n)$ is a maximal compact in $\mathrm{GL}(n,\mathbb{C})$, and we can select a torsion-free $\mathrm{U}(n)$-structure $B'\subset B$ over an open set $U\subset M$ by choosing a smooth function $u\in C^\infty(U)$ such that $\omega = i\partial\bar\partial u$ is a positive $(1,1)$-form. Locally, this is how all of the torsion-free (i.e., Kähler) structures subordinate to the $\mathrm{GL}(n,\mathbb{C})$-structure $B$ are described. Of course, every torsion-free $\mathrm{U}(n)$-structure $B'$ on $M$ underlies a unique torsion-free $\mathrm{GL}(n,\mathbb{C})$-structure $B = B'{\cdot}\mathrm{GL}(n,\mathbb{C})$. Now, the nice thing in this case is that, modulo diffeomorphism, locally at least, integrable $\mathrm{GL}(n,\mathbb{C})$-structures are all the same, i.e., they have no local geometry, whereas the torsion-free $\mathrm{U}(n)$-structures depend locally on $1$ function of $2n$-variables (which is represented exactly by the Kähler potential as above).

Consider what you would need to be true in either the hypercomplex case or the quaternionic case in order to have such an analogue:

First, consider the hypercomplex case: Now, the group is $\mathrm{GL}(n,\mathbb{H})\subset \mathrm{GL}(4n,\mathbb{R})$, with maximal compact $\mathrm{Sp}(n)\subset \mathrm{GL}(n,\mathbb{H})$. Thus, on a $4n$-manifold $M^{4n}$ you'd want a way to start with a torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure $B$ on $M$ (i.e., a hypercomplex structure) and specify the torsion-free $\mathrm{Sp}(n)$-structures (i.e., hyperKähler structures) that it contains. As in the complex/Kähler case, the reverse operation is automatic: Each torsion-free $\mathrm{Sp}(n)$-structure $B'$ lies in a unique torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure $B= B'{\cdot}\mathrm{GL}(n,\mathbb{H})$. However, you immediately see the problem: Modulo diffeomorphism, the torsion-free $\mathrm{GL}(n,\mathbb{H})$-structures in dimension $4n$ depend on $4n^2$ functions of $2n{+}1$ variables, while the the torsion-free $\mathrm{Sp}(n)$-structures in dimension $4n$ depend on only $2n$ functions of $2n{+}1$ variables, so the 'generic' torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure in this dimension does not contain any torsion-free $\mathrm{Sp}(n)$-structures, so there can't be a potential theory like the complex/Kähler potential theory in this case. You might think that you could get around this by considering the underlying $\mathrm{SL}(n,\mathbb{H})$-structures instead, but, modulo diffeomorphism, the torsion-free $\mathrm{SL}(n,\mathbb{H})$-structures in dimension $4n$ depend on $4n^2{-}2n$ functions of $2n{+}1$ variables, which is still too high (except when $n=1$, when $\mathrm{SL}(1,\mathbb{H}) = \mathrm{Sp}(1)$, so there is nothing to do). (This is why they punt and go for HKT structures instead in this case.)

Second, in the quaternionic case, the group is the group is $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)\subset \mathrm{GL}(4n,\mathbb{R})$, with maximal compact $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)\subset \mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$. Thus, on a $4n$-manifold $M^{4n}$ you'd want a way to start with a torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure $B$ on $M$ (i.e., a quaternionic structure) and specify the torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures (i.e., quaternion-Kähler structures) that it contains. As in the complex/Kähler case, the reverse operation is automatic: Each torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structure $B'$ lies in a unique torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure $B= B'{\cdot}\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$. However, again, there's a problem: Modulo diffeomorphism, the torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structures in dimension $4n$ depend on $2n(2n{+}1)$ functions of $2n{+}1$ variables, while the the torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures in dimension $4n$ depend on only $2n$ functions of $2n{+}1$ variables, so, again, the 'generic' torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure in this dimension does not contain any torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures, so, again, there can't be a potential theory like the complex/Kähler potential theory in this case.

For these function counts for the generality of torsion-free structures modulo diffeomorphism, in case you are curious, you might consult my paper: R. Bryant - Classical, exceptional, and exotic holonomies: a status report, in Actes de la Table Ronde de Géométrie Différentielle en l’Honneur de Marcel Berger, Soc. Math. France, 1996, 93–166.