# Joyce structures and their twistor spaces

Tom Bridgeland

###### Abstract.

Joyce structures are a class of geometric structures which first arose in
relation to holomorphic generating functions for Donaldson-Thomas invariants.
They can be thought of as non-linear analogues of Frobenius structures, or as
special classes of complex hyperkähler manifolds. We give a detailed
introduction to Joyce structures, with particular focus on the geometry of the
associated twistor space. We also prove several new results.

##  1\. Introduction

The aim of this paper111which is an expansion of the first half of the arXiv
preprint ‘Tau functions from Joyce structures’. is to give an introduction to
a class of geometric structures known as Joyce structures. These structures
have appeared in several contexts recently, including integrable systems [13,
20, 21] and topological string theory [1, 2]. A Joyce structure on a complex
manifold $M$ involves a one-parameter family of flat and symplectic non-linear
connections on the tangent bundle $T_{M}$, and gives rise to a complex
hyperkähler structure on the total space $X=T_{M}$. The precise definition
first appeared in [12], but the essential features are standard in twistor
theory (see e.g. [19]), and go back to work of Plebański [40].

Joyce structures take their name from a line of research initiated in [32]
which aims to encode the Donaldson-Thomas (DT) invariants [33, 36] of a three-
dimensional Calabi-Yau (CY3) category in a geometric structure on its space of
stability conditions [9, 10]. From this point of view a Joyce structure should
be thought of as a non-linear analogue of a Frobenius structure [17, 18], in
which the linear structure group $\operatorname{GL}_{n}(\mathbb{C})$ has been
replaced by the group of Poisson automorphisms of a complex torus
$(\mathbb{C}^{*})^{n}$. The wall-crossing formula shows that the DT invariants
can be viewed as the Stokes data for an isomonodromic family of irregular
connections on $\mathbb{P}^{1}$ taking values in this group. This perspective
is the subject of [12] and is summarised in [15, Appendix A].

Relations between the wall-crossing formula in DT theory and real hyperkähler
manifolds were first discovered in the celebrated work of Gaiotto, Moore and
Neitzke [25, 26]. The connection with complex hyperkähler manifolds is
somewhat different and was made in [15]. In physical terms the two stories are
related by the conformal limit [1, 2, 24].

The geometry of a Joyce structure is often clearer when viewed through the
lens of the associated twistor space $p\colon Z\to\mathbb{P}^{1}$. Each fibre
of the map $p$ is the leaf space of a half-dimensional foliation on $X=T_{M}$.
There are essentially three distinct fibres, $Z_{0}$, $Z_{1}$ and
$Z_{\infty}$, of which $Z_{0}$ is naturally identified with $M$. It is
intriguing to note that in simple examples associated to the DT theory of a
quiver, the fibre $Z_{0}=M$ is a quotient of the space of stability
conditions, whereas $Z_{1}$ is closely related to the cluster Poisson variety.
The fibre $Z_{\infty}$ remains rather mysterious.

An interesting class of examples of Joyce structures was constructed in [16].
The base $M$ parameterises pairs consisting of a Riemann surface of some fixed
genus $g$ equipped with a quadratic differential with simple zeroes. The
extension to spaces of quadratic differentials with poles of fixed orders will
appear in [41]. These Joyce structures are expected to arise from the DT
theory of the CY3 categories considered in [14, 28], and relate in physics [8]
to supersymmetric gauge theories of class $S[A_{1}]$. We give explicit
descriptions of the two simplest examples of this type, which are naturally
associated to the DT theory of the A1 and A2 quivers respectively. The A2
example is particularly interesting, and is closely related to the Painlevé I
equation.

### Plan of the paper

We begin in Section 2 with the notion of a pre-Joyce structure on a complex
manifold $M$. It gives rise to a complex hyperkähler structure on the total
space $X=T_{M}$. A Joyce structure is a pre-Joyce structure with certain
additional symmetries. These symmetries are discussed in Section 3 and are
controlled by a special type of integral affine structure on $M$ which we call
a period structure.222Some of this material appears also in [15] but we have
chosen to tell the story again from the beginning because a more detailed
treatment of several points seems worthwhile, and experience with examples has
suggested a few small changes in the definitions.

The twistor space $p\colon Z\to\mathbb{P}^{1}$ associated to a Joyce structure
is introduced in Section 4. In Section 5 we establish several new results on
its structure. In Section 6 we show how to associate a strongly-integrable
Hamiltonian system to a Joyce structure equipped with certain extra data,
namely a Lagrangian submanifold $R\subset Z_{\infty}$, and an identification
of $M$ with an open subset of the cotangent bundle of a complex manifold $B$.

In the rest of the paper we discuss examples. In Section 7 we first recall
from [16] the construction of Joyce structures of class $S[A_{1}]$ on spaces
of holomorphic quadratic differentials. We then explain how some of the
constructions from previous sections play out in this setting. In Section 8
and 9 we give explicit formulae in the two simplest cases, which are
associated to the DT theory of the A1 and A2 quivers respectively.

### Conventions

We work throughout in the category of complex manifolds and holomorphic maps.
All symplectic forms, metrics, bundles, connections, sections etc. are
holomorphic. The holomorphic tangent bundle of a complex manifold $M$ is
denoted $T_{M}$, and the derivative of a map of complex manifolds $f\colon
M\to N$ is denoted $f_{*}\colon T_{M}\to f^{*}(T_{N})$. The map $f$ is called
étale if $f_{*}$ is an isomorphism. We use the symbol $\mathcal{L}$ to denote
the Lie derivative.

### Acknowledgements

The author is very grateful for conversations and correspondence with Sergey
Alexandrov, Anna Barbieri, Maciej Dunajski, Dominic Joyce, Dimitry Korotkin,
Davide Masoero, Boris Pioline, Ian Strachan, Jörg Teschner and Iván Tulli.

##  2\. Pre-Joyce structures

Recall [17, 18] that a Frobenius structure on a complex manifold $M$ consists
of a pencil of flat, torsion-free connections on the tangent bundle of $M$
with certain additional properties (the existence of the identity and Euler
vector fields, potentiality of the multiplication, etc). In a similar way, a
Joyce structure on $M$ consists of a pencil of flat, symplectic _non-linear_
connections on the tangent bundle of $M$, again admitting certain additional
symmetries. In this section we focus on the pencil of connections, which we
refer to as a pre-Joyce structure, leaving discussion of the required
symmetries to the next section.

###  2.1. Non-linear connections

We begin by briefly summarising some basic facts about non-linear connections
in the sense of Ehresmann (see e.g. [34, Chapter 3]). Recall that we are
always working in the category of complex manifolds, so all connections will
be holomorphic.

Let $\pi\colon X\to M$ be a holomorphic submersion of complex manifolds. We
denote the fibres by $X_{m}=\pi^{-1}(m)$. The derivative of $\pi$ gives rise
to a short exact sequence of bundles

$0\longrightarrow T_{X/M}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}T_{X}\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}\pi^{*}(T_{M})\longrightarrow
0.$ (1)

######  Definition 2.1.

A non-linear connection on the map $\pi$ is a map of bundles
$h\colon\pi^{*}(T_{M})\to T_{X}$ satisfying $\pi_{*}\circ h=1$.

Writing $H=\operatorname{im}(h)$ and $V=T_{X/M}$, the tangent bundle of $X$
decomposes as a direct sum $T_{X}=H\oplus V$. We call tangent vectors and
vector fields horizontal or vertical if they lie in $H$ or $V$ respectively.
Note that a vector field $u\in H^{0}(M,T_{M})$ can be lifted to a horizontal
vector field $h(u)\in H^{0}(X,T_{X})$ by composing the pullback $\pi^{*}(u)\in
H^{0}(X,\pi^{*}(T_{M}))$ with the map $h$.

Consider a smooth path $\gamma\colon[0,1]\to M$. Given a point $x\in
X_{\gamma(0)}$ we can look for a lifted path $\alpha\colon[0,\delta]\to X$
satisfying $\alpha_{*}(\frac{d}{dt})=h(\gamma_{*}(\frac{d}{dt}))$ and
$\alpha(0)=x$. Since we have not assumed that $\pi$ is proper, such a lift
will exist only for small enough $\delta>0$. For $t\in[0,\delta]$ we call
$\alpha(t)\in X_{\gamma(t)}$ the time $t$ parallel transport of the point $x$
along the path $\gamma$. Given a point $x_{0}\in X_{\gamma(0)}$ we can find a
$\delta>0$ and open subsets $U_{t}\subset X_{\gamma(t)}$ with $x_{0}\in
U_{0}$, such that time $t$ parallel transport along $\gamma$ defines an
isomorphism $\operatorname{PT}_{\gamma}(t)\colon U_{0}\to U_{t}$ for each
$t\in[0,\delta]$.

Given complex manifolds $M,N$ there is a connection on the projection map
$\pi_{M}\colon M\times N\to M$ induced by the canonical splitting $T_{M\times
N}=\pi_{M}^{*}(T_{M})\oplus\pi_{N}^{*}(T_{N})$. The connection $h$ is called
flat if it is locally isomorphic to a connection of this form. More precisely:

######  Definition 2.2.

The connection $h$ is flat if the following equivalent conditions hold:

  * (i)

for every $x\in X$ there are local co-ordinates $(x_{1},\cdots,x_{n})$ on $X$
at $x$, and $(y_{1},\cdots,y_{d})$ on $M$ at $\pi(x)$, such that
$x_{i}=\pi^{*}(y_{i})$ and $h(\frac{\partial}{\partial
y_{i}})=\frac{\partial}{\partial x_{i}}$ for $1\leqslant i\leqslant d$,

  * (ii)

the sub-bundle $H=\operatorname{im}(h)\subset T_{X}$ is closed udner Lie
bracket: $[H,H]\subset H$.

Suppose given a relative symplectic form $\Omega_{\pi}\in
H^{0}(X,\wedge^{2}\,\,T^{*}_{X/M})$ on the map $\pi$. It restricts to a
symplectic form $\Omega_{m}\in H^{0}(X_{m},\wedge^{2}\,\,T^{*}_{X_{m}})$ on
each fibre $X_{m}$. We say that the connection $h$ preserves $\Omega_{\pi}$ if
for any path $\gamma\colon[0,1]\to M$ the partially-defined parallel transport
maps $\operatorname{PT}_{\gamma}(t)\colon X_{\gamma(0)}\to X_{\gamma(t)}$ take
$\Omega_{\gamma(0)}$ to $\Omega_{\gamma(t)}$. This is equivalent to the
statement that for any horizontal vector field $u$ on $X$ the Lie derivative
$\mathcal{L}_{u}(\Omega_{\pi})=0$.

Using the decomposition $T_{X}=H\oplus V$ the relative form $\Omega_{\pi}$ can
be lifted uniquely to a form $\Omega\in H^{0}(X,\wedge^{2}\,T^{*}_{X})$
satisfying $\ker(\Omega)=H$.

######  Lemma 2.3.

  * (i)

The connection $h$ preserves $\Omega_{\pi}$ precisely if
$i_{v_{1}}i_{v_{2}}(d\Omega)=0$ for any two vertical vector fields
$v_{1},v_{2}\in H^{0}(X,T_{X/M})$.

  * (ii)

If the connection $h$ is flat then it preserves $\Omega_{\pi}$ precisely if
$d\Omega=0$.

###### Proof.

Part (i) is [27, Theorem 4]. For part (ii), note that one implication follows
from (i), so let us assume that $h$ is flat and preserves $\Omega_{\pi}$ and
prove that $d\Omega=0$.

Take three vector fields $u_{1},u_{2},u_{3}$ on $X$ and consider the
expression defining $d\Omega(u_{1},u_{2},u_{3})$. We can assume that each
$u_{i}$ is either horizontal or vertical. Note that both horizontal and
vertical vector fields are closed under Lie bracket, and $i_{h}(\Omega)=0$ for
any horizontal vector field $h$. Thus $d\Omega(u_{1},u_{2},u_{3})=0$ as soon
as two of the $u_{i}$ are horizontal. In the remaining cases two of the
$u_{i}$ are vertical, and the claim follows from part (i).∎

Suppose that a discrete group $G$ acts freely and properly on $X$ preserving
the map $\pi$. Then $Y=X/G$ is a complex manifold and the quotient map
$q\colon X\to Y$ is étale. There is an induced submersion $\eta\colon Y\to M$
and a factorisation $\pi=\eta\circ q$. A connection $h\colon\pi^{*}(T_{M})\to
T_{X}$ will be called $G$-invariant if $g_{*}\circ h=h$ for all $g\in G$.
There is then an induced connection $j\colon\eta^{*}(T_{M})\to T_{Y}$ on
$\eta$ defined uniquely by the condition that $q_{*}\circ h=q^{*}(j)$. We say
that the connection $h$ descends along the quotient map $q$.

###  2.2. Pre-Joyce structures

Let $M$ be a complex manifold and let $\pi\colon X=T_{M}\to M$ be the total
space of the tangent bundle of $M$. There is a canonical isomorphism
$\nu\colon\pi^{*}(T_{M})\to T_{X/M}$ obtained by composing the chain of
identifications

$\pi^{*}(T_{M})_{x}=T_{M,\pi(x)}=T_{T_{M,\pi(x)},x}=T_{X_{\pi(x)},x}=T_{X/M,x}.$
(2)

We set $v=i\circ\nu$. A connection $h\colon\pi^{*}(T_{M})\to T_{X}$ on $\pi$
then defines a family of connections $h_{\epsilon}=h+\epsilon^{-1}v$
parameterised by $\epsilon\in\mathbb{C}^{*}$.333At this stage it might seem
more sensible to parameterise the pencil by $t=\epsilon^{-1}$. In later
applications however it is the parameter $\epsilon$ which appears most
naturally.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{X/M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{T_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{\pi^{*}(T_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\epsilon}}$$\scriptstyle{\nu}$$\textstyle{0}$
(3)

Suppose that $M$ is equipped with a holomorphic symplectic form $\omega\in
H^{0}(M,\wedge^{2}\,T^{*}_{M})$. Via the isomorphism $\nu$ we obtain a
relative symplectic form $\Omega_{\pi}\in H^{0}(X,\wedge^{2}T_{X/M}^{*})$
which restricts to a linear symplectic form $\omega_{m}$ on each fibre
$X_{m}=T_{M,m}$. We say that a connection on $\pi$ is symplectic if it
preserves $\Omega_{\pi}$ in the sense defined above.

######  Definition 2.4.

A pre-Joyce structure $(\omega,h)$ on a complex manifold $M$ consists of

  * (i)

a holomorphic symplectic form $\omega$ on $M$,

  * (ii)

a non-linear connection $h$ on the tangent bundle $\pi\colon X=T_{M}\to M$,

such that for each $\epsilon\in\mathbb{C}^{*}$ the connection
$h_{\epsilon}=h+\epsilon^{-1}v$ is flat and symplectic.

Take a local co-ordinate system $(z_{1},\cdots,z_{n})$ on $M$ which are
Darboux, in the sense that

$\omega=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot dz_{p}\wedge dz_{q},$ (4)

with $\omega_{pq}$ a constant skew-symmetric matrix. We denote by $\eta_{pq}$
the inverse matrix.

There are associated linear co-ordinates $(\theta_{1},\cdots,\theta_{n})$ on
the tangent spaces $T_{M,m}$ obtained by writing a tangent vector in the form
$\sum_{i}\theta_{i}\cdot{\partial}/{\partial z_{i}}$. We thus get induced
local co-ordinates $(z_{i},\theta_{j})$ on the total space $X=T_{M}$. In these
co-ordinates

$v_{i}:=v\Big{(}\frac{\partial}{\partial
z_{i}}\Big{)}=\frac{\partial}{\partial\theta_{i}}.$ (5)

The fact that the connection $h$ is flat and symplectic ensures that we can
write

$h_{i}:=h\Big{(}\frac{\partial}{\partial
z_{i}}\Big{)}=\frac{\partial}{\partial
z_{i}}+\sum_{p,q}\eta_{pq}\cdot\frac{\partial
W_{i}}{\partial\theta_{p}}\cdot\frac{\partial}{\partial\theta_{q}},$ (6)

for locally-defined functions $W_{i}=W_{i}(z,\theta)$. Note that $W_{i}$ is
only well-defined up to the addition of functions $a_{i}(z)$ independent of
the $\theta$ co-ordinates. We can fix these integration constants by insisting
that $W_{i}$ vanishes along the zero section $M\subset X=T_{M}$, i.e. that

$W_{i}(z_{1},\cdots,z_{n},0,\cdots,0)=0.$ (7)

The connection $h_{\epsilon}$ is flat precisely if
$\big{[}h_{i}+\epsilon^{-1}v_{i},h_{j}+\epsilon^{-1}v_{j}\big{]}=0$ for all
$1\leqslant i,j\leqslant n$. A short calculation shows that this holds for all
$\epsilon\in\mathbb{C}^{*}$ precisely if

$\displaystyle\frac{\partial}{\partial\theta_{k}}\Big{(}\frac{\partial
W_{i}}{\partial\theta_{j}}-\frac{\partial
W_{j}}{\partial\theta_{i}}\Big{)}=0,$ (8)
$\displaystyle\frac{\partial}{\partial\theta_{k}}\Big{(}\frac{\partial
W_{i}}{\partial z_{j}}-\frac{\partial W_{j}}{\partial
z_{i}}-\sum_{p,q}\eta_{pq}\cdot\frac{\partial
W_{i}}{\partial\theta_{p}}\cdot\frac{\partial
W_{j}}{\partial\theta_{q}}\Big{)}=0.$ (9)

for all $1\leqslant i,j,k\leqslant n$.

######  Remark 2.5.

So as to be able to include certain interesting examples it is sometimes
useful to weaken the axioms of a (pre-) Joyce structure to allow the
connection $h\colon\pi^{*}(T_{M})\to T_{X}$ to have poles. In precise terms
this means that $h$ should be defined by a bundle map
$h\colon\pi^{*}(T_{M})\to T_{X}(D)$ satisfying

$(\pi_{*}\otimes\mathscr{O}_{X}(D))\circ h=1_{\pi^{*}(T_{M})}\otimes s_{D},$
(10)

where $D\subset X$ is an effective divisor, and
$s_{D}\colon\mathscr{O}_{X}\to\mathscr{O}_{X}(D)$ is the canonical inclusion.
When expressed in terms of local co-ordinates as above, this just means that
the functions $W_{i}(z,\theta)$ are meromorphic. We will refer to the
resulting structures as meromorphic (pre-) Joyce structures.

###  2.3. Complex hyperkähler structures

The following structures have appeared before in the literature under various
different names (see e.g. [4, 20, 30, 31]). We emphasise that, as throughout
the paper, all quantities appearing are holomorphic.

######  Definition 2.6.

A complex hyperkähler structure $(g,I,J,K)$ on a complex manifold $X$ consists
of

  * (i)

a metric $g$, i.e. a symmetric, non-degenerate bilinear form $g\colon
T_{X}\otimes T_{X}\to\mathscr{O}_{X}$,

  * (ii)

endomorphisms $I,J,K\in\operatorname{End}_{X}(T_{X})$,

such that:

  * (HK1)

the quaternion relations hold: $I^{2}=J^{2}=K^{2}=IJK=-1$,

  * (HK2)

$I,J,K$ are compatible with $g$ and are parallel for the associated Levi-
Civita connection $\nabla$:

$g(R(u_{1}),R(u_{2}))=g(u_{1},u_{2}),\qquad\nabla(R)=0,\qquad
R\in\\{I,J,K\\}.$ (11)

Let $M$ be a complex manifold with a holomorphic symplectic form $\omega$. A
non-linear connection $h$ on the tangent bundle $\pi\colon X=T_{M}\to M$ gives
a decomposition

$T_{X}=\operatorname{im}(h)\oplus\operatorname{im}(v)\cong\pi^{*}(T_{M})\otimes_{\mathbb{C}}\mathbb{C}^{2}.$
(12)

We can then define an action of the quaternions on $T_{X}$ by choosing an
identification of the complexification of the quaternions
$\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}$ with the algebra of $2\times 2$
matrices $\operatorname{End}_{\mathbb{C}}(\mathbb{C}^{2})$. We can also define
a metric $g$ by taking the tensor product of $\pi^{*}(\omega)$ with a linear
symplectic form on $\mathbb{C}^{2}$.

With appropriate conventions this leads to the formulae444Compared to [15] we
have changed the signs of $I$ and $K$, and divided the metric by 2.

$\displaystyle I\circ h$ $\displaystyle=i\cdot h,\qquad\quad$ $\displaystyle
J\circ h$ $\displaystyle=-v,$ $\displaystyle\qquad\quad K\circ h$
$\displaystyle=i\cdot v,$ (13) $\displaystyle I\circ v$ $\displaystyle=-i\cdot
v,$ $\displaystyle J\circ v$ $\displaystyle=h,$ $\displaystyle K\circ v$
$\displaystyle=i\cdot h,$

which should be interpreted as equalities of maps $\pi^{*}(T_{M})\to T_{X}$,
and

$g(h(u_{1}),v(u_{2}))=\tfrac{1}{2}\omega(u_{1},u_{2}),\qquad
g(h(u_{1}),h(u_{2}))=0=g(v(u_{1}),v(u_{2})).$ (14)

It is easily checked that $g$ is preserved by the endomorphisms $I,J,K$.

The following result implies in particular that a pre-Joyce structure on a
complex manifold $M$ induces a complex hyperkähler structure on the total
space $X=T_{M}$.

######  Theorem 2.7.

The operators $I,J,K$ are parallel for the Levi-Civita connection $\nabla$
associated to $g$ precisely if the connection $h_{\epsilon}=h+\epsilon^{-1}v$
is flat and symplectic for all $\epsilon\in\mathbb{C}^{*}$.

###### Proof.

We begin with a general remark. Let $g\colon T_{X}\times
T_{X}\to\mathscr{O}_{X}$ be a metric on a complex manifold $X$ with associated
Levi-Civita connection $\nabla$. Let $R\in\operatorname{End}_{X}(T_{X})$ be an
endomorphism which is compatible with $g$ and satisfies $R^{2}=-1$. We can
then define a 2-form $\Omega$ on $X$ by setting
$\Omega_{R}(u_{1},u_{2})=g(R(u_{1}),u_{2})$. Let $H\subset T_{X}$ denote the
$+i$ eigenbundle of $R$. Then standard proofs from Kähler geometry apply
unchanged in this holomorphic context to give implications

$\nabla(R)=0\implies[H,H]\subset H,\qquad\nabla(R)=0\iff d\Omega_{R}=0.$ (15)

Return now to the setting above. For $\epsilon\in\mathbb{C}^{*}$ we introduce
the endomorphism

$J_{\epsilon}=I-i\epsilon^{-1}(J+iK).$ (16)

A simple calculation using the definitions (13) shows that
$J_{\epsilon}^{2}=-1$, and that the $+i$ eigenbundle of $J_{\epsilon}$
coincides with $H_{\epsilon}=\operatorname{im}(h_{\epsilon})$.

As in Section 2.2, the symplectic form $\omega$ on $M$ induces a relative
symplectic form $\Omega_{\pi}$ on the projection $\pi\colon X\to M$. Moreover,
as explained before Lemma 2.3, there is then a unique 2-form
$\Omega_{\epsilon}$ on $X$ satisfying the conditions

$\ker(\Omega_{\epsilon})=H_{\epsilon},\qquad\Omega_{\epsilon}(v(u_{1}),v(u_{2}))=\omega(u_{1},u_{2}),$
(17)

where $u_{1},u_{2}$ are arbitrary vector fields on $M$. Another calculation
using (13) and (14) shows that this form is given explicitly by the formula

$\Omega_{\epsilon}=\epsilon^{-2}\cdot\Omega_{+}+2i\epsilon^{-1}\cdot\Omega_{I}+\Omega_{-},\qquad\Omega_{\pm}=\Omega_{J\pm
iK}.$ (18)

We can now prove the Theorem. Suppose first that $I,J,K$ are parallel. Then
$J_{\epsilon}$ is parallel for all $\epsilon\in\mathbb{C}^{*}$, and applying
(15) with $R=J_{\epsilon}$ we find that $[H_{\epsilon},H_{\epsilon}]\subset
H_{\epsilon}$ and hence that $h_{\epsilon}$ is flat. Since $\Omega_{\epsilon}$
is also parallel and hence closed, applying Lemma 2.3 shows that
$h_{\epsilon}$ is symplectic. Conversely suppose that for all
$\epsilon\in\mathbb{C}^{*}$ the connection $h_{\epsilon}$ is flat and
symplectic. Then by Lemma 2.3 again, $d\Omega_{\epsilon}=0$ for all
$\epsilon\in\mathbb{C}^{*}$, and this easily implies that $d\Omega_{R}=0$ for
$R\in\\{I,J,K\\}$. By (15) we conclude that $I,J,K$ are parallel. ∎

###  2.4. Associated 2-forms

The complex hyperkähler structure $(g,I,J,K)$ gives rise to closed 2-forms on
$X$

$\Omega_{I}(w_{1},w_{2})=g(I(w_{1}),w_{2}),\qquad\Omega_{\pm}(w_{1},w_{2})=g((J\pm
iK)(w_{1}),w_{2}).$ (19)

Note that $\Omega_{I}$ is non-degenerate, but since $(J\pm iK)^{2}=0$ the
forms $\Omega_{\pm}$ have kernels.

Let us express these forms in terms of a local co-ordinate system $(z,\theta)$
on $X$ as in Section 2.2. We denote by $(h^{i},v^{j})$ the basis of covector
fields dual to the basis of vector fields $(h_{i},v_{j})$ defined by (5) –
(6). Thus $(h^{j},v_{i})=0=(v^{j},h_{i})$ and
$(h^{j},h_{i})=\delta_{ij}=(v^{j},v_{i})$. Explicitly we have

$h^{j}=dz_{j},\qquad v^{j}=d\theta_{j}+\sum_{r,s}\eta_{jr}\cdot\frac{\partial
W_{s}}{\partial\theta_{r}}\cdot dz_{s}.$ (20)

The definition (13) of the operators $I,J,K$ immediately give

$(J+iK)\circ h=-2v,\qquad(J+iK)\circ v=0=(J-iK)\circ h,\qquad(J-iK)\circ
v=2h,$ (21)

and so we have

$\displaystyle\Omega_{+}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
h^{p}\wedge h^{q},\qquad\Omega_{I}=\frac{i}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
v^{p}\wedge h^{q},$ (22)
$\displaystyle\Omega_{-}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
v^{p}\wedge v^{q}.$ (23)

Using the formulae (20) these expressions can be rewritten as

$\displaystyle\Omega_{+}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
dz_{p}\wedge dz_{q},$ (24) $\displaystyle
2i\Omega_{I}=\frac{1}{2}\cdot\sum_{p,q}\bigg{(}\frac{\partial
W_{p}}{\partial\theta_{q}}-\frac{\partial
W_{q}}{\partial\theta_{p}}\bigg{)}\cdot dz_{p}\wedge
dz_{q}-\sum_{p,q}\omega_{pq}\cdot d\theta_{p}\wedge dz_{q},$ (25)
$\displaystyle\Omega_{-}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
d\theta_{p}\wedge d\theta_{q}+\sum_{p,q}\frac{\partial
W_{q}}{\partial\theta_{p}}\cdot d\theta_{p}\wedge
dz_{q}-\frac{1}{2}\cdot\sum_{p,q,r,s}\eta_{rs}\cdot\frac{\partial
W_{p}}{\partial\theta_{r}}\frac{\partial W_{q}}{\partial\theta_{s}}\cdot
dz_{p}\wedge dz_{q}.$ (26)

Note that by (8) the first term in (25) is independent of the co-ordinates
$\theta_{k}$ and hence descends to $M$. In the case of a Joyce structure the
identity (47) shows that this term vanishes.

##  3\. Joyce structures

A Joyce structure on a complex manifold $M$ is a pre-Joyce structure with
certain additional symmetries. These symmetries are controlled by a special
kind of integral affine structure on $M$ which we call a period structure.
After introducing the necessary definitions we derive some consequences of the
extra symmetries, both for the associated complex hyperkähler structure on
$X=T_{M}$, and for the local generating functions $W_{i}$.

###  3.1. Period structures

Let $\mathcal{H}$ be a holomorphic vector bundle on a complex manifold $M$. By
a lattice in $\mathcal{H}$ we mean a locally-constant subsheaf of abelian
groups $\mathcal{H}^{\mathbb{Z}}\subset\mathcal{H}$ such that the
multiplication map
$\mathcal{H}^{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathscr{O}_{M}\to\mathcal{H}$ is
an isomorphism. There is an induced flat (linear) connection $\nabla$ on
$\mathcal{H}$ whose flat sections are $\mathbb{C}$-linear combinations of the
sections of $\mathcal{H}^{\mathbb{Z}}$.

######  Definition 3.1.

A period structure $(T_{M}^{\mathbb{Z}},\nabla,Z)$ on a complex manifold $M$
consists of

  * (P1)

a lattice $T_{M}^{\mathbb{Z}}\subset T_{M}$ whose associated flat connection
is denoted $\nabla$,

  * (P2)

a vector field $Z\in\Gamma(M,T_{M})$ satisfying $\nabla(Z)=\operatorname{id}$.

Given a point $p\in M$, a basis of the free abelian group
$T_{M,p}^{\mathbb{Z}}$ extends uniquely to a basis of $\nabla$-flat sections
$\phi_{1},\cdots,\phi_{n}$ of $T_{M}$ over a contractible open neighbourhood
$p\in U\subset M$. Writing the vector field $Z$ in the form
$Z=\sum_{i}z_{i}\cdot\phi_{i}$ then defines holomorphic functions $z_{i}\colon
U\to\mathbb{C}$. Given a local co-ordinate system $(u_{1},\cdots,u_{n})$ on
$M$, condition (P2) shows that

$\frac{\partial}{\partial u_{j}}=\nabla_{\frac{\partial}{\partial
u_{j}}}(Z)=\sum_{i}\frac{\partial z_{i}}{\partial u_{j}}\cdot\phi_{i},$ (27)

for all $1\leqslant j\leqslant n$, from which it follows that
$(z_{1},\cdots,z_{n})$ is also a local co-ordinate system. Applying (27) with
$u_{j}=z_{j}$ then shows that $\phi_{i}=\frac{\partial}{\partial z_{i}}$. This
implies in particular that the connection $\nabla$ is torsion-free.

Recall [35] that an integral affine structure on a complex manifold $M$ is a
lattice $T_{M}^{\mathbb{Z}}\subset T_{M}$ whose associated flat connection
$\nabla$ is torsion-free . A local co-ordinate system $(z_{1},\cdots,z_{n})$
is then called integral affine if the tangent vectors
$\frac{\partial}{\partial z_{i}}$ lie in the lattice $T_{M}^{\mathbb{Z}}$.
Such co-ordinate systems are uniquely defined up to affine transformations of
the form $z_{i}\mapsto\sum_{j}a_{ij}z_{j}+v_{i}$ with
$(a_{ij})\in\operatorname{GL}_{n}(\mathbb{Z})$ and $(v_{i})\in\mathbb{C}^{n}$.

Given a period structure on a complex manifold $M$ we obtain an integral
affine structure by forgetting the vector field $Z$. A system of integral
affine co-ordinates $(z_{1},\cdots,z_{n})$ will be called integral linear if
$Z=\sum_{i}z_{i}\cdot\frac{\partial}{\partial z_{i}}$. Such co-ordinate
systems are uniquely defined up to linear transformations of the form
$z_{i}\mapsto\sum_{j}a_{ij}z_{j}$ with
$(a_{ij})\in\operatorname{GL}_{n}(\mathbb{Z})$. Thus a period structure can be
thought of as an integral _linear_ structure.

######  Definition 3.2.

A period structure will be called homogeneous if the vector field $Z$
generates an action of the multiplicative group $\mathbb{C}^{*}$ on $M$.

We can use the connection $\nabla$ on $T_{M}$ to lift the vector field $Z$ on
$M$ to a horizontal vector field $E$ on $X=T_{M}$. If we take a system of
integral linear co-ordinates $(z_{1},\cdots,z_{n})$ on $M$, and associated co-
ordinates $(z_{i},\theta_{j})$ on $X=T_{M}$ as in Section 2.2, then

$Z=\sum_{i}z_{i}\cdot\frac{\partial}{\partial z_{i}},\qquad
E=\sum_{i}z_{i}\cdot\frac{\partial}{\partial z_{i}}.$ (28)

In what follows we refer to both $Z$ and $E$ as Euler vector fields. This will
hopefully not cause confusion since they live on different spaces.

######  Lemma 3.3.

Let $(T_{M}^{\mathbb{Z}},\nabla,Z)$ be a homogeneous period structure on a
complex manifold $M$. Then the lifted Euler vector field $E$ generates an
action of $\mathbb{C}^{*}$ on $X=T_{M}$. This action is obtained by combining
the derivative of the action map $a\colon\mathbb{C}^{*}\times M\to M$, with a
rescaling of weight $-1$ on the linear fibres of the projection $\pi\colon
T_{M}\to M$.

###### Proof.

Take local co-ordinates $(z_{i})$ on $M$ and $(z_{i},\theta_{j})$ on $X$ as
above. The $\mathbb{C}^{*}$-action on $M$ is given by
$t\cdot(z_{i})=(tz_{i})$. The derivative of the action map
$a\colon\mathbb{C}^{*}\times M\to M$ defines a $\mathbb{C}^{*}$-action on $X$
given by $t\cdot(z_{i},\theta_{j})=(tz_{i},t\theta_{j})$. Composing with the
weight $-1$ rescaling action on the fibres gives the action
$t\cdot(z_{i},\theta_{j})=(tz_{i},\theta_{j})$ whose generating vector field
is $E$. ∎

###  3.2. Joyce structures

We define a Joyce structure by combining a pre-Joyce structure with a
compatible period structure. Given a symplectic form $\omega\colon T_{M}\times
T_{M}\to\mathscr{O}_{M}$ there is an induced pairing $\eta\colon
T_{M}^{*}\times T_{M}^{*}\to\mathscr{O}_{M}$ defined by the condition that the
induced maps $\omega^{\flat}\colon T_{M}\to T_{M}^{*}$ and $\eta^{\flat}\colon
T_{M}^{*}\to T_{M}$ are mutually inverse. We refer to $\eta$ as the inverse of
$\omega$.

######  Definition 3.4.

A Joyce structure on a complex manifold $M$ consists of

  * (a)

a period structure $(T_{M}^{\mathbb{Z}},Z,\nabla)$ on $M$,

  * (b)

a pre-Joyce structure $(\omega,h)$ on $M$,

satisfying the following compatibility conditions:

  * (J1)

if $\eta$ is the inverse of $\omega$ then $(2\pi i)^{-1}\eta$ takes integral
values on the dual lattice $(T_{M}^{\mathbb{Z}})^{*}\subset T_{M}^{*}$,

  * (J2)

the connection $h$ is invariant under translations by the lattice $(2\pi
i)\,T_{M}^{\mathbb{Z}}\subset T_{M}$,

  * (J3)

if $E$ is the $\nabla$-horizontal lift of the vector field $Z$, then for any
vector field $u$ on $M$

$h([Z,u])=[E,h(u)],$ (29)

  * (J4)

the connection $h$ is invariant under the action of the involution $-1\colon
X\to X$ which acts by multiplication by $-1$ on the fibres of $\pi\colon
X=T_{M}\to M$.

We say that a Joyce structure is homogeneous if the underlying period
structure is.

Given a period structure on a complex manifold $M$ we introduce the quotient

$X^{\\#}=T_{M}^{\\#}=T_{M}/(2\pi i)\,T_{M}^{\mathbb{Z}}.$ (30)

Axiom (J2) is the statement that the connection $h$ descends to a connection
on the $(\mathbb{C}^{*})^{n}$-bundle over $M$ given by the projection
$\pi\colon X^{\\#}\to M$.

######  Remark 3.5.

In the context of DT theory, in which $M$ is a space of numerical stability
conditions on a CY3 category, the period structure
$(T_{M}^{\mathbb{Z}},Z,\nabla)$ and symplectic form $\omega$ are well-known
and immediate: the integral linear co-ordinates are given by central charges,
and the symplectic form is induced by the inverse of the Euler form. The extra
content of the Joyce structure is then just the non-linear connection $h$,
which however is required to satisfy a complicated system of symmetry and
curvature conditions.

###  3.3. Associated hyperkähler structures

Given a Joyce structure on a complex manifold $M$, the associated complex
hyperkähler structure $(g,I,J,K)$ on $X=T_{M}$ defined in Section 2.3 has
certain extra symmetry properties. For example, axiom (J2) ensures that it is
invariant under the translations by the lattice $(2\pi
i)\,T_{M}^{\mathbb{Z}}\subset T_{M}$, and hence descends to the quotient
manifold $X^{\\#}$ introduced above. The other axioms give the following
result.

######  Lemma 3.6.

There are identities

$\displaystyle\mathcal{L}_{E}(I)$ $\displaystyle=0,$
$\displaystyle\mathcal{L}_{E}(J\pm iK)$ $\displaystyle=\pm(J\pm iK),$
$\displaystyle\mathcal{L}_{E}(g)$ $\displaystyle=g.$ (31)
$\displaystyle(-1)^{*}(I)$ $\displaystyle=I,$ $\displaystyle(-1)^{*}(J\pm iK)$
$\displaystyle=-(J\pm iK),$ $\displaystyle(-1)^{*}(g)$ $\displaystyle=-g.$
(32)

###### Proof.

We will evaluate both sides of each identity on vector fields of the form
$h(u)$ and $v(u)$, where $u$ is an arbitrary vector field on $M$. For the
identities (31) note first that we have

$\mathcal{L}_{E}(h(u))=h(\mathcal{L}_{Z}(u)),\qquad\mathcal{L}_{E}(v(u))=v(\mathcal{L}_{Z}(u)+u)$
(33)

The first of these equations is axiom (J3), and the second follows from Lemma
3.3.

If $u$ is a vector field on $M$ then using (21) we have

$\displaystyle(\mathcal{L}_{E}(J-iK))\,h(u)$
$\displaystyle=\mathcal{L}_{E}((J-iK)\,h(u))-(J-iK)\,\mathcal{L}_{E}(h(u))=0,$
(34)
$\displaystyle\begin{split}(\mathcal{L}_{E}(J-iK))\,v(u)&=\mathcal{L}_{E}((J-iK)\,v(u))-(J-iK)\,\mathcal{L}_{E}(v(u))\\\
&=2h(\mathcal{L}_{Z}(u))-2h(\mathcal{L}_{Z}(u)+u)=-2h(u),\end{split}$ (35)

and it follows that $\mathcal{L}_{E}(J-iK)=-(J-iK)$. The claim
$\mathcal{L}_{E}(J+iK)=J+iK$ follows in the same way. The statement that
$\mathcal{L}_{E}(I)=0$ holds because by (33)

$(\mathcal{L}_{E}(I))\,h(u)=\mathcal{L}_{E}(I(h(u)))-I(\mathcal{L}_{E}(h(u)))=\mathcal{L}_{E}(ih(u))-i\mathcal{L}_{E}(h(u))=0,$
(36)

with an analogous identity for $v(u)$.

For the statement on the metric note first that (J1) implies that
$\mathcal{L}_{Z}(\omega)=2\omega$. Then using the definition (14) of the
metric $g$ we have

$\begin{split}\mathcal{L}_{E}(g)(h(u_{1}),v(u_{2}))&=E\cdot
g(h(u_{1}),v(u_{2}))-g(\mathcal{L}_{E}(h(u_{1})),v(u_{2}))-g(h(u_{1}),\mathcal{L}_{E}(v(u_{2})))\\\
&=E\cdot\tfrac{1}{2}\pi^{*}(\omega(u_{1},u_{2}))-\tfrac{1}{2}\pi^{*}(\omega(\mathcal{L}_{Z}(u_{1}),u_{2}))-\tfrac{1}{2}\pi^{*}(\omega(u_{1},\mathcal{L}_{Z}(u_{2})+u_{2}))\\\
&=\tfrac{1}{2}\pi^{*}(\mathcal{L}_{Z}(\omega)(u_{1},u_{2}))-\tfrac{1}{2}\pi^{*}(\omega(u_{1},u_{2}))=\tfrac{1}{2}\pi^{*}(\omega(u_{1},u_{2}))\\\
&=g(h(u_{1}),v(u_{2})),\end{split}$ (37)

where we used the fact that $\pi_{*}(E_{x})=Z_{\pi(x)}$ for all $x\in X$ to
write $E\cdot\pi^{*}(f)=\pi^{*}(Z\cdot f)$ for any function $f$ on $M$. An
easier argument gives

$\mathcal{L}_{E}(g)(h(u_{1}),h(u_{2}))=0=\mathcal{L}_{E}(g)(v(u_{1}),v(u_{2})),$
(38)

so we find that $\mathcal{L}_{E}(g)=g$ as required.

The identities (32) follow from the definitions (13) and (14) together with

$(-1)^{*}(h(u))=h(u),\qquad(-1)^{*}(v(u))=-v(u).$ (39)

The first of these equations is axiom (J4), and the second is immediate from
the definition of the involution $-1$. ∎

###  3.4. Plebański function

Consider a local system of integral linear co-ordinates $(z_{1},\cdots,z_{n})$
on $M$. Taking associated co-ordinates $(z_{i},\theta_{j})$ on $X=T_{M}$ we
can express the connections $h_{\epsilon}$ using locally-defined functions
$W_{i}=W_{i}(z,\theta)$ as in Section 2.2. As before we fix the integration
constants by imposing the condition that $W_{i}$ vanishes on the zero section
$M\subset X=T_{M}$.

######  Lemma 3.7.

There is a unique locally-defined function $W=W(z,\theta)$ which vanishes on
the zero section $M\subset X=T_{M}$ and satisfies $W_{i}=\frac{\partial
W}{\partial\theta_{i}}$ for all $1\leqslant i\leqslant n$. It also satisfies
the relations

$\displaystyle\frac{\partial^{2}W}{\partial\theta_{i}\partial
z_{j}}-\frac{\partial^{2}W}{\partial\theta_{j}\partial
z_{i}}=\sum_{p,q}\eta_{pq}\cdot\frac{\partial^{2}W}{\partial\theta_{i}\partial\theta_{p}}\cdot\frac{\partial^{2}W}{\partial\theta_{j}\partial\theta_{q}}$
(40)
$\displaystyle\frac{\partial^{2}W}{\partial\theta_{i}\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1}+2\pi
ik_{1},\cdots,\theta_{n}+2\pi
ik_{n})=\frac{\partial^{2}W}{\partial\theta_{i}\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}),$
(41) $\displaystyle W(\lambda z_{1},\cdots,\lambda
z_{n},\theta_{1},\cdots,\theta_{n})=\lambda^{-1}\cdot
W(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}),$ (42) $\displaystyle
W(z_{1},\cdots,z_{n},-\theta_{1},\cdots,-\theta_{n})=-W(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}),$
(43)

for all $1\leqslant i,j\leqslant n$, where
$(k_{1},\cdots,k_{n})\in\mathbb{Z}^{n}$ and $\lambda\in\mathbb{C}^{*}$.

###### Proof.

Using the expressions (28) the axioms (J2) – (J4) become the conditions

$\displaystyle\frac{\partial
W_{i}}{\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1}+2\pi
ik_{1},\cdots,\theta_{n}+2\pi ik_{n})=\frac{\partial
W_{i}}{\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}),$
(44) $\displaystyle\frac{\partial W_{i}}{\partial\theta_{j}}(\lambda
z_{1},\cdots,\lambda
z_{n},\theta_{1},\cdots,\theta_{n})=\lambda^{-1}\cdot\frac{\partial
W_{i}}{\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}),$
(45) $\displaystyle\frac{\partial
W_{i}}{\partial\theta_{j}}(z_{1},\cdots,z_{n},-\theta_{1},\cdots,-\theta_{n})=-\frac{\partial
W_{i}}{\partial\theta_{j}}(z_{1},\cdots,z_{n},\theta_{1},\cdots,\theta_{n}).$
(46)

Note that (46) implies that both sides vanish along the zero section $M\subset
X=T_{M}$ where all $\theta_{k}=0$. It then follows that for all $1\leqslant
i,j\leqslant n$

$\frac{\partial W_{i}}{\partial\theta_{j}}-\frac{\partial
W_{j}}{\partial\theta_{i}}=0.$ (47)

Indeed, (8) shows that this expression is independent of the co-ordinates
$\theta_{k}$, and so if it vanishes on the zero section it must be identically
zero. It follows that there is a locally-defined function $W=W(z,\theta)$ such
that $W_{i}=\partial W/\partial\theta_{i}$. We again fix the integration
constants by assuming that $W$ vanishes on the zero section $M\subset
X=T_{M}$. In view of (7) we therefore have

$W(z_{1},\cdots,z_{n},0,\cdots,0)=0=\frac{\partial
W}{\partial\theta_{i}}(z_{1},\cdots,z_{n},0,\cdots,0).$ (48)

The equations (40) then follow from (9). Indeed (9) shows that the difference
of the two sides is independent of the co-ordinates $\theta_{k}$, but by (46)
and (48) both sides vanish along the zero section. Similarly, we deduce (42)
from (45), and (43) from (46). ∎

The function $W$ is called the Plebański function, and the partial
differential equations (40) are known as Plebański’s second heavenly
equations.

##  4\. Twistor space

In this section we define the twistor space $p\colon Z\to\mathbb{P}^{1}$
associated a pre-Joyce structure on a complex manifold $M$. It is defined as
the space of leaves of a foliation on $\mathbb{P}^{1}\times X$, where as
before $X=T_{M}$ denotes the total space of the tangent bundle of $M$. The
construction coincides with the Penrose twistor space construction [39]
applied to the complex hyperkähler manifold on $X$. We explain some special
properties enjoyed by the twistor space of a Joyce structure, and derive an
equation describing the twistor lines. In the final part we use this equation
to revisit the analogy between Joyce and Frobenius structures which was the
main topic of [12].

###  4.1. Construction of the twistor space

Let $M$ be a complex manifold equipped with a pre-Joyce structure, and set
$X=T_{M}$. For each point
$(\epsilon_{0},\epsilon_{1})\in\mathbb{C}^{2}\setminus\\{0\\}$ there is a
bundle map

$\epsilon_{0}v+\epsilon_{1}h\colon\pi^{*}(T_{M})\to T_{X},$ (49)

whose image, which depends only on the corresponding point
$\epsilon=[\epsilon_{0}:\epsilon_{1}]\in\mathbb{P}^{1}$, is a half-rank sub-
bundle $H(\epsilon)\subset T_{X}$. The definition of a pre-Joyce structure
ensures that this sub-bundle is closed under Lie bracket, and hence defines a
foliation on $X$. The twistor space $Z$ defined below will have a map $p\colon
Z\to\mathbb{P}^{1}$ whose fibre over the point $\epsilon\in\mathbb{P}^{1}$ is
the space of leaves of this foliation.

To give a global definition of the twistor space $Z$ we first recall that the
tangent bundle of the product $\mathbb{P}^{1}\times X$ has a canonical direct
sum decomposition

$T_{\mathbb{P}^{1}\times
X}=\pi_{1}^{*}(T_{\mathbb{P}^{1}})\oplus\pi_{2}^{*}(X),$ (50)

where $\pi_{1},\pi_{2}$ denote the projections from $\mathbb{P}^{1}\times X$
onto the two factors. There is a half-rank sub-bundle
$H\subset\pi_{2}^{*}(T_{X})$ which when restricted to a fibre
$\pi_{1}^{-1}(\epsilon)$ is the sub-bundle $H(\epsilon)\subset T_{X}$.
Composing with the canonical inclusion $\pi_{2}^{*}(T_{X})\subset
T_{\mathbb{P}^{1}\times X}$ we can view $H$ as a sub-bundle of
$T_{\mathbb{P}^{1}\times X}$ and this is easily seen to be closed under Lie
bracket. The twistor space $Z$ is then defined to be the space of leaves of
the associated foliation on $\mathbb{P}^{1}\times X$.

We denote by $q\colon\mathbb{P}^{1}\times X\to Z$ the quotient map. There is
an induced projection $p\colon Z\to\mathbb{P}^{1}$ which satisfies $p\circ
q=\pi_{1}$. We denote by $Z_{\epsilon}=p^{-1}(\epsilon)\subset Z$ the twistor
fibre over $\epsilon\in\mathbb{P}^{1}$, and $q_{\epsilon}\colon X\to
Z_{\epsilon}$ the induced quotient map. For each $\epsilon\in\mathbb{P}^{1}$
there is a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\epsilon}}$$\textstyle{\mathbb{P}^{1}\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{\pi_{1}}$$\textstyle{Z_{\epsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\\{\epsilon\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}}$
(51)

in which the horizontal arrows are the obvious closed embeddings.

Each point $x\in X$ determines a section of the map $p\colon
Z\to\mathbb{P}^{1}$

$\sigma_{x}\colon\mathbb{P}^{1}\to Z,\qquad\epsilon\mapsto q(\epsilon,x),$
(52)

which we refer to as a twistor line.

Recall the complex hyperkähler structure $(g,I,J,K)$ on $X$ and the associated
closed 2-forms $\Omega_{\pm}$ and $\Omega_{I}$ defined by (19). As in the
proof of Theorem 2.7, an easy calculation using (13) shows that

$H(\epsilon)=\operatorname{im}\left(\epsilon_{0}v+\epsilon_{1}h\right)=\ker\left(\epsilon_{0}^{2}(J+iK)+2i\epsilon_{0}\epsilon_{1}I+\epsilon_{1}^{2}(J-iK)\right).$
(53)

Since the right-hand side is precisely the kernel of the closed 2-form

$\epsilon_{0}^{2}\,\Omega_{+}+2i\epsilon_{0}\epsilon_{1}\,\Omega_{I}+\epsilon_{1}^{2}\,\Omega_{-},$
(54)

this 2-form descends along the map $q_{\epsilon}\colon X\to Z_{\epsilon}$, and
defines a symplectic form $\Omega_{\epsilon}$ on $Z_{\epsilon}$.

In more global terms, the formula (54) defines a twisted relative 2-form on
$\mathbb{P}^{1}\times X$, namely a section of the bundle
$\pi_{1}^{*}(\mathscr{O}(2))\otimes\pi_{2}^{*}(\wedge^{2}\,T_{X}^{*})$. This
descends along the quoitient map $q\colon X\to Z$ to give a twisted relative
symplectic form on the twistor space $p\colon Z\to\mathbb{P}^{1}$. By
definition, this is the section $\Omega$ of the bundle
$p^{*}(\mathscr{O}(2))\otimes\wedge^{2}\,T_{Z/\mathbb{P}^{1}}^{*}$ uniquely
defined by the condition

$q^{*}(\Omega)=\epsilon_{0}^{2}\,\Omega_{+}+2i\epsilon_{0}\epsilon_{1}\Omega_{I}+\epsilon_{1}^{2}\,\Omega_{-}.$
(55)

From now on we will take the usual embedding $\mathbb{C}\subset\mathbb{P}^{1}$
with affine co-ordinate $\epsilon=\epsilon_{1}/\epsilon_{0}$, and write
$\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$ with $\infty=[0:1]$. When
$\epsilon\in\mathbb{C}\cup\\{\infty\\}$, the twistor fibre $Z_{\epsilon}$ is
the space of leaves of the foliation on $X$ associated to the integrable
distribution $\operatorname{im}(h_{\epsilon})\subset T_{X}$, where as before
we write $h_{\epsilon}=h+\epsilon^{-1}v$. On the other hand the twistor fibre
$Z_{0}$ is the space of leaves of the foliation on $X$ associated to the
vertical sub-bundle $\operatorname{im}(v)=\ker(\pi_{*})$, and is therefore
identified with $M$.

Restricting $\Omega$ to a twistor fibre $Z_{\epsilon}$ gives a complex
symplectic form $\Omega_{\epsilon}$, well-defined up to multiplication by a
nonzero constant. When $\epsilon\in\mathbb{C}\cup\\{\infty\\}$ we fix this
scale by taking

$q_{\epsilon}^{*}(\Omega_{\epsilon})=\epsilon^{-2}\,\Omega_{+}+2i\epsilon^{-1}\,\Omega_{I}+\Omega_{-},$
(56)

whereas for $\epsilon=0$ we take $\Omega_{0}=\omega$. We then have relations

$q_{0}^{*}(\Omega_{0})=\Omega_{+},\qquad
q_{\infty}^{*}(\Omega_{\infty})=\Omega_{-}.$ (57)

######  Remark 4.1.

Unfortunately, to obtain a well-behaved twistor space $p\colon
Z\to\mathbb{P}^{1}$ we cannot in general just take the space of leaves of the
foliation on $\mathbb{P}^{1}\times X$. Rather, we should consider the holonomy
groupoid, which leads to the analytic analogue of a Deligne-Mumford stack
[38]. We will completely ignore these subtleties here, and essentially pretend
that $Z$ is a complex manifold. In fact, for what we do here, nothing useful
would be gained by a more abstract point of view, because we are only really
using the twistor space as a useful and suggestive shorthand. All statements
we make about the space $Z$ can be easily translated into statements only
involving objects on $X$. For example, a symplectic form on the twistor fibre
$Z_{\epsilon}$ is nothing but a closed 2-form on $X$ whose kernel coincides
with the sub-bundle $H(\epsilon)\subset T_{X}$. Similarly, an étale map from
the twistor fibre $Z_{\epsilon}$ to some complex manifold $Y$ is just a
holomorphic map $f\colon X\to Y$ such that $\ker(f_{*})=H(\epsilon)$.

###  4.2. Twistor space of a Joyce structure

Suppose now that we have a Joyce structure on a complex manifold $M$ and
consider the associated twistor space $p\colon Z\to\mathbb{P}^{1}$. Recall the
vector field $Z$ on $M$ and the horizontal lift $E$ on $X$. Using the
decomposition (50) we can define a vector field on the product
$\mathbb{P}^{1}\times X$ as the sum

$\tilde{E}=\epsilon\,\frac{d}{d\epsilon}+E.$ (58)

######  Lemma 4.2.

The vector field $\tilde{E}$ descends along the quotient map
$q\colon\mathbb{P}^{1}\times X\to Z$.

###### Proof.

Let $u$ be a vector field on $M$. Writing
$h_{\epsilon}(u)=h(u)+\epsilon^{-1}v(u)$ and using (33) we have

$\begin{split}\big{[}\tilde{E},h_{\epsilon}(u))\big{]}&=[E,h(u)]+\epsilon^{-1}[E,v(u)]-\epsilon^{-1}v(u)\\\
&=h([Z,u])+\epsilon^{-1}v([Z,u])=h_{\epsilon}([Z,u]).\end{split}$ (59)

Thus $\mathcal{L}_{\tilde{E}}$ preserves the sub-bundle
$H_{\epsilon}=\ker(q_{\epsilon})_{*}$, and the claim follows. ∎

Let us specialise to the case of a homogeneous Joyce structure. The
$\mathbb{C}^{*}$-action on $M$ generated by the vector field $Z$ induces a
$\mathbb{C}^{*}$-action on $X$ with generating vector field $E$ as in Lemma
3.3. Combining this with the standard action of $\mathbb{C}^{*}$ on
$\mathbb{P}^{1}$ rescaling $\epsilon$ with weight 1, we can then consider the
diagonal action on $X\times\mathbb{P}^{1}$. It follows from Lemma 4.2 that
this descends to an action on $Z$. Thus we obtain $\mathbb{C}^{*}$-actions on
each of the spaces in the right-hand column of (51), and the vertical maps $q$
and $p$ intertwine these actions.

Since the map $p\colon Z\to\mathbb{P}^{1}$ is $\mathbb{C}^{*}$-equivariant,
there are at most three distinct twistor fibres: $Z_{0}$, $Z_{1}$ and
$Z_{\infty}$, and as above there is an identification $Z_{0}=M$.555Note that,
unlike in the case of real hyperkähler manifolds, there is no requirement for
an involution of the twistor space $Z$ lifting the antipodal map on
$\mathbb{P}^{1}$. In particular, there need be no relation between the twistor
fibres $Z_{0}$ and $Z_{\infty}$. Using the definition (19) and the identities
(31) we have

$\mathcal{L}_{E}(\Omega_{+})=2\Omega_{+},\qquad\mathcal{L}_{E}(\Omega_{I})=\Omega_{I},\qquad\mathcal{L}_{E}(\Omega_{-})=0.$
(60)

The $\mathbb{C}^{*}$-action on $Z$ induces a $\mathbb{C}^{*}$-action on
$Z_{0}$ and it follows from (60) that the symplectic form $\Omega_{0}$ is
homogeneous of weight 2 for this action. Similarly there is an induced
$\mathbb{C}^{*}$-action on $Z_{\infty}$, but in this case the symplectic form
$\Omega_{\infty}$ is invariant.

We can use the $\mathbb{C}^{*}$-action to trivialise the map $p\colon
Z\to\mathbb{P}^{1}$ over the open subset
$\mathbb{C}^{*}\subset\mathbb{P}^{1}$. We obtain a commutative diagram

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.18193pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{C}^{*}\times
Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
29.39241pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{m}$}}}\kern
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(61)

where $\pi_{1}$ is the projection onto the first factor, and the isomorphism
$m$ is the restriction of the action map $m\colon\mathbb{C}^{*}\times Z\to Z$.

Consider the map $y\colon\mathbb{C}^{*}\times X\to Z_{1}$ given by

$y(\epsilon,x)=\epsilon^{-1}\cdot q(\epsilon,x)=q(1,\epsilon^{-1}\cdot x).$
(62)

Note that, under the trivialisation (61), the restriction of the twistor line
$\sigma_{x}\colon\mathbb{P}^{1}\to Z$ becomes the section of $\pi_{1}$ given
by $\epsilon\mapsto(\epsilon,y(\epsilon,x))$. Let us take a system of integral
linear co-ordinates $(z_{1},\cdots,z_{n})$ on $M$ and consider the induced co-
ordinates $(z_{i},\theta_{j})$ on $X=T_{M}$ as in Section 2.2.

######  Lemma 4.3.

The map $y(\epsilon,x)$ satisfies the equation

$\frac{\partial}{\partial\epsilon}\,y(\epsilon,x)=\bigg{(}\frac{1}{\epsilon^{2}}\cdot\sum_{i}z_{i}\cdot\frac{\partial}{\partial\theta_{i}}+\frac{1}{\epsilon}\cdot\sum_{i,p,q}\eta_{pq}\cdot
z_{i}\cdot\frac{\partial^{2}W}{\partial\theta_{i}\partial\theta_{p}}\cdot\frac{\partial}{\partial\theta_{q}}\bigg{)}\,y(\epsilon,x),$
(63)

###### Proof.

The formula $y(\epsilon,x)=q(1,\epsilon^{-1}\cdot x)$ shows that $y$ is
invariant under the diagonal $\mathbb{C}^{*}$-action on
$X\times\mathbb{C}^{*}$, and is therefore annhilated by the vector field

$\tilde{E}=\sum_{i}z_{i}\cdot\partial/\partial
z_{i}+\epsilon\cdot\partial/\partial\epsilon.$ (64)

On the other hand, the formula $y(\epsilon,x)=\epsilon^{-1}\cdot
q(\epsilon,x)=\epsilon^{-1}\cdot q_{\epsilon}(x)$ shows that $y$ factors via
the map

$1\times q_{\epsilon}\colon\mathbb{C}^{*}\times X\to\mathbb{C}^{*}\times
Z_{\epsilon}.$ (65)

By definition of the twistor space it is therefore also annhilated by the
vector fields $h_{\epsilon}(\partial/\partial
z_{i})=h_{i}+\epsilon^{-1}v_{i}$. The result then follows from the formulae
(5) and (6). ∎

###  4.3. Preferred co-ordinates and Stokes data

The definition of a Joyce structure was first identified in [12] by
considering an analogy between the wall-crossing formula in DT theory and a
class of iso-Stokes deformations familiar in the theory of Frobenius
manifolds. This analogy can be re-expressed in terms of the geometry of the
twistor space $p\colon Z\to\mathbb{P}^{1}$, and leads to interesting
conjectural properties of $Z$ which we attempt to explain here. This section
is speculative, and can be safely skipped. The basic point is that one should
expect preferred systems of co-ordinates on the twistor fibre $Z_{1}$, in
terms of which the twistor lines, viewed as maps $\mathbb{C}^{*}\to Z_{1}$ via
the trivialisation (61), have good asymptotic properties as $\epsilon\to 0$.

Let us fix a point $m\in M$ and restrict attention to points $x\in
T_{M,m}\subset X$. Changing the notation slightly, we obtain a family of maps
$y(\epsilon)\colon T_{M,m}\to Z_{1}$. Thus for each point $x\in T_{M,m}$, the
restriction of the twistor line $\sigma_{x}\colon\mathbb{P}^{1}\to Z$
corresponds under the trivialisation (61) to the map
$\epsilon\mapsto(\epsilon,y(\epsilon)(x))$. It is then interesting to observe
that the equation (63) controlling the variation of $y(\epsilon)$ is formally
analogous to the linear equation

$\frac{d}{d\epsilon}\,y(\epsilon)=\bigg{(}\frac{U}{\epsilon^{2}}+\frac{V}{\epsilon}\bigg{)}y(\epsilon)$
(66)

occurring in the theory of Frobenius manifolds. In the equation (66) the
matrices $U$ and $V$ are infinitesimal linear automorphisms of the tangent
space $T_{m}M$ at a point $m\in M$ of a Frobenius manifold, whereas the
corresponding quantities in (63) are infinitesimal automorphisms of the vector
space $T_{M,m}$ preserving the linear symplectic form $\omega_{m}$. This
analogy was the main topic of [12].

Suppose that the matrix $U$ in (66) has distinct eigenvalues
$u_{i}\in\mathbb{C}$. The Stokes rays of the equation (66) are then defined to
be the rays
$\ell_{ij}=\mathbb{R}_{>0}\cdot(u_{i}-u_{j})\subset\mathbb{C}^{*}$. A result
of Balser, Jurkat and Lutz [3] shows that in any half-plane
$H(\varphi)\subset\mathbb{C}^{*}$ centred on a non-Stokes ray
$r=\mathbb{R}_{>0}\cdot\exp({i\pi\varphi})\subset\mathbb{C}^{*}$, there is a
unique fundamental solution $\Phi\colon
H(\varphi)\to\operatorname{GL}_{n}(\mathbb{C})$ to (66) satisfying

$\Phi(\epsilon)\cdot\exp(U/\epsilon)\to\operatorname{id}\text{ as }\epsilon\to
0.$ (67)

Comparing these canonical solutions for half-planes $H(\varphi_{\pm})$ centred
on small perturbations $r_{\pm}$ of a Stokes ray $\ell\subset\mathbb{C}^{*}$
defines the Stokes factors
$\mathbb{S}(\ell)\in\operatorname{GL}_{n}(\mathbb{C})$.

Turning to the equation (63), note that the first term on the right-hand-side
is the translation-invariant vector field
$U=\sum_{i}z_{i}\cdot\partial/\partial\theta_{i}$ on $T_{M,m}$ corresponding
to the value $Z_{m}\in T_{M,m}$ of the vector field $Z$ defined by the period
structure on $M$. Exponentiating this vector field yields well-defined linear
automorphisms $\exp(U/\epsilon)$ of the vector space $T_{M,m}$ given in co-
ordinates by $\theta_{i}\mapsto\theta_{i}+\epsilon^{-1}z_{i}$. The above
analogy then suggests that there should be a countable collection of Stokes
rays $\ell\subset\mathbb{C}^{*}$, and for any half-plane
$H(\varphi)\subset\mathbb{C}^{*}$ centred on a non-Stokes ray
$r=\mathbb{R}_{>0}\cdot\exp(i\pi\varphi)\subset\mathbb{C}^{*}$, a
symplectomorphism $F\colon Z_{1}\to T_{M,m}$, such that
$\Phi(\epsilon):=F\circ y(\epsilon)\colon T_{M,m}\to T_{M,m}$ satisfies666The
factors in (67) and (68) appear in different orders because we are working in
the group of symplectic automorphisms of the space $\mathbb{C}^{n}$ rather
than the opposite group of Poisson automorphisms of its ring of functions:
compare [12, Section 6.6].

$\exp(U/\epsilon)\circ\Phi(\epsilon)\to\operatorname{id}\text{ as }\epsilon\to
0.$ (68)

The maps $F$ corresponding to different half-planes $H(\varphi)$ then differ
by compositions of symplectic automorphisms
$\mathbb{S}(\ell)\in\operatorname{Aut}_{\omega_{p}}(T_{M,m})$ associated to
rays $\ell\subset\mathbb{C}^{*}$. These automorphisms $\mathbb{S}(\ell)$
should be viewed as non-linear Stokes factors.

In practice the above analogy should be considered a guiding principle rather
than a precise statement. In particular we cannot expect the map $F$ to be
defined on the whole of $Z_{1}$. Nonetheless the basic point is that once we
have chosen a system of integral linear co-ordinates $(z_{1},\cdots,z_{n})$
near the point $m\in M$, there should be preferred systems of Darboux co-
ordinates $(t_{1},\cdots,t_{n})$777Unfortunately these co-ordinates are
usually denoted $(x_{1},\cdots,x_{n})$ since in certain examples they are the
logarithms of cluster $X$ co-ordinates. We can only apologise for this excess
of $x$’s. on open subsets of $Z_{1}$, depending on a choice of half-plane
$H(\varphi)\subset\mathbb{C}^{*}$. Given a point $x\in X$ with local co-
ordinates $(z_{i},\theta_{j})$ these should satisfy

$t_{i}(y(\epsilon)(x))\sim-\epsilon^{-1}z_{i}+\theta_{i}+O(\epsilon),$ (69)

as $\epsilon\to 0$ in the half-plane $H(\varphi)$.

We observed in Section 3.2 that axiom (J2) of Definition 3.4 implies that the
connections $h_{\epsilon}=h+\epsilon^{-1}v$ descend to the quotient manifold
$X^{\\#}=T_{M}/(2\pi i)\,T_{M}^{\mathbb{Z}}.$ We can then define a twistor
space $Z^{\\#}$ in exactly the same way as before by considering the space of
leaves of the resulting foliation on $\mathbb{P}^{1}\times X^{\\#}$. We can
then repeat the above discussion replacing $Z_{1}$ with $Z^{\\#}_{1}$ and the
vector spaces $T_{M,m}$ with the tori

$T_{M,m}^{\\#}=T_{M,m}/(2\pi
i)\,T_{M,m}^{\mathbb{Z}}\cong(\mathbb{C}^{*})^{n}.$ (70)

This leads to an anaalogy between Frobenius structures and Joyce structures in
which the group of linear automorphisms of the vector space $T_{M,m}$ is
replaced by the group of symplectic automorphisms of the torus
$T^{\\#}_{M,m}$.

##  5\. Further properties of twistor space

Let $M$ be a complex manifold equipped with a Joyce structure. In this section
we gather some miscellaneous results about the associated twistor space
$p\colon Z\to\mathbb{P}^{1}$ associated to a Joyce structure. Firstly, we
investigate Hamiltonian generating functions for the $\mathbb{C}^{*}$-action
on $Z_{\infty}$. Secondly, we observe that the zero section $M\subset X=T_{M}$
is contracted by the quotient map $q_{\infty}\colon X\to Z_{\infty}$, and
hence defines a distinguished point $0\in Z_{\infty}$. Structures on the
tangent space $T_{0}\,Z_{\infty}$ can then be pulled back to give linear
structures on the tangent bundle $T_{M}$. Finally we explain an alternative
way to describe a Joyce structure in terms of local co-ordinates and a
generating function.

###  5.1. Joyce function

From the Cartan formula and (60) we have

$di_{E}(\Omega_{-})-=\mathcal{L}_{E}(\Omega_{-})-i_{E}(d\Omega_{-})=0.$ (71)

Let us consider a function $F$ on $X$ satisfying $dF=-i_{E}(\Omega_{-})$. Then
$F$ descends along the quotient map $q_{\infty}\colon X\to Z_{\infty}$.
Indeed, if a vector field $u$ on $X$ is horizontal for the connection
$h=h_{\infty}$ then

$u(F)=i_{u}(dF)=-i_{u}i_{E}(\Omega_{-})=i_{E}i_{u}(\Omega_{-})=0,$ (72)

since $\Omega_{-}=q_{\infty}^{*}(\Omega_{\infty})$. We call the resulting
function $F\colon Z_{\infty}\to\mathbb{C}$, or its pullback to $X$, a Joyce
function.888In [12] we used the term Joyce function as a synonym for the
Plebański function $W$ of Section 3.4. Following [1] we now prefer to use it
for the function introduced here, which was also considered by Joyce [32]. In
the case of a homogeneous Joyce structure, $F$ is a Hamiltonian generating
function for the symplectic action of $\mathbb{C}^{*}$ on $Z_{\infty}$.

Note that $F$ is only well-defined up to the addition of a constant. It
follows from Lemma 5.2 below that the pullback $q_{\infty}^{*}(F)$ is constant
along the zero section $M\subset T_{M}=X$. We can therefore normalise $F$ by
insisting that it vanishes on this locus. Assuming $M$ is connected this is
equivalent to the statement that $F$ vanishes on the distinguished point $0\in
Z_{\infty}$ of Lemma 5.2.

To write an explicit expression for the Joyce function let us choose a local
system of co-ordinates on $M$ as in Section 3.4, and denote by $W=W(z,\theta)$
be the resulting Plebański function.

######  Lemma 5.1.

The Joyce function $F$ is given by the expression

$F(z_{i},\theta_{j})=v(E)(W)=\sum_{q}z_{q}\cdot\frac{\partial
W}{\partial\theta_{q}}$ (73)

###### Proof.

Using (26) and (40), and recalling that $W_{i}=\partial W/\partial\theta_{i}$,
we have

$\begin{split}i_{E}(\Omega_{-})&=-\sum_{p,q}z_{q}\,\frac{\partial^{2}W}{\partial\theta_{p}\partial\theta_{q}}\,d\theta_{p}+\sum_{p,q}\frac{\partial^{2}W}{\partial
z_{p}\partial\theta_{q}}(z_{p}\,dz_{q}-z_{q}\,dz_{p})\\\
&=-\sum_{p,q}z_{q}\,\frac{\partial^{2}W}{\partial\theta_{p}\partial\theta_{q}}\,d\theta_{p}-\sum_{p,q}z_{q}\,\frac{\partial^{2}W}{\partial
z_{p}\partial\theta_{q}}\,dz_{p}-\sum_{q}\frac{\partial
W}{\partial\theta_{q}}\,dz_{q}\\\ &=-\sum_{p}\Big{(}\frac{\partial
F}{\partial\theta_{p}}\,d\theta_{p}+\frac{\partial F}{\partial
z_{p}}\,dz_{p}\Big{)}=-dF,\end{split}$ (74)

where we used the homogeneity property (42) in the form
$\sum_{p}z_{p}\cdot\frac{\partial W}{\partial z_{p}}=-W$. The expression (73)
vanishes along the zero section $M\subset T_{M}=X$ by the relation (48). ∎

###  5.2. Distinguished point of $Z_{\infty}$

In this section we will assume that the base $M$ of our Joyce structure is
connected. For simplicity we also assume that the Joyce structure is
homogeneous, although this is not strictly necessary.

######  Lemma 5.2.

The map $q_{\infty}\colon X\to Z_{\infty}$ contracts the zero section
$M\subset X=T_{M}$ to a single point $0\in Z_{\infty}$ which is a fixed point
for the induced $\mathbb{C}^{*}$-action on $Z_{\infty}$.

###### Proof.

Note that the parity property (43) and the formula (6) implies that along the
zero section $M\subset X=T_{M}$ we have $h_{i}=\partial/\partial z_{i}$. The
first claim follows immediately from this. The second claim holds because the
action of $\mathbb{C}^{*}$ on $X$ preserves the zero section. ∎

The operator $J\colon T_{X}\to T_{X}$ maps vertical tangent vectors to
horizontal ones, and hence identifies the normal bundle to the zero section
$M\subset T_{M}$ with the tangent bundle $T_{M}$. The derivative of the
quotient map $q_{\infty}\colon X\to Z_{\infty}$ identifies this normal bundle
with the trivial bundle with fibre $T_{Z_{\infty},0}$. The combination of
these two maps gives an isomorphism

$T_{M,p}\stackrel{{\scriptstyle J}}{{\longrightarrow}}N_{M\subset
X,p}\stackrel{{\scriptstyle
q_{\infty,*}}}{{\longrightarrow}}T_{Z_{\infty},0},$ (75)

for each point $p\in M\subset T_{M}$, and hence a flat connection on the
tangent bundle $T_{M}$. This is the linear Joyce connection from [12, Section
7], which appeared in the original paper of Joyce [32]. In co-ordinates it is
given by the formula

$\nabla^{J}_{\frac{\partial}{\partial z_{i}}}\Big{(}\frac{\partial}{\partial
z_{j}}\Big{)}=-\sum_{l,m}\eta_{lm}\cdot\frac{\partial^{3}W}{\partial\theta_{i}\,\partial\theta_{j}\,\partial\theta_{l}}\Big{|}_{\theta=0}\cdot\frac{\partial}{\partial
z_{m}}.$ (76)

It was shown in [15, Section 3.2] that $\nabla^{J}$ coincides with the Levi-
Civita connection of the complex hyperkähler structure on $X$ restricted to
the zero-section $M\subset X=T_{M}$.

The following result follows immediately from the definitions.

######  Lemma 5.3.

The weight space decomposition for the action of $\mathbb{C}^{*}$ on
$T_{Z_{\infty},0}$ defines via the identification (75) a decomposition
$T_{M}\cong\bigoplus_{i\in\mathbb{Z}}V_{i}$ into $\nabla^{J}$-flat sub-bundles
$V_{i}\subset T_{M}$.∎

If the distinguished point $0\in Z_{\infty}$ is an isolated fixed point for
the $\mathbb{C}^{*}$ action, there are some additional consequences described
in the following result. An example when this condition holds is described in
Section 9 below.

######  Lemma 5.4.

Suppose the point $0\in Z_{\infty}$ is an isolated fixed point for the action
of $\mathbb{C}^{*}$.

  * (i)

The Hessian of the Joyce function $F$ defines a non-degenerate symmetric
bilinear form on $T_{Z_{\infty},0}$. Via the identification (75) this induces
a metric on $M$ whose Levi-Civita connection is the linear Joyce connection
$\nabla^{J}$.

  * (ii)

The positive and negative weight spaces of the $\mathbb{C}^{*}$-action on
$T_{Z_{\infty},0}$ define via the identification (75) a decomposition
$T_{M}=V_{-}\oplus V_{+}$ into $\nabla^{J}$-flat sub-bundles. These are
Lagrangian for the symplectic form $\omega$.

###### Proof.

Part (i) is immediate from the result of Lemma 5.1 that $F$ is the Hamiltonian
for the $\mathbb{C}^{*}$-action on $Z_{\infty}$. For part (ii), note that
since $\Omega_{-}$ is $\mathbb{C}^{*}$-invariant, the positive and negative
weight spaces in $T_{Z_{\infty},0}$ are Lagrangian for the form $\Omega_{-}$.
The result then follows by noting that the operator $J$ exchanges the forms
$\Omega_{-}=q_{\infty}^{*}(\omega_{\infty})$ and $\Omega_{+}=\pi^{*}(\omega)$,
so the composite (75) takes the form $\omega_{p}$ to $\Omega_{\infty,0}$. ∎

The metric $g$ of Lemma 5.4 is given in co-ordinates by the formula

$g\Big{(}\frac{\partial}{\partial z_{i}},\frac{\partial}{\partial
z_{j}}\Big{)}=\frac{\partial^{2}F}{\partial\theta_{i}\partial\theta_{j}}\Big{|}_{\theta=0}.$
(77)

This is the Joyce metric of [12, Section 7], which also appeared in the
original paper [32].

###  5.3. Another Plebański function

In Section 3.4 we described a Joyce structure in local co-ordinates using the
Plebański function $W=W(z,\theta)$. For the sake of completeness we briefly
discuss here an alternative generating function, also introduced by Plebański
[40]. In the literature (see e.g. [19]) the function $W$ is called the
Plebański function of the second kind, whereas the function $U$ introduced
below is the Plebański function of the first kind.

Consider a Joyce structure on a complex manifold $M$. We can introduce local
co-ordinates on $X=T_{M}$ by combining local Darboux co-ordinates
$(z_{1},\cdots,z_{n})$ on the twistor fibre $Z_{0}=M$ as in Section 2.2, with
the pullback of local Darboux co-ordinates $(\phi_{1},\cdots,\phi_{n})$ on the
twistor fibre fibre $Z_{\infty}$. Then by definition

$\Omega_{+}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot dz_{p}\wedge
dz_{q},\qquad\Omega_{-}=\frac{1}{2}\cdot\sum_{p,q}\omega_{pq}\cdot
d\phi_{p}\wedge d\phi_{q}.$ (78)

To find an expression for $\Omega_{I}$ in the co-ordinate system
$(z_{i},\phi_{j})$, note that since $Z_{\infty}$ is the quotient of $X$ by the
distribution spanned by the vector fields $h_{i}$ of (6), we can write

$\frac{\partial}{\partial z_{i}}\bigg{|}_{\phi}=\frac{\partial}{\partial
z_{i}}\bigg{|}_{\theta}+\sum_{p,q}\eta_{pq}\cdot\frac{\partial^{2}W}{\partial\theta_{i}\partial\theta_{p}}\cdot\frac{\partial}{\partial\theta_{q}},$
(79)

where the subscripts indicate which variables are being held fixed. Then

$\frac{\partial}{\partial
z_{r}}\bigg{|}_{\phi}\Big{(}\sum_{k}\omega_{ks}\theta_{k}\Big{)}=\frac{\partial^{2}W}{\partial\theta_{r}\partial\theta_{s}},$
(80)

and the symmetry of the right-hand side ensures that there is a locally-
defined function $U=U(z_{i},\phi_{j})$ on $X$ satisfying the equations

$\frac{\partial U}{\partial
z_{s}}=\sum_{k}\omega_{ks}\theta_{k},\qquad\frac{\partial^{2}U}{\partial
z_{r}\partial
z_{s}}=\frac{\partial^{2}W}{\partial\theta_{r}\partial\theta_{s}}.$ (81)

The formula (25) then becomes

$2i\Omega_{I}=-d\Big{(}\sum_{p,q}\omega_{pq}\theta_{p}\,dz_{q}\Big{)}=-\sum_{p,q}\frac{\partial^{2}U}{\partial\phi_{p}\partial
z_{q}}\cdot d\phi_{p}\wedge dz_{q}.$ (82)

After restricting to a fibre of $\pi\colon X\to M$ the formula (26) shows that

$\sum_{p,q}\omega_{pq}\cdot d\theta_{p}\wedge
d\theta_{q}=\sum_{p,q}\omega_{pq}\cdot d\phi_{p}\wedge d\phi_{q},$ (83)

which implies that

$\sum_{p,q}\omega_{pq}\cdot\frac{\partial\theta_{p}}{\partial\phi_{i}}\cdot\frac{\partial\theta_{q}}{\partial\phi_{j}}=\omega_{ij}.$
(84)

Using (81) we then obtain the relations

$\sum_{r,s}\eta_{rs}\cdot\frac{\partial^{2}U}{\partial\phi_{i}\partial
z_{r}}\cdot\frac{\partial^{2}U}{\partial\phi_{j}\partial z_{s}}=-\omega_{ij}$
(85)

which are known as Plebański’s first heavenly equations.

##  6\. Hamiltonian systems

In this section we show how to use a Joyce structure to define a time-
dependent Hamiltonian system. The construction depends on two additional
pieces of data: a cotangent bundle structure on $Z_{0}$, and a Lagrangian
submanifold $R\subset Z_{\infty}$. We give some motivation in the next
section, where, in the particular case of Joyce structures of class
$S[A_{1}]$, we relate our Hamiltonian systems to isomonodromy connections.

###  6.1. Hamiltonian systems

We begin with the following definition.

######  Definition 6.1.

A time-dependent Hamiltonian system consists of the following data:

  * (i)

a submersion $f\colon Y\to B$ with a relative symplectic form $\Omega\in
H^{0}(Y,\wedge^{2}\,T^{*}_{Y/B})$,

  * (ii)

a flat, symplectic connection $k$ on $f$,

  * (iii)

a section $\varpi\in H^{0}(Y,f^{*}(T_{B}^{*}))$.

For each vector field $u\in H^{0}(B,T_{B})$ there is an associated function

$H_{u}=(f^{*}(u),\varpi)\colon Y\to\mathbb{C}.$ (86)

There is then a pencil $k_{\epsilon}$ of symplectic connections on $f$ defined
by

$k_{\epsilon}(u)=k(u)+\epsilon^{-1}\cdot\Omega^{\sharp}(dH_{u}).$ (87)

The system is called strongly-integrable if these connections are all flat.

These definitions becomes more familiar when expressed in local co-ordinates.
Take co-ordinates $t_{i}$ on the base $B$, which we can think of as times, and
$k$-flat Darboux co-ordinates $(q_{i},p_{i})$ on the fibres of $f$, so that
$\Omega=\sum_{i}dq_{i}\wedge dp_{i}$. We can then write
$\varpi=\sum_{i}H_{i}\,dt_{i}$ and view the functions $H_{i}\colon
Y\to\mathbb{C}$ as time-dependent Hamiltonians. The connection $k_{\epsilon}$
is then given by the flows

$k_{\epsilon}\Big{(}\frac{\partial}{\partial
t_{i}}\Big{)}=\frac{\partial}{\partial
t_{i}}+\frac{1}{\epsilon}\cdot\sum_{j}\Big{(}\frac{\partial H_{i}}{\partial
p_{j}}\frac{\partial}{\partial q_{j}}-\frac{\partial H_{i}}{\partial
q_{j}}\frac{\partial}{\partial p_{j}}\Big{)}.$ (88)

The condition that the system is strongly-integrable is that for all
$1\leqslant i,j\leqslant n$

$\sum_{r,s}\Big{(}\frac{\partial H_{i}}{\partial q_{r}}\cdot\frac{\partial
H_{j}}{\partial p_{s}}-\frac{\partial H_{i}}{\partial
q_{s}}\cdot\frac{\partial H_{j}}{\partial p_{r}}\Big{)}=0,\qquad\frac{\partial
H_{i}}{\partial t_{j}}=\frac{\partial H_{j}}{\partial t_{i}}.$ (89)

For a nice exposition of Definition 6.1 see [5, Section 5]. Note that Boalch
works in the real $C^{\infty}$ setting, whereas we assume, as elsewhere in the
paper, that all structures are holomorphic. Note also that Boalch assumes that
$Y=M\times B$ is a global product, with $M$ a fixed symplectic manifold, and
$k$ the canonical connection on the projection $f\colon M\times B\to B$. We
can always reduce to this case by passing to an open subset of $Y$.

###  6.2. Hamiltonian systems from Joyce structures

Let $M$ be a complex manifold with a holomorphic symplectic form $\omega$. By
a cotangent bundle structure on $M$ we mean the data of a complex manifold $B$
and an open embedding $M\subset T_{B}^{*}$, such that $\omega$ is the
restriction of the canonical symplectic form on $T_{B}^{*}$. We denote by
$\rho\colon M\to B$ the induced projection map.

The Liouville 1-form on $T^{*}B$ restricts to a 1-form $\lambda\in
H^{0}(M,T^{*}_{M})$ on the open subset $M$. It will be convenient to
distinguish $\lambda$ from the tautological section $\beta\in
H^{0}(M,\rho^{*}(T_{B}^{*}))$. The two are identified via the inclusion
$\rho^{*}(T_{B}^{*})\hookrightarrow T_{M}^{*}$ induced by $\rho$.

Given local co-ordinates $(t_{1},\cdots,t_{d})$ on $B$, there are induced
linear co-ordinates $(s_{1},\cdots,s_{d})$ on the cotangent spaces
$T_{B,b}^{*}$ obtained by writing a 1-form as $\sum_{i}s_{i}\,dt_{i}$. In the
resulting co-ordinates $(s_{i},t_{j})$ on $M$ we have

$\omega=\sum_{i}dt_{i}\wedge
ds_{i},\qquad\beta=\sum_{i}s_{i}\cdot\rho^{*}(dt_{i}),\qquad\lambda=\sum_{i}s_{i}\,dt_{i}.$
(90)

Let $M$ be a complex manifold equipped with a Joyce structure. Thus there is a
pencil of flat, symplectic connections $h_{\epsilon}=h+\epsilon^{-1}v$ on the
projection $\pi\colon X=T_{M}\to M$, and closed 2-forms $\Omega_{I}$ and
$\Omega_{\pm}$ on $X$. We denote by $p\colon Z\to\mathbb{P}^{1}$ the
associated twistor space. Suppose also given:

  * (i)

a cotangent bundle structure $M\subset T^{*}_{B}$,

  * (ii)

a Lagrangian submanifold $R\subset Z_{\infty}$.

Set $Y=q_{\infty}^{-1}(R)\subset X$, and denote by $i\colon Y\hookrightarrow
X$ the inclusion. There are maps

$\textstyle{Y\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{B}$
(91)

Define $p\colon Y\to M$ and $f\colon Y\to B$ as the composites $p=\pi\circ i$
and $f=\rho\circ\pi\circ i$. We make the transversality assumption:

  * $(\star$)

For each $b\in B$ the restriction of $q_{1}\colon X\to Z_{1}$ to the fibre
$f^{-1}(b)\subset Y\subset X$ is étale.

The following result will be proved in the next section.

######  Theorem 6.2.

Given the above data there is a strongly-integrable time-dependent Hamiltonian
system on the map $f\colon Y\to B$ uniquely specified by the following
conditions:

  * (i)

the relative symplectic form $\Omega$ is induced by the closed 2-form
$i^{*}(2i\Omega_{I})$ on $Y$;

  * (ii)

for each $\epsilon\in\mathbb{C}^{*}$ the connection $k_{\epsilon}$ on $f\colon
Y\to B$ satisfies

$\operatorname{im}(k_{\epsilon})=T_{Y}\cap\operatorname{im}(h_{\epsilon})\subset
T_{X};$ (92)

  * (iii)

the Hamiltonian form is $\varpi=p^{*}(\beta)\in H^{0}(Y,f^{*}(T_{B}^{*}))$.

To make condition (iii) more explicit, take a co-ordinates system
$(t_{1},\cdots,t_{d})$ on $B$, and extend to a co-ordinate system
$(s_{i},t_{j})$ on $M$ as above. We can also extend to a co-ordinate system
$(t_{i},q_{j},p_{j})$ on $Y$ as in Section 6.1. Then writing
$H_{i}=p^{*}(s_{i})$, we have $\varpi=\sum_{i}H_{i}\cdot f^{*}(dt_{i})$, and

$k_{\epsilon}\Big{(}\frac{\partial}{\partial
t_{i}}\Big{)}=\frac{\partial}{\partial
t_{i}}+\frac{1}{\epsilon}\cdot\sum_{j}\bigg{(}\frac{\partial H_{i}}{\partial
q_{j}}\frac{\partial}{\partial p_{j}}-\frac{\partial H_{i}}{\partial
p_{j}}\frac{\partial}{\partial q_{j}}\bigg{)}.$ (93)

The main claim of Theorem 6.2 is then that the connections $k_{\epsilon}$ are
all flat, so that the conditions (89) hold for the Hamiltonians $H_{i}$.

###  6.3. Proof of Theorem 6.2

Take a point $y\in Y\subset X$ with $\pi(y)=m\in M$ and set $b=\rho(m)$. Given
a tangent vector $u\in T_{B,b}$ let us choose a lift $w\in T_{M,m}$ satisfying
$\rho_{*}(w)=u$. Using the maps $h_{\epsilon},v\colon\pi^{*}(T_{M})\to T_{X}$
as in Section 2.2 we then obtain tangent vectors $h_{\epsilon}(w),v(w)\in
T_{X,y}$. Recall that $h_{\epsilon}(w)=\epsilon^{-1}v(w)+h(w)$, and note that
$h(w)\in T_{Y,y}$, since $Y=q_{\infty}^{-1}(R)$, and $q_{\infty}$ contracts
the leaves of $h=h_{\infty}$. This implies that for any
$\epsilon\in\mathbb{C}^{*}$ the following two conditions are equivalent:

  * (a)

$v(w)\in T_{Y,y}\subset T_{X,y}$,

  * (b)

$h_{\epsilon}(w)\in T_{Y,y}\subset T_{X,y}$.

Thus when (b) holds for some $\epsilon\in\mathbb{C}^{*}$ it holds for all such
$\epsilon$.

Consider next the transversality statement

  * $(\star)_{\epsilon}$

The restriction of $q_{\epsilon}\colon X\to Z_{\epsilon}$ to the fibre
$f^{-1}(b)\subset Y\subset X$ is étale at the point $y\in Y$.

Since the kernel of the derivative of $q_{\epsilon}$ at the point $y\in X$ is
the image of $h_{\epsilon}(T_{M,m})$, the condition $(\star)_{\epsilon}$ is
equivalent to

$h_{\epsilon}^{-1}(T_{Y,y})\cap\ker(\rho_{*})=(0)\subset T_{M,m}.$ (94)

But this is also the condition for the existence of a unique lift $w\in
T_{M,m}$ satisfying condition (b). Thus since we assumed $(\star)_{\epsilon}$
for $\epsilon=1$, it holds for all $\epsilon\in\mathbb{C}^{*}$.

We can now construct connections $k_{\epsilon}$ on $\pi\colon Y\to B$ for all
$\epsilon\in\mathbb{C}^{*}$ by setting $k_{\epsilon}(u)=h_{\epsilon}(w)$,
where $w$ is the unique lift satisfying condition (b). In more geometric
terms, the condition $(\star)_{\epsilon}$ ensures that the map
$(f,q_{\epsilon})\colon Y\to B\times Z_{\epsilon}$ is étale, and the
connection $k_{\epsilon}$ is then pulled back from the trivial connection on
the projection $B\times Z_{\epsilon}\to B$. This second description shows in
particular that the resulting connection $k_{\epsilon}$ is flat.

The closed 2-form $q_{\epsilon}^{*}(\Omega_{\epsilon})$ defines a relative
symplectic form on $f$ by the condition $(\star_{\epsilon})$. Recall the
identity of closed 2-forms

$q_{\epsilon}^{*}(\Omega_{\epsilon})=\epsilon^{-2}q_{0}^{*}(\Omega_{0})+2i\epsilon^{-1}\Omega_{I}+q_{\infty}^{*}(\Omega_{\infty})$
(95)

from Section 4. On restricting to $Y$ the last term on the right-hand side
vanishes, since $R\subset Z_{\infty}$ is Lagrangian. On further restricting to
a fibre $f^{-1}(b)\subset Y$ the first term also vanishes, since
$\rho^{-1}(b)\subset M$ is Lagrangian. Thus for $\epsilon\in\mathbb{C}^{*}$
the forms $\epsilon\cdot q_{\epsilon}^{*}(\Omega_{\epsilon})$ define the same
relative symplectic form $\Omega$ on $f$, and this is also induced by
$2i\Omega_{I}$. Note that the the kernel of the restriction
$\Omega_{\epsilon}|_{Y}$ clearly contains the subspace
$\operatorname{im}(k_{\epsilon})$, and hence coincides with it. This implies
that the connection $k_{\epsilon}$ on $f$ is symplectic [27, Theorem 4].

Observe next that there is a commutative diagram

$\textstyle{T_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-(2i\Omega_{I})^{\flat}}$$\textstyle{T_{X}^{*}}$$\textstyle{T_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\flat}}$$\scriptstyle{v}$$\textstyle{T_{M}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$
(96)

since the relations (13) show that for tangent vectors $w_{1}$ to $M$ and
$w_{2}$ to $X$

$\begin{split}-2i\Omega_{I}(v(w_{1}),w_{2})&=-g(2iI(v(w_{1})),w_{2})=-2g(v(w_{1}),w_{2})=g((J+iK)(h(w_{1})),w_{2})\\\
&=\Omega_{+}(h(w_{1}),w_{2})=(\pi^{*}\omega)(h(w_{1}),w_{2})=\omega(w_{1},\pi_{*}(w_{2})).\end{split}$
(97)

The fact that the closed 2-form $i^{*}(2i\Omega_{I})$ induces a relative
symplectic form on the map $f\colon Y\to B$ is the statement that the
composite of the bundle maps

$\textstyle{\ker(f_{*})\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2i\Omega_{I})^{\flat}}$$\textstyle{T_{Y}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{Y}^{*}/f^{*}(T_{B}^{*})}$
(98)

is an isomorphism. The relative Hamiltonian flow $r$ corresponding to a
function $H\colon Y\to\mathbb{C}$ is then the unique vertical vector field on
the map $f\colon Y\to B$ which is mapped to $dH$ viewed as a section of
$T_{Y}^{*}/f^{*}(T_{B}^{*})$. In symbols we can write
$(2i\Omega_{I})^{\flat}(r)=dH+f^{*}(\alpha)$ for some covector field $\alpha$
on $B$.

Consider the canonical section $\varpi=p^{*}(\beta)\in
H^{0}(Y,f^{*}(T^{*}_{B}))$. Note that given a vector field $u\in
H^{0}(B,T_{B})$ the corresponding Hamiltonian
$H_{u}=(f^{*}(u),\varpi)=p^{*}(\rho^{*}(u),\beta)$ on $Y$ is pulled back from
$M$. Let us lift $u$ to a vector field $w$ on $M$ as above. Then
$k_{\epsilon}(u)=h_{\epsilon}(w)=h(w)+\epsilon^{-1}\cdot v(w)$. We claim that
$v(w)$ is the relative Hamiltonian flow $r$ defined by the function $H_{u}$.
To prove this we must show that the 1-form
$(2i\Omega_{I})^{\flat}(v(w))-dH_{u}$ on $Y$ is a pullback from $B$. By the
commutative diagram (96) this is equivalent to showing that the 1-form
$\omega^{\flat}(w)-d(\rho^{*}(u),\beta)$ on $M$ is a pullback from $B$.

This final step is perhaps most easily done in local co-ordinates
$(s_{i},t_{i})$ on $M$ as above in which $\beta=\sum_{i}s_{i}dt_{i}$ and
$\omega=\sum_{i}dt_{i}\wedge ds_{i}$. If we take $u=\frac{\partial}{\partial
t_{i}}$ then $(\rho^{*}(u),\beta)=s_{i}$ and the lift $w$ has the form
$w=\frac{\partial}{\partial t_{i}}+\sum_{j}a_{j}\frac{\partial}{\partial
s_{j}}$ for locally-defined functions $a_{j}\colon M\to\mathbb{C}$. But then
$\omega^{\flat}(w)=ds_{i}-\sum_{j}a_{j}\,dt_{j}$, and since the $t_{i}$ are
pulled back via $\rho$ this proves the claim. We have now defined the
symplectic connections $k_{\epsilon}$ for $\epsilon\in\mathbb{C}^{*}$ and
proved the relation (88). We can then define a symplectic connection
$k=k_{\infty}$ by the same relation. Since the $k_{\epsilon}$ are flat for all
$\epsilon\in\mathbb{C}^{*}$ the relations (89) hold, and it follows that $k$
is also flat.

##  7\. Joyce structures of class $S[A_{1}]$

In this section we discuss an interesting class of examples of Joyce
structures. They are related to supersymmetric gauge theories of class
$S[A_{1}]$, and were first constructed in the paper [16]. The base $M$
parameterises pairs $(C,Q)$ consisting of an algebraic curve $C$ of some fixed
genus $g\geq 2$, and a quadratic differential $Q\in
H^{0}(C,\omega_{C}^{\otimes 2})$ with simple zeroes. The generalisation to the
case of meromorphic quadratic differential with poles of fixed orders will be
treated in the forthcoming work [41]. After reviewing the construction of
these Joyce structures, we relate the twistor fibre $Z_{1}$ to the associated
character variety, and the Hamiltonian systems of Section 6 to isomonodromy
connections.

###  7.1. Construction

For each point $(C,Q)\in M$ there is a branched double cover $p\colon\Sigma\to
C$ defined via the equation $y^{2}=Q(x)$, and equipped with a covering
involution $\sigma\colon\Sigma\to\Sigma$. Taking periods of the form $y\,dx$
on $\Sigma$ identifies the tangent space $T_{(C,Q)}M$ with the anti-invariant
cohomology group $H^{1}(\Sigma,\mathbb{C})^{-}$. The intersection pairing on
$H_{1}(\Sigma,\mathbb{C})^{-}$ then induces a symplectic form on $M$, and the
dual of the integral homology groups $H_{1}(\Sigma,\mathbb{Z})^{-}$ defines an
integral affine structure $T_{M}^{\mathbb{Z}}\subset T_{M}$. There is a
natural $\mathbb{C}^{*}$-action on $M$ which rescales the quadratic
differential $Q$ with weight 2. Taken together this defines a period structure
on $M$.

The usual spectral correspondence associates to a $\sigma$-anti-invariant line
bundle $L$ on $\Sigma$, a rank 2 vector bundle
$E=p_{*}(L)\otimes\sqrt{\omega_{C}}\,$ on $C$ with a Higgs field $\Phi$. A key
ingredient in [16] is an extension of this correspondence which relates anti-
invariant connections $\partial$ on $L$ to connections $\nabla$ on $E$. Given
this, we can view the space $X^{\\#}$ appearing in (30) as parameterising the
data $(C,E,\nabla,\Phi)$. The pencil of non-linear connections $h_{\epsilon}$
is then obtained by requiring that the monodromy of the connection
$\nabla-\epsilon^{-1}\Phi$ is constant as the pair $(C,Q)$ varies.

To explain the construction in a little more detail, let us fix a parameter
$\epsilon\in\mathbb{C}^{*}$ and contemplate the following diagram of moduli
spaces:

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(99)

Each moduli space parameterises the indicated objects, and the maps
$\rho,\rho^{\prime}$ and $\pi_{i}$ are the obvious projections. The map
$\alpha$ is the above-mentioned extension of the spectral correspondence, and
the map $\beta_{\epsilon}$ is defined by the rule

$\beta_{\epsilon}(C,E,\nabla,\Phi)=(C,-\det(\Phi),E,\nabla-\epsilon^{-1}\Phi).$
(100)

An important point is that $\alpha$ is birational, and $\beta_{\epsilon}$ is
generically étale.

Given a point of $M=\mathcal{M}(C,Q)$, an anti-invariant line bundle with
connection $(L,\partial)$ on the spectral curve $\Sigma$ has an associated
holonomy representation $H_{1}(\Sigma,\mathbb{Z})^{-}\to\mathbb{C}^{*}$. This
determines $(L,\partial)$ up to an action of the group of 2-torsion line
bundles on $C$. We therefore obtain an étale map from
$\mathcal{M}(C,Q,L,\partial)$ to the space $X^{\\#}$. The isomonodromy
connection on the map $\pi_{1}$ is a flat symplectic connection whose leaves
consist of connections $(E,\nabla_{\epsilon})$ with fixed monodromy. Pulling
this connection through (99) gives a family of non-linear symplectic
connections $h_{\epsilon}$ on the projection $\pi\colon X^{\\#}\to M$. This
gives rise to a meromorphic Joyce structure on $M$, with the poles arising
because $\alpha$ is only birational rather than an isomorphism, and
$\beta_{\epsilon}$ is only generically étale.

###  7.2. Twistor space

Consider the twistor space $p\colon Z^{\\#}\to\mathbb{P}^{1}$ associated to
the Joyce structure of Section 7.1. It could be an interesting problem to try
to give a direct moduli-theoretic construction of this space. For now we can
at least describe the fibre $Z_{1}$ up to an étale cover. Note that since
these Joyce structures are meromorphic, the connections $h_{\epsilon}$ are
only well-defined on the projection $\pi\colon X^{0}\to M$ of an open subset
$X^{0}\subset X=T_{M}$. We define the twistor fibres as the spaces of leaves
of the reuslting foliations of $X^{0}$.

Choose a reference surface $S_{g}$ of genus $g$, set
$G=\operatorname{PGL}_{2}(\mathbb{C})$, and define

$\operatorname{MCG}(g)=\pi_{0}(\operatorname{Diff}^{+}(S_{g})),\qquad\mathcal{X}(g)=\operatorname{Hom}_{\rm
grp}(\pi_{1}(S_{g}),G)/G.$ (101)

Then the mapping class group $\operatorname{MCG}(g)$ acts on the character
stack $\mathcal{X}(g)$ in the usual way, and sending a quadruple
$(C,E,\nabla)$ to the monodromy of the connection $\nabla$ defines a map

$\mu\colon\mathcal{M}(C,E,\nabla)/J^{2}(C)\to\mathcal{X}(g)/\operatorname{MCG}(g),$
(102)

whose fibres are by definition the horizontal leaves of the isomonodromy
connection. Transferring this map across the diagrams (99), and passing to the
leaf space yields an étale map

$\mu\colon Z_{\epsilon}^{\\#}\to\mathcal{X}(g)/\operatorname{MCG}(g).$ (103)

It would be interesting to test the speculations of Section 4.3 in this
context. It has been suggested that the appropriate Darboux co-ordinates to
consider on the character variety $\mathcal{X}(g)$ are the Bonahon-Thurston
shear co-ordinates [7, 22]. Let is instead consider spaces of quadratic
differentials with poles of fixed orders. The corresponding Joyce structures
will appear in [41]. It is natural to expect that there is a similar map to
(103) in which $\mathcal{X}(g)$ is replaced with a space of framed local
systems [23]. It is then expected that the preferred Darboux co-ordinates
associated to a general half-plane $H(\varphi)$ are the Fock-Goncharov co-
ordinates for the WKB triangulation determined by the horizontal trajectories
of the quadratic differential $e^{-2\pi i\varphi}\cdot Q$. It follows from the
work of Gaiotto, Moore and Neitzke [26], and the results of [14], that as
$\varphi$ varies these satisfy the jumps (76) determined by the DT theory of
the corresponding category. The asymptotic property (69) should follow from
existing results in exact WKB analysis. The whole story has been treated in
full detail [13] in the special case discussed in Section 9 below.

###  7.3. Hamiltonian systems

Let us consider the Hamiltonian system of Theorem 6.2 in the case of Joyce
structures of Section 7.1. Recall that a crucial feature of the construction
of these Joyce structures is the isomondromy connection on the map

$\pi_{1}\colon\mathcal{M}(C,E,\nabla_{1})\to\mathcal{M}(C),$ (104)

which is both flat and symplectic. To construct a Hamiltonian system we need a
whole one-parameter family of such connections. In the notation of Section 6,
the isomonodromy connection is $k_{1}$, but we also need the connection
$k_{\infty}$ before we can write (87). This issue is often a little hidden in
the literature because in many examples there is a natural choice for the
reference connection $k_{\infty}$ which is then taken without further comment.
It is discussed explicitly in [29], and is also mentioned for example in [6,
Remark 7.1].

One way to try to define a Hamiltonian system on the map (104) is to choose
for each bundle $E$ a distinguished ‘reference’ connection $\nabla_{\infty}$.
Then we can define a Higgs field $\Phi=\nabla_{\infty}-\nabla_{1}$ and for
$\epsilon\in\mathbb{C}^{*}$ a connection
$\nabla_{\epsilon}=\nabla_{\infty}-\epsilon^{-1}\Phi$. We can then define
$k_{\epsilon}$ to be the isomonodromy connection for the family of connections
$(E,\nabla_{\epsilon})$. Many interesting examples of isomonodromic systems in
the literature involve bundles with meromorphic connections on a genus 0
curve. Since the generic such bundle $E$ is trivial, it is then natural to
take $\nabla_{\infty}=d$. But for bundles with connection on higher genus
curves there is no such canonical choice.

Consider the meromorphic Joyce structures of Section 7.1. Note that the base
$M=\mathcal{M}(C,Q)$ has a natural cotangent bundle structure, with
$B=\mathcal{M}(C)$ being the moduli space of curves of genus $g$, and
$\rho\colon M\to B$ the obvious projection
$\rho\colon\mathcal{M}(C,Q)\to\mathcal{M}(C)$. Indeed, the tangent spaces to
$\mathcal{M}(C)$ are the vector spaces
$\mathcal{T}_{C}\mathcal{M}(C)=H^{1}(C,T_{C})$, and Serre duality gives
$H^{0}(C,\omega_{C}^{\otimes 2})=H^{1}(C,T_{C})^{*}$. Thus $T_{B}^{*}$
parameterises pairs $(C,Q)$ of a curve $C$ together with a quadratic
differential $Q\in H^{0}(C,\omega_{C}^{\otimes 2})$, and $M\subset T_{B}^{*}$
is the open subset where $Q$ has simple zeroes.

Let us choose a Lagrangian $R\subset\mathcal{X}(g)/\operatorname{MCG}(g)$.
Pulling back by the étale map gives a Lagrangian in $Z_{\infty}$ which we also
denote by $R$. The subspace of the quotient
$\mathcal{X}(g)/\operatorname{MCG}(g)$ consisting of monodromy representations
of connections on a fixed bundle $E$ is known to be Lagrangian. Then for a
generic bundle $E$ we can expect these two Lagrangians to meet in a finite set
of points, and so locally on the moduli of bundles we can define
$\nabla_{\infty}$ by insisting that its monodromy lies in $R$.

Using the natural cotangent bundle structure $\rho\colon M\to B$ and the
Lagrangian $R\subset Z_{\infty}$ we can now apply the construction of Section
6.2 to these examples. We obtain a diagram

|
---|---
$\textstyle{Y=q_{\infty}^{-1}(R)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{\mathcal{M}(C,E,\nabla,\Phi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{\mathcal{M}(C,E,\nabla_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\mathcal{M}(C)}$
(105)

where the map $\beta_{1}$ is defined by setting $\nabla_{1}=\nabla-\Phi$.

The isomonodromy connection defines a flat connection $h_{1}$ on the map
$\pi_{1}$. The Hamiltonian system of Theorem 6.2 defines a whole pencil of
flat connections $k_{\epsilon}$ on the map $f$. The transversality assumption
ensures that the map $\beta_{1}\circ i$ is étale, and the pullback of the
connection $h_{1}$ then coincides with $k_{1}$. Thus by choosing the
Lagrangian $R\subset Z_{\infty}$ and using it to define reference connections,
we have upgraded the isomonodromy connection to a Hamiltonian system.

##  8\. The doubled A1 example

In this section we discuss a simple and rather degenerate example of a pre-
Joyce structure which does not quite satisfy the conditions to be a Joyce
structure. It is related to the DT theory of the A1 quiver.

###  8.1. Pre-Joyce structure

We take $M=\mathbb{C}^{*}\times\mathbb{C}$ with co-ordinates $(z,z^{\vee})$,
and consider the total space of the tangent bundle $X=T_{M}$ with
corresponding co-ordinates $(z,z^{\vee},\theta,\theta^{\vee})$ as in Section
2.2. We define a pre-Joyce structure on $M$ by taking the symplectic form

$\omega=\frac{1}{2\pi i}\cdot dz\wedge dz^{\vee},$ (106)

and the pencil of connections $h+\epsilon^{-1}v$ defined by

$h_{\epsilon}\bigg{(}\frac{\partial}{\partial
z}\bigg{)}=\frac{\partial}{\partial
z}+\frac{1}{\epsilon}\cdot\frac{\partial}{\partial\theta}+\frac{\theta}{2\pi
iz}\cdot\frac{\partial}{\partial\theta^{\vee}},\qquad
h_{\epsilon}\bigg{(}\frac{\partial}{\partial
z^{\vee}}\bigg{)}=\frac{\partial}{\partial
z^{\vee}}+\frac{1}{\epsilon}\cdot\frac{\partial}{\partial\theta^{\vee}}.$
(107)

The corresponding Plebański function is

$W(z,z^{\vee},\theta,\theta^{\vee})=-\frac{\theta^{3}}{6(2\pi i)^{2}z}.$ (108)

The formulae (24) - (26) become

$\displaystyle\Omega_{+}=\frac{1}{2\pi i}\cdot dz\wedge dz^{\vee},\qquad
2i\Omega_{I}=\frac{1}{2\pi i}\big{(}d\theta\wedge
dz^{\vee}-d\theta^{\vee}\wedge dz\big{)},$ (109)
$\displaystyle\Omega_{-}=\frac{1}{2\pi i}\cdot d\theta\wedge
d\theta^{\vee}-\frac{\theta}{(2\pi i)^{2}z}\cdot d\theta\wedge dz.$ (110)

The above pre-Joyce structure was extracted from the DT theory of the A1
quiver in [12, Section 8] by first applying a doubling procedure and then
solving the resulting Riemann-Hilbert problem. The doubling procedure is
required because the Euler form of the category is degenerate, and in fact
identically zero.

###  8.2. Period structure

We can define a period structure on $M$ by declaring the co-ordinates
$(z,z^{\vee})$ to be integral linear. The resulting lattice
$T_{M}^{\mathbb{Z}}\subset T_{M}$ is spanned by integral linear combinations
of the vector fields $\frac{\partial}{\partial z}$ and
$\frac{\partial}{\partial z^{\vee}}$. This is a homogeneous period structure,
since the vector field

$Z=z\cdot\frac{\partial}{\partial z}+z^{\vee}\cdot\frac{\partial}{\partial
z^{\vee}}$ (111)

generates the $\mathbb{C}^{*}$-action $t\cdot(z,z^{\vee})=(tz,tz^{\vee})$. The
inverse $\eta\colon T_{M}^{*}\times T_{M}^{*}\to\mathscr{O}_{M}$ of the
symplectic form $\omega$ satisfies $\eta(dz^{\vee},dz)=2\pi i$.

Combining the above pre-Joyce structure and period structure does not quite
yield a Joyce structure. The problem is with axiom (J2): the connection (107)
is not invariant under translations by the lattice $(2\pi
i)\,T_{M}^{\mathbb{Z}}$ since it depends on $\theta$ rather than its
exponential. This problem seems to be related to the doubling procedure
referred to above.

###  8.3. Alternative co-ordinates

There is another integral affine structure on $M$ with integral affine co-
ordinates

$v=z,\qquad v^{\vee}=z^{\vee}-\frac{z}{2\pi i}\bigg{(}\log\Big{(}\frac{z}{2\pi
i}\Big{)}-1\bigg{)}.$ (112)

The associated co-ordinates $(v,v^{\vee},\phi,\phi^{\vee})$ on $X=T_{M}$ are

$\phi=\theta,\qquad\phi^{\vee}=\theta^{\vee}-\frac{\theta}{2\pi
i}\cdot\log\Big{(}\frac{z}{2\pi i}\Big{)}.$ (113)

In this co-ordinate system the connection $h_{\epsilon}$ takes the form

$h_{\epsilon}\bigg{(}\frac{\partial}{\partial
v}\bigg{)}=\frac{\partial}{\partial
v}+\frac{1}{\epsilon}\cdot\frac{\partial}{\partial\phi},\qquad
h_{\epsilon}\bigg{(}\frac{\partial}{\partial
v^{\vee}}\bigg{)}=\frac{\partial}{\partial
v^{\vee}}+\frac{1}{\epsilon}\cdot\frac{\partial}{\partial\phi^{\vee}},$ (114)

and hence the corresponding Plebański function
$W(v,v^{\vee},\phi,\phi^{\vee})=0$.

Note however that since

$Z=v\cdot\frac{\partial}{\partial v}+\Big{(}v^{\vee}-\frac{v}{2\pi
i}\Big{)}\cdot\frac{\partial}{\partial v^{\vee}},$ (115)

we cannot combine this integral affine structure with the vector field $Z$ to
form a period structure.

###  8.4. First Plebański function

It follows from (114) that the fibre co-ordinates $(\phi,\phi^{\vee})$ are
preserved by the connection $h=h_{\infty}$. By (26) they satisfy

$\Omega_{-}=\frac{1}{2\pi i}\cdot d\phi\wedge d\phi^{\vee}.$ (116)

They therefore descend to Darboux co-ordinates on the twistor fibre
$Z_{\infty}$. The corresponding first Plebański function defined as in Section
5.3 is

$U(z,z^{\vee},\phi,\phi^{\vee})=\frac{1}{2\pi
i}\bigg{(}z^{\vee}\phi-\phi^{\vee}z+\frac{\phi z}{2\pi i}-\frac{\phi z}{2\pi
i}\cdot\log\Big{(}\frac{z}{2\pi i}\Big{)}\bigg{)}.$ (117)

It satisfies

$\frac{\partial U}{\partial z^{\vee}}=\frac{\theta}{2\pi
i},\qquad\frac{\partial U}{\partial z}=\frac{-\theta^{\vee}}{2\pi i}.$ (118)

The Jacobian matrix has entries

$\frac{\partial^{2}U}{\partial z\partial\phi}=-\frac{1}{2\pi
i}\log\Big{(}\frac{z}{2\pi i}\Big{)},\qquad\frac{\partial^{2}U}{\partial
z^{\vee}\partial\phi}=\frac{1}{2\pi i}=-\frac{\partial^{2}U}{\partial
z\partial\phi^{\vee}},\qquad\frac{\partial^{2}U}{\partial
z^{\vee}\partial\phi^{\vee}}=0,$ (119)

and has determinant $-1/(2\pi i)^{2}$ as required by the first Plebański
equation (85).

######  Remark 8.1.

In this case the Jacobian matrix (119) is skew-symmetric. This means we can
find $\mathcal{F}=\mathcal{F}(z,z^{\vee})$ such that

$\frac{\partial\mathcal{F}}{\partial z}=\frac{\partial
U}{\partial\phi},\qquad\frac{\partial\mathcal{F}}{\partial
z^{\vee}}=-\frac{\partial U}{\partial\phi^{\vee}}.$ (120)

Then $\mathcal{F}$ is closely related to the prepotential of [12, Section 7].
Explicitly we have

$\mathcal{F}(z,z^{\vee})=\frac{zz^{\vee}}{2\pi i}-\frac{1}{(2\pi
i)^{2}}\bigg{(}\frac{1}{2}z^{2}\log\Big{(}\frac{z}{2\pi
i}\Big{)}-\frac{3}{4}z^{2}\bigg{)}.$ (121)

##  9\.  The A2 example

Here we consider a very interesting Joyce structure which is perhaps the
simplest non-trivial example. It is related to the DT theory of the A2 quiver,
and is an example of a Joyce structure of class $S[A_{1}]$. For more details
on this example we refer the reader to [13].

###  9.1. Period structure

The base of the Joyce structure is

$M=\\{(a,b)\in\mathbb{C}^{2}:4a^{3}+27b^{2}\neq 0\\}.$ (122)

Associated to a pair $(a,b)\in M$ is a quadratic differential

$Q_{0}(x)\,dx^{\otimes 2}=(x^{3}+ax+b)\,dx^{\otimes 2}$ (123)

on $\mathbb{P}^{1}$ which has a single pole of order 7 at $x=\infty$ and
simple zeroes. There is an associated double cover $\Sigma\to\mathbb{P}^{1}$
which is the projectivization of the affine elliptic curve $y^{2}=x^{3}+ax+b$.

Take a basis of cycles $(\gamma_{1},\gamma_{2})\subset
H_{1}(\Sigma,\mathbb{Z})$ with intersection $\gamma_{1}\cdot\gamma_{2}=1$. We
shall need the periods and quasi-periods of the elliptic curve $\Sigma$. They
are given by

$\omega_{i}=\int_{\gamma_{i}}\frac{dx}{2y},\qquad\eta_{i}=-\int_{\gamma_{i}}\frac{x\,dx}{2y},$
(124)

and satisfy the Legendre relation $\omega_{2}\eta_{1}-\omega_{1}\eta_{2}=2\pi
i$.

There are local co-ordinates on $M$ given by

$z_{i}=\int_{\gamma_{i}}y\,dx=\int_{\gamma_{i}}\sqrt{x^{3}+ax+b\,}\,dx,$ (125)

and relations

$\frac{\partial}{\partial a}=-\eta_{1}\frac{\partial}{\partial
z_{1}}-\eta_{2}\frac{\partial}{\partial z_{2}},\qquad\frac{\partial}{\partial
b}=\omega_{1}\frac{\partial}{\partial
z_{1}}+\omega_{2}\frac{\partial}{\partial z_{2}}.$ (126)

We define a period structure on $M$ by declaring $(z_{1},z_{2})$ to be the
integral linear co-ordinates. We take the symplectic form

$\omega=-\frac{1}{2\pi i}dz_{1}\wedge dz_{2}=da\wedge db.$ (127)

Note that the inverse satisfies

$\eta\big{(}dz_{1},dz_{2}\big{)}=2\pi i$ (128)

and so axiom (J1) of Definition 3.4 is satisfied.

The Euler vector field is

$Z=z_{1}\frac{\partial}{\partial z_{1}}+z_{2}\frac{\partial}{\partial
z_{2}}=\frac{4a}{5}\frac{\partial}{\partial
a}+\frac{6b}{5}\frac{\partial}{\partial b}.$ (129)

Note that $Z$ does not generate a $\mathbb{C}^{*}$ action on $M$. In fact the
moduli space of quadratic differentials of the form (123) is the quotient
$M/\mu_{5}$, where the group $\mu_{5}\subset\mathbb{C}^{*}$ acts via
$\zeta\cdot(a,b)=(\zeta^{4}a,\zeta^{6}b)$. The vector field $Z$ generates an
action of $\mathbb{C}^{*}$ on this quotient.

###  9.2. Joyce structure

We have local co-ordinates $(z_{i},\theta_{j})$ on the tangent bundle
$X=T_{M}$. We consider the quotient $X^{\\#}=T_{M}/(2\pi
i)\,T_{M}^{\mathbb{Z}}$. The fibre over a point $(a,b)\in M$ is the cohomology
group $H^{1}(\Sigma,\mathbb{C}^{*})\cong(\mathbb{C}^{*})^{2}$. We introduce
some alternative co-ordinates on $X^{\\#}$ by the relation

$\theta_{i}=-\int_{\gamma_{i}}\bigg{(}\frac{p}{x-q}+r\bigg{)}\frac{dx}{2y},$
(130)

where $p^{2}=q^{3}+aq+b$ and $r\in\mathbb{C}$. The integral is well-defined up
to multiples of $2\pi i$. The numbers $\xi_{i}=\exp(\theta_{i})$ are the
holonomies of a connection on the line bundle on $\Sigma$ with divisor
$(q,p)-\infty$.

In terms of the fibre co-ordinates $(\theta_{a},\theta_{b})$ associated to the
co-ordinates $(a,b)$ we then have

$\theta_{i}=-\eta_{i}\theta_{a}+\omega_{i}\theta_{b}.$ (131)

The functions $(\theta_{a},\theta_{b})$ are given explicitly by

$\theta_{a}=-\frac{1}{4}\int^{(q,p)}_{(q,-p)}\frac{dx}{y},\qquad\theta_{b}=\frac{1}{4}\int^{(q,p)}_{(q,-p)}\frac{xdx}{y}-r.$
(132)

The Joyce structure on $X=T_{M}$ is obtained by taking the connection
$h_{\epsilon}$ given by

$\displaystyle h_{\epsilon}\bigg{(}\frac{\partial}{\partial a}\bigg{)}$
$\displaystyle=-\frac{2p}{\epsilon}\frac{\partial}{\partial
q}-\frac{q}{\epsilon}\frac{\partial}{\partial
r}+\bigg{(}\frac{\partial}{\partial a}-\frac{r}{p}\frac{\partial}{\partial
q}-\frac{r^{2}(3q^{2}+a)-qpr}{2p^{3}}\frac{\partial}{\partial r}\bigg{)},$
(133) $\displaystyle h_{\epsilon}\bigg{(}\frac{\partial}{\partial b}\bigg{)}$
$\displaystyle=-\frac{1}{\epsilon}\frac{\partial}{\partial
r}+\bigg{(}\frac{\partial}{\partial
b}+\frac{r}{2p^{2}}\frac{\partial}{\partial r}\bigg{)}.$ (134)

This is the isomonodromy connection for a pencil of connections
$\nabla-\epsilon^{-1}\Phi$ on the trivial rank 2 bundle on $\mathbb{P}^{1}$
with

$\nabla=d-\begin{pmatrix}r&0\\\
0&-r\end{pmatrix}\frac{dx}{2p},\qquad\Phi=\begin{pmatrix}p&x^{2}+xq+q^{2}+a\\\
x-q&-p\end{pmatrix}dx.$ (135)

Equivalently we can consider the deformed cubic oscillator

$y^{\prime\prime}(x)=Q(x)y(x),\qquad
Q(x)=\epsilon^{-2}Q_{0}(x)+\epsilon^{-1}Q_{1}(x)+Q_{2}(x),$ (136)

where the terms in the potential are

$Q_{1}(x)=\frac{p}{x-q}+r,\qquad
Q_{2}(x)=\frac{3}{4(x-q)^{2}}+\frac{r}{2p(x-q)}+\frac{r^{2}}{4p^{2}}.$ (137)

There is a rational expression for the Plebański function

$W=\frac{1}{4(4a^{3}+27b^{2})p}\Big{(}2apr^{3}-(6aq^{2}-9bq+4a^{2})r^{2}-3p(3b-2aq)r-2ap^{2}\Big{)},$
(138)

although note that this does not satisfy the second of the normalisation
conditions (48). In other words, the uniquely-defined Plebański function of
Lemma 3.4 differs from the above expression by a function of the form
$\sum_{i}a_{i}(z)\cdot\theta_{i}$.

###  9.3. Further properties

In the co-ordinates $(a,b,q,r)$ the Euler vector field $E$ is

$E=\frac{4a}{5}\frac{\partial}{\partial
a}+\frac{6b}{5}\frac{\partial}{\partial
b}+\frac{2q}{5}\frac{\partial}{\partial q}+\frac{r}{5}\frac{\partial}{\partial
r}.$ (139)

The expressions (126) and (131) and a short calculation using (132) gives

$2i\Omega_{I}=-da\wedge d\theta_{b}+db\wedge d\theta_{a}=dq\wedge dp+da\wedge
dr.$ (140)

The twistor fibre $Z_{\infty}$ is the space of leaves of the foliation defined
by the connection $h=h_{\infty}$. The functions

$\phi_{1}=q+\frac{ar}{p},\qquad\phi_{2}=\frac{r}{2p},$ (141)

descend to $Z_{\infty}$ because they are constant along the flows (133) and
(134) with $\epsilon=\infty$. Moreover, if we restrict to a fibre $F$ of the
projection $\pi\colon X=T_{M}\to M$ by fixing $(a,b)$, then

$-\frac{1}{2\pi i}\cdot d\theta_{1}\wedge
d\theta_{2}\big{|}_{F}=d\theta_{a}\wedge
d\theta_{b}\big{|}_{F}=-\frac{dr}{2p}\wedge dq\big{|}_{F}=d\phi_{1}\wedge
d\phi_{2}\big{|}_{F},$ (142)

Thus we have

$\Omega_{\infty}=d\phi_{1}\wedge d\phi_{2}.$ (143)

The first Plebański function $U\colon X\to\mathbb{C}$ of Appendix 5.3 is given
by

$U=\frac{1}{2}\int_{(q,-p)}^{(q,p)}(x^{3}+ax+b)^{1/2}\,dx,$ (144)

because a simple calculation using the flows (133) and (134) shows that

$\frac{\partial U}{\partial b}\bigg{|}_{\phi}=h\Big{(}\frac{\partial}{\partial
b}\Big{)}U=-\theta_{a},\qquad\frac{\partial U}{\partial
a}\bigg{|}_{\phi}=h\Big{(}\frac{\partial}{\partial a}\Big{)}U=\theta_{b}.$
(145)

Since the $\mathbb{C}^{*}$-action rescales $\phi_{1}$ and $\phi_{2}$ with
weights $-\tfrac{2}{5}$ and $\tfrac{2}{5}$ respectively, we have

$i_{E}(\Omega_{\infty})=i_{E}(d\phi_{1}\wedge
d\phi_{2})=\frac{2}{5}d(\phi_{1}\phi_{2}).$ (146)

The Joyce function $F$ of Section 5.1 is given by

$F=\frac{1}{5}(1-2\phi_{1}\phi_{2}),$ (147)

where for the constant normalisation we used the limiting behaviour of $q,p,r$
along the zero section $M\subset T_{M}$ as discussed in [13, Section 4.3].

The distinguished fixed point of $Z_{\infty}$ is defined by
$(\phi_{1},\phi_{2})=(0,\infty)$. It is an isolated fixed point. The linear
Joyce connection on $M$ is the one whose flat co-ordinates are $(a,b)$, and
the negative and positive weight spaces $V_{-}$ and $V_{+}$ of Lemma 5.4 are
spanned by $\frac{\partial}{\partial a}$ and $\frac{\partial}{\partial b}$
respectively. The Joyce metric is

$g=\frac{1}{5}\cdot(da\otimes db+db\otimes da).$ (148)

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