affil0affil0affiliationtext: Fachbereich Mathematik
Universität Hamburg
Bundesstraße 55, 20146 Hamburg, Germany
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# Symmetries of one-loop deformed q-map spaces

Vicente Cortés  Alejandro Gil-García  and Danu Thung

###### Abstract

Q-map spaces form an important class of quaternionic Kähler manifolds of
negative scalar curvature. Their one-loop deformations are always
inhomogeneous and have been used to construct cohomogeneity one quaternionic
Kähler manifolds as deformations of homogeneous spaces. Here we study the
group of isometries in the deformed case. Our main result is the statement
that it always contains a semidirect product of a group of affine
transformations of $\mathbb{R}^{n-1}$ with a Heisenberg group of dimension
$2n+1$ for a q-map space of dimension $4n$. The affine group and its action on
the normal Heisenberg factor in the semidirect product depend on the cubic
affine hypersurface which encodes the q-map space.

_Keywords: quaternionic Kähler manifolds, special geometry, isometry groups_

_MSC classification: 53C26_

###### Contents

  1. 1 Introduction
  2. 2 Symmetries of rigid c-map spaces
    1. 2.1 Special Kähler manifolds
    2. 2.2 The rigid c-map
    3. 2.3 Canonical lifts
    4. 2.4 Translations in the fibers
    5. 2.5 The semidirect product
  3. 3 Symmetries under the HK/QK correspondence
    1. 3.1 The HK/QK correspondence as a twist
    2. 3.2 Infinitesimal description
    3. 3.3 Global description

##  1 Introduction

Quaternionic Kähler manifolds are Riemannian $4n$-manifolds $(n>1)$ whose
holonomy group is contained in $\mathrm{Sp}(n)\mathrm{Sp}(1)$, hence they are
Einstein. In this paper we are interested in a class of quaternionic Kähler
manifolds of negative scalar curvature arising from physics, namely
_supergravity c-map spaces_. The construction of these spaces have its origin
in supergravity. It was shown in [FS90] (see also [Hit09]) that to any
projective special Kähler manifold $\bar{M}$ of (real) dimension $2n-2$ one
can associate a quaternionic Kähler manifold $\bar{N}$ of dimension $4n$. This
corresponds to dimensional reduction of $\mathcal{N}=2$ supergravity coupled
to vector multiplets from 4 to 3 space-time dimensions. It was moreover shown
in [RSV06] that the supergravity c-map metric (sometimes referred to as the
Ferrara-Sabharwal metric) admits a one-parameter deformation by quaternionic
Kähler metrics of negative scalar curvature. This metric is known as the _one-
loop deformed supergravity c-map metric_ (this deformation can be interpreted
as a perturbative quantum correction in the string coupling in type IIB string
theory).

A mathematical proof that the (one-loop) deformed supergravity c-map metric is
indeed quaternionic Kähler of negative scalar curvature can be described as
the following two-step process. Given a projective special Kähler manifold
$\bar{M}$, one has a conical affine special Kähler manifold $M$ on a
$\mathbb{C}^{*}$-bundle over $\bar{M}$. The cotangent bundle $N=T^{*}M$ of $M$
is a pseudo-hyper-Kähler manifold [CFG89, ACD02] equipped with a rotating
circle symmetry [ACM13], so one can apply the HK/QK correspondence [Hay08,
ACM13] to obtain a quaternionic Kähler manifold $\bar{N}$. Finally, one checks
that the metric obtained by this method is the deformed supergravity c-map
metric [ACDM15]. Summarizing we have the following diagram:

${M^{2n}}$${N^{4n}}$${\bar{M}^{2n-2}}$${\bar{N}^{4n}}$rigid
c-mapHK/QKsupergravity c-map$\scriptstyle{\mathbb{C}^{*}}$

Therefore the deformed supergravity c-map produces a one-parameter family of
quaternionic Kähler metrics (depending on a real parameter $c\in\mathbb{R}$)
from a projective special Kähler manifold. Note that the case $c=0$ is the
Ferrara-Sabharwal metric. For a fixed projective special Kähler manifold, the
metrics in the image of the deformed supergravity c-map are locally isometric
for different values of $c>0$ [CDS17]. Moreover, any deformed supergravity
c-map space is non-locally homogeneous [CGS23].

There is a special class of supergravity c-map spaces known as _supergravity
q-map spaces_. These spaces arise as the composition of the supergravity r-map
and the supergravity c-map. The _supergravity r-map_ produces a projective
special Kähler manifold $\bar{M}$ of (real) dimension $2n-2$ from a projective
special real manifold $\mathcal{H}$ of dimension $n-2$. This construction was
introduced in [dWVP92] and it is induced by dimensional reduction of
$\mathcal{N}=2$ supergravity theories from 5 to 4 space-time dimensions.
Therefore, applying the supergravity q-map to a projective special real
manifold $\mathcal{H}$ of dimension $n-2$, we obtain a quaternionic Kähler
manifold of dimension $4n$. Since the supergravity c-map admits a one-
parameter deformation, so does the supergravity q-map. It is known that for
any $c\geq 0$, a supergravity q-map space is complete provided that the
projective special real manifold is complete [CDS17]. From the resolution of
the Alekseevsky conjecture [BL23] we know that all homogeneous quaternionic
Kähler manifolds of negative scalar curvature are Alekseevsky spaces [Ale75,
Cor96], that is admit a simply transitive solvable Lie group of isometries.
Moreover, except $\operatorname{\mathbb{H}\mathrm{H}}^{n}$ and
$\mathrm{SU}(n,2)/\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(2))$, all of these
spaces are in the image of the supergravity q-map [dWVP92].

The purpose of this paper is to determine explicitly a universal Lie group
acting effectively and isometrically on any deformed supergravity q-map space
$(\bar{N},g_{\bar{N}}^{c})$ for $c\geq 0$, see Theorem 3.6. For the undeformed
case $c=0$ the group was described in [CDJL21, Appendix A], compare with
[dWVVP93, dWVP95].

The only space that is in the image of the supergravity c-map but is not
obtained as a supergravity q-map space is
$(\bar{N}=\mathrm{SU}(n,2)/\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(2)),g_{\bar{N}}^{c})$.
For this particular case, a Lie group playing a similar role for supergravity
c-map spaces (rather than supergravity q-map spaces) acting effectively and
isometrically was obtained in [CRT21] by integrating the corresponding Killing
vector fields.

In this paper we directly work on the level of group actions rather than Lie
algebra actions. First we recall how the group
$\operatorname{Aut}(M)\ltimes\mathbb{R}^{2n}$, $n=\dim_{\mathbb{C}}M$, acts
globally on the rigid c-map space $N=T^{*}M$, see Propositions 2.11 and 2.12.
Then, we describe how to lift this action to an action of
$\operatorname{Aut}(M)\ltimes\operatorname{Heis}_{2n+1}$ on the trivial circle
bundle $P$ over $N$ that recovers the known infinitesimal action described in
[CST21], see (7) and Proposition 3.2. Finally, when the conical affine special
Kähler manifold $M$ is determined by a projective special real manifold
$\mathcal{H}$, we show that the subgroup

$\big{(}(\mathbb{R}_{>0}\times\operatorname{Aut}(\mathcal{H}))\ltimes\mathbb{R}^{n-1}\big{)}\ltimes(\operatorname{Heis}_{2n+1}/\mathcal{F}),$

where $\mathcal{F}$ is an infinite cyclic subgroup of the Heisenberg center,
acts isometrically and effectively on the quaternionic Kähler manifold
$\bar{N}\subset P$ viewed as a hypersurface of the circle bundle $P$, yielding
our main result. Here $\operatorname{Aut}(\mathcal{H})$ denotes the
automorphism group of the projective special real manifold $\mathcal{H}$. The
cyclic group $\mathcal{F}$ included to ensure effectiveness can be removed by
considering the universal covering of the quaternionic Kähler manifold, which
amounts to replacing the circle bundle $P$ by an $\mathbb{R}$-bundle.

Notation: We will use the Einstein summation convention throughout the text
for the computations in coordinates.

#### Acknowledgements

V. C. and A. G. are supported by the German Science Foundation (DFG) under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306.

##  2 Symmetries of rigid c-map spaces

In this section we recall some relevant background material. In particular, we
recall the definition and some properties of conical affine special Kähler
manifolds, as well as the rigid c-map construction, which assigns a pseudo-
hyper-Kähler manifold with a rotating Killing vector field to any conical
affine special Kähler manifold. Moreover, we review how to construct
Hamiltonian automorphisms of the rigid c-map structure.

###  2.1 Special Kähler manifolds

######  Definition 2.1.

An _affine special Kähler (ASK) manifold_ $(M,g,J,\nabla)$ is a (pseudo-)
Kähler manifold $(M,g,J)$ endowed with a flat torsion-free connection $\nabla$
such that $\nabla\omega=0$ and $\mathrm{d}^{\nabla}J=0$, where
$\omega=gJ:=g(J\cdot,\cdot)$ denotes the Kähler form.

######  Definition 2.2.

A _conical affine special Kähler (CASK) manifold_ $(M,g,J,\nabla,\xi)$ is an
ASK manifold $(M,g,J,\nabla)$ endowed with a complete vector field $\xi$,
called the _Euler vector field_ , such that

  * $\bullet$

$g$ is negative-definite on $\operatorname{span}\\{\xi,J\xi\\}$ and positive-
definite on its orthogonal complement,

  * $\bullet$

$D\xi=\nabla\xi=\operatorname{Id}$, where $D$ is the Levi-Civita connection of
$g$.

Moreover, $M$ is endowed with a principal $\mathbb{C}^{*}$-action generated by
$\xi$ and $J\xi$.

We collect in the following proposition some well-known facts about the
interaction between the vector fields $\xi,J\xi$ and the affine special Kähler
structure (see e.g. [CM09, Proposition 3] and [CHM12, page 652] for a proof).

######  Proposition 2.3.

Let $(M,g,J,\nabla,\xi)$ be a CASK manifold. Then

  1. (a)

The vector field $\xi$ is holomorphic and homothetic: $\mathcal{L}_{\xi}g=2g$.

  2. (b)

The vector field $J\xi$ is holomorphic and Killing.

  3. (c)

The function $f=\frac{1}{2}g(\xi,\xi)$ is a Kähler potential for $g$ and a
Hamiltonian function for $J\xi$, where our convention is
$\mathrm{d}f=-\omega(J\xi,\cdot)$.

Given a CASK manifold $(M,g,J,\nabla,\xi)$, the quotient space
$\bar{M}:=M/\mathbb{C}^{*}$ inherits a (positive-definite) Kähler metric
$\bar{g}$ and $(\bar{M},\bar{g})$ is called a _projective special Kähler (PSK)
manifold_. The metric is obtained by Kähler reduction exploiting the fact that
$J\xi$ is a Hamiltonian Killing vector field. Indeed,

$\bar{M}=M/\mkern-5.0mu/S^{1}=f^{-1}(-\tfrac{1}{2})/S^{1}$

is the Kähler quotient of the CASK manifold $M$ by the $S^{1}$-action
generated by $J\xi$.

Following [CST21] we introduce the natural notion of symmetry in this setting.

######  Definition 2.4.

  * $\bullet$

An _automorphism_ of a CASK manifold $(M,g,J,\nabla,\xi)$ is a diffeomorphism
of $M$ which preserves $g$, $J$, $\nabla$ and $\xi$.

  * $\bullet$

An _automorphism_ of a PSK manifold $\bar{M}=M/\mkern-5.0mu/S^{1}$ is a
diffeomorphism of $\bar{M}$ induced by an automorphism of the CASK manifold
$M$.

The corresponding groups of automorphisms are denoted by
$\operatorname{Aut}(M)$ and $\operatorname{Aut}(\bar{M})$, respectively. At
the infinitesimal level we have the following notion.

######  Definition 2.5.

  * $\bullet$

An _infinitesimal automorphism_ of a CASK manifold $(M,g,J,\nabla,\xi)$ is a
vector field $X\in\Gamma(TM)$ such that its local flow preserves the CASK data
on $M$. The Lie algebra of such vector fields is denoted by
$\mathfrak{aut}(M)$.

  * $\bullet$

An _infinitesimal automorphism_ of a PSK manifold is a vector field $\bar{X}$
induced by an infinitesimal automorphism of the corresponding CASK manifold
(which always projects since it commutes with $\xi$ and $J\xi$). The
corresponding Lie algebra is denoted by $\mathfrak{aut}(\bar{M})$.

It was shown in [CST21, Proposition 2.18] that $\mathfrak{aut}(M)$ and
$\mathfrak{aut}(\bar{M})$ are isomorphic when the Levi-Civita connection $D$
and the special connection $\nabla$ are not equal.

###  2.2 The rigid c-map

Let us consider a CASK manifold $(M,g,J,\nabla,\xi)$. We will discuss the
natural geometric structure that exists on its cotangent bundle $N=T^{*}M$.
For this, it will be useful to recall some basic facts about vector bundles
(see e.g. [MS22]).

Let $\pi:E\longrightarrow M$ be a vector bundle over a manifold $M$. We can
pull $E$ back to a bundle over its total space: $\pi^{*}E\longrightarrow E$.
This bundle always admits a tautological section $\Phi\in\Gamma(\pi^{*}E)$,
which assigns to the point $e\in E$ the value $e$. In the case where
$E=T^{*}M$, this is precisely (one interpretation of) the tautological one-
form $\lambda$. The tangent vectors to the fibers of $E$ determine a canonical
vertical distribution $T^{\mathrm{V}}E\subset TE$, which is moreover
canonically isomorphic to $\pi^{*}E$. The corresponding isomorphism is denoted
by

$\mathcal{V}:\pi^{*}E\longrightarrow T^{\mathrm{V}}E.$

In the case where $E=T^{*}M$, local coordinates $\\{q^{j}\\}$ on $M$ induce
canonical coordinates $\\{q^{j},p_{j}\\}$ on $T^{*}M$ and the isomorphism
$\mathcal{V}:\pi^{*}(T^{*}M)\longrightarrow T^{\mathrm{V}}(T^{*}M)$ is
implemented by mapping $\pi^{*}(\mathrm{d}q^{j})$ to $\frac{\partial}{\partial
p_{j}}$.

Now assume that $E$ comes equipped with some connection $\nabla$. Then we may
use the pullback connection on $\pi^{*}E$ to compute $(\pi^{*}\nabla)\Phi$.
The assignment $X\longmapsto(\pi^{*}\nabla)_{X}\Phi$, where $X\in\Gamma(TE)$,
then provides a left inverse to $\mathcal{V}$. Thus, a vector field $X$ is
determined by $(\pi^{*}\nabla)_{X}\Phi\in\Gamma(\pi^{*}E)$ and
$\pi_{*}(X)\in\Gamma(\pi^{*}(TM))$. This is just another way of phrasing the
fact that a connection on $E$ induces a splitting
$TE\cong\pi^{*}(TM)\oplus\pi^{*}E$. In particular, for $E=T^{*}M$ we obtain
the splitting $T(T^{*}M)\cong\pi^{*}(TM)\oplus\pi^{*}(T^{*}M)$. With respect
to this splitting, we may define the following tensor fields:

$g_{N}=\begin{pmatrix}g&0\\\ 0&g^{-1}\end{pmatrix},\qquad
I_{1}=\begin{pmatrix}J&0\\\ 0&J^{*}\end{pmatrix},\qquad
I_{2}=\begin{pmatrix}0&-\omega^{-1}\\\ \omega&0\end{pmatrix},\qquad
I_{3}=I_{1}I_{2}.$ (1)

It is well-known that these tensor fields define a pseudo-hyper-Kähler
structure on $N=T^{*}M$ if and only if $(M,g,J,\nabla)$ is an ASK manifold
[CFG89, ACD02]. Furthermore, the conical structure gives rise to the vector
field $Z:=-\widetilde{J\xi}$, where the tilde denotes the horizontal lifting
with respect to $\nabla$ (i.e. $(\pi^{*}\nabla)_{Z}\lambda=0$ where $\lambda$
is the tautological one-form). This vector field is Killing and
$\omega_{1}$-Hamiltonian, but satisfies $\mathcal{L}_{Z}\omega_{2}=\omega_{3}$
and $\mathcal{L}_{Z}\omega_{3}=-\omega_{2}$ [ACM13, Proposition 2]. It is
therefore called a _rotating_ Killing field, and the circle action it
generates is called rotating as well. The Hamiltonian function for $Z$ with
respect to $\omega_{1}$ is denoted by $f_{Z}^{c}$, and is given by
$f_{Z}^{c}=-\frac{1}{2}g_{N}(Z,Z)-\frac{1}{2}c$, $c\in\mathbb{R}$.

Taken together, the above hyper-Kähler structure with rotating circle action
is referred to as the _rigid c-map structure_ on $N=T^{*}M$ and
$(N,g_{N},I_{1},I_{2},I_{3})$ as the _rigid c-map space_.

######  Definition 2.6.

A diffeomorphism $\varphi:N\longrightarrow N$ is called an _automorphism of
the rigid c-map structure_ , or equivalently of the hyper-Kähler structure
with rotating circle action, if it preserves $g_{N}$, $I_{1}$, $I_{2}$,
$I_{3}$ and $f_{Z}^{c}$.

Note that an automorphism in the above sense automatically commutes with the
rotating circle action. The group of all automorphisms of the rigid c-map
structure is denoted by $\operatorname{Aut}_{S^{1}}(N)$. The subgroup of
$\omega_{1}$-Hamiltonian automorphisms is denoted by
$\operatorname{Ham}_{S^{1}}(N)$.

Given a hyper-Kähler structure with rotating circle action, there is a
canonical, closed two-form
$\omega_{\mathrm{H}}:=\omega_{1}+\mathrm{d}\iota_{Z}g_{N}$ of type (1,1) with
respect to each $I_{k}$ [CST22, Lemma 2.7] that plays a central role in the
following. Note that any vector field which is $\omega_{1}$-Hamiltonian and
preserves $Z$ and $g_{N}$ is automatically $\omega_{\mathrm{H}}$-Hamiltonian
as well. This applies, in particular, to the rotating Killing field $Z$, whose
Hamiltonian with respect to $\omega_{\mathrm{H}}$ we denote by
$f_{\mathrm{H}}^{c}$, and is given by
$f_{\mathrm{H}}^{c}=f_{Z}^{c}+g_{N}(Z,Z)=\frac{1}{2}g_{N}(Z,Z)-\frac{1}{2}c$.
In the following, we will focus on Hamiltonian functions with respect to
$\omega_{\mathrm{H}}$ rather than $\omega_{1}$.

There are two important sources of Hamiltonian automorphisms of the rigid
c-map structure: canonical lifts of CASK automorphisms and translations in the
fibers.

For completeness of the exposition we mention that in the case of the rigid
c-map the pre-symplectic form $\omega_{\mathrm{H}}$ is in fact symplectic.

######  Proposition 2.7.

Let $(N,g_{N},I_{1},I_{2},I_{3})$ be a rigid c-map space. Then the canonical
two-form $\omega_{\mathrm{H}}=\omega_{1}+\mathrm{d}\iota_{Z}g_{N}$ associated
with the rotating Killing field $Z$ is a symplectic structure.

###### Proof.

Since the one-form $\iota_{Z}g_{N}$ is the pull-back of
$\alpha=-\iota_{J\xi}g$ we can calculate its differential as
$\pi^{*}\mathrm{d}\alpha$, where $\mathrm{d}\alpha$ can be expressed as twice
the skew-symmetric part of the Levi-Civita covariant derivative

$D\alpha=-g(D(J\xi),\cdot)=-g(J\cdot,\cdot)=-\omega.$

Since this is skew-symmetric, we see that
$\mathrm{d}\iota_{Z}g_{N}=\pi^{*}\mathrm{d}\alpha=-2\pi^{*}\omega$. This shows
that

$\omega_{\mathrm{H}}=\begin{pmatrix}-\omega&0\\\ 0&\omega^{-1}\end{pmatrix}.$
(2)

∎

Incidentally, the above expression for $\omega_{\mathrm{H}}$ shows that the
endomorphism

$I_{\mathrm{H}}:=g_{N}^{-1}\omega_{\mathrm{H}}=\begin{pmatrix}-J&0\\\
0&J^{*}\end{pmatrix}$

is a $g_{N}$-skew-symmetric almost complex structure, which obviously commutes
with $I_{1}$, $I_{2}$ and $I_{3}$. In this way we recover the statement that
$\omega_{\mathrm{H}}$ is of type $(1,1)$ for the three complex structures.

######  Corollary 2.8.

Any rigid c-map space $(N,g_{N},I_{1},I_{2},I_{3})$ carries a canonical almost
Kähler structure $(g_{N},I_{\mathrm{H}})$.

###  2.3 Canonical lifts

Let $\operatorname{Aut}(M)$ be the group of CASK automorphisms of the CASK
manifold $M$. Canonically lifting to $N=T^{*}M$, we obtain a group of
$\omega_{\mathrm{H}}$-Hamiltonian automorphisms of the rigid c-map structure,
as proven in [CST21]. It was remarked in [CRT21, Proposition 2.10] that this
group even admits an equivariant moment map
$\mu:N\longrightarrow\mathfrak{g}^{*}$ with respect to $\omega_{\mathrm{H}}$.
We recall the result in Proposition 2.9 below adding details and fixing
notation.

Let $X$ denote a vector field on $M$ generating a one-parameter family of
automorphisms of the CASK structure, and $Y$ its canonical lift to $N$. Then,
with respect to the splitting $T(T^{*}M)=T^{\mathrm{H}}N\oplus
T^{\mathrm{V}}N\cong\pi^{*}(TM)\oplus\pi^{*}(T^{*}M)$, we may write

$Y=\pi^{*}X-\lambda\circ(\pi^{*}\nabla)(\pi^{*}X).$

Here $\lambda:\xi\longmapsto\lambda_{\xi}$ is the tautological section of the
vector bundle $\pi^{*}(T^{*}M)$, defined as

$\lambda_{\xi}(v)=\xi(v),\qquad\xi\in
N_{p}=T_{p}^{*}M,\,v\in(\pi^{*}(TM))_{\xi}=T_{p}M,\,\pi(\xi)=p,$

where $\pi^{*}X=X\circ\pi\in\Gamma(\pi^{*}(TM))$ and
$\lambda\circ(\pi^{*}\nabla)(\pi^{*}X)\in\Gamma(\pi^{*}(T^{*}M))\subset\Gamma(T^{*}N)$
is the one-form sending a vector field $A\in\Gamma(TN)$ to the smooth function
$\lambda((\pi^{*}\nabla)_{A}(\pi^{*}X))$, and where we have identified
$\pi^{*}(T^{*}M)=(\pi^{*}(TM))^{*}=(T^{\mathrm{H}}N)^{*}=(T^{\mathrm{V}}N)^{0}\subset
T^{*}N$.

Note that

$(\pi^{*}\nabla)_{Y}\lambda=-\lambda\circ(\pi^{*}\nabla)(\pi^{*}X).$ (3)

Indeed, for a vector field $Y\in\Gamma(TN)$ and
$\lambda\in\Gamma(\pi^{*}(T^{*}M))$ the tautological section,
$(\pi^{*}\nabla)_{Y}\lambda$ gives us the vertical component of $Y$, which is
precisely $-\lambda\circ(\pi^{*}\nabla)(\pi^{*}X)$.

In local coordinates $\\{q^{i},p_{i}\\}$ on $N$ induced by local
$\nabla$-affine coordinates $\\{q^{i}\\}$ on $M$, the tautological section
$\lambda$ on $N$ is given by $\lambda(\frac{\partial}{\partial q^{i}})=p_{i}$
and the vector field $Y\in\Gamma(TN)$ by

$Y=X^{j}\frac{\partial}{\partial q^{j}}-p_{i}\frac{\partial X^{i}}{\partial
q^{j}}\frac{\partial}{\partial p_{j}}$

in terms of the components $\\{X^{i}\\}$ of the vector field $X$ in the local
coordinates $\\{q^{i}\\}$.

######  Proposition 2.9.

Let $X$ be an infinitesimal CASK automorphism and $Y$ its canonical lift as
explained above. Then

$\mu_{Y}=\frac{1}{2}\left(g_{N}(Z,Y)+\pi^{*}\omega^{-1}(\lambda\circ(\pi^{*}\nabla)(\pi^{*}X),\lambda)\right)$

is a $\omega_{\mathrm{H}}$-Hamiltonian function for $Y$, where
$Z=-\widetilde{J\xi}$. This assignment determines an equivariant (co)moment
map $\mu:\mathfrak{aut}(M)\longrightarrow\mathcal{C}^{\infty}(N)$,
$X\longmapsto\mu_{X}:=\mu_{Y}$, for the action of $\mathfrak{aut}(M)$ on
$N=T^{*}M$.

###### Proof.

We have to show that $\iota_{Y}\omega_{\mathrm{H}}=-\mathrm{d}\mu_{Y}$. Recall
that $\omega_{\mathrm{H}}$ is given by (2) with respect to the splitting
$T(T^{*}M)\cong\pi^{*}(TM)\oplus\pi^{*}(T^{*}M)$. Thus we have

$\iota_{Y}\omega_{\mathrm{H}}=-\pi^{*}(\iota_{X}\omega)+\iota_{(\pi^{*}\nabla)_{Y}\lambda}\pi^{*}\omega^{-1}.$

Let us compute these two terms. First notice that, by Proposition 2.3 (a) we
have that $\mathcal{L}_{\xi}\omega=2\omega$ and, hence

$2\iota_{X}\omega=\iota_{X}\mathcal{L}_{\xi}\omega=\iota_{X}\mathrm{d}(\iota_{\xi}\omega)=\mathcal{L}_{X}(\iota_{\xi}\omega)-\mathrm{d}(\iota_{X}\iota_{\xi}\omega)=\mathrm{d}(g(-J\xi,X)),$

since $\mathcal{L}_{X}(\iota_{\xi}\omega)=0$ (recall that $X$ is an
infinitesimal CASK automorphism). So we see that

$\pi^{*}(\iota_{X}\omega)=\frac{1}{2}\mathrm{d}(g_{N}(Z,Y)).$ (4)

To compute the second term, let $A$ be an arbitrary section of
$\pi^{*}(T^{*}M)\cong T^{\mathrm{V}}(T^{*}M)$. Let $\\{q^{j}\\}$ be special
real coordinates on $M$ and $\\{q^{j},p_{j}\\}$ the corresponding canonical
coordinates on $N$. With respect to these, we can write
$A=A_{i}\mathrm{d}q^{i}$ and, using
$(\pi^{*}\nabla)_{Y}\lambda=-\lambda\circ(\pi^{*}\nabla)(\pi^{*}X)=-p_{i}\frac{\partial
X^{i}}{\partial q^{j}}\mathrm{d}q^{j}$, we get

$\pi^{*}\omega^{-1}((\pi^{*}\nabla)_{Y}\lambda,A)=-\omega^{jk}p_{i}\frac{\partial
X^{i}}{\partial q^{j}}A_{k}.$

Thus, $\iota_{(\pi^{*}\nabla)_{Y}\lambda}\pi^{*}\omega^{-1}$ is the one-form
on $T^{*}M$ which vanishes when applied to a horizontal vector field and
evaluates on a vertical vector field $A=A_{i}\frac{\partial}{\partial p_{i}}$
(corresponding to the $A$ consider earlier by the canonical isomorphism
$\pi^{*}(T^{*}M)\cong T^{\mathrm{V}}(T^{*}M)$) as above. This means that we
can write it, in local coordinates, as

$\iota_{(\pi^{*}\nabla)_{Y}\lambda}\pi^{*}\omega^{-1}=-\omega^{jk}p_{i}\frac{\partial
X^{i}}{\partial q^{j}}\mathrm{d}p_{k}.$ (5)

Now, we compute the differential of our proposed moment map:

$\mathrm{d}\mu_{Y}=\frac{1}{2}\left(\mathrm{d}(g_{N}(Z,Y))+\mathrm{d}(\pi^{*}\omega^{-1}(\lambda\circ(\pi^{*}\nabla)(\pi^{*}X),\lambda))\right),$

the second term of which can be computed (using (3) and (5)) in local
coordinates as follows:

$\displaystyle\frac{1}{2}\mathrm{d}(\pi^{*}\omega^{-1}(\lambda\circ(\pi^{*}\nabla)(\pi^{*}X),\lambda))$
$\displaystyle=\frac{1}{2}\mathrm{d}\left(\omega^{jk}p_{i}p_{k}\frac{\partial
X^{i}}{\partial q^{j}}\right)=\frac{1}{2}\omega^{jk}\frac{\partial
X^{i}}{\partial q^{j}}(p_{i}\mathrm{d}p_{k}+p_{k}\mathrm{d}p_{i})$
$\displaystyle=\omega^{jk}p_{i}\frac{\partial X^{i}}{\partial
q^{j}}\mathrm{d}p_{k}=-\iota_{(\pi^{*}\nabla)_{Y}\lambda}\pi^{*}\omega^{-1},$

where, in passing to the second line, we used that both $\omega^{jk}$ and
$\frac{\partial X^{i}}{\partial q^{j}}$ are anti-symmetric in their indices;
the latter fact is just the equation $\mathcal{L}_{X}\omega=0$ in
($\nabla$-affine) coordinates.

This computation, together with (4), concludes the proof that $\mu_{Y}$ is
indeed a Hamiltonian function for $Y$.

To show that the map $\mu$ is equivariant, let
$X_{1},X_{2}\in\mathfrak{aut}(M)$ and $Y_{1},Y_{2}\in\Gamma(TN)$ their
corresponding canonical lifts. Note that the canonical lift of $[X_{1},X_{2}]$
is precisely $[Y_{1},Y_{2}]$. Using this and the fact that the moment map is
constructed in a canonical way, we obtain

$\mathcal{L}_{Y_{1}}(\mu_{Y_{2}})=\frac{1}{2}\left(g_{N}(Z,[Y_{1},Y_{2}])+\pi^{*}\omega^{-1}(\lambda\circ(\pi^{*}\nabla)(\pi^{*}([X_{1},X_{2}])),\lambda)\right)=\mu_{[Y_{1},Y_{2}]}.$

∎

######  Remark 2.10.

  * $\bullet$

Note that the $\omega_{\mathrm{H}}$-Hamiltonian function $\mu_{Y}$ given by
Proposition 2.9 is the only moment map that is homogeneous, i.e.
$\mathcal{L}_{\Xi}\mu_{Y}=2\mu_{Y}$, where $\Xi\in\Gamma(TN)$ is the sum of
the $\nabla$-horizontal lifted Euler vector field $\xi$ with the fiberwise
Euler (or position) vector field of $T^{*}M$. In canonical coordinates
associated to (conical) special real coordinates, $\Xi$ takes the form
$\Xi=q^{j}\frac{\partial}{\partial q^{j}}+p_{j}\frac{\partial}{\partial
p_{j}}$.

  * $\bullet$

Note also that the equivariance of the moment map
$\mu:\mathfrak{aut}(M)\longrightarrow\mathcal{C}^{\infty}(N)$ (without
assuming homogeneity) fixes it uniquely up to adding a linear form
$c:\mathfrak{aut}(M)\longrightarrow\mathbb{R}$, $X\longmapsto c_{X}$,
invariant under the coadjoint representation. The space of such forms is
trivial if and only if the Lie algebra is perfect, that is, it coincides with
its derived ideal. This is the case, in particular, for semisimple Lie
algebras.

Thus, we have a canonical action of $\operatorname{Aut}(M)$ on $N$ which casts
$\operatorname{Aut}(M)$ as a subgroup of $\operatorname{Ham}_{S^{1}}(N)$, with
a canonical choice of equivariant moment map.

###  2.4 Translations in the fibers

One of the crucial features of the rigid c-map metric on $N$ is that it is
semi-flat. This means that it is foliated by half-dimensional flat
submanifolds. These are just the fibers of $\pi:N=T^{*}M\longrightarrow M$,
and the reason they are flat is that the metric in each fiber is constant with
respect to the affine structure induced by the vector space structure of the
fiber, as can be seen directly from (1).

An important consequence is the following result.

######  Proposition 2.11.

The cotangent bundle $N=T^{*}M$ carries locally an
$\omega_{\mathrm{H}}$-Hamiltonian action of the group $\mathbb{R}^{2n}$ by
automorphisms of the rigid c-map structure, which preserves the fibers and
acts on them by translations. If the holonomy group of the special connection
$\nabla$ on $M$ is trivial (in particular, if $M$ is simply connected), then
the action is global.

###### Proof.

We first give a description of the local action. Choosing special real
coordinates $\\{q^{j}\\}$ on $M$ and using the associated canonical
coordinates $\\{q^{j},p_{j}\\}$ on $N$, the action of $\mathbb{R}^{2n}$ is
generated by the locally defined vector fields $\frac{\partial}{\partial
p_{j}}$. A quick look at (1) reveals that the full hyper-Kähler structure is
preserved by these vector fields. Moreover, they commute with the rotating
circle action as well, since $Z=-\widetilde{J\xi}$ only depends on the
coordinates $\\{q^{j}\\}$ on the base $M$. This implies that they also
preserve the $\omega_{1}$-Hamiltonian function
$f_{Z}^{c}=-\frac{1}{2}g_{N}(Z,Z)-\frac{1}{2}c$. Finally, let us give a local
Hamiltonian with respect to $\omega_{\mathrm{H}}$. Writing (2) in the above
coordinates, we have

$\omega_{\mathrm{H}}=\frac{1}{2}(-\omega_{ij}\mathrm{d}q^{i}\wedge\mathrm{d}q^{j}+\omega^{ij}\mathrm{d}p_{i}\wedge\mathrm{d}p_{j}),$

where each $\omega_{ij}$ is constant (and hence so are the components
$\omega^{ij}$ of the inverse matrix), and we are omitting pullbacks when no
confusion can arise.

Hence, we find

$\iota_{\frac{\partial}{\partial
p_{k}}}\omega_{\mathrm{H}}=\omega^{kj}\mathrm{d}p_{j}=-\mathrm{d}(-\omega^{kj}p_{j}).$

We can thus assign the local Hamiltonian function

$\mu_{\frac{\partial}{\partial p_{k}}}=-\omega^{kj}p_{j}$ (6)

to $\frac{\partial}{\partial p_{k}}$. More generally, $\mu$ assigns to every
vertical vector field $v=v_{k}\frac{\partial}{\partial p_{k}}$ with constant
coefficients the function $\mu_{v}=-\omega^{kj}v_{k}p_{j}$. The vector fields
$\frac{\partial}{\partial p_{k}}$ generate an action of the group
$\mathbb{R}^{2n}$ with the claimed properties. In the local coordinates
$\\{q^{j},p_{j}\\}$ a vector $v=(v_{j})\in\mathbb{R}^{2n}$ acts by
$(q,p)\longmapsto(q,p+v)$.

To define a global moment map
$\mu:\mathbb{R}^{2n}\longrightarrow\mathcal{C}^{\infty}(N)$ and a global group
action of $\mathbb{R}^{2n}$ on $N$ it suffices to have a global frame of the
vertical bundle $T^{\mathrm{V}}N\cong\pi^{*}(T^{*}M)$ parallel with respect to
the connection $\pi^{*}\nabla$. Such a frame exists if and only if the
holonomy of the (flat) special connection is trivial. ∎

As a comment on the last step of the proof of Proposition 2.11 we note that
locally (over the preimage $\pi^{-1}(U)$ of a suitable open set $U\subset M$)
a parallel frame of $T^{\mathrm{V}}N\cong\pi^{*}(T^{*}M)$ can be chosen of the
form $\pi^{*}(\mathrm{d}q^{i})$, where $\\{q^{i}\\}$ are local $\nabla$-affine
coordinates.

###  2.5 The semidirect product

Under the assumption that the holonomy group of $\nabla$ is trivial, we have
constructed two subgroups of $\operatorname{Ham}_{S^{1}}(N)$ and the next
thing to do is to study how they interact. In this section, we check that the
generators of the two groups combine into a semidirect product. Let us start
by emphasizing that the group $\Gamma_{\nabla}(T^{*}M)\cong\mathbb{R}^{2n}$ of
$\nabla$-parallel sections acts by addition on $N=T^{*}M$, that is

$\alpha\cdot\beta=\alpha(p)+\beta,$

for all $\alpha\in\Gamma_{\nabla}(T^{*}M)$, $\beta\in T_{p}^{*}M$, $p\in M$.
The group $\operatorname{Aut}(M)$ acts naturally on $N=T^{*}M$:

$h\cdot\beta=h_{*}\beta=(h^{-1})^{*}\beta,$

for all $h\in\operatorname{Aut}(M)$, $\beta\in T_{p}^{*}M$, $p\in M$.

######  Proposition 2.12.

The subgroups $\mathbb{R}^{2n}$ and $\operatorname{Aut}(M)$ of
$\operatorname{Ham}_{S^{1}}(N)$ generate a group
$G\subset\operatorname{Ham}_{S^{1}}(N)$ which is a semidirect product
$\operatorname{Aut}(M)\ltimes\mathbb{R}^{2n}$.

###### Proof.

It is clear that the two subgroups have trivial intersection, as
$\operatorname{Aut}(M)$ lifts a subgroup of diffeomorphisms of $M$ whereas
$\mathbb{R}^{2n}$ preserves each fiber of $N=T^{*}M$. The action of
$\operatorname{Aut}(M)$ is linear on the fibers and preserves the space of
$\nabla$-parallel one-forms. We check explicitly that the elements
$h\in\operatorname{Aut}(M)$ normalize the group of translations:

$(h\cdot\alpha\cdot
h^{-1})\cdot\beta=h_{*}(\alpha(h^{-1}(p))+h^{*}\beta)=(h_{*}\alpha)(p)+\beta=(h_{*}\alpha)\cdot\beta,$

for all $\alpha\in\Gamma_{\nabla}(T^{*}M)$, $\beta\in T_{p}^{*}M$, $p\in M$.
This proves that $h\cdot\alpha\cdot
h^{-1}=h_{*}\alpha\in\Gamma_{\nabla}(T^{*}M)$ for all
$\alpha\in\Gamma_{\nabla}(T^{*}M)$. ∎

Infinitesimally, we can describe the semidirect product structure in terms of
structure constants if we choose a basis. Thus, let $\\{Y_{\alpha}\\}$ be
infinitesimal generators of the action of $\operatorname{Aut}(M)$ on $N$,
obtained by canonically lifting generators
$\\{X_{\alpha}\\}\subset\mathfrak{aut}(M)$. With respect to canonical
coordinates induced by (conical) special real coordinates on $M$, we can
express $Y_{\alpha}$ as

$Y_{\alpha}=X_{\alpha}^{j}\frac{\partial}{\partial q^{j}}-p_{i}\frac{\partial
X_{\alpha}^{i}}{\partial q^{j}}\frac{\partial}{\partial p_{j}},$

where the component functions $X_{\alpha}^{j}$ of $X_{\alpha}$ are linear
functions. The generators of the $\mathbb{R}^{2n}$-action are the vectors
$\frac{\partial}{\partial p_{k}}$. The structure constants are now easily
computed:

$\left[Y_{\alpha},\frac{\partial}{\partial p_{k}}\right]=\frac{\partial
X_{\alpha}^{k}}{\partial q^{j}}\frac{\partial}{\partial p_{j}}.$

Note that the coefficients multiplying $\frac{\partial}{\partial p_{j}}$ on
the right-hand side are indeed constants, since any $X\in\mathfrak{aut}(M)$ is
$\nabla$-affine (i.e. $\mathcal{L}_{X}\nabla=0$). Together with the structure
constants of $\operatorname{Aut}(M)$ this determines the structure of
$\mathfrak{aut}(M)\ltimes\mathbb{R}^{2n}$.

##  3 Symmetries under the HK/QK correspondence

Now that we have a completely explicit description of the group
$\operatorname{Aut}(M)\ltimes\mathbb{R}^{2n}\subset\operatorname{Ham}_{S^{1}}(N)$,
the next step is to transfer the group action to the quaternionic Kähler
manifold $\bar{N}$. Up to a so-called elementary modification explained below,
this is essentially done by first lifting to $P=N\times S^{1}$ and
subsequently studying the induced action on $\bar{N}$, which we realize as a
submanifold of $P$. We will continue to assume that the holonomy group of the
flat special connection $\nabla$ is trivial to ensure the global
$\mathbb{R}^{2n}$-action.

###  3.1 The HK/QK correspondence as a twist

The HK/QK correspondence [Hay08, ACM13] produces a quaternionic Kähler
manifold from a (pseudo-)hyper-Kähler manifold equipped with a rotating circle
action. This correspondence was studied in [MS15] using the twist formalism
introduced by Swann in [Swa10]. We briefly recall here this construction (see
[Swa10] for details).

Let $N$ be a manifold equipped with an $S^{1}$-action generated by a vector
field $Z\in\Gamma(TN)$, and let $\pi_{N}:P\longrightarrow N$ be a principal
circle bundle over $N$ with connection one-form $\eta\in\Omega^{1}(P)$ and
curvature $\omega\in\Omega^{2}(N)$ (i.e. $\mathrm{d}\eta=\pi_{N}^{*}\omega$).
We want to lift the vector field $Z$ to a vector field $Z_{P}\in\Gamma(TP)$ so
that it preserves the connection $\eta$, i.e. $\mathcal{L}_{Z_{P}}\eta=0$, and
it commutes with the principal circle action, i.e. $[Z_{P},X_{P}]=0$, where
$X_{P}\in\Gamma(TP)$ generates the principal circle action. It turns out that
such a lift exists if and only if $\iota_{Z}\omega=-\mathrm{d}f_{Z}^{c}$ for
some function $f_{Z}^{c}\in\mathcal{C}^{\infty}(N)$. The lift is given by

$Z_{P}=\tilde{Z}+\pi_{N}^{*}f_{Z}^{c}X_{P},$

where $\tilde{Z}$ denotes the horizontal lift with respect to $\eta$. The
triple $(Z,\omega,f_{Z}^{c})$ with the above properties is called _twist
data_. A manifold $N$ equipped with twist data produces a new smooth111We will
always assume that both $Z$ and $Z_{P}$ generate free and proper actions so
that the quotient space $\bar{N}$ is a smooth manifold. manifold
$\bar{N}:=P/\langle Z_{P}\rangle$ called the _twist_ of $N$ with respect to
the twist data $(Z,\omega,f_{Z}^{c})$. We then have a double fibration
structure on $P$:

${P}$${N}$${\bar{N}}$$\scriptstyle{\pi_{N}}$$\scriptstyle{\pi_{\bar{N}}}$$\scriptstyle{Z_{P}}$$\scriptstyle{X_{P}}$$\scriptstyle{Z}$$\scriptstyle{\mathrm{twist}}$$\scriptstyle{Z_{\bar{N}}}$

Recall that $X_{P}$ generates the principal circle action with respect to the
projection $\pi_{N}:P\longrightarrow N$ and note that $Z_{P}$ plays the same
role for the projection $\pi_{\bar{N}}:P\longrightarrow\bar{N}$. Note also
that the twist construction produces a circle action on $\bar{N}$ generated by
the vector field
$Z_{\bar{N}}:=\mathrm{d}\pi_{\bar{N}}(X_{P})\in\Gamma(T\bar{N})$.

Now let $(N,g_{N},I_{1},I_{2},I_{3})$ be a pseudo-hyper-Kähler manifold
equipped with a rotating Killing vector field $Z$ generating a circle action.
To obtain a quaternionic Kähler manifold from the twist construction, one
first needs to “deform” the pseudo-hyper-Kähler metric $g_{N}$ to another
metric $g_{\mathrm{H}}^{c}$ and then apply the twist construction to the
manifold $(N,g_{\mathrm{H}}^{c})$ with the appropriate twist data. This is the
idea behind an elementary deformation. Therefore, let
$\omega_{1}:=g_{N}(I_{1}\cdot,\cdot)$ and $f_{Z}^{c}$ a nowhere vanishing
function such that $\iota_{Z}\omega_{1}=-\mathrm{d}f_{Z}^{c}$. Define

$\omega_{\mathrm{H}}:=\omega_{1}+\mathrm{d}\iota_{Z}g_{N},\quad
f_{\mathrm{H}}^{c}:=f_{Z}^{c}+g_{N}(Z,Z),\quad
g_{\mathrm{H}}^{c}:=\frac{1}{f_{Z}^{c}}g_{N}|_{(\mathbb{H}Z)^{\perp}}+\frac{f_{\mathrm{H}}^{c}}{(f_{Z}^{c})^{2}}g_{N}|_{\mathbb{H}Z},$

where $\mathbb{H}Z:=\operatorname{span}\\{Z,I_{1}Z,I_{2}Z,I_{3}Z\\}$. One can
check that $(Z,\omega_{\mathrm{H}},f_{\mathrm{H}}^{c})$ is a twist data for
$N$. It was shown then in [MS15] that the twist of $(N,g_{\mathrm{H}}^{c})$
with respect to the twist data $(Z,\omega_{\mathrm{H}},f_{\mathrm{H}}^{c})$ is
a quaternionic Kähler manifold $(\bar{N},g_{\bar{N}}^{c})$. The upshot is that
we can use the twist formalism to study the geometry of the quaternionic
Kähler manifold $\bar{N}$ purely in terms of the geometry of $N$ and the twist
data. This will be the approach in the following subsections.

###  3.2 Infinitesimal description

The infinitesimal description of the transfer of symmetries under the HK/QK
correspondence appears explained in [CST21]: one performs an (elementary)
modification and then twists. More precisely, let
$\mathfrak{g}\subset\mathfrak{ham}_{S^{1}}(N)$ be a subalgebra with moment map
$\mu:N\longrightarrow\mathfrak{g}^{*}$ with respect to $\omega_{\mathrm{H}}$.
Denote the Hamiltonian function corresponding to $V\in\mathfrak{g}$ by
$\mu_{V}$. Then, the first step consists in modifying $V$ to

$V_{\mathrm{H}}:=V-\frac{\mu_{V}}{f_{\mathrm{H}}^{c}}Z\in\Gamma(TN).$

We will sometimes refer to $V_{\mathrm{H}}$ as the elementary deformation of
$V$. The second step is twisting $V_{\mathrm{H}}$ to a vector field
$\operatorname{tw}(V_{H})\in\Gamma(T\bar{N})$, which we will denote by
$V_{\mathrm{Q}}$. Twisting is done by lifting $V_{\mathrm{H}}$ horizontally
(with respect to a given connection $\eta$ whose curvature is
$\omega_{\mathrm{H}}$) to the trivial circle bundle $P=N\times S^{1}$ and then
projecting down to $\bar{N}$. In other words

$\operatorname{tw}(V_{\mathrm{H}})=V_{\mathrm{Q}}=\mathrm{d}\pi_{\bar{N}}(\tilde{V}_{\mathrm{H}})=\mathrm{d}\pi_{\bar{N}}\left(\tilde{V}-\frac{\mu_{V}}{f_{\mathrm{H}}^{c}}\tilde{Z}\right),$

where we have denoted the $\eta$-horizontal lift to $P$ by a tilde.

This procedure gives rise to an injective, linear map
$\varphi_{\mu}:\mathfrak{g}\longrightarrow\mathfrak{aut}_{S^{1}}(\bar{N})$,
dependent on the choice of moment map $\mu$. Here,
$\mathfrak{aut}_{S^{1}}(\bar{N})$ denotes the space of Killing fields of the
quaternionic Kähler manifold $(\bar{N},g_{\bar{N}}^{c})$, $c\geq 0$, which
commute with the canonical, isometric circle action on $\bar{N}$ generated by
$Z_{\bar{N}}=\operatorname{tw}(-\frac{1}{f_{\mathrm{H}}^{c}}Z)$.

It is shown in [CST21] that this linear map is a homomorphism of Lie algebras
if and only if the moment map $\mu$ is equivariant. Indeed, choosing a basis
$\\{V_{i}\\}$ of $\mathfrak{g}$ with corresponding structure constants
$\\{c_{ij}^{k}\\}$, the equivariance condition can be expressed as

$\omega_{\mathrm{H}}(V_{i},V_{j})-c_{ij}^{k}\mu_{V_{k}}=0.$

The left-hand side is precisely what measures the failure of $\varphi_{\mu}$
to be a homomorphism, according to [CST21, Theorem 3.8].

In the case at hand, we are considering
$\mathfrak{g}\cong\mathfrak{aut}(M)\ltimes\mathbb{R}^{2n}$, and we have a
canonical moment map. Since this canonical moment map is equivariant for the
action of the subgroup $\operatorname{Aut}(M)$ by Proposition 2.9, we obtain a
subalgebra of $\mathfrak{aut}_{S^{1}}(\bar{N})$ isomorphic to
$\mathfrak{aut}(M)$. However, the canonical choice of moment map we gave for
$\mathbb{R}^{2n}$ in (6) is not equivariant. Indeed

$\omega_{\mathrm{H}}\left(\frac{\partial}{\partial
p_{i}},\frac{\partial}{\partial p_{j}}\right)=\omega^{ij}\neq 0,$

while $\mathbb{R}^{2n}$ is of course abelian and therefore has vanishing
structure constants. Following [CST21, Theorem 3.8], this implies that
$\mathbb{R}^{2n}$ gives rise to a subalgebra of
$\mathfrak{aut}_{S^{1}}(\bar{N})$ which is a one-dimensional central extension
of $\mathbb{R}^{2n}$ by $Z_{\bar{N}}$, and whose non-trivial brackets are
given by the coefficients of $\omega^{-1}$. In other words,

$\left[\left(\frac{\partial}{\partial
p_{i}}\right)^{\mathrm{Q}},\left(\frac{\partial}{\partial
p_{j}}\right)^{\mathrm{Q}}\right]=\omega^{ij}Z_{\bar{N}}.$

Since $\omega^{-1}$ is the natural symplectic form on the fibers of
$N=T^{*}M$, the central extension in question is nothing but the Heisenberg
algebra $\mathfrak{heis}_{2n+1}$.

We thus obtain two algebras of Killing fields, isomorphic to
$\mathfrak{aut}(M)$ and $\mathfrak{heis}_{2n+1}$, respectively. Together, they
generate an algebra which is once again a semidirect product
$\mathfrak{aut}(M)\ltimes\mathfrak{heis}_{2n+1}$. To see that they indeed form
a semidirect product, it suffices to show that
$[\mathfrak{aut}(M),\mathfrak{heis}_{2n+1}]\subset\mathfrak{heis}_{2n+1}$. To
check this, consider
$Y_{\alpha}^{\mathrm{Q}}\in\mathfrak{aut}(M)\subset\mathfrak{aut}_{S^{1}}(\bar{N})$
and $(\frac{\partial}{\partial
p_{k}})^{\mathrm{Q}}\in\mathbb{R}^{2n}\subset\mathfrak{heis}_{2n+1}$ (we
already know that $Z_{\bar{N}}$ is central in the full algebra), where the
vector fields $Y_{\alpha}$ were introduced in Section 2.5. Then, again by
[CST21, Theorem 3.8], we have

$\left[Y_{\alpha}^{\mathrm{Q}},\left(\frac{\partial}{\partial
p_{k}}\right)^{\mathrm{Q}}\right]=\frac{\partial X_{\alpha}^{k}}{\partial
q^{j}}\left(\frac{\partial}{\partial
p_{j}}\right)^{\mathrm{Q}}+\operatorname{tw}\left(\omega_{\mathrm{H}}\left(Y_{\alpha},\frac{\partial}{\partial
p_{k}}\right)-\frac{\partial X_{\alpha}^{k}}{\partial
q^{j}}\mu_{\frac{\partial}{\partial p_{j}}}\right)Z_{\bar{N}},$

where $X_{\alpha}\in\Gamma(TM)$ lifts to $Y_{\alpha}$. Now we use local
coordinates to compute

$\omega_{\mathrm{H}}\left(Y_{\alpha},\frac{\partial}{\partial
p_{k}}\right)=-\frac{\partial X_{\alpha}^{k}}{\partial
q^{j}}\omega^{j\ell}p_{\ell}=\frac{\partial X_{\alpha}^{k}}{\partial
q^{j}}\mu_{\frac{\partial}{\partial p_{j}}}$

and conclude that

$\left[Y_{\alpha}^{\mathrm{Q}},\left(\frac{\partial}{\partial
p_{k}}\right)^{\mathrm{Q}}\right]=\frac{\partial X_{\alpha}^{k}}{\partial
q^{j}}\left(\frac{\partial}{\partial p_{j}}\right)^{\mathrm{Q}},$

so $\mathfrak{heis}_{2n+1}$ is an ideal inside the Lie algebra $\mathfrak{g}$
generated by $\mathfrak{aut}(M)$ and $\mathfrak{heis}_{2n+1}$ and we have
$\mathfrak{g}\cong\mathfrak{aut}(M)\ltimes\mathfrak{heis}_{2n+1}$.
Summarizing, we have obtained the following result.

######  Proposition 3.1.

There exists a subalgebra of $\mathfrak{aut}_{S^{1}}(\bar{N})$ which is
isomorphic to the semidirect product
$\mathfrak{aut}(M)\ltimes\mathfrak{heis}_{2n+1}$.

###  3.3 Global description

We would like to construct a global counterpart of the above construction, by
lifting the action of the group $\operatorname{Aut}(M)\ltimes\mathbb{R}^{2n}$
on $N$ to an action of the semidirect product
$\operatorname{Aut}(M)\ltimes\operatorname{Heis}_{2n+1}$ on $P$ and then
projecting to $\bar{N}$ directly, without passing to the generating vector
fields and having to integrate them as intermediate steps. Here the product in
the Heisenberg group $\operatorname{Heis}_{2n+1}$ is realized on the product
manifold $\mathbb{R}^{2n}\times\mathbb{R}$, where
$\mathbb{R}^{2n}\cong\Gamma_{\nabla}(T^{*}M)$,

$(\alpha_{1},\tau_{1})\cdot(\alpha_{2},\tau_{2})=(\alpha_{1}+\alpha_{2},\tau_{1}+\tau_{2}+\tfrac{1}{2}\omega^{-1}(\alpha_{1},\alpha_{2})),$

where the constant function $\omega^{-1}(\alpha_{1},\alpha_{2})$ is identified
with a number.

Let us consider $h\in\operatorname{Aut}(M)$ and
$(\alpha,\tau)\in\operatorname{Heis}_{2n+1}$. Then we have the following group
action of $\operatorname{Aut}(M)\ltimes\operatorname{Heis}_{2n+1}$ on the
trivial circle bundle $P$ over $N=T^{*}M$:

$\begin{split}h\cdot(\beta,s)&=(h_{*}\beta,s),\\\
(\alpha,\tau)\cdot(\beta,s)&=(\alpha(p)+\beta,s+[\tau+\tfrac{1}{2}\omega^{-1}(\alpha(p),\beta)]),\end{split}$
(7)

where $s\in S^{1}=\mathbb{R}/2\pi\mathbb{Z}$, $\beta\in T_{p}^{*}M$, $p\in M$
and $[r]=r\pmod{2\pi}$.

This action covers the action of $\operatorname{Aut}(M)\ltimes\mathbb{R}^{2n}$
on $N$ by means of the quotient homomorphism
$\operatorname{Heis}_{2n+1}\longrightarrow\mathbb{R}^{2n}\cong\operatorname{Heis}_{2n+1}/\mathbb{R}$.
Moreover, it induces the infinitesimal action on $\bar{N}$ from the previous
section.

######  Proposition 3.2.

The infinitesimal group action on $P$ corresponding to (7) descends to the
action of $\mathfrak{aut}(M)\ltimes\mathfrak{heis}_{2n+1}$ on $\bar{N}$ as
described above.

###### Proof.

Let $V\in\mathfrak{ham}_{S^{1}}(N)\subset\Gamma(TN)$ and $\tilde{V}$ its
$\eta$-horizontal lift to $P$. We define a lift of $V$ to $P$ by

$\hat{V}:=\tilde{V}+\varphi X_{P}\in\Gamma(TP),$

where $\varphi\in\mathcal{C}^{\infty}(P)$ and $X_{P}=\frac{\partial}{\partial
s}$ generates the principal circle action. We require that the lift $\hat{V}$
preserves the connection $\eta$. The condition $\mathcal{L}_{\hat{V}}\eta=0$
then implies that $\varphi=\pi_{N}^{*}f$ for some
$\omega_{\mathrm{H}}$-Hamiltonian function $f\in\mathcal{C}^{\infty}(N)$ for
$V$. Thus $\hat{V}=\tilde{V}+\pi_{N}^{*}fX_{P}$. Such lift automatically
commutes with $X_{P}$. We will omit the pullbacks in the notation from now on.

If we choose $f=\mu_{V}$, the canonical Hamiltonian, we recover the procedure
described in the previous section. Indeed, note that

$\tilde{V}_{\mathrm{H}}-\hat{V}=-\frac{\mu_{V}}{f_{\mathrm{H}}^{c}}Z_{P},$

where $Z_{P}=\tilde{Z}+f_{\mathrm{H}}^{c}X_{P}$. Since
$\mathrm{d}\pi_{\bar{N}}(Z_{P})=0$, by definition of $\bar{N}$, we get
$V_{\mathrm{Q}}=\mathrm{d}\pi_{\bar{N}}(\hat{V})$. This means that we obtain
the same Killing vector field on $\bar{N}$ projecting the lift $\hat{V}$ or
twisting the elementary deformation $V_{\mathrm{H}}$.

Now let us work out explicitly what the infinitesimal lift $\hat{V}$ looks
like. Since $P$ is trivial, we may regard any vector field on $N$ as a vector
field on $P$ which is horizontal with respect to the product structure (or
equivalently with respect to the trivial connection $\mathrm{d}s$, with $s$ a
local coordinate on $S^{1}$). Thus, we may write the $\eta$-horizontal lift
$\tilde{V}$ as

$\tilde{V}=V-\eta(V)X_{P},$

and similarly

$\hat{V}=\tilde{V}+\mu_{V}X_{P}=V-(\eta(V)-\mu_{V})X_{P}.$ (8)

A canonical choice of connection $\eta$ with curvature $\omega_{\mathrm{H}}$
is given by

$\eta=\mathrm{d}s+\frac{1}{2}\iota_{\Xi}\omega_{\mathrm{H}},$

where $\Xi\in\Gamma(TN)$ is the vector field expressed in coordinates by
$\Xi=q^{j}\frac{\partial}{\partial q^{j}}+p_{j}\frac{\partial}{\partial
p_{j}}$, thus

$\eta=\mathrm{d}s+\frac{1}{2}(-\omega_{ij}q^{i}\mathrm{d}q^{j}+\omega^{ij}p_{i}\mathrm{d}p_{j}).$

Recall that a canonically lifted automorphism of $M$ takes the form

$Y=X^{j}\frac{\partial}{\partial p_{j}}-\frac{\partial X^{i}}{\partial
q^{j}}p_{i}\frac{\partial}{\partial p_{j}}\in\mathfrak{ham}_{S^{1}}(N)$

with canonical Hamiltonian function (see Proposition 2.9)

$\mu_{Y}=\frac{1}{2}\left(-\omega_{ij}q^{i}X^{j}+\omega^{jk}p_{i}p_{k}\frac{\partial
X^{i}}{\partial q^{j}}\right).$

Using these expressions, we find
$\eta(Y)=\frac{1}{2}\iota_{Y}\iota_{\Xi}\omega_{\mathrm{H}}=\mu_{Y}$. The
upshot is that $\hat{Y}=Y$ (see (8)), i.e. the lifted action of
$\operatorname{Aut}(M)$ to $P=N\times S^{1}$ is trivial on the $S^{1}$-factor
and it corresponds to the action of $\operatorname{Aut}(M)$ described in the
first equation of (7).

Finally, we consider the lift of the group $\mathbb{R}^{2n}$. For an element
of its Lie algebra $v\in\mathbb{R}^{2n}$ we have the local expression
$v=v_{k}\frac{\partial}{\partial p_{k}}$ and corresponding moment map
$\mu_{v}=-\omega^{kj}v_{k}p_{j}$. This time, we find
$\eta(v)=\frac{1}{2}\mu_{v}$, and consequently
$\hat{v}=v+\frac{1}{2}\mu_{v}X_{P}$. These vector fields do not induce an
action of $\mathbb{R}^{2n}$ since they no longer commute. Indeed, we have

$\displaystyle[\hat{v},\hat{w}]$
$\displaystyle=\frac{1}{2}(v(\mu_{w})-w(\mu_{v}))X_{P}=\frac{1}{2}(\mathrm{d}\mu_{w}(v)-\mathrm{d}\mu_{v}(w))X_{P}$
$\displaystyle=\frac{1}{2}(-(\iota_{w}\omega_{\mathrm{H}})(v)+(\iota_{v}\omega_{\mathrm{H}})(w))=\omega_{\mathrm{H}}(v,w)X_{P},$

or, in local coordinates,
$[\hat{v},\hat{w}]=\omega^{ij}v_{i}w_{j}\frac{\partial}{\partial s}$. Since
$\omega^{ij}$ is constant and $\frac{\partial}{\partial s}$ is central, the
conclusion is that we are now dealing with an infinitesimal action of a one-
dimensional central extension of $\mathbb{R}^{2n}$ whose non-trivial
commutators are given by a symplectic form, i.e. a Heisenberg algebra
$\mathfrak{heis}_{2n+1}$. Integrating, we obtain the action of
$\operatorname{Heis}_{2n+1}$ described in the second line of (7). ∎

It is possible to describe the quaternionic Kähler manifold $\bar{N}$ as a
submanifold of the circle bundle $P$. For that we define the following tensor
fields on $P$222We are using a convention different from [ACDM15]. The
dictionary is given by $Z=-\frac{1}{2}Z^{\mathrm{ACDM}}$,
$f_{Z}^{c}=-\frac{1}{2}f^{\mathrm{ACDM}}$ and
$f_{\mathrm{H}}^{c}=-\frac{1}{2}f_{1}^{\mathrm{ACDM}}$.:

$g_{P}:=-\frac{1}{f_{\mathrm{H}}^{c}}\eta^{2}+\pi_{N}^{*}g_{N},\quad\theta_{0}^{P}:=\mathrm{d}f_{Z}^{c},\quad\theta_{1}^{P}:=\eta-\iota_{Z}g_{N},\quad\theta_{2}^{P}:=-\iota_{Z}\omega_{3},\quad\theta_{3}^{P}:=\iota_{Z}\omega_{2}.$

With them, define the $Z_{P}$-invariant tensor field

$\tilde{g}_{P}:=g_{P}+\frac{1}{f_{Z}^{c}}\sum_{j=0}^{3}(\theta_{j}^{P})^{2},$

where recall that $Z_{P}=\tilde{Z}+f_{\mathrm{H}}^{c}X_{P}$, and consider

$g_{\bar{N}}^{c}:=\frac{1}{4\lvert f_{Z}^{c}\rvert}\tilde{g}_{P}|_{\bar{N}},$

where

$\bar{N}:=\\{\arg(X^{0})=0\\}\subset P=N\times S^{1}$ (9)

is a codimension one submanifold of $P$ which is transversal to the vector
field $Z_{P}$ and $(X^{0},\ldots,X^{n-1})$ are special holomorphic coordinates
of the CASK manifold $M$. Then, by [ACDM15, Theorems 2 and 5],
$(\bar{N},g_{\bar{N}}^{c})$ is precisely the one-loop deformed c-map space.

To state Theorem 3.6, we have to focus on a particular class of CASK
manifolds, namely those coming from projective special real manifolds
$\mathcal{H}$. We now briefly describe them. We follow the notation and
conventions of [CT22]. For more details, see for instance [CHM12, CDJL21].

######  Definition 3.3.

A _projective special real (PSR) manifold_ is a Riemannian manifold
$(\mathcal{H},g_{\mathcal{H}})$ such that $\mathcal{H}\subset\mathbb{R}^{n-1}$
is a hypersurface and there is a homogeneous cubic polynomial
$h:\mathbb{R}^{n-1}\longrightarrow\mathbb{R}$ satisfying:

  * $\bullet$

$\mathcal{H}\subset\\{t\in\mathbb{R}^{n-1}\mid h(t)=1\\}$.

  * $\bullet$

$g_{\mathcal{H}}=-\partial^{2}h|_{T\mathcal{H}\times T\mathcal{H}}$.

We denote the coordinates of $\mathbb{R}^{n-1}$ by $t^{a}$. In particular we
write the cubic polynomial $h$ as

$h(t^{a})=\frac{1}{6}k_{abc}t^{a}t^{b}t^{c},$

where the coefficients $k_{acb}\in\mathbb{R}$ are symmetric in the indices.

Let $(\mathcal{H},g_{\mathcal{H}})$ be a PSR manifold defined by the real
cubic polynomial $h$, and let
$U:=\mathbb{R}_{>0}\cdot\mathcal{H}\subset\mathbb{R}^{n-1}\setminus\\{0\\}$.
We define $\bar{M}:=\mathbb{R}^{n-1}+iU\subset\mathbb{C}^{n-1}$ with the
canonical holomorphic structure, where the global holomorphic coordinates are
given by $z^{a}:=b^{a}+it^{a}\in\mathbb{R}^{n-1}+iU$. On $\bar{M}$ we consider
the metric

$\bar{g}=\frac{\partial^{2}\mathcal{K}}{\partial
z^{a}\partial\bar{z}^{b}}\mathrm{d}z^{a}\mathrm{d}\bar{z}^{b},$

where $\mathcal{K}:=-\log K(t)$ and
$K(t):=8h(t)=\frac{4}{3}k_{abc}t^{a}t^{b}t^{c}$. Then it can be shown that
$(\bar{M},\bar{g})$ is a PSK manifold [CHM12]. In fact, one can find the
following explicit expression for a PSK metric coming from a PSR manifold:

$\bar{g}=-\frac{1}{4}\frac{\partial^{2}\log h(t)}{\partial t^{a}\partial
t^{b}}(\mathrm{d}b^{a}\mathrm{d}b^{b}+\mathrm{d}t^{a}\mathrm{d}t^{b})=\left(-\frac{k_{abc}t^{c}}{4h(t)}+\frac{k_{acd}k_{bef}t^{c}t^{d}t^{e}t^{f}}{(4h(t))^{2}}\right)(\mathrm{d}b^{a}\mathrm{d}b^{b}+\mathrm{d}t^{a}\mathrm{d}t^{b}).$

######  Definition 3.4.

The construction of the PSK manifold $(\bar{M},\bar{g})$ from the PSR manifold
$(\mathcal{H},g_{\mathcal{H}})$ as explained above is known as the
_supergravity r-map_.

Given a PSK manifold $\bar{M}$ in the image of the supergravity r-map, its
corresponding CASK manifold is defined via the CASK domain $(M,\mathfrak{F})$,
where $M\subset\mathbb{C}^{n}$ is given by

$M:=\\{(X^{0},\ldots,X^{n-1})=X^{0}\cdot(1,z)\in\mathbb{C}^{n}\mid
X^{0}\in\mathbb{C}^{*},z\in\bar{M}\\}$

and

$\mathfrak{F}(X)=-\frac{h(X^{1},\ldots,X^{n-1})}{X^{0}}=-\frac{1}{6}k_{abc}\frac{X^{a}X^{b}X^{c}}{X^{0}}.$

We know recall what is a CASK domain and how to obtain a CASK manifold from
it.

######  Definition 3.5.

A _CASK domain_ is a pair $(M,\mathfrak{F})$ such that:

  * $\bullet$

$M\subset\mathbb{C}^{n}\setminus\\{0\\}$ is a $\mathbb{C}^{*}$-invariant
domain (with respect to the usual $\mathbb{C}^{*}$-action on
$\mathbb{C}^{n}\setminus\\{0\\}$ by multiplication).

  * $\bullet$

$\mathfrak{F}:M\longrightarrow\mathbb{C}$ is a holomorphic function,
homogeneous of degree 2 with respect to the $\mathbb{C}^{*}$-action, called
the _holomorphic prepotential_.

  * $\bullet$

With respect to the natural holomorphic coordinates $(X^{0},\ldots,X^{n-1})$
of $M$, the matrix

$(\tau_{ij}):=\left(\frac{\partial^{2}\mathfrak{F}}{\partial X^{i}\partial
X^{j}}\right)$

satisfies that $\operatorname{Im}(\tau_{ij})$ has signature $(n-1,1)$ and
$\operatorname{Im}(\tau_{ij})X^{i}\bar{X}^{j}<0$ for $X\in M$.

From this data, we obtain a CASK manifold $(M,g,\omega,\nabla,\xi)$ by

$g=\operatorname{Im}(\tau_{ij})\mathrm{d}X^{i}\mathrm{d}\bar{X}^{j},\quad\omega=\frac{i}{2}\operatorname{Im}(\tau_{ij})\mathrm{d}X^{i}\wedge\mathrm{d}\bar{X}^{j},\quad\xi=X^{i}\frac{\partial}{\partial
X^{i}}+\bar{X}^{i}\frac{\partial}{\partial\bar{X}^{i}}.$

The flat connection $\nabla$ is defined such that

$\mathrm{d}x^{i}:=\operatorname{Re}(\mathrm{d}X^{i})\quad\text{and}\quad\mathrm{d}y_{i}:=-\operatorname{Re}(\mathrm{d}(\frac{\partial\mathfrak{F}}{\partial
X^{i}}))$ (10)

is a flat frame of $T^{*}M$.

Note that for a CASK manifold $M$ determined by $\mathcal{H}$ as explained
above, the functions $x^{i}:=\operatorname{Re}(X^{i})$ and
$y_{i}:=-\operatorname{Re}(\frac{\partial\mathfrak{F}}{\partial X^{i}})$ are
globally defined and their differentials (10) give rise to a parallel frame of
$T^{*}M$, hence $\mathrm{Hol}(\nabla)$ is trivial and therefore the action of
$\mathbb{R}^{2n}$ given by Proposition 2.11 is global.

In this situation, where the CASK manifold $M$ is determined by a PSR manifold
$\mathcal{H}$, a group of isometries preserving the CASK structure was
described in [CDJL21, Appendix A]. More precisely, the group we are
considering is

$\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1}):=(\mathbb{R}_{>0}\times\operatorname{Aut}(\mathcal{H}))\ltimes\mathbb{R}^{n-1}\hookrightarrow\operatorname{Aut}(M)\subset\mathrm{Sp}(\mathbb{R}^{2n}),$

where

$\operatorname{Aut}(\mathcal{H}):=\\{A\in\mathrm{GL}(n-1,\mathbb{R})\mid
A\mathcal{H}=\mathcal{H}\\}$

and the arrow is a certain embedding [CDJL21, Proposition 23].

######  Theorem 3.6.

Let $M$ be a CASK manifold determined by the PSR manifold $\mathcal{H}$ as
explained above. Then the group
$\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1})\ltimes(\operatorname{Heis}_{2n+1}/\mathcal{F})$,
where $\mathcal{F}$ is an infinite cyclic subgroup of the Heisenberg center,
acts effectively and isometrically on $(\bar{N},g_{\bar{N}}^{c})$ for $c\geq
0$.

###### Proof.

From the explicit description of the action of
$\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1})$ on $M$ given in [CDJL21,
Appendix A] we see that the function $X^{0}\in\mathcal{C}^{\infty}(P)$ changes
under this action only by a real positive factor. It follows that the
submanifold $\bar{N}\subset P$ given by (9) is invariant under the action of
$\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1})\ltimes\operatorname{Heis}_{2n+1}$,
since the action of $\operatorname{Heis}_{2n+1}$ preserves the fibers of
$\pi\circ\pi_{N}:P\longrightarrow M$, where $\pi:N=T^{*}M\longrightarrow M$
and $\pi_{N}:P\longrightarrow N$. Hence the group action on $P$ described in
(7) restricts to $\bar{N}$. From the explicit description, it is also easy to
check that the action of $\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1})$ is
effective on $\bar{N}\subset P$.

Finally, the action of $\operatorname{Heis}_{2n+1}$ is not quite effective,
since its center acts by translations along the $S^{1}$-factor of $P$. To
obtain an effective action we need to divide out the infinite cyclic subgroup
of the center, whose elements correspond to shifting $s\in
S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ by $2\pi k$, $k\in\mathbb{Z}$.

The group
$\mathrm{Aff}_{\mathcal{H}}(\mathbb{R}^{n-1})\ltimes(\operatorname{Heis}_{2n+1}/\mathcal{F})$
acts by isometries on $\bar{N}$ by construction. ∎

######  Remark 3.7.

In [CRT21, Theorem 3.16] a similar result as in Theorem 3.6 is obtained in the
case of the CASK manifold $M$ determined by the PSK manifold
$\bar{M}=\operatorname{\mathbb{C}\mathrm{H}}^{n-1}$ with the transitive action
of $\operatorname{Aut}(M)=\mathrm{SU}(1,n-1)$. This example does not belong to
the above series of spaces determined by a PSR manifold $\mathcal{H}$ since
$\operatorname{\mathbb{C}\mathrm{H}}^{n-1}$ is not in the image of the
supergravity r-map. They obtain that
$\mathrm{SU}(1,n-1)\ltimes(\operatorname{Heis}_{2n+1}/\mathcal{F}^{\prime})$
acts effectively and isometrically on
$(\bar{N}^{\prime}/\mathcal{F}^{\prime},g_{\bar{N}}^{c})$, where
$\bar{N}^{\prime}\cong\mathbb{R}^{4n}$ is the universal covering of our
$\bar{N}$, and $\mathcal{F}^{\prime}$ is trivial if $c=0$ and infinite cyclic
for $c>0$. In the latter case $\bar{N}^{\prime}/\mathcal{F}^{\prime}=\bar{N}$.

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