# Implosion, Contraction and Moore-Tachikawa

Andrew Dancer  Andrew Dancer
Jesus College
Oxford OX1 3DW, United Kingdom<EMAIL_ADDRESS>,  Frances Kirwan
Frances Kirwan
New College
Oxford OX1 3BN, United Kingdom<EMAIL_ADDRESS>and  Johan Martens
Johan Martens
School of Mathematics and Maxwell Institute,University of Edinburgh,
James Clerk Maxwell Building, Edinburgh EH9 3FD, United Kingdom
<EMAIL_ADDRESS>_Dedicated to Oscar Garcia-Prada on the occasion of his
60th birthday._

###### Abstract.

We give a survey of the implosion construction, extending some of its aspects
relating to hypertoric geometry from type $A$ to a general reductive group,
and interpret it in the context of the Moore-Tachikawa category. We use these
ideas to discuss how the contraction construction in symplectic geometry can
be generalised to the hyperkähler or complex symplectic situation.

###### Key words and phrases:

Symplectic implosion, contraction, hyperkähler geometry

###### 2000 Mathematics Subject Classification:

53C26, 53D20, 14L24

##  1\. Introduction

The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar [18]
is an Abelianisation construction for Hamiltonian group actions. Given a
symplectic space $M$ with such an action of a compact Lie group $K$, the
implosion $M_{\textup{impl}}$ carries an action of the maximal torus $T$ of
$K$, such that the symplectic reduction of the implosion $M_{\textup{impl}}$
by $T$ at a level in the closed positive Weyl chamber is the same as the
reduction of $M$ by $K$.

If $M$ actually arises as a Kähler variety, projective over an affine,
equipped with a linearised action of $K_{{\mathbb{C}}}$ and compatible Kähler
structure, $M_{\textup{impl}}$ can also be understood as the non-reductive
geometric invariant theory quotient of $M$ by $U$, a maximal unipotent
subgroup of $G=K_{{\mathbb{C}}}$ [24].

Implosion was generalised to the hyperkähler situation in [9] in the
$\operatorname{\textsl{SU}}(n)$ case. In the general case, starting from an
arbitrary compact connected Lie group, work of Ginzburg and Riche [17] shows
we have at least a complex symplectic version of the implosion (though the
hyperkähler structure has not been shown yet, it is expected to exist, and we
will refer to it as such). The hyperkähler version takes inspiration from both
the (real) symplectic and algebro-geometric version, but is overall a step up
in complexity. This is parallel to other operations in equivariant geometry
that come in three flavours, such as symplectic reduction [27], geometric
invariant theory [29], and hyperkähler reduction [21], or (for abelian
actions) symplectic and algebraic cutting [26, 13] and hyperkähler
modifications [11].

In this note we expand this viewpoint by introducing the complex symplectic
version of contraction, which so far was understood only in an algebraic [30]
or real symplectic [20] context (see §5 for a survey of these constructions).
Unlike reduction or implosion, contraction does not change the dimension of
the space one starts from, but gives rise to an additional action of the
maximal torus, commuting with the actions of $K$ or $G$.

It is common to all these operations that they can be reduced to a universal
version, which is the result of applying them to the cotangent bundle $T^{*}K$
(in the real symplectic case), the complex group $G$ (in the algebraic case),
or the complex cotangent bundle $T^{*}G$ (in the complex symplectic case). The
resulting spaces are often familiar – the symplectic reductions of $T^{*}K$
are coadjoint $K$ orbits (or flag varieties for $G$), for example. For abelian
$K$ the symplectic cuts of $T^{*}K$, or algebraic cuts of $G$, give toric
varieties, and the hyperkähler modifications of $T^{*}G$ are the hypertoric
varieties.

This in turn means that they naturally fit in with the concept of Moore-
Tachikawa categories, whose objects are groups, morphisms are spaces with
group actions, and composition is done by reduction of products. Indeed, the
universal implosions are morphisms in this category between a group and its
maximal torus, and the universal contraction is the composition of this
morphism with itself, suitably interpreted.

##  2\. Hyperkähler implosion

In this section we survey symplectic and hyperkähler implosion.

Let $T$ be a maximal torus for a compact Lie group $K$, with Lie algebra
$\operatorname{\mathfrak{t}}\subset\operatorname{\mathfrak{k}}$. By using the
weight decomposition for the adjoint representation,
$\operatorname{\mathfrak{t}}$ has a canonical complement, and hence we have
${\operatorname{\mathfrak{t}}^{*}\subset\operatorname{\mathfrak{k}}^{*}}$, and
also
$\operatorname{\mathfrak{t}}^{*}_{+}\subset\operatorname{\mathfrak{k}}^{*}$
for the choice of a positive Weyl chamber
$\operatorname{\mathfrak{t}}^{*}_{+}$. If $M$ is a (real) Hamiltonian
$K$-manifold, with moment map
$\mu:M\rightarrow\operatorname{\mathfrak{k}}^{*}$, the symplectic implosion
$M_{\textup{impl}}$ is (as a set)
$\mu^{-1}(\operatorname{\mathfrak{t}}^{*}_{+})/\thicksim$, where

$x\sim y\Leftrightarrow\mu(x)=\mu(y)\ \text{and}\
y\in[K_{\mu(x)},K_{\mu(x)}].x.$

It is typically not a smooth manifold, but a stratified Hamiltonian space –
the residual map to
$\operatorname{\mathfrak{t}}^{*}_{+}\subset\operatorname{\mathfrak{t}}^{*}$ is
the moment map for a $T$-action. Applied to $T^{*}K$ (as a $K$-space with the
action induced from $K$ acting on itself as $k.h=hk^{-1}$), this operation
gives the universal implosion

(2.1)
$(T^{*}K)_{\textup{impl}}=K\times\operatorname{\mathfrak{t}}^{*}_{+}/\thicksim.$

Besides the $T$-action, $(T^{*}K)_{\textup{impl}}$ still has the residual
$K$-action that comes from multiplication on the left, since if a space has
two commuting actions by compact groups $K$ and $L$, the implosion by $K$
still has an action of $L$, and moreover implosion and symplectic reduction
commute:

(2.2) $M_{K-\textup{impl}}{\sslash}L\cong(M{\sslash}L)_{K-\textup{impl}}.$

This last property also explains the relevance of the universal implosion: for
any symplectic $K$-manifold, one has a canonical equivariant isomorphism

(2.3) $M\cong\left(M\times T^{*}K\right){\sslash}\Delta K,$

where the quotient in the right-hand side uses the diagonal action of $K$
acting on $M$ and on $T^{*}K$ by multiplication from the right – this quotient
space still inherits a $K$-action coming from the left action of $K$ on
$T^{*}K$.

Also for GIT quotients and complex symplectic reduction, one has an analogue
of (2.3), with the role of $T^{*}K$ now played by respectively
$G=K_{\mathbb{C}}$ or $T^{*}G$ (for simplicity of notation we shall indicate
both symplectic reduction and GIT quotients by ${\sslash}$, and hyperkähler
and complex symplectic reduction by ${\sslash\mkern-6.0mu/}$). Remark that
this is not quite true for the metric aspects in Kähler or hyperkähler
quotients: there is no known metric on $G$ or $T^{*}G$ that would make the
analogues of (2.3) an isometry.

Because of the commutativity of implosion and reduction, as in (2.2), when
applying the former to both sides of (2.3), we obtain

$M_{\textup{impl}}\cong(M\times(T^{*}K)_{\textup{impl}}){\sslash}\Delta K,$

which shows how to recover any implosion from the universal implosion.

The universal symplectic implosion $(T^{*}K)_{\textup{impl}}$ can be
identified (as a stratified $K\times T$ space) with
$G{\sslash}U=\overline{G/U}^{\operatorname{Aff}}=\operatorname{Spec}\left(k[G]^{U}\right)$,
for $U$ the unipotent radical of the Borel subgroup of $G$ determined by the
maximal torus $T_{{\mathbb{C}}}$ and the positive Weyl chamber
$\operatorname{\mathfrak{t}}_{+}^{*}$. Even though $U$ is not a reductive
group, the invariant ring $k[G]^{U}$ is still finitely generated, which makes
this definition sensible.

This is done as follows (see Section 6 in [18] and Appendix in [20]): given
the choice of a finite set of generators $\Pi$ for the semigroup of dominant
weights of $G$, one can embed

(2.4) $G{\sslash}U\hookrightarrow E=\oplus_{\varpi\in\Pi}V_{\varpi}.$

Here $V_{\varpi}$ is the irreducible representation of $G$ with highest weight
$\varpi$, and we moreover choose highest-weight vectors $v_{\varpi}\in
V_{\varpi}$ and $K$-invariant Hermitian inner products on the $V_{\varpi}$
such that $\lVert v_{\varpi}\rVert=1$. The image of $G{\sslash}U$ is now the
closure of the $G$-orbit of $\sum_{\varpi\in\Pi}v_{\varpi}$, and $G{\sslash}U$
inherits a Kähler structure from the embedding in $E$.

We can also consider an extra action of $T_{{\mathbb{C}}}$ on $E$ (commuting
with the $G$ action), acting diagonally on $V_{\varpi}$ with weight $-\varpi$,
and look at the affine toric variety inside $G{\sslash}U$ that is the closure
of the $T_{{\mathbb{C}}}$-orbit of $\sum_{\varpi\in\Pi}v_{\varpi}$. The image
of the $T$-moment map of this toric variety is
$-\operatorname{\mathfrak{t}}^{*}_{+}$, and the non-negative part of the toric
variety provides a section of this moment map. We can now use minus this
section to create a subjective map from
$K\times\operatorname{\mathfrak{t}}_{+}^{*}$ into $G{\sslash}U\subset E$,
which (using (2.1)) induces an isomorphism of symplectic stratified spaces
$(T^{*}K)_{\textup{impl}}$.

This identification is the basis for the correspondence between symplectic
reduction and quotients in non-reductive GIT, though this only concerns those
quotients on the algebraic side where the action of the group $U$ extends to
one of $G$, in which case the invariant ring will be finitely generated.

When it comes to a hyperkähler or complex symplectic analogue of implosion, as
it too commutes with complex symplectic reduction, the key case is again the
implosion of $T^{*}G$. We recall that this space is complex symplectic with
$G\times G$-action, and in fact, by the work of Kronheimer [25], is even known
to be hyperkähler. One way to view its implosion is as the complex symplectic
reduction of $T^{*}G$ by $U$, acting from the right. If we identify
$T^{*}G\cong G\times\operatorname{\mathfrak{g}}^{*}$, then the complex moment
map for the $U$-action is given by the projection of the second factor on
$\operatorname{\mathfrak{u}}^{*}$, where $\operatorname{\mathfrak{u}}$ is the
Lie algebra of $U$. The level set is the product affine variety
$G\times\operatorname{\mathfrak{u}}^{\circ}$ (where
$\operatorname{\mathfrak{u}}^{\circ}\subset\operatorname{\mathfrak{g}}^{*}$ is
the annihilator of $\operatorname{\mathfrak{u}}$), which has a $G\times
U$-action, but not a $G\times G$-action, and as a result it is not directly
clear that the invariant ring
$k[G\times\operatorname{\mathfrak{u}}^{\circ}]^{U}$ is finitely generated.
Alternatively, remark that if
$(G\times\operatorname{\mathfrak{u}}^{\circ}){\sslash}U=\operatorname{Spec}\left(k[G\times\operatorname{\mathfrak{u}}^{\circ}]^{U}\right)$
were well-defined as an affine variety, it would contain the cotangent space
$T^{*}(G/U)$, and hence the question is whether $T^{*}(G/U)$ is a quasi-affine
variety.

For $K=\operatorname{\textsl{SU}}(n)$, the question of finite generation was
answered affirmatively by Dancer, Kirwan and Swann in [9], where it was shown
that the resulting variety could also be constructed through quivers as a
hyperkähler quotient of a vector space by a compact group (which in particular
showed that the space is not just complex symplectic, but in fact hyperkähler
on its smooth locus). In particular, the following quiver diagram is
considered:

${{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}0}}$${{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}0}}$${\dots}$${{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color<EMAIL_ADDRESS>
(We follow the presentation of the quiver given in [39] here, including the
vacuous light-coloured part, to indicate this can be understood as a special
case of a framed and doubled quiver, as in the context of Nakajima quiver
varieties.) One can take the affine hyperkähler quotient (at zero for all
moment maps) of

$M=\\{(\alpha_{i},\beta_{i})\\}=\bigoplus_{i=1}^{r-1}\operatorname{Hom}({\mathbb{C}}^{i},{\mathbb{C}}^{i+1})\oplus\operatorname{Hom}({\mathbb{C}}^{i+1},{\mathbb{C}}^{i})$

by the natural action of
$\widetilde{H}=\prod_{i=1}^{n-1}\operatorname{\textsl{U}}(i)$, to obtain the
nilpotent cone inside
$\operatorname{\mathfrak{sl}}(n)\cong\operatorname{\mathfrak{sl}}(n)^{*}$.
When one takes the hyperkähler quotient by
$H=\prod_{i=1}^{r-1}\operatorname{\textsl{SU}}(i)$, however, one obtains an
affine variety containing $T^{*}(\operatorname{\textsl{SL}}(n)/U)$ as a dense
open subvariety of codimension at least two (Theorem 7.18 in [9]). As a
result,
$k[\operatorname{\textsl{SL}}(n)\times\operatorname{\mathfrak{u}}^{\circ}]^{U}$
is finitely generated, and

$M{\sslash\mkern-6.0mu/}H=\operatorname{Spec}\left(k[\operatorname{\textsl{SL}}(n)\times\operatorname{\mathfrak{u}}^{\circ}]^{U}\right)=\left(\operatorname{\textsl{SL}}(n)\times\operatorname{\mathfrak{u}}^{\circ}\right){\sslash}U,$

where the quotient is taken in the sense of non-reductive GIT.

The finite generation of $k[G\times\operatorname{\mathfrak{u}}^{\circ}]^{U}$
for general reductive $G$ was established by Ginzburg and Riche (Lemma 3.6.2
in [17]), and further studied by Ginzburg and Kazhdan in [15, 16]. We shall
refer to the resulting affine space, as a singular complex symplectic variety,
as the (right) universal implosion
$Q_{G}=(G\times\operatorname{\mathfrak{u}}^{\circ}){\sslash}U$. Remark that
$Q_{G}$ is always normal, following Proposition 1 in [36].

A key ingredient in the work of Ginzburg-Riche and Ginzburg-Kazhdan is an
action of the Weyl group $W_{G}$ of $G$ on $Q_{G}$. In the real symplectic or
algebraic case, we have an action of $K\times T$ (or its complexification) on
$T^{*}K_{\textup{impl}}\cong G{\sslash}U$, but in the complex symplectic case
$Q_{G}$ has an action of $G\times(W_{G}\ltimes T_{{\mathbb{C}}})$. From the
point of view of the quiver-construction in type $A_{n}$ of [9], this action
of $W_{G}$ was constructed recently by Wang in [39]. Ginzburg-Kazhdan also
conjectured that $Q_{G}$ has symplectic singularities (in the sense of [1]).
This was proved for type $A_{n}$ using the quiver construction by Jia [22,
23], and in the general case by Gannon [14].

Finally, Ginzburg and Kazhdan establish an affine embedding of $Q_{G}$
(Corollary 7.2.3 in [15], which follows from the proof of Lemma 3.6.2 in
[17]):

######  Proposition 2.1.

There is a $W_{G}$-equivariant closed embedding of affine varieties

(2.5)
$Q_{G}\rightarrow\left(\operatorname{\mathfrak{g}}^{*}\oplus\operatorname{\mathfrak{t}}_{\mathbb{C}}^{*}\right)\times\prod_{w\in
W_{G}}G{\sslash}U.$

Here the component going to
$\operatorname{\mathfrak{g}}^{*}\oplus\operatorname{\mathfrak{t}}_{{\mathbb{C}}}^{*}$
is the moment map for the $G\times T_{{\mathbb{C}}}$-action, and the component
going to each copy of $G{\sslash}U$ is given by the composition of the action
of $w\in W_{G}$ on $Q_{G}$, and the map ${Q_{G}\rightarrow G{\sslash}U}$ that
is the affinization of the natural projection $T^{*}(G/U)\rightarrow G/U$. The
$W_{G}$-action on the target is the coadjoint action on
$\operatorname{\mathfrak{t}}_{\mathbb{C}}$, and the permutation of the factors
of $\prod_{w\in W_{G}}G{\sslash}U$, while $\operatorname{\mathfrak{g}}$ is
left invariant. One can combine (2.5) with (2.4) for each $G{\sslash}U$ factor
to obtain an embedding into affine space that provides the counterpart to
$(\ref{G/Uembed})$

##  3\. A hypertoric variety related to $Q_{G}$

Inside each universal symplectic implosion $(T^{*}K)_{\textup{impl}}\cong
G{\sslash}U$ one can embed an affine toric variety $X_{G}$, preserved by the
action of $T_{{\mathbb{C}}}$, whose image under the $T$-moment map is
$\operatorname{\mathfrak{t}}^{*}_{+}$, and whose $K$-sweep is all of
$(T^{*}K)_{\textup{impl}}$, see $[\S 7]$ of [18] and the appendix to [20]. In
[10], an analogue of this was shown for the universal hyperkähler implosion
for $\operatorname{\textsl{SL}}(n)$: there exists a morphism of the affine
hypertoric variety $Y_{G}$, associated with the hyperplane arrangement given
by the Weyl chambers for $G$, to $Q_{G}$, which is generically an embedding.
We generalise this here to the case of arbitrary reductive groups.

We briefly recall the construction of hypertoric varieties, following [5, 32].
For our purpose, it suffices to restrict to affine ones. These are determined
by an arrangement $\mathcal{A}$ of $N$ distinct hyperplanes
$H_{1},\dots,H_{N}$ in $\operatorname{\mathfrak{t}}^{*}$, all containing the
origin. For each hyperplane, make a choice of a minimal normal vector
$\alpha_{i}\in\operatorname{Hom}(\mathbb{G}_{m},T_{{\mathbb{C}}})\subset\operatorname{\mathfrak{t}}$.
These choices determine a group homomorphism $\mathbb{G}_{m}^{N}\rightarrow
T_{\mathbb{C}}$, which we require to be surjective. It has kernel
$L\triangleleft\mathbb{G}_{m}^{N}$, and if we let $\mathbb{G}_{m}^{N}$ act
diagonally on ${\mathbb{C}}^{N}$, and via co-tangent lift on
$T^{*}{\mathbb{C}}^{N}\cong{\mathbb{C}}^{2N}$, then the hypertoric variety is
constructed as the complex symplectic reduction of ${\mathbb{C}}^{2N}$ by $L$:

$\mathfrak{M}(\mathcal{A})={\mathbb{C}}^{2N}{\sslash\mkern-6.0mu/}L.$

Up to isomorphism this is independent of the choices of the $\alpha_{i}$. We
will denote elements of ${\mathbb{C}}^{2N}$ as $N$-tuples of pairs
$(a_{i},b_{i})$, and will denote the equivalence class of such tuples
representing elements in $\mathfrak{M}(\mathcal{A})$ as
$[\underline{a},\underline{b}]$.

In the case of $\mathfrak{M}(\mathcal{A})=Y_{G}$, $N$ is half the number of
roots of the associated root system of $G$. It will be most convenient in this
case if the choice of the normal vectors $\alpha_{i}$ is induced by the choice
of a set of simple positive roots, such that the positive Weyl chamber
$\operatorname{\mathfrak{t}}^{*}_{+}$ is cut out by the equations
$\alpha_{i}(x)\geq 0,\forall i$. As could be expected by the symmetry of the
hyperplane arrangement for $Y_{G}$, this hypertoric variety also has an action
of $W_{G}\ltimes T_{{\mathbb{C}}}$ (which the toric variety $X_{G}$ does not
have). This action plays a key role in the construction of the desired map
$Y_{G}\rightarrow Q_{G}$, even though, surprisingly, this map will not be
$W_{G}$-equivariant.

######  Lemma 3.1.

The $T_{\mathbb{C}}$-action on the affine hypertoric variety $Y_{G}$ naturally
extends to a $W_{G}\ltimes T_{\mathbb{C}}$-action (where $W_{G}$ is the Weyl
group of $G$) such that all moment maps for the action of $T$ and
$T_{\mathbb{C}}$ on $Y_{G}$ are $W_{G}$-equivariant.

This fact, and the proof below, are valid for any affine hypertoric variety
determined by a hyperplane arrangement acted on by a finite group.

###### Proof.

The action of the Weyl group on $\operatorname{\mathfrak{t}}^{*}$ permutes the
hyperplanes in the arrangement, so we get a homomorphism
$\sigma:W_{G}\rightarrow S_{N}$. Moreover, for each $w\in W_{G}$ and
$i\in\\{1,\dots,N\\}$ we get a sign $s_{w,i}\in\\{\pm 1\\}$ (a cocycle for the
permutation action), determined by
$w(\alpha_{i})=s_{w,i}\alpha_{\sigma(w)(i)}$. We obtain an action $\phi$ of
$W_{G}$ on $\mathbb{G}_{m}^{N}$, given by

$\phi(w)(t_{1},\dots,t_{N})=(t_{\sigma(w)(1)}^{s_{w,1}},\dots,t_{\sigma(w)(N)}^{s_{w,N}}),$

which makes the morphism $\mathbb{G}_{m}^{N}\rightarrow T_{\mathbb{C}}$
(determined by the $\alpha_{i}$) $W_{G}$-equivariant. As a result, $\phi$
preserves $L$.

Finally, we can extend the action of $\mathbb{G}_{m}^{N}$ on
${\mathbb{C}}^{2N}$ to an action of $W_{G}\ltimes\mathbb{G}_{m}^{N}$ (note
that this does not work for the action of $\mathbb{G}_{m}^{N}$ on
${\mathbb{C}}^{N}$). Denoting elements of ${\mathbb{C}}^{2N}$ as $N$-tuples of
pairs $(a_{i},b_{i})\in{\mathbb{C}}^{2}$, then each $w\in W_{G}$ permutes the
pairs via $\sigma$, and moreover switches
$(a_{\sigma(w)(i)},b_{\sigma(w)(i)})$ to
$(b_{\sigma(w)(i)},a_{\sigma(w)(i)}))$ if $s_{w,i}=-1$.

We immediately get that, after taking the complex symplectic reduction of
${\mathbb{C}}^{2N}$ by $L$ (at zero-level of the moment maps), the action of
$W_{G}\ltimes\mathbb{G}_{m}^{N}$ descends to an action of
$\left(W_{G}\ltimes\mathbb{G}_{m}^{N}\right)\Big{/}L\cong W_{G}\ltimes
T_{\mathbb{C}}$ on the hypertoric variety. Moreover, all the moment maps for
the action of $T$ on $Y_{G}$ are $W_{G}$-equivariant. ∎

We now want to define a morphism from $Y_{G}$ into the target of (2.5), whose
image is contained in the image of $Q_{G}$. The components of this map into
both $\operatorname{\mathfrak{g}}^{*}$ and
$\operatorname{\mathfrak{t}}_{{\mathbb{C}}}^{*}$ are just given by the moment
map for the action of $T_{{\mathbb{C}}}$ on $Y_{G}$ – remark that such a map
from $Y_{G}$ into $\operatorname{\mathfrak{g}}^{*}$ is not $W_{G}$-invariant,
whereas the component into $\operatorname{\mathfrak{g}}^{*}$ of (2.5) is
$W_{G}$-invariant (this at the end will result in the map $Y_{G}\rightarrow
Q_{G}$ not being $W_{G}$-equivariant).

The components of the map into the various copies of $G{\sslash}U$ will land
in $X_{G}\subset G{\sslash}U$ for each $w$. If $Y_{G}$ were to be a smooth
hypertoric variety, it would be covered by cotangent bundles to the various
components of its extended core (see [32], Remark 2.1.6). These components are
Lagrangian subvarieties, each isomorphic to $X_{G}$, and permuted by the
action of $W_{G}$ on $Y_{G}$. Under the real moment map their images are given
by the (closed) Weyl chambers in $\operatorname{\mathfrak{t}}^{*}$. The
various projections $T^{*}X_{G}\rightarrow X_{G}$ would extend to give maps
$Y_{G}\rightarrow X_{G}\subset G{\sslash}U$, and these would give the
remaining components of the morphism from $Y_{G}$ into the target of (2.5).

However, the variety $Y_{G}$ will be singular in general, and therefore this
argument needs a bit more care. We can use a smooth stratification of affine
hypertoric varieties introduced in [31], §2. Given a hyperplane arrangement
$\mathcal{A}$, and a subset $F\subset\\{1,\dots,N\\}$, the affine subspace
$\cap_{i\in F}H_{i}$ is denoted $H_{F}$. If $F=\\{i|H_{i}\subset H_{F}\\}$,
$F$ is called a _flat_. For every flat $F$, $\mathcal{A}^{F}$ is defined to be
the hyperplane arrangement $\\{H_{i}\cap H_{F}|i\notin F\\}$ in the affine
space $H_{F}$. Similarly, $\mathcal{A}_{F}$ is defined to be the hyperplane
arrangement $\\{H_{i}/H_{F}|i\in F\\}$ in
$\operatorname{\mathfrak{t}}^{*}/H_{F}$. We have that the
$\mathfrak{M}(\mathcal{A}^{F})$ are all symplectic subvarieties of
$\mathfrak{M}(\mathcal{A}^{\emptyset})=\mathfrak{M}(\mathcal{A})$. Finally,
Proudfoot and Webster define $\overset{\circ}{\mathfrak{M}}(\mathcal{A}^{F})$
to be

$\overset{\circ}{\mathfrak{M}}(\mathcal{A}^{F})=\Big{\\{}[\underline{a},\underline{b}]\in\mathfrak{M}(\mathcal{A})\
\Big{|}\ a_{i}=b_{i}=0\ \Leftrightarrow\ i\in F\Big{\\}}$

(this is always a smooth subvariety). It is shown in Lemma 2.4 of [31] that
the decomposition

$\mathfrak{M}(\mathcal{A})=\bigsqcup_{F\
\text{flat}}\overset{\circ}{\mathfrak{M}}(\mathcal{A}^{F})$

is a stratification, with a normal slice to each
$\mathfrak{M}(\mathcal{A}^{F})$ provided by $\mathfrak{M}(\mathcal{A}_{F})$.
Since all of these strata are even-dimensional, and two-dimensional hypertoric
varieties (when hyperplanes are not repeated) are always smooth, this
immediately gives us

######  Lemma 3.2.

The singular locus of an affine hypertoric variety will have (complex)
codimension at least four.

In particular we can focus on
$\mathfrak{M}(\mathcal{A})^{g}=\overset{\circ}{\mathfrak{M}}(\mathcal{A})\cup\bigcup_{i\in\\{1,\dots,N\\}}\overset{\circ}{\mathfrak{M}}(\mathcal{A}^{\\{H_{i}\\}})$.

Furthermore, for each $V\subset\\{1,\dots,N\\}$, we can look at the cone
$\mathcal{C}_{V}$ cut out by $\alpha_{i}(x)\geq 0$ if $i\in V$ and
$\alpha_{i}(x)\leq 0$ if $i\notin V$. If this cone is top-dimensional, we say
$V$ is _broad_ , and we define

$\mathcal{X}(V)=\Big{\\{}\ [\underline{a},\underline{b}]\ \Big{|}\ b_{i}=0\
\text{if}\ i\in V\ \text{and}\ a_{j}=0\ \text{if}\ j\notin V\Big{\\}}.$

The $\mathcal{X}(V)$, for all broad $V$, are the components of the _extended
core_. They are Lagrangian subvarieties of $Y_{G}$, that are toric for the
action of $T_{{\mathbb{C}}}$, with at worst finite quotient singularities, and
whose image under the real moment map is given by $\mathcal{C}_{V}$. They will
intersect the singular locus of $Y_{G}$, but we can restrict our attention to
what happens in $\mathfrak{M}(\mathcal{A})^{g}$. We put
$\mathcal{X}(V)^{g}=\mathcal{X}(V)\cap\mathfrak{M}(\mathcal{A})^{g}$, which is
the subvariety of the toric variety $\mathcal{X}(V)$ consisting of the open
orbit and the codimension-one orbits.

######  Lemma 3.3.

The cotangent bundles $T^{*}\mathcal{X}(V)^{g}$, for all broad $V$, are
symplectic subvarieties that cover $\mathfrak{M}(\mathcal{A})^{g}$. Each of
these $T^{*}\mathcal{X}(V)^{g}$ has (complex) codimension at least two in
$\mathfrak{M}(\mathcal{A})$.

###### Proof.

It suffices to remark that

$T^{*}\mathcal{X}(V)^{g}=\Big{\\{}[\underline{a},\underline{b}]\,\Big{|}\,\text{at
most one of the}\ a_{i},i\in V,\ \text{and}\ b_{j},j\notin V,\,\text{is
zero}\Big{\\}}.$

∎

We now return our focus to the particular hypertoric variety
$\mathfrak{M}(\mathcal{A})=Y_{G}$. Here the cotangent bundles
$T^{*}\mathcal{X}(V)^{g}\subset Y_{G}$, for broad $V$, are permuted by the
$W_{G}$-action. Moreover, for $\Omega=\\{1,\dots,N\\}$, we have
$\mathcal{X}(\Omega)=X_{G}\subset Y_{G}$.

The natural projection $T^{*}\mathcal{X}(\Omega)^{g}\rightarrow X_{G}$ extends
to a morphism ${Y_{G}\rightarrow X_{G}}$ by the algebraic version of Hartogs’
theorem, since affine hypertoric varieties are always normal (see see [2],
Proof of 4.11, or [38], §4) and Lemma 3.3. Precomposing this map with the
action of $w\in W_{G}$ on $Y_{G}$ gives us the desired components into the
other copies of $X_{G}$, to finally obtain a morphism of affine varieties

(3.1)
$Y_{G}\rightarrow\left(\operatorname{\mathfrak{g}}^{*}\oplus\operatorname{\mathfrak{t}}_{{\mathbb{C}}}^{*}\right)\times\prod_{w\in
W_{G}}X_{G}.$

######  Lemma 3.4.

The map (3.1) restricted to $\overset{\circ}{Y_{G}}\subset Y_{G}$ is an
embedding.

Remark that the locus on which the map $Y_{G}\rightarrow Q_{G}$ is an
embedding is the same as the one described in [10] using the quiver
construction for $Q_{\operatorname{\textsl{SL}}(n,{\mathbb{C}})}$.

###### Proof.

Similar to Lemma 3.3, we have that $\overset{\circ}{Y_{G}}$ is covered
by$T^{*}\left(\mathcal{X}(V)\cap\overset{\circ}{Y_{G}}\right)$, for broad $V$.
Each of the latter, in turn, is

$T^{*}\left(\mathcal{X}(V)\cap\overset{\circ}{Y_{G}}\right)\ =\ \Big{\\{}\
[\underline{a},\underline{b}]\ \Big{|}\ a_{i}\neq 0\ \text{if}\ i\in V\
\text{and}\ b_{j}\neq 0\ \text{if}\ j\notin V\Big{\\}},$

which is isomorphic to $T^{*}T_{{\mathbb{C}}}$. It therefore suffices to
remark that, for any Lie group $H$, the map $T^{*}H\rightarrow
H\times\mathfrak{h}^{*}$, given by the natural projection onto $H$, and the
moment map for the cotangent lift of the action
$H\reflectbox{$\righttoleftarrow$}H$ by multiplication by inverses on the
right, is an isomorphism. Hence the map from
$T^{*}\left(\mathcal{X}(\Omega)\cap\overset{\circ}{Y_{G}}\right)$ to
${\left(T_{\mathbb{C}}\subset X_{G}\subset
G{\sslash}U\right)\times\operatorname{\mathfrak{t}}_{{\mathbb{C}}}^{*}}$ is an
embedding. As $W_{G}$ permutes the broad $V$, the full map
$(\ref{hypertoricemb})$ is an embedding on $\overset{\circ}{Y_{G}}$. ∎

######  Proposition 3.1.

The image of (3.1) is contained in the image of (2.5), hence we have a
morphism $Y_{G}\rightarrow Q_{G}$ that is an embedding on
$\overset{\circ}{Y_{G}}$.

###### Proof.

As (2.5) is a closed embedding, it suffices to show this for the dense open
subvariety
$T^{*}\left(\mathcal{X}(\Omega)\cap\overset{\circ}{Y_{G}}\right)\cong
T^{*}T_{\mathbb{C}}$. We can embed $T^{*}T_{\mathbb{C}}$ naturally into
$T^{*}(G/U)\subset Q_{G}$, by

$(t,\alpha)\in
T_{\mathbb{C}}\times\operatorname{\mathfrak{t}}_{\mathbb{C}}^{*}\mapsto\overline{(t,\alpha)}\in
T^{*}(G/U)\cong(G\times\mathfrak{u}^{\circ})/U.$

(We use here that $\operatorname{\mathfrak{t}}_{C}^{*}$ naturally is a
subspace of $\mathfrak{u}^{\circ}$.) As it is straightforward to verify that
the images of $T^{*}T_{{\mathbb{C}}}$ under (2.5) and (3.1) are identical, we
are done. ∎

##  4\. Implosion and the Moore-Tachikawa category

In this section, we discuss the Moore-Tachikawa category $HS$ and explain how
it gives a useful organising principle for implosion and contraction.

The objects of $HS$, as introduced in [28], are complex semisimple groups and
the morphisms from $G_{1}$ to $G_{2}$ are complex symplectic varieties with
linearised, Hamiltonian, $G_{1}\times G_{2}$-actions. In the setup of [28] one
also assumes a commuting circle action acting on the complex symplectic form
with weight 2, but we do not need that for our discussion, and we will also
allow for objects to be complex reductive groups.

The composition $Y\circ X$ of morphisms
$X\in{\operatorname{Hom}}_{HS}(G_{1},G_{2})$ and
$Y\in{\operatorname{Hom}}_{HS}(G_{2},G_{3})$ is given by the complex
symplectic quotient

(4.1) $Y\circ X=(X\times Y){\sslash\mkern-6.0mu/}\Delta G_{2}$

where $\Delta G_{2}$ denotes the diagonally embedded $G_{2}$ in the symmetry
group $G_{1}\times(G_{2}\times G_{2})\times G_{3}$ of $X\times Y$. This
quotient is complex symplectic with residual $G_{1}\times G_{3}$-action so
lies in $\operatorname{Hom}_{HS}(G_{1},G_{3})$ as required. If $K$ is a
compact Lie group with $G=K_{{\mathbb{C}}}$, then the Kronheimer space
$T^{*}G$ [25] is complex symplectic with $G\times G$-action and defines the
identity element in $\operatorname{Hom}_{HS}(G,G)$, due to the complex
analogue of (2.3).

More generally, we could consider a Moore-Tachikawa category in any setting
where one has a reduction theory for group actions on spaces of some kind,
provided that reduction by commuting actions commutes, and there exists an
identity. This is the case for Hamiltonian actions of compact groups on
symplectic manifolds and stratified spaces, and linearised actions of
reductive groups on varieties projective over an affine. As mentioned above,
an identity is missing in the case of Kähler reduction of Kähler manifolds by
compact groups, or hyper-Kähler reduction – in those cases one only has a
semi-category.

In [28] it was conjectured that a choice of group $G$ would give a functor
$\eta_{G}:Bo_{2}\rightarrow H$, where $Bo_{2}$ is the category of $2$-bordism,
mapping the cylinder to $T^{*}G$. The Riemann sphere
${\mathbb{C}}{\mathbb{P}}^{1}$ maps to the BFM space (or universal
centraliser) [3], while the once-punctured sphere maps to the product of $G$
and the Slodowy slice. This is actually a symmetric monoidal functor, where
the monoidal structure on HS is given by the product of groups and of spaces,
and the dual of a group is just itself. The functor thus defines a
2-dimensional TQFT. Recently Ginzburg and Kazhdan [15, 16] and Bielawski [4]
have defined the image under $\eta_{G}$ of the three-times punctured sphere —
together with the above data this determines the functor. Physically,
$\eta_{G}(C)$ represents the Higgs branch of a $N=2$ supersymmetric theory
associated to a punctured Riemann surface $C$ with defects located at the
punctures, associated to homomorphisms
$\operatorname{\mathfrak{su}}(2)\rightarrow\operatorname{\mathfrak{g}}$.

As mentioned, in the original Moore-Tachikawa discussion the groups are taken
to be semisimple. In what follows we shall find it useful to extend the
formalism to more general reductive groups.

As discussed in [12] and [7], the universal hyperkähler implosion for a
compact $K$ with maximal torus $T$ may be viewed as an element of
$\operatorname{Hom}_{HS}(G,T_{{\mathbb{C}}})$. The process of imploding a
complex symplectic manifold $M$ with $G$-action, viewed as an element of
$\operatorname{Hom}_{HS}(1,G)$, to obtain a manifold $M_{\textup{impl}}$ with
$T_{{\mathbb{C}}}$-action, is now exactly that of composition of morphisms
with the universal implosion.

$G=K_{\mathbb{C}}$$T_{\mathbb{C}}$1$(T^{*}G)_{\textup{impl}}$$M$$M_{\textup{impl}}$

Of course, one could also define a similar picture with compact groups
replacing complex reductive groups, and with (real) symplectic manifolds
playing the role of morphisms. Again, the universal symplectic implosion
$(T^{*}K)_{\textup{impl}}$ for a compact $K$ is an element of
$\operatorname{Hom}_{HS}(K,T)$, and implosion of a symplectic $K$-manifold as
defined in [18] is just composition with the universal implosion. Explicitly,
we have

$M_{\textup{impl}}=(M\times(T^{*}K)_{\textup{impl}}){\sslash}\Delta K.$

##  5\. Symplectic and algebraic contraction

Returning to our definition of composition in $HS$, we see that if the middle
group $G_{2}$ is _Abelian_ , then in fact $Y\circ X$ has an action not just of
$G_{1}\times G_{3}$, but of $G_{1}\times G_{2}\times G_{3}$, because the
_anti_ -diagonally embedded $G_{2}$ commutes with the diagonal $G_{2}$ and
hence descends to the quotient. In particular, if we have a space with
$G$-action, we may compose it with the (right) universal implosion (viewed as
a morphism from $G$ to $T_{{\mathbb{C}}}$) to obtain a space with
$T_{{\mathbb{C}}}$-action, i.e. we form the implosion. Then we may compose
again with the (left) universal implosion considered as a morphism in the
reverse direction, and get a space with $G\times T_{{\mathbb{C}}}$-action,
where the $T_{{\mathbb{C}}}$ factor arises because the middle group
$T_{{\mathbb{C}}}$ in the composition is Abelian.

$K_{\mathbb{C}}$$T_{\mathbb{C}}$$K_{\mathbb{C}}$1

In the setting of real symplectic manifolds with compact group actions, the
resulting space is the symplectic contraction of [20]. The dimension
calculation in the real case is as follows, denoting the contraction of $X$ by
$X_{\operatorname{sc}}$:

$\dim X_{\textup{impl}}=\dim X+\dim(T^{*}K)_{\textup{impl}}-2\dim K=\dim
X+\operatorname{rank}K-\dim K$ $\displaystyle\dim X_{\operatorname{sc}}$
$\displaystyle=$ $\displaystyle\dim
X_{\textup{impl}}+\dim(T^{*}K)_{\textup{impl}}-2\dim T$ $\displaystyle=$
$\displaystyle\dim X+\operatorname{rank}K-\dim K+(\dim
K+\operatorname{rank}K)-2\operatorname{rank}K$ $\displaystyle=$
$\displaystyle\dim X.$

So the contraction keeps the dimension the same but enhances the $K$-action to
a $K\times T$-action.

As with implosion, the key example is the universal contraction, i.e. the
contraction of $T^{*}K$ with $K\times K\times T$-action. We recover the
contraction of $X$ by taking the product with the universal contraction and
reducing by $K$. In the symplectic case

$(T^{*}K)_{\operatorname{sc}}=((T^{*}K)_{\rm
right\;\textup{impl}}\times(T^{*}K)_{\rm left\;\textup{impl}}){\sslash}T.$

Recall that the left $K$-action on $T^{*}K$ is

$h.(k,v)\mapsto(hk,v)$

with moment map

$\mu_{L}:(k,v)\mapsto-k.v$

where $k.v$ denotes the coadjoint action of $k$ on $v$. The right action is

$h.(k,v)\mapsto(kh^{-1},h.v)$

with the moment map

$\mu_{R}:(k,v)\mapsto v$

so that points of $(T^{*}K)_{\rm right\;\textup{impl}}$ are represented by
pairs $(k,v)$ with$k\in K$ and $v$ in the closed positive Weyl chamber
$\bar{C}$. Points of $(T^{*}K)_{\rm left\;impl}$ are represented by pairs
$(k,v)$ with $k\in K$ and $-k.v\in\bar{C}$. The involution
$(k,v)\mapsto(k^{-1},-k.v)$ intertwines the left and right actions and moment
maps, and induces an identification of the corresponding implosions.

In [20], a map $\Phi:T^{*}K\rightarrow(T^{*}K)_{\operatorname{sc}}$ is defined
by

(5.1) $(k,v)\mapsto[(kh^{-1},h.v),(h,v)]$

where $h$ is such that $h.v$ lies in the chosen closed positive Weyl chamber.
We use square brackets to denote the fact we are passing to the quotient by
$T$. Note that $h$ is uniquely defined up an element of $K_{\sigma}$, where
$K_{\sigma}$ is the stabiliser corresponding to the fact $\sigma$ of $\bar{C}$
containing $h.v$. But we collapse by $[K_{\sigma},K_{\sigma}]$ in forming the
implosion, and by $T$ in taking the symplectic quotient to form
$(T^{*}K)_{\operatorname{sc}}$, so the map (5.1) is well-defined.

This map is surjective, as proven in [20], because given any$[(k,w),(h,v)]$
$\in(T^{*}K)_{\operatorname{sc}}$, we have $w=h.v\in\bar{C}$ by the 0-level
set condition for the $T$-moment map. Now we see that

$\Phi:(kh,v)\mapsto[(k,h.v),(h,v)]=[(k,w),(h,v)].$

As a result, we can understand $(T^{*}K)_{\operatorname{sc}}$ or, in fact, any
$M_{\operatorname{sc}}$ set-theoretically as
$M_{\operatorname{sc}}=M/\thicksim$, where

$x\sim y\Leftrightarrow\mu(x)=\mu(y)\ \text{and}\
y\in[K_{\mu(x)},K_{\mu(x)}].x,$

where $K_{\mu(x)}$ is the stabiliser in $K$ of $\mu(x)$ for the coadjoint
action.

The algebraic counterpart of symplectic contraction is given byPopov’s notion
of horospherical contraction [30] for varieties with an action of a reductive
group. This was initially defined for an affine $G$-variety $X$ as follows:
the ring $k[X]$ is equipped with a poly-filtration obtained from the
decomposition into isotypical components $k[X]_{(\lambda)}$ for $G$ (where
$\lambda$ runs over the dominant weights). By choosing a regular dominant co-
weight
$\rho\in\operatorname{Hom}(\operatorname{\textsl{U}}(1),T)\cap\operatorname{\mathfrak{t}}_{+}$,
we can make this into an $\mathbb{N}$-filtered ring by putting

$k[X]_{\leq n}=\bigoplus_{\rho(\lambda)\leq n}k[X]_{(\lambda)}.$

The horospherical contraction of $X$ is now determined by the associated
graded of this algebra:

$X_{\operatorname{hc}}=\operatorname{Spec}(\operatorname{gr}(k[X])),$

which is independent of the choice of $\rho$. (This definition naturally
extends to a broader setting of varieties that are projective over an affine.)
The Rees construction moreover naturally creates a
$\mathbb{G}_{m}$-equivariant flat family
${\pi:\chi\rightarrow\mathbb{A}^{1}}$, the generic fiber of which is
isomorphic to $X$, and the fiber over $0$ is given by $X_{\operatorname{hc}}$.

The universal horospherical contraction is the asymptotic semigroup
$\operatorname{As}(G)$, as defined by Vinberg [37]. This is the central fiber
in a degeneration of $G$ known as the Vinberg monoid $S_{G}$ [35]. This is
obtained via the Rees construction as above, but without switching to an
$\mathbb{N}$-filtration by a choice of $\rho$ as above – this results in a
flat fibration $\pi_{G}:S_{G}\rightarrow\mathbb{A}_{G}$ over a higher
dimensional base, which is the (smooth) affine toric variety for the torus
$T/Z_{K}$ (where $Z_{K}$ is the center of $K$) given by the cone spanned by
the positive simple roots. Vinberg shows that $S_{G}$ is a reductive monoid,
with group of units $S_{G}^{\times}\cong(G\times T_{{\mathbb{C}}})/Z_{G}$. A
choice of a regular dominant $\rho$ as before gives rise to a morphism of
monoids $\mathbb{A}^{1}\rightarrow\mathbb{A}_{G}$, and the flat family over
$\mathbb{A}^{1}$ induced by this choice is the base-change of
$\pi:S_{G}\rightarrow\mathbb{A}_{G}$ by this morphism.

With a suitable (but not unique) choice of Kähler structure, we can identify
$X_{\operatorname{sc}}$ and $X_{\operatorname{hc}}$, see §5 and 6 in [20].
This is done, following Harada-Kaveh [19], by means of the gradient-
Hamiltonian vector field

(5.2) $V_{\pi}=-\frac{\nabla\Re(\pi)}{\lVert\nabla\Re(\pi)\rVert^{2}}$

on the smooth locus of the total space of the degeneration $\chi$, whose flow
induces a continuous surjective map $X\rightarrow X_{\operatorname{hc}}$ that
generically is a symplectomorphism. In general the results of [19] just
guarantee existence of this map, but in the case of a degeneration
corresponding to a horospherical contraction, as we are considering, it can be
made explicit. What enables this is a reduction to the universal
case${\pi_{G}:S_{G}\rightarrow\mathbb{A}_{G}}$, where the monoid structure of
$S_{G}$, and Lie group decompositions for $S_{G}^{\times}$, make the flow for
$V_{\pi}$ tractable. This allows the identification of $\operatorname{As}(G)$
and $(T^{*}K)_{\operatorname{sc}}$, which in turn implies the same for
$X_{\operatorname{hc}}$ and $X_{\operatorname{sc}}$.

##  6\. Complex symplectic contraction

Given the interpretation for the contraction in the real symplectic and
algebraic situations in terms of implosion, it is now natural to take the
product of the complex symplectic implosion of a space $M$ and the left
implosion of $T^{*}G$, and reduce this by the diagonal complex torus action.
We shall refer to this as the complex symplectic contraction, and denote it as
$M_{\operatorname{csc}}$.

The right implosion is the complex symplectic quotient, in the GIT sense, of
$T^{*}G=G\times\operatorname{\mathfrak{g}}^{*}$ by the maximal unipotent
subgroup $U$ of $G$, that is, the GIT quotient
$Q^{R}=(G\times\operatorname{\mathfrak{u}}^{\circ}){\sslash}U$. This can also
be viewed as $(G\times\operatorname{\mathfrak{b}}){\sslash}U$ where
$\operatorname{\mathfrak{b}}$ denotes the Borel algebra. We could also form
the left implosion

$Q^{L}=\Big{\\{}(g,v)\in G\times\operatorname{\mathfrak{g}}\ \Big{|}\
g.v\in\operatorname{\mathfrak{b}}\Big{\\}}\Big{/}\mkern-11.0mu\Big{/}U$

where $U$ now acts on the left, i.e. by translation in the $G$-factor. As
before, the involution $(g,v)\mapsto(g^{-1},-g.v)$ gives an identification
between left and right implosions.

######  Definition 6.1.

We now define the universal complex symplectic contraction to be the complex-
symplectic quotient by the diagonal complex torus $T_{{\mathbb{C}}}$ of the
product of right and left implosions:

$(T^{*}G)_{\operatorname{csc}}=(Q^{R}\times
Q^{L}){\sslash\mkern-6.0mu/}_{0}T_{{\mathbb{C}}}.$

Now $M_{\operatorname{csc}}$ can be obtained by taking the product of $M$ with
the universal example $(T^{*}G)_{\operatorname{csc}}$ and reducing by $G$.

The above discussion shows the following.

######  Proposition 6.1.

$(T^{*}G)_{\operatorname{csc}}$ has the same dimension as $T^{*}G$ but the
complex-symplectic $G\times G$-action is enhanced to a $G\times
T_{{\mathbb{C}}}\times G$-action.

Similarly $M_{\operatorname{csc}}$ has the same dimension as $M$ but its
$G$-action is enhanced to a $G\times T_{{\mathbb{C}}}$-action.

######  Remark 6.1.

The complex-symplectic reduction of $(T^{*}G)_{\operatorname{csc}}$ by the
$T_{{\mathbb{C}}}$-action is the same as the product of two copies of the
reduction of the implosion by $T_{{\mathbb{C}}}$ – in particular if we reduce
at level zero we obtain the product of two copies of the nilpotent cone.

Let us discuss some properties of the universal example. How could we mimic
the map (5.1) from the symplectic case? We want a map
$T^{*}G\rightarrow(T^{*}G)_{\operatorname{csc}}=(Q^{R}\times
Q^{L}){\sslash\mkern-6.0mu/}_{0}T_{\mathbb{C}}$. Given $(g,v)\in T^{*}G$, we
can choose $h\in G$ with $h.v\in\operatorname{\mathfrak{b}}$ as every element
of $\operatorname{\mathfrak{g}}$ is conjugate via the adjoint action to an
element of the fixed Borel algebra $\operatorname{\mathfrak{b}}$. So we could
try to define

(6.1) $\Psi:(g,v)\mapsto[(gh^{-1},h.v),(h,v)]$

and the right-hand side then lies in the 0-level set of the
$T_{\mathbb{C}}$-action as required.

However, the problem here is that the condition
$h.v\in\operatorname{\mathfrak{b}}$ does not specify $h$ up to an element of
the Borel subgroup $B$ in general. It is therefore natural to replace the
domain $T^{*}G$ by $G\times\tilde{\operatorname{\mathfrak{g}}}$ where
$\tilde{\operatorname{\mathfrak{g}}}$ is the Grothendieck simultaneous
resolution. That is, we consider

$G\times\tilde{\operatorname{\mathfrak{g}}}=\Big{\\{}(g,v,\operatorname{\mathfrak{b}}_{1})\
\Big{|}\ g\in
G,v\in\operatorname{\mathfrak{b}}_{1}:\;\operatorname{\mathfrak{b}}_{1}\in{\mathcal{B}}\Big{\\}}$

where $\mathcal{B}=G/B$ is the variety of Borel algebras in
$\operatorname{\mathfrak{g}}$.

######  Proposition 6.2.

The formula (6.1) above defines a map from
$G\times\tilde{\operatorname{\mathfrak{g}}}$ to the complex-symplectic
contraction $(T^{*}G)_{\operatorname{csc}}$.

###### Proof.

Given $(g,v\operatorname{\mathfrak{b}}_{1})\in
G\times\tilde{\operatorname{\mathfrak{g}}}$, there is now $h\in G$ with
$h.\operatorname{\mathfrak{b}}_{1}=\operatorname{\mathfrak{b}}$ (so
$h.v\in\operatorname{\mathfrak{b}}$), and $h$ _is_ unique up to an element of
$B$. Therefore the right-hand side of (6.1) is well-defined, as the terms in
round brackets are elements of the quotients by the maximal unipotent $U$, and
the square bracket denotes the equivalence class under the
$T_{\mathbb{C}}$-action. ∎

######  Remark 6.2.

We could generalise this construction to any $X$ with Hamiltonian
$G_{\mathbb{C}}$-action, and form

$\Big{\\{}(x,\operatorname{\mathfrak{b}}_{1})\ \Big{|}\
\mu(x)\in\operatorname{\mathfrak{b}}_{1}\big{\\}}\subset X\times\mathcal{B}.$

If $X=T^{*}G$ then the moment map is projection to the second factor, and
identifying $\operatorname{\mathfrak{g}}^{*}$ with
$\operatorname{\mathfrak{g}}$ we obtain
$G\times\tilde{\operatorname{\mathfrak{g}}}$ as above.

Recall from [9] that every orbit of $U$ in the open set

$\Big{\\{}(g,v)\in G\times\operatorname{\mathfrak{b}}\ \Big{|}\
\operatorname{\mathfrak{t}}-{\rm component\;of\;}v\;{\rm
is}\;\in\operatorname{\mathfrak{t}}_{\rm reg}\Big{\\}}$

(where $\operatorname{\mathfrak{t}}_{\rm reg}$ consists of the regular
elements in $\operatorname{\mathfrak{t}}$) has a unique representative with
$v\in\operatorname{\mathfrak{t}}_{\rm reg}$. We therefore have a set
$Q^{R}_{\circ}=G\times\operatorname{\mathfrak{t}}_{\rm reg}$ contained in
$Q^{R}$, and of the same dimension as $Q^{R}$, as well as an analogous subset
$Q^{L}_{\circ}\subset Q^{L}$. We may consider the locus $Q^{R}_{\circ}\times
Q^{L}_{\circ}$ and its projection $(T^{*}G)_{\operatorname{csc},\circ}$ to the
symplectic quotient by $T_{{\mathbb{C}}}$. Points in this locus are
represented by pairs

$((g_{1},v),(g_{2},w))\;:\;g_{1},g_{2}\in
G\;:\;v,\;g_{2}.w\in\operatorname{\mathfrak{t}}_{\rm reg}$

subject to the zero-level set condition for the $T_{\mathbb{C}}$-moment map

$v-g_{2}.w=0.$

######  Proposition 6.3.

The locus $(T^{*}G)_{\operatorname{csc},\circ}$ is contained in the image of
the map $\Psi$.

###### Proof.

For such points, we can proceed as in the real symplectic case, for now
$v=g_{2}.w\in\operatorname{\mathfrak{t}}\subset\operatorname{\mathfrak{b}}$
and $(g_{1}g_{2},w,\operatorname{\mathfrak{b}}_{1})\in
G\times\tilde{\operatorname{\mathfrak{g}}}$ maps to

$[(g_{1},g_{2}.w),(g_{2},w)]=[(g_{1},v),(g_{2},w)],$

where we choose $\operatorname{\mathfrak{b}}_{1}$ to be the Borel algebra
containing $w$ such that
$g_{2}.\operatorname{\mathfrak{b}}_{1}=\operatorname{\mathfrak{b}}$. So we
have surjectivity onto the locus represented by ${Q^{R}_{\circ}\times
Q^{L}_{\circ}}$. ∎

######  Remark 6.3.

We could also see this by recalling that we have an identification
$G\times_{B}\operatorname{\mathfrak{b}}\cong\tilde{\operatorname{\mathfrak{g}}}$
given by

$(g^{\prime},x)\mapsto(g^{\prime}.x,g^{\prime}B/B).$

So $\Psi$ can be viewed as a map from $G\times
G\times_{B}\operatorname{\mathfrak{b}}$ to the contraction given by the
composition

$(g,g^{\prime},x)\mapsto(g,g^{\prime}.x,g^{\prime}B/B)\mapsto[(gg^{\prime},x),((g^{\prime})^{-1},g^{\prime}.x)]$

as we can take $h=(g^{\prime})^{-1}$ in the formula (6.1).

######  Remark 6.4.

As the complex symplectic implosion has an action of $W_{G}\ltimes
T_{{\mathbb{C}}}$, and

$\Delta T_{{\mathbb{C}}}\mathrel{\unlhd}\Delta
W_{G}\ltimes(T_{{\mathbb{C}}}\times T_{{\mathbb{C}}})\subset(W_{G}\ltimes
T_{{\mathbb{C}}})\times(W_{G}\ltimes T_{{\mathbb{C}}}),$

we see that the contraction inherits an action of the diagonally embedded Weyl
group also. In particular the universal complex symplectic contraction
$T^{*}G_{\operatorname{csc}}$ has an action of $G\times G\times(W_{G}\ltimes
T_{{\mathbb{C}}})$.

######  Proposition 6.4.

The real symplectic contraction sits naturally inside the complex-symplectic
contraction.

###### Proof.

Recall from §4 of [6] that the real symplectic implosion $G{\sslash}U$ arises
as the fixed point locus of the ${\mathbb{C}}^{*}$-action on the hyperkähler
implosion induced from scaling in the $\operatorname{\mathfrak{u}}^{\circ}$
factor. The $T_{{\mathbb{C}}}$-moment map is equivariant for this action, so
the product of left and right real symplectic implosions lies in the zero
locus of the $T_{{\mathbb{C}}}$-moment map in $Q^{R}\times Q^{L}$, and taking
$T_{{\mathbb{C}}}$-quotients the statement follows. ∎

##  7\. Examples

Let us consider the simplest case, where $K=\operatorname{\textsl{SU}}(2)$.

We first look at the universal symplectic contraction. The universal
symplectic implosion is ${\mathbb{C}}^{2}$ with the flat Kähler structure and
$U(2)$-action, so the contraction is the symplectic quotient

$({\mathbb{C}}^{2}\times{\mathbb{C}}^{2}){\sslash}U(1),$

where $U(1)$ acts on the copies of ${\mathbb{C}}^{2}$ by scalar multiplication
by $e^{i\theta}$ and $e^{-i\theta}$ respectively. Equivalently, this is the
GIT quotient by ${\mathbb{C}}^{*}$ with action

$\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right)\mapsto\left(\begin{array}[]{c}tv_{1}\\\
tv_{2}\end{array}\right),\;\;\;:\;\;\;(w_{1}\;\;w_{2})\mapsto(t^{-1}w_{1}\;\;t^{-1}w_{2}).$

The invariants are given by the entries of

$\left(\begin{array}[]{cc}X&Y\\\ Z&W\end{array}\right)=v\otimes
w=(v_{i}w_{j})_{i,j=1}^{2}.$

This is the affine variety in ${\mathbb{C}}^{4}$ defined by $XW-YZ=0$, which
is the symplectic contraction for $\operatorname{\textsl{SU}}(2)$. Indeed, in
this case the Vinberg monoid $S_{\operatorname{\textsl{SL}}(2)}$ is just the
space of $2\times 2$ matrices, fibering over ${\mathbb{C}}$ by the
determinant, so that the central fiber, the asymptotic semigroup, consists of
the singular matrices.

To understand the flow for the vectorfield (5.2), where $\pi=\det$, we can
argue as follows (cf.[20]). Since both $\pi$ and the Kähler metric on
$S_{\operatorname{\textsl{SL}}(2)}$ are invariant under left and right
multiplication by matrices in $\operatorname{\textsl{SU}}(2)$, the flow for
$V_{\pi}$ is equivariant for these actions as well. Now consider the
$KAK$-type Cartan decomposition for
$S_{\operatorname{\textsl{SL}}(2)}^{\times}=\operatorname{\textsl{GL}}(2,{\mathbb{C}})$
($A$ here consists of the diagonal matrices in
$\operatorname{\textsl{GL}}(2,{\mathbb{R}})$ with positive entries). If
$x\in\operatorname{\textsl{SL}}(2,{\mathbb{C}})$ is contained in $k_{1}Ak_{2}$
for some $k_{1},k_{2}\in\operatorname{\textsl{SU}}(2)$, by the equivariance,
the entire flow of $x$ under $V_{\pi}$ will be as well. So it suffices to just
look at elements $\begin{pmatrix}x&0\\\
0&x^{-1}\end{pmatrix}\in(A\cap\operatorname{\textsl{SL}}(2,{\mathbb{C}}))$.
Level sets of $\pi$ restricted to $A$ are just given by components of
hyperbola

$\left\\{\begin{pmatrix}x&0\\\ 0&y\end{pmatrix}\ \Big{|}\
x,y\in{\mathbb{R}}_{+}\ \text{with}\ xy=\lambda\right\\},$

and hence the flow lines for $V_{\pi}$, which are everywhere orthogonal to
them, lie on hyperbola given by $\begin{pmatrix}x&0\\\ 0&y\end{pmatrix}$ with
$x^{2}-y^{2}=c$. This in turn allows us to determine exactly what the flow for
time $t=1$ does to elements in
$A\cap\operatorname{\textsl{SL}}(2,{\mathbb{C}})$:

$\begin{pmatrix}x&0\\\
0&x^{-1}\end{pmatrix}\mapsto\begin{cases}\begin{pmatrix}\sqrt{x^{2}-x^{-2}}&0\\\
0&0\end{pmatrix}&\text{if}\ \ x\geq 1,\\\ \ &\\\ \begin{pmatrix}0&0\\\
0&\sqrt{x^{-2}-x^{2}}\end{pmatrix}&\text{if}\ \ x\leq 1.\end{cases}$

We can translate this cleanly to all of
$\operatorname{\textsl{SL}}(2,{\mathbb{C}})$ using the polar decomposition as
follows:

(7.1) $\operatorname{\textsl{SL}}(2,{\mathbb{C}})\ni
B=UP=U\sqrt{B^{*}B}\mapsto U\sqrt{B^{*}B-\gamma\operatorname{Id}},$

where $\gamma$ is the smallest eigenvalue of $B^{*}B$. From this one
immediately sees that the moment maps for the actions of
$\operatorname{\textsl{SU}}(2)$ given by multiplication on the left and the
right are invariant under this map. Moreover, the stabilisers for these
actions remain the same, except when $B^{*}B=\operatorname{Id}$, i.e. when
$B\in\operatorname{\textsl{SU}}(2)$ (all of which get sent to the zero
matrix). So (7.1) just collapses
$\operatorname{\textsl{SU}}(2)\subset\operatorname{\textsl{SL}}(2.{\mathbb{C}})$,
which, using the Cartan decomposition, we can think of as the zero section of
$T^{*}\operatorname{\textsl{SU}}(2)\cong\operatorname{\textsl{SL}}(2,{\mathbb{C}})$.
This is of course the exactly the same as what happens for
$T^{*}\operatorname{\textsl{SU}}(2)_{\operatorname{sc}}$.

In the hyperkähler case, the universal implosion is

$Q_{\operatorname{\textsl{SL}}(2,{\mathbb{C}})}={\mathbb{H}}^{2}=\operatorname{Hom}({\mathbb{C}},{\mathbb{C}}^{2})\oplus\operatorname{Hom}({\mathbb{C}}^{2},{\mathbb{C}})$

with $\operatorname{\textsl{SU}}(2)$-action
$(\alpha,\beta)\mapsto(g\alpha,\beta g^{-1})$ and
$U(1)$-action$(\alpha,\beta)\mapsto(e^{-i\theta}\alpha,e^{i\theta}\beta)$.

To see the action of the Weyl group
$W_{\operatorname{\textsl{SL}}(2,{\mathbb{C}})}=\\{1,\gamma\\}$, it is easiest
to use the identification
$\operatorname{\textsl{SL}}(2,{\mathbb{C}})=Sp(2,{\mathbb{C}})$. If we take
$J=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$, then for any
$g\in\operatorname{\textsl{SL}}(2,{\mathbb{C}})$ we have $g^{T}J=Jg^{-1}$. We
can now put

$\gamma.(\alpha,\beta)=\left((\beta J)^{T},(J\alpha)^{T}\right),$

i.e.

$\gamma.\left(\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\end{array}\right),(\beta_{1}\
\beta_{2})\right)=\left(\left(\begin{array}[]{c}-\beta_{2}\\\
\beta_{1}\end{array}\right),(\alpha_{2}\ -\alpha_{1})\right).$

This all indeed combines to given an action of
$\operatorname{\textsl{SL}}(2,{\mathbb{C}})\times\left(W_{\operatorname{\textsl{SL}}(2,{\mathbb{C}})}\ltimes{\mathbb{C}}^{*}\right)$
on $\mathbb{H}^{2}$ (where $\gamma.\lambda=\lambda^{-1}$ for
$\lambda\in{\mathbb{C}}^{*})$. Note that the
$\operatorname{\textsl{SL}}(2,{\mathbb{C}})$-moment map $(\alpha\beta)_{0}$
(ie. the tracefree part of $\alpha\beta$) is Weyl-invariant, while the
${\mathbb{C}}^{*}$-moment map $\beta\alpha$ changes sign under the Weyl
action. The hypertoric variety $Y_{\operatorname{\textsl{SL}}(2,{\mathbb{C}}}$
is now just ${\mathbb{H}}\cong{\mathbb{C}}^{2}$, which embeds into
$Q_{\operatorname{\textsl{SL}}(2,{\mathbb{C}})}$ by
$(\alpha,\beta)\mapsto\left(\left(\begin{array}[]{c}\alpha\\\
0\end{array}\right),(\beta\ 0)\right)$ – remark that this is not
$W_{\operatorname{\textsl{SL}}(2,{\mathbb{C}})}$-equivariant.

For the universal complex symplectic contraction, we need to consider the
hyperkähler quotient of two copies of the implosion by the diagonal
$U(1)$-action. We can view this as
${\mathbb{C}}^{4}\times({\mathbb{C}}^{4})^{*}$ with
action$(v,w)\mapsto(t^{-1}v,tw)$. So we must impose the complex moment map
condition $\mu_{{\mathbb{C}}}:=\langle v,w\rangle=0$ and factor out the
complexified action.

Again we can form the invariants $v\otimes w=(v_{i}w_{j})_{i,j=1}^{4}$. We
obtain a hyperplane in the cone over the Segre variety defined by the
vanishing of all $2\times 2$ minors. If, for example, we set
$v_{1}=v_{2}=w_{3}=w_{4}=0$ then the condition $\mu_{\mathbb{C}}=0$ is
automatically satisfied and we recover the (real) symplectic contraction given
by coordinates $v_{3}w_{1},v_{3}w_{2},v_{4}w_{1},v_{4}w_{2}$ with one
quadratic relation.

More concretely, it is well-known that our space can be identified with the
Swann bundle ${\mathcal{U}}(Gr_{2}{\mathbb{C}}^{4})$ of the Grassmannian of
complex 2-planes in ${\mathbb{C}}^{4}$. Explicitly, $v\otimes w$ defines an
element of the closure of the highest root (or minimal) nilpotent orbit in
$\operatorname{\mathfrak{sl}}(4,{\mathbb{C}})$, and this is the total space of
the Swann bundle, see page 432 in [34]. If we reduce at a generic nonzero
level we instead obtain the Calabi space $T^{*}{\mathbb{C}}{\mathbb{P}}^{3}$.

In each case the hyperkähler space has an
$\operatorname{\textsl{SU}}(4)$-action preserving the hyperkähler structure –
in the case of ${\mathcal{U}}(Gr_{2}{\mathbb{C}}^{4})$ this is induced from
the $\operatorname{\textsl{SU}}(4)$-action on the quaternionic Kähler Wolf
space$Gr_{2}({\mathbb{C}}^{4})=\operatorname{\textsl{SU}}(4)/S(\operatorname{\textsl{U}}(2)\times\operatorname{\textsl{U}}(2))$.
This $\operatorname{\textsl{SU}}(4)$-action includes two commuting
$\operatorname{\textsl{SU}}(2)$’s, as well as a circle action commuting with
each $\operatorname{\textsl{SU}}(2)$ factor, in accordance with our
interpretation of the space as a contraction for
$\operatorname{\textsl{SU}}(2)$.

In this case the Weyl group action on the complex symplectic contraction had
already been observed by Swann, see Proposition 4.4.1 in [33], as the Galois
action for the covering of a nilpotent orbit closure in
$\operatorname{\mathfrak{sp}}(2,{\mathbb{C}})$ by the highest root nilpotent
orbit closure in $\operatorname{\mathfrak{sl}}(4,{\mathbb{C}})$. Indeed, if we
identify ${\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$ with
$\operatorname{\mathfrak{gl}}(4)\cong{\mathbb{C}}^{4}\otimes({\mathbb{C}}^{4})^{*}$
by means of ${v\otimes w\mapsto v\otimes(\widetilde{J}w)^{T}}$ (where
$\widetilde{J}=\left(\begin{array}[]{cc}J&0\\\ 0&J\end{array}\right)$ and $J$
is as above), then $\operatorname{\mathfrak{sp}}(2)$ corresponds to
$\operatorname{Sym}^{2}{\mathbb{C}}^{4}\subset{\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$.
The action of $\gamma$ is then just the restriction of the transposition of
the factors of ${\mathbb{C}}^{4}\otimes{\mathbb{C}}^{4}$, which on
$\operatorname{\mathfrak{gl}}(4)$ translates to
$A\mapsto\widetilde{J}A^{T}\widetilde{J}$.

The locus $Q^{R}_{\circ}$ of §6 corresponds to the locus where the
$T_{{\mathbb{C}}}={\mathbb{C}}^{*}$-moment map
$\beta\alpha=\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}$ is nonzero. The
associated locus in the contraction represented by points of
$Q^{R}_{\circ}\times Q^{L}_{\circ}$ is therefore the set of points where the
sum of the moment maps from the right and left implosions is zero, but the
values of the individual maps are nonzero.

######  Remark 7.1.

As described in Section 2, for $K=\operatorname{\textsl{SU}}(n)$ we have a
quiver description of the implosion, as the space of full flag quivers

$0\stackrel{{\scriptstyle[}}{{\beta}}_{0}]{\alpha_{0}}{\rightleftarrows}{\mathbb{C}}\stackrel{{\scriptstyle[}}{{\beta}}_{1}]{\alpha_{1}}{\rightleftarrows}{\mathbb{C}}^{2}\stackrel{{\scriptstyle[}}{{\beta}}_{2}]{\alpha_{2}}{\rightleftarrows}\dots\stackrel{{\scriptstyle[}}{{\beta}}_{n-2}]{\alpha_{n-2}}{\rightleftarrows}{\mathbb{C}}^{n-1}\stackrel{{\scriptstyle[}}{{\beta}}_{n-1}]{\alpha_{n-1}}{\rightleftarrows}{\mathbb{C}}^{n},$

with $\alpha_{0}=\beta_{0}=0$, satisfying the equations

(7.2)
$\alpha_{i-1}\beta_{i-1}-\beta_{i}\alpha_{i}=\lambda^{\mathbb{C}}_{i}I\qquad(i=1,\dots,n-1),$

for free complex scalars
$\lambda^{\mathbb{C}}_{1},\dots,\lambda^{\mathbb{C}}_{n-1}$, modulo the
natural action of
$\prod_{i=1}^{n-1}\operatorname{\textsl{SL}}(i,{\mathbb{C}})$

(7.3) $\alpha_{i}\mapsto g_{i+1}\alpha_{i}g_{i}^{-1},\;\;\;\beta_{i}\mapsto
g_{i}\beta_{i}g_{i+1}^{-1}\;\;\;(i=1,\ldots,n-2)$
$\alpha_{n-1}\mapsto\alpha_{n-1}g_{n-1}^{-1},\;\;\;\beta_{n-1}\mapsto
g_{n-1}\beta_{n-1}.$

This can be identified with the hyperkähler quotient of the space of full flag
quivers by the action of $\prod_{i=1}^{n-1}\operatorname{\textsl{SU}}(i)$.
Note that if $n=2$ this just reduces to
$\operatorname{Hom}({\mathbb{C}},{\mathbb{C}}^{2})\oplus\operatorname{Hom}({\mathbb{C}}^{2},{\mathbb{C}})$
as in the example above since the remaining equation 7.2) is automatic. For
higher $n$ we do not in general have a simple description of the space,
although for $n=3$ it is the Swann bundle of the Grassmannian
$SO(8)/(SO(4)\times SO(4))$, that is the closure of the minimal nilpotent
orbit in $\operatorname{\mathfrak{so}}(8,{\mathbb{C}})$.

We have a residual action of $\operatorname{\textsl{SL}}(n)$ by extending
(7.3) to $i=n$, as well as a complex torus action complexifying the
hyperkähler action of of
$T=\prod_{i=1}^{n-1}\operatorname{\textsl{U}}(i)/\operatorname{\textsl{SU}}(i)$,
whose moment map gives the $\lambda_{i}$. In general, elements of the
contraction may therefore be represented by $T$-equivalence classes of pairs
of quivers satisfying the equations, but with opposite $\lambda_{i}$.

######  Remark 7.2.

In [8] an alternative approach to implosion was introduced using Nahm’s
equations

$\frac{dT_{i}}{dt}+[T_{0},T_{i}]=[T_{j},T_{k}]\;\;\;:\;\;\;(ijk)\;{\rm
cyclic\;permutation\;of}\;(123).$

The idea here is to consider Nahm data on the half line $[0,\infty)$
asymptotic to a commuting triple $(\tau_{1},\tau_{2},\tau_{3})$ of elements of
a fixed Cartan algebra. We further collapse by gauge transformations
asymptotic at infinity to the commutator of the common centraliser $C$ of the
triple (so no collapsing occurs on the open dense set where $C$ is the maximal
torus). We obtain a stratified pseudo-hyperkähler space with an action of $T$,
represented by gauge transformations asymptotic to values in $T$ at infinity.
Moreover, the moment map for the $T$-action is just evaluation of the Nahm
data at infinity, i.e. the triple $(\tau_{1},\tau_{2},\tau_{3})$. The metric
becomes a genuine positive definite hyperkähler metric on the level sets of
the moment map, and the quotient gives the Kostant variety as required.

This construction will in general give a rather different space from the
implosion that we have discussed – in particular the structure as an algebraic
variety is still unclear. However, we may mimic the contraction construction
and take the hyperkähler quotient by $T$ of two copies of this space, and
obtain a space with the correct dimension and action. Being in the zero-level
set of the $T$-action means the two sets of Nahm data are equal and opposite
at infinity.

Using the symmetry $T_{i}(t)\mapsto-T_{i}(-t)$ of the Nahm equations, we may
regard such configurations as giving Nahm data on two half lines that match at
the respective points at infinity. Note that in general, we have a scaling
symmetry $T_{i}(t)\mapsto cT_{i}(ct)$ of the Nahm equations that expands the
interval of definition by a factor of $c^{-1}$ if $0<c<1$, and under this
symmetry the Nahm matrices scale by $c$ but their derivatives scale by
$c^{2}$. Recall that $T^{*}K_{{\mathbb{C}}}$ may be identified with the space
of $\operatorname{\mathfrak{k}}$-valued solutions to Nahm’s equations smooth
on a finite interval. Intuitively, we may view the above construction as a
limiting case of stretching out the interval so that the Nahm matrices are
almost commuting in the interior.

## Acknowledgements

We thank the organisers of the conference "Moduli spaces and geometric
structures" held in ICMAT Madrid in September 2022 for providing such a
stimulating atmosphere. We thank Victor Ginzburg and David Kazhdan for sending
us a copy of [15]. We also thank the referee for several valuable suggestions.

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# An Empirical Study on Learning Fairness Metrics for COMPAS Data with Human
Supervision

Hanchen Wang
Department of Engineering
University of Cambridge
<EMAIL_ADDRESS>
&Nina Grgić-Hlača
Max Planck Institute for
Software Systems
<EMAIL_ADDRESS>&Preethi Lahoti
Max Planck Institute
for Informatics
<EMAIL_ADDRESS>&Krishna P. Gummadi
Max Planck Institute for
Software Systems
<EMAIL_ADDRESS>&Adrian Weller
University of Cambridge
& The Alan Turing Institute
<EMAIL_ADDRESS>

###### Abstract

The notion of individual fairness requires that similar people receive similar
treatment. However, this is hard to achieve in practice since it is difficult
to specify the appropriate similarity metric. In this work, we attempt to
learn such similarity metrics from human annotated data. We gather a new
dataset of human judgments on a criminal recidivism prediction (COMPAS) task.
Assuming that people’s judgments encode the fairness metric they adhere to, we
leverage prior work on metric learning and attempt to learn people’s
similarity metrics from these judgments.

##  1 Introduction

Bias in automated decision making systems has raised many concerns. One
approach to address these concerns is to enforce individual fairness (Dwork et
al., 2012), which requires treating similar people similarly. However, it is
not straightforward to quantify the appropriate similarity of individuals.
There have been some noteworthy subsequent works on this topic, such as (Zemel
et al., 2013) and (Lahoti et al., 2019a). In our paper, we study the problem
of learning an individual fairness metric from human annotated data. We
leverage the intuition that human judgments might implicitly encode an
underlying fairness metric they adhere to. We gather a dataset of human
judgments about criminal recidivism risk predictions, and utilize this dataset
to test the performance of several different metric learning algorithms on
this data.

##  2 Related Work

Algorithmic Fairness. Most prior work on algorithmic fairness has focused on
group notions of fairness, which require that protected groups as indicated by
sensitive attributes receive similar treatment to others (Zafar et al., 2017b,
a; Hardt et al., 2016). On the other hand, individual fairness, introduced by
Dwork et al. (2012), requires that similar individuals be treated similarly.
To achieve this, one must first define a similarity metric that can be used to
compare the individuals. This similarity metric needs to be either given or
learned from data. Some recent work on individual fairness (Lahoti et al.,
2019b; Jung et al., 2019) elicit human judgments for individual fairness, and
incorporate them as pairwise constraints in the learning objective, without
explicitly learning a fair distance metric. Other works like (Speicher et al.,
2018; Kearns et al., 2017; Liu et al., 2017) use the objective of the learning
algorithm itself as an implicit similarity metric in the optimization problem.
Instead of specifying a similarity metric directly, we attempt to learn it
from human annotated data.

Metric Learning. We leverage the rich literature on metric learning, which has
been studied and applied in various domains, ranging from image processing
(Fei-Fei and Perona, 2005) to recommendation systems (McFee et al., 2012). In
the past years, it has increasingly been applied on human annotated data, in
order to model human notions of similarity (Tamuz et al., 2011). In our work,
we also take this approach. We refer the reader to Bellet et al. (2013) and
Kulis et al. (2013) for a more in depth overview of the relevant literature on
metric learning.

Learning Fairness Metrics. The recent work of Ilvento (2019) is closest to
ours. They propose an approach for approximating an individual fairness metric
from human judgments about the relative distance between inputs. While prior
work did suggest that humans find it easier to make relative judgments than
absolute ones (Stewart et al., 2005), data on relative judgments might be more
difficult to obtain than data on absolute judgments. For instance,
recommendation systems collect data about which articles and movies users
click on, and not about the users’ relative preferences between these articles
or movies. Therefore when learning fairness metrics in real world
applications, it may be easier to utilize existing data about people’s past
absolute judgments, than to gather new data about relative judgments. Hence,
in our work, we study the possibility of learning fairness metrics from
absolute ratings.

##  3 Methodology

###  3.1 Gathering Human Judgments

Scenario. In our experiments, we focus on the task of predicting criminal
recidivism risk. We use a dataset related to the COMPAS tool – a tool used
across the United States to help judges make bail decisions by predicting
defendants’ criminal recidivism risk on a 10 point scale (Angwin et al.,
2016). This dataset, gathered by ProPublica (Angwin et al., 2016), contains
information about the recidivism risk predicted by the COMPAS tool, as well as
the ground truth recidivism rates, for 7214 defendants who were arrested in
Broward County, Florida, in 2013 and 2014.

Survey Instrument. To gather human judgments, we conducted an online survey in
which we asked participants to estimate the likelihood of criminal recidivism
of a fixed set of 200 defendants from the ProPublica dataset.111These 200
defendants were selected uniformly at random without replacement from the 1000
ProPublica cases which were studied by Dressel and Farid (2018), in order to
allow us to compare the performance of our respondents with the performance of
their respondents, on the same set of cases. To mitigate the effects of order
bias (Redmiles et al., 2017), the defendants were shown in random order. For
each defendant, participants were shown information about the defendant’s
demographics and criminal history, in the same format as by Dressel and Farid
(2018) and Grgić-Hlača et al. (2019), and were asked to answer three
questions: (Q1) _How likely do you think it is that this person will commit
another crime within 2 years?_ , (Q2) _Do you think this person should be
granted bail?_ , and (Q3) _How confident are you in your answer about granting
this person bail?_. Our participants were asked to respond to question (Q2)
with _yes_ or _no_ , and to questions (Q1) and (Q3) using a 5-point Likert
scale. Even though the COMPAS tool provides criminal recidivism risk
predictions on a 10-point scale, we opted for this design choice in order to
minimize the duration of our 200-question survey, since providing answers
using 10-point Likert scales was found to be more time consuming than using
5-point scales (Matell and Jacoby, 1972).

Procedure. We recruited participants through the online crowdsourcing platform
Prolific (Palan and Schitter, 2018). Utilizing Prolific’s advanced pre-
screening options, we recruited 29 participants from the US who self-reported
to have served on a jury. On average, the respondents took approximately 71
minutes to complete the survey, and were paid a base fee of £8.50. In order to
increase response quality (Vaughan, 2017), we also provided a performance-
based bonus.222Performance-based payments have been found to increase the
quality of responses in effort-responsive tasks (Vaughan, 2017). Hence, in
order to incentivize participants to provide high-quality survey responses, we
increased their bonus fee by $0.10 for each correct bail decision (i.e., when
their response to (Q2) coincided with the ground truth data about the
defendant’s two year recidivism), and decreased it by the same amount for each
incorrect bail decision. As additional quality control measures, we discarded
responses of participants who (i) did not respond to our 5 attention check
questions correctly, or (ii) completed the survey in less than 45 minutes.
After discarding the responses of these participants, our final sample
consisted of 20 participants. These 20 participants had an average criminal
recidivism prediction accuracy of 62.4%, close to the 62.1% and 60.2% that
Dressel and Farid (2018) and Grgić-Hlača et al. (2019) reported that their
participants achieved on the same task.

Dataset. The final dataset $\mathcal{D}$ consists of (i) the criminal
recidivism risk scores $\mathcal{S}^{i}_{j}\in\\{1,2,..,5\\}$ provided by our
20 respondents $i$ for 200 defendants $j$, as well as (ii) the COMPAS tool
risk scores $\mathcal{S}^{C}_{j}\in\\{1,2,..,10\\}$ for 7214 defendants from
the ProPublica dataset. The dataset can be found at
https://github.com/hansen7/LearnFairMetric_Empirical.

###  3.2 Distance Metric Learning

In our experiments, we evaluate the performance of several different
Mahalanobis metric learning approaches on our criminal recidivism dataset
$\mathcal{D}$. To cover a broad range of learning methods, we considered one
from each of the three learning paradigms discussed by (Bellet et al., 2013):
(i) fully supervised: Large Margin Nearest Neighbor (_LMNN_ , (Weinberger et
al., 2006)); (ii) weakly supervised: Mahalanobis Metric for Clustering (_MMC_
, (Xing et al., 2003)); and (iii) semi-supervised: Least Squared-residual
Metric Learning (_LSML_ , (Liu et al., 2012)).

LMNN. (Weinberger et al., 2006) The fully supervised _LMNN_ method can be
directly applied on the labeled data provided by the ProPublica dataset and
our respondents. It attempts to minimize the distance between training
instances and neighbors of same class, while keeping instances of other
classes out of the neighborhood. During the implementation, we consider that
instances with the same rating in our dataset belong to the same class. In
other words, in this approach, even though our data consists of Likert scale
ratings, we treat these ratings as categorical values, thereby losing some
information.

MMC. (Xing et al., 2003) The weakly supervised _MMC_ method is designed to
work in scenarios when rich labeled data is not readily available, and takes
pairwise relative comparisons as inputs instead. The algorithm maximizes the
sum of pairwise distances between dissimilar pairs while keeping that of
similar pairs relatively small. The metric learned by _MMC_ can be constrained
either in a diagonal form (weighted Euclidean) or as a full matrix. We
consider instances which have equal ratings in our dataset to be similar
pairs, and the others to be dissimilar pairs. Again, as for _LMNN_ , this
approach treats our Likert scale data as categorical values, disregarding some
info.

LSML. (Liu et al., 2012) Unlike the _LMNN_ and _MMC_ methods, which can only
utilize the 200 labeled instances, the semi-supervised metric learning
algorithm _LSML_ allows the use of the remaining ~7000 unlabeled instances
from the ProPublica dataset as well333We ran the algorithm both with and
without using the 7000 unlabeled COMPAS data points. The results were
qualitatively similar and we report the results of running the algorithm
without using the unlabeled data.. It learns the metric from a set of triplet
relative comparisons of the form "$a$ and $b$ are more similar than $a$ and
$c$". The triplet constraint set $\mathcal{C}$ is constructed as
$\mathcal{C}=\\{a,b,c|S_{a}\leq S_{b}+\sigma<S_{c}\\}$ , where $\sigma>0$ and
$S_{a,b,c}$ are the recidivism scores from our dataset. Unlike _LMNN_ and
_MMC_ , _LSML_ uses relative triplets, which allow us to capture more nuanced
information available from our Likert scale judgments, such as "2 is closer to
3 than 5".

We implement an adapted version of the algorithm, by adding a trade-off
coefficient $\alpha=0.01$ on the logdet regularization term. This adaptation
allowed us to reduce the weight of the regularizer, thereby increasing the
weight for satisfying the relative triplet constraints. As suggested by (Liu
et al., 2012), we randomly subsampled the training inputs, instead of
utilizing the full set $\mathcal{C}$ whose size is $\mathcal{O}(N^{3})$,
$N=140$.

Procedure. Each metric (_LMNN_ , _MMC_ , _LSML_) is trained on 140 inputs and
evaluated on 60 inputs. These 200 inputs were randomly selected. In our
evaluation, we repeat this process 10 times and report the average results.

##  4 Experiments

###  4.1 High-level Analysis of our Dataset

As described in Section 3.1, we gathered data about a subset of 200 defendants
from the ProPublica dataset in our survey. These 200 defendants have a similar
distribution of demographic features as the full set of 7214 defendants from
the ProPublica dataset. Below, we report a high-level analysis of our
respondent’s recidivism predictions (Q1), bail decisions (Q2), and decision
confidence (Q3).

Recidivism Predictions and Bail Decisions. In Table 1, we investigate the
relationship between the respondents’ recidivism predictions and their bail
decisions. In the first row, we see the fraction of defendants that were
granted bail by our respondents. As expected, as the predicted risk of
criminal recidivism increases, the propensity for granting bail decreases.
However, in the second and third row, we find that different respondents have
different thresholds for granting bail. For example, on average, our
respondents granted bail to 18.9% of defendants that they perceived as
extremely likely to recidivate within 2 years. However, some respondents
decided to grant bail to 76.5% of such respondents, while others to 0%.

|  |  | Recidivism Prediction |  |
---|---|---|---|---|---
Bail Rate | Extremely Unlikely | Unlikely | Neither | Likely | Extremely Likely
Mean | 99.6% | 96.6% | 84.7% | 43.1% | 18.9%
Max | 100% | 100% | 100% | 95.6% | 76.5%
Min | 96.8% | 77.9% | 50.0% | 0% | 0%
Table 1: Relationship between recidivism predictions and bail decisions.

Bail Decisions and Decision Confidence. In Table 2, we explore the
relationship between our respondents’ bail decisions and their confidence in
their decisions. The average accuracy of our respondents’ bail decisions is
62.4%. In the first column, we observe that our respondents tend to be
slightly more accurate for decisions in which they have a higher confidence.
However, in columns 2-4, we see that this relationship between confidence and
accuracy varies between respondents – some respondents have better calibrated
confidence assessments than others.

Accuracy | All Judges | Judge 1 | Judge 10 | Judge 18 | Judge 8 | Judge 17
---|---|---|---|---|---|---
Overall |  0.624$\pm$0.027 | 0.635 | 0.580 | 0.580 | 0.536 | 0.640
High Confidence | 0.649$\pm$0.061 | 0.828 | 0.750 | 0.714 | 0.580 | 0.620
Low Confidence |  0.619$\pm$0.037 | 0.594 | 0.526 | 0.544 | 0.621 | 0.660
Table 2: Relationship between bail decisions and decision confidence.

###  4.2 Evaluation

In addition to the distance metric learning approaches discussed in Section
3.2, we compare the three aforementioned metric learning approaches against
two baselines: (i) the trivial baseline of the _Euclidean_ metric (i.e., the
$\ell_{2}$ distance in the feature space), as well as (ii) the non-trivial,
but naïve _precision_ matrix (i.e., the inverse of the covariance matrix),
which is a standard approach for removing correlation between
features(Mahalanobis, 1936). We evaluate the performance of the learned
metrics with respect to the three loss functions defined below:

Relative Comparisons. This loss calculates the percentage of relative
comparison triplets (constructed as described in the previous section) from
the test set $\mathcal{C}_{t}$ which violate the constraints
$d_{\mathbf{M}}(a,b)\leq d_{\mathbf{M}}(a,c)$. For the distance metric
$\mathbf{M}$, it is similar to the loss defined by (Hoffer and Ailon, 2015),
where we assign equal weight to each instance:

$L(\mathbf{M}|\mathcal{C}_{t})=\frac{H(d_{\mathbf{M}}(\mathbf{x}_{a},\mathbf{x}_{c})-d_{\mathbf{M}}(\mathbf{x}_{a},\mathbf{x}_{b}))}{|\mathcal{C}_{t}|}\qquad\qquad\text{for
}\forall\\{\mathbf{x}_{a},\mathbf{x}_{b},\mathbf{x}_{c}\\}\in\mathcal{C}_{t}$
(1)

where $H(\cdot)$ is the Heaviside step function.

kNN L1. This loss calculates the $\ell_{1}$ divergence between the test
instance ground truth label and the weighted rating of its neighbors, defined
by the metric $\mathbf{M}$:

$L(\mathbf{M}|\mathcal{C}_{t})=\sum_{\mathbf{x}_{i}\in\mathcal{C}_{t}}|\hat{y}_{i}-y_{i}|=\sum_{\mathbf{x}_{i}\in\mathcal{C}_{t}}|\sum_{j}w_{ij}y_{j}-y_{j}|$
(2)

where the normalised weight factor $w_{ij}$ is proportional to the inverse of
the distance between $i$ and its k nearest neighbors $j$: $w_{ij}\propto
1/{d_{\mathbf{M}}(\mathbf{x}_{i},\mathbf{x}_{j})}$.

kNN L2. This loss is similar to kNN $\ell_{1}$ but instead calculates the
$\ell_{2}$ distance. Compared to the $\ell_{1}$ norm, it introduces more
penalty for large prediction errors.

###  4.3 Metric Learning on Human Judgments

For the collected human survey, we implemented _LMNN, MMC, LSML_ in Section
3.2 following the procedure described in Section 3.3. The trade off
coefficient $\alpha$ of the regularizer in our adapted _LSML_ is set to 0.01,
and the number of neighbors for calculating _kNN L1, kNN L2_ is chosen to be
five444We tried varying these parameters. They do not affect the high-level
takeaways of our results.. The results are shown in Figure 1.

Figure 1: Performance of the learned metrics, evaluated on our survey data.

In Figure 1, we observe that the learned metrics slightly outperform the
Euclidean and Precision metrics with respect to the kNN L1 and L2 loss. For
the triplet relative comparison loss, which incorporates the relative order
between nearby ratings instead of treating them as categorical variables, the
learned metrics have significantly outperformed the Euclidean and Precision
metrics.

###  4.4 Relative Comparisons on COMPAS

In this section we evaluate the sensitivity of our adapted _LSML_ method to
the hyperparameter $\sigma$, which controls the minimum required distance
between inputs $b$ and $c$. To this end, we compare the loss of the learned
metric with the Euclidean metric on the COMPAS tool’s predictions, from the
ProPublica dataset, instead of the human judgments we gathered. Recall Section
3.2, which describes how we construct the sets of triplet constraints based on
the choice of $\sigma$. The loss based on the relative comparisons is then
calculated on the resulting $\mathcal{C}_{t}$.

$\sigma_{t}$ | Euclidean | Ours($\sigma=0$) | Ours($\sigma=2$)
---|---|---|---
0 |  0.40$\pm$0.037 |  0.39$\pm$ 0.041 | 0.38$\pm$ 0.035
2 |  0.40$\pm$0.027 |  0.39$\pm$ 0.032 | 0.37$\pm$ 0.037
4 |  0.35$\pm$0.031 |  0.31$\pm$ 0.045 | 0.30$\pm$ 0.034
6 |  0.31$\pm$0.033 | 0.29$\pm$ 0.060 | 0.29$\pm$ 0.060
Table 3: Loss of the metric learned using the adapted _LSML_ algorithm on the
ProPublica dataset. Here $\sigma$ and $\sigma_{t}\geq 0$ represent the
threshold values for constructing triplet constraints from the train and test
set respectively.

Our learned metrics outperform the Euclidean distance by a large margin. As
$\sigma_{t}$ increases, the loss decreases for all three metrics, since the
difference between $b$ and $c$ increases.

##  5 Discussion

In this work, we conducted a user study, in which we gathered a set of human
judgments about recidivism risk. We initiated work to examine various methods
for learning an individual fairness metric from human annotated data.
Surprisingly, we observed similar performance across methods when considering
predictive performance of ratings, though we saw differences when considering
triplet consistency of unseen data. In future work, it would be interesting to
understand this better and consider the interplay between the metric-learning
methods and the consistency of human ratings.

## Acknowledgements

HW acknowledges support from Cambridge Trust CSC Scholarship. AW acknowledges
support from the David MacKay Newton research fellowship at Darwin College,
The Alan Turing Institute under EPSRC grant EP/N510129/1 & TU/B/000074, and
the Leverhulme Trust via the CFI. This research was supported in part by a
European Research Council (ERC) Advanced Grant for the project "Foundations
for Fair Social Computing", funded under the European Union’s Horizon 2020
Framework Programme (grant agreement no. 789373).

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