# Kähler geometry of quiver varieties and machine learning

George Jeffreys  Department of Mathematics and Statistics, Boston University,
111 Cummington Mall, Boston MA 02215, USA<EMAIL_ADDRESS>and  Siu-Cheong Lau
Department of Mathematics and Statistics, Boston University, 111 Cummington
Mall, Boston MA 02215, USA<EMAIL_ADDRESS>
###### Abstract.

We develop an algebro-geometric formulation for neural networks in machine
learning using the moduli space of framed quiver representations. We find
natural Hermitian metrics on the universal bundles over the moduli which are
compatible with the GIT quotient construction by the general linear group, and
show that their Ricci curvatures give a Kähler metric on the moduli. Moreover,
we use toric moment maps to construct activation functions, and prove the
universal approximation theorem for the multi-variable activation function
constructed from the complex projective space.

##  1\. Introduction

Machine learning by artificial neural networks has made exciting developments
and has been applied to many branches of science in recent years.
Mathematically, stochastic gradient flow over a matrix space (or called the
weight space) is the central tool. The non-convex nature of the cost function
has made the problem very interesting. Current research has focused on
different types of stochastic gradient flows and finding new types of
networks, which have brought great improvements of computational efficiency.

In geometry and physics, the applications of gradient flow and Morse theory
have a long history and have brought numerous fundamental breakthroughs. For
instance, the gradient flow of the Yang-Mills functional is used to find
Hermitian Yang-Mills connections, whose existence in a stable holomorphic
vector bundle is proved by Donaldson [Don85] and Uhlenbeck-Yau [UY86]. The
celebrated Ricci flow found by Hamilton [Ham82], which is a crucial tool to
solve the three-dimensional Poincaré conjecture, is essentially a gradient
flow [Per02, Per03]. Its Kähler analog has been an important tool in finding
Kähler-Einstein metrics on Fano manifolds [Yau96, Tia97, Don12, CSW18, CDS15a,
CDS15b, CDS15c]. In these works, GIT quotients and finite-dimensional models
have provided important motivations and guidelines [Don99]. Hamiltonian Floer
theory [Flo89], which is essentially Morse theory on the loop space, was
invented to solve the Arnold conjecture [FHS95, Ono95, FO99]. Various versions
of Floer theory have been crucial ingredients in the study of mirror symmetry.

In this paper, we would like to develop a foundational _algebro-geometric
formulation_ for neural networks in machine learning. The theory of _quiver
representations_ , which is a well-developed branch of mathematics motivated
from Lie theory and has been an important tool in mathematical physics, will
be well suited for this purpose.

A quiver representation assigns to a directed graph $Q$ a bunch of vector
spaces for the vertices and a bunch of linear maps for the arrows. Such a
construction is in common with neural networks. However, in order to use
quiver theory to formulate machine-learning neural networks, there are two
main differences between these two subjects that needs to be addressed.

  1. (1)

Compactness of moduli space. A moduli space of quiver representations [Kin94]
is defined by identifying _isomorphic_ quiver representations using GIT
quotients. As a result, the moduli space is _compact_ when the quiver has no
oriented cycle. On the other hand, the matrix space used in neural networks is
non-compact. In machine learning, isomorphic quiver representations may
correspond to physically different input or output information and in general
cannot be identified.

  2. (2)

Non-linearity. _Activation functions_ , which are non-linear maps on the
vector spaces over the vertices, serve as a crucial ingredient to achieve
machine learning of non-linear functions. Such non-linearity jumps out of the
category of quiver representations. This is also related to the first point
above. Namely, such non-linear maps are not necessarily equivariant under the
group of automorphisms of quiver representations.

For the first point, we shall use framed quiver representations, which were
first found by Nakajima [Nak94] in the study of affine Lie algebras. A framed
representation assigns to each vertex a vector space together with a choice of
‘framing’ (for instance it is a basis in the simplest situation). In the
applications considered here, such a decoration makes sure that isomorphic
framed quiver representations correspond to the same physical state. Note that
framed quiver moduli $\mathcal{M}$ are also compact when $Q$ has no oriented
cycle. _Compactness_ is one of the main advantages of our algebro-geometric
formulation, which makes sure the convergence of a gradient flow.

In this formulation, the weight matrices are encoded as morphisms between the
_universal vector bundles_ (over the framed quiver moduli) associated to the
vertices. The data flow is encoded by sections of the universal bundles, which
are sent from one to another bundles by the morphisms associated to the arrows
of $Q$. The cost function, and hence its gradient flow, is defined on the
framed quiver moduli $\mathcal{M}$.

In particular, the _critical points and the gradient flow are controlled by
the topology of $\mathcal{M}$_ (for instance, the Morse inequalities). The
topology of a framed quiver moduli is well-understood by the work of Reineke
[Rei08] when $Q$ has no oriented cycle. $\mathcal{M}$ is an iterated Grassmann
bundle, and its Poincaré polynomial is a product of that of the Grassmannians.

For the purpose of gradient flow, one needs to choose a _Kähler metric_ on
$\mathcal{M}$, and also Hermitian metrics on the universal vector bundles. As
a result, we have found metrics that are defined by explicit beautiful
formulae. These metrics are not just $U_{\vec{d}}$-equivariant so that they
descend to symplectic quotients, but are also
$\mathrm{GL}_{\vec{d}}$-equivariant and hence compatible with the GIT
construction of $\mathcal{M}$. Moreover, they are compatible with the iterated
Grassmann structure found by Reineke. In application, such metrics would
simplify the actual computational algorithm over the quiver moduli. They are
summarized as follows.

######  Theorem 1.1 (Combining Theorem 3.7, 3.15,3.18).

Let $Q$ be an arbitrary quiver. Fix a vertex $i\in Q_{0}$. Let $\rho$ be the
row vector whose entries are $V_{\gamma}e^{\left(t(\gamma)\right)}$, where
$\gamma$ is any path whose head $h(\gamma)$ is $i$ (including the trivial
path), $t(\gamma)$ denotes its tail, and
$V_{\gamma}\in\mathrm{Hom}(\mathbb{C}^{d_{t(\gamma)}},\mathbb{C}^{d_{h(\gamma)}})$
is the representing matrix of $\gamma$. Then

$(\rho_{i}\rho_{i}^{*})^{-1}=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}$

is $\mathrm{GL}_{\vec{d}}$-equivariant, and it descends to a metric on the
universal bundle $\mathcal{V}_{i}$ over a certain domain of convergence
$\mathcal{M}^{\circ}$.

When $Q$ has no oriented cycle, $\mathcal{M}^{\circ}=\mathcal{M}$. Moreover,
the Ricci curvature of the induced metric on $\bigotimes_{i\in
Q_{0}}\mathcal{V}_{i}$ gives a Kähler metric on $\mathcal{M}$.

The precise definition of $M^{\circ}$ is given in Section 3.3.

In this paper, we focus on the framed quiver moduli defined over complex
numbers. In actual applications, we can also restrict to real coefficients.
Then the above formula defines a bundle metric over
$\mathcal{M}_{\mathbb{R}}$, and the Ricci curvature gives a Riemannian metric
on $\mathcal{M}_{\mathbb{R}}$.

Now let us address the second point. Namely, we need to introduce _non-
linearity_ in addition to the usual theory of quiver representations. By
definition, morphisms between universal vector bundles over $\mathcal{M}$ are
linear along fibers. They correspond to weight matrices in neural networks. To
introduce non-linearity, we shall treat the universal bundles as fiber bundles
and construct suitable _fiber-bundle maps_ that play the role of activation
functions.

One of the commonly-used activation functions is

$\frac{e^{2x}}{1+e^{2x}}:\mathbb{R}\to(0,1).$

We observe that this function also appears in the base of the symplectic
trivialization of the open dense toric orbit of $\mathbb{P}^{1}$ as a toric
variety:

$(\mathbb{C}^{\times},\omega_{\mathbb{C}\mathbb{P}^{1}})\cong((0,1)\times\mathbb{S}^{1},\omega_{\mathrm{std}}),$

or lifted to the universal cover:

$(\mathbb{C},\pi^{*}\omega_{\mathbb{C}\mathbb{P}^{1}})\cong((0,1)\times\mathbb{R},\omega_{\mathrm{std}}).$

Here, $\omega_{\mathbb{C}\mathbb{P}^{1}}$ denotes the Fubini-Study metric of
$\mathbb{C}\mathbb{P}^{1}$, that is, the standard area form of the unit
sphere; $\omega_{\mathrm{std}}=dx\wedge dy$ is the standard symplectic form.

Similarly, another activation function

$\frac{z}{\sqrt{1+|z|^{2}}}:\mathbb{C}\to\mathbb{C}$

also arises as a symplectic trivialization:
$(\mathbb{C},\omega_{\mathbb{C}\mathbb{P}^{1}})\cong(\\{w\in\mathbb{C}:|w|<1\\},\omega_{\mathbb{C}}).$

Motivated from these observations, we consider

$\sigma(x)=\left(\frac{e^{2x_{i}}}{1+\sum_{j=1}^{n}e^{2x_{j}}}\right)_{i=1}^{n}:\mathbb{R}^{n}\to\Delta$

and

$\psi(\vec{z})=\left(\frac{z_{i}}{\sqrt{1+\|\vec{z}\|^{2}}}\right)_{i=1}^{n}:\mathbb{C}^{n}\to\\{\vec{w}\in\mathbb{C}^{n}:\|\vec{w}\|<1\\}$

as multi-variable activation functions, where $\Delta$ denotes the standard
simplex with vertices $0$ and $\epsilon_{1},\ldots,\epsilon_{n}$, the standard
basis of $\mathbb{R}^{n}$. They arise from symplectomorphisms
$(\mathbb{C}^{n},\omega_{\mathbb{C}\mathbb{P}^{n}})\cong(\\{\vec{w}\in\mathbb{C}^{n}:\|\vec{w}\|<1\,\omega_{\mathbb{C}^{n}}\\})$.
More generally, these come from moment maps of _toric varieties_ [Gui94,
Abr98]. We note that $\psi$ has an advantage of being $U(n)$-equivariant.

The _universal approximation theorem_ (see for instance [Cyb89, Pet99, MM92,
Pin99]) provides a theoretical foundation for the success of neural networks.
In existing literature, the theorem was proved for single-variable activation
functions.

In this paper, we prove the universal approximation theorem for the above
multivariable function $\sigma=\sigma_{\mathbb{R}^{n}}$. Note that $\sigma$ is
the softmax function
$\left(\frac{e^{2x_{i}}}{\sum_{j=0}^{n}e^{2x_{j}}}\right)_{i=0}^{n}$
restricted to the hyperplane $x_{0}=0$ and composed with the projection along
$x_{0}$-direction. We shall restrict to real coefficients in this theorem.

######  Theorem 1.2 (same as Theorem 5.2).

Let $K$ be a compact set of $\mathbb{R}^{d_{1}}$, and
$f:K\to\mathbb{R}^{d_{3}}$ a continuous function. For any $\epsilon>0$, there
exists $d_{2}>0$ and $W_{1}\in\operatorname{Mat}(d_{2},d_{1})$,
$W_{2}\in\operatorname{Mat}(d_{3},d_{2})$, $b\in\mathbb{R}^{d_{2}}$ such that
$\|f^{U}_{W_{1},W_{2},b}-f\|_{L^{2}(K)}<\epsilon$. Here,
$f^{U}_{W_{1},W_{2},b}(x)=W_{2}\cdot\sigma_{\mathbb{R}^{d_{1}}}(W_{1}\cdot
x+b)$ is the function coming from the $A_{3}$ quiver.

The $A_{3}$ quiver corresponds to the feed-forward network with one input
layer, one middle layer and one output layer. See Figure 1.

${a}$${b}$${c}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$ Figure 1. The
$A_{3}$ quiver.

The above theorem is proved by using the tropical limit of the toric manifold
$\mathbb{P}^{n}$, and a geometric object that we call a centered polyhedral
web, which is an analog of a tropical variety in an integral affine manifold.
Since we do not have integral structure in the context here, we need to invent
this new notion.

In above, we have focus on explaining non-linearity for a single vector space.
We shall _globalize them as non-linear fiber-bundle morphisms_ for the
universal bundles over $\mathcal{M}$. This can be achieved with the help of
Hermitian metrics on the universal bundles, so that the Fubini-Study metric on
$\mathbb{P}^{n}$ can be globalized as a fiberwise symplectic structure on
projective bundles over $\mathcal{M}$. Actually, the globalization from a
single framing vector space $V$ to the universal fiber bundle over
$\mathcal{M}$ _works for any continuous function_ $V\to V$ (and in particular
for a symplectomorphism from $V$ to its image). Combining the ingredients
explained above together, we can construct a gradient flow over the framed
quiver moduli to achieve machine learning. The detail is given in Section 4.

Such an algebro-geometric formulation has several _advantages_. First, the
gradient flow under consideration runs in a _compact_ manifold. This ensures
the existence of absolute extrema, convergence of the flow, and upper bound
for the norm of the gradient vector field. Second, because of compactness, the
flow is constrained by _topology_ of the manifold due to Morse theory. See
Section 4.4. Finally, the moduli space has extra _symmetry_ coming from
framing. If we use activation functions that respect this symmetry (for
instance $\psi$ above enjoys $U(n)$-equivariance), we can perform dimension
reduction which improves the effectiveness of the network. (See Proposition
4.16 and 4.17.)

In summary, from this point of view, the success of neural network is resulted
from the interplay between algebraic morphisms and (transcendental)
symplectomorphisms. Interestingly, such an interplay is also an important
feature that occurs in the study of complete integrable systems and mirror
symmetry for toric manifolds and flag varieties, see for instance [Gui94,
FLTZ12, Abo06, CLL12, NNU10, HKL18].

### Some related works

The relation between neural network and quiver representation was investigated
in the recent paper [AJ20]. Their work considered the quotient space by
$(\mathbb{C}^{\times})^{N}$ of pairs $(W,f)$, where $W$ is a quiver
representation of $Q$ with the dimension vector $\vec{1}$, and $f$ associates
each vertex a function $\mathbb{C}\to\mathbb{C}$ (playing the role of an
activation function). Moreover, in dimension $\vec{1}$ (which is a typical
case for machine learning), they invented an interesting way of encoding the
data flow as a quiver representation. (In our work, the data flow is given as
sections of universal bundles over the quiver moduli.)

The approach and the goal of this paper is rather different. We aim at
formulating machine learning as a gradient flow over a compact quiver moduli.
In [AJ20], quiver representations were used in encoding the data in the
network; however, the machine learning process was independent of the quiver
moduli. Moreover, ‘double-framing’ was used, and the corresponding moduli
space is non-compact.

The map $(W,f)\mapsto W$ gives an infinite-dimensional fiber bundle over the
quiver moduli $\mathcal{M}$, whose fibers are the spaces of choices of
activation functions. In a typical program of machine learning, the activation
functions are fixed during the optimization process. In order to formulate the
program as a gradient flow over the compact moduli $\mathcal{M}$, we found a
non-trivial way by equipping intermediate vertices with additional framings
and metrics, so that we can lift $f$ to be a well-defined fiber-bundle map
over $\mathcal{M}$. Note that $f$ is not equivariant under the group action of
$(\mathbb{C}^{\times})^{N}$ ($\mathrm{GL}_{\vec{d}}$ in the higher rank case).
Such a lifting is an important non-trivial step.

Furthermore, we have dealt with representations of general rank $\vec{d}$, and
a class of activation functions coming from toric symplectomorphisms.
Different functions (on the same domain and target) are obtained if we deform
the toric Kähler metric. To also optimize the activation functions during the
learning process (see also [GGL19]), we may consider a gradient flow on
$\mathcal{M}\times\mathcal{K}$ where $\mathcal{K}$ denotes the moduli of toric
Kähler metrics in the same class. By the celebrated works of [Don99, Sem92],
$\mathcal{K}$ is an infinite-dimensional negatively curved symmetric space.

Recently, there is a rising interest of applying geometric techniques to the
study of neural networks. For instance, in the works [GBH18, CYRL19],
hyperbolic spaces are applied to machine learning in graphs and achieved great
performance.

Moreover, the applications of symmetry and group equivariance in neural
networks were studied and developed in [CW16, CGW19, CGKW18, CWKW19, CAW+19,
dHCW20]. Overall, these works aim at capturing symmetry of the input data and
designing networks that are adapted to such symmetry. Moreover, homogeneous
spaces (in place of vector spaces) have been employed in layers of
convolutional neural networks.

In comparison, our paper aims at revealing the geometric nature of neural
networks and build a connection with algebraic geometry. The resulting framed
quiver moduli, which has interesting topology and metrics, is the main
geometric object of interest. Furthermore, we study activation functions that
respects the ‘intrinsic symmetry’ over the quiver moduli, which can provide a
more effective algorithm by dimension reduction.

In the reverse direction, there are interesting applications of machine
learning in frontier geometry and physics. For instance, [HY20] used machine
learning to solve problems in computing graph Laplacians, such as recognizing
graph Ricci-flatness and predicting the spectral gap. In physics, [HSTT18a,
HSTT18b] used deep learning to study AdS/CFT correspondence by discretizing
the equation of motion. Since we have formulated machine learning using quiver
representations, it will be interesting to find direct relations between
quiver gauge theory and these problems that can be attacked via machine
learning.

### Organization of this paper

In Section 2, we will take a quick review on quiver representations and their
moduli spaces. In Section 3, we will construct nice Hermitian metrics on
universal bundles over the moduli. For readers who are mainly interested in
machine learning, Section 3 can be skipped for the first reading. Then we give
an algebro-geometric formulation of neural network using quiver
representations in Section 4. In Section 5, we prove the universal
approximation theorem for the multivariable activation function $\sigma$.

## Acknowledgment

We are grateful to Marco Antonio Armenta for informing us about the work
[AJ20] and the further useful discussions. We express our gratitude to Shing-
Tung Yau for his generous encouragement. The work of S.C. Lau in this paper is
partially supported by the Simons collaboration grant.

##  2\. Review of framed quiver moduli

Let $Q$ be a directed graph. Denote by $Q_{0},Q_{1}$ the set of vertices and
arrows respectively. A quiver representation $V$ with dimension vector
$\vec{d}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$ associates each arrow $a$ with a
matrix $V(a)$ of size $d_{h(a)}\times d_{t(a)}$ (where $h(a),t(a)$ denote the
head and tail vertices of $a$ respectively). The set of complex quiver
representations with dimension $\vec{d}$ form a vector space denoted by
$R_{\vec{d}}(Q)$. The moduli space of quiver representations is a GIT quotient
of $R_{\vec{d}}(Q)$ by the group of isomorphisms
$\mathrm{GL}(\vec{d})=\prod_{i\in Q_{0}}\mathrm{GL}(d_{i},\mathbb{C})$
[Kin94], where $\mathrm{GL}(\vec{d})$ acts on $R_{\vec{d}}(Q)$ via

$g\cdot(V(a):a\in Q_{1})=(g_{h(a)}\cdot V(a)\cdot g_{t(a)}^{-1}:a\in Q_{1}).$
(2.1)

In the applications we consider in this paper, since the vector space over the
input and output vertices are equipped with fixed basis with physical
meanings, we need to use framed quiver representations [Nak94, Nak01, CB03,
Rei08].

Let $\vec{d},\vec{n}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$. $\vec{n}$ will be the
dimension vector for the framing, which is a linear map
$e^{(i)}:\mathbb{C}^{n_{i}}\to V_{i}$ at each $i\in Q_{0}$ (where
$V_{i}=\mathbb{C}^{d_{i}}$). Since we will take a quotient by
$\mathrm{GL}(\vec{d})$, we shall think of $V_{i}$ as a vector space without a
preferred basis, while $\mathbb{C}^{n_{i}}$ is equipped with the standard
basis.

######  Definition 2.1.

The vector space of framed representations is given by

$R_{\vec{n},\vec{d}}=R_{\vec{d}}\times\bigoplus_{i\in
Q_{0}}\mathrm{Hom}(\mathbb{C}^{n_{i}},\mathbb{C}^{d_{i}}).$

It carries a natural action of $\mathrm{GL}(\vec{d})$ given by
$g\cdot(V,e)=(g\cdot V,(ge^{(i)}:i\in Q_{0}))$, where $g\cdot V$ is given by
Equation (2.1).

We need to remove unstable framed representations from $R_{\vec{n},\vec{d}}$
in order to get a nice quotient by $\mathrm{GL}(\vec{d})$.

######  Theorem 2.2 ([Nak96]).

$(V,e)\in R_{\vec{n},\vec{d}}$ is called stable if there is no proper
subrepresentation $U$ of $V$ which contains $\mathrm{Im}\,e$. The set of all
stable points of $R_{\vec{n},\vec{d}}$ is denoted by
$R_{\vec{n},\vec{d}}^{s}$. Then the quotient
$\mathcal{M}_{\vec{n},\vec{d}}:=R_{\vec{n},\vec{d}}^{s}/\mathrm{GL}(\vec{d})$
is a smooth variety, which is called to be a framed quiver moduli.

Actually $\mathcal{M}_{\vec{n},\vec{d}}$ can be formulated as a GIT quotient
[CB03, Rei08]. Namely, by adding an extra vertex labeled as $\infty$ to the
quiver and $n_{i}$ arrows from the vertex $\infty$ to the vertex $i$, $(V,e)$
can be identified as a usual representation of this bigger quiver with the
dimension vector $(\vec{d},1)$. The above stability condition can be rewritten
as slope stability, and hence it is a GIT quotient [Kin94].

Since $(d,1)$ is a primitive vector, $\mathcal{M}_{\vec{n},\vec{d}}$ is a
smooth fine moduli. There are universal vector bundles $\mathcal{V}_{i}$ over
$\mathcal{M}_{\vec{n},\vec{d}}$ corresponding to each vertex $i$, with fibers
$\mathcal{V}_{i}|_{[V,e]}=V_{i}$.

######  Example 2.3.

For the quiver with a single vertex and no arrow, and $n>d$,

$\mathcal{M}_{n,d}=\operatorname{Gr}(n,d)=\\{e\in\mathrm{Hom}(\mathbb{C}^{n},\mathbb{C}^{d}):e\textrm{
is surjective}\\}/\mathrm{GL}_{\vec{d}}$

is the (dual) Grassmannian. We have the tautological bundle $\mathcal{V}$ over
$\operatorname{Gr}(n,d)$. (Note that this tautological bundle is dual to the
one on $\operatorname{Gr}(d,n)\cong\operatorname{Gr}(n,d)$.)

The topology of $\mathcal{M}_{\vec{n},\vec{d}}$ is well-understood. Let’s make
an ordering of the vertices. Namely the vertices are labeled by
$\\{1,\ldots,N\\}$, such that $i<j$ implies there is no arrow going from $j$
to $i$. Such a labeling exists if $Q$ has no oriented cycle.

######  Theorem 2.4 (Reineke [Rei08]).

Assume $Q$ has no oriented cycle. Consider the chain of iterated Grassmann
bundles $M^{(N)}\stackrel{{\scriptstyle
p_{N}}}{{\to}}M^{(N-1)}\stackrel{{\scriptstyle
p_{N-1}}}{{\to}}\ldots\stackrel{{\scriptstyle
p_{2}}}{{\to}}M^{(1)}\stackrel{{\scriptstyle p_{1}}}{{\to}}\mathrm{pt}$ (where
$\mathrm{pt}$ denotes a singleton) defined by induction:

$M^{(i)}=\operatorname{Gr}_{M^{(i-1)}}\left(\underline{\mathbb{C}^{n_{i}}}\oplus\bigoplus_{j\to
i}p_{i-1}^{*}\dots p_{j+1}^{*}(S_{j}),d_{i}\right)\to M^{(i-1)},$

where $S_{i}$ denotes the tautological bundle on $M_{i}$ (as a Grassmann
bundle over $M_{i-1}$). (The direct sum is over each arrow $j\to i$.) Then
$\mathcal{M}_{\vec{n},\vec{d}}\cong M^{(N)}$, with universal bundles
$\mathcal{V}_{i}\cong p_{N}^{*}\dots p_{i+1}^{*}S_{i}$ for all $i\in Q_{0}$.

######  Corollary 2.5 (Reineke [Rei08]).

The Poincare polynomial of nonempty $\mathcal{M}_{\vec{n},\vec{d}}$ is given
by

$\prod_{i\in Q_{0}}\binom{n_{i}+\sum_{j\to i}d_{j}}{d_{i}}_{q^{2}}$

where

$\binom{n}{d}_{q}=\prod_{k=1}^{d}\frac{q^{n-d+k}-1}{q^{k}-1}.$

######  Remark 2.6.

In [Rei08], the framing $e$ goes in the other direction (from
$\mathbb{C}^{d_{i}}$ to $\mathbb{C}^{n_{i}}$). The above theorem is stated in
the dual way, which is the convention we take for the rest of this paper.

######  Example 2.7.

Consider the $A_{3}$-quiver which has three vertices $i=1,2,3$ and two arrows
$a_{1}:1\to 2,a_{2}:2\to 3$. Suppose $n_{1}=d_{1}$, $n_{2}=d_{2}+1$ and
$n_{3}=d_{3}$. Then the iterated Grassmann bundle is $M^{(3)}\to M^{(2)}\to
M^{(1)}$, where $M^{(1)}=\operatorname{Gr}(d_{1},d_{1})=\mathrm{pt}$ (and its
tautological bundle is the vector space $\mathbb{C}^{d_{1}}$);
$M^{(2)}=\operatorname{Gr}(d_{2}+1+d_{1},d_{2})$ is equipped with the
tautological bundle $S_{2}$ of rank $d_{2}$;
$M^{(3)}=\operatorname{Gr}_{\operatorname{Gr}(d_{2}+1+d_{1},d_{2})}(\underline{\mathbb{C}}^{d_{3}}\oplus
S_{2},d_{3})$ is a Grassmannian bundle over
$\operatorname{Gr}(d_{2}+1+d_{1},d_{2})$ with fibers
$\operatorname{Gr}(d_{2}+d_{3},d_{3})$. The corresponding Poincare Polynomial
will be

$\left(\prod_{k=1}^{d_{3}}\frac{q^{2(d_{2}+k)}-1}{q^{2k}-1}\right)\left(\prod_{k=1}^{d_{2}}\frac{q^{2(d_{1}+1+k)}-1}{q^{2k}-1}\right).$

See Figure 2.

${1}$${Gr(d_{1},d_{1})=\mathrm{pt}}$${2}$${Gr(d_{2}+1+d_{1},d_{2})}$${3}$${\operatorname{Gr}_{\operatorname{Gr}(d_{2}+1+d_{1},d_{2})}(\underline{\mathbb{C}}^{d_{3}}\oplus
S_{2},d_{3})}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$ Figure 2. The
iterated Grassmann bundles associated to the $A_{3}$ quiver.

##  3\. Hermitian Metric over framed quiver moduli

In constructing fiber-bundle endomorphisms, it will be crucial to consider
Kähler metrics on universal bundles. In this section, we find a beautiful
formula for the canonical metric on the universal bundle $\mathcal{V}_{i}$
over $\mathcal{M}_{n,d}$ written in homogeneous coordinates. Using this
formula, we then show that the sum of Ricci curvatures over the vertices $i$
give a Kähler metric on $\mathcal{M}_{\vec{n},\vec{d}}$.

First, let us begin by recalling the typical example $\operatorname{Gr}(n,k)$.

###  3.1. The Grassmannian

Consider

$\operatorname{Gr}\left(n,k\right)=\operatorname{Mat}^{\mathbb{C}}_{n,k}\sslash_{\chi=1}U(k)=\left\\{e\in\operatorname{Mat}^{\mathbb{C}}_{n,k}:ee^{*}=I_{k}\right\\}\big{/}U\left(k\right)$

for $n\geq k$. Here we have used the dual description which better matches the
frame convention used in this paper. Namely,
$\operatorname{Gr}\left(n,k\right)$ parametrizes $k$-dimensional quotient
vector spaces of a fixed $n$-dimensional vector space as opposed to
$k$-dimensional subspaces. The moment map for the standard $U(k)$-action on
$\operatorname{Mat}^{\mathbb{C}}_{n,k}$ is
$ee^{*}:\operatorname{Mat}^{\mathbb{C}}_{n,k}\to\mathbf{i}\mathfrak{u}_{k}$.
We have taken the moment map level $\chi=1$ in the above symplectic reduction.
Note that $U(k)$ is acting on the left, although in the above expression
$U(k)$ appears on the right.

Writing $e=\left(b,p\right)$ where $b\in\operatorname{Mat}^{\mathbb{C}}_{k,k}$
and $p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}$, the moment-map equation
$ee^{*}=I_{k}$ becomes

$bb^{*}+pp^{*}=I_{k}.$

We shall consider the chart defined by

$U=\\{[b,p]\in\operatorname{Gr}(n,k):\det
b\not=0\\}\cong\operatorname{Mat}^{\mathbb{C}}_{n-k,k}$

where the identification is given by the holomorphic coordinates

$\zeta^{h}=b^{-1}p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}.$

We also have the symplectic coordinates

$\zeta^{u}=\left(b^{*}b\right)^{\frac{1}{2}}b^{-1}p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}.$

The entries of $\zeta^{u}$ are not meromorphic functions. On the other hand,
$\zeta^{u}$ has the advantage that it satisfies the moment-map equation

$b^{*}b+\zeta^{u}(\zeta^{u})^{*}=I_{k}.$ (3.1)

(Note that the first term is $b^{*}b$ instead of $bb^{*}$.)

The construction of $\zeta^{u}$ uses the polar decomposition

$b=\left(b\left(b^{*}b\right)^{-\frac{1}{2}}\right)\left(b^{*}b\right)^{\frac{1}{2}}$

where $\left(b\left(b^{*}b\right)^{-\frac{1}{2}}\right)\in U(k)$ and
$\left(b^{*}b\right)^{\frac{1}{2}}\in\mathbf{i}\mathfrak{u}(k)$ is positive
definite. We obtain the coordinates $\zeta^{u}$ by observing

$[b,p]=\left[\left(b^{*}b\right)^{\frac{1}{2}},\zeta^{u}\right]$

using the left-$U(k)$-action, such that the first component
$\left(b^{*}b\right)^{\frac{1}{2}}$ is Hermitian, and is determined
$\zeta^{u}$ due to the moment-map equation (3.1).

The two coordinate systems are related by

$\zeta^{u}=\left(b^{*}b\right)^{\frac{1}{2}}\cdot\zeta^{h}.$ (3.2)

Let $S$ be the tautological vector bundle whose fibers are the quotient vector
spaces. (This is dual to the tautological bundle of
$\operatorname{Gr}(k,n)\cong\operatorname{Gr}(n,k)$.) It can be written as the
quotient of the trivial bundle:

$S=\left(\left\\{ee^{*}=I_{k}\right\\}\times\mathbb{C}^{k}\right)\big{/}U\left(k\right)$

where the left action of $U(k)$ on $\mathbb{C}^{k}$ is the standard one.

We now take the standard metric on $\mathbb{C}^{k}$, which is preserved by
$U(k)$ and hence descends to a metric $H$ of $S$.

Denote the standard basis of $\mathbb{C}^{k}$ by $\epsilon_{j}$ for
$j=1,\ldots,k$. Under this metric, we have the lifting of a local Hermitian
frame over the chart $U=\\{\det b\not=0\\}$ being

$u_{i}=b\left(b^{*}b\right)^{-\frac{1}{2}}\cdot\epsilon_{i}$

since
$\left[b,p,b\left(b^{*}b\right)^{-\frac{1}{2}}\epsilon_{i}\right]\sim\left[\left(b^{*}b\right)^{\frac{1}{2}},\zeta^{u},\epsilon_{i}\right]$.

We also have the lifting of a local holomorphic frame

$h_{i}=b\epsilon_{i}=b\left(b^{*}b\right)^{\frac{1}{2}}b^{-1}u_{i}$

since
$\left[b,p,b\epsilon_{i}\right]\sim\left(I_{k},\zeta^{h},\epsilon_{i}\right)$.
The two frames are related as follows.

######  Lemma 3.1.

$h_{i}=u_{j}a_{i}^{j}$ where
$\left(a_{i}^{j}\right)=\left(b^{*}b\right)^{\frac{1}{2}}$, $i$ is indexing
the colomns and $j$ is indexing the rows.

###### Proof.

Consider

$\left(h_{1}\ldots
h_{k}\right)=b\left(b^{*}b\right)^{\frac{1}{2}}b^{-1}\left(u_{1}\ldots
u_{k}\right)=\left(u_{1}\ldots u_{k}\right)\left(a_{i}^{j}\right).$

Thus $\left(a_{i}^{j}\right)=\left(u_{1}\ldots
u_{k}\right)^{-1}b\left(b^{*}b\right)^{\frac{1}{2}}b^{-1}\left(u_{1}\ldots
u_{k}\right)$.

$\left(u_{1}\ldots u_{k}\right)=b\left(b^{*}b\right)^{-\frac{1}{2}}$

since $u_{i}=b\left(b^{*}b\right)^{-\frac{1}{2}}\epsilon_{i}$. Result follows.
∎

######  Proposition 3.2.

The metric $H$ defined above on the tautological bundle $S$ is represented by
the matrix $\left(I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{*}\right)^{-1}$ in
the local holomorphic frame $h_{i}$ and the local coordinates $\zeta^{h}$.

###### Proof.

Using Lemma 3.1,

$\left(H\left(h_{i},h_{p}\right)\right)=\sum_{j,l}\left(H\left(a_{i}^{j}u_{j},a_{p}^{l}u_{l}\right)\right)=\sum_{j,l}\left(\overline{a_{i}^{j}}a_{p}^{l}H\left(u_{j},u_{l}\right)\right)=\sum_{j}\left(\overline{a_{i}^{j}}a_{p}^{j}\right)=b^{*}b$

where $i$ is indexing the rows and $p$ is indexing the columns. By the moment
map equation (3.1) and the relation (3.2),

$b^{*}b+\left(b^{*}b\right)^{\frac{1}{2}}\cdot\zeta^{h}\left(\zeta^{h}\right)^{*}\left(b^{*}b\right)^{\frac{1}{2}}=I_{k}.$

Then

$I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{*}=\left(b^{*}b\right)^{-1}.$

Hence

$H=\left(I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{*}\right)^{-1}.$

∎

######  Example 3.3.

Let’s consider the simplest example:

$\mathbb{P}^{1}=\operatorname{Gr}(2,1)=(\mathbb{C}^{2}-\\{0\\})/\mathbb{C}^{\times}=\mathbb{S}^{3}/\mathrm{U}\left(1\right).$

The tautological bundle for $\operatorname{Gr}(2,1)$ is

$S=\left(\mathbb{S}^{3}\times\mathbb{C}\right)/U\left(1\right)$

where $U(1)$ acts on $\mathbb{C}$ in the standard way, and it acts on both
factors on the left. (Note that this is dual to the usual notion of the
tautological bundle of $\operatorname{Gr}(1,2)=\mathbb{P}^{1}$, since we are
now considering the family of quotient lines of $\mathbb{C}^{2}$, which are
dual to subspaces of $\mathbb{C}^{2}$.)

Let’s take the standard metric on $\mathbb{C}$. We have the local Hermitian
frame (over $z_{1}\neq 0$) $u$ given by

$\left(z_{1},z_{2},z_{1}/\left|z_{1}\right|\right)\stackrel{{\scriptstyle
U(1)}}{{\sim}}\left(\left|z_{1}\right|,\zeta^{u},1\right)$

where $\zeta^{u}$ is the coordinate of $\mathbb{P}^{1}$ which belongs to the
open unit disc, and $|z_{1}|$ is determined by the moment-map equation

$\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}=\left|z_{1}\right|^{2}+\left|\zeta^{u}\right|^{2}=1.$

We also have the local holomorphic frame $h$ defined by

$\left(z_{1},z_{2},z_{1}\right)\stackrel{{\scriptstyle\mathbb{C}^{\times}}}{{\sim}}\left(1,\zeta^{h},1\right).$

The unitary and holomorphic coordinates are related by
$\zeta^{u}=\left|z_{1}\right|\cdot\zeta^{h}=(1-|\zeta^{u}|^{2})\cdot\zeta^{h}$
for $\zeta^{h}\in\mathbb{C}$. The frames are related by

$h=\left|z_{1}\right|\cdot u.$

The Hermitian frame $u$ always have length one. Writing the metric in the
holomorphic frame $h$:

$\displaystyle\left|h\right|^{2}$
$\displaystyle=\left|z_{1}\right|^{2}=1-\left|\zeta^{u}\right|^{2}=1-\left|z_{1}\right|^{2}\left|\zeta^{h}\right|^{2}$
$\displaystyle=1-\left(1-\left|z_{1}\right|^{2}\left|\zeta^{h}\right|^{2}\right)\left|\zeta^{h}\right|^{2}=\ldots$
$\displaystyle=\frac{1}{1+\left|\zeta^{h}\right|^{2}}.$

This is the standard metric on $\mathcal{O}_{\mathbb{P}^{1}}(1)$, whose
curvature gives the Fubini-Study metric on $\mathbb{P}^{1}$.

###  3.2. Metric on framed quiver moduli

We have seen that the standard metric on the trivial bundle over
$\operatorname{Mat}^{\mathbb{C}}_{n,k}$ descends to give the standard metric
on $\operatorname{Gr}(n,k)$. However, it turns out that for the framed quiver
moduli, the standard metric on the trivial bundle over $R_{\vec{n},\vec{d}}$
is not good from the GIT quotient point of view, namely it is not equivariant
under $\mathrm{GL}_{\vec{d}}$. In this section, we find a nice metric over
$R_{\vec{n},\vec{d}}$ which is equivariant under $\mathrm{GL}_{\vec{d}}$.

Recall from the last section that
$\mathcal{M}_{\vec{n},\vec{d}}=R_{\vec{n},\vec{d}}^{s}/\mathrm{GL}_{\vec{d}}$.
The universal bundle over the vertex $i$ is given by

$\mathcal{V}_{i}=\left(R_{\vec{n},\vec{d}}^{s}\times\mathbb{C}^{d_{i}}\right)\big{/}\mathrm{GL}_{\vec{d}}$

where $\mathrm{GL}_{\vec{d}}$ acts diagonally on the left, the factor
$\mathrm{GL}(d_{i},\mathbb{C})$ of $\mathrm{GL}_{\vec{d}}$ acts on
$\mathbb{C}^{d_{i}}$ in the standard way, and other factors of
$\mathrm{GL}_{\vec{d}}$ act trivially on $\mathbb{C}^{d_{i}}$.

There is an equivalent description of
$\mathcal{M}=\mathcal{M}_{\vec{n},\vec{d}}$ and the universal bundle
$\mathcal{V}_{i}$ in terms of symplectic quotient. Namely, let
$\mu:R_{\vec{n},\vec{d}}\to\mathbf{i}\mathfrak{u}_{\vec{d}}$ be the moment
map. Explicitly, $\mu=(\mu_{i})_{i\in Q_{0}}$ where

$\mu_{i}=e^{(i)}(e^{(i)})^{*}-\sum_{t(a)=i}V_{a}^{*}V_{a}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{*}.$

Then define

$\mathcal{M}_{\vec{n},\vec{d}}=\mu^{-1}\left\\{-c\right\\}\big{/}U_{\vec{d}}$

for the following level $c$.

######  Lemma 3.4.

The slope stability condition
$\left(1,\vec{0}\right)\in\mathbb{C}^{\hat{Q}_{0}}$ corresponds to the moment-
map level $c=\left(-I_{d_{i}}\right)_{i\in
Q_{0}}\in\mathbf{i}\mathfrak{u}_{\vec{d}}$, where $I_{k}$ denotes the identity
matrix of rank $k$.

###### Proof.

The character taken in King’s stability [Kin94] corresponding to
$\left(1,\vec{0}\right)$ is

$\left(1+\Sigma\vec{d}\right)\left(\left(1,\vec{0}\right)-\frac{1}{1+\Sigma\vec{d}}\left(1,\vec{1}\right)\right)=\left(\Sigma\vec{d},-1,\ldots,-1\right)$

where $\Sigma\vec{d}=\sum_{i\in Q_{0}}d_{i}$, and the first entry is over the
root vertex. (Note that there is no group action over the root vertex.) Thus
we should take $c$ to be $-I_{d_{i}}$ over each vertex $i$. ∎

The universal bundle over the vertex $i$ is then given by

$\mathcal{V}_{i}=\left(\mu^{-1}\left\\{I_{\vec{d}}\right\\}\times\mathbb{C}^{d_{i}}\right)\big{/}U_{\vec{d}}.$

Let’s review some very basic definitions about group actions.

######  Definition 3.5.

Suppose a Lie group $G$ acts on a vector bundle
$V\stackrel{{\scriptstyle\pi}}{{\to}}M$ equivariantly, namely,
$g\circ\pi=\pi\circ g$ for all $g\in G$, and the action is fiberwise linear. A
metric $H$ on $V$ is said to be $G$-equivariant if

$H_{x}\left(v,w\right)=H_{g\cdot x}\left(g\cdot v,g\cdot w\right).$

Writing in matrix form when $G=\mathrm{GL}(n,\mathbb{C})$, the above equation
is $v^{*}\cdot H_{x}\cdot w=v^{*}\cdot\left(g^{*}\cdot H_{g\cdot x}\cdot
g\right)\cdot w,$ that is,

$(g^{*})^{-1}\cdot H_{x}\cdot g^{-1}=H_{g\cdot x}.$ (3.3)

The following easily follows from the definition.

######  Lemma 3.6.

Suppose a Lie group $G$ acts on a vector bundle
$V\stackrel{{\scriptstyle\pi}}{{\to}}M$ equivariantly and fiberwise linearly,
and the action of $G$ on $M$ is free and proper. A Hermitian form $H$ on $V$
descends to the corresponding bundle over the quotient $M/G$ if and only if
$H$ is $G$-equivariant.

For framed quiver varieties, we have the framing map $e^{(j)}\colon
R_{\vec{n},\vec{d}}^{s}\rightarrow\mathrm{Hom}\left(\mathbb{C}^{n_{j}},\mathbb{C}^{d_{j}}\right)$
for each vertex $j$. Using this, we cook up a
$\mathrm{GL}_{\vec{d}}$-invariant Hermitian form on the trivial bundle
$\underline{\mathbb{C}^{d_{i}}}\to R_{\vec{n},\vec{d}}$, which descends to a
metric on $\mathcal{V}_{i}\to\mathcal{M}_{\vec{n},\vec{d}}$.

######  Theorem 3.7.

Suppose $Q$ has no oriented cycle. Fix $i\in Q_{0}$. Let $\rho$ be the row
vector whose entries are $V_{\gamma}e^{\left(t(\gamma)\right)}$, where
$\gamma$ is any path whose head $h(\gamma)$ is $i$ (including the trivial
path), $t(\gamma)$ denotes its tail, and
$V_{\gamma}\in\mathrm{Hom}(\mathbb{C}^{d_{t(\gamma)}},\mathbb{C}^{d_{h(\gamma)}})$
is the representing matrix of $\gamma$. This defines a map

$V_{\gamma}e^{\left(t(\gamma)\right)}:R_{\vec{n},\vec{d}}\to\mathrm{Hom}(\mathbb{C}^{n_{t(\gamma)}},\mathbb{C}^{d_{i}}).$

Take

$\rho\rho^{*}=\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}:R_{\vec{n},\vec{d}}\to\mathrm{End}(\mathbb{C}^{d_{i}}).$

Then $H=(\rho\rho^{*})^{-1}$ is $\mathrm{GL}_{\vec{d}}$-equivariant, and it
descends to a metric on $\mathcal{V}_{i}$ over
$\mathcal{M}_{\vec{n},\vec{d}}$.

###### Proof.

$(\rho\rho^{*})^{-1}$ is $\mathrm{GL}_{\vec{d}}$-equivariant:

$\displaystyle H_{\left(g_{h(a)}V_{a}g^{-1}_{t(a)},g_{j}e^{(j)}\right)_{a\in
Q_{1},j\in Q_{0}}}$
$\displaystyle=\left(\sum_{\gamma}\left(g_{h(\gamma)}V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(g_{h(\gamma)}V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}$
$\displaystyle=(g^{*}_{i})^{-1}\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}g_{i}^{-1}$

for all $g\in\mathrm{GL}_{\vec{d}}$. By Lemma 3.6, it descends to the bundle
$\mathcal{V}_{i}$ of the quotient.

Then we prove that the matrix-valued function $H_{(V_{a},e^{(j)})_{a\in
Q_{1},j\in Q_{0}}}=(\rho\rho^{*})^{-1}$ defines a positive-definite metric on
the trivial bundle $\underline{\mathbb{C}^{d_{i}}}$ _over the moment map level
$\mu^{-1}(I_{\vec{d}})$_ (rather than the whole $R_{\vec{n},\vec{d}}$). We
prove by induction on the vertices that $\rho\rho^{*}=I_{d_{i}}+B$ where $B$
is a semi-positive-definite Hermitian matrix, and hence $\rho\rho^{*}$ is
positive definite (and so does $(\rho\rho^{*})^{-1}$).

Since the quiver does not have oriented cycle, $Q_{0}$ can be ordered such
that $i<j$ whenever there is an arrow $i\to j$. Let $i_{0}$ be the minimal
vertex.

At $i_{0}$, there is no incoming arrow (other than the framing), and the
moment-map equation reads

$e^{(i_{0})}(e^{(i_{0})})^{*}=I_{d_{i_{0}}}+\sum_{t(a)=i_{0}}V_{a}^{*}V_{a}.$

$\sum_{t(a)=i_{0}}V_{a}^{*}V_{a}$ is semi-positive definite: $v^{*}\cdot
V_{a}^{*}V_{a}\cdot v=\|V_{a}\cdot v\|^{2}\geq 0$ for any column vector $v$.
Thus the statement is true for $\rho\rho^{*}=e_{i_{0}}e_{i_{0}}^{*}$.

Suppose the statement is true for all vertices less than $i\in Q_{0}$. At $i$,
the moment-map equation is

$e^{(i)}(e^{(i)})^{*}=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{*}V_{a}-\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{*}.$

Then

$\displaystyle\rho\rho^{*}$
$\displaystyle=e^{(i)}(e^{(i)})^{*}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{*}V_{a}^{*}$
$\displaystyle=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{*}V_{a}-\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{*}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}(I_{d_{t(a^{\prime})}}+B_{t(a^{\prime})})V_{a^{\prime}}^{*}$
$\displaystyle=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{*}V_{a}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{*}$

where
$\rho_{(t(a))}=\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)_{h(\gamma)=t(a)}$,
which by inductive assumption can be written as
$I_{d_{t(a^{\prime})}}+B_{t(a^{\prime})}$ where $B_{t(a^{\prime})}$ is semi-
positive definite. The matrices $V_{a}^{*}V_{a}$ and
$V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{*}$ are semi-positive
definite:

$v^{*}V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{*}v=(V_{a^{\prime}}^{*}v)^{*}B_{t(a^{\prime})}(V_{a^{\prime}}^{*}v)\geq
0$

for all $v$. This proves the statement for the vertex $i$. ∎

The expression $(\rho\rho^{*})^{-1}$ can be understood as follows. $\rho^{*}$
embeds the dual $V_{i}^{*}$ into the dual frame which is a trivial bundle
equipped with the standard metric. This gives an induced metric on
$V_{i}^{*}$, which is $\rho\rho^{*}$ written in matrix form. Taking the dual,
we get the metric $H_{i}=(\rho\rho^{*})^{-1}$ on $V_{i}$.

By construction, the metrics on the dual $\mathcal{V}_{i}^{*}$ (still denoted
as $H_{i}$) have the following nice property. Inductively, it gives nice
expressions of $H_{i}$ in terms of _holomorphic coordinates_.

######  Proposition 3.8.

Suppose $Q$ has no oriented cycle. For $v,w\in(\mathcal{V}_{i})^{*}$,

$H_{i}(v,w)=H_{0}((e^{(i)})^{*}(v),(e^{(i)})^{*}(w))+\sum_{h(a)=i}H_{t(a)}(a^{*}(v),a^{*}(w))$

where $H_{0}$ denotes the trivial metric on the trivial bundle, and
$e^{(i)},a$ are denoting the holomorphic bundle maps corresponding to the
framing and arrow maps respectively.

###### Proof.

The metric on $\mathcal{V}_{i}^{*}$ is given by the matrix $\rho\rho^{*}$.
Then the above equation follows from

$\rho\rho^{*}=e^{(i)}(e^{(i)})^{*}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{*}V_{a}^{*}.$

∎

######  Remark 3.9.

As we have seen, the Grassmannian $\operatorname{Gr}(n,k)$ can be understood
as the framed moduli for the quiver which has one vertex and no arrow. The
matrix $e\in\mathrm{Hom}(\mathbb{C}^{n},\mathbb{C}^{k})$ is the framing map.
Then the moment map equation implies

$\rho\rho^{*}=ee^{*}=I_{k}$

in the above proposition. This is the standard metric on the trivial bundle
$\underline{\mathbb{C}^{k}}$ that we have used in the last subsection. In
particular $\rho\rho^{*}=(\rho\rho^{*})^{-1}$ in this case. But this is not
true for other quivers.

######  Remark 3.10.

Note that the above becomes an infinite sum if the quiver has oriented cycles.
The $\mathrm{GL}_{\vec{d}}$-equivariance still holds. We should restrict to
the open subset of $R^{s}_{n,d}$ that $(\rho\rho^{*})^{-1}$ is convergent. In
the next subsection, we will prove that the same expression defines a metric
for any given quiver.

The GIT description will be important to the proof of Theorem 3.14.

There is a residual symmetry $U_{\vec{n}}=\prod_{i\in Q_{0}}U({n_{i}})$ acting
on $\mathcal{M}_{\vec{n},\vec{d}}$. Actually, there is a bigger symmetry by
the non-compact group $\mathrm{GL}(\hat{W})$ [Rei08]. $U_{\vec{n}}$ is
considered here since this is the symmetry of the metric $H$ on
$\mathcal{V}_{i}$ as we shall see.

######  Definition 3.11.

The right residual action of $U_{\vec{n}}$ on $\mathcal{M}$ is defined as

$\left[(V_{a},e^{(j)})_{a\in Q_{1},j\in Q_{0}}\right]\cdot
g=\left[(V_{a},e^{(j)}\circ g_{j})_{a\in Q_{1},j\in Q_{0}}\right]$

for $g=(g_{j}\in U(d_{i}))_{j\in Q_{0}}\in U_{\vec{n}}$.

Since the above commutes with the left action of $\mathrm{GL}_{\vec{d}}$, the
action is well-defined on $\mathcal{M}$.

######  Lemma 3.12.

There is a canonical lift of the action of $U_{\vec{n}}$ on
$\mathcal{M}_{\vec{n},\vec{d}}$ to the universal bundle $\mathcal{V}_{i}$, so
that the bundle map $\mathcal{V}_{i}\rightarrow\mathcal{M}_{\vec{n},\vec{d}}$
is equivariant.

###### Proof.

$\mathcal{V}_{i}$ is the $\mathrm{GL}_{\vec{d}}$-quotient of the trivial
bundle $R^{s}_{n,d}\times V_{i}$. $U_{\vec{n}}$ acts on this by acting on the
component $V_{i}$ trivially. This action commutes with the left action of
$\mathrm{GL}_{\vec{d}}$ on $R^{s}_{n,d}\times V_{i}$, and hence descends to
act on $\mathcal{V}_{i}$. ∎

######  Lemma 3.13.

The metric defined in Theorem 3.7 are $U_{\vec{n}}$-invariant.

###### Proof.

For any $g\in U_{\vec{n}}$, since $g_{j}g_{j}^{*}=I_{n_{j}}$ for any $j\in
Q_{0}$,

$\displaystyle H_{\left(V_{a},e^{(j)}\right)_{a\in Q_{1},j\in Q_{0}}\cdot g}$
$\displaystyle=\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\cdot
g_{t(\gamma)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\cdot
g_{t(\gamma)}\right)^{*}\right)^{-1}$
$\displaystyle=\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}=H_{\left(V_{a},e^{(j)}\right)_{a\in
Q_{1},j\in Q_{0}}}.$

∎

Recall from Theorem 2.4 that $\mathcal{M}_{\vec{n},\vec{d}}$ is the total
space of an iterated Grassmann bundle $M^{(N)}\stackrel{{\scriptstyle
p_{N}}}{{\to}}M^{(N-1)}\stackrel{{\scriptstyle
p_{N-1}}}{{\to}}\ldots\stackrel{{\scriptstyle
p_{2}}}{{\to}}M^{(1)}\stackrel{{\scriptstyle p_{1}}}{{\to}}\mathrm{pt}$.
Moreover, $\mathcal{V}_{i}$ is the pull-back of the tautological bundle
$S_{i}$ of the Grassmann bundle
$M^{(i)}=\operatorname{Gr}_{M^{(i)}}(\underline{\mathbb{C}^{n_{i}}}\oplus\bigoplus_{j\to
i}p_{i}^{*}\ldots p_{j+1}^{*}(S_{j}),d_{i})\to M^{(i-1)}$. The tautological
bundle of the Grassmannian is equipped with a standard metric as illustrated
in Section 3.1. Inductively, $\mathcal{V}_{i}$ is also equipped with a pull-
back metric. We show that this equals to the metric $H$ we defined by an
explicit formula.

######  Theorem 3.14.

For all $i\in Q_{0}$, the metric
$H=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}$
equals to the metric on $\mathcal{V}_{i}$ constructed from the iterated
Grassmann bundle.

###### Proof.

As the proof of Theorem 3.7, we do induction on the vertices, which are
totally ordered such that $i<j$ whenever there is an arrow $i\to j$.

First, we have the GIT fiber-bundle map $\mathcal{M}\to M^{(j)}$, where
$M^{(j)}$ is the framed moduli of the quiver $Q^{(j)}$ (which is obtained by
removing all vertices $k>j$ and the corresponding arrows from $Q$). This map
$(V,e)\mapsto(V^{\prime},e^{\prime})$ is simply forgetting all the irrelevant
arrow maps and frame maps that are not supported on the subquiver $Q^{(j)}$.
Note that the stability condition is preserved: any subrepresentation
$R^{\prime}\subset V^{\prime}$ can be extended to a subrepresentation
$R\subset V$ by assigning the whole $V_{k}$ to the additional vertices $k>j$
(and the arrow maps just come from restriction). If
$\mathrm{Im}(e^{\prime})\subset R^{\prime}$, then $\mathrm{Im}(e)\subset R$.

Note that here we use the GIT description instead of symplectic reduction
since $(V^{\prime},e^{\prime})$ no longer satisfies the moment-map equation in
defining $M^{(j)}$ (even when $(V,e)$ satisfies the moment-map equation for
$\mathcal{M}$).

We start with the minimal vertex $i_{0}$. The quiver $Q^{(i_{0})}$ is simply a
single vertex, and the corresponding framed moduli is
$M^{(i_{0})}=\operatorname{Gr}(n_{i_{0}},d_{i_{0}}).$ The universal bundle
$\mathcal{V}_{i_{0}}^{Q^{(i_{0})}}$ is the tautological bundle of
$M^{(i_{0})}=\operatorname{Gr}(n_{i_{0}},d_{i_{0}})$. From the last
subsection, the standard metric of $\mathcal{V}_{i_{0}}^{Q^{(i_{0})}}$ is
descended from $I_{d_{i_{0}}}$, which is exactly
$(e^{Q^{(i_{0})}}(e^{Q^{(i_{0})}})^{*})^{-1}$ by the moment map equation for
$Q^{(i_{0})}$. The statement is trivial in this case.

Now consider the vertex $i$. Denote the vertex right before $i$ by $i-1$. For
the quiver $Q^{(i-1)}$, assume that the two metrics on the universal bundle
$\mathcal{V}^{Q^{(i-1)}}_{j}$ agree for every $j\in Q^{(i-1)}_{0}$. We have
the bundle map $\pi:M^{(i)}\to M^{(i-1)}$, and
$\mathcal{V}^{Q^{(i)}}_{j}=\pi^{*}\mathcal{V}^{Q^{(i-1)}}_{j}$ for all $j<i$.
Moreover, the metric on $\mathcal{V}^{Q^{(i)}}_{j}$ is pull-back from
$\mathcal{V}^{Q^{(i-1)}}_{j}$, which equals to
$\left(\sum_{h(\gamma)=j}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}$
by inductive assumption. The pull-back map does not change the arrow and
framing maps of $Q^{(i-1)}$. This proves the statement for $S^{Q^{(i)}}_{j}$
for $j<i$.

Consider $\mathcal{V}^{Q^{(i)}}_{i}$, which is the tautological bundle
associated to the Grassmannian bundle over $M^{(i-1)}$ parametrizing quotients
of

$(e^{(i)},(V_{a})_{h(a)=i}):\left.\left(\underline{\mathbb{C}^{d_{i}}}\oplus\left(\bigoplus_{h(a)=i}\mathcal{V}^{Q^{(i-1)}}_{t(a)}\right)\right)\right|_{x\in
M^{(i-1)}}\to\mathbb{C}^{d_{i}}.$

The metric on $(\mathcal{V}^{Q^{(i)}}_{i})^{*}$ is induced by the embedding
$(e^{(i)},(V_{a})_{h(a)=i})^{*}$ to
$\left(\underline{\mathbb{C}^{d_{i}}}\oplus\left(\bigoplus_{h(a)=i}\mathcal{V}_{t(a)}\right)\right)^{*}$,
whose metric is given by

$I_{d_{i}}\oplus\bigoplus_{h(a)=i}\left(\sum_{h(\gamma)=t(a)}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)$

by inductive assumption. Thus the induced metric on
$(\mathcal{V}^{Q^{(i)}}_{i})^{*}$ is

$\displaystyle(e^{(i)},(V_{a})_{h(a)=i})\cdot\left(I_{d_{i}}\oplus\bigoplus_{h(a)=i}\left(\sum_{h(\gamma)=t(a)}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)\right)\cdot(e^{(i)},(V_{a})_{h(a)=i})^{*}$
$\displaystyle=e^{(i)}(e^{(i)})^{*}+\sum_{\begin{subarray}{c}h(a)=i\\\
h(\gamma)=t(a)\end{subarray}}V_{a}\cdot\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\cdot
V_{a}^{*}$
$\displaystyle=\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}.$

Taking reciprocal gives the metric on $\mathcal{V}^{Q^{(i)}}_{i}$. This proves
the metric has the given expression. ∎

######  Theorem 3.15.

The Ricci curvature of the metric on $\bigotimes_{i\in Q_{0}}\mathcal{V}_{i}$
given in Theorem 3.7 defines a Kähler metric on $\mathcal{M}$.

###### Proof.

As in Theorem 3.7, denote
$\rho=\rho^{\left(i\right)}=\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)_{\gamma:h(\gamma)=i}$
which is a matrix-valued function on the vector space $R_{\vec{n},\vec{d}}$.
At each point of $R_{\vec{n},\vec{d}}$, $\rho$ is a linear map from

$\widehat{W}_{i}:=\bigoplus_{\gamma:h(\gamma)=i}\mathbb{C}^{n_{t(\gamma)}}$
(3.4)

to $V_{i}$. The Ricci curvature of the metric $(\rho\rho^{*})^{-1}$ is given
by $i\partial\overline{\partial}\log\det\rho\rho^{\mathrm{*}}$. We have

$\displaystyle\partial\overline{\partial}\log\det\rho\rho^{\mathrm{*}}$
$\displaystyle=\partial\left(\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{*}}\right)^{-1}\overline{\partial}\left(\rho\rho^{\mathrm{*}}\right)\right)\right)$
$\displaystyle=\mathrm{tr}\left(\partial\left(\left(\rho\rho^{\mathrm{*}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{*}}\right)\right)$
$\displaystyle=\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\left(\partial\rho\right)^{\mathrm{*}}+\left(\partial\left(\rho\rho^{\mathrm{*}}\right)^{-1}\right)\rho\left(\partial\rho\right)^{\mathrm{*}}\right)$
$\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{*}}\right)^{-1}\left(\partial\left(\rho\rho^{\mathrm{*}}\right)\right)\left(\rho\rho^{\mathrm{*}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{*}}\right)$
$\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{*}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\left(\left(\partial\rho\right)\rho^{\mathrm{*}}\right)\right)$
$\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\partial\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\left(\partial\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)\right)$

where $\overline{\partial}\rho=0$ since the matrix $\rho$ has polynomial
entries in holomorphic coordinates.

We can take the singular value decomposition

$\rho=U\cdot\left(\text{diag}\left(\lambda_{1},\ldots,\lambda_{d_{i}}\right)\,\,\,0\right)\cdot
V^{\mathrm{*}}$

where $U\in U\left(d_{i}\right),~{}V\in U\left(\dim\widehat{W}_{i}\right)$,
and $\lambda_{i}>0$. ($\lambda_{i}\neq 0$ since $\rho$ is surjective.) Then

$\displaystyle\rho\rho^{\mathrm{*}}$
$\displaystyle=U\left(\text{diag}\left(\lambda_{1}^{2},\ldots,\lambda_{d_{i}}^{2}\right)\right)U^{\mathrm{*}}.$
$\displaystyle\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}$
$\displaystyle=V\left(\begin{array}[]{c}\text{diag}\left(\lambda_{1},\ldots,\lambda_{\alpha\left(i\right)}\right)\\\
0\end{array}\right)\left(\text{diag}\left(\lambda_{1}^{-1},\ldots,\lambda_{\alpha\left(i\right)}^{-1}\right)\right)U^{\mathrm{*}}=V\left(\begin{array}[]{c}I_{\alpha\left(i\right)}\\\
0\end{array}\right)U^{\mathrm{*}}.$

In other words,
$\rho^{\mathrm{*}}=\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\left(\rho\rho^{\mathrm{*}}\right)^{\frac{1}{2}}$
is decomposed into the rescaling $\rho\rho^{\mathrm{*}}$ and the orthogonal
embedding
$\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)$
to $\mathrm{Im}\rho^{\mathrm{*}}\subset\widehat{W}_{i}$.

Now take a vector $v\in T^{1,0}R_{\alpha,d}\cong TR_{\alpha,d}$, and evaluate
the above two-form by $(v,\bar{v})$. The first term
$\mathrm{tr~{}}\left(\left(\partial_{v}\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial_{v}\rho\right)$
is the square norm of the linear map

$\partial_{v}\rho\colon\left(\widehat{W}_{i},h_{\mathrm{std}}\right)\rightarrow\left(V_{i},~{}h_{{\left(\rho\rho^{\mathrm{*}}\right)^{-1}}}\right).$

Namely we take the standard basis in $\widehat{W}_{i}$ (which is orthonormal
under the standard metric $h_{\mathrm{std}}$), map it to $V_{i}$ by
$\partial_{v}\rho$, and take the sum of their square norms with respect to the
metric $h_{{\left(\rho\rho^{\mathrm{*}}\right)^{-1}}}$.

The second term

$\mathrm{tr~{}}\left(\left(\partial_{v}\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\left(\partial_{v}\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)\right)$

is the square norm of the following component
$\left(\partial_{v}\rho\right)_{1}$ of $\partial_{v}\rho$. Namely, we
decompose

$\widehat{W}_{i}=\left(\mathrm{Im}~{}\rho^{\mathrm{*}}\right)\oplus\left(\mathrm{Im}~{}\rho^{\mathrm{*}}\right)^{\bot}$

and write
$\partial_{v}\rho=\left(\left(\partial_{v}\rho\right)_{1},\left(\partial_{v}\rho\right)_{2}\right)$
where
$\left(\partial_{v}\rho\right)_{1}\colon\mathrm{Im}~{}\rho^{\mathrm{*}}\rightarrow
V_{i}$ and
$\left(\partial_{v}\rho\right)_{2}\colon\left(\mathrm{Im}~{}\rho^{\mathrm{*}}\right)^{\bot}\rightarrow
V_{i}$.

We have

$\partial_{v}\overline{\partial_{v}}\log\det\rho\rho^{\mathrm{*}}=\left\|\partial_{v}\rho\right\|_{H}^{2}-\left\|\left(\partial_{v}\rho\right)_{1}\right\|_{H}^{2}=\left\|\left(\partial_{v}\rho\right)_{2}\right\|_{H}^{2}\geq
0$ (3.5)

for all $v\in T^{1,0}$. ($H$ stands for the metric
$(\rho\rho^{\mathrm{*}})^{-1}$.) This proves that the Ricci curvature of the
metric for each $i$ is semi-positive definite.

Now consider

$H_{T}=\sum_{i\in
Q_{0}}\partial\overline{\partial}\log\det\rho^{(i)}\left(\rho^{(i)}\right)^{\mathrm{*}}.$

Suppose it is zero when evaluated at $(v,\bar{v})$. Then each individual term
equals to zero. This forces $\left(\partial_{v}\rho^{(i)}\right)_{2}=0$ for
all $i$, that is, image of
$\left(\partial_{v}\rho^{(i)}\right)^{\mathrm{*}}=\partial_{v}(\rho^{(i)})^{*}$
sits in the image of $(\rho^{(i)})^{\mathrm{*}}$. This exactly means $v$
descends to the zero tangent vector in the quotient $\mathcal{M}$: $v$ does
not alter the subspaces given by $(\rho^{(i)})^{*}:V_{i}\to\hat{W}_{i}$ for
all $i$. By the identification of $\mathcal{M}$ as a quiver Grassmannian
[Rei08], it means $v$ does not change the position of the point
$((\rho^{(i)})^{*}:i\in Q_{0})$ in the quiver Grassmannian, and hence must be
the zero tangent vector. This proves the above expression is positive
definite. ∎

By Equation (3.5), the metric on $T\mathcal{M}$ produced from the Ricci
curvature of $\mathcal{V}_{i}$ is

$H_{T}(v,v)=\sum_{i\in Q_{0}}\|(\partial_{v}\rho^{(i)})_{2}\|_{H_{i}}^{2}.$

We have the tautological exact sequence of vector bundles over $\mathcal{M}$:

$0\to\bigoplus_{i\in Q_{0}}\mathrm{End}(\mathcal{V}_{i})\to\bigoplus_{a\in
Q_{1}}\mathrm{Hom}(\mathcal{V}_{t_{a}},\mathcal{V}_{h_{a}})\oplus\bigoplus_{i\in
Q_{0}}\mathcal{V}_{i}^{n_{i}}\to T\mathcal{M}\to 0$

where the second arrow is given by sending
$X|_{[\phi,e]\in\mathcal{M}}\in\bigoplus_{i\in
Q_{0}}\mathrm{End}(\mathcal{V}_{i})$ to

$\left((X_{h_{a}}\phi_{a}-\phi_{a}X_{t_{a}})_{a\in Q_{1}},(X_{i}e^{(i)})_{i\in
Q_{0}}\right)$

(which is the derivative of the action of $\mathrm{GL}_{\vec{d}}$), and
$T\mathcal{M}$ is obtained as the quotient bundle (of the middle one by the
first one). $H(v,v)$ can be defined for $v\in\bigoplus_{a\in
Q_{1}}\mathrm{Hom}(\mathcal{V}_{t_{a}},\mathcal{V}_{h_{a}})\oplus\bigoplus_{i\in
Q_{0}}\mathcal{V}_{i}^{n_{i}}$. $H_{TM}$ is zero on $\bigoplus_{i\in
Q_{0}}\mathrm{End}(\mathcal{V}_{i})$: the action of
$\mathrm{GL}(d_{j},\mathbb{C})$ does not change $\rho^{(i)}$ for $j\not=i$;
for $X\in\mathrm{End}(\mathcal{V}_{i})$, $(X\cdot\rho^{(i)}\cdot
w)^{*}v=w^{*}(\rho^{(i)})^{*}(X^{*}\cdot v)=0$ for all
$v\in\mathcal{V}_{i},w\in(\operatorname{Im}(\rho^{(i)})^{*})^{\perp}$, and so
$(\partial_{X}\rho^{(i)})_{2}=0$.

###  3.3. Quiver with oriented cycles

When the quiver $Q$ has an oriented cycle, the framed moduli
$\mathcal{M}_{\vec{n},\vec{d}}$ is no longer projective. Examples of such
quivers were studied algebraically by [Fed13, ER09] along the line of Reineke.

On the other hand, the metric given in Theorem 3.7 still makes sense for
quiver with oriented cycles, as long as we stay in the domain of convergence
and prove that it is positive-definite. Below we will prove this for any given
quiver.

Denote the moment-map level by $R^{\mu=I}_{n,d}=\\{\mu=1\\}\subset
R_{\vec{n},\vec{d}}$. Let $\|A\|=\sup_{\|v\|=1}\|A\cdot v\|$ be the operator
norm of a matrix $A$. We take the following open subset of $R^{\mu=I}_{n,d}$.

######  Definition 3.16.

Define

$R_{\vec{n},\vec{d}}^{\mu=I,\circ}:=\\{(V,e)\in
R^{\mu=I}_{n,d}:\|V(\gamma)\|<1\textrm{ for every oriented cycle }\gamma\\}$

where $V(\gamma)=V(a_{k})\ldots V(a_{1})$ for $\gamma=a_{k}\ldots a_{1}$.

$R_{\vec{n},\vec{d}}^{\circ}:=\mathrm{GL}_{\vec{d}}\cdot
R_{\vec{n},\vec{d}}^{\mu=I,\circ}.$

and

$\mathcal{M}_{\vec{n},\vec{d}}^{\circ}:=R_{\vec{n},\vec{d}}^{\mu=I,\circ}/U_{\vec{d}}=R_{\vec{n},\vec{d}}^{\circ}/\mathrm{GL}_{\vec{d}}.$

The above definition of $\mathcal{M}_{\vec{n},\vec{d}}^{\circ}$ makes sense
because of the following.

######  Lemma 3.17.

$R_{\vec{n},\vec{d}}^{\mu=I,\circ}$ is invariant under $U_{\vec{d}}$.

###### Proof.

For every oriented cycle $\gamma$ at $i\in Q_{0}$, $(g\cdot
V)(\gamma)=g_{i}V(\gamma)g_{i}^{-1}$, and hence the condition
$\|V(\gamma)\|<1$ is respected for $g\in U_{\vec{d}}$. ∎

The main theorem in this section is the following.

######  Theorem 3.18.

Let $Q$ be an arbitrary quiver. As in Theorem 3.7, for each $i\in Q_{0}$, set

$H_{i}=(\rho_{i}\rho_{i}^{*})^{-1}=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right)^{-1}$

which is an infinite sum, whose terms are ordered by the length of the path
$\gamma$. (There are just finitely many paths under each fixed length.) This
gives a convergent function
$H_{i}:R_{\vec{n},\vec{d}}^{\circ}\to\mathrm{End}(\mathbb{C}^{d_{i}})$.
$H_{i}$ is $\mathrm{GL}_{\vec{d}}$-equivariant, and it descends to a metric on
$\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}^{\circ}$.

We break into several steps to prove the above theorem. First, consider the
convergence.

######  Lemma 3.19.

$\rho\rho^{*}$ is absolutely convergent over
$R_{\vec{n},\vec{d}}^{\mu=I,\circ}$. Hence $(\rho\rho^{*})^{-1}$ is well-
defined and $\mathrm{GL}_{\vec{d}}$-equivariant on
$R_{\vec{n},\vec{d}}^{\circ}$.

###### Proof.

For $(V,e)\in R_{\vec{n},\vec{d}}^{\mu=I,\circ}$, we consider the expression

$\sum_{h(\gamma)=i}\left\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\|\left\|\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{*}\right\|\leq\sum_{h(\gamma)=i}\left\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\|^{2}.$

There are only finitely many paths $\gamma_{1},\ldots,\gamma_{k}$ with
$h(\gamma_{l})=i$ which do not contain any oriented cycle. Any other path
(with $h(\gamma)=i$) can be written as concatenation of one of these
$\gamma_{l}$ and some oriented cycles at some vertices. Thus

$\sum_{h(\gamma)=i}\left\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\|^{2}\leq\sum_{l=1}^{k}\|\gamma_{l}e^{(t(\gamma_{l}))}\|^{2}\sum_{p=0}^{\infty}(1-\epsilon)^{p}=\sum_{l=1}^{k}\|\gamma_{l}e^{(t(\gamma_{l}))}\|^{2}\sum_{p=0}^{\infty}(1-\epsilon)^{p}=\sum_{l=1}^{k}\frac{\|\gamma_{l}e^{(t(\gamma_{l}))}\|^{2}}{\epsilon}<\infty$

where given $V$, there is a fixed $\epsilon\in(0,1)$ such that
$\|V_{\gamma}\|^{2}<1-\epsilon$ for all oriented cycles $\gamma$. Hence
$\rho\rho^{*}$ is absolutely convergent for every $(V,e)\in
R_{\vec{n},\vec{d}}^{\mu=I,\circ}$.

Every element in $R_{\vec{n},\vec{d}}^{\circ}$ can be written as $g\cdot(V,e)$
for $g\in\mathrm{GL}_{\vec{d}}$ and $(V,e)\in
R_{\vec{n},\vec{d}}^{\mu=I,\circ}$.
$(\rho\rho^{*})^{-1}|_{g\cdot(V,e)}=(g_{i}^{*})^{-1}(\rho\rho^{*})^{-1}|_{(V,e)}g_{i}^{-1}$
where $(\rho\rho^{*})^{-1}|_{(V,e)}$ is convergent. ∎

It remains to prove positive definiteness of $H_{i}$. First we consider the
following specific quiver which is simply a single oriented cycle.

######  Lemma 3.20.

If $Q$ is a single oriented cycle with $N$ vertices, then for each vertex $i$,
$\rho_{i}\rho_{i}^{*}$ is positive definite. (In particular, when $N=1$, $Q$
consists of one vertex and a self loop.)

###### Proof.

By symmetry, we just need to prove for $i=N$. Let $l=a_{N}\ldots a_{1}$, where
$a_{k}$ is the arrow $(k-1)\to k$.

$\rho_{N}\rho_{N}^{*}=\sum_{k=1}^{N}\left(a_{N}\ldots
a_{k+1}e_{k}e_{k}^{*}a_{k+1}^{*}\ldots a_{N}^{*}+\sum_{p>0}l^{p}a_{N}\ldots
a_{k+1}e_{k}e_{k}^{*}a_{k+1}^{*}\ldots a_{N}^{*}(l^{p})^{*}\right).$

The moment map equation at the vertex $k$ is
$e_{k}e_{k}^{*}=I-a_{k}a_{k}^{*}+a_{k+1}^{*}a_{k+1}$. Then the first term
gives

$\displaystyle\sum_{k=1}^{N}a_{N}\ldots
a_{k+1}(I-a_{k}a_{k}^{*}+a_{k+1}^{*}a_{k+1})a_{k+1}^{*}\ldots a_{N}^{*}$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{N}a_{N}\ldots
a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}-\sum_{k=1}^{N}a_{N}\ldots
a_{k+1}a_{k}a_{k}^{*}a_{k+1}^{*}\ldots a_{N}^{*}+\sum_{k=1}^{N}a_{N}\ldots
a_{k+1}a_{k+1}^{*}a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}$ $\displaystyle=$
$\displaystyle I-ll^{*}+\sum_{k=1}^{N}a_{N}\ldots
a_{k+1}a_{k+1}^{*}a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}.$

Similarly, the second term gives

$\displaystyle\sum_{k=1}^{N}\sum_{p>0}l^{p}a_{N}\ldots
a_{k+1}(I-a_{k}a_{k}^{*}+a_{k+1}^{*}a_{k+1})a_{k+1}^{*}\ldots
a_{N}^{*}(l^{p})^{*}$ $\displaystyle=$
$\displaystyle\sum_{p>0}l^{p}(l^{p})^{*}-\sum_{p>0}l^{p}ll^{*}(l^{p})^{*}+\sum_{p>0}\sum_{k=1}^{N}l^{p}a_{N}\ldots
a_{k+1}a_{k+1}^{*}a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}(l^{p})^{*}$
$\displaystyle=$ $\displaystyle
ll^{*}+\sum_{p>0}\sum_{k=1}^{N}l^{p}a_{N}\ldots
a_{k+1}a_{k+1}^{*}a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}(l^{p})^{*}.$

Combining the two terms,

$\rho_{N}\rho_{N}^{*}=I+\sum_{p\geq 0}\sum_{k=1}^{N}l^{p}a_{N}\ldots
a_{k+1}a_{k+1}^{*}a_{k+1}a_{k+1}^{*}\ldots a_{N}^{*}(l^{p})^{*}$

and the second term is semi-positive-definite. Hence $\rho_{N}\rho_{N}^{*}$ is
positive definite. ∎

The following is the key lemma to prove positive-definiteness for a general
quiver.

######  Lemma 3.21.

Suppose $Q$ has the property that for every $i\in Q_{0}$, restricted to the
intersection of the moment map locus and $R^{Q,\circ}_{n,d}$,
$\rho^{Q}_{i}(\rho^{Q}_{i})^{*}=I+B_{i}$ for some semi-positive definite
matrix $B_{i}$. Let $Q^{\prime}$ be obtained by concatenating to $Q$ a chain
$x_{1}\stackrel{{\scriptstyle
a_{1}^{\prime}}}{{\to}}\ldots\stackrel{{\scriptstyle
a_{k-1}^{\prime}}}{{\to}}x_{k}$ where $x_{1}$ and $x_{k}$ are certain vertices
in $Q$. ($x_{1}$ can be equal to $x_{k}$, meaning what we have added is an
oriented cycle. When $k=1$, we have added a loop;
$a^{\prime}_{0}=a^{\prime}_{1}$. When $k=2$, there is no intermediate vertex
in the chain.) Then $Q^{\prime}$ has the same property. Namely, for every
$i\in Q^{\prime}_{0}$, restricted to the intersection of the moment map locus
and $R^{Q^{\prime},\circ}_{n,d}$,
$\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{*}=I+B_{i}^{\prime}$ for some
semi-positive-definite matrix $B_{i}^{\prime}$.

###### Proof.

First, consider the case that the given vertex $i\in Q_{0}^{\prime}$ belongs
to $Q_{0}$. For the original quiver $Q$,
$\rho^{Q}_{i}(\rho^{Q}_{i})^{*}=I+B_{i}$ where $B_{i}$ is semi-positive
definite. After concatenating the chain, the terms in $\rho^{Q}_{i}$,

$\gamma^{x_{1}\to i}_{Q}e^{(1)}(e^{(1)})^{*}(\gamma^{x_{1}\to
i}_{Q})^{*}\textrm{ and }\gamma^{x_{k}\to
i}_{Q}e^{(k)}(e^{(k)})^{*}(\gamma^{x_{k}\to i}_{Q})^{*}$

where $\gamma^{x_{1}\to i}_{Q}$ ($\gamma^{x_{k}\to i}_{Q}$ resp.) is a path
from $x_{1}$ (from $x_{k}$ resp.) to $i$, get affected. Namely, the moment map
equation for $e^{(1)}(e^{(1)})^{*}$ (or $e^{(k)}(e^{(k)})^{*}$) gets an extra
term $(a_{1}^{\prime})^{*}a_{1}^{\prime}$
($-a_{k-1}^{\prime}(a_{k-1}^{\prime})^{*}$ resp.). (If $k=1$, $x_{1}=x_{k}$
and $a^{\prime}_{1}=a^{\prime}_{0}$, and the moment map equation for
$e^{(1)}(e^{(1)})^{*}$ gets both the extra terms
$(a_{0}^{\prime})^{*}a_{0}^{\prime}$ and
$-a_{0}^{\prime}(a_{0}^{\prime})^{*}$.) As a result, we have an extra negative
term

$-\gamma^{x_{k}\to
i}_{Q}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{*}(\gamma^{x_{k}\to i}_{Q})^{*}$
(3.6)

for each path $\gamma^{x_{k}\to i}_{Q}$. We shall show that these negative
terms can be canceled.

We also have additional paths in $Q^{\prime}$ heading to $i$, which can be
divided into the following types:

  1. (1)

$\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for
$l=2,\ldots,k-1$. (This is an empty case when $k=1,2$.)

  2. (2)

$\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}$
where $\gamma_{\mathrm{new}}:=a_{k-1}^{\prime}\ldots a_{1}^{\prime}$ and
$\gamma_{Q}^{j\to x_{1}}$ is any path in $Q$ from $j\in Q_{0}$ to $x_{1}$.
($\gamma_{Q}^{x_{1}\to x_{1}}$ can be the trivial path.
$\gamma_{\mathrm{new}}=a_{0}^{\prime}$ when $k=1$.)

  3. (3)

$\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for some $p>0$
and $l=2,\ldots,k-1$.

  4. (4)

$\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}$ for some $p>0$
and $j\in Q_{0}$.

For (1), by the moment-map equation
$e^{(x_{l})}(e^{(x_{l})})^{*}=I+(a^{\prime}_{l})^{*}a^{\prime}_{l}-a^{\prime}_{l-1}(a^{\prime}_{l-1})^{*}$,
we have

$\displaystyle\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a_{l}^{\prime}\cdot e^{(x_{l})}(e^{(x_{l})})^{*}\cdot(\gamma_{Q}^{x_{k}\to
i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{*}$ $\displaystyle=$
$\displaystyle\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a_{l}^{\prime}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a_{l}^{\prime})^{*}+\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a_{l}^{\prime}(a^{\prime}_{l})^{*}a^{\prime}_{l}(\gamma_{Q}^{x_{k}\to
i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{*}$
$\displaystyle-\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a_{l}^{\prime}a^{\prime}_{l-1}(a^{\prime}_{l-1})^{*}(\gamma_{Q}^{x_{k}\to
i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{*}.$

The first term above for $l=k-1$ cancel with the extra negative term (3.6) for
$\rho_{i}^{Q}$. The third term (which is negative) for
$l\in\\{3,\ldots,k-1\\}$ cancel with the first term of $l-1$. As a result,
after combining (1) with the modified $\rho_{i}^{Q}$, the remaining negative
terms are

$-\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a^{\prime}_{1}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots
a^{\prime}_{1})^{*}=-\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})^{*}.$
(3.7)

(For the case $k=1$, this trivially holds since
$\gamma_{\mathrm{new}}=a^{\prime}_{0}$, and the above equals to (3.6).)

Now consider (2): $\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to
x_{1}}e_{j}e_{j}^{*}(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}})^{*}$. The moment map equation
for $e_{j}e_{j}^{*}$ when $j\not=x_{1},x_{k}$ are the same for $Q$ and
$Q^{\prime}$. Summing $\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to
x_{1}}e_{j}e_{j}^{*}(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}})^{*}$ over arbitrary
$\gamma_{Q}^{j\to x_{1}}$ and $j\in Q_{0}$, we obtain

$(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}})\cdot\rho^{Q}_{x_{1}}\cdot(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}})^{*}=(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}})\cdot(I+B_{1})\cdot(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}})^{*}$

plus

$\sum_{\gamma_{Q}^{x_{1}\to x_{1}}}\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{1}\to
x_{1}}(a_{1}^{\prime})^{*}a_{1}^{\prime}(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{1}\to
x_{1}})^{*}-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to
x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{*}(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}})^{*}$

which is due to the additional terms in the moment map equations for
$e^{(x_{1})}(e^{(x_{1})})^{*}$ and $e^{(x_{k})}(e^{(x_{k})})^{*}$. (When
$k=1$, $\gamma_{Q}^{x_{0}\to x_{0}}$ being the trivial path at $x_{0}$ is one
of the possibilities.) In above, $B_{1}$ is semi-positive-definite by the
assumption on $Q$. Then the negative terms (3.7) cancel with the first term.
After combining the paths in $Q$ and (1) and (2), the remaining negative terms
are

$-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to
x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{*}(\gamma_{Q}^{x_{k}\to
i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}})^{*}.$ (3.8)

(If there is no path $\gamma_{Q}^{x_{k}\to x_{1}}$ in $Q$ from $x_{k}$ to
$x_{1}$, then this is zero, and we do not have (3) nor (4). We stop here and
get that $\rho_{i}^{Q^{\prime}}(\rho_{i}^{Q^{\prime}})^{*}=I+B_{i}^{\prime}$
for a semi-positive-definite matrix $B_{i}^{\prime}$.)

The terms in (3) for $p=1$ and $l=k-1$ cancel with the above (3.8):

$\displaystyle\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\cdot
e^{(x_{l})}(e^{(x_{l})})^{*}\cdot\left(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{*}$
$\displaystyle=$ $\displaystyle\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\left(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{*}$
$\displaystyle+\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots
a_{l}^{\prime}(a^{\prime}_{l})^{*}a^{\prime}_{l}\left(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{*}$
$\displaystyle-\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots
a_{l}^{\prime}a^{\prime}_{l-1}(a^{\prime}_{l-1})^{*}\left(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{*}.$

Like in (1), for each $p$, the third term (which is negative) for
$l\in\\{3,\ldots,k-1\\}$ cancel with the first term for $l-1$. (When $p=0$ and
$l=k-1$, the third term is exactly (3.8).) Then the remaining negative terms
are (3.7) modified by inserting the loops
$\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)$ for $p>0$, that is, $-\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\gamma_{\mathrm{new}}(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\gamma_{\mathrm{new}})^{*}$. Then like in (2), these negative
terms cancel with terms in (4). Summing up to finite $p$, the only negative
terms left are (3.8) modified by inserting the loops
$\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)$:

$-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to
x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{*}\left(\gamma_{Q}^{x_{k}\to
i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}}\right)^{*}$

which cancel with terms in (3) for $(p+1)$. As $p\to\infty$,
$\|\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to
x_{1}}\right)\|\to 0$. This finishes the proof that
$\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{*}=I+B_{i}$, for $i\in Q_{0}$.

For the case that $i=x_{j}$ for $j=2,\ldots,k-1$, the proof is similar. (We do
not need to consider this case when $k=1,2$.) The paths in $Q^{\prime}$
heading to $x_{j}$ are divided into the following types:

  1. (1)

$a_{j-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for $l=2,\ldots,j$.

  2. (2)

$a_{j-1}^{\prime}\ldots
a_{1}^{\prime}\left(\prod_{r=1}^{p}\gamma_{Q,r}^{x_{k}\to
x_{1}}\gamma_{\mathrm{new}}\right)\gamma_{Q}^{j\to x_{1}}e_{j}$ for some
$p\geq 0$, $j\in Q_{0}$.

  3. (3)

$a_{j-1}^{\prime}\ldots
a_{1}^{\prime}\left(\prod_{r=1}^{p}\gamma_{Q,r}^{x_{k}\to
x_{1}}\gamma_{\mathrm{new}}\right)\gamma_{Q}^{x_{k}\to
x_{1}}a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for some $p\geq 0$,
$l=2,\ldots,k-1$.

The cancellation is similar and we do not repeat here. ∎

Similarly, adding a chain at a single vertex of $Q$ preserves the positive-
definiteness property.

######  Lemma 3.22.

Suppose $Q$ as in Lemma 3.21. Let $Q^{\prime}$ be obtained by concatenating to
$Q$ a chain $x_{1}\stackrel{{\scriptstyle
a_{1}^{\prime}}}{{\to}}\ldots\stackrel{{\scriptstyle
a_{k-1}^{\prime}}}{{\to}}x_{k}$ at either $x_{1}$ or $x_{k}$ in $Q_{0}$. Then
for every $i\in Q^{\prime}_{0}$,
$\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{*}=I+B_{i}^{\prime}$ for some
semi-positive-definite matrix $B_{i}^{\prime}$.

###### Proof.

The proof in this case is simpler than that of Lemma 3.21, since there is no
new oriented cycle.

Consider the case that $x_{k}\in Q_{0}$. If $i$ belongs to the chain, then the
only paths that head to $i$ are contained in the chain. Since no oriented
cycle is involved in all such paths, Theorem 3.7 already gives the result.

If $i\in Q_{0}$, then

$\displaystyle\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{*}=$
$\displaystyle\sum_{j\in Q_{0}-\\{x_{k}\\}}\sum_{\gamma_{Q}^{j\to
i}}(\gamma_{Q}^{j\to i}e^{(j)})(\gamma_{Q}^{j\to
i}e^{(j)})^{*}+\sum_{\gamma_{Q}^{x_{k}\to i}}\gamma_{Q}^{x_{k}\to i}\cdot
e^{(x_{k})}(e^{(x_{k})})^{*}\cdot(\gamma_{Q}^{x_{k}\to i})^{*}$
$\displaystyle+\sum_{\gamma_{Q}^{x_{k}\to
i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a_{k-1}\ldots
a_{r})e_{x_{r}}e_{x_{r}}^{*}\cdot(\gamma_{Q}^{x_{k}\to
i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{*}$ $\displaystyle=$
$\displaystyle\rho_{i}^{Q}(\rho_{i}^{Q})^{*}-\sum_{\gamma_{Q}^{x_{k}\to
i}}\gamma_{Q}^{x_{k}\to
i}(a^{\prime}_{k-1}(a^{\prime}_{k-1})^{*})(\gamma_{Q}^{x_{k}\to i})^{*}$
$\displaystyle+\sum_{\gamma_{Q}^{x_{k}\to
i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots
a^{\prime}_{r})(I-a^{\prime}_{r-1}(a^{\prime}_{r-1})^{*}+(a^{\prime}_{r})^{*}a^{\prime}_{r})\cdot(\gamma_{Q}^{x_{k}\to
i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{*}$ $\displaystyle=$
$\displaystyle\rho_{i}^{Q}(\rho_{i}^{Q})^{*}+\sum_{\gamma_{Q}^{x_{k}\to
i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots
a^{\prime}_{r})(a^{\prime}_{r})^{*}a^{\prime}_{r}\cdot(\gamma_{Q}^{x_{k}\to
i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{*}.$

Since $\rho_{i}^{Q}(\rho_{i}^{Q})^{*}=I+B_{i}$ for some semi-positive-definite
matrix $B_{i}$, and the second term is semi-positive-definite,
$\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{*}$ satisfies the requirement.

The case that $x_{1}\in Q_{0}$ is similar and the proof is omitted. ∎

###### Proof of Theorem 3.18.

Without loss of generality, suppose $Q$ is connected. (Otherwise
$\mathcal{M}_{\vec{n},\vec{d}}$ and $\mathcal{V}_{i}$ decompose into products
coming from the connected components, and we just need to study each
component.) The case without oriented cycle is given in Theorem 3.7. Suppose
$Q$ has at least one oriented cycle. By Lemma 3.20, the statement is true for
this oriented cycle as a quiver. There must be additional arrows if this
single oriented cycle is not yet the whole $Q$. Then we can either add a chain
as in Lemma 3.21 or 3.22, and the statement still holds. (Both the cases of
loop at a vertex or multiple edge are covered by Lemma 3.21.) Inductively the
statement holds for $Q$. ∎

######  Example 3.23.

For the $A_{2}$-quiver, $\rho_{i}\rho_{i}^{*}$ gives a metric for all $i$. By
Lemma 3.21, this is still true if we add an oriented cycle with arrows
$l_{1},\ldots,l_{p}$. See Figure 3.

${1}$${4}$${2}$${5}$${3}$$\scriptstyle{e^{(1)}}$$\scriptstyle{a_{1}}$$\scriptstyle{e_{2,1}}$$\scriptstyle{e_{2}}$$\scriptstyle{a_{2}}$$\scriptstyle{\ell_{1}}$$\scriptstyle{e_{3}}$$\scriptstyle{\ell_{p}}$$\scriptstyle{e_{2,p-1}}$
Figure 3. $A_{2}$ modified by adding an oriented cycle.

Note that the following equality still holds over $R^{\circ}_{n,d}$:

$\rho\rho^{*}=e^{(i)}(e^{(i)})^{*}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{*}V_{a}^{*}.$

Thus Proposition 3.8 still holds for quivers with oriented cycles.

######  Proposition 3.24.

For any quiver $Q$ and every $v,w\in(\mathcal{V}_{i})^{*}$,

$H_{i}(v,w)=H_{0}((e^{(i)})^{*}(v),(e^{(i)})^{*}(w))+\sum_{h(a)=i}H_{t(a)}(a^{*}(v),a^{*}(w)).$

Now we consider a version of Theorem 3.15 in this case. We define
$\hat{W}_{i}$ by Equation (3.4). But this time, it is an infinite direct sum
of Hilbert spaces (meaning that it consists of infinite sequence
$w=(w_{\gamma}:h(\gamma)=i)$ with
$\|w\|^{2}=\sum_{\gamma}\|w_{\gamma}\|^{2}<\infty$).

######  Lemma 3.25.

For each $(V,e)\in R^{\circ}_{n,d}$, $\rho_{i}(V,e)$ defines a bounded linear
map $\hat{W}_{i}\to V_{i}$. Its adjoint
$\rho_{i}(V,e)^{*}:V_{i}\to\hat{W}_{i}$ has a singular-value decomposition.

###### Proof.

For $w=(w_{\gamma}:h(\gamma)=i)\in\hat{W}_{i}$, $\rho_{i}(V,e)$ maps it to

$\sum_{\gamma:h(\gamma)=i}V_{\gamma}e^{(t(\gamma))}w_{\gamma}.$

Like in the proof of Lemma 3.19, consider
$\sum_{\gamma:h(\gamma)=i}\|V_{\gamma}e^{(t(\gamma))}\|\|w_{\gamma}\|\leq\|w\|\sum_{\gamma:h(\gamma)=i}\|V_{\gamma}e^{(t(\gamma))}\|<+\infty.$
This also shows that if $\|w\|=1$, then the image of $w$ is also bounded.
$\rho_{i}(V,e)^{*}$ has image being finite-dimensional (since $V_{i}$ is
finite-dimensional), and hence is a compact operator. Thus it has a singular-
value decomposition. ∎

######  Proposition 3.26.

The Ricci curvature of the metric given by $(\rho_{i}\rho_{i}^{*})^{-1}$ is
semi-positive definite on $\mathcal{M}^{\circ}$.

###### Proof.

By the previous lemma, the proof of Theorem 3.15 on semi-positive definiteness
still works. Namely,

$\partial\overline{\partial}\log\det\rho\rho^{\mathrm{*}}=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\partial\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\left(\partial\rho\cdot\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\right)\right).$

Note that the two terms on the RHS are finite:
$\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\right)=\mathrm{tr~{}}\left(\rho\rho^{\mathrm{*}}\right)^{-1}\partial\rho\left(\left(\partial\rho\right)^{\mathrm{*}}\right)=\sum_{j=1}^{d_{i}}\langle\partial\rho\left(\partial\rho\right)^{\mathrm{*}}\epsilon_{j},\left(\rho\rho^{\mathrm{*}}\right)^{-1}\epsilon_{j}\rangle_{V_{i}}$
which is a finite sum, and similar for the second term.
$\rho^{\mathrm{*}}=\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)\left(\rho\rho^{\mathrm{*}}\right)^{\frac{1}{2}}$
is decomposed into the rescaling $(\rho\rho^{\mathrm{*}})^{\frac{1}{2}}$ and
the orthogonal embedding
$\left(\rho^{\mathrm{*}}\left(\rho\rho^{\mathrm{*}}\right)^{-\frac{1}{2}}\right)$
to $\mathrm{Im}\,\rho^{\mathrm{*}}\subset\widehat{W}_{i}$. Then the above
equals to $\|(\partial\rho)_{2}\|_{H}^{2}\geq 0$ as in Theorem 3.15. ∎

##  4\. Fiberwise Nonlinearity

In the mathematical study of quivers, we mostly focused on linear
representations. In particular, the morphisms between universal vector bundles
are linear along fibers. On the other hand, nonlinear ‘activation functions’
play a key role in machine learning. In this section, we construct some
natural non-linear fiber-bundle endomorphisms of the universal bundles
$\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}$ by using fiberwise
symplectomorphisms.

For simplicity, we shall take $n_{i}=d_{i}+1$ for all $i\in Q_{0}$ in this
section.

###  4.1. Activation functions arising from toric moment maps and
symplectomorphisms

In this section, we make the observation that several activation functions
commonly used in machine learning actually belong to a much bigger class,
namely the $T$-equivariant symplectomorphisms on open subsets of a symplectic
toric variety.

First, let’s recall the basic setup for toric varieties. Let’s equip
$\mathbb{C}^{m}$ with the standard Kähler structure. We obtain a symplectic
toric variety $(X,\omega_{X})$ as a symplectic quotient by the real torus
$T^{m-d}$. We assume $X$ is smooth. The $T^{m-d}$-action can be specified by
an injective homomorphism $\mathbb{Z}^{m-d}\to\mathbb{Z}^{m}$, which induces a
map $T^{m-d}\to T^{m}$, and $T^{m}$ acts on $\mathbb{C}^{m}$ by coordinate-
wise multiplication. We assume that the quotient of $\mathbb{Z}^{m}$ by the
image of $\mathbb{Z}^{m-d}$ is again a lattice, which we identify as
$\mathbb{Z}^{d}$. We denote by $v_{i}\in\mathbb{Z}^{d}$ the images of the
standard basic vectors of $\mathbb{Z}^{m}$ under the quotient map
$\mathbb{Z}^{m}\to\mathbb{Z}^{d}$.

The residual action of $T^{d}$ on $X$ gives a moment-map fibration over a
polytope $P$, which is given by the intersection of $m$ half-spaces in
$\mathbb{R}^{d}$:

$\\{x\in\mathbb{R}^{d}:\ell_{j}(x):=v_{j}\cdot x-c_{j}\geq 0\\}$

where the constants $c_{j}\in\mathbb{R}^{d}$ are determined by the level taken
in the symplectic quotient. We assume that the level is chosen such that for
all $j$, $\\{\ell_{j}(x)=0\\}\cap P$ is a (non-empty) codimension-one boundary
of the polytope $P$.

Consider the open toric orbit of $X$, which can be identified as
$(\mathbb{C}^{\times})^{d}$ by fixing a basis of $\mathbb{Z}^{d}$. Denote by
$\omega_{X}$ the Kähler form induced on the symplectic quotient. Let
$\omega_{\mathrm{std}}=\sum_{i=1}^{d}dx_{i}\wedge d\theta_{i}$ be the standard
symplectic form on $\mathbb{R}^{d}\times T^{d}$ (where $T^{d}$ denotes the
real $d$-torus). The symplectic form $\omega_{X}|_{(\mathbb{C}^{\times})^{d}}$
has an explicit description by the following beautiful formula.

######  Theorem 4.1 ([Gui94, Abr98]).

$\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\mathrm{Id}\right):(P^{\circ}\times
T^{d},\omega_{\mathrm{std}}|_{P^{\circ}\times T^{d}})\to\mathbb{R}^{d}\times
T^{d}\stackrel{{\scriptstyle\exp}}{{\cong}}((\mathbb{C}^{\times})^{d},\omega_{X}|_{(\mathbb{C}^{\times})^{d}})$

is a symplectomorphism.

Taking the universal cover $\mathbb{R}^{d}\to T^{d}$ and lifting the above,
one obtains the following.

######  Corollary 4.2.

The inverse of
$\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\,\mathrm{Id}\right)$
gives a symplectomorphism

$\sigma_{\mathbb{C}}=(\sigma(\mathrm{Re}(\vec{z})),\mathrm{Im}(\vec{z})):(\mathbb{C}^{d},\exp^{*}\omega_{X})\to(P^{\circ}\times\mathbb{R}^{d},\omega_{\mathrm{std}})$

where $\exp:\mathbb{C}^{d}\to(\mathbb{C}^{\times})^{d}\subset X$, and
$\sigma:\mathbb{R}^{d}\to P^{\circ}$ is the inverse of
$\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\,\mathrm{Id}\right)$.

######  Example 4.3.

For $\mathbb{C}^{d}$, the moment polytope $P$ is $\mathbb{R}_{\geq
0}^{d}=\\{x_{i}\geq 0:i=1,\ldots,d\\}$. $v=(1,\ldots,1)$. The above map is
simply $\left(\frac{\log
x_{i}}{2}\right)_{i=1}^{d}:\mathbb{R}_{>0}^{d}\to\mathbb{R}^{d}$. The
symplectomorphism
$((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{n}}|_{\mathbb{C}^{\times})^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\mathbb{R}_{>0}^{d}\times
T^{d},\omega_{\mathrm{std}}|_{P^{\circ}\times T^{d}})$ is
$\left(|z_{i}|^{2},\frac{\log z_{i}-\log\bar{z_{i}}}{2i}\right)_{i=1}^{d}$.

######  Example 4.4.

For the complex projective space $\mathbb{P}^{d}$, the corresponding moment
polytope $P$ (for a chosen level) is the $d$-simplex given by $\ell_{i}\geq 0$
where $\ell_{i}(x)=x_{i}$ for $i=1,\ldots,d$, and
$\ell_{d+1}(x)=1-x_{1}-x_{2}-\dots-x_{d}$. The generators are
$v_{i}=\epsilon_{i}$ for $i=1,\ldots,d$ (the standard basis) and
$v_{d+1}=-\sum_{i=1}^{d}v_{i}$. We have

$d\left(\frac{1}{2}\sum_{i=1}^{d+1}\ell_{i}(x)\log\ell_{i}(x)\right)=\frac{1}{2}\sum_{i=1}^{d}\log\left(\frac{x_{i}}{1-\sum_{j=1}^{d}x_{j}}\right)dx_{i}$

as a map $P^{\circ}\to\mathbb{R}^{d}$. By direct computation, the inverse of
this map equals to

$\sigma(\vec{r})=\left(\frac{e^{2r_{i}}}{1+\sum_{j=1}^{d}e^{2r_{j}}}\right)_{i=1}^{d}:\mathbb{R}^{d}\to
P^{\circ}.$ (4.1)

Written in terms of the complex coordinates
$\vec{z}\in(\mathbb{C}^{\times})^{d}$, the symplectomorphism
$((\mathbb{C}^{\times})^{d},\omega_{\mathbb{P}^{n}}|_{(\mathbb{C}^{\times})^{d}})\to(P^{\circ}\times
T^{d},\omega_{\mathrm{std}}|_{P^{\circ}\times T^{d}})$ is given by

$\left(\frac{|z_{i}|^{2}}{1+\sum_{j=1}^{d}|z_{j}|^{2}},\frac{\log
z_{i}-\log\bar{z_{i}}}{2i}\right)_{i=1}^{d}.$

Pulling back by $\mathbb{C}^{d}\to(\mathbb{C}^{\times})^{d}$, we have the
symplectomorphism
$\sigma_{\mathbb{C}}=\left(\left(\frac{e^{2r_{i}}}{1+\sum_{j=1}^{d}e^{2r_{j}}}\right)_{i=1}^{d},\mathrm{Id}\right):(\mathbb{C}^{d},\exp^{*}\omega_{X})\to(P^{\circ}\times\mathbb{R}^{d},\omega_{\mathrm{std}})$.

When $d=1$, $\frac{e^{2r}}{1+e^{2r}}$ is a commonly-used activation function.
See Figure 4. By taking a direct product,
$\left(\frac{e^{2r_{i}}}{1+e^{2r_{i}}}\right)_{i=1}^{d}:\mathbb{R}^{d}\to[0,1]^{d}$
corresponds to $(\mathbb{P}^{1})^{d}$.

$/2\pi i\mathbb{Z}$$\cong$$\left(\frac{e^{2x}}{1+e^{2x}},y\right)$ Figure 4.
$\mathbb{C}$ as a Covering Space Mapped to an Open Strip

######  Remark 4.5.

In above, we have taken the quotient Kähler structure from $\mathbb{C}^{m}$.
For a general toric Kähler structure, the symplectomorphism in Theorem 4.1 is
given by

$\left(d\left(\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}\log\ell_{i}(x)\right)+h\right),\mathrm{Id}\right)$

where $h$ is a smooth function on the closed polytope $P$ such that the
Hessian of
$\left(\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}\log\ell_{i}(x)\right)+h\right)$
is positive definite in $P^{\circ}$ [Abr98].

In particular, for a general projective toric variety $X$, we can take an
embedding of $X$ to $\mathbb{P}^{N}$ by toric holomorphic sections of a very
ample line bundle $L$, and use the induced toric Kähler structure from
$\mathbb{P}^{N}$. Then the symplectomorphism $\sigma_{\mathbb{C}}$ is given by
$(\sigma,\mathrm{Id})$ where

$\sigma(\vec{r})=\frac{\sum_{i=1}^{d}e^{2(\vec{u_{i}},\vec{r})}\vec{u_{i}}}{\sum_{j=1}^{d}e^{2(\vec{u_{j}},\vec{r})}}:\mathbb{R}^{d}\to
P^{\circ},$

$\vec{u_{i}}$ are points such that their convex hull equal to $P$, and
$(\vec{u},\vec{r})$ is the standard dot product on $\mathbb{R}^{d}$. See
[Ful93, Section 4.2].

Now we have the symplectomorphisms $\phi_{X}:(P^{\circ}\times
T^{d},\omega_{\mathrm{std}}|_{P^{\circ}\times
T^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}((\mathbb{C}^{\times})^{d},\omega_{X}|_{(\mathbb{C}^{\times})^{d}})$
and $\phi_{\mathbb{C}^{d}}:\mathbb{R}_{>0}^{d}\times
T^{d}\stackrel{{\scriptstyle\cong}}{{\to}}((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{d}}|_{(\mathbb{C}^{\times})^{d}})$
(Example 4.3). For the toric structure of $X$, let’s arrange the order of the
indices such that the first $d$ vectors $v_{i}$ for $i=1,\ldots,d$ form a
basis of $\mathbb{Z}^{d}$. (We assume $m\geq d$.) Moreover, we take the first
$d$ constants $c_{j}=0$ for $j=1,\ldots,d$. Then
$P^{\circ}\subset\mathbb{R}_{>0}^{d}$.

Consider the composition
$\phi_{\mathbb{C}^{d}}\circ\phi_{X}^{-1}:((\mathbb{C}^{\times})^{d},\omega_{X}|_{(\mathbb{C}^{\times})^{d}})\to((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{d}}|_{(\mathbb{C}^{\times})^{d}})$.
It is a symplectomorphism onto the image
$\phi_{\mathbb{C}^{d}}(P^{\circ}\times T^{d})$.

######  Proposition 4.6.

$\phi_{\mathbb{C}^{d}}\circ\phi_{X}^{-1}$ extends to a $T$-equivariant
symplectomorphism

$\psi:(\mathbb{C}^{d},\omega_{X}|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\pi_{\mathbb{C}^{d}}^{-1}(P-B),\omega_{\mathbb{C}^{d}}|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)})$

where $B=\bigcup_{i=d+1}^{m}\\{\ell_{i}(x)=0\\}$, and
$\pi_{\mathbb{C}^{d}}=(|z_{i}|^{2})_{i=1}^{d}:\mathbb{C}^{d}\to\mathbb{R}_{\geq
0}^{d}$ is the moment map for $\mathbb{C}^{d}$.

###### Proof.

$\phi_{X}$ is given by $2r^{X}_{i}=\log
x_{i}+\sum_{j=d+1}^{m}(v_{j}^{(i)}\log\ell_{j}(x))$, where
$v_{j}=(v_{j}^{(1)},\ldots,v_{j}^{(d)})$. $\phi_{\mathbb{C}^{d}}$ is given by
$2r^{\mathbb{C}^{d}}_{i}=\log x_{i}$. Hence
$e^{2r^{X}_{i}}=e^{2r^{\mathbb{C}^{d}}_{i}}\prod_{j=d+1}^{m}\ell_{j}^{v_{j}^{(i)}}(e^{2r^{\mathbb{C}^{d}}_{1}},\ldots,e^{2r^{\mathbb{C}^{d}}_{d}})$.
In terms of the complex coordinates, this gives

$z^{X}_{i}=z^{\mathbb{C}^{d}}_{i}\left(\prod_{j=d+1}^{m}\ell_{j}^{v_{j}^{(i)}}\left(|z^{\mathbb{C}^{d}}_{1}|^{2},\ldots,|z^{\mathbb{C}^{d}}_{d}|^{2}\right)\right)^{1/2}.$

It is obviously well-defined over $\pi_{\mathbb{C}^{d}}^{-1}(P-B)$. We need to
show that it has inverse, which gives the required extension of
$\phi_{\mathbb{C}^{n}}\circ\phi_{X}^{-1}$. Since
$(\phi_{\mathbb{C}^{n}}\circ\phi_{X}^{-1})^{*}(\omega_{\mathbb{C}^{n}})=\omega_{X}$
on $(\mathbb{C}^{\times})^{d}$, this still holds over $\mathbb{C}^{d}$ as the
equality is a closed condition.

Consider the Jacobian of
$\vec{z}^{X}(\vec{z}^{\mathbb{C}^{d}},\overline{\vec{z}^{\mathbb{C}^{d}}})$.
We shall show it is positive definite, and hence invertible. To simplify, we
write $z=z^{\mathbb{C}^{d}}$. Denote
$G=\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}(x)\log\ell_{i}(x)-v\cdot x\right)$.
For any non-zero vector $(a_{1},\ldots,a_{d})$,

$\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}z_{j}^{X}$
$\displaystyle=$
$\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}\left(\exp(\partial_{x_{j}}|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\frac{z_{j}}{|z_{j}|}\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\exp(\partial_{x_{j}}|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\partial_{z_{i}}\left(\frac{z_{j}}{|z_{j}|}\right)+\sum_{i,j}\overline{a_{j}}a_{i}\frac{z_{j}}{|z_{j}|}\cdot\exp(\partial_{x_{j}}|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\frac{\partial(z_{i}\bar{z_{i}})}{\partial
z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$
$\displaystyle=$
$\displaystyle\sum_{i}|a_{i}|^{2}|z_{i}^{X}|\cdot\left(\frac{1}{2|z_{i}|}\right)+\sum_{i,j}|z_{j}|^{-1}\cdot|z_{j}^{X}|\cdot\overline{a_{j}}z_{j}\cdot
a_{i}\bar{z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}.$

Note that
$|z_{j}|^{-1}|z_{j}^{X}|=\left(\prod_{k=d+1}^{m}\ell_{k}^{v_{k}^{(j)}}\left(|z^{\mathbb{C}^{d}}_{1}|^{2},\ldots,|z^{\mathbb{C}^{d}}_{d}|^{2}\right)\right)^{1/2}$
which is positive. Let $c=\min\\{|z_{j}|^{-1}|z_{j}^{X}|:j=1,\ldots,d\\}$.
Then the second term is no less than
$ca_{i}\bar{z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$.
Since $\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$ is positive
definite on $P^{\circ}$, it is semi-positive definite on $P-B$. Thus this term
is non-negative. The first term is positive. Thus
$\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}z_{j}^{X}>0$. Similarly
$\sum_{i,j}a_{j}\overline{a_{i}}\overline{\partial_{z_{i}}}z_{j}^{X}$. Hence
the Jacobian is positive-definite and hence invertible. ∎

######  Example 4.7.

We continue to consider $\mathbb{P}^{d}$. From Example 4.4,
$x_{i}=\frac{|z^{\mathbb{P}^{d}}_{i}|^{2}}{1+\sum_{j=1}^{d}|z^{\mathbb{P}^{d}}_{j}|^{2}}$.
$\pi^{-1}_{\mathbb{C}^{d}}(P-B)=\\{\|\vec{z}^{\mathbb{C}^{d}}\|<1\\}$. From
Example 4.3, $x_{i}=|z^{\mathbb{C}^{d}}_{i}|^{2}$. Hence the symplectomorphism
$(\mathbb{C}^{d},\omega_{\mathbb{P}^{d}}|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\\{\|\vec{z}^{\mathbb{C}^{d}}\|<1\\},\omega_{\mathbb{C}^{n}}|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)})$
is

$z^{\mathbb{C}^{d}}_{i}=\frac{z^{\mathbb{P}^{d}}_{i}}{\sqrt{1+\|\vec{z}^{\mathbb{P}^{d}}\|^{2}}}.$
(4.2)

When $d=1$, this gives $\frac{z}{\sqrt{1+|z|^{2}}}$ which is another
activation function used in machine learning. ($z$ is restricted in
$\mathbb{R}$ in most algorithms.) The symplectomorphism can be easily checked
in this case: ($z=z^{\mathbb{P}^{1}}$ for simplicity)

$\displaystyle dz^{\mathbb{C}}\wedge\overline{dz^{\mathbb{C}}}=$
$\displaystyle d\frac{z}{\sqrt{1+|z|^{2}}}\wedge
d\frac{\overline{z}}{\sqrt{1+|z|^{2}}}$ $\displaystyle=$
$\displaystyle\frac{(1+z\bar{z})dz-(|z|^{2}dz+z^{2}\overline{dz})/2}{(1+z\bar{z})^{3/2}}\wedge\frac{(1+z\bar{z})\overline{dz}-(|z|^{2}\overline{dz}+\bar{z}^{2}dz)/2}{(1+z\bar{z})^{3/2}}=\frac{dz\wedge\overline{dz}}{(1+z\bar{z})^{2}}$

giving the Fubini-Study metric. See Figure 5.

By taking the direct product, the symplectomorphism for the case
$X=(\mathbb{P}^{1})^{d}$ is
$\left(\frac{z_{j}}{\sqrt{1+|z_{j}|^{2}}}\right)_{i=1}^{d}$.

$\frac{z}{\sqrt{1+|z|^{2}}}$ Figure 5. $\mathbb{C}$ as a Chart in
$\mathbb{P}^{1}$ Mapped to an Open Disk

Due to the nice fact that the Fubini-Study metric on $\mathbb{P}^{d}$ is
$U(d)$-invariant (and so does the standard metric on $\mathbb{C}^{d}$), we
have the following (which is not true for $(\mathbb{P}^{1})^{d}$ nor general
toric manifolds).

######  Lemma 4.8.

For $X=\mathbb{P}^{d}$, the symplectomorphism
$\psi:(\mathbb{C}^{d},\omega_{\mathbb{P}^{d}}|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\pi_{\mathbb{C}^{d}}^{-1}(P-B),\omega_{\mathbb{C}^{n}}|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)}\right)$
in Proposition 4.6 is $U(d)$-equivariant.

###### Proof.

Using the explicit expression (4.2),

$\psi(U\cdot\vec{z}^{\mathbb{P}^{d}})=\frac{U\cdot\vec{z}^{\mathbb{P}^{d}}}{\sqrt{1+\|U\cdot\vec{z}^{\mathbb{P}^{d}}\|^{2}}}=\frac{U\cdot\vec{z}^{\mathbb{P}^{d}}}{\sqrt{1+\|\vec{z}^{\mathbb{P}^{d}}\|^{2}}}=U\cdot\psi(\vec{z}^{\mathbb{P}^{d}})$

for all $U\in U(d)$. ∎

As explained in Remark 4.5, we can also equip $X$ with another $T$-invariant
Kähler form (that do not come from the standard Kähler structure on
$\mathbb{C}^{m}$). The symplectomorphism is given by

$\left(\frac{1}{2}d\left(\sum_{i=1}^{m}\ell_{i}(x)\log\ell_{i}(x)+h(x)\right),\mathrm{Id}\right):(P^{\circ}\times
T^{d},\omega_{\mathrm{std}}|_{P^{\circ}\times T^{d}})\to\mathbb{R}^{d}\times
T^{d}\stackrel{{\scriptstyle\exp}}{{\cong}}((\mathbb{C}^{\times})^{d},\omega^{h}|_{(\mathbb{C}^{\times})^{d}}).$

Thus the toric construction is rather flexible.

######  Example 4.9.

The ‘softplus’ function $x=\log(1+e^{2y}):\mathbb{R}\to\mathbb{R}_{>0}$ gives
an example of such a Kähler structure on $\mathbb{C}$ (by identifying it with
$\mathbb{R}_{>0}\times\mathbb{R}$ with the standard symplectic structure). The
inverse is $y=\frac{1}{2}\log(e^{x}-1)$, whose difference with
$\frac{1}{2}\log x$ is
$h^{\prime}=\frac{1}{2}\log\frac{e^{x}-1}{x}=\frac{1}{2}\log\left(1+\sum_{k=1}^{\infty}\frac{x^{k}}{(k+1)!}\right)$
which is indeed a smooth function on $\mathbb{R}_{\geq 0}$. Moreover,
$y^{\prime}=\frac{e^{x}}{2(e^{x}-1)}>0$ on $\mathbb{R}_{>0}$.

###  4.2. Symplectomorphisms of fiber bundles over the moduli

In the last subsection, we have exhibited various symplectic embeddings for
$\mathbb{C}^{n}$. Now we want to make a family version of these maps over the
framed quiver moduli $\mathcal{M}_{\vec{n},\vec{d}}$. The last subsection can
be understood as constructing self-maps on a fiber of a vector bundle over
$\mathcal{M}_{\vec{n},\vec{d}}$.

To globalize (4.2), we consider the universal bundle $\mathcal{V}_{i}$
equipped with a Hermitian metric $H_{i}$. (We have constructed a nice
Hermitian metric on $\mathcal{V}_{i}$ in Section 3.1.) We have a fiberwise
symplectic structure $\omega_{\mathcal{V}_{i}}$ induced from the Hermitian
metric. Moreover, we have the projective bundle
$\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})$ which is a
fiberwise compactification of $\mathcal{V}_{i}$. Then the fiber bundle
$\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})$ is equipped with
a fiberwise Kähler metric
$\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}$ induced
from $H_{i}$ (namely, $\frac{i}{2}\partial\bar{\partial}\log(H_{i}\oplus
H_{0})$ where $H_{0}$ is the trivial metric on $\mathcal{O}_{M}$).

######  Proposition 4.10.

There is a fiberwise symplectomorphism

$\psi_{\mathcal{V}_{i}}:(\mathcal{V}_{i},\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}|_{\mathcal{V}_{i}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\\{v\in\mathcal{V}_{i}:H_{i}(v,v)<1\\},\omega_{\mathcal{V}_{i}}|_{\\{H_{i}(v,v)<1\\}}\right).$

###### Proof.

For each $p\in\mathcal{M}$, we have computed the symplectomorphism

$(\mathcal{V}_{i}|_{p},\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}|_{\mathcal{V}_{i}|_{p}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\\{v\in\mathcal{V}_{i}|_{p}:H_{i}(v,v)<1\\},\omega_{\mathcal{V}_{i}}|_{\\{H_{i}(v,v)<1\\}}\right)$

in (4.2), with the metric given by $H_{i}|_{p}$ here. Thus

$\psi_{\mathcal{V}_{i}}(v)=\frac{v}{\sqrt{1+H_{i}(v,v)}}$ (4.3)

gives a fiberwise symplectomorphism whose image is ${\\{H_{i}(v,v)<1\\}}$. ∎

Recall that the universal bundle $\mathcal{V}_{i}\to\mathcal{M}$ admits an
action of $U_{\vec{n}}$ coming from framing (Definition 3.11). One advantage
of $\psi_{\mathcal{V}_{i}}$ is that it is equivariant under this action.

######  Lemma 4.11.

For $g\in U_{\vec{n}}$,

$\psi_{\mathcal{V}_{i}}\circ g=g\circ\psi_{\mathcal{V}_{i}}$

if we used the metric $H_{i}$ given in Theorem 3.7.

###### Proof.

This follows from (4.3) and $H_{i}(g\cdot v,g\cdot v)=H_{i}(v,v)$ by Lemma
3.13. ∎

######  Remark 4.12.

Equation (4.3) has an alternative derivation using the framing. Namely, we
have the surjective morphism $\rho:\underline{\hat{W}_{i}}\to\mathcal{V}_{i}$
(see Equation (3.4)), whose dual give a fiberwise-linear embedding
$\rho^{*}:\mathcal{V}_{i}\stackrel{{\scriptstyle
H_{i}}}{{\cong}}\mathcal{V}_{i}^{*}\to\underline{\hat{W}_{i}}^{*}$. (The
underline means the trivial bundle over $\mathcal{M}$ associated with the
vector space.) Then $\mathbb{P}(\underline{\hat{W}_{i}}^{*}\oplus\mathcal{O})$
(with the standard metric) induces a fiberwise Kähler form on
$\mathcal{V}_{i}^{*}$, and we have a fiberwise symplectic embedding
$(\mathcal{V}_{i},\omega_{\mathbb{P}(\underline{\hat{W}_{i}}^{*}\oplus\mathcal{O})})\hookrightarrow(\mathcal{V}_{i},\omega_{\mathcal{V}_{i}})$.
This gives

$\frac{\rho\rho^{*}\cdot H_{i}\cdot v}{\sqrt{1+H_{0}(\rho^{*}\cdot H_{i}\cdot
v,\rho^{*}\cdot H_{i}\cdot v)}}=\frac{\rho\rho^{*}\cdot H_{i}\cdot
v}{\sqrt{1+v^{*}\cdot H_{i}^{*}\cdot\rho\rho^{*}\cdot H_{i}\cdot v}}.$

Now if we use the metric $H_{i}=(\rho\rho^{*})^{-1}$ given by Theorem 3.7,
then the above equals to the expression in (4.3).

We also have a more flexible construction using the framing, which globalize
_any given non-linear continuous map_
$\sigma_{\mathbb{C}}:\mathbb{C}^{n_{i}}\to\mathbb{C}^{n_{i}}$. Namely,
$\sigma_{\mathbb{C}}$ can be regarded as a fiberwise non-linear self-map on
the trivial bundle
$\underline{\mathbb{C}^{n_{i}}}\to\underline{\mathbb{C}^{n_{i}}}$ (still
denoted by $\sigma_{\mathbb{C}}$). Then we take the composition

$\mathcal{V}_{i}\stackrel{{\scriptstyle
H_{i}}}{{\cong}}\mathcal{V}_{i}^{*}\stackrel{{\scriptstyle(e^{(i)})^{*}}}{{\to}}\underline{\mathbb{C}^{n_{i}}}\stackrel{{\scriptstyle\sigma_{\mathbb{C}}}}{{\to}}\underline{\mathbb{C}^{n_{i}}}\stackrel{{\scriptstyle
e^{(i)}}}{{\to}}\mathcal{V}_{i}$

and denote it by $\sigma_{\mathcal{V}_{i}}$. See Figure 6.

${\underline{\mathbb{C}^{n}}}$${\underline{\mathbb{C}^{n}}^{*}}$${\mathcal{V}_{i}}$${\mathcal{V}_{i}^{*}}$$\scriptstyle{e^{(i)}}$$\scriptstyle{\sigma_{\mathbb{C}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(e^{(i)})^{*}}$
Figure 6. Using the framing to globalize an activation function
$\sigma_{\mathbb{C}}:\mathbb{C}^{n}\to\mathbb{C}^{n}$ over $\mathcal{M}$.

It is easy to get the following explicit expression in terms of
$\sigma_{\mathbb{C}}$.

######  Lemma 4.13.

The above fiber-bundle map
$\sigma_{\mathcal{V}_{i}}:\mathcal{V}_{i}\to\mathcal{V}_{i}$ equals to

$\sigma_{\mathcal{V}_{i}}(v)=\sum_{k=1}^{n_{i}}(\sigma_{\mathbb{C}})_{k}\left(H_{i}(e^{(i)}_{1},v),\ldots,H_{i}(e^{(i)}_{n_{i}},v)\right)\cdot
e^{(i)}_{k}$ (4.4)

where we write
$\sigma_{\mathbb{C}}=((\sigma_{\mathbb{C}})_{1},\ldots,(\sigma_{\mathbb{C}})_{n_{i}})$
and $e^{(i)}=(e^{(i)}_{1}\ldots e^{(i)}_{n_{i}})$.

For instance, we can take $\sigma$ to be the one coming from the
symplectomorphism in Corollary 4.2. We shall prove the universal approximation
theorem for such $\sigma$ in Section 5.

###  4.3. A machine learning program using the framed quiver moduli

Let $Q$ be a digraph and denote by $\mathbb{C}\cdot Q$ its path algebra over
$\mathbb{C}$. Let $\vec{d}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$ be a dimension
vector. We take the framing dimension vector to be $\vec{n}=\vec{d}+\vec{1}$,
where $\vec{1}_{i}:=1$ for all $i\in Q_{0}$. The one additional framing vector
is used for translation (called a ‘bias’ vector).

We fix a collection of input vertices and a collection of output vertices
$I_{\mathrm{in}},I_{\mathrm{out}}\subset Q_{0}$, and

$\gamma\in I_{\mathrm{out}}\cdot(\mathbb{C}\cdot Q)\cdot
I_{\mathrm{in}}=\bigoplus_{i\in I_{\mathrm{in}},j\in
I_{\mathrm{out}}}j\cdot(\mathbb{C}\cdot Q)\cdot i.$

(The trivial path at a vertex $i$ is again denoted by $i$.)

####  4.3.1. Machine learning using a flat space.

Let’s first formulate a typical machine learning program in the quiver setup.
The following flat space

$U\cong\prod_{a\in Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{i\in
Q_{0}}V_{i}$

is used frequently in the subject.

######  Lemma 4.14.

The open subset

$U=\\{[V,e]\in\mathcal{M}_{\vec{d}+\vec{1},\vec{d}}:(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})=I_{d_{i}}\textrm{
for all }i\in Q_{0}\\}\subset\mathcal{M}$

gives a coordinate chart of $\mathcal{M}$. ($I_{d_{i}}$ denotes the identity
matrix of rank $d_{i}$.)

###### Proof.

First, such $(V,e)$ are stable: $\mathrm{Im}(e)$ is the whole $V$. Second, for
distinct $(V,e),(V^{\prime},e^{\prime})$ satisfying the above condition,
$[V,e]\not=[V^{\prime},e^{\prime}]$: since they are stable, their orbits are
closed. Suppose $g\cdot(V,e)=(V^{\prime},e^{\prime})$ for some
$g\in\mathrm{GL}_{\vec{d}}$. Then $g_{i}\cdot e^{(i)}=(e^{\prime})^{(i)}$. But
since
$(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})=I_{d_{i}}=((e^{\prime})^{(i)}_{1},\ldots,(e^{\prime})^{(i)}_{d_{i}})$,
this forces $g=\mathrm{Id}$ and so $(V,e)=(V^{\prime},e^{\prime})$,
contradicting that they are distinct.

Then we have the chart map $U\stackrel{{\scriptstyle\cong}}{{\to}}\prod_{a\in
Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{i\in Q_{0}}V_{i}$ defined by

$W_{a}:=(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})^{-1}V(a),\,\,b_{i}:=(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})^{-1}e^{(i)}_{d_{i}+1}.$

∎

Now we fix a path $\gamma$ from the input vertices $I_{\mathrm{in}}$ to the
output vertices $I_{\mathrm{out}}$. For each element $[V,e]\in U$, by
composing the affine linear maps $V_{a}(\cdot)+e^{(t(a))}_{d_{t(a)}+1}$
attached to arrows $a$ in the path $\gamma$, together with some non-linear
functions $\sigma_{i}:V_{i}\to V_{i}$ that are called ‘activation functions’,
one obtains a non-linear function

$f^{U}_{[V,e]}:\bigoplus_{i\in
I_{\mathrm{in}}}\mathbb{C}^{d_{i}}\stackrel{{\scriptstyle(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})}}{{\cong}}\bigoplus_{i\in
I_{\mathrm{in}}}V_{i}\to\bigoplus_{j\in
I_{\mathrm{out}}}V_{j}\stackrel{{\scriptstyle(e^{(j)}_{1},\ldots,e^{(j)}_{d_{i}})}}{{\cong}}\bigoplus_{j\in
I_{\mathrm{out}}}\mathbb{C}^{d_{j}}$

which is used to approximate a non-explicitly given function $f$. A stochastic
gradient flow of the error function on $U$ is employed to find the optimal
point in $U$.

Let’s write down the symmetry in Lemma 4.11 in the chart $U$.

######  Lemma 4.15.

The chart $U$ is invariant under the right action by

$\prod_{i\in Q_{0}}U(d_{i})\subset\prod_{i\in Q_{0}}U(d_{i}+1)=U_{\vec{n}}$

where $U(d_{i})\subset U(d_{i}+1)$ is embedded as
$\left(\begin{array}[]{cc}U(d_{i})&0\\\ 0&1\end{array}\right)$.

Consider the trivialization $\mathcal{V}_{i}|_{U}\cong\left(\prod_{a\in
Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{j\in
Q_{0}}V_{j}\right)\times V_{i}$. The right action of $g\in U(d_{i})$ on
$\mathcal{V}_{i}|_{U}$ is given by

$(W_{a},b_{j},v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in
V_{j}\\\ v\in V_{i}\end{subarray}}\cdot g=(g^{-1}\cdot W_{a},g^{-1}\cdot
b_{j},g^{-1}v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in V_{j}\\\
v\in V_{i}\end{subarray}}$

where $g^{-1}\cdot W_{a}$ equals to $g^{-1}W_{a}$ if $h(a)=i$, $W_{a}g$ if
$t(a)=i$, and $W_{a}$ otherwise; $g^{-1}\cdot b_{j}$ equals to $g^{-1}b_{j}$
if $j=i$, and $b_{j}$ if $j\not=i$.

###### Proof.

$U$ consists of points $[V,e]$ where $(e^{(j)}_{1},\ldots,e^{(j)}_{d_{i}})$
are invertible for all $j\in Q_{0}$. This property is invariant under the
action of $\prod_{i\in Q_{0}}U(d_{i})$. Hence $U$ is an invariant subset.

$(W_{a},b_{j},v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in
V_{j}\\\ v\in V_{i}\end{subarray}}$ corresponds to the point
$[W,e,v]\in\mathcal{V}_{i}=(R^{s}_{n,d}\times V_{i})/\mathrm{GL}_{\vec{d}}$.
where $e^{(j)}=(I_{d_{j}}\,\,b_{j})$.

$\displaystyle[W,e,v]\cdot g=[W,e\cdot g,v]=[W,(g\,\,b_{j})_{j\in Q_{1}},v]$
$\displaystyle=$ $\displaystyle[g^{-1}\cdot W,(I_{d_{j}}\,\,g^{-1}\cdot
b_{j})_{j\in Q_{1}},g^{-1}\cdot v]$

where the left action by $g^{-1}$ (as an element in $\mathrm{GL}_{\vec{d}}$)
is as specified by definition. ∎

Now we prove an important symmetric property that the activation function
(4.2) enjoys, which can be used to reduce the dimensions.

######  Proposition 4.16.

Suppose the activation functions $\sigma_{i}:V_{i}\to V_{i}$ are taken to be
the one given in Equation (4.2). Then $f^{U}$ is invariant under
$\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\subset
U_{\vec{n}}$. (See the embedding in Lemma 4.15.)

###### Proof.

The terms of $f^{U}$ are of the form
$\sigma_{h(a_{k})}\left(W_{a_{k}}\ldots\left(\sigma_{h(a_{1})}\left(W_{a_{1}}(v)+b_{h(a_{1})}\right)\ldots\right)+b_{h(a_{k})}\right)$.
By the above lemma, for $g\in U(d_{h(a_{i})})$ where $h(a_{i})\not\in
I_{\mathrm{in}}\cup I_{\mathrm{out}}$, the action of $g$ results in
$\sigma_{h(a_{i})}\mapsto g\cdot\sigma_{h(a_{i})}g^{-1}$ in the above
expression and does not affect any other part. By Lemma 4.8, $\sigma_{i}$ is
$U(d_{i})$-equivariant, and hence $f^{U}$ remains invariant. ∎

By the above proposition, we can descend $f^{U}$ to the orbit space
$U/\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})$ to reduce
the dimensions. However, the quotient space will be highly singular. Instead,
we can realize the dimension reduction by restricting $f^{U}$ to a submanifold
$U^{\prime}$ of $U$ whose orbit occupies the whole $U$, and do stochastic
gradient flow on $U^{\prime}$ instead of $U$. The following gives one simple
possibility.

######  Proposition 4.17.

Consider a subset $\\{a_{1},\ldots,a_{p}\\}$ of arrows whose heads and tails
do not belong to $I_{\mathrm{in}}\cup I_{\mathrm{out}}$, and for any two
distinct arrows $a_{1},a_{2}$ in the subset, $t(a_{1})\not=h(a_{2})$. Then the
vector subspace

$U^{\prime}:=\\{(W_{a},b_{j})_{\begin{subarray}{c}a\in Q_{1}\\\ j\in
Q_{0},b_{j}\in V_{j}\end{subarray}}\in U:W_{a_{i}}\textrm{ is of the form
}\left(D\,\,W_{a_{i}}^{\prime}\right)\,\,\forall i=1,\ldots,p\\}$

has its orbit being the whole $U$, that is, $\left(\prod_{j\not\in
I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\right)\cdot U^{\prime}=U$. In
above, $D$ is a diagonal matrix (of the maximum possible size) and
$W_{a_{i}}^{\prime}$ is any matrix occupying the rest.

###### Proof.

This follows from the singular-value decomposition of a matrix $W$ as
$A\cdot\left(D\,\,W^{\prime}\right)\cdot B$ where $A$ and $B$ are unitary
matrices of appropriate sizes. ∎

####  4.3.2. Machine learning using the quiver moduli.

The quiver moduli $\mathcal{M}=\mathcal{M}_{\vec{n},\vec{d}}$ gives a
compactification of $U$. Compactness is important for the formulation of Morse
theory and convergence of a gradient flow. We would like to use the whole
$\mathcal{M}$ in application of machine learning. Non-trivial metrics over the
moduli will play a crucial role.

First, consider the situation before adding in activation functions. Each
arrow $a\in Q_{1}$ is associated with a vector-bundle morphism
$a_{\mathcal{M}}:\mathcal{V}_{t(a)}\to\mathcal{V}_{h(a)}$. For each path
$a_{k}\ldots a_{1}$, we take the map

$(a_{k})_{\mathcal{M}}\left(\ldots\left((a_{2})_{\mathcal{M}}\left((a_{1})_{\mathcal{M}}(v)+e^{(h(a_{1}))}_{d_{h(a_{1})}+1}\right)+e^{(h(a_{2}))}_{d_{h(a_{2})}+1}\right)\ldots\right)+e^{(h(a_{k}))}_{d_{h(a_{k})}+1}$
(4.5)

which is fiberwise affine linear. Thus a path $\gamma$ gives an affine bundle
morphism

$\gamma_{\mathcal{M}}:\mathcal{V}_{I_{\mathrm{in}}}:=\bigoplus_{i\in
I_{\mathrm{in}}}\mathcal{V}_{i}\to\mathcal{V}_{I_{\mathrm{out}}}:=\bigoplus_{j\in
I_{\mathrm{out}}}\mathcal{V}_{j}.$

Then we have

$L_{\gamma}\left(\left(s^{(i)}_{k}\right)_{\begin{subarray}{c}i\in
I_{\mathrm{in}}\\\
k\in\\{1,\ldots,d_{i}\\}\end{subarray}}\right)=\left(H_{j}\left(e^{(j)}_{p},\sum_{i\in
I_{\mathrm{in}}}\gamma_{\mathcal{M}}\cdot\sum_{k=1}^{d_{i}}s^{(i)}_{k}e^{(i)}_{k}\right)\right)_{\begin{subarray}{c}j\in
I_{\mathrm{out}},\\\ p\in\\{1,\ldots,d_{j}\\}\end{subarray}}:\bigoplus_{i\in
I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}\to\bigoplus_{j\in
I_{\mathrm{out}}}\underline{\mathbb{C}^{d_{j}}}.$ (4.6)

Suppose a continuous function $f=K\to\bigoplus_{j\in
I_{\mathrm{out}}}\mathbb{C}^{d_{j}}$ is given, where $K$ is a compact subset
of $\bigoplus_{i\in I_{\mathrm{in}}}\mathbb{C}^{d_{i}}$. Then the fiberwise
integral

$\mathcal{E}:=\int_{K}\left\|f-L_{\gamma}\right\|^{2}_{\mathcal{V}_{I_{\mathrm{out}}}}d\mu_{K}$
(4.7)

gives a smooth function on $\mathcal{M}$.

######  Remark 4.18.

Alternatively, we can define the fiber-bundle morphism

$f_{\mathcal{M}}:\sum_{j\in
I_{\mathrm{out}}}\sum_{l=1}^{d_{j}}f^{(j)}_{l}\cdot
e^{(j)}_{l}:\underline{K}\to\mathcal{V}_{I_{\mathrm{out}}}$

and take

$\int_{K}\left\|f_{\mathcal{M}}\left(\left(s^{(i)}_{k}\right)_{\begin{subarray}{c}i\in
I_{\mathrm{in}}\\\ k\in\\{1,\ldots,d_{i}\\}\end{subarray}}\right)-\sum_{i\in
I_{\mathrm{in}}}\gamma_{\mathcal{M}}\cdot\sum_{k=1}^{d_{i}}s^{(i)}_{k}e^{(i)}_{k}\right\|^{2}_{\mathcal{V}_{I_{\mathrm{out}}}}d\mu_{K}.$

However, with such a definition, we need to worry that
$(e^{(j)}_{1},\ldots,e^{(j)}_{d_{j}})$ for some output $j$ degenerates (as a
frame), in which case approximating $f$ and approximating $f_{\mathcal{M}}$
are different.

We have a gradient flow $r:\mathbb{R}\to\mathcal{M}$ which can be used to
minimize $\mathcal{E}$:

$\frac{dr}{dt}=-(\nabla\mathcal{E})(r(t))$

where $\nabla\mathcal{E}=(d\mathcal{E})^{\\#_{g}}$ where
$(\cdot)^{\\#}_{g}:T^{*}\mathcal{M}\stackrel{{\scriptstyle
g}}{{\cong}}T\mathcal{M}$ is the identification by a metric $g$ on
$\mathcal{M}$. ($(\nabla\mathcal{E})^{p}=g^{pq}\partial_{q}\mathcal{E}$ in
local coordinates.) $g$ can be taken to be the induced metric from the trivial
metric on the vector space $R_{\vec{n},\vec{d}}$ via symplectic reduction.
Alternatively, $g$ can be taken to be the metric given by the Ricci curvature
in Theorem 3.15 (when $Q$ has no oriented cycle), which has a better
expression in homogeneous coordinates.

Note that $(L_{\gamma})|_{[V,e]}$ is affine linear on $\bigoplus_{i\in
I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}$ for every
$[V,e]\in\mathcal{M}$, which is not good enough for the purpose of
approximating $f$. We introduce fiberwise non-linearity below.

######  Definition 4.19.

Let $A$ be a finite set whose every element is associated with two vertices
(head $h$ and tail $t$) in $Q_{0}$. Elements in $A$ are called activation
arrows. (These are not arrows in $Q_{1}$.) The semiring generated by $Q_{1}$
and $A$, denoted by $\Gamma(Q,A)$, has the underlying vector space spanned by
the independent set $\coprod_{p=0}^{\infty}S_{p}$ where $S_{p}$ is defined
inductively as follows.

  1. (1)

$S_{0}$ consists of all paths of $Q$.

  2. (2)

Suppose $S_{p}$ has been defined, and each element in $S_{p}$ has a head $h$
and a tail $t$. $S_{p+1}$ consists of $\gamma\cdot\alpha\cdot\tilde{\gamma}$,
where $\alpha\in A$, $\gamma$ is any path of $Q$ with $h(\alpha)=t(\gamma)$,
and $\tilde{\gamma}\in t(\alpha)\cdot(\mathbb{C}\cdot S_{p})$.
($\mathbb{C}\cdot S_{p}$ denotes the vector space generated by $S_{p}$;
$t(\alpha)\cdot(\mathbb{C}\cdot S_{p})$ is the subspace generated by elements
of $S_{p}$ with head being $t(\alpha)$.) The above element has the head
$h(\gamma)$ and the tail $t(\tilde{\gamma})$.

The above vector space $\Gamma(Q,A)$ has an obvious product by concatenation.
($s_{1}\cdot s_{2}=0$ if $h(s_{2})\not=t(s_{1})$.) Note that for $\alpha\in
A$, $(s_{1}+cs_{2})\cdot\alpha=s_{1}\cdot\alpha+cs_{2}\cdot\alpha$, but
$\alpha\cdot(s_{1}+cs_{2})\not=\alpha\cdot s_{1}+c\alpha\cdot s_{2}$.

Now, suppose each element $\alpha\in A$ is associated with a fiber-bundle
morphism

$\alpha_{\mathcal{M}}:\mathcal{V}_{t(\alpha)}\to\mathcal{V}_{h(\alpha)}.$

Then by composing the corresponding affine linear morphisms (as in Equation
(4.5)) and $\alpha_{\mathcal{M}}$, an element $\tilde{\gamma}\in\Gamma(Q,A)$
induces a fiber-bundle morphism

$\tilde{\gamma}_{\mathcal{M}}:\mathcal{V}_{t(\tilde{\gamma})}\to\mathcal{V}_{h(\tilde{\gamma})}.$

In particular, if we fix $\tilde{\gamma}\in
I_{\mathrm{out}}\cdot\Gamma(Q,A)\cdot I_{\mathrm{in}}$, then we define

$f_{\tilde{\gamma}}:\bigoplus_{i\in
I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}\to\bigoplus_{j\in
I_{\mathrm{out}}}\underline{\mathbb{C}^{d_{j}}}$

like in the definition of $L_{\gamma}$ in (4.6) by replacing
$\gamma_{\mathcal{M}}$ by $\tilde{\gamma}_{\mathcal{M}}$. Then a stochastic
gradient flow for the corresponding error function (by replacing $L_{\gamma}$
by $f_{\tilde{\gamma}}$ in (4.7)) can be carried out.

For now, we set $A$ to be $Q_{0}$ as a set, and each element
$\mathfrak{o}_{i}$ has the head and tail being $i\in Q_{0}$. We can associate
$\mathfrak{o}_{i}$ with the fiber-bundle morphism $\psi_{\mathcal{V}_{i}}$
given in (4.3), or $\sigma_{\mathcal{V}_{i}}$ in (4.4). Then we obtain
$f_{\tilde{\gamma}}$ above which is non-linear along fibers for the purpose of
machine learning.

If we associate $\mathfrak{o}_{i}$ with $\psi_{\mathcal{V}_{i}}$ given in
(4.3), then the symmetry of $\prod_{j\not\in I_{\mathrm{in}}\cup
I_{\mathrm{out}}}U(d_{j})\subset U_{\vec{n}}$ is respected, by Lemma 4.11. The
proof is similar to that for Proposition 4.16 and is omitted here.

######  Proposition 4.20.

Suppose $\mathfrak{o}_{i}$ is assigned as $\psi_{\mathcal{V}_{i}}$ given in
(4.3), and $H_{i}$ is taken to be the metric in Theorem 3.7. Then
$f_{\tilde{\gamma}}$ is invariant under $\prod_{j\not\in I_{\mathrm{in}}\cup
I_{\mathrm{out}}}U(d_{j})\subset U_{\vec{n}}$. (See the embedding in Lemma
4.15.)

######  Remark 4.21.

Since we are taking affine linear morphisms
$(a_{i})_{\mathcal{M}}(v)+e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$ for the arrows
which involves the term $e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$, only the symmetry
$U(d_{j})$ rather than $U(d_{j}+1)=U(n_{j})$ is respected. If the bias vector
$e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$ is not used in the program, then
$f_{\tilde{\gamma}}$ will be invariant under the bigger group $\prod_{j\not\in
I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(n_{j})$.

####  4.3.3. A simple example

Recall the quiver in Example 2.7. Let’s take
$n_{1}=d_{1},n_{2}=d_{2}+1,n_{3}=d_{3}$ instead of $n_{i}=d_{i}+1\,\forall i$,
since we do not need to use bias vectors at the input and output vertices. The
path $\gamma$ is simply $a_{2}a_{1}$, and
$\tilde{\gamma}=a_{2}\cdot\mathfrak{o}_{1}\cdot a_{1}$.

We have the universal bundles
$\mathcal{V}_{1}\cong\underline{\mathbb{C}}^{d_{1}}$, $\mathcal{V}_{2}$ and
$\mathcal{V}_{3}$. We have $\rho^{(1)}=e^{(1)}$,
$\rho^{(2)}=(e^{(2)}\,\,a_{1}e^{(1)})$,
$\rho^{(3)}=(e^{(3)}\,\,a_{2}e^{(2)}\,\,a_{2}a_{1}e^{(1)})$. In terms of
homogeneous coordinates (namely the coordinates
$(e^{(1)},e^{(2)},e^{(3)},a_{1},a_{2})$ on the vector space
$R_{\vec{n},\vec{d}}$, where each entry is a matrix of suitable size), the
metrics given in Theorem 3.7 are

$\displaystyle H_{1}$
$\displaystyle=\left(e^{(1)}(e^{(1)})^{*}\right)^{-1},\,H_{2}=\left(e^{(2)}(e^{(2)})^{*}+a_{1}e^{(1)}(e^{(1)})^{*}a_{1}^{*}\right)^{-1},\,$
$\displaystyle H_{3}$
$\displaystyle=\left(e^{(3)}(e^{(3)})^{*}+a_{2}e^{(2)}(e^{(2)})^{*}a_{2}^{*}+a_{2}a_{1}e^{(1)}(e^{(1)})^{*}a_{1}^{*}a_{2}^{*}\right)^{-1}.$

on $\mathcal{V}_{i},i=1,2,3$ respectively. The activation functions we
constructed in the previous subsection are

$\psi_{i}(v)=\frac{v}{\sqrt{1+v^{*}H_{i}v}}\textrm{ and
}\sigma_{i}(v)=\sum_{k=1}^{d_{i}+1}(\sigma_{\mathbb{C}})_{k}\left((e^{(i)}_{1})^{*}H_{i}v,\ldots,(e^{(i)}_{d_{i}+1})^{*}H_{i}v\right)\cdot
e^{(i)}_{k},$

where we can take
$(\sigma_{\mathbb{C}})_{k}(\vec{z})=\frac{e^{2\mathrm{Re}(z_{k})}}{1+\sum_{j=1}^{d_{i}+1}e^{2\mathrm{Re}(z_{j})}}+\mathbf{i}\,\mathrm{Im}(z_{k})$
for instance. Both have the $\mathrm{GL}$-equivariance property
$\psi_{i}(g\cdot v)=g\cdot\psi_{i}(v)$ and $\sigma_{i}(g\cdot
v)=g\cdot\sigma_{i}(v)$.

The function (over $\mathcal{M}$) cooked up from this quiver is
$f_{\tilde{\gamma}}:\mathbb{C}^{d_{1}}\to\mathbb{C}^{d_{3}}$,

$f_{\tilde{\gamma}}(s_{1},\ldots,s_{d_{1}})=\left(H_{3}\left(e^{(3)}_{p},a_{2}\cdot\sigma_{2}\left(a_{1}\cdot\sum_{k=1}^{d_{1}}s_{k}e^{(1)}_{k}+e^{(2)}_{d_{2}+1}\right)\right)\right)_{p=1}^{d_{3}}$

if we use $\sigma_{2}$ as the activation function, or the same expression with
$\sigma_{2}$ replaced by $\psi_{2}$. Then we run a stochastic gradient flow on
$\mathcal{M}$ (or on the vector space $R_{\vec{n},\vec{d}}$ upstairs) to
minimize the distance of $f_{\tilde{\gamma}}$ and a function $f$ coming from
reality.

To run the gradient flow, we need to take a metric on $\mathcal{M}$. Recall
that we have the metric on the tangent bundle of $\mathcal{M}$ coming from the
Ricci curvatures of $H_{i}$:

$H_{T}=\sum_{i=1}^{3}\mathrm{tr~{}}\left(\left(\partial_{v}\rho^{(i)}\right)^{\mathrm{*}}\cdot
H_{i}\cdot\partial_{w}\rho^{(i)}\right)-\sum_{i=1}^{3}\mathrm{tr~{}}\left(\left(\partial_{v}\rho^{(i)}\cdot(\rho^{(i)})^{\mathrm{*}}\cdot
H_{i}^{\frac{1}{2}}\right)^{\mathrm{*}}\cdot
H_{i}\cdot\left(\partial_{w}\rho^{(i)}\cdot(\rho^{(i)})^{\mathrm{*}}\cdot
H_{i}^{\frac{1}{2}}\right)\right).$ (4.8)

Consider the open subset of the vector space $R_{\vec{n},\vec{d}}$ in which
$(e_{1}^{(i)},\ldots,e_{d_{i}}^{(i)})$ is invertible. (This is the preimage of
the chart $U\subset\mathcal{M}$.) A tangent vector $v$ of $\mathcal{M}$ is
lifted as $(\delta a_{1},\delta a_{2},\delta e_{d_{2}+1}^{(2)})$ (and all
other components are set to be zero). Since $\rho^{(1)}=e^{(1)}$,
$\rho^{(2)}=(e^{(2)},a_{1}e^{(1)})$,
$\rho^{(3)}=(e^{(3)},a_{2}e^{(2)},a_{2}a_{1}e^{(1)})$, we have
$\partial_{v}\rho^{(1)}=0$, $\partial_{v}\rho^{(2)}=\left((0\,\,\delta
e_{d_{2}+1}^{(2)}),\,\,(\delta a_{1})e^{(1)}\right)$, and

$\partial_{v}\rho^{(3)}=\left(0,\,\,(\delta a_{2})e^{(2)}+(0\,\,a_{2}\delta
e_{d_{2}+1}^{(2)}),\,\,\,(\delta a_{2})a_{1}e^{(1)}+a_{2}(\delta
a_{1})e^{(1)}\right).$

Then the above metric $H_{T}$ can be computed explicitly in terms of the
homogeneous coordinates.

######  Remark 4.22.

In above, if we use trivial metrics over the vector space
$R_{\vec{n},\vec{d}}$ instead of $H_{i}$ and $H_{T}$, the expressions will get
simpler; however they will only be $U_{\vec{d}}$-equivariant rather than
$\mathrm{GL}_{\vec{d}}$-equivariant. Then we need to restrict to the moment-
map level $\mu^{-1}(I)\subset R_{\vec{n},\vec{d}}$ and its tangent bundle, in
order to stay in the same moduli $\mathcal{M}$ downstairs. This would increase
the computational complexity.

We can also write in inhomogeneous coordinates $(W_{1},W_{2},b)$ in the chart
$U\subset\mathcal{M}$, where $(e_{1}^{(i)},\ldots,e_{d_{i}}^{(i)})=I_{d_{i}}$
for all $i=1,2,3$. $W_{1},W_{2}$ are the matrices of the arrows $a_{1},a_{2}$
and $e_{d_{2}+1}^{(2)}=b$ is the bias vector. Then $H_{1}=I$,
$H_{2}=(I+bb^{*}+W_{1}W_{1}^{*})^{-1}$, and
$H_{3}=(I+W_{2}W_{2}^{*}+W_{2}bb^{*}W_{2}^{*}+W_{2}W_{1}W_{1}^{*}W_{2}^{*})^{-1}$.
Then we can run the gradient flow in $U\subset\mathcal{M}$ (which has lower
dimensions than $R_{\vec{n},\vec{d}}$).

Note that when $W_{1},W_{2},b$ are close to zero, $H_{i}$ are close to the
identity matrix, and $\sigma_{i}$ is close to $\sigma_{\mathbb{C}}$. Moreover,
the second term of (4.8) is close to zero, and the first term is close to the
standard metric. Thus when $W_{1},W_{2},b$ are small, the function
$f_{\tilde{\gamma}}$ is close to the commonly used one

$f^{U}_{W_{1},W_{2},b}(v)=W_{2}\cdot(\sigma_{\mathbb{C}}(W_{1}\cdot v+b)),$
(4.9)

and the gradient flow is close to the usual one on the flat space $U$ (see
Section 4.3.1). The additional terms can be understood as modifications to
ensure the flow converges in $\mathcal{M}$.

###  4.4. A discussion on Morse inequalities

By the work of Reineke [Rei08], the framed quiver moduli $\mathcal{M}$ is a
tower of Grassmannians (Theorem 2.4), and hence its Poincaré polynomial is a
product of that of Grassmannians (Corollary 2.5). Such topological invariants
give important information about a gradient flow on $\mathcal{M}$.

In particular the Morse inequalities for a Morse function $\mathcal{E}$ on a
compact manifold $\mathcal{M}$ state as follows. Let $c^{j}(\mathcal{E})$ be
the number of critical points of index $j$ for $\mathcal{E}$. Then for every
$j$,

$c^{j}(\mathcal{E})\geq h^{j}(\mathcal{M})$

where $h^{j}(\mathcal{M})$ denotes the cohomological numbers (which are
coefficients of the Poincaré polynomial).

Given a gradient flow, which is a path $\gamma:\mathbb{R}\to\mathcal{M}$
satisfying the gradient flow equation, $\lim_{t\to\pm\infty}\gamma(t)$ are
critical points. Moreover, critical points carry important effect to the rate
of the gradient flow. Namely, when the flow $\gamma$ gets close to a critical
point with index being $1,\ldots,\dim\mathcal{M}-1$,
$\|\gamma^{\prime}(t)\|=\|\mathrm{grad}\,\mathcal{E}(\gamma(t))\|$ becomes
small. In other words the flow slows down when it passes through a
neighborhood of a critical point. Such a slowing-down effect of saddle points
was studied in machine learning in [PDGB14, DPG+14].

The cohomological numbers $h^{j}(\mathcal{M})$ give the minimum number of
critical points and hence are important invariants of a neural network (which
simply means a directed graph $Q$ together with a dimension vector $\vec{d}$
here). Over $\mathbb{C}$, $h^{j}(\mathcal{M})=0$ when $j$ is odd. Thus the
Euler characteristic $\chi_{Q,\vec{d}}$ equals to
$\sum_{j}h^{j}(\mathcal{M})$, which is the minimal total number of critical
points. It is computed by simply setting $q=1$ in Corollary 2.5. Another
important invariant is $\dim\mathcal{M}$ (that is, the number of training
parameters of the network), which is simply

$\mathcal{D}_{Q,\vec{d}}=\sum_{i\in
Q_{0}}d_{i}\left(n_{i}+\sum_{\begin{subarray}{c}j\to i\\\
j\not=i\end{subarray}}d_{j}\right)$

using the notation of Corollary 2.5. (We take $n_{i}=d_{i}+1$ in this
section.) These are illustrated in the two practical examples below.

######  Example 4.23.

Consider $Q$ being the $A_{k+2}$ quiver, which has $k+2$ vertices labeled by
$0,\ldots,k+1$, and there is exactly one arrow from $i$ to $i+1$ for
$i=0,\ldots,k$, and no arrow otherwise. Set $d_{-1}:=0$. Then the minimal
total number of critical points is

$\chi_{Q,\vec{d}}=\prod_{i=-1}^{k}\binom{d_{i}+d_{i+1}+1}{d_{i+1}}$

and

$\mathcal{D}_{Q,\vec{d}}=\sum_{i=-1}^{k}d_{i+1}(d_{i}+1).$

######  Example 4.24.

Now consider the following quiver $A_{k+2}^{\prime}$, which has vertices
labeled by $0,\ldots,k+1$, and there is one arrow from vertex $i$ to vertex
$j$ for every $i<j$. Set $d_{-1}:=0$. Then

$\chi_{Q,\vec{d}}=\prod_{i=-1}^{k}\binom{\sum_{j=0}^{i+1}d_{j}+1}{d_{i+1}}$

and

$\mathcal{D}_{Q,\vec{d}}=\sum_{i=-1}^{k}d_{i+1}\left(\sum_{j=0}^{i}d_{j}+1\right).$

Figure 7 shows the graph of $\log\chi_{Q,\vec{d}}$ versus
$\mathcal{D}_{Q,\vec{d}}$ for the two examples, where we set $k=3$,
$d_{1}=600,d_{5}=10$, and $d_{2}=d_{3}=d_{4}$.

Note that the quiver denoted by $A_{k+2}^{\prime}$ in Example 4.24 is a simple
analog of the network known as ResNet, which adds arrows to the
$A_{k+2}$-quiver that skip the middle vertices to get around with the
‘gradient-vanishing problem’. Namely, in the $A_{k+2}$ case, the derivatives
of $\mathcal{E}$ with respect to matrix entries for arrows in the early stage
are typically very small by chain rule, which is not good for the flow rate.
Arrows that skip the middle vertices are added, so that there are short paths
which involve the early arrows.

From Figure 7, we see that in the same dimensions, the minimal number of
critical points in $\mathcal{M}$ is smaller for $A_{5}^{\prime}$ than that for
$A_{5}$. (We have numerically verified this for general $k$.) This gives a
supporting evidence that $\chi_{Q,\vec{d}}$ is an important invariant in
applications to machine learning.

Figure 7. A plot of $\log\chi_{Q,\vec{d}}$ (y-axis) versus
$\mathcal{D}_{Q,\vec{d}}$ (x-axis) for $A_{5}$ and $A_{5}^{\prime}$.

###  4.5. A remark on Abelianization

In many basic neural networks, each vertex of $Q$ is associated with a vector
space of only dimension one. When $\vec{d}=\vec{1}$, that is, all entries of
the dimension vector equal to one, $\mathcal{M}_{\vec{n},\vec{1}}$ is a
quotient by the Abelian group $(\mathbb{C}^{\times})^{\Sigma\vec{d}}$, and