# A refined multiplication formula for cluster characters

Bernhard Keller  B. K. : Université Paris de Paris
UFR de Mathématiques
CNRS
Institut de Mathématiques de Jussieu–Paris Rive Gauche, IMJ-PRG
Bâtiment Sophie Germain
75205 Paris Cedex 13
France<EMAIL_ADDRESS>https://webusers.imj-prg.fr/
bernhard.keller/ ,  Pierre-Guy Plamondon  P.-G.P. : Laboratoire de
Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, Institut
Universitaire de France (IUF<EMAIL_ADDRESS>https://www.imo.universite-paris-saclay.fr/ plamondon/ and  Fan Qin  F.Q.: The
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan
RD, Shanghai, 200240 China<EMAIL_ADDRESS>https://sites.google.com/site/qinfanmath/

###### Abstract.

We obtain a multiplication formula for cluster characters on (stably)
$2$-Calabi–Yau (Frobenius or) triangulated categories. This formula
generalizes those known for arbitrary pairs of objects and for
Auslander–Reiten triangles. As an application, we show that for cluster
algebras of acyclic types, specialization of a cluster variable to $1$ sends
all cluster variables to elements of a cluster algebra of smaller rank. We
also obtain applications to the reduction of friezes of acyclic type.

###### Contents

  1. 1 Introduction
  2. 2 Refined multiplication formula: triangulated case
    1. 2.1 Recollections on $2$-Calabi–Yau triangulated categories
      1. 2.1.1 $2$-Calabi–Yau categories
      2. 2.1.2 Cluster-tilting objects and associated cluster characters
      3. 2.1.3 Constructible cones
    2. 2.2 The refined multiplication formula
  3. 3 Refined multiplication formula: Frobenius case
    1. 3.1 Recollections on Frobenius categories
      1. 3.1.1 2-Calabi–Yau Frobenius categories
      2. 3.1.2 Cluster-tilting objects
    2. 3.2 The formula
  4. 4 Applications
    1. 4.1 Specialization of cluster variables in cluster algebras
    2. 4.2 Reduction of friezes
    3. 4.3 A formula for Auslander–Reiten triangles
    4. 4.4 Another restricted formula

##  1\. Introduction

The additive categorification of cluster algebras has been an important tool
in their study almost from their inception (see for instance the survey papers
[Kel09, Rei10, Ami11, Pla18]). Such a categorification is given by a category
$\mathcal{C}$ (usually triangulated or exact) and a _cluster character_
sending objects of $\mathcal{C}$ to Laurent polynomials in several variables
so that suitable objects of $\mathcal{C}$ are sent to cluster variables in a
cluster algebra. The key property that a cluster character satisfies is a
_multiplication formula_ which recovers the exchange relations in a cluster
algebra. Such formulas at various levels of generality have been obtained in
[CC06, CK08, CK06, GLS08, Pal08, FK10, DWZ10, Pal12, Pla11b, XX10, Xu10, DG14,
Rup15, GLS18, CIEFR21] and more.

The main result of this paper is a multiplication formula generalizing most
previously known ones in the following context. Let $\mathcal{C}$ be a small
$\operatorname{{\rm Hom}}$-finite Krull–Schmidt $2$-Calabi–Yau triangulated
category over $\mathbb{C}$, together with a basic cluster tilting object $T$
(definitions are recalled in Section 2.1.1). Let

$CC_{T}:Obj(\mathcal{C})\to\mathbb{Z}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$

be the corresponding cluster character (Definition 2.6). For any objects $L$
and $M$ of $\mathcal{C}$, let

$\beta_{L,M}:\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M)\times\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma L)\longrightarrow k$

be the non-degenerate bifunctorial bilinear form conferring to $\mathcal{C}$
its $2$-Calabi–Yau structure. For an object $Y$, let $\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}$ be the set of those
morphisms $\varepsilon:L\to\Sigma M$ such that, if we have a triangle

$M\xrightarrow{}Y^{\prime}\xrightarrow{}L\xrightarrow{\varepsilon}\Sigma M,$

the objects $Y$ and $Y^{\prime}$ have the same index (see Definition 2.5) and
for each dimension vector $\mathbf{e}$, the submodule Grassmannians
$\mbox{{\rm Gr}}_{\mathbf{e}}(\operatorname{{\rm Hom}}_{\mathcal{C}}(T,Y))$
and $\mbox{{\rm Gr}}_{\mathbf{e}}(\operatorname{{\rm
Hom}}_{\mathcal{C}}(T,Y^{\prime}))$ have the same Euler characteristic. It is
easy to check that this set is invariant under multiplication by a non-zero
scalar. For a subset $X$ of $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M)$, let $X_{\langle Y\rangle}$ be the intersection of $X$ with
$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}$. Let
$\mathcal{Y}_{L,M}$ be a set of representatives of equivalence classes for the
equivalence relation defined by $\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}=\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y^{\prime}\rangle}$.

Our main result is the following refined multiplication formula.

######  Theorem (2.10).

Let $\mathcal{C}$ be a small $\operatorname{{\rm Hom}}$-finite Krull–Schmidt
$2$-Calabi–Yau triangulated category over $\mathbb{C}$ with constructible
cones (see Section 2.1.3) together with a basic cluster tilting object $T$.
Let $L$ and $M$ be objects of $\mathcal{C}$ such that $\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)$ is non-zero. Finally, let $V$ be a non-zero
vector subspace of $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$. Then

$\chi(\mathbb{P}V)CC_{T}(L)CC_{T}(M)=\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y)+\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y),$

where $\mathcal{R}=\mathbb{P}\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma
L)\setminus\mathbb{P}\mbox{\rm Ker\,}\beta_{L,M}(V,?)$.

If $V$ is the full space, then this formula specializes to the one proved in
[Pal12, Theorem 1.1]. Our main result also has a counterpart for exact
categories.

######  Theorem (3.6).

Let $\mathcal{E}$ be an $\operatorname{\rm Ext}$-finite 2-Calabi–Yau Frobenius
category with a cluster tilting object $T$. Assume that the triangulated
category $\underline{\mathcal{E}}$ has constructible cones. Let $L$ and $M$ be
two objects of $\mathcal{E}$ such that $\operatorname{\rm
Ext}^{1}_{\mathcal{E}}(L,M)$ is non-zero, and let $V$ be a non-zero vector
subspace of $\operatorname{\rm Ext}^{1}_{\mathcal{E}}(L,M)$. Then

$\chi(\mathbb{P}V)CC_{T}(L)CC_{T}(M)=\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y)+\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y).$

This generalizes a result of [FK10]. We expect that the refined multiplication
formula generalizes to the setting of suitable extriangulated categories such
as the Higgs category of [Wu21], in which the classical multiplication formula
can be proved [KW].

We apply our main results to the specialization of cluster variables to $1$.
Let $Q$ be finite quiver without loops or $2$-cycles and let $Q^{\prime}$ be
the quiver obtained from $Q$ by removing a vertex $i$. Let $\sigma$ be the
specialization of $x_{i}$ at $1$. In the case where $Q$ is mutation-equivalent
to an acyclic quiver, it was proved in [ADS14] that the image of the cluster
algebra $\mathcal{A}_{Q}$ by $\sigma$ is contained in
$\mathcal{A}_{Q^{\prime}}\otimes_{\mathbb{Z}}\mathbb{Q}$. Using our refined
multiplication formula, we can improve on this result.

######  Corollary (4.4).

Assume that $Q$ is mutation-equivalent to an acyclic quiver. Then the image of
the cluster algebra $\mathcal{A}_{Q}$ by $\sigma$ is
$\mathcal{A}_{Q^{\prime}}$.

More generally, we have the following results.

######  Corollary (4.3).

Assume that the quiver $Q$ admits a non-degenerate Jacobi-finite potential. If
the upper cluster algebra $\mathcal{U}_{Q^{\prime}}$ is equal to the cluster
algebra $\mathcal{A}_{Q^{\prime}}$, then
$\sigma(\mathcal{A}_{Q})=\mathcal{A}_{Q^{\prime}}$.

######  Corollary (4.5).

Assume that the quiver $Q$ admits a non-degenerate Jacobi-finite potential. If
the upper cluster algebra $\mathcal{U}_{Q^{\prime}}$ is spanned by the cluster
characters of objects of the associated generalized cluster category
$\mathcal{C}$, then $\sigma(\mathcal{U}_{Q})=\mathcal{U}_{Q^{\prime}}$.

Note that in the above results the variable that gets specialized to $1$ is
not frozen.

Our formula also finds applications in the reduction of friezes. A _frieze_ is
ring morphism $f:\mathcal{A}_{Q}\to\mathbb{Z}$ that sends all cluster
variables to positive integers. Friezes originated in work of Conway and
Coxeter [Cox71, CC73a], but have been vastly generalized using cluster
algebras, see for instance [BM09, ARS10, BFG+21, MG19] and the survey paper
[MG15]. Our result on friezes is the following.

######  Corollary (4.6).

Let $Q$ be an acyclic quiver without loops or $2$-cycles, let $Q^{\prime}$ be
the quiver obtained by removing the vertex $i$ in $Q$, and let
$\sigma:\mathcal{A}_{Q}\to\mathcal{A}_{Q^{\prime}}$ be the specialization of
$x_{i}$ to $1$. Let $f^{\prime}:\mathcal{A}_{Q^{\prime}}\to\mathbb{Z}$ be a
frieze. Then there exists a unique frieze $f:\mathcal{A}_{Q}\to\mathbb{Z}$
such that $f^{\prime}\circ\sigma=f$.

The non-trivial part of the above result is the existence.

Finally, in Section 4.3, we give a new proof of a multiplication formula for
Auslander–Reiten triangles first obtained in [DG14], and in Section 4.4, we
obtain a formula reminiscent of the one stated in [DX].

##  2\. Refined multiplication formula: triangulated case

###  2.1. Recollections on $2$-Calabi–Yau triangulated categories

The setting in which the multiplication formula holds is that of
$\operatorname{{\rm Hom}}$-finite, Krull-Schmidt, triangulated, $2$-Calabi–Yau
categories with a cluster tilting object and constructible cones. The aim of
this section is to recall the main definitions and properties of this setting.

####  2.1.1. $2$-Calabi–Yau categories

Let $\mathcal{C}$ be a small $\operatorname{{\rm Hom}}$-finite triangulated
category over a field $k$, with suspension functor $\Sigma$.

######  Definition 2.1.

The category $\mathcal{C}$ is _$2$ -Calabi–Yau_ if, for any objects $L$ and
$M$ of $\mathcal{C}$, it is equipped with a bilinear form

$\beta_{L,M}:\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M)\times\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma L)\longrightarrow k$

which is non-degenerate and bifunctorial. Here, bifunctorial means that if
$L$, $M$, $N$ and $P$ are objects of $\mathcal{C}$, and if
$\varepsilon\in\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma N)$,
$\eta\in\operatorname{{\rm Hom}}_{\mathcal{C}}(N,\Sigma L)$,
$\delta\in\operatorname{{\rm Hom}}_{\mathcal{C}}(P,\Sigma M)$,
$f\in\operatorname{{\rm Hom}}_{\mathcal{C}}(L,M)$ and $g\in\operatorname{{\rm
Hom}}_{\mathcal{C}}(N,P)$, then

$\displaystyle\beta_{L,N}(\varepsilon\circ f,\eta)$ $\displaystyle=$
$\displaystyle\beta_{M,N}(\varepsilon,\Sigma f\circ\eta)\quad\textrm{and}$
$\displaystyle\beta_{M,P}(\Sigma g\circ\varepsilon,\delta)$ $\displaystyle=$
$\displaystyle\beta_{M,N}(\varepsilon,\delta\circ g).$

Equivalently, $\mathcal{C}$ is $2$-Calabi–Yau if it is equipped with an
isomorphism of bifunctors

$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)\longrightarrow
D\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma L),$

where $D=\operatorname{{\rm Hom}}_{k}(?,k)$ is the usual duality for vector
spaces.

####  2.1.2. Cluster-tilting objects and associated cluster characters

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite $2$-Calabi–Yau
triangulated category.

######  Definition 2.2 ([BMR+06]).

An object $T$ of $\mathcal{C}$ is a _cluster-tilting object_ if the following
hold:

  1. (1)

$T$ is _rigid_ , that is, the space $\operatorname{{\rm
Hom}}_{\mathcal{C}}(T,\Sigma T)$ vanishes;

  2. (2)

for any object $X$ of $\mathcal{C}$, if $\operatorname{{\rm
Hom}}_{\mathcal{C}}(T,\Sigma X)$ vanishes, then $X$ lies in $\mbox{{\rm add
\\!}}T$ (that is, $X$ is a direct factor of a direct sum of copies of $T$).

We will usually assume that $T$ is _basic_ , and write
$T=T_{1}\oplus\ldots\oplus T_{n}$, where the $T_{i}$’s are pairwise non-
isomorphic indecomposable objects.

######  Examples 2.3.

  1. (1)

The cluster categories of [BMR+06] are triangulated $\operatorname{{\rm
Hom}}$-finite $2$-Calabi–Yau categories with a cluster-tilting object.

  2. (2)

The generalized cluster categories of [Ami09] also have these properties.

  3. (3)

The stable categories of all the Frobenius categories of Example 3.3 also have
these properties.

Cluster-tilting objects are essential in the categorification of cluster
algebras via triangulated categories. This is done via _cluster characters_ ,
whose definition we recall in Definition 2.6.

######  Proposition 2.4 ([KR07]).

Let $T$ be a basic cluster-tilting object of $\mathcal{C}$.

  1. (1)

The functor $F=\operatorname{{\rm Hom}}_{\mathcal{C}}(T,\Sigma?)$ induces an
equivalence of categories

$\mathcal{C}/(T)\xrightarrow{F}\mbox{{\rm mod \\!}}\operatorname{\rm
End}_{\mathcal{C}}(T),$

where $(T)$ is the ideal of all morphisms factoring through an object of
$\mbox{{\rm add \\!}}T$.

  2. (2)

Any object $X$ of $\mathcal{C}$ sits in a triangle

$T_{1}^{X}\rightarrow T_{0}^{X}\rightarrow X\rightarrow\Sigma T_{1}^{X},$

where $T_{1}^{X}$ and $T_{0}^{X}$ lie in $\mbox{{\rm add \\!}}T$.

######  Definition 2.5 ([DK08]).

Let $T$ be a basic cluster-tilting object of $\mathcal{C}$. The _index_ of an
object $X$ of $\mathcal{C}$ _with respect to $T$_ is the element of the
Grothendieck group $K_{0}(\mbox{{\rm add \\!}}T)$ defined by

$\operatorname{\rm ind}_{T}X=[T_{0}^{X}]-[T_{1}^{X}],$

where $T_{0}^{X}$ and $T_{1}^{X}$ are as in Proposition 2.4(2).

Note that, while the triangle in Proposition 2.4(2) is not unique, the index
of $X$ does not depend on the one we choose [Pal08, Lemma 2.1]. Moreover, it
was shown in [Pal08] (in the proof of Lemma 1.3) that for any object $X$ of
$\mathcal{C}$, the value of $\operatorname{\rm ind}_{T}\Sigma
X+\operatorname{\rm ind}_{T}X$ only depends on the dimension vector $e$ of
$FX$. We will denote this value by $\iota(e)$. Note that $\iota$ extends to a
linear map defined on all of $\mathbb{Z}^{n}$.

Let us now assume that the field $k$ is the field $\mathbb{C}$ of complex
numbers.

######  Definition 2.6 ([CC06][Pal08]).

Let $T$ be a basic cluster-tilting object of $\mathcal{C}$. The _cluster
character associated with $T$_ is the map

$CC_{T}:Obj(\mathcal{C})\longrightarrow\mathbb{Q}(x_{1},\ldots,x_{n})$

defined by

$CC_{T}(M)=x^{\operatorname{\rm
ind}_{T}M}\sum_{e\in\mathbb{N}^{n}}\chi\Big{(}\mbox{{\rm
Gr}}_{e}\big{(}FM\big{)}\Big{)}x^{-\iota(e)},$

where

  * •

$n$ is the number of indecomposable direct factors of $T$ in a decomposition
$T=\bigoplus_{i=1}^{n}T_{i}$;

  * •

$x^{a}=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$, for any
$a=\sum_{i=1}^{n}a_{i}[T_{i}]\in K_{0}(\mbox{{\rm add \\!}}T)$;

  * •

$\chi$ is the Euler characteristic for topological spaces;

  * •

$FM=\operatorname{{\rm Hom}}_{\mathcal{C}}(T,\Sigma M)$ is considered as a
right module over $\operatorname{\rm End}_{\mathcal{C}}(T)$;

  * •

for any module $R$, $\mbox{{\rm Gr}}_{e}(R)$ is the submodule Grassmannian
[CC06, Section 2.3], a projective variety whose points parametrize the
submodules of $R$ of dimension vector $e$;

  * •

$\iota(e)$ is as defined below Definition 2.5.

####  2.1.3. Constructible cones

The coefficients in the multiplication formula are Euler characteristics of
subsets of certain algebraic varieties. For the formula to be well-defined, we
must ensure that the Euler characteristics of these subsets are well-defined
integers. In [Pal12], this is done by proving that the subsets in question are
constructible. In order to do so, we need to assume that the category
$\mathcal{C}$ _has constructible cones_. Although we will not recall the
definition of a category with consctructible cones (and simply refer to
[Pal12, Section 1.3]), we will list the properties of such categories that we
will need.

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite triangulated category
with a basic cluster-tilting object $T$. Fix two objects $L$ and $M$. For any
object $Y$ of $\mathcal{C}$, let $\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}$ be the subset of
$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$ of all morphisms
$\varepsilon$ such that if

$M\rightarrow Y^{\prime}\rightarrow
L\stackrel{{\scriptstyle\varepsilon}}{{\rightarrow}}\Sigma M$

is a triangle, then

  * •

$\operatorname{\rm ind}_{T}Y^{\prime}=\operatorname{\rm ind}_{T}Y$, and

  * •

for all dimension vectors $e$, we have $\chi(\mbox{{\rm
Gr}}_{e}(FY))=\chi(\mbox{{\rm Gr}}_{e}(FY^{\prime}))$.

For any subset $V$ of $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$,
let $V_{\langle Y\rangle}$ be the intersection of $V$ with $\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}$.

Note that the condition

$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle
Y\rangle}=\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle
Y^{\prime}\rangle}$

induces an equivalence relation on the set of objects of $\mathcal{C}$. Let
$\mathcal{Y}_{L,M}$ be a set of representatives for this equivalence relation.

######  Proposition 2.7 (Proposition 2.8 of [Pal12]).

If $\mathcal{C}$ has constructible cones, then

$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M)=\coprod_{Y\in\mathcal{Y}_{L,M}}\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)_{\langle Y\rangle}$

is a partition of $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$ into a
finite number of constructible subsets.

######  Corollary 2.8.

If $\mathcal{C}$ has constructible cones, and if $V$ is a constructible subset
of $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$, then

$V=\coprod_{Y\in\mathcal{Y}_{L,M}}V_{\langle Y\rangle}$

is a decomposition of $V$ into a finite number of pairwise disjoint
constructible subsets.

######  Example 2.9.

All the triangulated categories mentioned in this paper have constructible
cones, thanks to these two facts proved in [Pal12, Sections 2.4-2.5]: stable
categories of $\operatorname{{\rm Hom}}$-finite Frobenius categories and the
generalized cluster categories of [Ami09] have constructible cones.

###  2.2. The refined multiplication formula

This section is devoted to the proof of the refined multiplication formula.
The proof follows the lines of [Pal12] and relies heavily on results obtained
there.

For any vector space $E$ and for any subset $U$ which is stable by scalar
multiplication, we denote by $\mathbb{P}U$ the subset of the projective space
$\mathbb{P}E$ consisting of elements $[u]$ with $u\in U$, where $[u]$ denotes
the class of $u$ in $\mathbb{P}E$.

######  Theorem 2.10.

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite Krull–Schmidt
$2$-Calabi–Yau triangulated category over $\mathbb{C}$ with constructible
cones and admitting a basic cluster-tilting object $T$. Let $L$ and $M$ be two
objects of $\mathcal{C}$, and let $V$ be a non-zero vector subspace of
$\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma M)$. Then the following
equality holds:

$\chi(\mathbb{P}V)CC_{T}(L)CC_{T}(M)=\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y)+\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y),$

where $\mathcal{R}=\mathbb{P}\operatorname{{\rm Hom}}_{\mathcal{C}}(M,\Sigma
L)\setminus\mathbb{P}\mbox{\rm Ker\,}\beta_{L,M}(V,?)$.

######  Remark 2.11.

If $V$ is the whole space $\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M)$, then the formula recovers that of Y. Palu [Pal12, Theorem 1.1].

We assume for the rest of this section that $\mathcal{C}$ has constructible
cones.

The first step into proving the formula is by replacing $CC_{T}(L)$ and
$CC_{T}(M)$ by their definitions in the left-hand side of the formula. Doing
this, we get rCl 3lχ(PV)CC_T(L)CC_T(M)
&= χ(PV)( x^ind_TL∑_eχ( Gr_e(FL) ) x^-ι(e) ) (x^ind_TM ∑_fχ( Gr_f(FM) )x^-ι(f)
)
= x^ind_T(L⊕M)∑_e,fχ(PV ×Gr_e(FL) ×Gr_f(FM) ) x^-ι(e+f). We will refine this
sum by replacing $\mathbb{P}V\times\mbox{{\rm Gr}}_{e}(FL)\times\mbox{{\rm
Gr}}_{f}(FM)$ by another constructible set with the same Euler characteristic.
Let us construct this set.

Define $W_{L,M}^{V}$ to be the subset of
$\mathbb{P}V\times\coprod_{d,g}\prod_{i=1}^{n}\mbox{{\rm
Gr}}_{g_{i}}(\mathbb{C}^{d_{i}})$ consisting of pairs $([\varepsilon],E)$
where $E$ is a subrepresentation of $FY$, where $Y$ is the middle term of a
triangle
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\Sigma
M}$.

Furthermore, define rCl W_L,M^V(e,f,g) &= { ([ε], E)∈W_L,M^V — dim E = g, dim
Fp(E) = e, dim F i^-1(E) = f } ;
W_L,M^V(e,f) = { ([ε], E)∈W_L,M^V — dim Fp(E) = e, dim F i^-1(E) = f } ;
W_L,M^V,Y(e,f,g) = {([ε], E)∈W_L,M^V — ε∈PV_⟨Y ⟩, dim E = g, dim Fp(E) = e,
dim F i^-1(E) = f };
W_L,M^V,Y(e,f) = {([ε], E)∈W_L,M^V — ε∈PV_⟨Y ⟩, dim Fp(E) = e, dim F i^-1(E) =
f }. Then $W_{L,M}^{V}$ and all the sets defined above are finite disjoint
unions of subsets of the form $W_{L,M}^{V,Y}(e,f,g)$. Moreover, since we
assumed that $\mathcal{C}$ has constructible cones, then the results of Y.
Palu give us the following.

######  Lemma 2.12 (Lemma 3.1 of [Pal12]).

The sets $W_{L,M}^{V,Y}(e,f,g)$, $W_{L,M}^{V,Y}(e,f)$, $W_{L,M}^{V}(e,f)$,
$W_{L,M}^{V}(e,f,g)$ and $W_{L,M}^{V}$ are constructible.

Proof. In [Pal12, Lemma 3.1], it is shown that certain sets
$W_{LM}^{Y}(e,f,g)$ are constructible. Our sets $W_{L,M}^{V,Y}(e,f,g)$ are the
intersection of these $W_{LM}^{Y}(e,f,g)$ with
$\mathbb{P}V\times\coprod_{d,g}\prod_{i=1}^{n}\mbox{{\rm
Gr}}_{g_{i}}(\mathbb{C}^{d_{i}})$; thus they are constructible. Since all the
other sets are finite unions of sets of the form $W_{L,M}^{V,Y}(e,f,g)$, they
must also be constructible. $\Box$

Now, consider the constructible map

$\displaystyle\Psi_{L,M}(e,f):W_{L,M}^{V}(e,f)$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{P}V\times\mbox{{\rm Gr}}_{e}(FL)\times\mbox{{\rm
Gr}}_{f}(FM)$ $\displaystyle([\varepsilon],E)$ $\displaystyle\longmapsto$
$\displaystyle([\varepsilon],Fp(E),Fi^{-1}(E)).$

Let $L^{V}_{1}(e,f)$ be the image of this map; let $L_{2}^{V}(e,f)$ be the
complement of the image. Then $\chi(\mathbb{P}V\times\mbox{{\rm
Gr}}_{e}(FL)\times\mbox{{\rm
Gr}}_{f}(FM))=\chi(L_{1}^{V}(e,f))+\chi(L_{2}^{V}(e,f))$.

Therefore our equation becomes

$\displaystyle(\star)\quad\chi(\mathbb{P}V)CC_{T}(L)CC_{T}(M)$
$\displaystyle=$ $\displaystyle x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}L_{1}^{V}(e,f)\big{)}x^{-\iota(e+f)}$
$\displaystyle\quad+x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}L_{2}^{V}(e,f)\big{)}x^{-\iota(e+f)}.$

We will now study the two terms of the right-hand side of ($\star$).

#### The first term of the RHS of ($\star$)

######  Lemma 2.13.

We have an equality

$x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}L_{1}^{V}(e,f)\big{)}x^{-\iota(e+f)}=\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y).$

Proof. It is proved in [CC06] (see also Section 3 of [Pal12]) that the fibers
of $\Psi_{L,M}(e,f)$ are affine spaces. As a consequence, we have that
$\chi(L^{V}_{1}(e,f))=\chi(W_{L,M}^{V}(e,f))$. Thus

$\displaystyle x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}L_{1}^{V}(e,f)\big{)}x^{-\iota(e+f)}$ $\displaystyle=$
$\displaystyle x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}W_{L,M}^{V}(e,f)\big{)}x^{-\iota(e+f)}$
$\displaystyle=$ $\displaystyle\sum_{e,f,g,\langle
Y\rangle}\chi\big{(}W_{L,M}^{V,Y}(e,f,g)\big{)}x^{-\iota(e+f)+\operatorname{\rm
ind}_{T}(L\oplus M)}.$

Now, by [Pal08, Lemma 5.1], if $([\varepsilon],E)$ lies in
$W_{L,M}^{V,Y}(e,f,g)$, then it implies that $\operatorname{\rm
ind}_{T}(L\oplus M)-\iota(e+f)=\operatorname{\rm ind}_{T}(Y)-\iota(g)$.

Moreover, for a fixed $g$, consider the map

$\coprod_{e,f}W_{L,M}^{V,Y}(e,f,g)\longrightarrow\mathbb{P}V_{\langle
Y\rangle}$

sending a pair $([\varepsilon],E)$ to $[\varepsilon]$. This map is obviously
surjective if the left-hand side is non-empty. Moreover, the preimage of any
$[\varepsilon^{\prime}]$ is isomorphic to
$\\{[\varepsilon^{\prime}]\\}\times\mbox{{\rm Gr}}_{g}(FY^{\prime})$, where
$Y^{\prime}$ sits in a triangle $M\rightarrow Y^{\prime}\rightarrow
L\xrightarrow{\varepsilon^{\prime}}\Sigma M$. By definition of
$\mathbb{P}V_{\langle Y\rangle}$, the Euler characteristic of all the fibers
is the same and is equal to $\chi(\mbox{{\rm Gr}}_{g}(FY))$. Thus

$\chi(\coprod_{e,f}W_{L,M}^{V,Y}(e,f,g))=\chi(\mbox{{\rm
Gr}}_{g}(FY))\chi(\mathbb{P}V_{\langle Y\rangle}).$

So the sum becomes

$\displaystyle...$ $\displaystyle=$
$\displaystyle\sum_{e,f,g,Y}\chi\big{(}W_{L,M}^{V,Y}(e,f,g)\big{)}x^{-\iota(g)+\operatorname{\rm
ind}_{T}(Y)}$ $\displaystyle=$
$\displaystyle\sum_{g,Y}\chi\big{(}\coprod_{e,f}W_{L,M}^{V,Y}(e,f,g)\big{)}x^{-\iota(g)+\operatorname{\rm
ind}_{T}(Y)}$ $\displaystyle=$ $\displaystyle\sum_{g,Y}\chi(\mbox{{\rm
Gr}}_{g}(FY))\chi(\mathbb{P}V_{\langle
Y\rangle})x^{-\iota(g)+\operatorname{\rm ind}_{T}(Y)}$ $\displaystyle=$
$\displaystyle\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y).$

This finishes the proof of the lemma. $\Box$

#### The second term of the RHS of ($\star$)

######  Lemma 2.14.

We have an equality

$x^{\operatorname{\rm ind}_{T}(L\oplus
M)}\sum_{e,f}\chi\big{(}L_{2}^{V}(e,f)\big{)}x^{-\iota(e+f)}=\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y).$

Proof. Recall that

$\mathcal{R}=\\{[\eta]\in\mathbb{P}\operatorname{{\rm
Hom}}_{\mathcal{C}}(M,\Sigma L)\ |\ \exists\varepsilon\in V\textrm{ with
}\beta_{L,M}(\varepsilon,\eta)\neq 0\\}.$

Define $W_{M,L}^{\mathbb{P}\operatorname{{\rm Hom}}_{\mathcal{C}}(L,\Sigma
M),Y}(f,e,g)$ as before Lemma 2.12 and let $W_{M,L}^{\mathcal{R},Y}(f,e,g)$ be
the constructible subset of all pairs $([\eta],E)$ with $\eta\in\mathcal{R}$.
For fixed $e$, $f$ and $g$, let $C_{L,M}^{\mathcal{R},Y}(e,f,g)$ be the subset
of $L_{2}^{V}(e,f)\times W_{M,L}^{\mathcal{R},Y}(f,e,g)$ consisting of pairs
$\big{(}([\varepsilon],R,S),([\eta],E)\big{)}$ such that
$\beta_{L,M}(\varepsilon,\eta)\neq 0$, $Fi^{-1}(E)=S$ and $Fp(E)=R$. Finally,
let
$C_{L,M}^{\mathcal{R}}(e,f)=\coprod_{g,Y\in\mathcal{Y}_{M,L}}C_{L,M}^{\mathcal{R},Y}(e,f,g)$.

Consider the two projections

$\displaystyle C_{L,M}^{\mathcal{R}}(e,f)$ $\displaystyle\xrightarrow{p_{1}}$
$\displaystyle L_{2}^{V}(e,f)$ $\displaystyle C_{L,M}^{\mathcal{R},Y}(e,f,g)$
$\displaystyle\xrightarrow{p_{2}}$ $\displaystyle
W_{M,L}^{\mathcal{R},Y}(f,e,g).$

By [Pal12, Proposition 3.3], $p_{1}$ and $p_{2}$ are surjective. Moreover, by
[Pal12, Proposition 3.4], the fibers of $p_{1}$ are extensions of affine
spaces, and those of $p_{2}$ are affine spaces.

Therefore $\chi(C_{L,M}^{\mathcal{R}}(e,f))=\chi(L_{2}^{V}(e,f))$ and
$\chi(C_{L,M}^{\mathcal{R},Y}(e,f,g))=\chi(W_{M,L}^{\mathcal{R},Y}(f,e,g))$.
Thus the left-hand side in the statement is equal to

$\displaystyle...$ $\displaystyle=$ $\displaystyle x^{\operatorname{\rm
ind}_{T}(L\oplus M)}\sum_{e,f}\chi(L_{2}^{V}(e,f))x^{-\iota(e+f)}$
$\displaystyle=$
$\displaystyle\sum_{e,f}\chi(C^{\mathcal{R}}_{L,M}(e,f))x^{\operatorname{\rm
ind}_{T}(L\oplus M)-\iota(e+f)}$ $\displaystyle=$
$\displaystyle\sum_{e,f,g,Y}\chi(C^{\mathcal{R},Y}_{L,M}(e,f,g))x^{\operatorname{\rm
ind}_{T}(L\oplus M)-\iota(e+f)}$ $\displaystyle=$
$\displaystyle\sum_{e,f,g,Y}\chi(W^{\mathcal{R},Y}_{M,L}(f,e,g))x^{\operatorname{\rm
ind}_{T}(L\oplus M)-\iota(e+f)}.$

Again, by [Pal08, Lemma 5.1], we have that if $([\varepsilon],E)$ lies in
$W^{\mathcal{R},Y}_{M,L}(f,e,g)$, then $\operatorname{\rm ind}_{T}(L\oplus
M)-\iota(e+f)=\operatorname{\rm ind}_{T}(Y)-\iota(g)$. Moreover, the map

$\coprod_{e,f}W^{\mathcal{R},Y}_{M,L}(f,e,g)\longrightarrow\mathcal{R}_{\langle
Y\rangle}$

sending $([\varepsilon],E)$ to $[\varepsilon]$ is surjective (if the left-hand
side is non-empty), and its fibers have the form
$\\{[\varepsilon^{\prime}]\\}\times\mbox{{\rm Gr}}_{g}(FY^{\prime})$, where
$Y^{\prime}$ sits in a triangle $L\rightarrow Y^{\prime}\rightarrow
M\xrightarrow{\varepsilon^{\prime}}\Sigma L$. Thus

$\chi(\coprod_{e,f}W^{\mathcal{R},Y}_{M,L}(f,e,g))=\chi(\mathcal{R}_{\langle
Y\rangle})\chi(\mbox{{\rm Gr}}_{g}(FY)).$

Therefore the above sequence of equalities continues:

$\displaystyle\ldots$ $\displaystyle=$
$\displaystyle\sum_{e,f,g,Y}\chi(W^{\mathcal{R},Y}_{M,L}(f,e,g))x^{\operatorname{\rm
ind}_{T}Y-\iota(g)}$ $\displaystyle=$
$\displaystyle\sum_{g,Y}\chi(\coprod_{e,f}W^{\mathcal{R},Y}_{M,L}(f,e,g))x^{\operatorname{\rm
ind}_{T}Y-\iota(g)}$ $\displaystyle=$
$\displaystyle\sum_{g,Y}\chi(\mathcal{R}_{\langle
Y\rangle})\chi(Gr{g}(FY))x^{\operatorname{\rm ind}_{T}Y-\iota(g)}$
$\displaystyle=$
$\displaystyle\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y).$

This finishes the proof.

$\Box$

Theorem 2.10 then follows directly from Lemma 2.13 and Lemma 2.14.

##  3\. Refined multiplication formula: Frobenius case

We will now follow the ideas of [FK10] (see also [Pal12, Section 4]). The main
difference will be that our Frobenius categories can be $\operatorname{{\rm
Hom}}$-infinite; we will only assume that they are $\operatorname{\rm
Ext}$-finite. We will also assume that they are Krull–Schmidt, and that their
stable categories have constructible cones. Since the proofs are very similar
to the ones in the triangulated case, in this section we only provide a
detailed outline for the Frobenius case.

###  3.1. Recollections on Frobenius categories

####  3.1.1. 2-Calabi–Yau Frobenius categories

A _Frobenius category_ is an exact category $\mathcal{E}$ in the sense of
Quillen with enough projectives and enough injectives, in which projectives
and injectives coincide. It is _$\operatorname{\rm Ext}$ -finite_ if for any
objects $X$ and $Y$ of $\mathcal{E}$, the space $\operatorname{\rm
Ext}^{1}_{\mathcal{E}}(X,Y)$ is finite-dimensional.

It was proved in [Hap87, Theorem 9.4] that if $\mathcal{E}$ is a Frobenius
category, then its stable category $\underline{\mathcal{E}}$ is triangulated
($\underline{\mathcal{E}}$ is the quotient of $\mathcal{E}$ by the ideal of
all morphisms factoring through a projective-injective object). Note that if
$\mathcal{E}$ is $\operatorname{\rm Ext}$-finite, then
$\underline{\mathcal{E}}$ is $\operatorname{{\rm Hom}}$-finite.

######  Definition 3.1 (Section 2.7 of [FK10]).

An $\operatorname{\rm Ext}$-finite Frobenius category is _2-Calabi–Yau_ if its
stable category is 2-Calabi–Yau as a triangulated category.

####  3.1.2. Cluster-tilting objects

Let $\mathcal{E}$ be an $\operatorname{\rm Ext}$-finite Krull–Schmidt
2-Calabi–Yau Frobenius category.

######  Definition 3.2 (Section 2.7 of [FK10]).

An object $T$ of $\mathcal{E}$ is a _cluster-tilting object_ if

  1. (1)

$T$ is rigid, that is, the space $\operatorname{\rm
Ext}^{1}_{\mathcal{E}}(T,T)$ vanishes;

  2. (2)

for any object $X$ of $\mathcal{E}$, if $\operatorname{\rm
Ext}^{1}_{\mathcal{E}}(T,X)$ vanishes, then $X\in\mbox{{\rm add \\!}}T$; and

  3. (3)

each object $X$ of $\mathcal{E}$ admits a right $\mbox{{\rm add
\\!}}T$-approximation $T^{X}\to X$ and a left $\mbox{{\rm add
\\!}}T$-approximation $X\to T_{X}$ (in other words, the functors
$\operatorname{{\rm Hom}}_{\mathcal{E}}(X,?)|_{\mbox{{\rm add \\!}}T}$ and
$\operatorname{{\rm Hom}}_{\mathcal{E}}(?,X)|_{(\mbox{{\rm add \\!}}T)^{op}}$
are finitely generated).

Note that if $T$ is a cluster-tilting object, then every indecomposable
projective-injective object is isomorphic to a direct summand of $T$.

######  Examples 3.3.

  1. (1)

The module category of a preprojective algebra of Dynkin type is a
$\operatorname{{\rm Hom}}$-finite stably $2$-Calabi–Yau Frobenius category
with a cluster tilting object. It was used in [GLS06] in the categorification
of cluster algebras. Its stable category has constructible cones by [Pal12,
Section 2.4].

  2. (2)

More generally, subcategories $\mathcal{C}_{w}$ of modules over preprojective
algebras were were used in [BIRS09, GLS12a] to category cluster algebras.
These categories have the same properties as those of the previous example.

  3. (3)

Let $0<k<n$ be integers, and put $\hat{R}=\mathbb{C}[[x,y]]/(x^{k}-y^{n-k})$.
The group $G=\langle\zeta\rangle$ of $n$-th roots of unity acts on $\hat{R}$
by $\zeta.x=\zeta x$ and $\zeta.y=\zeta^{-1}y$. Then the category
$\mathcal{E}=CM_{G}(\hat{R})$ of $G$-equivariant Cohen–Macaulay
$\hat{R}$-modules is a (not necessarily $\operatorname{{\rm Hom}}$-finite)
$\operatorname{\rm Ext}$-finite Frobenius category with a cluster tilting
object. Its stable category has constructible cones, since it is equivalent to
categories from the previous example. This was used in [JKS16] to give a
categorification of the cluster algebra structure of the homogeneous
coordinate ring $\mathbb{C}[G_{k,n}]$ of the Grassmannian $G_{k,n}$.

Let $T$ be a basic cluster-tilting object of $\mathcal{E}$. Write
$T=T_{1}\oplus\ldots\oplus T_{n}$, where each $T_{i}$ is indecomposable, and
let $C=\operatorname{\rm End}_{\mathcal{E}}(T)$. Since $T$ is cluster-tilting,
we have a functor

$F=\operatorname{{\rm
Hom}}_{\mathcal{E}}(T,?):\mathcal{E}\longrightarrow\mbox{{\rm f.g.mod \\!}}C,$

where $\mbox{{\rm f.g.mod \\!}}C$ is the category of finitely generated right
$C$-modules. For any finitely generated $C$-modules $L$ and $N$ such that $N$
is finite-dimensional, define

$\langle L,N\rangle_{\tau}=\dim\operatorname{{\rm
Hom}}_{C}(L,N)-\dim\operatorname{\rm Ext}^{1}_{C}(L,N),$ $\langle
L,N\rangle_{3}=\sum_{i=0}^{3}(-1)^{i}\dim\operatorname{\rm Ext}^{i}_{C}(L,N).$

Note that these expressions are well-defined integers, since
$\operatorname{\rm Ext}^{i}_{C}(L,N)$ is finite-dimensional because
$\mathcal{E}$ is $\operatorname{\rm Ext}$-finite, and since
$\operatorname{{\rm Hom}}_{C}(L,N)$ is finite-dimensional because $L$ is
finitely generated and $N$ is finite-dimensional.

Finally, let $\underline{C}=\operatorname{\rm
End}_{\underline{\mathcal{E}}}(T)$. As in [KR07, Section 4] and [FK10, Section
3], we view $\underline{C}$-modules as $C$-modules with no composition factors
isomorphic to the simple modules corresponding to the projective-injective
direct summands of $T$.

######  Proposition 3.4 (Proposition 3.2 of [FK10]).

If $L$ and $N$ are finite-dimensional $\underline{C}$-modules of the same
dimension vector, then for any finite-dimensional $C$-module $Y$, we have that

$\langle L,Y\rangle_{3}=\langle N,Y\rangle_{3}.$

In view of this proposition, if $e$ is a dimension vector, we can write
$\langle e,Y\rangle_{3}$ for the value of $\langle L,Y\rangle_{3}$ for any
$\underline{C}$-module $L$ of dimension vector $e$.

######  Definition 3.5 ([FK10]).

The _cluster character associated with $T$_ is the map

$CC_{T}:Obj(\mathcal{E})\longrightarrow\mathbb{Q}(x_{1},\ldots,x_{n})$

defined by

$CC_{T}(M)=\prod_{i=1}^{n}x_{i}^{\langle
FM,S_{i}\rangle_{\tau}}\sum_{e}\chi\big{(}\mbox{{\rm
Gr}}_{e}(\operatorname{\rm
Ext}^{1}_{\mathcal{E}}(T,M))\big{)}\prod_{i=1}^{n}x_{i}^{\langle
e,S_{i}\rangle_{3}}.$

###  3.2. The formula

######  Theorem 3.6.

Let $\mathcal{E}$ be a $\operatorname{{\rm Hom}}$-finite 2-Calabi–Yau
Frobenius category with a cluster tilting object $T$. Assume that the
triangulated category $\underline{\mathcal{E}}$ has constructible cones. Let
$L$ and $M$ be two objects of $\mathcal{E}$, and let $V$ be a vector subspace
of $\operatorname{\rm Ext}^{1}_{\mathcal{E}}(L,M)$. Then

$\chi(\mathbb{P}V)CC_{T}(L)CC_{T}(M)=\sum_{Y\in\mathcal{Y}_{L,M}}\chi(\mathbb{P}V_{\langle
Y\rangle})CC_{T}(Y)+\sum_{Y\in\mathcal{Y}_{M,L}}\chi(\mathcal{R}_{\langle
Y\rangle})CC_{T}(Y).$

The proof follows the lines of that of [Pal12, Theorem 4.1]; it is similar to
that of Theorem 2.10, but uses [FK10, Lemma 3.4] instead of [Pal08, Lemma
5.1].

##  4\. Applications

###  4.1. Specialization of cluster variables in cluster algebras

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite 2-Calabi–Yau
triangulated category with a basic cluster tilting object
$T=\bigoplus_{i=1}^{n}T_{i}$. Following [CILFS15], let the _Caldero-Chapoton
algebra_ $\mathcal{A}_{\mathcal{C}}$ be the subring of the ring
$\mathbb{Z}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ generated by the set of all
$CC_{T}(X)$, as $X$ spans all objects of $\mathcal{C}$.

Motivated by the reduction of friezes (see Section 4.2) and the study of
morphisms of rooted cluster algebras of [ADS14], we wish to study the algebra
obtained from $\mathcal{A}_{\mathcal{C}}$ by specializing $x_{n}$ to $1$. To
fix notation, let

$\sigma:\mathbb{Z}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]\to\mathbb{Z}[x_{1}^{\pm
1},\ldots,x_{n-1}^{\pm 1}]$

be the morphism sending each of $x_{1},\ldots,x_{n-1}$ to itself and sending
$x_{n}$ to $1$.

The main result of this section is stated in terms of Calabi–Yau reduction: it
was proved in [IY08] that the category

$\mathcal{C}^{\prime}=\left(\Sigma^{-1}T_{n}\right)^{\perp}/\left(T_{n}\right)$

is a $\operatorname{{\rm Hom}}$-finite 2-Calabi–Yau triangulated category with
a basic cluster tilting object $T^{\prime}$, where $T^{\prime}$ is the image
of $T$ under the projection functor.

######  Theorem 4.1.

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite 2-Calabi–Yau
triangulated category with constructible cones and a basic cluster tilting
object $T=\bigoplus_{i=1}^{n}T_{i}$. Then
$\sigma(\mathcal{A}_{\mathcal{C}})=\mathcal{A}_{\mathcal{C}^{\prime}}$.

######  Remark 4.2.

The proof of [ADS14, Theorem 6.13] shows that
$\sigma(\mathcal{A}_{\mathcal{C}})\subseteq\mathcal{A}_{\mathcal{C}^{\prime}}\otimes_{\mathbb{Z}}\mathbb{Q}$.
It uses the multiplication formula of Palu [Pal12]. Our proof of Theorem 4.1
follows the same lines using our refined multiplication formula (Theorem 2.10)
instead. We nonetheless include the complete argument below.

###### Proof.

(of Theorem 4.1). It suffices to prove that
$\sigma\left(CC_{T}(X)\right)\in\mathcal{A}_{\mathcal{C}^{\prime}}$ for all
objects $X$ of $\mathcal{C}$. The proof is by induction on the dimension of
the space $\operatorname{{\rm Hom}}_{\mathcal{C}}(T_{n},\Sigma X)$. Assume
first that $\dim\operatorname{{\rm Hom}}_{\mathcal{C}}(T_{n},\Sigma X)=0$.
Then $X\in(\Sigma^{-1}T_{n})^{\perp}$. Denote by $\pi$ the projection

$\pi:\left(\Sigma^{-1}T_{n}\right)^{\perp}\to\mathcal{C}^{\prime}.$

Then $\sigma\left(CC_{T}(X)\right)=CC_{T^{\prime}}(\pi
X)\in\mathcal{A}_{\mathcal{C}^{\prime}}$. Note that this implies that
$\mathcal{A}_{\mathcal{C}^{\prime}}\subseteq\sigma(\mathcal{A}_{\mathcal{C}})$,
since all objects of $\mathcal{C}^{\prime}$ have the form $\pi(X)$ with $X$ an
object of $\mathcal{C}$.

Assume now that $\dim\operatorname{{\rm Hom}}_{\mathcal{C}}(T_{n},\Sigma
X)=d>0$. Choose any non-split triangle

$X\xrightarrow{}E\xrightarrow{}T_{n}\xrightarrow{\xi}\Sigma X$

and let $V$ be the span on $\xi$ in $\operatorname{{\rm
Hom}}_{\mathcal{C}}(T_{n},\Sigma X)$. Applying Theorem 2.10, we get that

$CC_{T}(X)CC_{T}(T_{n})=CC_{T}(E)+\sum_{Y\in\mathcal{Y}_{X,T_{n}}}\chi\big{(}\mathcal{R}_{\langle
Y\rangle}\big{)}CC_{T}(Y).$

Since $CC_{T}(T_{n})=x_{n}$, applying the specialization $\sigma$ to the left-
hand side yields $\sigma\left(CC_{T}(X)\right)$. Since all
$\chi\big{(}\mathcal{R}_{\langle Y\rangle}\big{)}$ are integers, it thus
suffices to prove that $CC_{T}(E)$ and all $CC_{T}(Y)$ on the right-hand side
are in $\mathcal{A}_{\mathcal{C}^{\prime}}$. We do this by showing that the
dimensions of $\operatorname{{\rm Hom}}_{\mathcal{C}}(T_{n},\Sigma E)$ and
$\operatorname{{\rm Hom}}_{\mathcal{C}}(T_{n},\Sigma Y)$ are strictly smaller
than $d$ and by applying induction.

To see this, first apply the functor $\operatorname{{\rm
Hom}}_{\mathcal{C}}(T_{n},?)$ to the triangle defined by $\xi$. We obtain an
exact sequence

$(T_{n},T_{n})\xrightarrow{\xi_{*}}(T_{n},\Sigma
X)\xrightarrow{f}(T_{n},\Sigma E)\xrightarrow{}(T_{n},\Sigma T_{n}),$

where we write $(U,V)$ instead of $\operatorname{{\rm
Hom}}_{\mathcal{C}}(U,V)$ to save space. Since $T_{n}$ is rigid,
$(T_{n},\Sigma T_{n})$ vanishes, so $f$ is surjective; thus,

$(T_{n},\Sigma E)\cong(T_{n},\Sigma X)/\xi_{*}\big{(}(T_{n},T_{n})\big{)}.$

Lastly, $\xi_{*}\big{(}(T_{n},T_{n})\big{)}$ is non-zero, since it contains
$\xi_{*}(id_{T_{n}})=\xi$. Therefore,

$\dim(T_{n},\Sigma E)<\dim(T_{n},\Sigma X)=d,$

and by induction,
$\sigma\left(CC_{T}(E)\right)\in\mathcal{A}_{\mathcal{C}^{\prime}}$.

Now let $Y\in\mathcal{Y}_{T_{n},X}$. By definition, there exists a non-split
triangle

$T_{n}\xrightarrow{}Y\xrightarrow{}X\xrightarrow{\delta}\Sigma T_{n}.$

Applying the functor $\operatorname{{\rm Hom}}_{\mathcal{C}}(?,T)$ and
repeating the above argument, we get that

$\dim(Y,\Sigma T_{n})<\dim(X,\Sigma T_{n})=d.$

By the $2$-Calabi–Yau property, $\dim(Y,\Sigma T_{n})=\dim(T_{n},\Sigma Y)$.
Thus, by induction, we also have that
$\sigma\left(CC_{T}(Y)\right)\in\mathcal{A}_{\mathcal{C}^{\prime}}$. This
finishes the proof. ∎

Theorem 4.1 has an interesting application to cluster algebras.

######  Corollary 4.3.

Let $Q$ be a quiver without loops or $2$-cycles, $i$ be a vertex of $Q$ and
$Q^{\prime}$ be the quiver obtained from $Q$ by removing the vertex $i$.
Assume that there exists a non-degenerate potential $W$ such that the
generalized cluster category $\mathcal{C}_{Q,W}$ is $\operatorname{{\rm
Hom}}$-finite.

If the cluster algebra $\mathcal{A}_{Q^{\prime}}$ is equal to its upper
cluster algebra $\mathcal{U}_{Q^{\prime}}$ (see [BFZ05] for details), then the
specialization $\sigma$ sending $x_{i}$ to $1$ satisfies
$\sigma(\mathcal{A}_{Q})=\mathcal{A}_{Q^{\prime}}$.

###### Proof.

Let $\mathcal{C}=\mathcal{C}_{Q,W}$ and
$\mathcal{C}^{\prime}=\mathcal{C}_{Q^{\prime},W^{\prime}}$, where $W^{\prime}$
is the potential obtained by removing the terms of $W$ involving the vertex
$i$. We know from [Pal08] that
$\mathcal{A}_{Q}\subset\mathcal{A}_{\mathcal{C}}$, and from [Pla11a, Corollary
4.14] that $\mathcal{A}_{\mathcal{C}}\subset\mathcal{U}_{Q}$. The same is true
if we replace $Q$ with $Q^{\prime}$; thus, by our assumption,
$\mathcal{A}_{Q^{\prime}}=\mathcal{A}_{\mathcal{C}^{\prime}}=\mathcal{U}_{Q^{\prime}}$.
Applying $\sigma$, we get that
$\mathcal{A}_{Q^{\prime}}\subset\sigma(\mathcal{A}_{Q})\subset\sigma(\mathcal{A}_{\mathcal{C}})$,
and this last set is $\mathcal{A}_{\mathcal{C}^{\prime}}$ by Theorem 4.1. This
finishes the proof. ∎

######  Corollary 4.4.

If $Q$ is mutation-equivalent to an acyclic quiver, then we have that
$\sigma(\mathcal{A}_{Q})=\mathcal{A}_{Q^{\prime}}$.

######  Corollary 4.5.

Assume that the quiver $Q$ admits a non-degenerate Jacobi-finite potential,
and let $\mathcal{C}$ be its generalized cluster category. If the upper
cluster algebra $\mathcal{U}_{Q^{\prime}}$ is spanned the cluster characters
of some objects in $\mathcal{C}$, then we have that
$\sigma(\mathcal{U}_{Q})=\mathcal{U}_{Q^{\prime}}$.

###### Proof.

Notice that we have
$\mathcal{A}_{\mathcal{C}^{\prime}}\subset\mathcal{U}_{Q^{\prime}}$ and
$\mathcal{A}_{\mathcal{C}}\subset\mathcal{U}_{Q}$ by the universal Laurent
property of cluster characters. Since $\mathcal{U}_{Q^{\prime}}$ is spanned by
some cluster characters, we have
$\mathcal{U}_{Q^{\prime}}\subset\mathcal{A}_{\mathcal{C}^{\prime}}$.
Consequently, Theorem 4.1 implies
$\mathcal{U}_{Q^{\prime}}=\mathcal{A}_{\mathcal{C}^{\prime}}=\sigma(\mathcal{A}_{\mathcal{C}})\subset\sigma(\mathcal{U}_{Q})$.

Notice that every cluster for $Q^{\prime}$ (see [BFZ05]) is the image of some
cluster for $Q$ under $\sigma$. So we have
$\sigma(\mathcal{U}_{Q})\subset\mathcal{U}_{Q^{\prime}}$. The desired claim
follows. ∎

Many upper cluster algebras are known to possess a generic basis, see [GLS12b,
Pla13, Qin19]. They satisfy the assumption in Corollary 4.5.

###  4.2. Reduction of friezes

Let $Q$ be a quiver without loops or $2$-cycles. A _frieze_ is a morphism of
rings $f:\mathcal{A}_{Q}\to\mathbb{Z}$ sending every cluster variable of
$\mathcal{A}_{Q}$ to a positive integer. This definition generalizes the
originial one of Conway and Coxeter [CC73b], and has been an area of active
interest in recent years.

In [BFG+21, Section 5], an operation of reduction on friezes is considered.
The purpose of this section is to show that this reduction operation can be
“reversed” by adding a $1$ in a frieze.

######  Corollary 4.6.

Let $Q$ be an acyclic quiver without loops or $2$-cycles, and let $Q^{\prime}$
be the quiver obtained by removing the vertex $i$ in $Q$, and let
$\sigma:\mathcal{A}_{Q}\to\mathcal{A}_{Q^{\prime}}$ be the specialization of
$x_{i}$ to $1$ (this is well-defined thanks to Corollary 4.4). Let
$f^{\prime}:\mathcal{A}_{Q^{\prime}}\to\mathbb{Z}$ be a frieze. Then there
exists a unique frieze $f:\mathcal{A}_{Q}\to\mathbb{Z}$ such that
$f^{\prime}\circ\sigma=f$.

###### Proof.

If $f$ exists, then it is unique, since it is determined by its action on the
initial cluster variables of $\mathcal{A}_{Q}$. Let us prove that such an $f$
exists for any frieze $f^{\prime}$. By Corollary 4.4,
$f=f^{\prime}\circ\sigma$ is a well-defined morphism of rings from
$\mathcal{A}_{Q}$ to $\mathbb{Z}$. We only need to check that all cluster
variables of $\mathcal{A}_{Q}$ are sent to positive values by $f$; this
follows from the positivity theorem [LS15]: any cluster variable of
$\mathcal{A}_{Q}$ is a Laurent polynomial with nonnegative coefficients in the
initial cluster variables, and these are sent to positive values by $f$. ∎

######  Remark 4.7.

Corollary 4.6 can be deduced for friezes where $Q$ is of type $A_{n}$ from the
results of [CC73b], and was shown to be true in types $A_{n},D_{n}$ and
$E_{6}$ in [BFG+21, Section 5], where it was also observed to be true for all
known friezes of types $E_{7}$ and $E_{8}$ by a direct check. The total number
of possible friezes in these types is still unknown and was conjectured in
[FP16].

###  4.3. A formula for Auslander–Reiten triangles

In this section, we will show how Theorem 2.10 allows for a new proof of the
following formula of S. Dominguez and C. Geiss when $\mathcal{C}$ has
constructible cones.

######  Theorem 4.8 (Theorem 1 of [DG14]).

Let $\mathcal{C}$ be a $\operatorname{{\rm Hom}}$-finite 2-Calabi–Yau category
with constructible cones and a cluster tilting object $T$. Let $Z$ be an
indecomposable object of $\mathcal{C}$, and assume that it sits in an
Auslander–Reiten triangle

$\Sigma
Z\xrightarrow{\alpha}Y\xrightarrow{\beta}Z\xrightarrow{\varepsilon}\Sigma^{2}Z.$

Then

$CC_{T}(Z)CC_{T}(\Sigma Z)=CC_{T}(Y)+1.$

We will give a proof of this theorem under the additionnal assumption that
_$\mathcal{C}$ has constructible cones_.

Let $V$ be the one-dimensional subspace of $\operatorname{{\rm
Hom}}_{\mathcal{C}}(Z,\Sigma^{2}Z)$ generated by $\varepsilon$. Applying
Theorem 2.10, we get

$CC_{T}(Z)CC_{T}(\Sigma Z)=CC_{T}(Y)+\sum_{E\in\mathcal{Y}_{\Sigma
Z,Z}}\chi(\mathcal{R}_{\langle E\rangle})CC_{T}(E).$

Here $\mathcal{R}=\\{[\eta]\in\mathbb{P}\operatorname{{\rm
Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)\ |\ \beta_{Z,\Sigma
Z}(\varepsilon,\eta)\neq 0\\}$. Let us show that $[\eta]$ lies in
$\mathcal{R}$ if and only if $\eta$ is an isomorphism.

Since $Z$ is indecomposable, $\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma
Z,\Sigma Z)$ is a local ring. Hence $\eta$ is an isomorphism if and only if it
does not lie in the radical of $\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma
Z,\Sigma Z)$. Thus we need to show that $\eta$ lies in the radical of
$\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$ if and only if
$\beta_{Z,\Sigma Z}(\varepsilon,\eta)=0$.

Assume that $\eta$ is in the radical (assume $\eta\neq 0$; the case $\eta=0$
is trivial). Then it is not an isomorphism, and since $Z$ is indecomposable,
it is not a retraction. Then, by definition of an Auslander–Reiten triangle,
we must have that there exists a morphism $f:Z\rightarrow Y$ such that
$\Sigma^{-1}\eta=\beta f$. But then

$\displaystyle\beta_{Z,\Sigma Z}(\varepsilon,\eta)$ $\displaystyle=$
$\displaystyle\beta_{Z,\Sigma Z}(\varepsilon,\Sigma\beta\Sigma f)$
$\displaystyle=$ $\displaystyle\beta_{Y,\Sigma Z}(\varepsilon\beta,\Sigma f)$
$\displaystyle=$ $\displaystyle\beta_{Y,\Sigma Z}(0,\Sigma f)$
$\displaystyle=$ $\displaystyle 0.$

Assume next that $\eta$ is an isomorphism. Then $\eta$ and $\mbox{ \,\rm
rad\,}\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$ generate
$\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$ as a vector space.
If $\beta_{Z,\Sigma Z}(\varepsilon,\eta)$ were to vanish, it would thus vanish
for any $\eta^{\prime}$ in $\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma
Z,\Sigma Z)$, contradicting the fact that $\beta_{Z,\Sigma Z}$ is non-
degenerate. Thus $\beta_{Z,\Sigma Z}(\varepsilon,\eta)\neq 0$.

This proves that $\mathcal{R}$ is the set of $[\eta]$, with $\eta$ an
isomorphism. But then $\mathcal{R}=\mathcal{R}_{\langle 0\rangle}$ (since the
middle term of a triangle associated with an isomorphism is 0). Moreover,
$\mathcal{R}=\mathbb{P}\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma
Z)\setminus\mathbb{P}\\!\mbox{ \,\rm rad\,}\operatorname{{\rm
Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$ is an affine space, since $\mbox{
\,\rm rad\,}\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$ is a
hyperplane in $\operatorname{{\rm Hom}}_{\mathcal{C}}(\Sigma Z,\Sigma Z)$.
Thus $\chi(\mathcal{R})=1$.

Therefore

$\displaystyle CC_{T}(Z)CC_{T}(\Sigma Z)$ $\displaystyle=$ $\displaystyle
CC_{T}(Y)+\sum_{E\in\mathcal{Y}_{Z,\Sigma Z}}\chi(\mathcal{R}_{\langle
E\rangle})CC_{T}(E)$ $\displaystyle=$ $\displaystyle
CC_{T}(Y)+\chi(\mathcal{R}_{\langle 0\rangle})CC_{T}(0)$ $\displaystyle=$
$\displaystyle CC_{T}(Y)+1.$

This finishes the proof.

###  4.4. Another restricted formula

Theorem 2.10 allows us to obtain (always assuming contructibility of cones)
the following formula, reminiscent of the one stated in [DX]. For two objects
$L$ and $M$ of $\mathcal{C}$, let $(T)(L,M)$ be the space of morphisms from
$L$ to $M$ factoring through an object of $\mbox{{\rm add \\!}}T$.

######  Proposition 4.9.

Under the hypotheses of Theorem 2.10, we have that rCl
χ(P(T)(L,ΣM))CC_T(L)CC_T(M) &= ∑_Y∈Y_L,Mχ(P(T)(L,ΣM)_⟨Y ⟩) CC_T(Y)
\+ ∑_Y∈Y_M,Lχ(PHom_C(M,ΣL)_⟨Y⟩ ∖P(T)(M,ΣL)_⟨Y⟩ ) CC_T(Y).

Proof. This follows from Theorem 2.10 by taking $V=(T)(L,\Sigma M)$. To see
this, we only need to prove that $\mbox{\rm
Ker\,}\beta_{L,M}(V,?)=(T)(M,\Sigma L)$.

Notice first that $(T)(M,\Sigma L)$ is contained in $\mbox{\rm
Ker\,}\beta_{L,M}(V,?)$; indeed, if $f\in V$ and $g\in(T)(M,\Sigma L)$, then
$\Sigma g\circ f=0$ (since $T$ is rigid), so

$\beta_{L,M}(f,g)=\beta_{L,\Sigma L}(\Sigma g\circ f,id_{\Sigma L})=0.$

Moreover, $\dim\mbox{\rm Ker\,}\beta_{L,M}(V,?)=\dim\operatorname{{\rm
Hom}}_{\mathcal{C}}(L,\Sigma M)/V$, and this last vector space is isomorphic
to the dual of $(T)(M,\Sigma L)$ thanks to [Pal08, Lemma 3.3]. Thus $\mbox{\rm
Ker\,}\beta_{L,M}(V,?)$ and $(T)(M,\Sigma L)$ have the same (finite)
dimension, and so they are equal. $\Box$

## Acknowledgements

The first and second authors were supported by the French ANR grant CHARMS
(ANR-19-CE40-0017-02). The second author was supported by the Institut
Universitaire de France (IUF). The third author was supported by the National
Natural Science Foundation of China (Grant No. 12271347). The final stages of
this project were completed while the first and second author were
participating in a trimester programme at the Isaac Newton Institute. The
authors would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme _Cluster
algebras and representation theory_ where work on this paper was undertaken.
This work was supported by EPSRC grant no EP/R014604/1.

We would like to thank Karin Baur, Eleonore Faber, Ana Garcia Elsener,
Alastair King, Matthew Pressland and Khrystyna Serhiyenko for discussions
about applications to cluster algebras and friezes.

## References

  * [ADS14] Ibrahim Assem, Grégoire Dupont, and Ralf Schiffler.  On a category of cluster algebras.  J. Pure Appl. Algebra, 218(3):553–582, 2014.
  * [Ami09] Claire Amiot.  Cluster categories for algebras of global dimension 2 and quivers with potential.  Ann. Inst. Fourier (Grenoble), 59(6):2525–2590, 2009.
  * [Ami11] Claire Amiot.  On generalized cluster categories.  In Representations of algebras and related topics, EMS Ser. Congr. Rep., pages 1–53. Eur. Math. Soc., Zürich, 2011.
  * [ARS10] Ibrahim Assem, Christophe Reutenauer, and David Smith.  Friezes.  Adv. Math., 225(6):3134–3165, 2010.
  * [BFG+21] Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko, and Gordana Todorov.  Friezes satisfying higher ${\rm SL}_{k}$-determinants.  Algebra Number Theory, 15(1):29–68, 2021.
  * [BFZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky.  Cluster algebras. III. Upper bounds and double Bruhat cells.  Duke Math. J., 126(1):1–52, 2005.
  * [BIRS09] A. B. Buan, O. Iyama, I. Reiten, and J. Scott.  Cluster structures for 2-Calabi-Yau categories and unipotent groups.  Compos. Math., 145(4):1035–1079, 2009.
  * [BM09] Karin Baur and Robert J. Marsh.  Frieze patterns for punctured discs.  J. Algebraic Combin., 30(3):349–379, 2009.
  * [BMR+06] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov.  Tilting theory and cluster combinatorics.  Adv. Math., 204(2):572–618, 2006.
  * [CC73a] J. H. Conway and H. S. M. Coxeter.  Triangulated polygons and frieze patterns.  Math. Gaz., 57(400):87–94, 1973.
  * [CC73b] J. H. Conway and H. S. M. Coxeter.  Triangulated polygons and frieze patterns.  Math. Gaz., 57(400):87–94, 1973.
  * [CC06] Philippe Caldero and Frédéric Chapoton.  Cluster algebras as Hall algebras of quiver representations.  Comment. Math. Helv., 81(3):595–616, 2006.
  * [CIEFR21] Giovanni Cerulli Irelli, Francesco Esposito, Hans Franzen, and Markus Reineke.  Cell decompositions and algebraicity of cohomology for quiver Grassmannians.  Adv. Math., 379:Paper No. 107544, 47, 2021.
  * [CILFS15] Giovanni Cerulli Irelli, Daniel Labardini-Fragoso, and Jan Schröer.  Caldero-Chapoton algebras.  Trans. Amer. Math. Soc., 367(4):2787–2822, 2015.
  * [CK06] Philippe Caldero and Bernhard Keller.  From triangulated categories to cluster algebras. II.  Ann. Sci. École Norm. Sup. (4), 39(6):983–1009, 2006.
  * [CK08] Philippe Caldero and Bernhard Keller.  From triangulated categories to cluster algebras.  Invent. Math., 172(1):169–211, 2008.
  * [Cox71] H. S. M. Coxeter.  Frieze patterns.  Acta Arith., 18:297–310, 1971.
  * [DG14] Salomón Dominguez and Christof Geiss.  A Caldero-Chapoton formula for generalized cluster categories.  J. Algebra, 399:887–893, 2014.
  * [DK08] Raika Dehy and Bernhard Keller.  On the combinatorics of rigid objects in 2-Calabi-Yau categories.  Int. Math. Res. Not. IMRN, (11):Art. ID rnn029, 17, 2008.
  * [DWZ10] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.  Quivers with potentials and their representations II: applications to cluster algebras.  J. Amer. Math. Soc., 23(3):749–790, 2010.
  * [DX] Ming Ding and Fan Xu.  Notes on the cluster multiplication formulas for $2$-Calabi–Yau categories.  arXiv:1001.4365v2 [math.RT].
  * [FK10] Changjian Fu and Bernhard Keller.  On cluster algebras with coefficients and 2-Calabi-Yau categories.  Trans. Amer. Math. Soc., 362(2):859–895, 2010.
  * [FP16] Bruce Fontaine and Pierre-Guy Plamondon.  Counting friezes in type $D_{n}$.  J. Algebraic Combin., 44(2):433–445, 2016.
  * [GLS06] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Rigid modules over preprojective algebras.  Invent. Math., 165(3):589–632, 2006.
  * [GLS08] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Preprojective algebras and cluster algebras.  In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 253–283. Eur. Math. Soc., Zürich, 2008\.
  * [GLS12a] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Generic bases for cluster algebras and the Chamber Ansatz.  J. Amer. Math. Soc., 25:21–76, 2012.
  * [GLS12b] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Generic bases for cluster algebras and the Chamber ansatz.  J. Amer. Math. Soc., 25(1):21–76, 2012.
  * [GLS18] Christof Geiß, Bernard Leclerc, and Jan Schröer.  Quivers with relations for symmetrizable Cartan matrices V: Caldero-Chapoton formulas.  Proc. Lond. Math. Soc. (3), 117(1):125–148, 2018.
  * [Hap87] Dieter Happel.  On the derived category of a finite-dimensional algebra.  Comment. Math. Helv., 62:339–389, 1987.
  * [IY08] Osamu Iyama and Yuji Yoshino.  Mutation in triangulated categories and rigid Cohen-Macaulay modules.  Invent. Math., 172(1):117–168, 2008.
  * [JKS16] Bernt Tore Jensen, Alastair D. King, and Xiuping Su.  A categorification of Grassmannian cluster algebras.  Proc. Lond. Math. Soc. (3), 113(2):185–212, 2016.
  * [Kel09] Bernhard Keller.  Algèbres amassées et applications.  November 2009.  Séminaire Bourbaki, Exposé 1014, 27 pages.
  * [KR07] Bernhard Keller and Idun Reiten.  Cluster-tilted algebras are Gorenstein and stably Calabi-Yau.  Adv. Math., 211(1):123–151, 2007.
  * [KW] Bernhard Keller and Yilin Wu.  Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces.  In preparation.
  * [LS15] Kyungyong Lee and Ralf Schiffler.  Positivity for cluster algebras.  Ann. of Math. (2), 182(1):73–125, 2015.
  * [MG15] Sophie Morier-Genoud.  Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics.  Bull. Lond. Math. Soc., 47(6):895–938, 2015.
  * [MG19] Sophie Morier-Genoud.  Symplectic frieze patterns.  SIGMA Symmetry Integrability Geom. Methods Appl., 15, 2019.
  * [Pal08] Yann Palu.  Cluster characters for 2-Calabi-Yau triangulated categories.  Ann. Inst. Fourier (Grenoble), 58(6):2221–2248, 2008.
  * [Pal12] Yann Palu.  Cluster characters II: a multiplication formula.  Proc. Lond. Math. Soc. (3), 104(1):57–78, 2012.
  * [Pla11a] Pierre-Guy Plamondon.  Cluster algebras via cluster categories with infinite-dimensional morphism spaces.  Compos. Math., 147:1921 – 1954, 2011.
  * [Pla11b] Pierre-Guy Plamondon.  Cluster characters for cluster categories with infinite-dimensional morphism spaces.  Advances in Mathematics, 227(1):1 – 39, 2011.
  * [Pla13] Pierre-Guy Plamondon.  Generic bases for cluster algebras from the cluster category.  International Mathematics Research Notices, 2013(10):2368–2420, 2013.
  * [Pla18] Pierre-Guy Plamondon.  Cluster characters.  In Homological methods, representation theory, and cluster algebras, CRM Short Courses, pages 101–125. Springer, Cham, 2018.
  * [Qin19] Fan Qin.  Bases for upper cluster algebras and tropical points.  Journal of the European Mathematical Society, 2019.  to appear.
  * [Rei10] Idun Reiten.  Cluster categories.  In Proceedings of the International Congress of Mathematicians. Volume I, pages 558–594. Hindustan Book Agency, New Delhi, 2010.
  * [Rup15] Dylan Rupel.  Quantum cluster characters for valued quivers.  Trans. Amer. Math. Soc., 367(10):7061–7102, 2015.
  * [Wu21] Yilin Wu.  Relative cluster categories and Higgs categories.  2021\.  arXiv:2109.03707.
  * [Xu10] Fan Xu.  On the cluster multiplication theorem for acyclic cluster algebras.  Trans. Amer. Math. Soc., 362(2):753–776, 2010.
  * [XX10] Jie Xiao and Fan Xu.  Green’s formula with $\mathbb{C}^{*}$-action and Caldero-Keller’s formula for cluster algebras.  In Representation theory of algebraic groups and quantum groups, volume 284 of Progr. Math., pages 313–348. Birkhäuser/Springer, New York, 2010.


New instanton on a warped Kerr spacetime
Reinoud Jan Slagtercor1label2
[label2]Asfyon, Astronomisch Fysisch Onderzoek Nederland
[cor1]Former: University of Amsterdam

We find an exact time-dependent instanton solution on a vacuum Kerr-like warped spacetime in conformal dilaton gravity. Remarkably, the metric solution results from a first-order PDE, allowing the connection with self-duality.
The antipodal boundary condition on the hypersurface of a Klein bottle $\sim \mathbb{C}^1\times\mathbb{C}^1$ is applied to describe the Hawking particles. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Klein bottle. The twist fits very well with the antipodal identification of the point on the horizon. No "cut and past" is necessary, so the Hawing particles remain pure without instantaneous information transport.
A local observer passing the horizon will not notice a central singularity in suitable coordinates.
The black hole paradoxes are also revisited in our new black hole model.
A connection is made with the geomeric quantization of $\mathbb{C}^1\times\mathbb{C}^1\sim S^3$, by considering the symplectic 2-form.
The model can be easily extended to the non-vacuum situation by including a scalar field. Both the dilaton and the scalar field can be treated as quantum fields when approaching the Planck area.

Kerr black hole conformal invariance brane-world models Klein surface antipodicity Hawking radiation instantons
04.20.-q 04.50.-h 04.62.+v 02.40.-k 04.70.Dy 02.40.Dr 03.65.Ud 03.65.Ta 03.65.Vf 02.30.Jr

§ INTRODUCTION

Following the pioneering work of [Hawking, 1975] on the evaporation of black holes, many scientists began to study the quantum-gravitational properties of black holes. However, there are some profound unsolved problems as one approaches the Planck scale. Due to the extremely high curvature, quantum effects can no longer be denied. It is thought that knowledge of the quantum effects near the horizon of a black hole will eventually lead to a consistent quantum gravity model. The properties of the black hole, i.e. its entropy, mass and emission, need to be revised because there are some serious paradoxes.  When fields are quantized on a manifold, one needs a one-to-one map between the entire asymptotic domain
of the maximally extended Penrose diagram.
We will follow here the approach of ['t Hooft, 2016], i.e. an antipodal topological boundary condition, already proposed by [Schrödinger, 1957].
This approach differs significantly from other approaches to attack the paradoxes, i.e. the information paradox, the firewall paradox, the complementarity paradox and the unitarity of the S-matrix.
The Randall-Sundrum (RS) warped brane world model ([Randall, et al., 1999], [Randall, et al., 1999]) is a simplification of the full string model. The extra-dimensional bulk space has a negative cosmological constant. The dilution of gravity by the presence of the bulk effectively weakens gravity on the brane. The result is that the true higher dimensional Planck scale can be much lower than the effective 4D Planck scale, possibly down to the electroweak scale. It also extends the range of graviton modes that can be felt on the brane, the so-called Kaluza-Klein (KK) modes. The model does not rely on compactification to localize gravity on the brane, but on the curvature of the bulk.
The notion of an instanton, or pseudoparticle, was introduced by 't Hooft and represents a localized finite action topological invariant in field theory. It was first developed in Yang-Mills theory, in particular in $SU(n)$ gauge theories.
The classical field equations with self-dual field strength then gave rise to a new non-perturbative treatment. The Minkowski spacetime was replaced by a Wick rotation $t\rightarrow i\tau$, where $\tau$ is a special parameterization of the favoured tunneling path through the configuration space. Feynman's path-integral approach to quantization is then unambiguous. Solutions are concentrated around a point in four-dimensional Euclidean space, and at one instant there is a "bump" in the field strength, which quickly dies away. There are striking similarities between these instantons and their gravitational counterparts. They are obtained in general relativity by a Wick rotation on Riemannian spacetime. The field equations then lead to first order differential equations. For an overview see [Gibbons, et al., 1993].
The structure of this letter is as follows. In section two we present the model under consideration. In section three we treat the Hopf fibration needed to describe the embedding of the Klein surface in the 5D model.
In section four we make the connection with the instanton and in section five a possible treatment of the Hawking particles.

§ THE MODEL

In an earlier study [Slagter, 2023] we studied the conformally invariant warped spacetime.

\begin{eqnarray}
ds^2=\omega(t,r,y_5)^{4/3}y_0\Bigl[-N(t,r)^2dt^2+\frac{1}{N(t,r)^2}dr^2+dz^2\cr +r^2(d\varphi+N^\varphi(t,r)dt)^2+dy_5^2\Bigr] \label{2-1}
\end{eqnarray}

where $y_5$ is the fifth coordinate, $y_0$ the bulk dimension and $\omega$ is a warp factor, reinterpreted as a dilaton field. We write $^{(5)}{g_{\mu\nu}}=\omega^{4/(d-2)} {^{(5)}{\tilde g_{\mu\nu}}}$ and $\tilde g_{\mu\nu}=^{(4)}{\tilde g_{\mu\nu}}+n_\mu n_\nu$. A "unphysical" spacetime is thus split off. Here $n_\mu$ is the unit normal to the brane. Again we write $^{(4)}{\tilde g_{\mu\nu}}=\bar\omega^2{^{(4)}{\bar g_{\mu\nu}}}$.
The Lagranian under consideration is

\begin{eqnarray}
S=\int d^dx\sqrt{-\tilde g}\Bigl[-\frac{1}{2}\xi (\Phi\Phi^*+\omega^2)\tilde R\cr-\frac{1}{2}\tilde g^{\mu\nu} \Bigl({\cal D}_\mu\Phi({\cal D}_\mu\Phi)^*+\partial_\mu\omega\partial_\nu\omega\Bigr)\cr
\end{eqnarray}

which is invariant under (['t Hooft, 2015])

\begin{equation}
\tilde g_{\mu\nu}\rightarrow \Omega^{\frac{4}{d-2}} \tilde g_{\mu\nu},\quad \omega \rightarrow \omega^{-\frac{d-2}{2}}\omega, \quad \Phi \rightarrow \omega^{-\frac{d-2}{2}}\Phi\label{2-3}
\end{equation}

where ${\cal D}$ is the gauge covariant derivative and $F$  the Abelian field strength. Furthermore, $\xi=(d-2)/(4(d-1)$ and we applied the Wick rotations $\omega^2\rightarrow -6\frac{\omega^2}{\kappa^2}$, with $\kappa=8\pi G_N$, to ensure that the field $\omega$ has the same unitarity and positivity properties as the scalar field.
The $\omega$ freedom is used by the in-falling observer to describe his experience of the vacuum.
Since we are working in a RS warped brane world model, one solves the 5D and the effective 4D equations simultaneously, with the contribution of the projected bulk Weyl tensor appearing in the latter.
We are mainly interested in the vacuum case. For the inclusion of the scalar field we refer to [Slagter, 2022, Slagter, 2023, Slagter, 2024].
It turns out that one can isolate the field equations for $N, \omega$ and $ N^\varphi$. The exact solution of $N$, without the scalar field, becomes

\begin{equation}
N^2=\frac{C_2}{r^2(t -b)^4+C_3}\Bigl[\frac{(r-a)^{k+1}\Bigl(r(k+1)+a\Bigr)}{k+2}+C_1 \Bigr]\label{2-4}
\end{equation}

where $C_i$ are constant. The solution is valid for both the 5D and the effective field equations (except for the constants). The constant $k\in \mathbb{N}$. For $k=3$ we get our metric solution, a quintic equation with in general five zero's. The solution for $N^\varphi$ becomes

\begin{equation}
N^\varphi=\int \frac{dr}{r^3\omega^{\frac{n-1}{n-3}}}+F_n(t)\label{2-5}
\end{equation}

where $F_n(t)$ is a arbitrary function determined by a constraint equation.
Since the solution for $N$ is separable, we write $N^2=N_1(r)^2 N_2(t)^2$. The solution for $\omega$ is[On a FLRW spacetime, the dilaton field is then considered as a warp factor, which can also be expressed exactly in $(r,t,y_5)$, where the bulk part becomes $e^{\sqrt{\frac{\Lambda_5}{6}(y_5-y_0)}}$ ([Slagter, et.al., 2016]). You will need these solutions when the Hawking particles become hard.]

\begin{equation}
\omega = \Bigl(\frac{c}{(r-a)(t-b)}\Bigr)^{\frac{1}{2}n-1}\label{2-6}
\end{equation}

with $n=4,5$ for the effective and 5D situation respectively.
The $r$-dependent part, $N_1$ of equation (<ref>) is determined by a first-order differential equation

\begin{equation}
rN_1\frac{\partial N_1}{\partial r}+N_1^2=\frac{k+1}{2}(r-a)^k\label{2-7}
\end{equation}

For $C_1=0$ the zeros are at $r_H=a_2$ with multiplicity $(k+1)$ and at $r_H=-\frac{a_2}{k+1}$. For $k=3$ and $C_1=-a^5/5$ there is one real solution, $r_H\approx 1.606a$, while the others are complex.
For $k=2$ and $C_1 < a^4/4$ there are 4 zeros, as in the BTZ solution, two of them negative.
For $k=1$ and $C_1 < -a^3/3$ there is still a real solution.
For $k=0$ we  surprisingly get two horizons $r_H=\pm a_2$.
The ergo region can be determined by solving $g_{tt}=0$, with (for k=3)

\begin{eqnarray}
\bar g_{tt}=\frac{(t-b)^6}{4r^2}\Bigl(6ar^2\log r-a^2+6a^2r-2r^3\Bigr)^2\cr
\end{eqnarray}

One can make the spacetime free of an ergo region by suitable values of the parameters and $F_k(t)$.
The effective 4d spacetime  in suitable coordinates becomes

\begin{equation}
\end{equation}

\begin{equation}
r^*=\frac{1}{4}\sum_{r^H_i}\frac{r^H_i \log(r-r^H_i)}{(r^H_i+a_2)^3}, t^*=\frac{1}{4C_2}\sum_{t^H_i}\frac{\log(t-t^H_i)}{(t^H_i+a_3)^3}\label{2-10}.
\end{equation}

The sum it taken over the roots of $(10a_2^3r^2+20a_2^2r^3+15a_2r^4+4r^5+C_1)$ and $C_2(t+a_3)^4+C_3$, i. e. $r^H_i$ and $t^H_i$.  We also defined the azimuthal angle coordinate $d\varphi^*=d\varphi+(N^\varphi/N_2^2) dt^{*}$, which can be used when an incoming null geodesic falls on the horizon, i.e. in a coordinate system that rotates about the z-axis relative to the Boyer-Lindquist coordinates.
In Kruskal–Szekeres  lightcone coordinates $(U,V)$ we have

\begin{equation}
ds^2=\omega^{4/3}\bar\omega^2\Bigl[\frac{N_1^2}{N_2^2}\frac{dUdV}{\epsilon^2 UV}+dz^2+r^2d\varphi^{*2}\Bigr]\label{2-11}
\end{equation}

with $\epsilon$ a constant related to the surface gravity at the horizon.

Penrose diagram. The antipodal points $P(X)$ and $P(\bar X)$ are identified. Particles entering the black hole will create waves that approach the horizon from the outside, and those that pass the horizon will emerge from the "other side" of the black hole. Note that $t^*=log(U/V)\sim log(t-t_H)$. In this approach, I and II are CPT invariant. This means that time runs backwards in II.

If we omit the contribution of the projected Weyl tensor in the effective 4D Einstein equations, we obtain the Ban̆ados-Teitelboim-Zanelli 3D solution ([Ban̆ados, et al., 1992], [Compère, 2018]). This is equivalent to the substitution $k=2$ in Eq. (<ref>).
We also found a remarkable connection with the icosahedral Möbius group $A_5$ when inscribed in $S^2$. The isometry group is of order 5 and has three orbits which are invariant under the antipodal map.
Our general polynomial of $N^2$ is a quintic, which  cannot be a coincident ([Slagter, 2023]).
If we include a complex scalar-gauge $(\Phi,A_\mu)$, we find from the superfluous dilaton equation, that the scalar-dilaton potential $V(\omega,\Phi)$ satisfies

\begin{equation}
V(\omega,\Phi)=\beta_1 \Phi^{\beta_2}\omega^{\frac{2}{3}-\frac{\eta\beta_2}{y_0}}\label{2-12}
\end{equation}

So we get a quartic conformal invariant matter coupling $\sim \Phi^2\omega^2$ for $|y_0|=3/2\eta$. Where $\eta$ is the vacuum expectation and  we have used a constant gauge field.

§ HOPF FIBRATING $\MATHBB{C}^1\OTIMES \MATHBB{C}^1$ AND THE KLEIN BOTTLE

We have used the embedding of the Klein bottle in the warped 5D spacetime in our model. To get back to our black hole in the effective 4D spacetime, we can use the Hopf fibration of $S^3\sim \mathbb{C}^1\otimes\mathbb{C}^1$ and apply the antipodal map (['t Hooft, 2016]) to get the $S^2$ of our black hole horizon.
The Hawking particle created near the horizon at $P(U,V,z,\varphi,y_5)$, leaves the horizon at the antipode $\tilde P(-U,-V,-z,\varphi +\pi,-y_5)$ after a finite proper time.

Fibering the $S^3$.

In our situation we define  $W=x+iy, Z=z+iy_5$[or $W$ in polar coordinates $re^{i\varphi}$.]. We write the 3-sphere $S^3=\{x,y,z,y_5\in \mathbb{R}^4|x^2+y^2+z^2+y_5^2=r^2+R^2\}=\{(Z,W)\in \mathbb{C}^2|Z\bar Z+W\bar W=r^2+R^2\}$.
See figure (<ref>) and the diagram of Eq. (<ref>).

\begin{eqnarray}
\begin{tikzcd}
& (Z, W)\:\:\:\:\: \in \hspace{-1.5cm}  \arrow[d, "H"]
& \mathbb{C}^2/\{0\} \arrow[d, "\mathbb{C}P^1"]
& S^3 \arrow[l]\arrow[d, "p" ]\\
& \:\:\:\:\:u\equiv\frac{Z}{W} \:\:\:\:\:\: \:\: \in\hspace{-1cm}
&\:\:\:\: \mathbb{C}\cup\{\infty\}\:\:\:\:  =\hspace{-1cm}
& S^2
\end{tikzcd}\label{3-1}
\end{eqnarray}

There are two basic compact self-dual manifolds,  the round metric on $S^4$  and the Fubini-Study metric on $\mathbb{C}P^2$. In our situation we are mainly interested in the quantization on the latter. It is possible to construct a complex two-dimensional Hilbert space ${\cal H}$. If we  write on $\mathbb{C}^2$, $\ket{{\cal U}}=(Z,W)$ with density matrix $\rho=\ket{{\cal U}}\bra{{\cal U}}$, we can  construct a quantization of $\mathbb{C}P^2$ ([Urbantke, 1990, Slagter, 2023].
On the non-orientable Klein surface, meromorphic functions remain constant and our quintic is meromorphic. It is also homeomorphic to the connected sum of two projected planes.

Geodesics on the torus and the Klein bottle. Note the difference: because of the $\mathbb{Z}_2$ symmetry of our warped spacetime, we need a non-closed geodesic for the torus.

It is also possible to construct geodesics and a minimum surface needed to follow the particles. You can use the torus, on which the orbits are easy to find. Note that the orbit is still in 4D space. The geodesics on the Klein surface will never intersect each other because there is no "inside".

§ THE INSTANTON

It turns out that self-duality in the Yang-Mills model, leads to first-order differential equations. They played a fundamental role in non-abelian gauge theories. See, for example, the textbook of [Felsager, 1998].

Plot of the instanton for $\tilde g_{\mu\nu}$ for $a=\pm 1$ respectively. We have chosen $C_1=0$.

It is now known that there are Euclidean self-dual instanton solutions in gravity theory. The self-duality is then expressed in terms of the Riemann curvature tensor. They are thought to play a role in understanding quantum gravitational effects near the horizon of BH's, such as the evaporation process and the formation of Planck-sized BH's[Recently, primordial BHs have been observed, which could not be formed by contracting matter.].
Well-known examples are the Eguchi-Hanson solution ([Eguchi, et al., 1978]) and the Fubini study, which can be described on a $(\mathbb{C}^1\times\mathbb{C}^1)$ Kähler manifold ([Joyce, 1995]).

If we switch to Riemannian space with $t\rightarrow i\tau$, we get the same solution as in Eq.(<ref>) - (<ref>).
The line element can be written in the stationary case and for $C_1=0, k=3$ as

\begin{eqnarray}
\end{eqnarray}

Left: Plot of $r^*$ against $r$ and the inverse for the case $a=1$, with range $-\frac{a}{4}<r<a$. If $r\rightarrow a$, then $r^*\rightarrow \pm\infty$. Right: Surface of revolution around the $r$ axis. A possible geodesic is drawn. The surface is not orientable, but there is a cusp singularity at $r=-1/4a$..

\begin{eqnarray}
r^*=\frac{1}{5\sqrt{a}}\Bigl[ 6 arctanh\Bigl(\sqrt{\frac{a+4r}{5a}}\Bigr)
\end{eqnarray}

with $-\frac{a}{4}<r<a$. It  describes the interior of the black hole. In figure (<ref>) we have plotted $r^*$, valid only inside the horizon $r_h=1$, for $a=1$.
We have also plotted the surface of revolution ($y_5$ suppressed).

Left: Plot of $z(r)$ of the Euclidean 3-space for our model. Right: The  surface of revolution over $\varphi$. A particle arriving at the horizon $r=a$ "jumps" to the antipode $\varphi\rightarrow \varphi +\pi$.

Note the difference from the wormhole case[In our case we have no explicit expression for the surface of revolution.]. The region $r<\frac{a}{4}$ is inaccessible. It would be challenging to consider large values of $k$. We will then get a horizon and the singularity for $\rightarrow 0$ will never be reached.
It became quite clear that Euclidean self-dual solutions in Einstein's theory play a role in quantum gravity. These solutions have vanishing classical action and non-trivial topological invariants. Moreover, there is a similarity with the YM Euclidean self-dual solutions [Eguchi, et al., 1978].
Despite their Euclidean origin, instantons, like solitons, play a fundamental role in quantum theories. Their importance arises, for example, in tunneling effects using the WKB approximation.
Instanton-like solutions are expected to be all self-dual, localized in Euclidean spacetimes and free of singularities.
Asymptotically locally Euclidean self-dual metrics were also important for the path integral calculations.
They give a dominant contribution to the path integral that is as important as that of the flat metric itself. They also have time-parity inversion.
However, gravitational instantons do not contribute to the zero-frequency modes of a spin-1/2 Dirac particle, while they produce zero-frequency solutions of spin-3/2 particles.
It is conjectured ([Gibbons, et al., 1993], [Atiyah, et al., 1978]) that there is a deep connection between the gravitational instanton and the alternating discrete symmetry group ${\cal A}_5$ which serves as a discrete subgroup of $SO(4)$ of its boundary.
A gravitational instanton is a four-dimensional regular spacetime $M$, self-dual Ricci curvature and asymptotically local Euclidean, whose boundary  has a ${\cal A}_k,D_k, T, 0, I$ symmetry[These are the dihedral, Möbius, tetrahedral, octahedral and icosahedral groups respectively.]. In fact, every finite subgroup of $S^3\sim SU(2)$, is either cyclic or conjugate to  these binary subgroups.
For our 5D warped spacetime, our boundary is the Klein surface, embedded in $\mathbb{R}^4\sim \mathbb{C}^1\times \mathbb{C}^1$ and the symmetry the double cover group  $I$ of the  icodahedral.
An interesting study has also been done by  [Gibbons, et al., 1993]. They studied, on $\mathbb{C}P^2$,  a gravitational instanton surrounded by an event horizon and compared the solution with the $SU(2)$ Yang-Mills instanton.
This study is based on the Riemannian solution of the Einstein equations with a cosmological constant on $\mathbb{C}P^2$ by applying the Hopf-fibering. The Weyl tensor is then anti-self dual and they conclude that there is no Lorentzian counterpart connection, i. e. it cannot be related to a real 4-dimensional manifold. Our model does.

§ TREATMENT OF THE HAWKING PARTICLES

We have taken a slightly different approach to the antipodal map used by ['t Hooft, 2015]. To revise the information, firewall and complementarity paradoxes, 't Hooft uses the antipodal map on $S^2$, where the non-orientability is provided by the Möbius strip. His method restores quantum purity and CPT invariance.  The particles remain entangled. The incoming particles return at the antipode and the particles remain entangled. There is no inside of the black hole, i.e. region II is not "really there" for the observer inside by the cut-and-paste procedure.

Treatment of the Hawking particles on the Klein bottle.

In our model we use the Klein surface $\mathbb{K}$.
Our conjecture is that our solution can be seen as a dynamical solution embedded in our $(4+1)$ dimensional spacetime. More precisely, if ${\cal M}$ is the sub-manifold with metric $g$, i.e. the metric on our effective 4D spacetime, then ${\cal M}$ is diffeomorphic to the hyperbolic 5D spacetime: $\chi :\mathbb{R}^5\rightarrow {\cal N}$.
The Riemannian manifold $({\cal M},g)$ must be conformally flat, which it is.
Now physicists are interested in the topology of moduli spaces of self-dual connections on vector bundles over Riemannian manifolds. One reason was that on these spaces the instanton approximation to the Green functions of Euclidean quantum gravity, Yang-Mills theory, can be expressed in terms of integrals over moduli spaces. One then needs the metric and volume form of the moduli spaces.
We suspect that we can consider $\mathbb{K}$ as a 4-sphere in our hyperbolic pseudo-Riemannian spacetime.
A note must be made about the $\mathbb{Z}_2$ symmetry in the original description of the antipodal mapping (['t Hooft, 2021]).
At the boundary of regions I and II in the Penrose diagram, the antipodes were glued together on a 3-sphere, and the transverse $(\theta,\varphi)$ part is a projected 2-sphere.
In our model it is replaced by the projected 3-sphere $(z,\varphi ,y_5)$ using the $\mathbb{Z}_2$ symmetry of the bulk space $(U,V,z,\varphi,y_5)$. We "blow up" the 4-manifold to 5D to handle the singularities in the curvature and to apply the antipodal map.
One can mathematically formulate the topology of a 4-manifold using self-dual connections over de Riemannian $S^4$ ([Freed, 1984]). It depends only on the conformal class of the Riemannian metric. This self-dual connection can be interpreted by the conformal map: $\mathbb{R}^4\rightarrow S^4/\{0\}$ as a self-dual connection or "instanton". We interpret the Riemannian 5D warped spacetime, an open 5-ball, as an instanton on $\mathbb{R}^4$.
In the pseudo-Riemannian spacetime, the boundary is the non-orientable Klein surface, which we used for the antipodicity to preserve the pure states of the Hawking particles. It is known that a static solution in $m$ space dimensions is completely equivalent to an instanton in $m$ space-time dimensions.
An outside observer will register the Hawking radiation as thermal, i.e. in a mixed state, while a local observer will doubt the vacuum state.
Furthermore, the evolution of the wave function of the incoming particle must be unitary, i.e. it must satisfy the Schrödinger equation $\ket{\chi(t_1)}=U(t_1,t_2)\ket{\chi(t_2)}$ and be bijective. It is thought that information about the quantum state is preserved during black hole evaporation. The loss of information is incompatible with unitarity.
Thus, 't Hooft's method resolves the controversy between the pure quantum state of the incoming particles and the mixed state property of the Hawking radiation.
The information is retrieved by the outgoing particles. They carry the information of the incoming particles.
In quantum mechanical terms, the incoming position of the wave packet is imprinted on the outgoing momentum. The positions are then highly de-localised. This is how long-wavelength Hawking particles are produced from short-wavelength waves.
In our model, the Hawking particles don't jump infinitely fast to the antipode, but stay on the Klein surface for a while. Moreover, the complementarity in the different notion of the dilaton $\omega$ used by the inside and outside observer is formulated by the invariance of Eq.(<ref>).
Then we have the problem of handing over the "hard" gravitons. 't Hooft uses the Shapiro delay method to describe the gravitational interaction near the horizon. Hard particles become "soft" by taking into account the equations in the transverse spherical harmonics ${\bf Y}_{l,m}(\theta,\varphi)$. The dynamics is factorized in $(l,m)$. The dilaton is written as

\begin{equation}
\omega={\bf J}(r,t)e^{in\varphi}{\bf Y}_m(\varphi)\label{5-1}
\end{equation}

where ${\bf J}$ is the elliptic Jacobi function and ${\bf Y}_m$ is the cylindrical harmonic.
The firewall decays due to the identification $(\theta,\varphi)\rightarrow (\pi-\theta,\varphi+\pi)$.
In our model, the shape of the probability function makes the gravitons hard in the bulk ([Randall, et al., 1999]) and soft on the brane. The result is that the presence of the bulk causes the KK contribution of the Weyl tensor on the brane, leading to our new solution.
The spherical harmonics are replaced by the cylindrical ${\bf Y}_m(\varphi)$. The second quantum number is provided by the "winding number", because the dilaton behaves as an ordinary quantum field with axial symmetry, just like the scalar field, when it is included. Furthermore, we obtained PDE's in $(r,t)$ with exact solution.
There will be a time delay between the incoming and outgoing waves or particles depending on $y_0$, which can be expressed in the parameters of our model. Thus, experimental data could provide an estimate of the bulk dimension.
Furthermore, 't Hooft's method requires high wave numbers when approaching the Planck region, and the cut-off is then determined by the number of microstates. In our model we don't need a cut-off. We have the elapsed time on the Klein surface in the bulk, which gives the degree of "hardness".
Furthermore, in our model the wavelength increases with time. This would imply that the particles gain momentum and the quanta become blue-shifted. However, our process takes place in the bulk, so it is not visible.
One can also apply Schrödinger's original treatment of the Klein-Gordon equation on a hypersurface ([Schrödinger, 1957]). He finds conditions for the values of the quantum numbers $(l,m)$[Schrödinger applies the case of an expanding universe.] when the lines of constant phase on the hypersurface will close. In our case we can have the situation where the phase lines spiral around the Klein surface many times without the possibility of intersecting.
It should be noted that the antipodal map is obligatory in order to make the mapping of the coordinates in regions I and II one-to-one, and yet they are distinct because they are spatially separated. There is no summation over "invisible" quantum states to form the density matrix, we still have pure states.
If you sum over the other states, which are far away in the antipodal region and unobservable to the local observer, you would conclude that these particles have the Hawking temperature. But the state is not thermal at all.
The Hawking particles on one hemisphere are entangled with the antipode.
To the outside observer, the Hartle-Hawking wave function is no longer a thermally mixed state, but a single pure state.
In general, the antipodes are completely entangled. From the outside, the black hole is not in a stationary Hartle-Hawking quantum state. In other words, there is no complete heat bath.
Finally, we can study the surface gravity of our model. In the $\varphi^*$ coordinate it becomes

\begin{equation}
\kappa=\frac{2}{5}\Bigl(\frac{10C_2(t-b)^3+6r^2+10a^2-15ar-\frac{5C+a^5}{r^3}}{C_2(t-b)^4+C_3}\Bigr)
\end{equation}

Our solution becomes an extremal black hole, i.e. the surface gravity is zero, for

\begin{equation}
\end{equation}

In general, the black hole never becomes extremal in $(t,r,\varphi)$ coordinates.

§ CONCLUSIONS

We formulate our results in the following  proposition:

1. The vacuum solution found on the conformal invariant 5D warped spacetime, where the metric is determined by a quintic polynomial, can be described by a Laplace-type equation on a Klein surface. The zeros of this quintic polynomial depend on the  constraints on the parameters of the model under consideration. In a special case, the zeros can be expressed  analytically  as $\sim (r-r_H)^{k+1}[r(k+1)+r_H]+C(k+2)$ where $k$ is an integer and $C$ a constant. They are related to the eigenvalues of the Laplacian on the Klein surface. The minimal  area is determined by an elliptic integral of the second kind.

2. The antipodal mapping  of regions I and II in the Penrose diagram is accompanied by the continuous transition in the extra bulk dimension by means of the non-orientable Klein surface embedded in $\mathbb{R}^4\sim \mathbb{C}^1\times \mathbb{C}^1$ by means of a Hopf fibration. Measuring the elapsed time that the Hawking particles spend on the Klein surface could give an estimate of the dimension of the bulk.

3. In the conformal dilaton-scalar field model on a warped spacetime, the fields equations for the dilaton $\omega$, which is part of the gravitational metric, and the scalar field obey comparable Klein-Gordon-type PDE's. The interaction of these quantum fields on the "un-physical"  conformally flat spacetime also describes the time evolution of the horizon.

4. The information problem could be  solved by the antipodal boundary condition, i.e. by removing the inside of the black hole. The result was that the black hole is not thermal. The reason is that there is no "hidden sector" anymore to produce the thermodynamically mixed state of the Hawking radiation. All states remain pure for the outside observer.
In our model, the antipodal map on the brane is accompanied by the $\mathbb{Z}_2$ symmetry in the bulk in the Randall-Sundrum model. No "cut-and-past" is required.  This means that there is no  infinitely fast transport to the other side of the hemisphere. The Hawking particles remain  on the Klein surface for a while, as seen by the outside observer. A local observer will not notice the $r=0$ singularity in a suitably chosen coordinate system for the $S^3$, which is conformal flat.
The vacuum solution described an instanton in Riemannian space. In suitable coordinates, the inside of the black hole shows no central singularity.

5. The geometric quantization of our Klein surface can be constructed using a two-level Hilbert space on the complex hyper-surface $\mathbb{C}^1\times\mathbb{C}^1$.

[Atiyah, et al., 1978]
M. F. Atiyah, N. J.  Hitchin and I. M. Singer, Proc. R. Soc. Lond. A  362 (1978) 425.
[Ban̆ados, et al., 1992]
M. Ban̆ados, M. C. Teitelboim and T. Zanelli, Phys. Rev. Lett. 69, (1992) 1849.
[Compère, 2018]
G. Compère, Advanced Lectures on General Relativity, Lecture notes in Physics,  Springer, 2019.
[Eguchi, et al., 1978]
T. Eguchi and A. J. Hanson, Ann. of Phys. 120 (1979) 82.
[Felsager, 1998]
B. Felsager, Geometry, Particles and Fields, Springer, New York, 1998.
[Freed, 1984]
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Springer-Verlag: New York, 1984.
[Gibbons, et al., 1993]
G. W. Gibbons and S. W. Hawking, Euclidean Quantum Gravity,World Scientific, Singapore, 1993.
[Hawking, 1975]
S. Hawking, Comm. Math. Phys. 43 (1975) 199.
['t Hooft, 2016]
G. 't Hooft, Found. Phys., 46 (2016) 1185.
['t Hooft, 2015]
G. 't Hooft,  Int. J. of Mod. Phys. D, 24 (2015) 1543001.
['t Hooft, 2021]
G. 't Hooft, Universe, 7 (2021) 298.
[Joyce, 1995]
D. D. Joyce, Duke Math. J. 77 (1995)  519.
[Randall, et al., 1999]
L. Randall, R. Sundrum, Phys. Rev Lett 83 (1999) 3370.
[Randall, et al., 1999]
L. Randall, R. Sundrum, Phys. Rev Lett 83 (1999) 4690.
[Schrödinger, 1957]
E. Schrödinger, Expanding Universe, Cambridge University. Press, Cambridge U.K. 1957.
[Slagter, et.al., 2016]
R. J. Slagter and S. Pan, Found. Phys.  46 (2016) 1075.
[Slagter, 2022]
R. J. Slagter, Int. J. Mod. Phys. A, 37 (2022) 2250176.
[Slagter, 2023]
R. J. Slagter, Universe, 9 (2023) 383.
[Slagter, 2024]
R. J. Slagter, Black holes and Cosmic Strings Revisited, (2024). To appear.
[Urbantke, 1990]
H. Urbantke,  Am. J. Phys.  59 (1990) 503.