# On an injectivity lemma in the proof of Tannakian duality

Phùng Hô Hai<EMAIL_ADDRESS>Institute of Mathematics, Vietnam Academy of
Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

###### Abstract.

In this short work we give a very short and elementary proof of the
injectivity lemma, which plays an important role in the Tannakian duality for
Hopf algebras over a field. Based on this we provide some generalizations of
this fact to the case of flat algebras over a noetherian domain.

###### 2010 Mathematics Subject Classification:

16T15, 18A22

This research is funded by Vietnam National Foundation for Science and
Technology Development(NAFOSTED) under grant number 101.01-2011.34. Part of
this work has been carried out when the author was visiting the Vietnam
Institute for Advanced Study in Mathematics.

## Introduction

Let $k$ be a field. The neutral Tannakian duality establishes a dictionary
between $k$-linear tensor abelian categories, equipped with a fiber functor to
the category of $k$-vector spaces, and affine group schemes over $k$. The
duality was first obtained by Saavedra in [5], among other important results.
In [2], Deligne and Milne gave a very readable self-contained account on the
result.

The main part of the proof of Tannakian duality was to establish the duality
between abelian category equipped with fiber functors to ${\rm vect}_{k}$ and
$k$-coalgebras. Here, one first proves the claim for those categories which
have a (pseudo-) generator. Such categories are in correspondence to finite
dimensional coalgebras. The injectivity lemma claims that, under this
correspondence, fully faithful exact functors, which preserves subobjects,
correspond to injective homomorphisms of coalgebras (see Lemma 1.2 for the
precise formulation). This lemma was implicitly used in the proof of Prop.
2.21 in [2]. In the original work of Saavedra this claim was obtained as a
corollary of the duality, cf. [5, 2.6.3 (f)]. In his recent book Szamuely gave
a more direct proof of the injectivity lemma, cf. [8, Prop. 6.4.4]. Szamuely’s
proof is nice but still quite involved. Similar treatment and some
generalizations was also made in Hashimoto’s book [3, Lem. 3.6.10].

In this short work we give a very short and elementary proof of the
injectivity lemma. Based on this, we provide some generalizations of this fact
to the case of flat coalgebras over a noetherian domain. We believe that these
results will find applications in the Tannakian duality for Hopf algebras over
noetherian domains.

### Notations

Throughout this work we shall fix a commutative noetherian domain $R$. The
quotient field of $R$ will be denoted by $K$.

For an algebra (or more general a ring) $A$ (commutative or not), ${\rm
mod}(A)$ denotes the category of left $A$-modules and ${\rm mod}_{\rm f}(A)$
denotes the subcategory finitely generated modules.

For a coalgebra $C$ over $R$, ${\rm comod}(C)$ denotes the category of right
$C$-comodules, and ${\rm comod}_{\rm f}(C)$ denotes the subcategory of
comodules which are finite over $R$.

##  1\. A simple proof of the injectivity lemma

Let $k$ be a field and $f:A\to B$ be a homomorphism of finite dimensional
$k$-algebras. Then $f$ induces a functor $\omega:{\rm mod}_{\rm f}(B)\to{\rm
mod}_{\rm f}(A)$ between the categories of finite modules over $A$ and $B$,
which is identity on the underlying vector spaces. In particular, $\omega$ is
a faithfully exact functor.

######  Lemma 1.1.

A homomorphism $f$ given as above is surjective if and only if the induced
functor $\omega$ is full and has the property: for any $B$-module $X$ and any
$A$-submodule $Y$ of $\omega(X)$, there exists a $B$-submodule $X^{\prime}$ of
$X$ such that $\omega(X^{\prime})=Y$. In other words $f$ is surjective if and
only if ${\rm mod}_{\rm f}(B)$, by means of $f$, is a full (abelian)
subcategory of ${\rm mod}_{\rm f}(A)$, closed under taking submodules.

###### Proof.

If $f$ is surjective then obviously $\omega$ has the claimed properties. We
prove the converse statement. Thus for any $B$-module $X$ and any submodule
$Y$ of $X$, considered as modules over $A$, we know that $Y$ is also stable
under the action of $B$ (obtained by restricting the action of $B$ on $X$).
Assume the contrary that $f$ is not surjective, i.e., $B_{0}:={\rm im}(f)$ is
a strict subalgebra of $B$. Then $B_{0}\subset B$ is an inclusion of
$A$-modules, $B$ it self is a $B$-module, but $B_{0}$ is not stable under the
action of $B$ as it contains the unit of $B$. A contradiction. ∎

By duality we have the following result for comodules.

######  Lemma 1.2.

Let $f:C\to D$ be a homomorphism of finite dimensional $k$-coalgebras. Then
the category ${\rm comod}_{\rm f}(C)$, considered by means of $f$ as a
subcategory of ${\rm comod}_{\rm f}(D)$, is full and closed under taking
subobjects if and only if $f$ is injective.

######  Remark 1.3.

In the proof of Lemma 1.1, there is no need to assume that $A$ is finite
dimensional. Therefore, in Lemma 1.2 there is no need to assume that $D$ is
finite dimensional. On the other hand, it is known that each coalgebra is the
union of its finite dimensional subcoalgebras. Therefore there is no need to
impose the dimension condition on $C$ either.

##  2\. Generalizations

We give here several generalizations of the lemmas in Section 1 to the case
when $k$ is a noetherian domain.

###  2.1. The case of algebras

Let $R$ be a noetherian domain. We consider $R$-algebras. Modules over such an
algebra are automatically $R$-modules, we call such a module $R$-finite (resp.
torsion-free, flat, projective, free) if it is finite (resp. torsion-free,
flat, projective, free) over $R$. Let $R\to S$ be a homomorphism of
commutative algebras. Then the base change $R\to S$ will be denoted by the
subscript $()_{S}$. For instance $M_{S}:=M\otimes_{R}S$,
$f_{S}:=f\otimes_{R}S$ for an $R$-linear map $f$.

Let $f:A\to B$ be a homomorphism of finite, torsion-free $R$-algebras. It
induces a functor $\omega:{\rm mod}_{\text{f}}(B)\to{\rm mod}_{\text{f}}(A)$,
which is identity functor on the underlying $R$-modules, therefore it is
faithful and exact. The following lemma is a straightforward generalization of
Lemma 1.1.

######  Lemma 2.1.

The map $f$ as above is surjective if and only if ${\rm mod}_{\rm f}(B)$ when
considered by means of $f$ as a subcategory of ${\rm mod}_{\rm f}(A)$ is full
and closed under taking subobjects.

###### Proof.

Only the “if” claim needs verification. Let $\mathfrak{m}\in R$ be a maximal
ideal and let $k(\mathfrak{m}):=R/\mathfrak{m}$ be the residue field. Then the
full subcategory of ${\rm mod}_{\rm f}(A)$ annihilated by $\mathfrak{m}$ is
equivalent to ${\rm mod}_{\rm f}(A_{k(\mathfrak{m})})$. Note that this
subcategory is also closed under taking subobjects.

Thus, by assumption, ${\rm mod}_{\rm f}(B_{k(\mathfrak{m})})$ is a full
subcategory of ${\rm mod}_{\rm f}(A_{k(\mathfrak{m})})$, closed under taking
subobjects. Therefore the map $f_{k(\mathfrak{m})}:A_{k(\mathfrak{m})}\to
B_{k(\mathfrak{m})}$ is surjective, by means of Lemma 1.1. This holds for any
maximal ideal $\mathfrak{m}$ of $R$, hence $(B/f(A))_{k(\mathfrak{m})}=0$ for
all maximal ideals $\mathfrak{m}$. According to [4, Thm 4.8], we conclude that
$B/f(A)$ itself is zero. ∎

Lemma 2.1 can also be proved in the same way as Lemma 1.1. We prefer to
present the above proof as its main idea (using base change) will be exploited
further.

An $A$-submodule $N$ of $M$ is said to be saturated iff $M/N$ is $R$-torsion-
free. A homomorphism $f:A\to B$ is said to be dominant if $f_{K}:A_{K}\to
B_{K}$ is surjective, or equivalently $B/f(A)$ is $R$-torsion. Here $K$
denotes the quotient field of $R$.

######  Proposition 2.2.

Let $f:A\to B$ be a homomorphism of $R$-torsion free algebras and assume that
$B$ is $R$-finite. Let $\omega:{\rm mod}_{\rm f}(B)\to{\rm mod}_{\rm f}(A)$ be
the induced functor. Then:

  1. (1)

The image of $\omega$ is closed under taking saturated subobjects of
$R$-torsion-free objects iff $f$ is dominant. In this case $\omega$ is also
closed under taking saturated submodules of any modules and its restriction to
the subcategory of $R$-torsion-free modules is full.

  2. (2)

The image of $\omega$ is closed under taking subobjects of $R$-torsion-free
objects iff $f$ is surjective. (In this case $\omega$ is also obviously full.)

###### Proof.

(1). Assume that $\omega$ has the required property. We show that $f$ is
dominant, i.e. $f_{K}$ is surjective. It suffices to show that the functor
$\omega_{K}:{\rm mod}(B_{K})\to{\rm mod}(A_{K})$ induced from $f_{K}$
satisfies the condition of Lemma 1.1. Let $X$ be a finite $B_{K}$-module and
$Y\subset X$ an $A_{K}$-submodule. Consider $X$ as a $B$-module, take a
$K$-basis of $X$ such that a part of it is a basis of $Y$ and let $M$ be the
$B$-submodule generated by this basis, let $N:=M\cap Y$. Then $N_{K}=Y$ (as it
contains a $K$-basis of $Y$). By the diagram below $M/N$ is $R$-torsion-free.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X/Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Thus $N$ is a saturated submodule of $M$, hence, by assumption, $N$ is stable
under $B$, consequently $Y=N_{K}$ is stable under $B_{K}$.

Conversely, assume that $f$ is dominant, i.e., $f_{K}$ is surjective. Then
$\omega_{K}$ is fully faithful and closed under taking submodules. Let $M$ be
an $R$-torsion-free $B$-module, $N\subset M$ be a saturated $A$-submodule.
Then $M/N$ is also $R$-torsion-free, hence $N=M\cap N_{K}$. Now $N_{K}$ is
stable under $B_{K}$ and $M$ is stable under $B$, showing that $N$ is stable
under $B$.

Let $\varphi:M\to N$ be an $A$-linear map, where $M,N$ are both $R$-torsion-
free then $\varphi$ is determined by $\varphi_{K}:M_{K}\to N_{K}$. Since
$f_{K}$ is surjective, we know that $\varphi_{K}$ is $A_{K}$-linear, hence
also $B_{K}$-linear, implying that $f$ is $B$-linear. Thus $\omega$ restricted
to $R$-torsion free modules is full.

Let now $M$ be a finite $A$-module, $N$ be a finite $B$-module and
$\varphi:M\to N$ be an injective $A$-linear map with $M/\varphi(N)$ being
$R$-torsion free. Consider a finite free $B$-linear cover $\psi:N^{\prime}\to
N$. Let $\varphi^{\prime}:M^{\prime}\to N^{\prime}$ be the pull-back of
$\varphi$ along $\psi$ (as $A$-modules).

$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N.}$

Then $\varphi^{\prime}$ is injective and $\psi^{\prime}:M^{\prime}\to M$ is
surjective, moreover $N^{\prime}/\varphi^{\prime}(M^{\prime})\cong
N/\varphi(N)$ hence is $R$-torsion free. Consequently, $M^{\prime}$ is
$B$-stable and, since $\varphi\psi^{\prime}=\psi\varphi^{\prime}$ is
$B$-linear, there is a $B$-action on $M$ making $\varphi$ $B$-linear.

(2). According to the proof of Lemma 2.1, it suffices to show that for any
maximal ideal $\mathfrak{m}$ of $R$, the image of $\omega_{k(\mathfrak{m})}$
is closed under taking submodules. Let $V$ be a $B_{k(\mathfrak{m})}$-module
and let $\varphi:U\to V$ be an inclusion of $A_{k(\mathfrak{m})}$-modules.
Represent $V$ as a quotient of some (free) $B$-modules $M$, then $U$ will be a
quotient of some $A$-submodule $N$ of $M$. We have $N$ is $R$-torsion-free,
hence, by assumption, $N$ is stable under $B$, so that $U$ must be
$B_{k(\mathfrak{m})}$-stable. ∎

###  2.2. The case of coalgebras

In this subsection we consider $R$-flat coalgebras. For such coalgebras the
comodule categories are abelian, see, e.g. [3, Lem. 3.3.3] or [1, Thm. 3.13].
Moreover, the kernel and image of a comodule homomorphism are the same as
those of the underlying $R$-module homomorphism. A homomorphism of flat
$R$-coalgebras $f:C\to D$ induces the restriction functor $\omega:{\rm
comod}(C)\to{\rm comod}(D)$ which is the identity functor on the underlying
$R$-module. Hence $\omega$ is faithful and exact.

We say that a homomorphism of flat $R$-coalgebras $f:C\to D$ is pure if it is
a pure homomorphism of $R$-modules. This condition is the same as requiring
$D/f(C)$ be $R$-flat. Note also that over a noetherian domain, finite flat
modules are projective.

For the case $C$ and $D$ are $R$-projective and $C$ is $R$-finite, the desired
results can be deduced from the previous subsection by means of the following
lemma.

######  Lemma 2.3.

Let $C$ be an $R$-finite flat (hence projective) module and $D$ be an
$R$-projective module. Then:

  1. (1)

$f$ is injective iff $f^{\vee}:D^{\vee}\to C^{\vee}$ is dominant;

  2. (2)

$f$ is injective and pure iff $f^{\vee}:D^{\vee}\to C^{\vee}$ is surjective,

where $C^{\vee}:={\rm Hom}_{R}(C,R)$.

###### Proof.

Embedding $D$ as a direct summand into a free module does not change the
properties of $f$ and $f^{\vee}$, hence we can assume that $D$ is free. Since
$C$ is finite, there exists a finite direct summand of $D$ which contains the
image of $f$ and we can replace $D$ by this summand, that means we can assume
that $D$ is finite. The claims for finite projective modules are obvious. For
(1), it involves only the generic fibers. For (2), $f:C\to D$ is pure iff
$D/f(C)$ projective, and iff the sequence $0\to C\to D\to D/f(C)\to 0$ is
split exact, iff the sequence $0\to(D/f(C))^{\vee}\to D^{\vee}\to C^{\vee}\to
0$ is split exact. ∎

A subcomodule $N$ of a $C$-comodule $M$ said to be saturated if $M/N$ is
$R$-torsion free.

######  Proposition 2.4.

Let $C,D$ be $R$-projective coalgebras. Let $f:C\to D$ be a homomorphism of
$R$-coalgebras and $\omega:{\rm comod}_{\rm f}(C)\to{\rm comod}_{\rm f}(D)$ be
the induced functor. Assume that $C$ is $R$-finite. Then:

  1. (1)

The image of functor $\omega$ is closed under taking saturated subcomodules of
$R$\- torsion-free comodules iff $f$ is injective. In this case $\omega$ is
also closed under taking saturated subcomodules of any comodules and its
restriction to the subcategory of $R$-torsion-free comodules is full.

  2. (2)

The image of functor $\omega$ is closed under taking subcomodules of
$R$-torsion-free comodules iff $f$ is injective and pure. (In this case
$\omega$ is also obviously full.)

###### Proof.

Since $D$ is projective, the natural functor ${\rm comod}(D)\to{\rm
mod}(D^{\vee})$ is fully faithful, exact with image closed under taking
subobjects [3, 3.10]. Thus the functor, induced from $\omega$, ${\rm
mod}(D^{\vee})\to{\rm mod}(C^{\vee})$ is fully faithful, exact and has image
closed under taking subobjects iff $\omega$ is. The claim follows from
Proposition 2.2 and Lemma 2.3. ∎

We say that a flat $R$-coalgebra $C$ is locally finite if $C$ is the union of
its finite $R$-projective pure subcoalgebras $C_{\alpha}$, $\alpha\in A$. This
property is called IFP (ind-finite projective property) in [3]. As a corollary
of Proposition 2.4, we have

######  Corollary 2.5.

Let $C,D$ be projective $R$-coalgebras. Assume that $C$ has IFP, $C=\bigcup
C_{\alpha}$. Let $f:C\to D$ be a homomorphism of $R$-coalgebras. Then $f$ is
injective iff the induced functor $\omega:{\rm comod}_{\rm f}(C)\to{\rm
comod}_{f}(D)$ has image closed under taking subobjects. In particular, ${\rm
comod}_{\rm f}(C)$ is the union of its full subcategories ${\rm comod}_{\rm
f}(C_{\alpha})$, which are closed under taking subobjects.

Notice that there exist $R$-flat coalgebras which contains almost no finite
pure subcoalgebras, as shown in the examples below.

######  Example 2.6 ([6]).

(1) Assume that $R$ is a Dedekind ring with characteristic equal to 2 and
$\pi\neq 0$ is a non-unit in $R$. Consider the algebra $H:=R[T]/(\pi T^{2}+T)$
with the coalgebra structure given by

$\Delta(T)=T\otimes 1+1\otimes T.$

Then $H$ is torsion-free, hence flat over $R$. Let $M\subset H$ be a saturated
$R$-finite subcomodule of $H$. Then $M$ is free over $R$, more over its rank
is at most 2, as $H_{K}$ has dimension 2 over $K$. If $M$ has rank 2 then
$H/M$ is $R$-torsion, contradiction, hence $M$ has rank 1 over $R$. The
coaction of $H$ on $M$ is thus given by a group-like element in $H$. On the
other hand, the $R$-flatness of $H$ implies $H\otimes H$ is a submodule of
$H_{K}\otimes_{K}H_{K}$, hence the coproduct on $H$ is the restriction of the
coproduct on $H_{K}$:

$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\quad}$$\textstyle{H\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\qquad}$$\textstyle{H_{K}\otimes_{K}H_{K}.}$

Consequently, a group-like element of $H$ is a group-like element of $H_{K}$.
But in $H_{K}$ the unique group-like element is 1. Thus $M$ is a trivial
comodule of $H$.

(2) Similarly, assume that $R$ is a Dedekind ring, in which 2 is invertible.
Consider the algebra $H:=R[T]/(\pi T^{2}+2T)$ where $\pi\neq 0$ is a non-unit.
$H$ is a Hopf algebra with the coaction given by

$\Delta(T)=T\otimes 1+1\otimes T+\pi T\otimes T.$

Similar discussion shows that there are only two finite saturated subcomodules
of $H$, the one generated by 1 and the other generated by the group-like
element $1+\pi T$.

To treat a general coalgebra homomorphism $f:C\to D$ we shall imitate the
proof of Lemma 1.1. Our condition on $\omega$ will be some what stronger.

######  Proposition 2.7.

Let $C,D$ be $R$-flat coalgebras and $f:C\to D$ be a homomorphism of
$R$-coalgebras. Let $\omega:{\rm comod}(C)\to{\rm comod}(D)$ be the induced
functor on comodule categories. Then:

  1. (1)

The image of functor $\omega$ is closed under taking saturated subcomodules of
$R$-torsion-free comodules iff $f$ is injective. In this case $\omega$ is also
closed under taking saturated subcomodules of any comodules and its
restriction to the subcategory of $R$-torsion-free comodules is full.

  2. (2)

The image of functor $\omega$ is closed under taking subcomodules of any
comodules iff $f$ is injective and pure. In this case $\omega$ is full.

###### Proof.

(1) Assume that $\omega$ has the required property. Let $C_{0}:={\rm ker}(f)$.
Then $C/C_{0}\cong{\rm im}(f)\subset D$ is $R$-torsion-free, hence $C_{0}$ is
a saturated subcomodule of $C$, considered as $D$-comodules (the big left
square with curved vertical arrows below is commutative). By assumption, the
coaction of $D$ on $C_{0}$ lifts to a coaction of $C$. That is, there exists a
coaction $C_{0}\to C_{0}\otimes C$ (the dotted arrow below) making the
following diagram commutative:

$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{C_{0}\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\otimes f}$$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\otimes f}$$\textstyle{C_{0}\otimes
D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\otimes
D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes{\rm
id}}$$\textstyle{D\otimes D.}$

In particular, the coaction of $C$ on $C_{0}$ is the restriction of that on
$C$ (the upper-left square). That is, for any $c\in C_{0}$ we have a
representation

$\Delta(c)=\sum_{(c)}c_{(1)}\otimes c_{(2)},$

with $c_{(1)}\in C_{0}$. On the other hand, as $f$ is a coalgebra
homomorphism, we have $\varepsilon\circ f=\varepsilon$. Consequently,
$\varepsilon(C_{0})=0$. Applying $\varepsilon\otimes{\rm id}$ to the above
equation we get

$c=\sum_{(c)}\varepsilon(c_{(1)})\otimes c_{(2)}=0.$

A contradiction. Thus ${\rm ker}(f)=0$.

Conversely, assume that $f:C\to D$ is injective, then the map $f_{K}:C_{K}\to
D_{K}$ is also injective, as the base change $R\to K$ is flat. Hence,
according to 1.2, 1.3, $\omega_{K}$ is fully faithful and is closed under
taking subcomodules. Thus $\omega$ is full when restricted to $R$-torsion free
comodules.

Finally we show that the image of $\omega$ is closed under taking saturated
subcomodules of any comodules. For an $R$-module $M$, let $M^{\text{tor}}$
denote its torsion part, i.e. those elements of $M$ killed by some non-zero
element of $R$. Then we have

$M^{\text{tor}}={\rm ker}(M\to M\otimes K).$

Therefore, for any flat $R$-module $P$ we have

$(M^{\text{tor}}\otimes P)\cong(M\otimes P)^{\text{tor}}.$

Since any $R$-linear map preserves the torsion part, we conclude that, if $M$
is a $C$-comodule then $M^{\text{tor}}$ is a subcomodule. Let now $N\subset M$
be a subcomodule with respect to the action of $D$. If $M/N$ is $R$-torsion
free then $M^{\text{tor}}=N^{\text{tor}}$ and hence is stable under the
coaction of $C$. Hence we can consider the saturated inclusion
$N/N^{\text{tor}}\subset M/M^{\text{tor}}$, which by assumption shows that
$N/N^{\text{tor}}$ is stable under the coaction of $C$. As $C$ is flat, we
conclude that $N$ itself is stable under $C$.

(2) Assume that $\omega$ is closed under taking subcomodules of any comodules.
Then according to (1), $f$ is injective. Assume $f$ is not pure, then there
exists an ideal $I$ of $R$ such that the induced map $R/I\otimes C\to
R/I\otimes D$ is not injective. Let $C_{0}$ be the kernel of this map. Repeat
the argument of the proof of (1) we conclude that $C_{0}$ is stable under the
coaction of $D$ but not under the coaction of $C$, a contradiction. Thus $f$
has to be pure.

For the converse, assuming that $f:C\to D$ is injective and pure and $N\subset
M$ be $R$-modules, then we have the equality of submodules of $M\otimes D$:

$N\otimes D\cap M\otimes C=N\otimes C,$

where $C$ is considered as an $R$-submodule of $D$ by means of $f$. Hence, if
$M$ is a $C$-comodule and $N$ is a $D$-subcomodule of $M$, then, denote by
$\delta$ the coaction, we have

$\delta(N)\subset N\otimes D\cap M\otimes C=N\otimes C.$

That is, $N$ is stable under the coaction of $C$. ∎

######  Remark 2.8.

According to Serre [7, Prop. 2], any object in ${\rm comod}(C)$ is the union
of its $R$-finite subcomodules (but generally not saturated). I don’t know if
one can prove Propsition 2.7 with ${\rm comod}(C)$, ${\rm comod}(D)$ replaced
by ${\rm comod}_{f}(C)$, ${\rm comod}_{f}(D)$, respectively.

## Acknowledgment

The author would like to thank Dr. Nguyen Chu Gia Vuong and Nguyen Dai Duong
for stimulating discussions and VIASM for providing a very nice working space
and the financial support. A special thank goes to the anonymous referee for
his careful and throughout reading. His comments and remarks have greatly
improved the work.

## References

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  * [3] M. Hashimoto, Auslander-Buchweitz approximations of equivariant modules, London Mathematical Society Lecture Note Series, 282. Cambridge University Press, Cambridge (2000), xvi+281 pp.
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Piecewise linear potentials for false vacuum decay and negative modes

Wen-Yuan Ai,*††*<EMAIL_ADDRESS>Jean Alexandre††††
<EMAIL_ADDRESS>and Sarben Sarkar‡††‡<EMAIL_ADDRESS>

Theoretical Particle Physics and Cosmology, King’s College London,
Strand, London WC2R 2LS, UK

###### Abstract

We study bounce solutions and associated negative modes in the class of
piecewise linear triangular-shaped potentials that may be viewed as
approximations of smooth potentials. In these simple potentials, the bounce
solution and action can be obtained analytically for a general spacetime
dimension $D$. The eigenequations for the fluctuations around the bounce are
universal and have the form of a Schrödinger-like equation with delta-function
potentials. This Schrödinger equation is solved exactly for the negative modes
whose number is confirmed to be one. The latter result may justify the
usefulness of such piecewise linear potentials in the study of false vacuum
decay.

###### Contents

  1. 1 Introduction
  2. 2 Polygonal-potential models for false vacuum decay and classical bounces
    1. 2.1 Lee-Weinberg potential
    2. 2.2 Triangular potential
      1. 2.2.1 $D=1$
      2. 2.2.2 $D\geq 3$
  3. 3 Eigenequations and negative modes
    1. 3.1 Lee-Weinberg potential with $D\geq 3$
    2. 3.2 Triangular potential with $D\geq 3$
    3. 3.3 Triangular potential with $D=1$
  4. 4 Conclusion and discussion
  5. A $D=2$ for the triangular potential

##  1 Introduction

In the semiclassical analysis of false vacuum decay [1, 2, 3, 4, 5] based on
Euclidean path integrals, a special classical solution to the Euclidean
equation of motion, called the bounce, plays a key role. The decay rate
$\Gamma$ of the false vacuum generally takes the form of

$\Gamma=A\exp\left(-B/\hbar\right)\left(1+O\left(\hbar\right)\right)\,$ (1)

where $A$ and $B$ are potential-dependent constants. At the Gaussian level
(taking $\hbar=1$), Callan and Coleman derived the decay formula per unit
volume as [2]

$\displaystyle\frac{\Gamma}{V}=\left(\frac{B}{2\pi}\right)^{D/2}\left|\frac{\det^{\prime}[-\partial^{2}+U^{\prime\prime}(\phi_{b})]}{\det[-\partial^{2}+U^{\prime\prime}(\phi_{+})]}\right|^{-1/2}\,\mathrm{e}^{-B}\,,$
(2)

where $\phi_{+}$ and $\phi_{b}$ are the false vacuum and the classical bounce,
respectively, $B\equiv S[\phi_{b}]-S[\phi_{+}]$ is the normalized bounce
action (given in terms of the classical action $S$), and $D$ is the spacetime
dimension. In Eq. (2) $\det^{\prime}$ means that the zero eigenvalues are
omitted from the determinant and a prefactor $\sqrt{B/2\pi}$ is included for
each of the $D$ collective coordinates that are related to the zero modes
corresponding to spacetime translations [6].

The Callan-Coleman formula for the decay rate is derived from the imaginary
part of the false-vacuum energy through $\Gamma=-2{\rm Im}E_{0}$. For the
imaginary part of the ground-state energy to be nonvanishing,
$\det^{\prime}[-\partial^{2}+U^{\prime\prime}(\phi_{b})]$ needs to be
negative. The imaginary part ${\rm Im}E_{0}$ is then obtained through the so-
called “potential deformation” method [2].111See Refs. [7, 8] for a more
rigorous explanation using the Picard-Lefschetz theory [9, 10, 11, 12]. For
this derivation to be valid, the number of the negative eigenfunctions in the
spectrum of the fluctuation operator
$-\partial^{2}+U^{\prime\prime}(\phi_{b})$ must be an odd number, and
typically it is believed to be just one, although there is no proof of this in
field theory.

In the case of a finite set of coupled oscillators, Coleman gave an intuitive
argument that there is only one negative eigenvalue [13] in the absence of
gravitational effects.222In the presence of gravity there have been cases
where the spectrum for bounces has more than one negative eigenvalues [14,
15]. Although at a formal level a scalar field theory may be represented as
the limiting behaviour of an infinite number of coupled oscillators, it is not
a controlled limit. Hence the arguments of Coleman are not immediately
applicable in a field theoretic context and it is a part of our purpose to
investigate the validity of Coleman’s conjecture in field theory within the
class of piecewise linear potentials.

In field theory, the determination of the number of negative eigenvalues
cannot be performed rigorously for general potentials since the bounce
solutions are not known analytically, except in very restricted cases [16, 17,
18, 3, 19, 20, 21, 5] and for some of these the prefactor $A$ may be
analytically computed at the Gaussian level [22, 23]. In general, only
numerical bounce solutions can be obtained. This makes analysing the
eigenvalue equations for the fluctuations around the bounce difficult. In many
cases, the number of the negative modes can be numerically confirmed to be odd
by using the Gel’fand-Yaglom method [24] (see e.g. Refs. [25, 26]). Even with
only a numerical bounce solution, the eigenvalue problem (including the
computation of negative eigenvalues) can, in principle, be handled numerically
upon a discretization of the fluctuation operator [27].

For scalar field theories, one way of generating potentials, which have
explicit closed-form bounce-like solutions, is to consider potentials from
piecewise linear functions (also referred to as polygonal potentials in
general) [28, 29]. Clearly, any smooth potential graph can be suitably sampled
at a discrete set of points, which can be sequentially connected by straight
lines to give a polygonal graph. Some simple polygonal potentials were
proposed in Refs. [28, 30, 31, 29]. In this work, we focus on the triangular
potential [28] and the Lee-Weinberg potential [19]. The latter is unbounded
from below and could be used as a model for non-Hermitian
$\mathcal{PT}$-symmetric theory [32, 33]. Smooth, exact analytic bounce
solutions can be found for these potentials. In addition, there is a crucial
and simple feature of such polygonal potentials; $U^{\prime\prime}(\phi)$ is
nonvanishing only at some finite points in the field space, making the
eigenequation extremely simple and universal. Hence such piecewise linear
potentials allow an analytic study of the negative eigenmodes associated with
the bounce for arbitrary $D$.

The outline of the paper is as follows. In Sec. 2, we introduce the two types
of potentials mentioned above and derive the analytic bounce solutions and
actions for general spacetime dimensions. In Sec. 3 we study the
eigenequations for the fluctuation operator evaluated at the bounce, focusing
on the number of negative modes. We find that for all the cases that we study,
there is one and only one negative mode in the spectrum. This may justify
Coleman’s conjecture [13] in a field theory setup for any spacetime
dimensions. We conclude in Sec. 4.

##  2 Polygonal-potential models for false vacuum decay and classical bounces

###  2.1 Lee-Weinberg potential

We first consider a potential introduced in Ref. [19], which we refer to as
the Lee-Weinberg potential below, reading

$\displaystyle U(\phi)=\begin{cases}0\,,\qquad&{\rm for\ }|\phi|<v\,,\\\
-K\,(|\phi|-v)\,,\qquad&{\rm for\ }|\phi|>v\,.\end{cases}$ (3)

This gives, for $\phi\geq 0$,

$\displaystyle U^{\prime}(\phi)=-K\Theta(\phi-v)\,,\qquad{\rm and}\qquad
U^{\prime\prime}(\phi)=-K\delta(\phi-v)\,,$ (4)

where $\Theta(x)$ is the Heaviside step function. The potential is shown in
Fig. 1.

Figure 1: The Lee-Weinberg potential studied in Ref. [19].

Even though there is no potential barrier in the Lee-Weinberg potential, there
are still bounce solutions as long as $D\geq 3$. These bounce solutions
describe “tunnelling without barrier”. Mathematically, a bounce solution is
possible because of the “friction term” in the equation of motion (see below).
This is reminiscent of the famous Fubini-Lipatov instanton [16, 17] for the
$-\lambda\phi^{4}$ (with $\lambda>0$) in four-dimensional spacetime, which is
also used as a non-Hermitian $\mathcal{PT}$-symmetric model [34, 33]. Below,
we consider the tunnelling from $\phi=0$ to a point $\phi_{*}>v$ where the
exit point $\phi_{*}$ is determined by the solution. The bounce solution has
$O(D)$ symmetry and satisfy

$\displaystyle-\frac{\mathrm{d}^{2}\phi(r)}{\mathrm{d}r^{2}}-\frac{D-1}{r}\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}+U^{\prime}(\phi(r))=0\,,\quad\left.\frac{\mathrm{d}\phi}{\mathrm{d}r}\right|_{r=0}=0\,,\quad\phi(r\rightarrow\infty)=0\,,$
(5)

where $r=\sqrt{x^{2}_{1}+x^{2}_{2}+\ldots+x^{2}_{D}}$. The second term is the
“friction term” we referred to above. For $D=2$, the above boundary conditions
cannot be satisfied for the Lee-Weinberg potential and thus one obtains a
bounce solution only for $D\geq 3$. We will return to the case of $D=2$ when
we discuss the triangular potential.

Figure 2: The bounce solution given in Eq. (6) for $D=4$.

For $D\geq 3$ we find the solution as

$\displaystyle\phi_{b}(r)=\Theta(r_{1}-r)v\left[\frac{D}{2}-\frac{(D-2)}{2}\frac{r^{2}}{r_{1}^{2}}\right]+\Theta(r-r_{1})v\left(\frac{r_{1}}{r}\right)^{D-2}\,,$
(6)

where

$\displaystyle r_{1}=\left(\frac{D(D-2)v}{K}\right)^{\frac{1}{2}}\,.$ (7)

One can read the exit point $\phi_{*}=\phi_{b}(0)=vD/2$. The classical bounce
action333The action $S[\phi]$ is given by
$S[\phi]=\frac{2\pi^{D/2}}{\Gamma\left(\frac{D}{2}\right)}\int^{\infty}_{0}\mathrm{d}r\,r^{D-1}\left(\frac{1}{2}\left(\frac{\mathrm{d}\phi}{\mathrm{d}r}\right)^{2}+U\left(\phi\right)\right)$
with $\Gamma[x]$ being the gamma function. is obtained as

$\displaystyle
B=\frac{4\pi^{2}(D-2)v^{2}}{D+2}\left(\frac{D(D-2)v}{K}\right)^{\frac{D-2}{2}}\,.$
(8)

For $D=4$, we recover the results given in Ref. [19]. The functional
determinant is

$\displaystyle
A=\frac{\det^{\prime}\left[-\partial^{2}+U^{\prime\prime}(\phi_{b}(r))\right]}{\det[-\partial^{2}]}=\frac{\det^{\prime}\left[-\partial^{2}-\frac{D}{r_{1}}\delta(r-r_{1})\right]}{\det[-\partial^{2}]}=r_{1}^{2D}\,\frac{\det^{\prime}[-\partial^{2}-D\delta(r-1)]}{\det[-\partial^{2}]}\,,$
(9)

where in the second step we have done the rescaling: $x\rightarrow r_{1}x$,
$\partial_{x}\rightarrow\partial_{x}/r_{1}$. The factor of $r_{1}^{2D}$ is a
consequence of subtracting $D$ zero modes in the functional determinant in the
numerator, which causes a mismatch in the total number of eigenmodes from the
numerator and denominator. Taking $D=4$ as an example, we plot the bounce,
which is smooth, in Fig. 2.

###  2.2 Triangular potential

Figure 3: Triangular potential studied in Ref. [28]. We have taken
$\phi_{+}=0$, $V_{+}=0$.

Another simple model that can give exact bounce solutions is the triangular
potential [28], which (when the sharp bends in the potential are smoothed out)
serves as a common paradigm for false vacuum decay. We illustrate this
potential in Fig. 3. The transition from the false vacuum $\phi_{+}$ to the
true vacuum $\phi_{-}$ is sensitive only to the barrier part (thick lines in
Fig. 3) and therefore this model is characterized by the three pair of
parameters $\\{(\phi_{+},V_{+}),(\phi_{T},V_{T}),(\phi_{-},V_{-})\\}$. Without
loss of generality, we take $\phi_{+}=0$, $V_{+}=0$. The potential thus reads

$\displaystyle U(\phi)=\begin{cases}\lambda_{+}\phi\,,\qquad&{\rm for\
}\phi_{+}<\phi<\phi_{T}\,,\\\ V_{T}-\lambda_{-}(\phi-\phi_{T})\,,\qquad&{\rm
for\ }\phi_{T}<\phi<\phi_{-}\,,\end{cases}$ (10)

where

$\displaystyle\lambda_{+}=\frac{V_{T}}{\phi_{T}}>0\,,\qquad\lambda_{-}=\frac{V_{T}-V_{-}}{\phi_{-}-\phi_{T}}>0\,.$
(11)

From the potential, one has (for $0\leq\phi\leq\phi_{-}$)

$\displaystyle
U^{\prime}(\phi)=\lambda_{+}\Theta(\phi_{T}-\phi)-\lambda_{-}\Theta(\phi-\phi_{T})\,,\quad{\rm
and}\quad
U^{\prime\prime}(\phi)=-(\lambda_{+}+\lambda_{-})\delta(\phi-\phi_{T})\,.$
(12)

For future use, we define the parameter

$\displaystyle c\equiv\frac{\lambda_{-}}{\lambda_{+}}\,.$ (13)

Note that the Lee-Weinberg potential discussed in the last subsection can be
viewed as a special case of this triangular potential, $\lambda_{+}=0$,
$\lambda_{-}=K$, which gives $c\rightarrow\infty$. Although
$c\rightarrow\infty$ is a physical limit, $c=0$ is a singular limit and is not
physical because in that case, $\phi_{+}$ cannot be a false vacuum as we have
assumed.

For $D=1$, the bounce solution is a function of the imaginary time $\tau\equiv
x_{1}\in(-\infty,\infty)$ while for $D>1$, it is a function of the
$D$-dimensional Euclidean radius $r\in[0,\infty)$. Below, we discuss $D=1$ and
$D\geq 3$ cases separately, while leaving the case $D=2$ for the appendix.

####  2.2.1 $D=1$

For $D=1$, the bounce satisfies (taking the “center” of the bounce to be
$\tau=0$)

$\displaystyle-\frac{\mathrm{d}^{2}\phi(\tau)}{\mathrm{d}\tau^{2}}+U^{\prime}(\phi(\tau))=0\,,\quad\left.\frac{\mathrm{d}\phi(r)}{\mathrm{d}\tau}\right|_{\tau=0}=0\,,\quad\phi(\tau\rightarrow\pm\infty)=0\,.$
(14)

First, it is easy to check, by energy conservation in the Euclidean space,
that the turning point $\phi_{*}$ is given by

$\displaystyle\phi_{*}=\phi_{T}+\frac{V_{T}}{\lambda_{-}}\,.$ (15)

The bounce solution is found to be

$\displaystyle\phi_{b}(\tau)=\begin{cases}-\frac{1}{2}\lambda_{-}\tau^{2}+\phi_{*}\,,\qquad&{\rm
for\ }|\tau|<\tau_{1}\,,\\\
\frac{1}{2}\lambda_{+}(|\tau|-\tau_{1})^{2}-\sqrt{2V_{T}}|\tau|+\phi_{T}+\frac{2V_{T}}{\lambda_{-}}\,,\qquad&{\rm
for\ }\tau_{1}\leq|\tau|<T\,,\\\ 0\,,\qquad&{\rm for\ }|\tau|\geq
T\,,\end{cases}$ (16)

where

$\displaystyle\tau_{1}=\frac{\sqrt{2V_{T}}}{\lambda_{-}}\,,\qquad
T=\tau_{1}+\frac{\sqrt{2V_{T}}}{\lambda_{+}}=\frac{\sqrt{2V_{T}}}{\lambda_{-}}+\frac{\sqrt{2V_{T}}}{\lambda_{+}}\,.$
(17)

We show the bounce solution in Fig. 4. Note that in the bounce solution, the
field reaches the false vacuum $\phi_{+}=0$ at finite times $\pm T$. The
bounce action is found to be

$\displaystyle
B=\frac{4V_{T}\sqrt{2V_{T}}}{3}\left(\frac{1}{\lambda_{+}}+\frac{1}{\lambda_{-}}\right)\,.$
(18)

Figure 4: The bounce solution, Eq. (16), for the triangular potential (10).
The bounce solution reaches the false vacuum at finite time $\pm T$.

To analyse the eigenequations, it is sufficient for us to know the derivative
of $\phi_{b}(\tau)$ at $\pm\tau_{1}$,
$\phi_{b}(\tau)^{\prime}|_{\tau=\pm\tau_{1}}=\mp\sqrt{2V_{T}}$. Therefore, the
fluctuation operator at the bounce reads

$\displaystyle-\partial_{\tau}^{2}+U^{\prime\prime}(\phi_{b}(\tau))=-\partial_{\tau}^{2}-\frac{(\lambda_{+}+\lambda_{-})}{\sqrt{2V_{T}}}\left[\delta(\tau+\tau_{1})+\delta(\tau-\tau_{1})\right]+C\left[\delta(\tau+T)+\delta(\tau-T)\right]\,,$
(19)

where $C$ is a positive parameter. The last two Dirac delta functions are due
to the discontinuity of the potential at the false vacuum ($\phi_{+}=0$).
These delta functions effectively restrict the range of $\tau$ to a finite
region $[-T,T]$ and the specific form of $C$ is not needed for analysing the
eigenequations. Thus the relevant fluctuation operator becomes

$\displaystyle-\partial_{\tau}^{2}-\frac{(\lambda_{+}+\lambda_{-})}{\sqrt{2V_{T}}}\left[\delta(\tau+\tau_{1})+\delta(\tau-\tau_{1})\right]\qquad{\rm
with}\quad\tau\in[-T,T]\,.$ (20)

To simplify further, we can rescale $\tau\rightarrow\tau_{1}\tau$. Then the
relevant fluctuation operator becomes

$\displaystyle\frac{1}{\tau_{1}^{2}}\left(-\partial_{\tau}^{2}-\left(1+\frac{1}{c}\right)\left[\delta(\tau+1)+\delta(\tau-1)\right]\right)\qquad{\rm
with}\quad\tau\in[-T/\tau_{1},T/\tau_{1}]\,,$ (21)

where we recall $c=\lambda_{-}/\lambda_{+}$. The total factor $1/\tau_{1}^{2}$
does not affect the analysis of the number of negative modes.

####  2.2.2 $D\geq 3$

Now we consider $D\geq 3$. The general bounce solution that we find reads

$\displaystyle\phi_{b}(\tau)=\begin{cases}-\frac{\lambda_{-}}{2D}r^{2}+\phi_{*}\,,\qquad&{\rm
for\ }0\leq r<r_{1}\,,\\\
\frac{\lambda_{+}}{2D}r^{2}+\frac{\lambda_{+}R^{D}}{D(D-2)}r^{2-D}-\frac{\lambda_{+}R^{2}}{2(D-2)}\,,\qquad&{\rm
for\ }r_{1}\leq r<R\,,\\\ 0\,,\qquad&{\rm for\ }r\geq R\,,\end{cases}$ (22)

where

$\displaystyle
R=\left(\frac{2D(D-2)\phi_{T}(1+c)^{\frac{2}{D}}}{\lambda_{+}\left[D+2c-D(1+c)^{\frac{2}{D}}\right]}\right)^{\frac{1}{2}}\,,$
(23a)
$\displaystyle\phi_{*}=\phi_{T}+\frac{c(D-2)\phi_{T}}{D+2c-D(1+c)^{\frac{2}{D}}}\,,$
(23b) $\displaystyle
r_{1}=\left(\frac{2D(D-2)\phi_{T}}{\lambda_{+}\left[D+2c-D(1+c)^{\frac{2}{D}}\right]}\right)^{\frac{1}{2}}\,.$
(23c)

The bounce action is

$\displaystyle
B=\frac{4\pi^{\frac{D}{2}}}{\Gamma\left(\frac{D}{2}\right)}\frac{(1+c)\lambda_{+}\phi_{T}}{D(D+2)}r_{1}^{D}\,.$
(24)

For $D=4$, we recover the results given in Ref. [28]. Taking $D=4$ as an
example, we show the bounce solution in Fig. 5. Similar to the $D=1$ case, the
bounce solution $\phi_{b}(r)$ reaches the false vacuum at finite $R$. And the
delta-potentials at $r=\pm R$ in the fluctuation operator effectively restrict
$r$ to $[0,R]$. The fluctuation operator at the bounce, after rescaling
$r\rightarrow r_{1}r$, reads

$\displaystyle\frac{1}{r_{1}^{2}}\left[-\partial^{2}-D\left(1+\frac{1}{c}\right)\delta(r-1)\right]\qquad{\rm
with}\quad r\in[0,R/r_{1}],\quad D\geq 3\,.$ (25)

Figure 5: The bounce solution (22) for the triangular potential (10).
Independently of $D$, the bounce reaches the false vacuum at a finite radius
$R$ because we have a nonvanishing slope at (or more precisely, infinitely
close to) the false vacuum.

##  3 Eigenequations and negative modes

In this section, we study the eigenequation for the fluctuations about the
bounce. We do not aim to solve the full eigenspectrum but focus on the
negative eigenmodes. For the analytic bounce solutions to faithfully describe
false vacuum decay, we expect that there should be one and only one negative
mode.

###  3.1 Lee-Weinberg potential with $D\geq 3$

The eigenequation reads (after rescaling)

$\left[-\partial^{2}-D\delta\left(r-1\right)\right]\Psi\left(\mathbf{x}\right)=\lambda\Psi\left(\mathbf{x}\right)$
(26)

We have suppressed the label for the eigenfunctions since we are only
interested in the eigenvalues. To be specific, we are interested in negative
modes and we shall take $\lambda=-\kappa^{2}$ with $\kappa>0$ below.

We write $\Psi(\mathbf{x})=\sum_{l}\psi_{l}(r)Y_{l}(\mathbf{\theta})$ where
$\Delta_{S^{D-1}}Y_{l}(\theta)=l(2-D-l)Y_{l}(\theta)$ with $\Delta_{S^{D-1}}$
being the Laplace-Beltrami operator. The eigenspectrum for fixed $l$ is
degenerate with the degree of degeneracy for $D>2$ given by444When
analytically continuing this formula to noninteger $D$, ${\rm deg}(l;D)$ is
noninteger except for $l=0$ when ${\rm deg}(0;D)=1$.

$\displaystyle{\rm deg}(l;D)=\frac{(D+l-3)!(D+2l-2)}{l!(D-2)!}\,.$ (27)

For $D=2$, ${\rm deg}\left(l;2\right)=2-\delta_{l0}$. Note also that a
degeneracy of one only occurs for $l=0$ when $D\geq 2$. Consequently, the
number of negative eigenvalues can be one, if and only if the negative
eigenvalue occurs only for $l=0$.

Using the decomposition we arrive at

$\displaystyle\left[-\frac{1}{r^{D-1}}\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{D-1}\frac{\mathrm{d}}{\mathrm{d}r}\psi_{l}\right)+\frac{l\left(l+D-2\right)}{r^{2}}\psi_{l}-D\delta\left(r-1\right)\psi_{l}\right]=-\kappa_{l}^{2}\psi_{l}\,.$
(28)

The solution of this equation is more evident in terms of

$\displaystyle\Phi_{l}(r)=r^{(D-2)/2}\psi_{l}(r)$ (29)

together with a change of variable $\rho=\kappa_{l}r$. Then we get

$\displaystyle\frac{1}{\rho}\frac{\mathrm{d}}{\mathrm{d}\rho}\left(\rho\frac{\mathrm{d}\Phi_{l}(\rho)}{\mathrm{d}\rho}\right)-\frac{1}{\rho^{2}}\left[\rho^{2}+\nu_{l,D}^{2}\right]\Phi_{l}(\rho)=-\frac{D}{\kappa_{l}}\delta(\rho-\kappa_{l})\Phi_{l}(\rho)\,.$
(30)

where

$\displaystyle\nu_{l,D}^{2}=\left(l+\frac{D-2}{2}\right)^{2}\,.$ (31)

Requiring $\Phi_{l}(0)=\Phi_{l}(\infty)=0$, we obtain the solution

$\displaystyle\Phi_{l}(\rho)=\Theta(\kappa_{l}-\rho)c_{1}I_{\nu_{l,D}}(\rho)+\Theta(\rho-\kappa_{l})c_{2}K_{\nu_{l,D}}(\rho)\,.$
(32)

where $I_{\nu}$ and $K_{\nu}$ are modified Bessel functions of the first and
third kind. Note that although Eq. (30) is the same for $\rho<\kappa_{l}$ and
$\kappa_{l}<\rho$, the solutions are not the same because of the different
boundary conditions. The matching conditions at $\rho=\kappa_{l}$ are

$\displaystyle
c_{1}I_{\nu_{l,D}}(\kappa_{l})=c_{2}K_{\nu_{l,D}}(\kappa_{l})\,,$ (33a)
$\displaystyle
c_{2}K^{\prime}_{\nu_{l,D}}(\kappa_{l})-c_{1}I^{\prime}_{\nu_{l,D}}(\kappa_{l})=-\frac{D}{\kappa_{l}}c_{1}I_{\nu_{l,D}}(\kappa_{l})\,.$
(33b)

Substituting the first equation into the second gives the transcendental
equation

$\displaystyle g_{l,D}(\kappa_{l})=1\,,$ (34)

where

$g_{l,D}(\kappa_{l})\equiv\frac{\kappa_{l}}{D}\left(\frac{I^{\prime}_{\nu_{l,D}}(\kappa_{l})}{I_{\nu_{l,D}}(\kappa_{l})}-\frac{K^{\prime}_{\nu_{l,D}}(\kappa_{l})}{K_{\nu_{l,D}}(\kappa_{l})}\right).$
(35)

The above equation can be solved numerically, showing a monotonically
increasing behaviour of $g_{l,D}(x)$ with $x$. We plot $g_{l,D}(x)$ for
$l=0,1,2$ and $D=3,4$ in Fig. 6. It can be seen that there is a nonvanishing
solution only for $l=0$. Since ${\rm deg}(0;D)=1$, this means that there is
only one negative mode. From the curve for $l=1$, one can also see that there
are $D$ zero modes in the sector $l=1$ as ${\rm deg}(1;D)=D$. The numerical
results may be understood analytically on noting the asymptotic relations of
the modified Bessel functions [35] and one can analytically confirm that there
is only one solution for $l=0$ and $D$ vanishing solutions for $l=1$ and none
for higher $l\geq 2$ for arbitrary $D$. Hence the bounce solution found here
can be used to describe tunnelling in higher dimensional field theory as well.

Figure 6: The function $g_{l,D}(\kappa_{l})$ for the Lee-Weinberg potential
for $D=3$ (left panel) and $D=4$ (right panel). A solution is indicated when
the coloured curves intersect with the dashed line.

###  3.2 Triangular potential with $D\geq 3$

For the triangular potential, let us first consider the case $D\geq 3$ as this
case is similar to what has been discussed in the last subsection. The
eigenequation reads

$\displaystyle\left[-\partial^{2}-D\left(1+\frac{1}{c}\right)\delta(r-1)\right]\Psi(r)=-\kappa^{2}\Psi(r)$
(36)

where $r\in[0,\mathcal{R}]$ with $\mathcal{R}=(1+c)^{\frac{1}{D}}$. Following
the previous procedure, we arrive at

$\displaystyle\frac{1}{\rho}\frac{\mathrm{d}}{\mathrm{d}\rho}\left(\rho\frac{\mathrm{d}\Phi_{l}(\rho)}{\mathrm{d}\rho}\right)-\frac{1}{\rho^{2}}\left[\rho^{2}+\nu_{l,D}^{2}\right]\Phi_{l}(\rho)=-\frac{D(1+\frac{1}{c})}{\kappa_{l}}\delta(\rho-\kappa_{l})\Phi_{l}(\rho)\,.$
(37)

Requiring $\Phi_{l}(0)=0$, the general solution is

$\Phi(\rho)=\begin{cases}c_{1}I_{\nu_{l,D}}(\rho)\,,\quad&0<\rho<\kappa_{l}\,,\\\
c_{2}I_{\nu_{l,D}}(\rho)+c_{3}K_{\nu_{l,D}}(\rho)\,,\quad&\kappa_{l}<\rho<\kappa_{l}\mathcal{R}\,.\end{cases}$
(38)

The matching conditions at $\rho=\kappa_{l}$ are

$\displaystyle
c_{1}I_{\nu_{l,D}}(\kappa_{l})=c_{2}I_{\nu_{l,D}}(\kappa_{l})+c_{3}K_{\nu_{l,D}}(\kappa_{l})\,,$
(39a) $\displaystyle
c_{2}I^{\prime}_{\nu_{l,D}}(\kappa_{l})+c_{3}K^{\prime}_{{\nu_{l,D}}}(\kappa_{l})-c_{1}I^{\prime}_{{\nu_{l,D}}}(\kappa_{l})=-\frac{D\left(1+\frac{1}{c}\right)}{\kappa_{l}}c_{1}I_{\nu_{l,D}}(\kappa_{l})\,.$
(39b)

The Dirichlet boundary condition at $\rho=\kappa_{l}\mathcal{R}$ gives

$\displaystyle
c_{2}I_{\nu_{l,D}}(\kappa_{l}\mathcal{R})+c_{3}K_{{\nu_{l,D}}}(\kappa_{l}\mathcal{R})=0\,.$
(40)

Combining all these equations, we find

$\displaystyle
h_{l,D}(\kappa_{l})\equiv\left[1-\epsilon_{\nu_{l,D}}\frac{I_{\nu_{l,D}}\left(\kappa_{l}\right)}{K_{\nu_{l,D}}\left(\kappa_{l}\right)}\right]^{-1}\frac{\kappa_{l}}{D(1+\frac{1}{c})}\left(\frac{I^{\prime}_{\nu_{l,D}}\left(\kappa_{l}\right)}{I_{\nu_{l,D}}\left(\kappa_{l}\right)}-\frac{K^{\prime}_{\nu_{l,D}}\left(\kappa_{l}\right)}{K_{\nu_{l,D}}(\kappa_{l})}\right)=1\,,$
(41)

where

$\displaystyle\epsilon_{\nu_{l,D}}\equiv\frac{K_{\nu_{l,D}}\left(\kappa_{l}\mathcal{R}\right)}{I_{\nu_{l,D}}(\kappa_{l}\mathcal{R})}\,.$
(42)

Note that in the limit $c\rightarrow\infty$, one has
$\mathcal{R}\rightarrow\infty$, $\epsilon_{\nu_{l,D}}\rightarrow 0$ and thus
that Eq. (41) coincides with Eq. (34).

Again, Eq. (41) can be solved numerically. We find that there is one solution
only for $l=0$, which confirms that there is only one negative mode. In Fig.
7, taking $c=1$, we plot $h_{l,D}(\kappa_{l})$ for $l=0,1,2$, $D=3$ (left
panel) and $D=4$ (right panel). To see how the parameter $c$ impacts the
negative eigenvalue, we plot $h_{0,4}(\kappa_{0})$ as a function of
$\kappa_{0}$ and $c$ in Fig. 8. It can be seen that the negative eigenvalue
becomes smaller, i.e. $\kappa_{0}$ becomes larger (recall
$\lambda=-\kappa_{0}^{2}$), for smaller $c$.

Figure 7: The function $h_{l,D}(\kappa_{l})$ for the triangular potential for
$D=3$ (left panel) and $D=4$ (right panel) and $c=1$. A solution is indicated
when the coloured curves intersect the dashed line. Figure 8: The function
$h_{l,D}(\kappa_{l})$ as a function of $\kappa_{l}$ and $c$ for $l=0$ and
$D=4$. Eq. (41) is satisfied on the white thick line.

###  3.3 Triangular potential with $D=1$

In this case, we have eigenequation

$\displaystyle\left(-\partial_{\tau}^{2}-\left(1+\frac{1}{c}\right)\left[\delta(\tau+1)+\delta(\tau-1)\right]\right)\Psi(\tau)=-\kappa^{2}\Psi(\tau)\,,$
(43)

where $\tau\in(-\Upsilon,\Upsilon)$ with $\Upsilon=1+c$. The general solution
is

$\displaystyle\Psi(t)=\begin{cases}A_{1}\,\mathrm{e}^{\kappa\tau}+A_{2}\,\mathrm{e}^{-\kappa
t},\qquad&-\Upsilon<\tau<-1\,,\\\
G_{1}\,\mathrm{e}^{\kappa\tau}+G_{2}\,\mathrm{e}^{-\kappa\tau},\qquad&-1<\tau<1\,,\\\
F_{1}\,\mathrm{e}^{\kappa t}+F_{2}\,\mathrm{e}^{-\kappa
t},\qquad&-1<\tau<\Upsilon\,.\end{cases}$ (44)

The boundary conditions at $\tau=\pm\Upsilon$ gives

$\displaystyle
A_{1}\,\mathrm{e}^{-\kappa\Upsilon}+A_{2}\,\mathrm{e}^{\kappa\Upsilon}=0\,,$
(45a) $\displaystyle
F_{1}\,\mathrm{e}^{\kappa\Upsilon}+F_{2}\,\mathrm{e}^{-\kappa\Upsilon}=0\,.$
(45b)

The matching conditions at $\tau=-1$ read

$\displaystyle
G_{1}\,\mathrm{e}^{-\kappa}+G_{2}\,\mathrm{e}^{\kappa}=A_{1}\,\mathrm{e}^{-\kappa}+A_{2}\,\mathrm{e}^{\kappa}\,,$
(46a) $\displaystyle
G_{1}\kappa\,\mathrm{e}^{-\kappa}-G_{2}\kappa\,\mathrm{e}^{\kappa}=A_{1}\left[\kappa-\left(1+\frac{1}{c}\right)\right]\mathrm{e}^{-\kappa}-A_{2}\left[\kappa+\left(1+\frac{1}{c}\right)\right]\mathrm{e}^{\kappa}\,,$
(46b)

while at $\tau=1$ are

$\displaystyle
F_{1}\,\mathrm{e}^{\kappa}+F_{2}\,\mathrm{e}^{-\kappa}=G_{1}\,\mathrm{e}^{\kappa}+G_{2}\,\mathrm{e}^{-\kappa}\,,$
(47a) $\displaystyle
F_{1}\kappa\,\mathrm{e}^{\kappa}-F_{2}\kappa\,\mathrm{e}^{-\kappa}=G_{1}\left[\kappa-\left(1+\frac{1}{c}\right)\right]\mathrm{e}^{\kappa}-G_{2}\left[\kappa+\left(1+\frac{1}{c}\right)\right]\mathrm{e}^{-\kappa}\,.$
(47b)

Combing all the above equations, we obtain

$\displaystyle f(\kappa)\equiv
f_{0}(\kappa)+f_{1}(\kappa)\epsilon(\kappa)+f_{2}(\kappa)\epsilon(\kappa)^{2}=0$
(48)

where $\epsilon=\exp(-2\kappa\Upsilon)$ and

$\displaystyle f_{0}$
$\displaystyle=-\alpha^{2}+\mathrm{e}^{4\kappa}\left(2\kappa-\alpha\right)^{2}\,,$
(49a) $\displaystyle f_{1}$
$\displaystyle=2\alpha\,\mathrm{e}^{2\kappa}\left[\alpha+2\kappa+\left(2\kappa-\alpha\right)\mathrm{e}^{4\kappa}\right]\,,$
(49b) $\displaystyle f_{2}$
$\displaystyle=\mathrm{e}^{8\kappa}\alpha^{2}-\mathrm{e}^{4\kappa}\left(\alpha+2\kappa\right)^{2}\,,$
(49c)

with $\alpha\equiv 1+1/c>1$. Note that $\epsilon$ is a function of $\kappa$.

We solve Eq. (48) numerically and find that there is one and only one positive
solution for $c\in(0,\infty)$. We plot $f(\kappa)$ in Fig. 9 for relatively
smaller values of $c$ (left panel), and bigger values of $c$ (right panel),
respectively. It can be seen that one has a larger positive root $\kappa$
(which gives a smaller negative eigenvalue $-\kappa^{2}$) for a smaller value
of $c$, in agreement with the higher-dimensional cases. The value of the
eigenvalue approaches a limit quickly when $c$ increases.

Figure 9: $f(\kappa)$ defined in Eq. (48) for small values of $c$ (left panel)
and bigger values of $c$ (right panel). The different curves intersect the
axis $f(\kappa)=0$ only once away from $\kappa=0$.

##  4 Conclusion and discussion

In this paper, we have discussed the piecewise linear potentials, specifically
the Lee-Weinberg potential introduced in Ref. [19] and the triangular-shaped
potential introduced in Ref. [28], as simplified false vacuum decay models. We
calculate the bounce solutions for general spacetime dimension $D$ and for
$D=4$ results in the literature are recovered. A condition for such analytic
bounce solutions to describe tunnelling is that there should be an odd number
of negative modes. We show that there is one and only one negative mode in the
spectrum of the fluctuation operator at the bounce for arbitrary $D$. This is
achieved by explicitly solving the eigenequation for the negative modes. An
advantage of such piecewise linear potentials is that the eigenequations take
a very simple Schrödinger-like equation with delta potentials in a universal
form.

Piecewise linear potentials considered here could have several applications.
For example, the triangular potential was used in studying metastable
supersymmetric minima in Refs. [36, 37], dark energy [38] and gravitational
wave production from cosmological first-order phase transitions [39]. The
unbounded-from-below Lee-Weinberg potential is similar to those that appear in
theories with $\mathcal{PT}$ symmetry [40, 41, 42, 43]. Potentials unbounded
from below lead to instability of the vacuum in the conventional Hermitian
framework. In the presence of $\mathcal{PT}$ symmetry, there is a “phase” [44]
without the instability, known as the $\mathcal{PT}$-symmetric phase [45]. For
$D=1$, using a WKB treatment in a quantum mechanical framework, it is known
that the eigenvalues for the Hamiltonian are real [34, 32]. It is believed
that higher dimensional generalisations of this result also hold. A way to
investigate this for $D>1$ is in terms of Euclidean path integrals and
fluctuations around bounce-like solutions [33] through the approach here. When
extending the present analysis to polygonal potentials with more segments, the
analysis may also justify the validity of using the latter for efficient
computation of bubble nucleation rates [29] (cf. other recent works [27, 46]).

## Acknowledgments

The authors of this work are supported by the Engineering and Physical
Sciences Research Council (grant No. EP/V002821/1). JA is also supported by
the Leverhulme Trust (grant No. RPG-2021-299) and by the Science and
Technology Facilities Council (grant No.STFC-ST/X000753/1).

##  Appendix A $D=2$ for the triangular potential

Unlike the case of the Lee-Weinberg potential, a bounce solution for the
triangular potential for $D=2$ exists. The bounce solution is found to be555We
have assumed that $\phi_{*}\leq\phi_{-}$, which should impose a condition for
the parameters via Eq. (52).

$\displaystyle\phi_{b}(r)=\begin{cases}-\frac{\lambda_{-}r^{2}}{4}+\phi_{*}\,,\qquad&{\rm
for\ }0\leq r<r_{1}\,,\\\
\frac{\lambda_{+}r^{2}}{4}-2\left[\left(\phi_{*}-\phi_{T}\right)\left(1+\frac{1}{c}\right)\right]\log\left(\frac{r}{R}\right)-\left[\left(\phi_{*}-\phi_{T}\right)\left(1+\frac{1}{c}\right)\right]\,,&{\rm
for\ }r_{1}\leq r<R\\\ 0\,,\qquad&{\rm for\ }r\geq R\,,\end{cases}$ (50)

where

$\displaystyle r_{1}=2\sqrt{\frac{\phi_{*}-\phi_{T}}{\lambda_{-}}}\,,\quad
R=2\sqrt{\frac{\left(\phi_{*}-\phi_{T}\right)\left(1+\frac{1}{c}\right)}{\lambda_{+}}}\,,$
(51)

and $\phi_{*}$ is determined implicitly by the matching conditions at
$r=r_{1}$, leading to

$\displaystyle\left(1+\frac{\lambda_{+}}{\lambda_{-}}\right)\log\left(\frac{4(\phi_{*}-\phi_{T})}{\lambda_{-}}\right)+\frac{\phi_{*}}{\phi_{*}-\phi_{T}}=4\left(1+\frac{\lambda_{+}}{\lambda_{-}}\right)\sqrt{(\phi_{*}-\phi_{T})\left(\frac{1}{\lambda_{+}}+\frac{1}{\lambda_{-}}\right)}\,.$
(52)

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