# On the minimal genus problem in four-manifolds

András I. Stipsicz  Rényi Institute of Mathematics
H-1053 Budapest
Reáltanoda utca 13–15, Hungary<EMAIL_ADDRESS>and  Zoltán Szabó
Department of Mathematics
Princeton University
Princeton, NJ, 08544<EMAIL_ADDRESS>
###### Abstract.

We study the minimal genus problem for some smooth four-manifolds.

##  1\. Introduction

For a closed, oriented, smooth four-manifold $X$ the _genus function_ (or
minimal genus function) $g_{X}\colon H_{2}(X;\mathbb{Z})\to\mathbb{N}$ assigns
to each homology class the minimal genus of an oriented smooth surface
representative of the class; that is, if $\alpha\in H_{2}(X;\mathbb{Z})$ then

$g_{X}(\alpha)=\min\\{g(\Sigma)\mid i\colon\Sigma\hookrightarrow X,\
i_{*}([\Sigma])=\alpha\\},$

where $\Sigma$ is an oriented, closed, connected surface of genus $g(\Sigma)$
and $i$ is a $C^{\infty}$ embedding. It is a standard fact, that every second
homology class in a four-manifold can be represented by an embedded, oriented,
closed, connected surface, hence the above minimum provides a finite value.

For some examples of exotic pairs of smooth four-manifolds the genus functions
show very different behavior, so it is expected that this function codes
important geometric information about the smooth structure of the four-
manifold. In order to formalize this notion we have the following definition:

######  Definition 1.1.

Let $X$ and $Y$ be smooth, closed, oriented four-manifolds with
$H_{1}(X;\mathbb{Z})=H_{1}(Y;\mathbb{Z})=0$. We say that $X$ and $Y$ have
_equivalent minimal genus functions_ if there exists an isomorphism $f\colon
H_{2}(X;\mathbb{Z})\longrightarrow H_{2}(Y;\mathbb{Z})$ with the properties
that

  * •

$f$ preserves the intersection form:

$Q_{X}(a,b)=Q_{Y}(f(a),f(b))$

for $a,b\in H_{2}(X;\mathbb{Z})$, and

  * •

$f$ also preserves the minimal genus function:

$g_{X}(a)=g_{Y}(f(a))$

for all $a\in H_{2}(X;\mathbb{Z})$.

A natural question regarding the above function is the following:

######  Question 1.2.

Let $X$ and $Y$ be smooth, closed, oriented and simply connected four-
manifolds and assume that $X$ and $Y$ have equivalent minimal genus functions.
Is there an orientation preserving diffeomorphism between $X$ and $Y$?

######  Remark 1.3.

Note that the above question only asks about the existence of an orientation
preserving diffeomorphism $G\colon X\to Y$, and in particular does not require
that the induced action $G_{\ast}\colon H_{2}(X)\to H_{2}(Y)$ agrees with a
fixed $f$. Indeed, for $X=Y=K3$ the map $f\colon H_{2}(K3)\to H_{2}(K3)$ given
by $f(a)=-a$ preserves both the intersection form and the minimal genus
function, but (by work of Donaldson, see [5, Corollary 9.1.4]), $f$ cannot be
induced by a diffeomorphism.

Despite its importance, very little is known in general about $g_{X}$. There
are only a handful of simply connected examples where the entire genus
function is known. When $X=S^{4}$ (or more generally, $X$ is a homotopy
$4$-sphere) the genus function is trivial; in fact, the positive answer for
Question 1.2 in this special case would imply the Smooth $4$-dimensional
Poincare Conjecture (S4PC).

For ${\mathbb{C}}{\mathbb{P}}^{2}$ (and similarly, for
${\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$) the resolution of the Thom
conjecture by Kronheimer and Mrowka [13] provides the answer: suppose that
$h\in H_{2}({\mathbb{C}}{\mathbb{P}}^{2};\mathbb{Z})\cong\mathbb{Z}$ is a
generator and $d\neq 0$, then

$g_{{\mathbb{C}}{\mathbb{P}}^{2}}(dh)=\frac{1}{2}(|d|-1)(|d|-2).$

Along similar lines, Ruberman gave a description of the minimal genus function
for
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
and for $S^{2}\times S^{2}$ (the two $S^{2}$-bundles over $S^{2}$) in [20]: if
$(a,b)\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})\cong\mathbb{Z}^{2}$
(where the $\mathbb{Z}$-factors are generated by the projective lines in the
factors) is a nontrivial homology class with $|a|>|b|$, then

$g_{{{\mathbb{C}}{\mathbb{P}}}^{2}\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}}(a,b)=\frac{1}{2}(|a|-1)(|a|-2)-\frac{1}{2}|b|(|b|-1);$

if $|a|<|b|$ then the same formula applies with the roles of $a$ and $b$
reversed, and
$g_{{{\mathbb{C}}{\mathbb{P}}}^{2}\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}}(a,\pm
a)=0$. In a similar manner, for $(u,v)\in H_{2}(S^{2}\times
S^{2};\mathbb{Z})\cong\mathbb{Z}^{2}$ (where the $\mathbb{Z}$ factors are
generated by the $S^{2}$-factors in $S^{2}\times S^{2}$) we have

$g_{S^{2}\times S^{2}}(u,v)=(|u|-1)(|v|-1)$

if $uv\neq 0$ and $g_{S^{2}\times S^{2}}(u,v)=0$ if $uv=0$.

Although these four-manifolds seem to be the simplest ones (definitely with
smallest Euler characteristics among simply connected closed four-manifolds),
it is not known if they support distinct smooth structures. Application of a
suitable sequence of Luttinger surgeries provide (not necessarily simply
connected) smooth four-manifolds $X_{n}$ ($n\in\mathbb{N}$) with homology
isomorphic to $H_{*}(S^{2}\times S^{2};\mathbb{Z})$, first constructed in [7]
by Fintushel, Park and Stern (see also [2]). In addition, the construction can
be carried out so that $X_{1}$ carries a symplectic structure. We will review
the construction of these manifolds in Section 2 and prove the following:

######  Theorem 1.4.

Suppose that $X_{n}$ is one of the homology $S^{2}\times S^{2}$’s constructed
in [7] (see also Section 2). Using the isomorphism
$H_{2}(X_{n};\mathbb{Z})=\mathbb{Z}^{2}$, where the two generators $x$ and $y$
satisfy $Q_{X_{n}}(x,x)=Q_{X_{n}}(y,y)=0$, $Q_{X_{n}}(x,y)=1$, the minimal
genus on the non-zero homology class $\alpha=ax+by$ is given by

$g_{X_{n}}(\alpha)=(|a|+1)(|b|+1).$

In particular, for all $n,k\in\mathbb{N}$ the manifolds $X_{n}$ and $X_{k}$
have equivalent minimal genus functions.

Notice that (as Proposition 2.2 will show), the Seiberg-Witten values do
distinguish these four-manifolds; it is still an open problem whether this
family contains simply connected examples. It is also open if there is a pair
$n,k\in\mathbb{N}$ so that $X_{n}$ and $X_{k}$ are homeomorphic — it might
happen that their fundamental groups distinguish them.

Regarding Question 1.2, as partial evidence for the opposite answer, we show
that the minimal genus function does distinguish (even infinite collections
of) exotic smooth four-manifolds. For such example we will rely on two
constructions on elliptic surfaces. First, recall that for given $n\geq 1$,
$p,q>0$, $(p,q)=1$, the manifolds $E(n)_{p,q}$ are constructed from a simply
connected genus $1$ Lefschetz fibration $E(n)$ over the sphere with Euler
characteristics $\chi(E(n))=12n$ using logarithmic transformations of
multiplicity $p$ and $q$. (For more on elliptic surfaces see [11, Chapter 3].)
For $q=1$ the manifold $E(n)_{p,q}$ will be denoted by $E(n)_{p}$.

A further construction of new four-manifolds out of old one is knot surgery
[8]: suppose that $X$ contains an embedded torus $T$ with trivial normal
bundle and with simply connected complement. For a knot $K\subset S^{3}$
replace the tubular neighbourhood $\nu(T)=T^{2}\times D^{2}$ of $T$ with
$(S^{3}\setminus\nu(K))\times S^{1}$ (using an appropriate gluing map along
the $T^{3}$ boundaries, where the meridian of the torus $T$ will be matched
with the longitude of the knot $K\subset\partial((S^{3}\setminus\nu(K))\times
S^{1})$). We denote the result by $X_{K}$. For an elliptic surface $E(n)$ we
can choose $T$ to be the generic fiber of the fibration; the result of the
above construction then will be denoted by $E(n)_{K}$. We will prove the
following

######  Theorem 1.5.

For every $n\geq 2$ there are infinitely many smooth four-manifolds all
homeomorphic to $E(n)$, distinguished by their genus function. Indeed, if
$K_{m}$ denotes the $(2,2m+1)$ torus knot $T_{2,2m+1}$, then
$\\{E(n)_{K_{m}}|m\in\mathbb{N}\\}$ have all distinct genus functions.

In the closing sections of the paper we investigate the minimal genus problem
for some simply connected four-manifolds with small Euler characteristics.
These include smooth four-manifolds that are homeomorphic to
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
or
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$,
constructed by Akhmedov and Park [1] and by Baldridge and Kirk [4],
respectively. For surveys on the genus function, see also [6, 17].

###  1.1. Tools used in the proofs

In determining the value of the genus function on a homology class, the
Symplectic Thom Conjecture (a theorem proved in [19]) turns out to be rather
convenient.

######  Theorem 1.6.

(Ozsváth-Szabó, [19]) Suppose that $(X,\omega)$ is a closed symplectic four-
manifold and $\Sigma\subset X$ is a connected, closed symplectic surface of
genus $g(\Sigma)$. Then $g_{X}([\Sigma])=g(\Sigma)$, i.e. symplectic surfaces
minimize genus in their homology classes.

The following inequality — called the _adjunction inequality_ — does not
determine the genus function on a specific homology class, but provides bounds
and is applicable in a greater generality. Recall that the Seiberg-Witten
function of a smooth, closed, oriented, simply connected four-manifold $X$ is
a map $SW_{X}\colon{\rm Spin}^{c}(X)\to\mathbb{Z}$ which counts solutions of a
certain geometric partial differential equation. (Here ${\rm Spin}^{c}(X)$
denoted the set of spinc structures on $X$.) For manifolds with $b_{2}^{+}>1$
(or $b_{2}^{+}=1$ and $b_{2}^{-}\leq 9$ and with the addition that the
equations are considered with small perturbation terms) this map is a smooth
invariant. $SW_{X}$ has finite support, and $\mathbf{s}\in{\rm Spin}^{c}(X)$
with $SW_{X}(\mathbf{s})\neq 0$ is called a Seiberg-Witten basic class. The
four-manifold $X$ is of Seiberg-Witten simple type if for a Seiberg-Witten
basic class $\mathbf{s}$ we have that
$c_{1}^{2}(\mathbf{s})=3\sigma(X)+2\chi(X)$ (where $\sigma(X)$ and $\chi(X)$
are the signature and the Euler characteristic of the underlying four-
manifold).

######  Theorem 1.7.

(Adjunction inequality, [19, Corollary 1.7]) Suppose that $X$ is a smooth
four-manifold of Seiberg-Witten simple type with $b_{1}(X)=0$ and either with
$b_{2}^{+}(X)>1$ or with $b_{2}^{+}(X)=1$ and $b_{2}^{-}(X)\leq 9$. Then for
an embedded surface $\Sigma\subset X$ with genus $g(\Sigma)>0$ and a Seiberg-
Witten basic class $\mathbf{s}\in{\rm Spin}^{c}(X)$ (i.e. with
$SW_{X}(\mathbf{s})\neq 0$) we have

(1.1) $|\langle c_{1}(\mathbf{s}),[\Sigma]\rangle|+[\Sigma]^{2}\leq
2g(\Sigma)-2.$

In our arguments we will frequently use the fact that for a self-
diffeomorphism $f\colon X\to X$ we have $g_{X}(\alpha)=g_{X}(f_{*}(\alpha))$
for all $\alpha\in H_{2}(X;\mathbb{Z})$. Furthermore, if $S\subset X$ is an
embedded sphere of self-intersection $(\pm 1)$ or $(\pm 2)$ in $X$, then it is
easy to construct a self-diffeomorphism $f\colon X\to X$, for which the
induced map $f_{*}$ is just reflection in the class represented by $S$.
Consequently, if $S\subset X$ is an embedded sphere with self-intersection
$(\pm 1)$ or $(\pm 2)$, then the map

$\alpha\mapsto\alpha-2\frac{[S]\cdot\alpha}{[S]\cdot[S]}[S]$

on $H_{2}(X;\mathbb{Z})$ is induced by a diffeomorphism.

The paper is organized as follows. In Section 2 we examine the homology
$S^{2}\times S^{2}$’s and verify Theorem 1.4. In Section 3 we study the genus
function of elliptic surfaces and of the manifolds we get by logarithmic
transformations or by knot surgeries on them. We also prove Theorem 1.5.
Section 4 is devoted to the partial understanding of the genus function on the
twice and three-times blown up complex projective plane. The exotic manifolds
$AP$ and $BK$ (and the further infinite families of exotic smooth structures
on
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
and on
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$),
and their genus functions are studied in Sections 5 and 6.

Acknowledgements: AS was partially supported by the _Élvonal (Frontier) grant_
KKP126683 of the NKFIH. He would also acknowledge the hospitality of
MSRI/SLMath in Berkeley, where part of this project was carried out. The
second author was partially supported by NSF grant DMS-1904628. We would like
to thank the referee for their suggestions which significantly improved the
presentation of the results in the paper.

##  2\. Homology $S^{2}\times S^{2}$’s

The application of torus (and in particular, Luttinger) surgery, or some
variation of it, leads to the construction of four-manifolds which have the
same homology as $S^{2}\times S^{2}$, but potentially different fundamental
group. Such examples were constructed in [7] (see also [2]).

First we briefly recall the concept of surgery along a torus in a four-
manifold from [7]. Assume that $X$ is a smooth four-manifold and $T\subset X$
is an oriented torus with trivial normal bundle (that is, $[T]^{2}=0$). Fix a
framing $f$ on $T$, i.e., a trivialization of its normal bundle. For an
oriented simple closed curve $\gamma\subset T$ therefore there is a unique
push-off ${\overline{\gamma}}_{f}$ of $\gamma$ to the boundary $T^{3}$ of a
small tubular neighbourhood of $T$. For a rational number $\frac{p}{q}$, the
manifold $(X,T,f,\gamma,\frac{p}{q})$ is defined by deleting a small tubular
neighbourhood of $T$, and gluing back $T^{2}\times D^{2}$ (which can be viewed
as the union of a four-dimensional 2-handle, two 3-handles and a 4-handle) by
attaching the 2-handle along a simple closed curve representing the homology
class $p[\mu_{T}]\pm q[{\overline{\gamma}}_{f}]$ (where $\mu_{T}$ is a
meridional curve to $T$, regarded as a subset of the boundary of the trivial
neighbourhood, oriented so that it has linking number 1 with $T$).

In the special case, when $(X,\omega)$ is a symplectic four-manifold and
$T\subset(X,\omega)$ is a Lagrangian torus (that is, $\omega|_{T}=0$), the
torus $T$ comes with a canonical framing. Indeed, by the Lagrangian
neighbourhood theorem an appropriate small normal neighbourhood of $T$ is
symplectomorphic to a neighbourhood of the zero-section of the cotangent
bundle $T^{*}T^{2}\to T^{2}$ of the 2-torus $T^{2}$ (when the total space of
this bundle is equipped with the tautological symplectic form coming from its
Liouville 1-form). As $T^{*}T^{2}$ is trivialized, this identification equips
$T\subset X$ with its _Lagrangian framing_. (Pushing $T$ along this framing we
get further Lagrangian tori in $(X,\omega)$.) As is explained in [3], using
this framing and applying $\frac{1}{k}$ surgery along $T$ (for any choice of
$\gamma$) the resulting four-manifold $(X,T,\gamma,\frac{1}{k})$ will admit a
symplectic structure which is equal to $\omega$ outside of a suitably chosen
neighbourhood of $T$. This symplectic construction is frequently referred to
as _Luttinger surgery_ [3, 14].

###  2.1. The construction

For the construction of the homology $S^{2}\times S^{2}$’s in [7, Section 4],
consider two genus 2 surfaces $\Sigma^{1}_{2},\Sigma_{2}^{2}$, and equip their
first integral homologies with the symplectic bases $a_{1},b_{1},a_{2},b_{2}$
and $c_{1},d_{1},c_{2},d_{2}$, respectively. Represent these homology classes
by oriented simple closed curves (denoted by the same letters). Equip the
direct product $\Sigma^{1}_{2}\times\Sigma^{2}_{2}$ with the direct product
symplectic form. Notice that in this symplectic four-manifold the product of
an oriented simple closed curve $\gamma_{1}$ in $\Sigma_{2}^{1}$ and
$\gamma_{2}$ in $\Sigma_{2}^{2}$ provides an oriented Lagrangian torus
$T_{\gamma_{1},\gamma_{2}}\subset X$, hence using the chosen bases we see 16
tori which are in 8 hyperbolic pairs. In addition, by considering small
disjoint displacements of the curves in $\Sigma^{i}_{2}$, tori with algebraic
intersection number 0 can be chosen to be disjoint. Apply now the following
eight Luttinger surgeries, all with the choice $k=\pm 1$:

  * •

$(T_{a_{1},c_{1}},a_{1}),(T_{b_{1},c_{1}},b_{1}),(T_{a_{2},c_{2}},a_{2}),(T_{b_{2},c_{2}},b_{2})$,

  * •

$(T_{a_{2},c_{1}},c_{1}),(T_{a_{2},d_{1}},d_{1}),(T_{a_{1},c_{2}},c_{2}),(T_{a_{1},d_{2}},d_{2})$;

the first four with coefficient $k=-1$ and the last four with $k=1$, resulting
in the four-manifold $X$. As each torus comes with a dual torus, the
meridional curves for each are homologically trivial. Therefore after the
Luttinger surgery the chosen curve will represent the trivial element in first
homology, hence $H_{1}(X;\mathbb{Z})=0$. By [3, Proposition 2.2] $X$ will
admit a symplectic structure so that the surfaces
$\Sigma_{2}^{1}\times\\{p\\},\\{q\\}\times\Sigma_{2}^{2}$ are symplectic
submanifolds. The intersection form (given in the basis
$\Sigma_{2}^{1}\times\\{p\\},\\{q\\}\times\Sigma_{2}^{2}$) is a hyperbolic
pair, hence the manifold is spin and its cohomology ring is isomorphic to
$H^{*}(S^{2}\times S^{2};\mathbb{Z})$. By changing the last Luttinger surgery
(on $(T_{a_{1},d_{2}},d_{2})$) to a torus surgery with coefficient $n$ we get
a four-manifold $X_{n}$, which for $n>1$ does not carry a natural symplectic
structure.

We claim that the four-manifold $-X$ (the manifold $X$ constructed above, with
its orientation reversed) also admits a symplectic structure. Indeed, consider
the map
$f\colon\Sigma_{2}^{1}\times\Sigma_{2}^{2}\to\Sigma_{2}^{1}\times\Sigma_{2}^{2}$
which is given by the identity on the first factor and by the orientation
reversing diffeomorphism on the second factor given by the reflection which
leaves $c_{1},c_{2}$ fixed, and sends $d_{1}$ and $d_{2}$ to $-d_{1}$ and
$-d_{2}$. The map $f$ is an orientation reversing self-diffeomorphism of
$\Sigma_{2}^{1}\times\Sigma_{2}^{2}$, and all tori in $X$ we used in the
Luttinger surgeries above are invariant under $f$: the tori not involving
$d_{1}$ and $d_{2}$ are fixed pointwise, while the tori $T_{a_{2},d_{1}}$ and
$T_{a_{1},d_{2}}$ stay the same as subsets. Let $W$ denote the complement of
the tubular neighborhood of the eight tori in $X$ along which we perform
Luttinger surgeries, and note that the above map $f$ restricted to $W$ gives a
map form $W$ to $-W$. Let $\delta_{i}$ for $1\leq i\leq 8$ denote the curves
in the boundary of $W$ that define the Luttinger surgeries on $X$. Then using
surgeries along $f(\delta_{i})$ in $-W$ will give $-X$. Since the map $f$
carries the meridians of the tori to meridians (with potentially reversing
their orientation) it follows that the surgeries on $-W$ are still Luttinger
surgeries, and $f$ extends to those surgeries (as an orientation reversing
map). In particular $-X$ can be presented as a sequence of Luttinger surgeries
along Lagrangian tori in $\Sigma_{2}^{1}\times\Sigma_{2}^{2}$, therefore it is
also symplectic.

######  Remark 2.1.

Note that the above argument shows that the manifold $X$ admits a symplectic
structure with both orientations, yet it is not diffeomorphic to any surface
bundle. Such examples were already known to exist.

In the same manner, it follows that $-X_{n}$ is given from the product of two
genus-2 surfaces by a sequence of seven Luttinger surgeries and a torus
surgery with some integral coefficient, similarly to the construction of
$X_{n}$.

The property of admitting symplectic structure with both orientation allows us
to compute the genus function $g_{X}$ of $X$: note that the Luttinger
surgeries are in the complement of the two genus 2 surfaces
$h_{1}=\Sigma_{2}^{1}\times\\{p\\}$ and $h_{2}=\\{q\\}\times\Sigma_{2}^{2}$ of
$\Sigma_{2}^{1}\times\Sigma_{2}^{2}$ (for suitably chosen $p\in\Sigma_{2}^{2}$
and $q\in\Sigma_{2}^{1}$), hence provide a basis of $H_{2}(X;\mathbb{Z})$ with
symplectic representatives.

###### Proof of Theorem 1.4 for $X=X_{1}$.

The two generators $h_{1},h_{2}$ can be represented by symplectic genus 2
surfaces. If for $\alpha=ah_{1}+bh_{2}$ both $a$ and $b$ are positive, then by
smoothing the intersection points of push-offs, we get a symplectic surface of
the claimed genus. If both $a$ and $b$ are negative, we can take the negative
of the homology class. If one coefficient is positive and the other is
negative, then by reversing the orientation on the four-manifold we are in one
of the previous cases. (Notice that here we use the fact that $-X$ is
constructed in a similar fashion; in particular, it is symplectic.) Finally if
one of the coefficients is zero, we represent the homology class near the
other surface with an appropriate cover, and choosing it within a small
neighbourhood, the cover can be assumed to be symplectic. Then simple Euler
characteristic computation verifies the claimed formula. ∎

By determining the Seiberg-Witten basic classes, the same result can be shown
to hold for all $X_{n}$.

######  Proposition 2.2.

The smooth four-manifold $X_{n}$ admits two Seiberg-Witten basic classes, and
so does $-X_{n}$ obtained by reversing the orientation on $X_{n}$.

###### Proof.

The argument uses repeatedly a surgery formula for the Seiberg-Witten
invariants, see [16, 22].

For the version we need, we fix a smooth closed oriented compact four-manifold
$M$ with three-torus boundary $\partial M=T^{3}$, and a spinc structure s
whose first Chern class restricts trivially to $\partial M$. Fix two homology
classes $a,b\in H_{1}(T^{3})$ given by $a=[S^{1}\times pt\times pt]$ and
$b=[pt\times S^{1}\times pt]$. Given a pair of integers $p,q$, where
$(p,q)\in\mathbb{Z}\times\mathbb{Z}$ is a primitive element, let $M_{p,q}$
denote the effect of torus surgery, where we glue $M$ and $D^{2}\times T^{2}$
along $\phi$ to get $M_{p,q}=M\cup_{\phi}D^{2}\times T^{2}$, so that in
homology $\phi_{\ast}$ maps $[\partial(D^{2})]$ to $pa+qb$. Let $S(p,q)$
denote the set of spinc structures on $M_{p,q}$ whose restriction to $M$
agrees with ${\bf s}$ and define $F(p,q)$ as the sum of the Seiberg-Witten
invariants of classes in $S(p,q)$. Then, according to [16]

(2.1) $F(p,q)=p\cdot F(1,0)+q\cdot F(0,1).$

The manifolds $X_{n}$ in the theorem are all constructed from
$\Sigma_{2}\times\Sigma_{2}$ by using torus surgery eight times. Let $s_{0}$
denote the spinc structure on $\Sigma_{2}\times\Sigma_{2}$, with
$c_{1}(s_{0})$ evaluating as $2$ on both $[\Sigma_{2}\times pt]$ and
$[pt\times\Sigma_{2}]$. Note that the Seiberg-Witten invariant of $s_{0}$ is
$\pm 1$. Now we use Formula (2.1) inductively. During these surgeries we use
$a$ to be the meridional circle. So by choosing $b$ appropriately we can
assume the effect of the torus surgery is given by $M(p,1)$, where $p=\pm 1$
in the first seven steps and $p=\pm n$ in the last step. The crucial
observation is that in all these steps the manifolds $M(0,1)$ contain an
embedded torus with self-intersection $0$ with the property that the first
Chern class of $s$ evaluates as 2 on it. It follows from the adjunction
inequality that $F(0,1)=0$, so

$F(p,1)=p\cdot F(1,0).$

Since all the $8$ surgeries are along tori that represent primitive second
homology classes that are killed under the surgery, it follows that there is a
unique spinc structure in $S(p,1)$. This means that if we look at the nine
different 4-manifolds $Y_{i}$, $i=0,..,8$ starting with
$\Sigma_{2}\times\Sigma_{2}$ and ending in $X_{n}$, there is always a unique
spinc structure $s_{i}$ that corresponds to $s_{0}$. During the inductive step
we see that $Y_{i}=M(1,0)$ and $Y_{i+1}=M(p,1)$. It follows that the Seiberg-
Witten invariant of $s_{8}$ in $X_{n}$ is equal to $\pm n$.

We proved that $s_{8}$ and $-s_{8}$ are two non-trivial Seiberg-Witten classes
on $X_{n}$. Since the surgery tori are all disjoint from $pt\times\Sigma_{2}$
and $\Sigma_{2}\times pt$, it follows from the adjunction inequality that
there are no other Seiberg-Witten basic classes. ∎

###### Proof of Theorem 1.4 for $X_{n}$.

The argument of the proof for $X=X_{1}$ applies almost verbatim: we consider
the same surfaces constructed in that proof, and apply the adjunction
inequality of Theorem 1.7 (providing lower bounds for values of the genus
function) to verify that the constructions indeed provide surfaces minimizing
genus. As the Seiberg-Witten calculations apply for $X_{n}$ and $-X_{n}$ as
well, minimality of the genus of the constructed representatives follows from
the respective adjunction formulae. ∎

##  3\. Elliptic surfaces

There are various ways to get invariants for an oriented, closed, connected
four-dimensional manifold $X$ from $g_{X}$ which are easier to compare than
the genus function itself. Below we consider one such example. Let $Q_{X}$
denote the intersection form of $X$ and $k=b_{2}^{+}(X)$.

######  Definition 3.1.

A $k$ element sequence ${\bf y}=(y_{1},\ldots,y_{k})$ with $y_{i}\in
H_{2}(X,{\bf Z})$ is called an _allowed sequence_ if the span of
$y_{1},\ldots,y_{k}$ is a positive definite $k$-dimensional subspace of
$H_{2}(X,{\bf Z})$, and for all $i\neq j$ we have $Q_{X}(y_{i},y_{j})=0$ and
$Q(y_{i},y_{i})$ is even.

For an allowed sequence we associate a tuple of $k$ integers

$Pr(X,{\bf y})=(a_{1},\ldots,a_{k})$

where $a_{i}=g_{X}(y_{i})$.

######  Definition 3.2.

The _genus profile_ of $X$ is defined as the smallest among all the
corresponding tuples of $k$ (with $k=b_{2}^{+}(X)$) integers in the
lexicographical ordering:

$Pr(X)=min_{\bf y}Pr(X,{\bf y}).$

As an example, the genus profile of the $K3$ surface is $(2,2,2)$. A further
example is given by $X=E(2)_{p}$ with $p$ odd, the $K3$ surface after a
multiplicity $p$ logarithmic transformation has been applied. The minimal
genus profile of $X$ will be $(2,2,f(p))$, since we can find two genus $2$
surfaces that are disjoint from each other and from the fiber of the fibration
and have self-intersection $2$. The function $f(p)$ can be estimated as
follows: There is a Seiberg-Witten basic class of the form $(p-1)\cdot z$
where $z\in H^{2}(X)$ has square zero and $z\neq 0$. Let ${\bf
y}=(y_{1},y_{2},y_{3})$ be an allowed sequence for which

$Pr(X)=Pr(X,{\bf y}).$

Using the adjunction inequalities we see that $z\cdot y_{1}=z\cdot y_{2}=0$,
implying $z\cdot y_{3}\neq 0$. Let $g$ denote the minimal genus of $y_{3}$.
Using the the adjunction inequality we have

$2g-2\geq Q_{X}(y_{3},y_{3})+p-1\geq p+1.$

It follows that the minimal genus is at least $(p+3)/2$. As $f(p)$ goes to
infinity as $p\to\infty$, this argument already verifies that there are
infinitely many $E(2)_{p}$’s with distinct genus functions. A similar, more
explicit computation of the genus profile provides the proof of Theorem 1.5:

###### Proof of Theorem 1.5.

Consider $E(n)_{K_{m}}$, the result of knot surgery along the generic fiber of
the elliptic fibration on $E(n)$ (a torus with trivial normal bundle), with
$K_{m}=T_{2,2m+1}$ the alternating torus knot with crossing number $2m+1$
($m\geq 1$). As the knot $K_{m}$ admits a Seifert surface of genus $m$, there
is a surface in $E(n)_{K_{m}}$ of genus $m$ intersecting the generic fiber
once and with self-intersection $-n$: we get this surface by gluing the
section of $E(n)$ with the Seifert surface of $K_{m}$. By the computation of
the Seiberg-Witten invariants of $E(n)_{K_{m}}$ in [8], the adjunction
inequality implies that this surface is of minimal genus in its homology
class. The same holds for the surface we get by adding
$k=\lceil\frac{n}{2}\rceil$ copies of the fiber, so the result will have genus
$m+k$ and positive self-intersection. This then shows that
$Pr(E(n)_{K_{m}})=(2,\ldots,2,k+m)$ (with $2n-2$ copies of $2$), verifying the
result. ∎

##  4\. The genus function for
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
and
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$

The genera of homology classes in
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
have been studied in [10] and [18], see also [23]. Indeed, the genera of
classes in
$H_{2}({\mathbb{C}}{\mathbb{P}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
with non-negative self-intersection can be determined by a simple algorithm.
To explain this algorithm, we introduce the following conventions. A homology
class $\alpha\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
will be denoted by a triple $(a,b_{1},b_{2})$, reflecting the decomposition

$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})=H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2};\mathbb{Z})\oplus
H_{2}({\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})\oplus
H_{2}({\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})\cong\mathbb{Z}^{3}.$

In
$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
the classes with one coordinate $\pm 1$ and the others 0 can be obviously
represented by spheres. Therefore we can assume that in computing
$g_{{{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}}$,
we have $a,b_{1},b_{2}\geq 0$: otherwise we apply a reflection to the
coordinate in question (which is generated by a self-diffeomorphism) to
reverse its sign. In the same way, by applying a diffeomorphism interchanging
the two ${\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$ components if necessary,
we can assume $b_{1}\geq b_{2}$.

Genera of classes with $\alpha^{2}\geq 0$. Suppose that $x=(a,b_{1},b_{2})$
has $x^{2}=a^{2}-b_{1}^{2}-b_{2}^{2}\geq 0$, and assume that
$a,b_{1},b_{2}\geq 0$. If $a\geq b_{1}+b_{2}$, then the homology class can be
represented by a symplectic surface. Indeed, take $b_{1}$ lines in
${\mathbb{C}}{\mathbb{P}}^{2}$ passing through a given point $P_{1}$, take
$b_{2}$ lines passing through another point $P_{2}$ ($P_{1}\neq P_{2}$), and
take $a-b_{1}-b_{2}$ further generic lines, all disjoint from $P_{1}$ and
$P_{2}$. Blowing up $P_{1}$ and $P_{2}$ we get a complex curve, which (after
smoothing the double points in a symplectic way) provides a smooth symplectic
surface representing the homology class in question. By the Symplectic Thom
Conjecture (Theorem 1.6) this minimizes genus in its homology class. The genus
of this curve is easy to determine: for such a triple $g(a,b_{1},b_{2})$ is
equal to

(4.1) $\frac{1}{2}((a-1)(a-2)-b_{1}(b_{1}-2)-b_{2}(b_{2}-1)).$

The following idea, determining the genus function for all classes with non-
negative self-intersection, already appeared in [10, 23]: Consider a class
$\alpha=(a,b_{1},b_{2})$ and let $c$ denote $a-b_{1}-b_{2}$. For $c\geq 0$ the
genus of $\alpha$ has been described above. Suppose now that $c<0$. Consider
the sphere representing the homology class $(1,1,1)$ (which has self-
intersection $(-1)$) we get by connecting the projective lines in the
projective plane factors, and apply a diffeomorphism $f$ inducing reflection
to this sphere. The induced map $f_{*}$ takes $(a,b_{1},b_{2})$ to
$(a+2c,b_{1}+2c,b_{2}+2c)$. By applying further diffeomorphisms we can assume
that these integers are non-negative, and (as $c<0$) it is easy to see that
$|a+2c|<a$. Therefore after finitely many steps of the above algorithm we get
a vector $\alpha^{\prime}=(a^{\prime},b_{1}^{\prime},b_{2}^{\prime})$ with
$c^{\prime}=a^{\prime}-b_{1}^{\prime}-b_{2}^{\prime}\geq 0$ and with
$g(\alpha)=g(\alpha^{\prime})$. For $\alpha^{\prime}$ the formula of Equation
(4.1) above applies and determines the value of the genus function.

If $\alpha^{2}=0$ (i.e., when $a^{2}-b_{1}^{2}-b_{2}^{2}=0$, that is,
$(a,b_{1},b_{2})$ is a Pythagorean triple), the assumption
$a^{2}-b_{1}^{2}-b_{2}^{2}=0$ together with $a-b_{1}-b_{2}\geq 0$ implies that
one of $b_{1}$ or $b_{2}$ is equal to zero, hence the above algorithm stops at
a vector of the form $(a,a,0)$ (or $(a,0,a)$). These classes can be
represented by complex curves which consist of $a$ copies of
${\mathbb{C}}{\mathbb{P}}^{1}$. By smoothly connect summing them, we get

######  Corollary 4.1.

If $\alpha\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
has $\alpha^{2}=0$ then $\alpha$ can be represented by a sphere (i.e.
$g(\alpha)=0$).

Classes with $\alpha^{2}<0$. For classes of negative self-intersection we do
not have a complete picture. There are a number of cases when one can find a
sphere representing the class, while in some cases we have a lower bound for
the genus. This latter case emerges mainly for characteristic elements.

A similar algorithm as before applies for classes of negative square and
(after applying diffeomorphisms) we need to examine only those triples
$(a,b_{1},b_{2})$ which satisfy $2a\geq b_{1}+b_{2}$. Indeed, by taking the
sphere of self-intersection 2 representing the class $(2,1,1)$, we can apply
reflection to a class with $d=2a-b_{1}-b_{2}$ negative (and further
diffeomorphisms to regain $a,b_{1},b_{2}\geq 0$), and this process will stop
at a representative of the diffeomorphism orbit satisfying $2a\geq
b_{1}+b_{2}$. This information is, however, not sufficient to determine the
genus function on the homology class. Indeed, in the above form (as $b_{1}\geq
b_{2}$) we can assume that $a\geq b_{2}$, so the second blow-up can be
performed in a point of ${\mathbb{C}}{\mathbb{P}}^{2}$ where $b_{2}$ lines
pass through, but for the rest of the surface representing the class
$(a,b_{1},b_{2})$ we need to apply ad hoc methods, which sometimes give better
results for those representatives of the diffeomorphism orbit of the class
which fail to satisfy $2a\geq b_{1}+b_{2}$.

There are simple examples of homology classes which can be represented by an
embedded sphere; for example:

######  Lemma 4.2.

Consider the homology classes $(a,a+1,c)$ with $c=1,2$. These classes can be
represented by embedded spheres.

###### Proof.

The class $(a,a+1)\in
H_{2}({\mathbb{C}}{\mathbb{P}}^{2}\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
can be represented by a sphere. Connect summing with the sphere representing
$c=1,2$ we find the sphere representative of the class. ∎

Lower bounds for the genus function for homology classes with negative self-
intersection are scarce; there are some bounds for characteristic homology
classes, though, which we present below. Note that the class
$\alpha=(a,b_{1},b_{2})\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
is characteristic if and only if the coordinates $a,b_{1},b_{2}$ are all odd.
Suppose now that the characteristic homology class $\alpha$ can be represented
by an embedded sphere. Consider the $(|\alpha^{2}|-1)$-fold antiblow-up (i.e.
connected sum with ${\mathbb{C}}{\mathbb{P}}^{2}$) of
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$,
where the (still characteristic) homology class $(a,b_{1},b_{2},1,\ldots,1)$
can be now represented by an embedded sphere of self-intersection $(-1)$. We
can blow that sphere down, and as the class was characteristic, its complement
(and hence the result of the blow-down) is a spin four-manifold. Applying
Furuta’s $\frac{10}{8}$-theorem [9] giving
$b_{2}(X)\geq\frac{5}{4}|\sigma(X)|+2$ and the divisibility of $\sigma(X)$ by
16 to this spin four-manifold, we get

######  Lemma 4.3.

Suppose that $\alpha=(a,b_{1},b_{2})\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
is a characteristic homology element with negative self-intersection
$\alpha^{2}=-n$ ($n\in\mathbb{N}$). If $\alpha$ can be represented by an
embedded sphere then $n$ is $1$. ∎

A more general lower bound of the genus function on characteristic elements
can be derived as follows. Suppose that $\alpha$ is a characteristic element
in
$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
with $\alpha^{2}=-8n-1$. (Recall that the square of a characteristic element
is congruent to the signature of the ambient four-manifold mod 8.) We claim
that

######  Proposition 4.4.

The smooth representative of a characteristic element $\alpha\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
with $\alpha^{2}=-8n-1$ has genus at least $\frac{n}{2}$.

###### Proof.

As the characteristic element represents the Poincare dual of the second
Stiefel-Whitney class $w_{2}$ of the ambient manifold
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$,
the complement of the normal disk bundle admits a spin structure.

Suppose now that the smooth representative of $\alpha$ is of genus
$g<\frac{n}{2}$. Consider the connected sum of $8n+1$ copies of
${\mathbb{C}}{\mathbb{P}}^{2}$ when $n$ is even, and $8n-7$ copies if $n$ is
odd. Below we concentrate on the case of even $n$: choose in every summand the
projective line, connect sum them, and add $g$ fake handles, resulting in a
surface $S$ in $\\#_{8n+1}{\mathbb{C}}{\mathbb{P}}^{2}$ of genus $g$ and self-
intersection $8n+1$, which represents a characteristic class (hence its
complement is spin).

In constructing the normal connected sum of the two four-manifolds along these
surfaces we can choose the gluing map so that the result is a spin manifold
$Z$. Indeed, the spin structures on a genus $g$ surface come in two types
depending on their Arf invariants (see the discussion in [12, page 103]), and
two spin structures are diffeomorphic iff they have equal Arf invariants. The
spin three-manifold $Y$ we get as the boundary of the complement of the open
neighborhood of the characteristic surface is a circle bundle over a surface
(of odd Euler number). Deleting a small tubular neighbourhood of a fiber, we
get a trivial bundle over a surface with boundary. On this surface the exact
same classification of spin structures holds as for the closed case, implying
that on the circle bundle there are two orbits of spin structures under the
diffeomorphism group. The two structures can be given either as above, or
considering the surface in the $(8n-7)$-fold connected sum of
${\mathbb{C}}{\mathbb{P}}^{2}$, but choosing the cubic curve in one of the
components as opposed to the projective line. As gluing with this latter spin
structure we would get a four-manifold violating Rokhlin’s theorem about
signatures of closed spin manifolds, it follows that the two spin structures
on the boundaries in our construction are diffeomorphic, hence (by choosing
the right gluing map) provide a spin structure on $Z$.

As $S$ has simply connected complement, the resulting manifold will have
$b_{1}(Z)=0$, hence its signature $\sigma(Z)=8n$ and its second Betti number
$b_{2}(Z)=\chi(Z)-2=8n+4g+2$ satisfies the $\frac{10}{8}$-theorem of Furuta
[9], implying

$b_{2}(Z)\geq\frac{5}{4}\sigma(Z)+2,$

eventually providing $g\geq\frac{n}{2}$. (An almost identical argument gives
the bound when $n$ is odd.) ∎

An upper bound for the genus can be given by the following simple algorithm.
First transform the given homology class to $(a,b_{1},b_{2})$ with all
nonnegative and $b_{1}\geq b_{2}$, and in addition $2a\geq b_{1}+b_{2}$. In
particular, we can assume that $a\geq b_{2}$.

As $a-b_{2}<b_{1}$ (which follows from $a^{2}-b_{1}^{2}-b_{2}^{2}<0$) and
$a\geq b_{2}$ for $\alpha=(a,b_{1},b_{2})$ with $\alpha^{2}<0$ and $2a\geq
b_{1}+b_{2}$, we can consider $b_{2}$ lines passing through a given point
$P_{1}$ and the remaining $a-b_{2}$ through another point $P_{2}$. Blowing up
the two points, we get a symplectic representative of the class
$(a,a-b_{2},b_{2})$. Addig $b_{1}+b_{2}-a$ copies of the first exceptional
divisor to it, we get the right homology class, while by resolving the
negative intersection points the result will not necessarily be symplectic
anymore. The genus of the resulting surface gives an upper bound for the
minimal genus $g(\alpha)$.

###  4.1. The case of
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$

Most of the above arguments provide (after some minor modifications) results
for the genus function of
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$.
As before, for a class $\alpha=(a,b_{1},b_{2},b_{3})\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
with $\alpha^{2}\geq 0$ there is a diffeomorphism under which the coordinates
of the image satisfy $a\geq b_{1}+b_{2}+b_{3}$ (and all non-negative). In this
step we use the $(-2)$-sphere representing the homology class $(1,1,1,1)\in
H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$.
Under the inequality $a\geq b_{1}+b_{2}+b_{3}$ the homology class can be
represented by a symplectic submanifold: as before, consider three distinct
points $P_{1},P_{2},P_{3}$ in ${\mathbb{C}}{\mathbb{P}}^{2}$, $b_{i}$ lines
through $P_{i}$ (and $a-b_{1}-b_{2}-b_{3}$ in generic position, avoiding all
the $P_{i}$) and blow up $P_{i}$. After symplectically smoothing the double
points of the resulting complex curve, we get the desired symplectic
submanifold. This procedure again gives an algorithm, rather than a formula to
determine the genus function
$g_{{{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}}$.
Once again, if $\alpha=(a,b_{1},b_{2},b_{3})$ satisfies $b_{1}\geq b_{2}\geq
b_{3}$ (which can be assumed without loss of generality) and $\alpha^{2}=0$,
then after some diffeomorphism $\alpha=(a,a,0,0)$, and this class can be
represented by a smoothly embedded sphere. Hence

######  Corollary 4.5.

All classes $\alpha\in
H_{2}(H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
with $\alpha^{2}=0$ have
$g_{{{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}}(\alpha)=0$.

As in the case of
${\mathbb{C}}{\mathbb{P}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$,
we were not able to determine the genus function on all elements with negative
self-intersection in
$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$.
The argument for finding a bound earlier for the genera of characteristic
elements applies in this case as well, providing some constraint, but in
general such results are not sufficient for getting exact values of the genus
function.

##  5\. The Akhmedov-Park exotic
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
manifold $AP$

In [1] an infinite family of pairwise non-diffeomorphic smooth four-manifolds
has been constructed, each homeomorphic to
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$.
One of these examples, which will be denoted by $AP$, was shown to carry a
symplectic structure; we recall the definition of these manifolds below. The
goal of this section is to partially determine the genus function of these
exotic manifolds.

The study of $AP$ (and indeed, all the exotic copies discussed below) is
different from
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
in two crucial aspects. First, we do not see the spheres which then provide
the diffeomorphisms inducing reflections in homology. (Indeed, we suspect that
$AP$ contains no homologically nontrivial sphere at all, cf. Conjecture 5.4.)
The other aspect, however, provides more tools to study the genus function:
the proof of exoticness shows that the symplectic form $[\omega]$ and its
first Chern class $c_{1}(AP,\omega)$ pair negatively, hence $AP$ has Kodaira
dimension $\kappa=2$ (a ’symplectic four-manifold of general type’). This
implies that the adjunction inequality holds true for $AP$: As this theorem
holds for negative self-intersections as well, we find more restrictions in
$AP$ for classes of negative square. In a similar manner, the further exotic
smooth manifolds admit nontrivial Seiberg-Witten invariants, hence the
adjunction inequality also provides bounds on the genera of homology classes
(no matter what square they have).

###  5.1. The construction of the manifold $AP$

The symplectic exotic example is given as a symplectic normal connected sum of
two symplectic manifolds along symplectic genus 2 surfaces, followed by a
sequence of appropriately chosen Luttinger surgeries. Indeed, consider
$S_{0}^{3}(T)$, the result of 0-surgery along the trefoil knot $T\subset
S^{3}$, and let $X_{1}=S^{3}_{0}(T)\times S^{1}$. This is a symplectic four-
manifold, as it admits a $T^{2}$-fibration structure over $T^{2}$ (with a
section) — resting on the fact that $T$ is a fibered knot of genus 1. Let
$t_{1}$ be an embedded torus representing the fiber and $t_{2}$ representing a
section of this bundle. Indeed, the homology class $2[t_{1}]$ also admits a
torus representative (by braiding the generic fiber), which we denote by
$t_{1}^{\prime}$, and all of these tori can be chosen to be symplectic. As
$t_{1}^{\prime}$ and $t_{2}$ intersect in two points, by blowing up one of
them and resolving the other, we get a genus 2 symplectic surface
$\Sigma\subset X$ with self-intersection 0 in $X=(S^{3}_{0}(T)\times
S^{1})\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$. Take the fiber connected
sum of $(X,\Sigma)$ with $(\Sigma_{2}\times T^{2},\Sigma_{2}\times\\{p\\})$
for the genus 2 surface $\Sigma_{2}$ and for some point $p$ in the 2-torus
$T^{2}$. The resulting manifold $Y$ will be a symplectic manifold. We can
identify a few interesting submanifolds in $Y$: for example, the symplectic
genus 2 surface $\Sigma$ along which the normal sum was taken (after pushed
off) provides a symplectic genus 2 surface in $Y$ with self-intersection 0,
which we will denote by $A$. In a similar manner, two further surfaces $B$ and
$C$ can be constructed as follows: take a torus in $\Sigma_{2}\times T^{2}$
intersecting the genus 2 surface we will use in the symplectic normal sum
transversely in a single point, and glue to a push-off of either $t_{1}$ or
$t_{2}$. (Notice that $t_{1}$ intersects $\Sigma$ once, so after the gluing we
get a genus 2 surface $B$, while $t_{2}$ intersects $\Sigma$ in two points, so
in the construction we need to use two copies of the torus in
$\Sigma_{2}\times T^{2}$, and the resulting surface $C$ will have genus 3.) A
fourth surface $D$ can be constructed by taking the exceptional divisor of the
blow-up in $(S^{3}_{0}(T)\times
S^{1})\\#{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$. This sphere is punctured
twice in the symplectic normal sum procedure, and gluing it to punctured tori
on both sides we get a genus 2 surface $D$. All the above surfaces can be
constructed to be symplectic.

The main result of Akhmedov and Park in [1] is that by applying Luttinger
surgeries in $Y$ we can get a simply connected four-manifold. (As Luttinger
surgery does not change signature and Euler characteristic, the result will be
necessarily homeomorphic to the two-fold blow-up of
${\mathbb{C}}{\mathbb{P}}^{2}$.)

######  Theorem 5.1.

([1]) There are four tori in $Y$ (originating from $\Sigma\times T^{2}$) along
which Luttinger surgeries (with appropriately chosen coefficients) provide a
simply connected four-manifold, and the symplectic structure of $Y$ extends to
the resulting four-manifold $AP$.

######  Remark 5.2.

In [1] the coefficients are carefully described, and the fundamental group
calculation is given in details.

The Luttinger surgeries performed in $\Sigma_{2}\times T^{2}$ are disjoint
from the surfaces introduced above in $Y$, hence $A,B,C,D$ give rise to
surfaces (also denoted by the same letters) in $AP$. It is not hard to see
that if $c_{1}$ denotes the first Chern class of the symplectic structure on
$AP$, then

  * •

$g(A)=g(B)=g(D)=2$, $g(C)=3$.

  * •

$c_{1}(A)=c_{1}(B)=2$, $c_{1}(C)=4$, $c_{1}(D)=3$.

  * •

$A\cdot A=B\cdot B=C\cdot C=0$, $D\cdot D=-1$.

  * •

$B\cdot D=C\cdot D=0$

  * •

$A\cdot B=1$, $A\cdot C=2$, $B\cdot C=1$, $A\cdot D=2$.

Indeed, $H_{2}(AP;\mathbb{Z})$ is generated by $B,C,D$, and we have a relation
$A=2B+C-2D$. As these surfaces are symplectic, Theorem 1.6 implies that the
above genera are indeed minimal. This observation then shows

######  Theorem 5.3.

[1] The four-manifold $AP$ is not diffeomorphic to
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$.

###### Proof.

The genus function is a diffeomorphism invariant, and it vanishes on the
subset of
$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
consisting of elements of zero square, while in $AP$ there are symplectic
submanifolds of square zero and nonzero genus (any of $A,B,C$ above will do).
∎

Indeed, we can determine the function $g_{AP}$ for a large class of elements.
Let $\alpha=t_{A}\cdot A+t_{B}\cdot B+t_{C}\cdot C$, where all $t_{i}\geq 0$
(and at least one of them is non-zero). This class can be represented by a
symplectic surface, and its genus can be determined by a simple (although not
very enlightening) formula. Indeed, the multiple of a class can be represented
by a cover of the 0-section in its (trivial) normal bundle, which can be
chosen to be symplectic. The sum of the terms then can be represented by the
union of these potential covers, and the positive intersections can be
symplectically resolved.

The lower bound provided by Equation (1.1) of the adjunction inequality cannot
provide sharp results in general — indeed, when the self-intersection of a
homology class is negative, then for sufficiently large multiples of the class
the inequality of Equation (1.1) does not provide any information. For this
reason, at the moment we do not have effective tools to estimate the genus
function for homology classes in general, and this implies that $g_{X}$ cannot
be determined in general.

Notice that the generic homology class in
$H_{2}({{\mathbb{C}}{\mathbb{P}}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2};\mathbb{Z})$
can be given as $t_{B}B+t_{C}C+t_{D}D$, but since $D$ has self-intersection
$-1$, even if all coefficients are non-negative, the direct argument given
above does not necessarily provide a symplectic submanifold. The same
phenomenon can happen if the symplectic surface representing a given homology
class happens to be disconnected: in this case smoothly we can tube the
components together, but this cannot necessarily be done symplectically. In
the following we show some cases where a better understanding of the
construction of $AP$ and a surface representing a particular homology class
provides results close to optimal (and sometimes even optimal).

Some examples:

  * •

Consider $\alpha=B+C+D$. As $B,C,D$ all can be represented by a symplectic
surface, we have a disconnected representative of $\alpha$ (with components of
genus 5 and 2); tubing them we get a surface of genus 7 representing $\alpha$.
As $c_{1}(\alpha)=9$, the adjunction formula of Theorem 1.7 for any smooth
representative gives $g\geq 6$ as a lower bound.

  * •

For $\alpha=B+C-D$ we have $c_{1}(\alpha)=3$ and $\alpha^{2}=1$, so adjunction
implies $g\geq 3$. A surface representing this class can be given by taking
the disconnected symplectic representative of $B+C+D$, reverse the orientation
on the component corresponding to $D$ and then tube. The result will be a
surface of genus 7. A better result can be achieved by the following
simplification: when we construct the individual surfaces $B$ and $D$, in the
$\Sigma_{2}\times T^{2}$-part we consider two punctured genus-1 surfaces (once
with coefficient 1 in $B$ and once with coefficient $-1$ in $D$). Substituting
these subsurfaces with an annulus we get a genus 5 representative of the class
$\alpha$.

  * •

Let $\alpha=B+2C+D$; then $\alpha^{2}=3$ and $c_{1}(\alpha)=13$. The lower
bound for the genus provided by the adjunction formula is 9. A genus 10
surface can be easily constructed by taking the disconnected symplectic
representative (with a genus-8 and a genus 2 components) and tube them.

  * •

For the class $\alpha=2B+2C-2D$ we have $\alpha^{2}=4$ and $c_{1}(\alpha)=6$,
so the adjunction inequality provides 6 as the minimal genus. Noticing that
$A=2B+C-2D$ is a symplectic surface of genus 2 intersecting $C$ in two points,
after resolving the intersection points, we find a genus 6 symplectic surface
representing the homology class at hand.

  * •

Let us take $\alpha=B+D$. In this case $\alpha^{2}=-1$ and $c_{1}(\alpha)=5$,
so the adjunction inequality provides 3 as a lower bound. The disconnected
representative (after tubing) provides a genus 4 surface. Now if we use
(instead of the two punctured tori, each appearing in $B$ and $D$ on the
$\Sigma_{2}\times T^{2}$-side) a twice-braided torus, which now intersects the
normal connect sum surface, we get a representative of genus 3.

After an extensive search we did not find a nontrivial homology class in
$H_{2}(AP;\mathbb{Z})$ which we could represent by a sphere. For classes of
self-intersection at least $-1$ there is no such class, as can be deduced from
Seiberg-Witten theory, but for more negative squares this is far from obvious.
(Compare with the examples of Lemma 4.2 for
${\mathbb{C}}{\mathbb{P}}^{2}\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$.)
Indeed, the search for embedded tori (representing non-trivial homology
classes) did not provide any result either. This led us to the following

######  Conjecture 5.4.

The four-manifold $AP$ does not contain homologically essential embedded
spheres or tori.

As it happened in the case of the homology $S^{2}\times S^{2}$ examples, the
last Luttinger surgery in the construction can be replaced by a torus surgery
with coefficient $n$, providing an infinite family $Z_{n}$ of four-manifolds,
distinguished by the value of the Seiberg-Witten function on the two basic
classes. (See the computation in Proposition 2.2; above $Z_{1}=AP$.) As the
above constructions all happened in the complement of all surgeries, the
constructions are still valid, and whenever the minimality is verified by the
adjunction inequality, the statement holds true for the entire family. (Notice
that these manifolds — with the exception of $AP$ — are not symplectic.) In
conclusion, whenever we were able to identify the value of the genus function
we get $g_{Z_{n}}(\alpha)=g_{AP}(\alpha)$, leading us to

######  Conjecture 5.5.

The genus functions of the exotic four-manifolds $Z_{n}$ are all equivalent to
$g_{AP}$.

######  Remark 5.6.

The Seiberg-Witten functions $SW_{Z_{n}}\colon{\rm
Spin}^{c}(Z_{n})\to\mathbb{Z}$ all have the same support (the two basic
classes of $Z_{n}$), but different values on their support. As the adjunction
formula (and any genus-related formula) is insensitive for the Seiberg-Witten
value, there is reason to believe that the genus functions for $Z_{n}$ are all
the same. At the same time, this observation raises the question how the
Seiberg-Witten value influences the geometry of the underlying four-manifold.

Conjecture 5.4 naturally extends to the four-manifolds $Z_{n}$: we expect that
these four-manifolds do not contain homologically essential embedded spheres
or tori.

##  6\. The Baldridge-Kirk exotic
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
manifold $BK$

A similar construction as in the previous section provides exotic structures
on
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$,
as it was demonstrated in [4] by Baldridge and Kirk. Indeed, the building
blocks are very similar to the ones used earlier: consider the four-manifold
$S^{3}_{0}(T)\times S^{1}$ for the trefoil knot $T$, which admits a
$T^{2}$-fibration over $T^{2}$ with a section. Resolve the union of a section
and a fiber and blow up the resulting genus 2 surface twice, getting a genus 2
surface $\Sigma$ with trivial normal bundle in $(S^{3}_{0}(T)\times
S^{1})\\#2{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$. Consider the symplectic
normal sum of the resulting symplectic surface in the symplectic four-manifold
with $\Sigma_{2}\times T^{2}$ along $\Sigma_{2}\times\\{p\\}$ for some $p\in
T^{2}$.

######  Theorem 6.1.

([4]) There are four tori in the above normal connected sum (originating from
$\Sigma\times T^{2}$) along which Luttinger surgeries (with appropriately
chosen coefficients) provide a simply connected, symplectic four-manifold,
which we call $BK$.

Using the pieces, and considering surfaces before the Luttinger surgeries, one
can find several symplectic submanifolds in $BK$. First, there is the genus 2
surface used in the symplectic normal sum, which provides a symplectic
submanifold with self-intersection 0; the corresponding homology class will be
denoted by $A$. There is a punctured symplectic torus in
$\Sigma_{2}\times(T^{2}\setminus\\{p\\})$, and as it can be chosen to be
disjoint from the Lagrangian tori of the Luttinger surgeries, it descends to
$BK$. From the other side, the section and the fiber of the $T^{2}$-fibration
on $S_{0}^{3}(T)\times S^{1}$ give punctured tori in the complement of the
genus 2 surface; when gluing them with the punctured torus in
$\Sigma_{2}\times(T^{2}\setminus\\{p\\})$, we get two surfaces $B$ and $C$ of
genus 2, self-intersection 0, intersecting each other and $A$ once. The
exceptional divisors of the blow-ups on $S_{0}^{3}(T)\times S^{1}$ provide two
punctured disks with relative framing $(-1)$, hence after gluing them to the
punctured tori in $\Sigma_{2}\times(T^{2}\setminus\\{p\\})$ we get two
disjoint tori of self-intersection $(-1)$, and call them $D_{1}$ and $D_{2}$.

A basis of $H_{2}(BK;\mathbb{Z})\cong\mathbb{Z}^{4}$ can be given by
$B,C,D_{1},D_{2}$; the genera and intersection patterns are described above,
and the first Chern class $c_{1}$ of the symplectic four-manifold $BK$
evaluates as $c_{1}(B)=c_{1}(C)=2,c_{1}(D_{1})=c_{1}(D_{2})=1$, while
$c_{1}(A)$ is also 2.

Obviously, the genus function $g_{BK}$ is nonzero on the subset of
$H_{2}(BK;\mathbb{Z})$ formed by elements of self-intersection 0 (as $g_{BK}$
takes the value 2 on $A,B,C$), verifying again that the diffeomorphism
invariant $g_{X}$ distinguishes
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
from $BK$, the straightforward extension of Theorem 5.3 for the manifold $BK$.
(Notice that in the partial determination of $g_{BK}$ we rely on the
Symplectic Thom Conjecture Theorem 1.6.)

Once again, homology classes of the form $t_{A}\cdot A+t_{B}\cdot B+t_{C}\cdot
C$ with $t_{A},t_{B},t_{C}\geq 0$ can be represented by symplectic
submanifolds, and hence the genus function on those classes can be
algorithmically determined, while in general the adjunction inequality of
Theorem 1.7 provides a lower bound on the genus (in terms of the value of
$c_{1}$ on the homology class and its self-intersection), while specific
constructions give upper bounds on $g_{BK}$. The two bounds, however, do not
meet in general.

As before, the last Luttinger surgery in the construction of $BK$ can be
replaced by an appropriate sequence of torus surgeries, providing a sequence
of four-manifolds $V_{n}$ (with $V_{1}=BK$), which then show the existence of
infinitely many distinct smooth structures on
${{\mathbb{C}}{\mathbb{P}}}^{2}\\#3{\overline{{\mathbb{C}}{\mathbb{P}}}}^{2}$
(see [4]). As in the previous section, these manifolds are distinguished by
the Seiberg-Witten values on the basic classes. We cannot compare their genus
functions, as the values of these functions on most homology classes of
negative self-intersection are unknown at the moment. We do expect, however

######  Conjecture 6.2.

The exotic four-manifolds $V_{n}$ have equivalent genus functions.

In contrary to $AP$, in the four-manifold $BK$ it is easy to find tori and
spheres. Indeed, the homology classes $D_{1},D_{2}$ admit toric
representatives. In addition, when applying the two blow-ups in
$S_{0}^{3}(T)\times S^{1}$, we can arrange the blow-ups so that the resulting
four-manifold has a $(-2)$-sphere disjoint from the genus 2 surface along
which the normal sum is taken. Therefore this sphere embeds into $BK$ (and
indeed to all $V_{n}$’s).

## References

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# Enhanced pair production in frequency modulated Sauter potential wells

Li Wang  Key Laboratory of Beam Technology of the Ministry of Education, and
College of Nuclear Science and Technology, Beijing Normal University, Beijing
100875, China  Binbing Wu  Key Laboratory of Beam Technology of the Ministry
of Education, and College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Lie-Juan Li  Key Laboratory of Beam
Technology of the Ministry of Education, and College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  B. S. Xie
111Corresponding author. Email address<EMAIL_ADDRESS>Key Laboratory of
Beam Technology of the Ministry of Education, and College of Nuclear Science
and Technology, Beijing Normal University, Beijing 100875, China  Beijing
Radiation Center, Beijing 100875, China

###### Abstract

Electron-positron pair production in frequency modulated Sauter potential
wells is investigated in the framework of the computational quantum field
theory. In combined potential wells with a static Sauter potential well and a
frequency modulated oscillating one, the modulated amplitude has a large
effect on the number of created pairs. The optimal modulation amplitude of
frequency at different center frequencies is obtained, which increases the
number of electrons at about two times. However, for a single oscillating
potential well with frequency modulation, chirp effect is sensitive to the
center frequency, and the number of electrons can be enhanced even to four
orders of magnitude at a regime of low center frequency. It implies that for a
slowly oscillating Sauter potential well, the chirp effect through the
frequency modulation is better than adding a static potential well to improve
the pair production.

###### pacs:

03.65.Sq, 11.15.Kc, 12.20.Ds

##  I Introduction

Since the positron was proposed by Dirac to explain the negative solution of
Dirac equation, the generation of electron-positron pairs in vacuum with
strong electric field aroused great concern Dirac1928 . Soon after that the
existence of positrons was proved experimentally Anderson1933 , many
theoretical studies were shown to explain the mechanism to produce positrons
Sauter1931 ; Heisenberg1936 ; Schwinger1951 ; Brezin1970 ; Marinov1977 . So
far, several methods were developed, such as the worldline instanton technique
Gies2005 ; Dunne2006 ; Ilderton2015 ; Schneider2016 , the Dirac-Heisenberg-
Wigner formalism Li2015 ; Blinne2016 ; Kohlfurst2018 ; Olugh2019 , the quantum
Vlasov equation solution method Kluger1991 ; Alkofer2001 ; JiangMin2013 ;
Li2014 ; Nuriman2012 ; Sitiwaldi2017 ; Xie2017 , the computational quantum
field theory Krekora2005 ; Lv2013 ; Tang2013 ; Jiang2013 ; Jiang2014 ; Liu2014
; Wang2016 ; Gong2018  and so on. Overall, there are two mechanisms proposed.
The Schwinger mechanism is due to the quantum tunneling effect and requires
the electric field to reach $10^{16}\rm{V/cm}$, which is beyond the current
experimental conditions Schwinger1951 . The multi-photon process is due to a
transition between the negative and positive states through photon absorption,
which depends on the frequency of the alternating field Brezin1970 ;
Marinov1977 . Due to the fact that current ultrafast laser can not produce a
considerable number of positrons, people proposed new ways to reduce the
threshold or increase the yield.

Taking both the static potential well and the alternating field into account,
the pair creation in a symmetric potential well was investigated Tang2013 . It
is found that the symmetric potential produces more electrons than the
asymmetric potential, and combined potential wells are more favorable for the
creation of pairs than a static one. Due to the dynamically assisted Schwinger
mechanism, the pair creation rate can also be dramatically enhanced by
combining a strong and slowly varying electric field with a weak and rapidly
changing one Li2014 ; Nuriman2012 ; Schutzhold2008 . When the photon energy
equals the distance between the bound state and the Dirac sea level, the pair
creation can be enhanced, which is the bound state resonance enhanced
mechanism Wu:2019weg . For different bound states, different frequencies are
needed to achieve resonance so that the frequency chirp field was also
considered to enhance the creation of pairs Dumlu2010 ; Jiang2013 ;
Abdukerim2017 ; Olugh2019 . Recently the process of pairs creation in a
spatially homogeneous but frequency modulation (FM) electric field is studied
Gong2019 , where the result shows that the number of pairs can be enhanced or
weakened by adjusting the parameter of modulation frequency.

In this paper, we study the enhancement of electron-positron pairs in FM
Sauter potential wells, and optimize the modulation parameters. Considering
chirp effect, we study the pair creation process in combined potential wells
composed of a static Sauter potential well and a FM one in the framework of
the computational quantum field theory. Moreover, the single oscillatory
potential well is also studied. It is found that chirp effect has a great
influence on the number of created pairs at low center frequencies, and the
detailed explanation and analysis are given.

This paper is organized as follows. In Sec. II, the scheme is introduced
briefly for the computational quantum field theory, which solves the Dirac
equation in operator formalism and get the required number of created
electron-positron pairs. In Sec.III, FM Sauter potential wells are presented,
in which the time evolution of created electrons under different FM amplitudes
and center frequencies are simulated. In Sec.IV, we summarize our work.

##  II Outline of computational quantum field theory

Let us first describe the computational quantum field theory. The time
evolution of the operator $\hat{\psi}(z,t)$ for the electron-positron yield in
a potential $V(z,t)$ is given by the Dirac equationKrekora2005 ,

$i\partial\hat{\psi}\left(z,t\right)/\partial{t}=\left[c\alpha_{z}\hat{p}_{z}+\beta
c^{2}+V\left(z,t\right)\right]\hat{\psi}\left(z,t\right),$ (1)

where $\alpha_{z}$ and $\beta$ are Dirac matrices, $c$ is the speed of light
in vacuum, $\hat{p}_{z}$ is the momentum operator of $z$ direction,
$V\left(z,t\right)$ is external field that varies with time $t$ in the $z$
direction. We use the atomic units (a.u.) as $\hbar=e=m_{e}=1$. By introducing
the creation and annihilation operators, the field operator $\hat{\psi}(z,t)$
can be decomposed as follows:

$\displaystyle\hat{\psi}(z,t)$
$\displaystyle=\sum_{p}\hat{b}_{p}(t)W_{p}(z)+\sum_{n}\hat{d}_{n}^{{\dagger}}(t)W_{n}(z)$
(2)
$\displaystyle=\sum_{p}\hat{b}_{p}W_{p}(z,t)+\sum_{n}\hat{d}_{n}^{\dagger}W_{n}(z,t),$

where $p$ and $n$ denote the momenta of positive and negative energy states,
$\sum_{p(n)}$ represents summation over all states with positive
$($negative$)$ energy, $W_{p}(z)=\langle z|p\rangle(W_{n}(z)=\langle
z|n\rangle)$ is field-free positive $($negative$)$ energy eigenstate. Note
that $W_{p}(z,t)=\langle z|p(t)\rangle$ and $W_{n}(z,t)=\langle z|n(t)\rangle$
satisfy the single-particle time-dependent Dirac equation (1). From Eq.(2), we
obtain

$\displaystyle\hat{b}_{p}(t)$
$\displaystyle=\sum_{p^{\prime}}\hat{b}_{p^{\prime}}U_{pp^{\prime}}(t)+\sum_{n^{\prime}}\hat{d}_{n^{\prime}}^{\dagger}U_{pn^{\prime}}(t),$
(3) $\displaystyle\hat{d}_{n}^{\dagger}(t)$
$\displaystyle=\sum_{p^{\prime}}\hat{b}_{p^{\prime}}U_{np^{\prime}}(t)+\sum_{n^{\prime}}\hat{d}_{n^{\prime}}^{\dagger}U_{nn^{\prime}}(t),$
$\displaystyle\hat{b}_{p}^{\dagger}(t)$
$\displaystyle=\sum_{p^{\prime}}\hat{b}_{p^{\prime}}^{\dagger}U_{pp^{\prime}}^{*}(t)+\sum_{n^{\prime}}\hat{d}_{n^{\prime}}U_{pn^{\prime}}^{*}(t),$
$\displaystyle\hat{d}_{n}(t)$
$\displaystyle=\sum_{p^{\prime}}\hat{b}_{p^{\prime}}^{\dagger}U_{np^{\prime}}^{*}(t)+\sum_{n^{\prime}}\hat{d}_{n^{\prime}}U_{nn^{\prime}}^{*}(t),$

where $U_{pp^{\prime}}(t)=\langle p|\hat{U}(t)|p^{\prime}\rangle$,
$U_{pn^{\prime}}(t)=\langle p|\hat{U}(t)|n^{\prime}\rangle$,
$U_{nn^{\prime}}(t)=\langle n|\hat{U}(t)|n^{\prime}\rangle$,
$U_{np^{\prime}}(t)=\langle n|\hat{U}(t)|p^{\prime}\rangle$ and the time-
ordered propagator
$\hat{U}(t)=\hat{T}\textrm{exp}\\{{-i\int_{0}^{t}d\tau[c\alpha_{z}\hat{p}_{z}+\beta
c^{2}+V(z,\tau)]}\\}$.

In Eq.(2), the electronic portion of the field operator is defined as
$\hat{\psi}_{e}(z,t)\equiv\sum_{p}\hat{b}_{p}(t)W_{p}(z)$. So we can obtain
the probability density of created electrons by

$\displaystyle\rho(z,t)$
$\displaystyle=\langle\text{vac}|\hat{\psi}_{e}^{\dagger}(z,t)\hat{\psi}_{e}(z,t)|\text{vac}\rangle$
(4) $\displaystyle=\sum_{n}|\sum_{p}U_{pn}(t)W_{p}(z)|^{2}$

By integrating this expression over space, the number of created electrons can
be obtained as

$N(t)=\int\rho(z,t)dz=\sum_{p}\sum_{n}|U_{pn}(t)|^{2}.$ (5)

The time-ordered propagator $U_{pn}(t)$ can be numerically calculated by
employing the split-operator technique Tang2013 . Therefore, according to
Eq.(4) and Eq.(5), we can compute various properties of the electrons produced
under the action of the external potential.

##  III Numerical results

In this work, we study combined potential wells with a static Sauter potential
well and a FM one, which is given by

$\displaystyle V(z,t)$
$\displaystyle=V_{1}S(z)f(t)+V_{2}\sin[\omega(t)t]S(z)\theta(t;t_{0},t_{0}+t_{1}),$
(6)

where $S(z)=\\{\tanh[(z-D/2)/W]-\tanh[(z+D/2)/W]\\}/2$, $D$ is the width of
the potential well, $W$ is the width of the potential edge. Here $V_{1}$ is
the depth of the static potential well, $V_{2}$ is the amplitude of the
oscillating potential well. The time dependent oscillation frequency is set to
$\omega(t)=\omega_{0}+\Delta\omega\sin[\Omega(t-t_{0})]$, where $\omega_{0}$
is the center frequency, $\Delta\omega$ is the FM amplitude, $\Omega$ reflects
the speed of frequency change. The function $f(t)=\sin[\pi
t/2t_{0}]\theta(t;0,t_{0})+\theta(t;t_{0},t_{0}+t_{1})+\cos[\pi(t-t_{0}-t_{1})/2t_{0}]\theta(t;t_{0}+t_{1},2t_{0}+t_{1})$
describes the turning on and off processes of the potential well, and
$\theta(t,t_{1},t_{2})$ is the step function. The static potential well is
opened during $(0,t_{0})$ and closed during $(t_{0}+t_{1},2t_{0}+t_{1})$. From
$t_{0}$ to $t_{0}+t_{1}$, the oscillating potential well exists. In this
paper, other parameters are set to $t_{0}=5/c^{2}$, $W=0.3\lambda_{C}$, and
$\lambda_{C}=1/c$ is the Compton wavelength.

Figure 1:  Number of created electrons as a function the FM amplitude
$\Delta\omega$. The simulation time is set to $t_{1}=40\pi/c^{2}$. Other
parameters are $D=10\lambda_{C}$, $V_{1}=V_{2}=1.47c^{2}$,
$\omega_{0}=0.5c^{2}$, $\Omega=0.2c^{2}$. The spatial size is $L=2.0$.

In Fig. 1, we represent the number of created electrons as a function of the
FM amplitude $\Delta\omega$. Other parameters are set to $D=10\lambda_{C}$,
$V_{1}=V_{2}=1.47c^{2}$, $\omega_{0}=0.5c^{2}$, $\Omega=0.2c^{2}$. The spatial
size is $L=2.0$. The simulation time of the numerical calculation is set to
$t=40\pi/c^{2}$, which is less than the time required for the electrons
generated in the potential well to leave the simulation regions. With the
increase of $\Delta\omega$, the number of created electrons first increases
rapidly, then decreases slowly. When the FM amplitude is set to
$\Delta\omega=0$, the potential well oscillates at a fixed frequency
$\omega_{0}=0.5c^{2}$, and the number of created electrons is $2.39$. For
$\Delta\omega=0.2c^{2}$, the peak value is $6.42$, which is larger than that
of the fixed frequency. It is obviously that chirp effect can promote the
generation of electron-positron pairs. The final number of electrons does not
increase monotonously with the increase of FM amplitude.

Figure 2:  Number of created electrons from $t=0$ to $t=(10+40\pi)/c^{2}$ for
$\Delta\omega=0$ (black dotted line), $\Delta\omega=0.1c^{2}$ (green dotted
and dashed line), $\Delta\omega=0.2c^{2}$ (red solid line) and
$\Delta\omega=0.45c^{2}$ (blue double-dashed line). Other parameters are same
as in Fig. 1.

In Fig. 2, the time evolution of the number of created electrons at different
FM amplitudes is plotted. For $\Delta\omega=0$, the number grows periodically
over time, and the final number is $N=2.39$, which conforms to Fig. 1. For
$\Delta\omega=0.1c^{2}$, $0.2c^{2}$ and $0.45c^{2}$, the number grows unevenly
over time. For $\Delta\omega=0.1c^{2}$, the number grows faster than that of
the fixed frequency. For $\Delta\omega=0.2c^{2}$, the number growth rate is
also larger than that of $\Delta\omega=0.1c^{2}$. However, when the parameter
$\Delta\omega$ grows to $0.45c^{2}$, the number is not always larger than that
of $\Delta\omega=0.2c^{2}$ with the increase of time. When the evolution time
reaches a certain point, the growth rate deceases.

Figure 3:  Instantaneous eigenvalues of combined potential wells over time.
Other parameters are the same as those in Fig. 2.

To understand above results, we exhibit the instantaneous bound state of
combined potential wells for different FM amplitudes in Fig. 3. Other
parameters are the same as those in Fig. 2. The evolution of the bound state
in processes of turning on and off the static potential is also included in
all insets. In Fig. 3(a), when the static potential well is completely turned
on, there are eight energy levels in the energy gap. With the increase of
time, the bottom five levels periodically dive into the Dirac sea, and then go
back to positive continuum states. Compared with the black dotted line in Fig.
2, we find that the number of electrons grows fastest when the energy level is
minimum. This conclusion is consistent with our previous work, in which we
introduced the efficient interaction time to explain the production of pairs
at a fixed frequency Wangli2019 . In the simulation time, the energy level
dives into the Dirac sea 10 times.

In Figs. 3(b), and (c), with the increase of time, bound states change
unevenly, and the efficient interaction time becomes more and more smaller.
Near $t=0.001$a.u., the efficient interaction time is lager than that of Fig.
3(a). Compared with the subgraph in Fig. 2, the lager efficient interaction
time leads to more electrons created. In Fig. 3(d), bound states change more
rapidly with time. The number of times that bound states dive into the Dirac
sea increases with the parameter $\Delta\omega$.

As can be seen above, the final number of electrons is related to the number
of times that bound states dive into the Dirac sea, and the efficient
interaction time. Therefore, we propose a function to roughly count the final
number of electrons, $N_{final}\varpropto\sum_{i=1}^{n}r\Delta t$, where $n$
is the number of times that bound states dive into the Dirac sea during the
simulation time, $\Delta t$ is the efficient interaction time, $r$ is the
growth rate during the efficient interaction time. According to Fig. 2 in Ref.
Krekora2005 , the number of electrons is quadratic growth when the time set to
$t<<1/c^{2}$, and linear growth for $1/c^{2}<t<1/c^{2}+W/c$, where the
parameter $W$ corresponds to the width of potential well $D$ in this paper. In
our work, the two parameters that determine the growth rate are about
$5.33\times 10^{-5}$a.u. and $5.86\times 10^{-4}$a.u.. In Fig. 3(a), the
efficient interaction time of each dive is located between the two parameters,
which means the growth rate is constant. For $\Delta\omega=0.1c^{2}$, although
the efficient interaction time is shortened a little, it also increases the
growth rate and the number of dives. When the modulation amplitude is
$0.45c^{2}$, although the number of dives and the growth rate are very large,
the efficient interaction time, especially in the later period of time
evolution, is so short that the generation of electrons is suppressed.
Therefore, under the competition mechanism of the growth rate, the number of
dives, and the efficient interaction time, there are optimal parameters that
make the output the most. For combined potential wells with
$\omega_{0}=0.5c^{2}$, the optimal FM amplitude is located at
$\Delta\omega_{\rm o}=\Delta\omega_{opt}=0.2c^{2}$.

Figure 4:  The optimal FM amplitude as a function of the center frequency.
Other parameters are the same as those in Fig. 1.

Next, the optimal FM amplitude as a function of the center frequency is shown
in Fig. 4. With the increase of the center frequency, the optimal FM amplitude
first increases linearly, and then decays nonlinearly. For low center
frequencies, the parameter $n$ plays a major role in the creation of
electrons. The greater the FM amplitude is, the larger the parameter $n$ is.
So, for $\omega_{0}<0.2c^{2}$, the optimal FM amplitude satisfies the relation
that $\Delta\omega_{\texttt{o}}=\omega_{0}$. For high center frequencies, the
generation of electrons is dominated by the efficient interaction time. Since
chirp effect reduces the efficient interaction time, the optimal FM amplitude
is smaller for a higher center frequency.

Figure 5:  The final number of electrons created in combined potential wells
as a function of the center frequency. The black line is for a fixed
frequency, and the red line is for the optimal FM amplitude. Other parameters
are the same as those in Fig. 1.

In Fig. 5, we represent the final number of electrons created in combined
potential wells as a function of the center frequency. For a fixed frequency
potential well, the final number first increases slowly, and then increases
quickly with the increase of the center frequency. When the FM amplitude is
optimized, the number of electrons first increases rapidly, and then increases
slowly. Compared with the two lines, it is found that chirp effect has a great
influence on the generation of electrons at low frequencies. For the center
frequency that $\omega_{0}>1.9c^{2}$, chirp effect does not promote the
electron creation.

Table 1: The center frequency and the ratio of the final number of the optimal FM amplitude to that of the fixed frequency in Fig.5 $\omega_{0}(c^{2})$ | 0.06 | 0.1 | 0.3 | 0.5 | 0.7 | 1.0 | 1.2 | 1.5 | 1.8 | 2.0
---|---|---|---|---|---|---|---|---|---|---
$R$ | 1.42 | 2.09 | 2.22 | 2.68 | 2.15 | 1.99 | 1.54 | 1.12 | 1.12 | 1.00

Table I represents the center frequency and the ratio of the final number of
the optimal FM amplitude to that of the fixed frequency in Fig.5. Taking into
account chirp effect, the number of created electrons is increased by about or
less than two times. For high frequencies, the enhancement by chirp effect is
very small.

Figure 6:  The final number created in a single oscillating potential well as
a function of the center frequency. The black line is for a fixed frequency,
and the red line is for the optimal FM amplitude. Other parameters are the
same as those in Fig. 1, except that $V_{1}=0$.

In Fig. 6, we represent the final number created in a single oscillating
potential well as a function of the center frequency. For a fixed frequency,
the number of created electrons is few at low frequencies, and increases a
little at high frequencies. In the FM potential well, the number of electrons
first increases quickly at low frequencies, and then stays around $N=5$, which
is larger than that of a fixed frequency. It is obvious that chirp effect
dramatically increases the number of electrons for an oscillating potential
well. In Fig. 5, the number of electrons created in combined potential wells
with a fixed frequency is no more than $3$ at low center frequencies, which is
smaller than $5$. By comparison, it is found that for an oscillating potential
well with a low fixed frequency, considering chirp effect is better than
adding a static potential well.

Table 2: The center frequency and the ratio of the final number of the optimal FM amplitude to that of a fixed frequency in Fig.6, where $a(n)$ stands for $a\times 10^{n}$. $\omega_{0}(c^{2})$ | 0.1 | 0.2 | 0.4 | 0.7 | 1.0 | 1.2 | 1.5 | 1.8 | 2.0
---|---|---|---|---|---|---|---|---|---
$R$ | 1.32(4) | 6.98(2) | 2.81(2) | 2.30(1) | 1.10(1) | 3.74 | 2.77 | 2.80 | 1.81

Table II represents the center frequency and the ratio of the final number of
the optimal FM amplitude to that of a fixed frequency in Fig.6. With the
increase of $\omega_{0}$, the ratio $R$ decreases from a very large number to
$1.81$. It is worth noting that the number of electrons created in the
potential well with a fixed frequency is very small, and increases by four
orders of magnitude with the increase of $\omega_{0}$, see the inset of Fig.
6. Therefore, the number of created electrons increased by up to four orders
of magnitude at low center frequencies, and the ratio is much lager than that
of table I. Compared with combined potential wells, chirp effect is more
sensitive to an oscillating Sauter potential well with low center frequencies.

Figure 7:  Number of created electrons for $\Delta\omega=0$ (black dotted
line) and $0.05c^{2}$ (red solid line) . Other parameters are the same as
those in Fig. 1, expect that $\omega_{0}=1.5c^{2}$ and $D=4\lambda_{C}$.

For the high center frequency regime, in Fig. 7, the number of created
electrons in FM combined potential wells is represented. The center frequency
is set to $\omega_{0}=1.5c^{2}$, in which the multi-photon process plays an
important role in the creation of electrons. In order to study the multi-
photon process under chirp effect more easily, we set the width of the
combined potential wells to $D=4\lambda_{C}$. When the frequency is fixed, the
number grows slowly over time. After considering chirp effect, the number
increases rapidly over time. It is found that chirp effect increases the pair
growth rate for a high center frequency.

Figure 8:  Snapshots of the electronic spatial densities at times
$t_{k}=k\times 1.4\times 10^{-4}$a.u. (k=1,2,3,…,50). Panel (a) is for the
fixed frequency potential well with $\omega_{0}=1.5c^{2}$, panel (b) is for
the FM one with $\omega_{0}=1.5c^{2}$ and $\Delta\omega=0.05c^{2}$. Other
parameters are the same as for Fig. 7.

In Fig. 8, snapshots of the electronic spatial densities are shown. Panel (a)
is for the fixed frequency potential well with $\omega_{0}=1.5c^{2}$. Panel
(b) is for the FM one with $\omega_{0}=1.5c^{2}$ and $\Delta\omega=0.05c^{2}$.
In Fig. 8(a), created electrons are all trapped in the static potential well
during the simulation time. When chirp effect is taken account, some of
electrons escape out of the static potential well, see that in Fig. 8(b).
Therefore, chirp effect reduces Pauli blocking and promotes the creation of
electrons.

Figure 9:  The energy spectrum density of electrons created at the end of the
simulation time for $\Delta\omega=0$ (black dotted line) and $0.05c^{2}$ (red
solid line) . Other parameters are the same as those in Fig. 7

In Fig. 9, we represent the energy spectrum density of created electrons at
the end of the simulation time for $\Delta\omega=0$ and $0.05c^{2}$. Other
parameters are the same as those in Fig. 7. The depth and the width of the
static potential well are set to $V_{1}=1.47c^{2}$ and $D=4\lambda_{C}$. The
static potential well provides three energy levels in the gap, which are
located at $E_{1}=-0.2631c^{2}$, $E_{2}=0.1762c^{2}$, and $E_{3}=0.6698c^{2}$,
respectively. In Fig. 9(a), the dotted black line is for the fixed frequency
with $\omega_{0}=1.5c^{2}$. These discrete peaks correspond to different
multi-photon absorption processes. The red solid line is for the FM potential
well with $\Delta\omega=0.05c^{2}$. The number of peaks is larger, and the
value of each peak is marked in Figs. 9(b), (c), and (d).

In Fig. 9(b), we show the one-photon process. For a fixed frequency, these
three peaks are located at $E_{P1}=1.24c^{2}$, $E_{P2}=1.74c^{2}$, and
$E_{P3}=2.21c^{2}$, respectively. These peak values satisfy that
$E_{pi}=E_{i}+\omega_{0}$ ($i=1,2,3$), which is consistent with that of Ref.
Tang2013 . For the FM one, the position of the peak moves, and the number of
peaks increases. Because the frequency changes from $\omega_{0}-\Delta\omega$
to $\omega_{0}+\Delta\omega$, the energy of created electrons also changes
from $E_{i}+(\omega_{0}-\Delta\omega)$ to $E_{i}+(\omega_{0}+\Delta\omega)$.
So, new peaks are located between $1.19c^{2}$ and $1.29c^{2}$, or $1.69c^{2}$
and $1.79c^{2}$, or $2.16c^{2}$ and $2.26c^{2}$. For example, the peak value
$1.21c^{2}$ is between $1.19c^{2}$ and $1.29c^{2}$, and corresponds to
$E_{1}+(\omega_{0}-0.03c^{2})$, which means the electron in the first bound
state escapes from the energy gap by absorbing one photon with
$\omega=1.47c^{2}$. And the frequency of the photon is located between
$\omega_{0}-\Delta\omega$ and $\omega_{0}+\Delta\omega$.

But some other peaks are not located at these regions, for example the peak
value $1.4c^{2}$. The reason is that frequencies are more widely distributed
in the frequency spectra. The peak value $1.4c^{2}$ corresponds to
$E_{1}+(\omega_{0}+0.16c^{2})$, it means the electron in the first bound state
escapes from the energy gap by absorbing one photon with $\omega=1.66c^{2}$.
Other peak values such as $1.47c^{2}$, $1.66c^{2}$, $1.95c^{2}$, $2.13c^{2}$,
and $2.42c^{2}$, correspond to $E_{1}+(\omega_{0}+0.23c^{2})$,
$E_{1}+(\omega_{0}+0.42c^{2})$, $E_{2}+(\omega_{0}+0.21c^{2})$,
$E_{2}+(\omega_{0}+0.39c^{2})$, and $E_{3}+(\omega_{0}+0.21c^{2})$,
respectively. These corresponding frequencies of absorbed photons are
respectively $1.73c^{2}$, $1.92c^{2}$, $1.71c^{2}$, $1.89c^{2}$, and
$1.71c^{2}$, which are confirmed in Fig. 10.

In Fig. 9(c) and (d), for a fixed frequency, peak values satisfy
$E_{pi}=E_{i}+n\omega_{0}$ ($n=2,3$). When chirp effect is considered, the
energy of created electrons are located between
$E_{i}+n(\omega_{0}-\Delta\omega)$ and $E_{i}+n(\omega_{0}+\Delta\omega)$. For
multi-photon processes, the energy of electrons is more overlapped than the
one-photon process. Therefore, more high-energy electrons are produced under
chirp effect. The growth rate of the number of electrons is enhanced by chirp
effect through multi-photon processes.

Figure 10:  The frequency spectra of the FM potential well with modulation
parameters (a) $\omega_{0}=1.5c^{2}$, $\Delta\omega=0.05c^{2}$, (b)
$\omega_{0}=0.3c^{2}$, $\Delta\omega=0.05c^{2}$. Other parameters are the same
as those in Fig. 7.

Now let us to see the frequency spectra of the FM potential well, which is
shown in Fig. 10. Panel (a) is to help understand those peaks in Fig. 9. Based
on the above analysis, the frequency is not limited between
$\omega_{0}-\Delta\omega$ and $\omega_{0}+\Delta\omega$, which is seen in Fig.
10 (a). The spectrum is centered at $1.5c^{2}$, and gradually increases to the
left and right within the width of $0.05c^{2}$. And what is interesting is
that it repeats every $0.2c^{2}$, which is related with the modulation
parameter $\Omega=0.2c^{2}$. As an illustration, for example, for the peak
value $1.66c^{2}$ of the one-photon process in Fig. 9 (b), one can see that it
does correspond to $E_{1}+(\omega_{0}+0.42c^{2})$ but not correspond to
$E_{2}+(\omega_{0}-0.08c^{2})$, since the frequency of
$\omega_{0}+0.42c^{2}=1.92c^{2}$ has a higher amplitude than that of
$\omega_{0}-0.08c^{2}=1.42c^{2}$ in the frequency spectra. Thus it can be
verified that the corresponding frequencies such as $1.73c^{2}$, $1.92c^{2}$,
$1.71c^{2}$, and $1.89c^{2}$ are all revealed in the frequency spectra shown
in Fig. 10 (a).

Moreover, as a comparison, in Fig. 10 (b), the relative low center frequency
case is plotted by setting $\omega_{0}=0.3c^{2}$. Strikingly for this low
center frequency case, frequency distribution is very similar to that of high
center frequency. Since the frequency spectra is not too dependent on the
center frequency, the final number of electrons for different center
frequencies are almost the same, which explains the result in Fig. 6.

From the results and analysis mentioned above, we can conclude that for a high
center frequency, extra low frequencies caused by chirp effect have little
effect on the generation of electrons through the quantum tunneling effect,
however, for a low center frequency, extra high frequencies caused by chirp
effect strongly enhance the generation of electrons through the multi-photon
process and the dynamically assisted Schwinger mechanism. That is why chirp
effect is more sensitive to the low frequency.

##  IV Summary and Conclusion

We have investigated the chirp effect of FM Sauter potential well on the
electron-positron pair creation by employing the computational quantum field
theory. First, we study combined Sauter potential wells with a static one and
a FM one to investigate the number of created electrons and its variation with
FM parameters. Second, we remove the static potential well and study chirp
effect on the pair creation in a single FM potential well. In order to
understand the number changes or/and enhancement, the energy spectrum density
and the spatial density of electrons, and the frequency spectra of fields are
given as the helpful explanation illustrations.

The main results of our work include:

1\. In combined potential wells, chirp effect enhances the number of created
electrons. The optimal FM amplitude varies from $0$ to $0.3c^{2}$. Compared
with a fixed frequency, chirp effect can enhance the number of created
electrons by about two times.

2\. In a single FM oscillating potential well, chirp effect is more sensitive
to low center frequencies, and increases the number of created electrons by up
to four orders of magnitude compared with the fixed frequency.

3\. For an oscillating potential well with a low center frequency, considering
chirp effect is better than adding a static potential well.

Chirp effect influences the change of the depth of the potential well with
time, causing electrons created in the potential well to escape from the trap,
thus promoting the electron pumping. The final number is related to the
efficient interaction time, the growth rate, and the number of times that the
bound state dives into the Dirac sea. The optimal FM parameters is the result
of the competition of these three factors. Because of chirp effect, the
frequency spectrum is widened, and energies of created electrons change from
discrete to continuous, which increases the number of electrons. Compared with
adding a static potential well, considering chirp effect is more conducive to
the generation of electrons in a single slowly oscillating potential well.

In this work, the frequency is sinusoidal with time. There are other shape of
frequencies that can be considered and studied. The innovation of this paper
is to introduce the relation of the final number to the efficient interaction
time, the growth rate, and the number of times that the bound state dives into
the Dirac sea, which can well explain the generation process of electrons. It
is significant to conclude that chirp effect is more effective at low center
frequencies.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(NSFC) under Grants No. 11875007 and No. 11935008. The computation was carried
out at the HSCC of the Beijing Normal University.

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