# More on Seiberg-Witten Theory and Monstrous Moonshine

Shun’ya<EMAIL_ADDRESS>Takumi
<EMAIL_ADDRESS>Hitomi
<EMAIL_ADDRESS>and Shotaro
<EMAIL_ADDRESS>11footnotemark: 1Theory Center, Institute of
Particle and Nuclear Studies, KEK
Tsukuba, Ibaraki, 305-0801, Japan  11footnotemark: 122footnotemark:
233footnotemark: 344footnotemark: 4SOKENDAI (The Graduate University for
Advanced Studies)
Tsukuba, Ibaraki, 305-0801, Japan

(November 14, 2022)

###### Abstract

We continue the study of a relationship between the instanton expansion of the
Seiberg-Witten (SW) prepotential of $D=4$, ${\cal N}=2$ $SU(2)$ SUSY gauge
theory and the Monstrous moonshine. Extending the previous results, we show
for the cases of $N_{f}=2$ and $3$ that $q=e^{2\pi i\tau}$, where $\tau$ is
the complex gauge coupling, again has an expansion whose coefficients are all
integer-coefficient polynomials of the moonshine coefficients of the modular
$j$-function in terms of an appropriate expansion variable. We also
demonstrate that the new method of calculating the SW prepotential developed
here is useful by performing some explicit computations.

††preprint: KEK-TH-2460

SWcurve

##  I Introduction

Recently, an explicit relationship between the instanton expansion of the
Seiberg-Witten (SW) prepotential SW1 ; SW2  of $D=4$, ${\cal N}=2$ $SU(2)$
SUSY gauge theory and the Monstrous moonshine McKay ; Thompson ;
MonstrousMoonshine ; FLM ; FLM2 ; DGH ; Borcherds  was studied SWMoonshine
(See Tachikawalecture  and McKayessay  for reviews on the respective topics.).
There, by a simple method to obtain SW prepotentials using the modular
$j$-function, it was shown that the coefficients of the expansion of
$q=e^{2\pi i\tau}$ in terms of $A^{2}\equiv\frac{\Lambda^{2}}{16a^{2}}$
$(N_{f}=0)$ or $\frac{\Lambda^{2}}{32a^{2}}$ $(N_{f}=1)$ are all integer-
coefficient polynomials of the moonshine coefficients of the modular
$j$-function111Here, as in the standard literature on the SW theory, $a$ is
the eigenvalue of the $SU(2)$ adjoint scalar playing the role of the Higgs,
and $\tau$ is the complex gauge coupling $\frac{\pi
i}{g^{2}}+\frac{\theta}{8\pi}$ identified as the modulus of the elliptic fiber
over the $u$-plane of the SW curve.. In this paper, we continue this study; we
further examine whether the same phenomenon is also observed for $N_{f}=2$ and
$3$ and show that it is indeed the case.

As already emphasized in SWMoonshine , the relation between the SW theory and
the Monstrous moonshine is not at all surprising in itself, since SW curves
are elliptic curves and the modular $j$-function is a bijection between the
fundamental domain and the complex plane, and thus describes the modulus of an
elliptic curve. On the other hand, it has been known for some time
1207.5139MalmendierOno  that the $u$-plane integral MooreWitten  for a certain
“mock modular form” arising in the elliptic genus of K3 yields the $SO(3)$
Donaldson invariants of $\hbox{\mybb P}^{2}$ (See Mockingtheuplaneintegral
and references therein for more recent discussions.). Since the former
exhibits a similar moonshine phenomenon called the Mathieu moonshine
MathieuMoonshine , while the latter are certain correlation functions of a
twisted 4D ${\cal N}=2$ $SO(3)$ supersymmetric gauge theory Wittentopological
, this observation may be the first example of a connection between a
moonshine phenomenon and a supersymmetric gauge theory (albeit a twisted one),
which is nontrivial. In contrast, the correspondence we will use is not that
somewhat indirect and intricate, but more direct and easy to understand.
Nonetheless, there are other reasons to focus on the relationship.

One of them is as an effective tool for calculating the prepotential. This
method can be applied to any theory as long as its SW curve is an elliptic
curve, as in E-string theory. We will demonstrate that the method developed in
SWMoonshine  for calculating the prepotentials is useful by implementing it in
$SU(2)$ SW theory with $N_{f}=2$ and $3$. To date, various ways of computing
the SW prepotential are available, but our method is more efficient than
A-model like computations such as the Nekrasov instanton counting Nekrasov ;
NekrasovOkounkov , and simpler than other B-model like approaches (e.g. KLT ;
MasudaSuzuki ; Ohta ; Ohta2 ; CDH ; DV ; EguchiMaruyoshi ; IOY ; MOT ) as we
do not need to compute the dual period $a_{D}$. Our approach is simply to
develop a series, so there is no way to go wrong (as long as one can use
computational software like Mathematica). For example, by using our method we
found four typos in the result of the $N_{f}=3$ prepotential computed by CDH
long time ago, as shown in the text 222A similar method for computing the
prepotential was already used in ClossetMagureanu2107.03509 . The reference to
the $j$-function in SW theory itself is of course much older, e.g., the
$q$-expansion for the $j$-function can be found in 1109.5728  (but no
reference to the Monstrous moonshine). .

Another nontrivial observation in the connection between the SW theory and
Monstrous moonshine is a possible relationship between two different 2d
conformal field theories (CFTs) with different central charges. We will show
(as was done in SWMoonshine ) that the quantity $q$ allows an expansion in
which the coefficients are integer-coefficient polynomials of the moonshine
Fourier coefficients. This means that $q$ is also expressed in terms of
integer-coefficient polynomials of the dimensions of representations of the
Monster group. As pointed out in SWMoonshine , since the dimensions of
representations of the Monster are related to the vertex operator CFT FLM ;
FLM2 ; DGH ; Borcherds  whereas each instanton contribution of the SW
prepotential is written as a correlation function of the Liouville theory AGT
, our observation suggests that those are related in some way. What exactly
that is remains as a problem to be solved in the future.

##  II SW prepotential from the modular $j$-function

Let us consider a SW curve given in the Weierstrass form

$Y^{2}=X^{3}+f(u)X+g(u).$ (1)

We can obtain the value of the modular $j$-function as

$j(\tau)=\frac{12^{3}\cdot 4f(u)^{3}}{4f(u)^{3}+27g(u)^{2}}.$ (2)

We expand the right hand side of this equation (2) by $u$ around $u=\infty$,
thereby we get $\frac{1}{u}$-expansion of $\frac{1}{j}$.

On the other hand, the modular $j$-function has an expansion in terms of
$q=e^{2\pi i\tau}$ as

$j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_{n}q^{n},$ (3)

whose coefficients $a_{n}$ are integers. Specifically, the first few terms of
the expansion (3) are:

$n$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$a_{n}$ | 744 | 196884 | 21493760 | 864299970 | 20245856256 | 333202640600

Thus we find $\frac{1}{j}$-expansion of $q$ as

$\begin{split}q&=\frac{1}{j}+a_{0}\quantity(\frac{1}{j})^{2}+\quantity(a_{0}^{2}+a_{1})\quantity(\frac{1}{j})^{3}+\quantity(a_{0}^{3}+3a_{1}a_{0}+a_{2})\quantity(\frac{1}{j})^{4}\\\
&\quad+\quantity(a_{0}^{4}+6a_{0}^{2}a_{1}+2a_{1}^{2}+4a_{0}a_{2}+a_{3})\quantity(\frac{1}{j})^{5}\\\
&\quad+\quantity(a_{0}^{5}+10a_{0}^{3}a_{1}+10a_{0}a_{1}^{2}+10a_{0}^{2}a_{2}+5a_{1}a_{2}+5a_{0}a_{3}+a_{4})\quantity(\frac{1}{j})^{6}+\order{\quantity(\frac{1}{j})^{7}}\end{split}$
(4)

by solving inversely the equation (3).

Substituting $\frac{1}{u}$-expansion of $\frac{1}{j}$ into this equation (4),
we can get $\frac{1}{u}$-expansion of $q$. Furthermore,
$\frac{1}{a}$-expansion of $\frac{1}{u}$ is known from the period integral on
the SW curve MasudaSuzuki  and we obtain $\frac{1}{a}$-expansion of $q$. Since
the complex structure modulus $\tau$ and the SW prepotential $\mathcal{F}$ are
related as

$\tau=\partialderivative[2]{\mathcal{F}}{a},$ (5)

we finally get the expression of the prepotential $\mathcal{F}$ by integrating
$\frac{1}{a}$-expansion of $q$ with respect to $a$ twice.

This procedure was performed in SWMoonshine  for $N_{f}=0$ and $1$, thereby an
explicit relationship between the instanton expansion and the moonshine was
obtained. Below we will do this for the cases $N_{f}=2$ and $3$. 333It is
known Nahm ; MooreWitten  that the monodromy groups in these cases are the
congruence subgroups $\Gamma(2)$ and $\Gamma_{0}(4)$ of $SL(2,\hbox{\mybbsmall
Z})$, respectively. The Hauptmoduln for these reduced monodromy groups, that
is, the weight-0 modular forms associated with these congruence subgroups,
also allow expansions in term of $q$ known as the McKay-Thompson series, whose
coefficients are a sum of traces of a particular element of the Monster group
MonstrousMoonshine . The analysis of such modular rational elliptic surfaces
was recently worked out in Elias2107.04600 ; Horia2203.03755 .

###  $N_{f}=2$ case

Let us consider the $\mathcal{N}=2$ $SU(2)$ SUSY gauge theory with $N_{f}=2$
hypermultiplets. The SW curve is given by

$y^{2}=\left(x^{2}-u+\frac{\Lambda^{2}}{8}\right)^{2}+\Lambda^{2}(x+m_{1})(x+m_{2})$
(6)

in the quartic-polynomial representation MasudaSuzuki . This is equivalent to
the Weierstrass form

$\displaystyle Y^{2}$
$\displaystyle=X^{3}+f(u,m_{1},m_{2})X+g(u,m_{1},m_{2}),$ (7) $\displaystyle
f(u,m_{1},m_{2})$
$\displaystyle=-\frac{\Lambda^{4}}{4}+4\Lambda^{2}{m_{1}}{m_{2}}-\frac{16}{3}u^{2},$
(8) $\displaystyle g(u,m_{1},m_{2})$
$\displaystyle=\Lambda^{4}\quantity(m_{1}^{2}+m_{2}^{2})-\frac{2}{3}\quantity(\Lambda^{4}+8\Lambda^{2}m_{1}m_{2})u+\frac{128}{27}u^{3}.$
(9)

Substituting (8) and (9) into the definition of the $j$-function (4), we have
an expansion of $\frac{1}{j}$

$\begin{split}\frac{1}{j}&=U^{2}-\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{3}+\quantity[\hat{m}_{1}\hat{m}_{2}\quantity(\hat{m}_{1}\hat{m}_{2}+64)-704]U^{4}\\\
&\quad-72(\hat{m}_{1}\hat{m}_{2}-16)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{5}\\\
&\quad-16\quantity[27\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-5\hat{m}_{1}^{3}\hat{m}_{2}^{3}-102\hat{m}_{1}^{2}\hat{m}_{2}^{2}+5376\hat{m}_{1}\hat{m}_{2}-18688]U^{6}\\\
&\quad-3456(\hat{m}_{1}\hat{m}_{2}-40)(\hat{m}_{1}\hat{m}_{2}-4)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{7}\\\
&\quad-2304(\hat{m}_{1}\hat{m}_{2}-4)\quantity[27\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-2\hat{m}_{1}^{3}\hat{m}_{2}^{3}+30\hat{m}_{1}^{2}\hat{m}_{2}^{2}+2880\hat{m}_{1}\hat{m}_{2}-11008]U^{8}\\\
&\quad-110592(\hat{m}_{1}\hat{m}_{2}-112)(\hat{m}_{1}\hat{m}_{2}-4)^{2}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{9}\\\
&\quad-221184(\hat{m}_{1}\hat{m}_{2}-4)^{2}\quantity[27\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-\hat{m}_{1}^{3}\hat{m}_{2}^{3}+66\hat{m}_{1}^{2}\hat{m}_{2}^{2}+2112\hat{m}_{1}\hat{m}_{2}-8576]U^{10}\\\
&\quad+955514880(\hat{m}_{1}\hat{m}_{2}-4)^{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{11}\\\
&\quad-1769472(\hat{m}_{1}\hat{m}_{2}-4)^{3}\quantity[270\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-5\hat{m}_{1}^{3}\hat{m}_{2}^{3}+816\hat{m}_{1}^{2}\hat{m}_{2}^{2}+17472\hat{m}_{1}\hat{m}_{2}-73984]U^{12}\\\
&\quad+382205952(\hat{m}_{1}\hat{m}_{2}-4)^{4}(\hat{m}_{1}\hat{m}_{2}+176)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})U^{13}\\\
&\quad-254803968(\hat{m}_{1}\hat{m}_{2}-4)^{4}\quantity[27\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-\hat{m}_{1}^{3}\hat{m}_{2}^{3}+450\hat{m}_{1}^{2}\hat{m}_{2}^{2}+7680\hat{m}_{1}\hat{m}_{2}-33536]U^{14}\\\
&\quad+\order{U^{15}},\end{split}$ (10)

where we have introduced $U\equiv\frac{\Lambda^{2}}{64u},\
\hat{m}_{i}\equiv\frac{8m_{i}}{\Lambda}.$

To relate $a$ and $u$, we use the fact MasudaSuzuki  that
$\partialderivative{a}{u}$ is given by

$\partialderivative{a}{u}=\frac{F\quantity(\frac{1}{12},\frac{5}{12},1;\frac{12^{3}}{j})}{\sqrt{2}\quantity(-3f(u,m_{1},m_{2}))^{\frac{1}{4}}},$
(11)

where $F(a,b,c;z)$ is a hypergeometric function. Since we have the
$U$-expansion of $\frac{1}{j}$ (10), we can obtain the $U$-expansion of $a$
after substituting (10) into (11) and integrating with respect to $U$.
Therefore, defining $A\equiv\frac{\Lambda}{8\sqrt{2}a}$, $U$ is conversely
expanded by this $A$ as

$\begin{split}U&=A^{2}-8(\hat{m}_{1}\hat{m}_{2}+1)A^{6}+24\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})A^{8}+24\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}-8\hat{m}_{1}\hat{m}_{2}+1)A^{10}\\\
&\quad+16\quantity(39+4\hat{m}_{1}\hat{m}_{2})\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})A^{12}\\\
&\quad-8\quantity[81\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})+56\hat{m}_{1}^{3}\hat{m}_{2}^{3}+660\hat{m}_{1}^{2}\hat{m}_{2}^{2}+1128\hat{m}_{1}\hat{m}_{2}+56]A^{14}+\order{A^{15}}.\end{split}$
(12)

By using this expression in (10), we obtain $A^{2}$-expansion of
$\frac{1}{j}$. Thus, substituting this expansion into (4), we find
$A^{2}$-expansion of $q$,

$\begin{split}q&=A^{4}-\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})A^{6}+\quantity[\quantity(a_{0}-720)+48\hat{m}_{1}\hat{m}_{2}+\hat{m}_{1}^{2}\hat{m}_{2}^{2}]A^{8}\\\
&\quad+\quantity[\quantity(1224-2a_{0})\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})-48\hat{m}_{1}\hat{m}_{2}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})]A^{10}\\\
&\quad+\left[\quantity(321648-1440a_{0}+a_{0}^{2}+a_{1})+4(a_{0}-120)\hat{m}_{1}^{2}\hat{m}_{2}^{2}+32(3a_{0}-2056)\hat{m}_{1}\hat{m}_{2}\right.\\\
&\quad\quad\left.+48\hat{m}_{1}^{3}\hat{m}_{2}^{3}+\quantity(a_{0}-504)\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})\right]A^{12}\\\
&\quad+\left[\quantity(-666024+3888a_{0}-3a_{0}^{2}-3a_{1})\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})-\quantity(744+2a_{0})\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{4}+\hat{m}_{2}^{2}\hat{m}_{1}^{4})\right.\\\
&\quad\quad\left.+\quantity(114944-192a_{0})\quantity(\hat{m}_{1}\hat{m}_{2}^{3}+\hat{m}_{2}\hat{m}_{1}^{3})\right]A^{14}+\order{A^{16}},\end{split}$
(13)

whose coefficients are integer-coefficient polynomials of the moonshine
Fourier coefficients $a_{n}$.

Next, we take logarithm of $q$ as the complex modulus $\tau$ is given by
$q=e^{2\pi i\tau}$. We have

$\begin{split}2\pi
i\tau&=\log\quantity(A^{4})-\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})A^{2}+\frac{1}{2}\quantity(2a_{0}-1440-\hat{m}_{1}^{4}-\hat{m}_{2}^{4}+96\hat{m}_{1}\hat{m}_{2})A^{4}\\\
&\quad-\frac{1}{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})\quantity(3a_{0}-1512+\hat{m}_{1}^{4}+\hat{m}_{2}^{4}-\hat{m}_{1}^{2}\hat{m}_{2}^{2})A^{6}\\\
&\quad+\frac{1}{4}\left[\quantity(2a_{0}^{2}-2880a_{0}+4a_{1}+249792)-\hat{m}_{1}^{8}-\hat{m}_{2}^{8}+\quantity(4a_{0}+384)\hat{m}_{1}^{2}\hat{m}_{2}^{2}\right.\\\
&\quad\quad\left.+\quantity(192a_{0}-124928)\hat{m}_{1}\hat{m}_{2}\right]A^{8}\\\
&\quad-\frac{1}{5}\left(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}\right)\left[\quantity(5a_{0}^{2}-6120a_{0}+10a_{1}-92520)+\hat{m}_{1}^{8}+\hat{m}_{2}^{8}\right.\\\
&\quad\quad\left.+\quantity(240a_{0}-124800)\hat{m}_{1}\hat{m}_{2}+\hat{m}_{1}^{4}\hat{m}_{2}^{4}-\hat{m}_{1}^{6}\hat{m}_{2}^{2}-\hat{m}_{1}^{2}\hat{m}_{2}^{6}\right]A^{10}\\\
&\quad+\frac{1}{6}\left[-\hat{m}_{1}^{12}-\hat{m}_{2}^{12}+\quantity(288a_{0}-87552)\hat{m}_{1}^{3}\hat{m}_{2}^{3}\right.\\\
&\quad\quad\left.+\quantity(288a_{0}^{2}-394752a_{0}+576a_{1}+22620672)\hat{m}_{1}\hat{m}_{2}\right.\\\
&\quad\quad\left.+\quantity(12a_{0}^{2}-2880a_{0}+24a_{1}-7466688)\hat{m}_{1}^{2}\hat{m}_{2}^{2}\right.\\\
&\quad\quad\left.+\quantity(3a_{0}^{2}-3024a_{0}+6a_{1}-390096)\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})\right.\\\
&\quad\quad\left.+\quantity(2a_{0}^{3}-4320a_{0}^{2}+12a_{0}a_{1}+1929888a_{0}-8640a_{1}+6a_{2}-53824512)\right]A^{12}\\\
&\quad+\order{A^{14}}.\end{split}$ (14)

Integrating (14) with respect to $a$ twice, we can finally derive $2\pi i$
times the $N_{f}=2$ prepotential.

Now let us show the results of our calculations. The $N_{f}=2$ prepotential
$\mathcal{F}^{N_{f}=2}$ takes the form

$\mathcal{F}^{N_{f}=2}=\frac{ia^{2}}{2\pi}\quantity[\log\quantity(\frac{a}{\Lambda})^{2}+(5\log
2-2+i\pi)-\frac{\log
a}{2a^{2}}\sum_{i}^{2}m_{i}^{2}-\sum_{k=2}^{\infty}\mathcal{F}^{N_{f}=2}_{k}\quantity(\frac{\Lambda}{a})^{2k}],$
(15)

where first few instanton expansion coefficients $\mathcal{F}_{k}^{N_{f}=2}$
is given by

$\displaystyle\mathcal{F}_{2}^{N_{f}=2}$
$\displaystyle=\frac{1}{4096}+\frac{1}{32}{\tilde{m}_{1}\tilde{m}_{2}}-\frac{1}{48}\quantity(\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4}),$
(16) $\displaystyle\mathcal{F}_{3}^{N_{f}=2}$
$\displaystyle=-\frac{3}{8192}\quantity(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2})-\frac{1}{480}\quantity(\tilde{m}_{1}^{6}+\tilde{m}_{2}^{6}),$
(17) $\displaystyle\mathcal{F}_{4}^{N_{f}=2}$
$\displaystyle=\frac{5}{134217728}+\frac{5}{16384}\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}+\frac{5}{196608}\tilde{m}_{1}\tilde{m}_{2}-\frac{1}{2688}\quantity(\tilde{m}_{1}^{8}+\tilde{m}_{2}^{8}).$
(18)

Note that we define $\hat{m}_{i}=\frac{8m_{i}}{\Lambda}=8\tilde{m}_{i}$ and
$\mathcal{F}_{k}^{N_{f}=2}$ in the above expression is related to
$\mathcal{F}_{k}^{2}$ in Ohta2  as
$\mathcal{F}_{k}^{N_{f}=2}$(here)$=-\frac{2}{\Lambda^{2k}}\mathcal{F}_{k}^{2}$,
$\Lambda$(here)$=\Lambda_{2}$.

In this notation, we can obtain the coefficients of the instanton expansion
$\mathcal{F}_{k}^{N_{f}=2}$ which include the moonshine Fourier coefficients
$a_{n}$ from the integration of (14). Here we show the coefficients of the
instanton expansion of $N_{f}=2$ prepotential up to order 4:

$\displaystyle\mathcal{F}_{2}^{N_{f}=2}$
$\displaystyle=\frac{1}{(8\sqrt{2})^{4}\cdot 3\cdot
2}\cdot\frac{1}{2}\left[\quantity(2a_{0}-1440)-\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})+96\hat{m}_{1}\hat{m}_{2}\right]$
(19) $\displaystyle\mathcal{F}_{3}^{N_{f}=2}$
$\displaystyle=-\frac{1}{(8\sqrt{2})^{6}\cdot 5\cdot
4}\cdot\frac{1}{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2})\quantity[\quantity(3a_{0}-1512)+\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4})-\hat{m}_{1}^{2}\hat{m}_{2}^{2}]$
(20) $\displaystyle\mathcal{F}_{4}^{N_{f}=2}$
$\displaystyle=\frac{1}{(8\sqrt{2})^{8}\cdot 7\cdot
6}\cdot\frac{1}{4}\left[\quantity(2a_{0}^{2}-2880a_{0}+4a_{1}+249792)-\quantity(\hat{m}_{1}^{8}+\hat{m}_{2}^{8})\right.$
$\displaystyle\quad\quad\left.+\quantity(4a_{0}+384)\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\quantity(192a_{0}-124928)\hat{m}_{1}\hat{m}_{2}\right]$
(21)

Substituting the coefficients of $j$-function $a_{n}$ into above expressions,
we can derive explicit results of the instanton expansion and we have
succeeded to reproduce all known results $\mathcal{F}_{k=2}^{N_{f}=2}$ for
$k=2,3,4$ from the coefficients of $A^{4},A^{6},A^{8}$ in (14), respectably.

Indeed, substituting $a_{0}=744$ and $\hat{m}_{i}=8\tilde{m}_{i}$ into (19),
then we get

$\displaystyle\mathcal{F}_{2}^{N_{f}=2}$
$\displaystyle=\dfrac{\frac{1}{2}\left[\quantity(2a_{0}-1440)-8^{4}\quantity(\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4})+96\cdot
8^{2}\tilde{m}_{1}\tilde{m}_{2}\right]}{(8\sqrt{2})^{4}\cdot 3\cdot 2}$
$\displaystyle=\quantity[\frac{1}{4096}+\frac{1}{32}{\tilde{m}_{1}\tilde{m}_{2}}-\frac{1}{48}\quantity(\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4})]$

which is consistent with the known $k=2$ expression of $\mathcal{F}^{N_{f}=2}$
in Ohta2 . Also, the expansion coefficients $\mathcal{F}_{k}^{N_{f}=2}$ for
larger $k$ can similarly be calculated from the coefficients of $A^{2k}$ in
(14).

###  $N_{f}=3$ case

A similar result can be obtained in the case of $N_{f}=3$. The SW curve in
this case is

$y^{2}=\quantity[x^{2}-u+\frac{\Lambda}{4}\quantity(x+\frac{m_{1}+m_{2}+m_{3}}{2})]^{2}+\Lambda(x+m_{1})(x+m_{2})(x+m_{3})$
(22)

in the quartic-polynomial representation MasudaSuzuki . This is equivalent to
the Weierstrass form

$\displaystyle Y^{2}$
$\displaystyle=X^{3}+f(u,m_{1},m_{2},m_{3})X+g(u,m_{1},m_{2},m_{3}),$ (23)
$\displaystyle f(u,m_{1},m_{2},m_{3})$
$\displaystyle=-\frac{\Lambda^{4}}{768}-\frac{\Lambda^{2}}{4}\quantity(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})+4\Lambda
m_{1}m_{2}m_{3}-\frac{16u^{2}}{3}+\frac{\Lambda^{2}u}{3},$ (24) $\displaystyle
g(u,m_{1},m_{2},m_{3})$
$\displaystyle=\frac{\Lambda^{6}}{55296}+\frac{\Lambda^{4}}{192}\quantity(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})+\Lambda^{2}\quantity(m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{3}^{2}m_{1}^{2})$
$\displaystyle\quad-u\quantity[\frac{\Lambda^{4}}{144}+\frac{2}{3}\Lambda^{2}\quantity(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})+\frac{16}{3}\Lambda
m_{1}m_{2}m_{3}]$
$\displaystyle\quad-\frac{\Lambda^{3}}{12}m_{1}m_{2}m_{3}+\frac{128}{27}u^{3}+\frac{5\Lambda^{2}}{9}u^{2}.$
(25)

From the definition of the $j$-function (2) and then using (24) and (25), we
get an expansion of $\frac{1}{j}$,

$\begin{split}\frac{1}{j}&=-U+\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2}-752)U^{2}\\\
&\quad+\quantity[1472\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})-56\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}-368640]U^{3}\\\
&\quad+\left[-704\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})-2656\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})+\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\right.\\\
&\quad\quad\left.+64\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+1057792\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-149094400\right]U^{4}\\\
&\quad+8\left[-123392\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+70123520\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-7548928\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right.\\\
&\quad\quad\left.+144\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{2}+\hat{m}_{2}^{4}\hat{m}_{3}^{2}+\hat{m}_{3}^{4}\hat{m}_{1}^{2})-337408\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+21376\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-9\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.-6748635136\right]U^{5}\\\
&\quad+16\left[18688\quantity(\hat{m}_{1}^{6}+\hat{m}_{2}^{6}+\hat{m}_{3}^{6})-48361472\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+5\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\right.\\\
&\quad\quad\left.+15678832640\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+116686848\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+136704\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{2}+\hat{m}_{2}^{4}\hat{m}_{3}^{2}+\hat{m}_{3}^{4}\hat{m}_{1}^{2})-27\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{4}+\hat{m}_{2}^{4}\hat{m}_{3}^{4}+\hat{m}_{3}^{4}\hat{m}_{1}^{4})\right.\\\
&\quad\quad\left.-2016542720\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}+11020288\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.-5376\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})-20544\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+102\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-1136991928320\right]U^{6}+\order{U^{7}}\end{split}$
(26)

where we have introduced $U\equiv\frac{\Lambda^{2}}{4096u},\
\hat{m}_{i}\equiv\frac{64m_{i}}{\Lambda}.$

Similarly to the $N_{f}=2$ case, we again use (11) and then get the
$U$-expansion of $a$. Defining $A\equiv\frac{\Lambda}{64\sqrt{2}a}$, $U$ is
conversely expanded by this $A$ as

$\begin{split}U&=A^{2}-8A^{4}-8\quantity[\left(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2}\right)+\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}-7]A^{6}\\\
&\quad+8\quantity[22\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+3\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})+40\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}-48]A^{8}\\\
&\quad+8\left[3\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})-312\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-132\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.-840\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}-24\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+3\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}+323\right]A^{10}\\\
&\quad+16\left[-48\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+1806\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+1616\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+39\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{2}+\hat{m}_{2}^{4}\hat{m}_{3}^{2}+\hat{m}_{3}^{4}\hat{m}_{1}^{2})+6528\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}+728\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.+4\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})+268\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}-1080\right]A^{12}\\\
&\quad-8\left[56\quantity(\hat{m}_{1}^{6}+\hat{m}_{2}^{6}+\hat{m}_{3}^{6})-1802\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+37160\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.+58704\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})+5496\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{2}+\hat{m}_{2}^{4}\hat{m}_{3}^{2}+\hat{m}_{3}^{4}\hat{m}_{1}^{2})+56\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\right.\\\
&\quad\quad\left.+81\quantity(\hat{m}_{1}^{4}\hat{m}_{2}^{4}+\hat{m}_{2}^{4}\hat{m}_{3}^{4}+\hat{m}_{3}^{4}\hat{m}_{1}^{4})+169928\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}+45504\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.+1128\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+6176\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+41832\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}+660\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-14344\right]A^{14}+\order{A^{16}}.\end{split}$
(27)

By using this expression in (26), we obtain $A^{2}$-expansion of
$\frac{1}{j}$. Thus, substituting this expansion into (4) and taking
logarithm, we find $A^{2}$-expansion of $q$,

$\begin{split}q&=-A^{2}+\quantity[\quantity(a_{0}-744)+\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})]A^{4}\\\
&\quad+\left[\quantity(-356664+1488a_{0}-a_{0}^{2}-a_{1})+\quantity(1464-2a_{0})\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.-\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})-48\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right]A^{6}\\\
&\quad\left[\quantity(-140379008+1266864a_{0}-2232a_{0}^{2}+a_{0}^{3}-2232a_{1}+3a_{0}a_{1}+a_{2})\right.\\\
&\quad\quad\left.+\quantity(1034496-4416a_{0}+3a_{0}^{2}+3a_{1})\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})+\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\right.\\\
&\quad\quad\left.+\quantity(a_{0}-720)\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+\quantity(4a_{0}-2688)\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+\quantity(96a_{0}-70144)\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}+48\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right]A^{8}+\order{A^{10}}.\end{split}$
(28)

Its logarithm yields

$\begin{split}2\pi
i\tau&=\log\left(-A^{2}\right)+\left[\quantity(744-a_{0})-\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right]A^{2}\\\
&\quad+\frac{1}{2}\left[\quantity(159792-1488a_{0}+a_{0}^{2}+2a_{1})+\quantity(2a_{0}-1440)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.\\\
&\quad\quad\left.-\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+96\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right]A^{4}\\\
&\quad+\frac{1}{3}\left[\quantity(36893760-1069992a_{0}+2232a_{0}^{2}-a_{0}^{3}+4464a_{1}-6a_{0}a_{1}-3a_{2})\right.\\\
&\quad\quad\left.+\quantity(4392a_{0}-3a_{0}^{2}-6a_{1}-426456)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-\quantity(\hat{m}_{1}^{6}+\hat{m}_{2}^{6}+\hat{m}_{3}^{6})\right.\\\
&\quad\quad\left.+\quantity(1512-3a_{0})\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})+\quantity(103296-144a_{0})\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right]A^{6}\\\
&\quad+\frac{1}{4}\left[\left(8515094496-561516032a_{0}+2533728a_{0}^{2}-2976a_{0}^{3}+a_{0}^{4}+5067456a_{1}\right.\right.\\\
&\quad\quad\left.\left.-17856a_{0}a_{1}+12a_{0}^{2}a_{1}+6a_{1}^{2}-8928a_{2}+12a_{0}a_{2}+4a_{3}\right)+4a_{0}\hat{m}_{1}^{2}\hat{m}_{2}^{2}\hat{m}_{3}^{2}\right.\\\
&\quad\quad\left.+\left(4137984a_{0}-8832a_{0}^{2}+4a_{0}^{3}-17664a_{1}+24a_{0}a_{1}+12a_{2}\right.\right.\\\
&\quad\quad\left.\left.-132871168\right)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})-\quantity(\hat{m}_{1}^{8}+\hat{m}_{2}^{8}+\hat{m}_{3}^{8})\right.\\\
&\quad\quad\left.+\quantity(249792-2880a_{0}+2a_{0}^{2}+4a_{1})\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})\right.\\\
&\quad\quad+\left.\quantity(26941440-280576a_{0}+192a_{0}^{2}+384a_{1})\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right.\\\
&\quad\quad\left.+\quantity(454656-10752a_{0}+8a_{0}^{2}+16a_{1})\quantity(\hat{m}_{1}^{2}\hat{m}_{2}^{2}+\hat{m}_{2}^{2}\hat{m}_{3}^{2}+\hat{m}_{3}^{2}\hat{m}_{1}^{2})\right.\\\
&\quad\quad\left.+\quantity(192a_{0}-124928)\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right]A^{8}+\order{A^{10}}.\end{split}$
(29)

Integrating (29) with respect to $a$ twice, we can finally derive $2\pi i$
times the $N_{f}=3$ prepotential.

In this case, the $N_{f}=3$ prepotential $\mathcal{F}^{N_{f}=3}$ is

$\mathcal{F}^{N_{f}=3}=\frac{ia^{2}}{2\pi}\quantity[\frac{1}{2}\log\quantity(\frac{a}{\Lambda})^{2}+\frac{1}{2}(9\log
2-2-i\pi)-\frac{\log
a}{2a^{2}}\sum_{i}^{2}m_{i}^{2}-\sum_{k=2}^{\infty}\mathcal{F}^{N_{f}=3}_{k}\quantity(\frac{\Lambda}{a})^{2k}],$
(30)

where first few instanton expansion coefficients $\mathcal{F}_{k}^{N_{f}=3}$
is given in Ohta2  by

$\displaystyle\mathcal{F}_{2}^{N_{f}=3}$
$\displaystyle=\frac{1}{33554432}+\frac{1}{4096}\left({\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}}\right)+\frac{1}{32}{\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}}+\frac{1}{48}\left({\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4}+\tilde{m}_{3}^{4}}\right),$
(31) $\displaystyle\mathcal{F}_{3}^{N_{f}=3}$
$\displaystyle=-\frac{3}{33554432}\left(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}\right)-\frac{1}{32768}\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}$
$\displaystyle\quad-\frac{3}{8192}\left(\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}+\tilde{m}_{2}^{2}\tilde{m}_{3}^{2}+\tilde{m}_{3}^{2}\tilde{m}_{1}^{2}\right)-\frac{1}{480}\left(\tilde{m}_{1}^{6}+\tilde{m}_{2}^{6}+\tilde{m}_{3}^{6}\right)$
(32) $\displaystyle\mathcal{F}_{4}^{N_{f}=3}$
$\displaystyle=\frac{5}{4503599627370496}+\frac{5}{103079215104}\left(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}\right)+\frac{7}{268435456}\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}$
$\displaystyle\quad+\frac{5}{134217728}\left(\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4}+\tilde{m}_{3}^{4}\right)+\frac{25}{33554432}\left(\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}+\tilde{m}_{2}^{2}\tilde{m}_{3}^{2}+\tilde{m}_{3}^{2}\tilde{m}_{1}^{2}\right)$
$\displaystyle\quad+\frac{5}{196608}\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}\left(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}\right)+\frac{5}{16384}\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}\tilde{m}_{3}^{2}-\frac{1}{2688}\left(\tilde{m}_{1}^{8}+\tilde{m}_{2}^{8}+\tilde{m}_{3}^{8}\right).$
(33)

Note that we define $\hat{m}_{i}=\frac{64m_{i}}{\Lambda}=64\tilde{m}_{i}$ and
$\mathcal{F}_{k}^{N_{f}=3}$ in the above expression is related to
$\mathcal{F}_{k}^{3}$ in Ohta2  as
$\mathcal{F}_{k}^{N_{f}=3}$(here)$=-\frac{2}{\Lambda^{2k}}\mathcal{F}_{k}^{3}$,
$\Lambda$(here)$=\Lambda_{3}$.

In this notation, we have succeeded to reproduce all known results
$\mathcal{F}_{k=2}^{N_{f}=3}$ for $k=2,3,4$ from the coefficients of
$A^{4},A^{6},A^{8}$ in Ohta2 , respectively.

For example, we take the third term of (29) and integrating with respect to
$a$ twice, we get the coefficient of $A^{4}$ term as

$\displaystyle\mathcal{F}_{2}^{N_{f}=3}$
$\displaystyle=\frac{1}{(64\sqrt{2})^{4}\cdot 3\cdot
2}\cdot\frac{1}{2}\left[\quantity(159792-1488a_{0}+a_{0}^{2}+2a_{1})+\quantity(2a_{0}-1440)\quantity(\hat{m}_{1}^{2}+\hat{m}_{2}^{2}+\hat{m}_{3}^{2})\right.$
$\displaystyle\quad\quad\left.-\quantity(\hat{m}_{1}^{4}+\hat{m}_{2}^{4}+\hat{m}_{3}^{4})+96\hat{m}_{1}\hat{m}_{2}\hat{m}_{3}\right]$
$\displaystyle=\frac{1}{33554432}+\frac{1}{4096}\left({\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}}\right)+\frac{1}{32}{\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}}+\frac{1}{48}\left({\tilde{m}_{1}^{4}+\tilde{m}_{2}^{4}+\tilde{m}_{3}^{4}}\right)$

which corresponds to $\mathcal{F}_{2}^{N_{f}=3}$ in (31) correctly.

##  III Comparison with Nekrasov Partition Function

As is well known, the SW prepotential is also obtained by Instanton counting
by Nekrasov Nekrasov ; NekrasovOkounkov . Our results are of course in full
agreement with those obtained that way. We will see this briefly below.

### $N_{f}=3$

The instanton partition function is calculated as a sum over all possible
Young tableaus parametrized as $Y=(\lambda_{1}\geq\lambda_{2}\geq\cdots)$,
where $\lambda_{\ell}$ is the height of the $\ell$-th column. In the case of
$D=4$, ${\cal N}=2$ $SU(2)$ gauge theory with $N_{f}=3$, the instanton
partition function is 444We use the notation used in EguchiMaruyoshi , where
the terms $\epsilon_{1}L_{Y_{j}}(t)$ and $-\epsilon_{2}A_{Y_{i}}(t)$ in the
second line of eq.(A.3) in EguchiMaruyoshi  should read
$\epsilon_{1}L_{Y_{i}}(t)$ and $-\epsilon_{2}A_{Y_{j}}(t)$, respectively, and
are corrected in (35).

$Z^{N_{f}=3}_{\text{inst}}=\sum_{(Y_{1},Y_{2})}{\Lambda^{|\vec{Y}|}_{3}Z_{\text{vector}}(\vec{a},\vec{Y})Z_{\text{antifund}}(\vec{a},\vec{Y},\mu_{1})Z_{\text{antifund}}(\vec{a},\vec{Y},\mu_{2})Z_{\text{fund}}(\vec{a},\vec{Y},-\mu_{3})}.$
(34)

Here

$\begin{split}Z_{\text{vector}}(a,\vec{Y})=&\prod_{i,j=1,2}\prod_{s\in
Y_{i}}(2a\delta_{ij}-\epsilon_{1}L_{Y_{j}}(s)+\epsilon_{2}(A_{Y_{i}}(s)+1))^{-1}\\\
&\quad\quad\times\prod_{t\in
Y_{j}}(-2a\delta_{ij}+\epsilon_{1}L_{Y_{i}}(t)-\epsilon_{2}(A_{Y_{j}}(t)+1)+\epsilon_{+})^{-1},\\\
Z_{\text{fund}}(a,\vec{Y},\mu)=&\prod_{i=1,2}\prod_{s\in
Y_{i}}(a\delta_{i}+\epsilon_{1}(l-1)+\epsilon_{2}(m-1)-\mu+\epsilon_{+}),\\\
Z_{\text{antifund}}(a,\vec{Y},\mu)=&\prod_{i=1,2}\prod_{s\in
Y_{i}}(a\delta_{i}+\epsilon_{1}(l-1)+\epsilon_{2}(m-1)+\mu),\end{split}$ (35)

where we define $\epsilon_{+}=\epsilon_{1}+\epsilon_{2}$, $\delta_{1}=+1$,
$\delta_{2}=-1$ and

$\delta_{ij}=\left\\{\begin{aligned} 0\ &\text{for}\ i=j,\\\ 1\ &\text{for}\
i=1\ \text{and}\ j=2,\\\ -1\ &\text{for}\ i=2\ \text{and}\
j=1.\end{aligned}\right.$ (36)

For a box $s$ at the coordinate $(\ell,m)$, the leg-length
$L_{Y}(s)=\lambda^{\prime}_{m}-\ell$ and the arm-length
$A_{Y}(s)=\lambda_{\ell}-m$ where $\lambda^{\prime}_{m}$ is the length of the
$m$-th row. The prepotential can be obtained in the limit where the
deformation parameters go to zero (with a fixed ratio
$\epsilon_{1}/\epsilon_{2}$):

$\mathcal{F}_{\text{inst}}=\lim_{\epsilon_{1,2}\rightarrow
0}(-\epsilon_{1}\epsilon_{2})\log Z_{\text{inst}}.$ (37)

Therefore, $F^{N_{f}=3}$ is expanded as

$\begin{split}\mathcal{F}^{N_{f}=3}=&\left(\frac{1}{33554432}+\frac{\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2}}{4096}+\frac{\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}}{32}\right)\frac{\Lambda^{4}}{a^{2}}\\\
&\quad-\left(\frac{3}{33554432}(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2})+\frac{\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}}{32768}+\frac{3}{8192}\left(\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}+\tilde{m}_{2}^{2}\tilde{m}_{3}^{2}+\tilde{m}^{2}_{3}\tilde{m}^{2}_{1}\right)\right)\frac{\Lambda^{6}}{a^{4}}\\\
&\quad+\left(\frac{5}{4503599627370496}+\frac{5}{134217728}(\tilde{m}^{2}_{1}+\tilde{m}^{2}_{2}+\tilde{m}^{2}_{3})+\frac{7}{268435456}\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}\right.\\\
&\quad+\frac{5}{134217728}(\tilde{m}^{4}_{1}+\tilde{m}^{4}_{2}+\tilde{m}^{4}_{3})+\frac{25}{33554432}(\tilde{m}_{1}^{2}\tilde{m}_{2}^{2}+\tilde{m}_{2}^{2}\tilde{m}_{3}^{2}+\tilde{m}_{3}^{2}\tilde{m}_{1}^{2})\\\
&\quad+\frac{5}{196608}\tilde{m}_{1}\tilde{m}_{2}\tilde{m}_{3}(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}+\tilde{m}_{3}^{2})\left.+\frac{5}{16384}\tilde{m}^{2}_{1}\tilde{m}^{2}_{2}\tilde{m}_{3}^{2}\right)\frac{\Lambda^{8}}{a^{6}}+\mathcal{O}(\Lambda^{9}).\end{split}$
(38)

Note that we have defined $\tilde{m}_{i}\equiv\frac{\Lambda}{\sqrt{2}}\mu_{i}$
$(i=1,2,3)$ and $\Lambda=\frac{\Lambda_{3}}{4\sqrt{2}}$.

### $N_{f}=2$

Next, let us consider the case of $N_{f}=2$. We take a limit where
$\mu_{2}\rightarrow\infty$ while keeping
$\mu_{2}\Lambda_{3}\equiv\Lambda^{2}_{2}$ fixed. In this limit, the partition
function becomes:

$Z^{N_{f}=2}_{\text{inst}}=\sum_{(Y_{1},Y_{2})}\Lambda^{2|\vec{Y}|}_{2}Z_{\text{vector}}(a,\vec{Y})Z_{\text{antifund}}(a,\vec{Y},\mu_{1})Z_{\text{fund}}(a,\vec{Y},-\mu_{3}).$
(39)

Therefore, the expansion of $\mathcal{F}^{N_{f}=2}$ is

$\begin{split}\mathcal{F}^{N_{f}=2}=&\frac{1}{16}\Lambda^{2}+\left(\frac{1}{4096}+\frac{1}{32}\tilde{m}_{1}\tilde{m}_{2}\right)\frac{\Lambda^{4}}{a^{2}}-\frac{3}{8192}\left(\tilde{m}_{1}^{2}+\tilde{m}_{2}^{2}\right)\frac{\Lambda^{6}}{a^{4}}\\\
&\quad+\left(\frac{5}{134217728}+\frac{5}{196608}\tilde{m}_{1}\tilde{m}_{2}+\frac{5}{16384}\tilde{m}^{2}_{1}\tilde{m}^{2}_{2}\right)\frac{\Lambda^{8}}{a^{6}}+\mathcal{O}(\Lambda^{9}),\end{split}$
(40)

where $\tilde{m}_{1}\equiv\frac{\Lambda}{\sqrt{2}}\mu_{1}$,
$\tilde{m}_{2}\equiv\frac{\Lambda}{\sqrt{2}}\mu_{3}$ and
$\Lambda\equiv\frac{\Lambda_{2}}{2\sqrt{2}}$.

Comparing (38) with (31)$\sim$(33) and (40) with (16)$\sim$(18), they
perfectly agree except for $\sum_{i}m^{2k}_{i}$. The first term of
eq(40):$\frac{\Lambda^{2}}{16}$ is known to be required from $\text{U}(1)$
factor, but this is beyond the scope of our study and will not be discussed
further here.

As we can see above, we need to compute to the 8th order term in the instanton
expansion in order to derive all terms of our 4th order $\mathcal{F}_{4}$ in
our method. This is because Nekrasov’s instanton partition function is a
$\Lambda$-expansion while our expansion is in terms of $a$, where in the
latter the coefficients are homogeneous polynomials of $\Lambda$ and
$m_{i}$’s, as is always the case in such B-model-like approaches.

##  IV Conclusions

To summarize, we have confirmed that for both cases of $N_{f}=2$ and $3$, $q$
allows an expansion in terms of $A\equiv\frac{\Lambda}{8\sqrt{2}a}$
$(N_{f}=2)$ or $\frac{\Lambda}{64\sqrt{2}a}$ $(N_{f}=3)$ in which the
coefficients are integer-coefficient polynomials of the moonshine
coefficients. As we mentioned at the beginning of this paper, this fact
suggests an unknown relationship between the vertex operator algebra CFT and
the Liouville CFT. The details of that relationship would be worth studying
and will be discussed elsewhere.

On the other hand, the method to compute the instanton expansion of the SW
prepotential developed in SWMoonshine  and this paper is efficient and simple.
We have not only re-derived all the known expansions of the $N_{f}=0,\ldots,3$
prepotentials but found some typos in the literature. For instance, in the
$N_{f}=3$ prepotential presented in section 6 of CDH : (1)
$-a^{2}(m_{1}^{2}m_{2}^{2}+\cdots)$ in $[\cdots]$ of ${\cal F}^{(2)}$ should
read $-3a^{2}(m_{1}^{2}m_{2}^{2}+\cdots)$. (2) In $\\{\cdots\\}$ of ${\cal
F}^{(5)}$, the term $-6600a^{6}m_{1}^{2}m_{2}^{2}m_{3}^{2}$ is missing. (3)
$330(m_{1}^{4}m_{2}^{2}+\cdots)$ in $\\{\cdots\\}$ of ${\cal F}^{(6)}$ should
read $3300(m_{1}^{4}m_{2}^{2}+\cdots)$. (4) Also in ${\cal F}^{(6)}$,
$2310(m_{1}^{2}m_{2}^{2}+\cdots)$ should be $2310(m_{1}^{6}m_{2}^{6}+\cdots)$.
It is also easy and straight-forward to derive higher order terms by our
method, and will also be applied to E-string theory.

There are several possible directions to extend the present study. One
direction is to consider a theory with $SU(3)$ or a higher rank gauge group
instead of $SU(2)$. This would generally lead to a high genus SW curve, which
would not fit with the elliptic functions involving the Monstrous moonshine.
However, it should be mentioned that ref.Elias2010.06598  considers special
loci of $SU(3)$ Coulomb branch that parametrize a family of elliptic curves.

The Hauptmoduln for various rational elliptic surfaces (see
e.g.Horia2203.03755 ) could also be similarly used as an analogue of the
$j$-function to compute prepotentials of the corresponding SW theories. For
example, the $N_{f}=2$ theory we considered in this paper has the congruence
subgroup $\Gamma(2)$ as the monodromy group, whose corresponding Hauptmodul is
known to be the modular $\lambda$-function. Thus, instead of the $j$-function,
we could use it to expand $q$ or the prepotential in some variable, in which
the coefficients would now be polynomials of the coefficients of the McKay-
Thompson series, and hence of the characters of the Monster.

Finally, let us comment on other moonshine phenomena MathieuMoonshine ;
UmbralMoonshine  than the Monstrous moonshine. Since we used the special
properties of the modular $j$-function to compute the instanton partition
functions, we were naturally led to think about the Monstrous moonshine. On
the other hand, in other moonshine phenomena, the Mathieu moonshine for
example, the expansion of the elliptic genus of K3 in terms of ${\cal N}=4$
superconformal characters, or equivalently the $q$-expansion of a certain mock
modular form, gives coefficients which are related to dimensions of
representations of $M_{24}$, the Mathieu group. As already mentioned in
Introduction, this connection between the mock modular form and the (twisted)
SUSY gauge theory is very different from that between the $j$-function and
SUSY gauge theories in our discussion.

We thank Kazuhiro Sakai for valuable discussions on the Mathieu moonshine and
E-string theory. The authors also thank K. Ishiguro and R. Kuramochi for
discussions.

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# Direct Antenna Frequency-Hopped M-FSK Modulation with Time-Modulated Arrays

Roberto Maneiro-Catoira, Julio Brégains,
José A. García-Naya, and Luis Castedo Manuscript received 15 October 2023;
accepted 2 November 2023. Date of publication 6 November 2023; date of current
version 5 February 2024. This work was supported in part by Grant ED431C
2020/15 funded by Xunta de Galicia and ERDF Galicia 2014-2020; in part by
Grant PID2022- 137099NB-C42 (MADDIE) and Grant TED2021-130240B-I00 (IVRY)
funded by MCIN/AEI/10.13039/501100011033; in part by the European Union
NextGenerationEU/PRTR; and funding for open access charge: Universidade da
Coruña/CISUG. (Corresponding author: José A. García-Naya.) The authors are
with the Universidade da Coruña (University of A Coruña), CITIC Research
Center, 15071 A Coruña, Spain (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>luis.castedo@udc.es). Digital Object
Identifier 10.1109/LAWP.2023.3330435

###### Abstract

We present an innovative approach that simultaneously enables direct antenna
frequency hopped M-ary frequency shift keying (DAFH-MFSK) modulation and
beamsteering through the use of time-modulated arrays (TMAs). The distinctive
feature of our approach lies in the modulation of the TMA excitations with
binary periodic sequences which can be easily frequency-adjusted and time-
delayed to simultaneously allow for DAFH-MFSK direct antenna modulation and
beamsteering. Notably, our TMA proposal offers a distinct advantage over
conventional architectures in terms of performance metrics, including reduced
insertion losses and enhanced phase resolution for beam steering, while also
simplifying hardware complexity.

###### Index Terms:

Time-modulated arrays, direct antenna modulation, frequency hopping, frequency
shift keying modulation, beamsteering.

pt This work is licensed under a Creative Commons Attribution 4.0 License. For
more information, see https://creativecommons.org/licenses/by/4.0/.
The final published version of this work is available open access, with the
same license, at https://doi.org/10.1109/LAWP.2023.3330435

##  I Introduction

Low-power and low-data-rate internet of things (IoT) wireless devices widely
use frequency-shift keying (FSK) modulation due to their simplicity and
resilience to noise and attenuation [1]. In standard M-ary FSK (MFSK) [2,
Chapter 5], a constant-amplitude sine carrier with frequency
$f_{c}{+}f^{m}_{\text{FSK}}=f_{c}{+}m\Delta f_{\text{FSK}}$,
$m{\in}\mathcal{M}{=}\\{1,2,\dots,M\\}$ is selected every symbol period
$T_{s}$, depending on which symbol is to be transmitted, being $f_{c}$ the
reference (or base) carrier frequency, and $\Delta f_{\text{FSK}}$ the
separation between two adjacent $f^{m}_{\text{FSK}}$ values. Accordingly,
$B_{\text{FSK}}{=}M\Delta f_{\text{FSK}}$ corresponds to the total FSK
bandwidth.

Despite their advantages, MFSK signals can be easily intercepted and,
moreover, can be seriously distorted by frequency selective channels. These
drawbacks can be overcome by means of frequency-hopping (FH) techniques [2,
Chapter 13], which randomly change the carrier frequency in every hop period
$T_{h}$. We assume hopping frequencies that are randomly selected from a set
of equally spaced frequencies $f_{\text{FH}}^{k}{=}(k{-}1)\Delta
f_{\text{FH}}$, $k{\in}\mathcal{K}{=}\\{1,2,\dots,K\\}$ with $\Delta
f_{\text{FH}}$ being the separation between any two adjacent
$f_{\text{FH}}^{k}$. We consider $\Delta f_{\text{FH}}{=}M\Delta
f_{\text{FSK}}$, hence the transmit frequency for every $T_{s}$ in FH-FSK is
$f_{\text{FH-
FSK}}^{mk}{=}f_{c}{+}f_{\text{FSK}}^{m}{+}f^{k}_{\text{FH}}{=}f_{c}{+}[m{+}(k{-}1)M]\Delta
f_{\text{FSK}}$, whereas $W{=}K\Delta f_{\text{FH}}{=}KB_{\text{FSK}}$ is the
total transmission bandwidth. We focus on slow FH where $T_{h}{=}LT_{s}$
($L{\in}\mathbb{N}$), as shown in Fig. 1, which contains the time-frequency
plot of an FH-FSK transmission. Since demodulation requires knowledge of the
pseudo-random FH pattern, FH-FSK prevents eavesdropping while increases
robustness to frequency-selective channels.

Figure 1: Time-frequency diagram for slow FH-FSK modulation with $M{=}4$,
$K{=}3$, and $L{=}4$. The transmitted MFSK symbols are denoted by $s_{m}$,
$m{\in}\mathcal{M}$. Figure 2: Proposed TMA architecture to jointly perform
DAFH-MFSK modulation and beamsteering. The schematic of a TM-FN module is
detailed in Fig. 3b). For each modulating waveform $h_{n}^{mk}(t)$,
$n{\in}\mathcal{N}$ denotes the corresponding antenna element,
$m{\in}\mathcal{M}$ is the FSK symbol transmitted, and $k{\in}\mathcal{K}$ is
the hop frequency slot. In the text, the values of $m$ and $k$ will be single
digits so that the notation with the superscript $mk$ is not misleading.
Figure 3: (a) Periodical ($T_{\text{TMA}}^{mk}$) six-level LP-TM signal which
allows for the direct transmission of the $m$-th FSK symbol over the $k$-th FH
slot using a TMA. Notice that only $\measuredangle h^{mk}(t)$ is sketched
since $|h^{mk}(t)|{=}1$ $\forall m,k$. To additionally perform beamsteering,
in the $n$-th antenna element, $h^{mk}(t)$ is subject to a time-delay
$D^{mk}_{n}$, i.e., $h_{n}^{mk}(t){=}h^{mk}(t{-}D_{n}^{mk})$. (b) Switched TM-
FN of the $n$-th TMA element, which consists of two SP3T switches and two SPDT
switches or, equivalently, six SPDT switches, to time modulate an input signal
$s(t)$ with $h^{mk}_{n}(t)$; and (c) simplified block diagram of (b).

Another concept, in line with IoT, is direct antenna modulation (DAM), which
consists in modulating the carrier in the antenna itself [3, 4]. DAM replaces
baseband modulation and significantly reduces transmission hardware (HW)
complexity, cost, and power consumption [5, 6, 7]. Furthermore, since
modulation occurs after amplification, power amplifiers only need to amplify a
single carrier, hence avoiding wideband PAs.

This letter combines FH-FSK and DAM into a transmission method termed DAFH-
MFSK and proposes an innovative approach for its implementation using a highly
efficient and versatile single sideband (SSB)111SSB TMAs remove unwanted
frequency-mirrored beam patterns produced by conventional TMAs to achieve high
efficiency levels. TMA [8, 9, 10, 11, 12, 13] which, in addition, is capable
of performing beamsteering.

##  II Joint DAFH-MFSK Modulation and Beamsteering: A TMA Approach

###  II-A SSB Time Modulating Feeding Network

Fig. 2 plots the proposed SSB TMA architecture for DAFH-MFSK, equipped with
periodic time-modulating feeding networks (TM-FNs) to jointly perform DAFH-
MFSK modulation and beam steering. We consider a linear array of $N$ isotropic
elements with unitary static excitations, $I_{n}{=}1$, $n{\in}\mathcal{N}$,
and $\mathcal{N}{=}\\{1,\dots,N\\}$. The $n$-th element excitation is
modulated during a symbol period $T_{s}$ by the periodic pulsed signal
$h_{n}^{mk}(t)$, which is a time-shifted version of the stair-step
approximation to $h^{mk}(t)$ (see Fig. 3a), an LP-TM waveform [12] with unit
amplitude and phase varying from $0$ to $5\pi/3$ (six steps). Therefore,
$h_{n}^{mk}(t){=}h^{mk}(t{-}D_{n}^{mk})$, being $D_{n}^{mk}$ a variable time-
delay. Notice that $h_{n}^{mk}(t)$ has a fundamental period
$T_{\text{TMA}}^{mk}{=}1/f_{\text{TMA}}^{mk}{\ll}T_{s}$, where
$m{\in}\mathcal{M}$ refers to the transmitted $m$-th FSK-modulated level, and
$k{\in}\mathcal{K}$ accounts for the hop frequency slot selected during
$T_{s}$.

The synthesis of $h_{n}^{mk}(t)$ is described using a switched TM-FN (see Fig.
3b). Considering the rectangular pulse signal $p^{mk}(t){=}1$ when
$0{\leq}t{<}T_{\text{TMA}}^{mk}/6$, and $p^{mk}(t){=}0$ otherwise, we can
express a single period of $h^{mk}(t)$ as
$\sum_{l=0}^{5}p^{mk}(t{-}\tfrac{lT_{\text{TMA}}^{mk}}{6})e^{j\frac{2\pi
l}{6}}$. Hence, the exponential Fourier series coefficients of the periodic
signal $h^{mk}(t)$ are:

$\displaystyle H_{q}^{mk}=\begin{cases}\frac{6}{\pi q}\sin\left(\frac{\pi
q}{6}\right)e^{j\frac{\pi q}{6}},&q{\in}\Psi,\\\
0,&q{\notin}\Psi;\Psi{=}\\{q{=}6i{+}1;i{\in}\mathbb{Z}\\}\end{cases}$ (1)

The Fourier coefficients $H_{q}^{mk}$ are the same for all values of $m$ and
$k$ because $h^{mk}(t)$ is always a time-scaled version of the waveform shown
in Fig. 3a. In view of Eq. 1, the Fourier coefficients of
$h_{n}^{mk}(t){=}h^{mk}(t{-}D_{n}^{mk})$ are given by

$\displaystyle H_{nq}^{mk}=\begin{cases}H_{q}^{mk}e^{-j2\pi
qf_{\text{TMA}}^{mk}D_{n}^{mk}},&q\in\Psi,\\\ 0,&q\notin\Psi\end{cases}$ (2)

and the exponential Fourier series expansion of $h_{n}^{mk}(t)$ is

$h_{n}^{mk}(t)=\sum_{q\in\Psi}H_{q}^{mk}\mathrm{e}^{-\mathrm{j}2\pi
qf_{\text{TMA}}^{mk}D_{n}^{mk}}\mathrm{e}^{\mathrm{j}2\pi
qf_{\text{TMA}}^{mk}t}.$ (3)

According to LABEL:eq:Fourier_coefficients_of_h^{mk}(t), Fig. 4 shows the
normalized Fourier series power spectrum of $h_{n}^{mk}(t)$ in dB, namely
$20\log_{10}\left|{H^{mk}_{nq}}/{H^{mk}_{n1}}\right|$. We can see, apart from
the SSB property of the waveform, that the most meaningful unwanted harmonic
is the one with order $q{=}-5$, whose relative level is at $-13.97$ dB with
respect to the useful harmonic $q{=}1$.

Figure 4: Normalized Fourier series power spectrum of $h_{n}^{mk}(t)$. All
unwanted harmonics have a minimum rejection level of $13.97$ dB with respect
to the exploited harmonic at $q{=}1$. Notice that $\left|H^{mk}_{nq}\right|$
is the same for all values of $m$, $k$, and $n$ for a given $q$ because
$h_{n}^{mk}(t)$ is either a time-scaled version (when $m$ and/or $k$ changes)
or a time-shifted version (when $n$ changes) of the same waveform.

TMA periodic modulating signals, $h_{n}^{mk}(t)$, have the following features:
(1) they have no frequency-mirrored harmonics (hence the term SSB) and their
first positive harmonic concentrates almost all the transmitted energy; (2)
this harmonic is located at
$f_{\text{TMA}}^{mk}{=}f_{\text{FH}}^{k}{+}f^{m}_{\text{FSK}}$, hence the TMA
transmits the $m$-th level within the $k$-th FH slot; and (3) the phase term
of the first positive Fourier coefficient of $h_{n}^{mk}(t)$ is proportional
to the time delay $D_{n}^{mk}$ (see
LABEL:eq:Fourier_coefficients_of_h^{mk}(t)), which is instrumental to
determine the steering direction of the TMA beampattern and, unlike digital
variable phase shifters in standard phased arrays, $D_{n}^{mk}$ can be
adjusted almost continuously [14, 15].

In addition to the rejection threshold of unwanted harmonics shown in Fig. 4,
the time-modulation efficiency of the TMA is
$\eta_{\text{TM}}{=}\mathcal{P}_{U}^{\text{TM}}/\mathcal{P}_{R}^{\text{TM}}$,
where $\mathcal{P}_{U}^{\text{TM}}$ and $\mathcal{P}_{R}^{\text{TM}}$ are the
respective useful and total average power radiated by the TMA. According to
[16, Eq. 16], $\eta_{\text{TM}}$ is given by

$\eta_{\text{TM}}={\left|H^{mk}_{n1}\right|^{2}}/{\sum_{q\in\Psi}\left|H^{mk}_{nq}\right|^{2}}.$
(4)

Since $h_{n}^{mk}(t)$ has unit amplitude, then

$\frac{1}{T_{\text{TMA}}^{mk}}\int_{0}^{T_{\text{TMA}}^{mk}}|h_{n}^{mk}(t)|^{2}dt=\sum_{q\in\Psi}|H_{nq}^{mk}|^{2}=1,$
(5)

thus obtaining (see LABEL:eq:Fourier_coefficients_of_h^{mk}(t) and Eq. 4)

$\eta_{\text{TM}}=|H_{n1}^{mk}|^{2}=[6/\pi\cdot\sin(\pi/6)]^{2}=0.912.$ (6)

This means that the proposed SSB TMA architecture ensures that more than $91$
% of the total energy is transmitted over the first positive harmonic. On the
basis of this result, the following approximation is applicable in Eq. 3

$h_{n}^{mk}(t)\approx H_{1}^{mk}\mathrm{e}^{-\mathrm{j}2\pi
f_{\text{TMA}}^{mk}D_{n}^{mk}}\mathrm{e}^{\mathrm{j}2\pi
f_{\text{TMA}}^{mk}t}$ (7)

###  II-B Signal Radiated During a Symbol Period

As shown in Fig. 2, the input to the proposed SSB TMA is the single-frequency
carrier signal $s(t){=}\mathrm{e}^{\mathrm{j}2\pi f_{c}t}$. During a symbol
period $T_{s}$, the TMA excitations are time modulated by $h_{n}^{mk}(t)$ and
the signal radiated by the TMA in the spatial direction $\theta$ is given by

$x^{mk}(\theta,t)=\sum_{n=1}^{N}h_{n}^{mk}(t)\mathrm{e}^{\mathrm{j}\beta_{c}z_{n}\sin\theta}s(t)$
(8)

where $z_{n}$ is the position of the $n$-th array element on the $z$ axis and
$\beta_{c}{=}2\pi/\lambda_{c}$ is the wavenumber for a carrier wavelength
$\lambda_{c}{=}\mathrm{c}/f_{c}$. Normalizing Equation 7 with respect to
$H_{1}^{mk}$, which is the same for all values of $m$ and $k$ (see Eq. 1), Eq.
Equation 8 can be rewritten as

$\displaystyle x^{mk}(\theta,t)$
$\displaystyle\approx\sum_{n=1}^{N}\mathrm{e}^{\mathrm{j}2\pi\left(\frac{z_{n}}{\lambda_{c}}\sin\theta{-}f_{\text{TMA}}^{mk}D_{n}^{mk}\right)}\mathrm{e}^{\mathrm{j}2\pi\left(f_{c}{+}f_{\text{TMA}}^{mk}\right)t}$
$\displaystyle\approx\underbrace{\mathrm{AF}^{mk}(\theta)}_{\text{BS}}\hskip
14.22636pt\cdot\underbrace{\vphantom{\mathrm{AF}^{mk}(\theta)}\mathrm{e}^{\mathrm{j}2\pi
f_{\text{FH-FSK}}^{mk}t}}_{\text{DAFH-MFSK modulation}}$ (9)

where the term
$\mathrm{AF}^{mk}(\theta){=}\sum_{n=1}^{N}\mathrm{e}^{\mathrm{j}2\pi\left(\frac{z_{n}}{\lambda_{c}}\sin\theta{-}f_{\text{TMA}}^{mk}D_{n}^{mk}\right)}$
is the spatial array factor during $T_{s}$ and provides the beamsteering
ability to the TMA. Indeed, the maximum of the radiation pattern can be
pointed to the direction $\theta_{0}$ by adjusting the delays $D_{n}^{mk}$,
$n{\in}\mathcal{N}$, so that the following equation is satisfied

$\mathrm{e}^{\mathrm{j}2\pi
f_{\text{TMA}}^{mk}D_{n}^{mk}}=\mathrm{e}^{\mathrm{j}2\pi\frac{z_{n}}{\lambda_{c}}\sin\theta_{0}}$
(10)

in which case the array factor is

$\mathrm{AF}^{mk}(\theta)=\sum_{n=1}^{N}\mathrm{e}^{\mathrm{j}2\pi\frac{z_{n}}{\lambda_{c}}\left(\sin\theta-\sin\theta_{0}\right)}$
(11)

On the other hand, the term $\mathrm{e}^{\mathrm{j}2\pi f_{\text{FH-
FSK}}^{mk}t}$ in Section II-B allows the TMA to transmit the $m$-th FSK level
over the $k$-th FH slot, and thus perform DAFH-MFSK modulation.

Figure 5: (a) Time delays $D_{n}^{m2}$ specified in Table I as a function of
$n{\in}\mathcal{N}$ and time. During $T_{h}$, the sequence of symbols $s_{m}$,
corresponding to the frequencies $f_{\text{FSK}}^{m}$, $m{=}\\{1,2,3,4\\}$,
are transmitted. (b) Relative power radiated pattern of the proposed TMA. When
its modulating waveforms are subject to $D_{n}^{m2}$ during the considered
$T_{h}$, the beampattern points towards $\theta_{0}{=}30^{\circ}$.

##  III Case Study and Comparative Analysis

This section has a twofold purpose: (1) Demonstrate the feasibility of the
proposed technique by means of numerical simulations, and (2) compare it with
conventional architectures performing FH-MFSK and beamsteering simultaneously.

###  III-A Numerical Example

Let us consider a DAFH-MFSK modulator architecture based on TM-FNs (refer to
Fig. 2). We assume the following parameters: carrier frequency: $f_{c}{=}2.5$
GHz; number of antenna elements: $N{=}4$, spaced $\lambda_{c}/2$ apart;
modulation scheme: 4-FSK ($M{=}4$); hopping frequencies: 6 possible values
($K{=}6$); symbol period: $T_{s}{=}10$ ms; hop duration: $L{=}4$, thus
$T_{h}{=}4T_{s}{=}4$ ms; frequency separation for FSK: $\Delta
f_{\text{FSK}}{=}50$ kHz, with $f^{1}_{\text{FSK}}{=}50$ kHz; and frequency
spacing for FH: $\Delta f_{\text{FH}}{=}200$ kHz, with $f^{1}_{\text{FH}}{=}0$
Hz. The minimum frequency of the TMA modulating waveforms is given by
$f_{\text{TMA}}^{11}{=}f_{\text{FH}}^{1}{+}f^{1}_{\text{FSK}}{=}50$ kHz,
corresponding to their maximum possible period
$T_{\text{TMA}}^{\text{max}}{=}T_{\text{TMA}}^{11}{=}1/f_{\text{TMA}}^{11}{=}20\,\text{\textmu
s}\ll T_{s}{=}10$ ms. Additionally, the TMA offers a minimum period of
$T_{\text{TMA}}^{\text{min}}{=}T_{\text{TMA}}^{46}{=}1/f_{\text{TMA}}^{46}{=}833$
ns.

As an example, let us consider that the TMA radiates sequentially the four
4-FSK levels $m{=}\\{1,2,3,4\\}$ over the second ($k{=}2$) FH slot of duration
$T_{h}$ and centered at frequency
$f_{\text{FH}}^{2}{=}f_{\text{FH}}^{1}{+}\Delta f_{\text{FH}}{=}250$ kHz. The
frequencies to be transmitted and the antenna elements time delays
$D_{n}^{m2}$, $n{\in}\mathcal{N}$, (determined according to Eq. 10) to point
the maximum of the radiation pattern towards $\theta_{0}{=}30^{\circ}$ are
shown in Fig. 5 and summarized in Table I.

Figure 6: Block diagram of a conventional FH-MFSK transmitter followed by a standard phased array with digitally tuned passive VPS to perform beamsteering. TABLE I: Frequencies and time delays $D_{n}^{m2}$ (nanoseconds) of the TMA modulating waveforms for the hop frequency $f_{\text{FH}}^{2}{=}200\,\mathrm{kHz}$ and the four 4-FSK levels $m=\\{1,2,3,4\\}$ to steer the beam towards $\theta_{0}{=}30^{\circ}$ (see Fig. 5)  $\boldsymbol{m}$ | $\boldsymbol{f_{\text{FSK}}^{m}{+}f_{\text{FH}}^{2}}$ | $\boldsymbol{D_{1}^{m2}}$ | $\boldsymbol{D_{2}^{m2}}$ | $\boldsymbol{D_{3}^{m2}}$ |  $\boldsymbol{D_{4}^{m2}}$[.5]()
---|---|---|---|---|---
[.5]() 1 | 250 kHz | 500 | 1000 | 1500 | 2000
[.5]() 2 | 300 kHz | 417 | 833 | 1250 | 1667
[.5]() 3 | 350 kHz | 357 | 714 | 1071 | 1429
[.5]() 4 | 400 kHz | 313 | 625 | 938 | 1250

###  III-B Comparison with Conventional Techniques

Fig. 6 shows the block diagram of a conventional FH-MFSK transmitter followed
by a standard phased array equipped with digitally tuned passive VPSs to
perform beamsteering. Every $T_{s}$, the incoming binary data bits are
employed, via a multiplexer (MUX), to select the transmitting carrier
frequency from a pool of $M$ possibilities. The FH-MFSK signal is generated by
mixing the resulting MFSK modulated signal with a carrier obtained from a
digital frequency synthesizer under the control of a code generator.

Given that the mixer produces both sum and difference frequency components,
but only the sum frequency is intended for radiation, a bandpass filter (BPF)
is placed after the mixer. Following amplification through the PA, the signal
is radiated to a given direction by adjusting the VPSs within the standard
phased array.

Compared with the conventional scheme in Fig. 6, our proposed TMA approach in
Fig. 2 offers several key advantages:

  1. 1.

Reduced hardware complexity: In our approach, there is no need for a MUX,
mixer, frequency synthesizer, or BPF, and the PA only needs to amplify a
single carrier.

  2. 2.

Minimal oscillator requirements: Unlike the conventional scheme illustrated in
Fig. 6, which necessitates multiple oscillators corresponding to the number of
MFSK levels, $M$, our approach requires just one oscillator. Refer to Table II
where we employ big $O$ notation [17] for a comprehensive hardware comparison.

  3. 3.

Reduced number of SPDT switches: The number of SPDT swtches in our approach is
only half that of the conventional scheme employing 6-bit VPSs (minimum
resolution comparable to that of the TMA [14]). This results in a linear
complexity, as detailed in Table II.

  4. 4.

Improved insertion losses: For both the conventional architecture and the TMA
approach, we consider off-the-shelf devices with the lowest insertion loss,
denoted as $\eta$, within the specified frequency band. Specifically,
$\eta_{\text{MUX}}{=}0.7$ dB ( single-pole four-throw (SP4T) switch) [18],
$\eta_{\text{mixer}}{=}4.5$ dB [19], $\eta_{\text{BPF}}{=}2$ dB [20],
$\eta_{\text{VPS}}{=}4$ dB [18], $\eta_{\text{SPDT}}{=}0.5$ dB [18], and ideal
$1{:}N$ power splitters are taken into account.

Under these circumstances, the conventional architecture exhibits an insertion
loss of
$\eta_{\text{conv}}=\eta_{\text{MUX}}+\eta_{\text{mixer}}+\eta_{\text{BPF}}+\eta_{\text{VPS}}=11.2$
dB. In contrast, the TMA approach [21] demonstrates significantly lower
insertion losses, with
$\eta_{\text{TMA}}=-10\log_{10}\eta_{\text{TM}}+6\eta_{\text{SPDT}}=3.4$ dB.
Consequently, assuming equal power levels in the respective carrier signals,
the same PA gain, and that the performance of all components remains
consistent across the entire bandwidth, the proposed TMA approach achieves a
substantial insertion loss reduction of
$\Delta\eta{=}\eta_{\text{TMA}}{-}\eta_{\text{conv}}{=}-7.8$ dB. This
insertion loss reduction leads to a significant performance improvement of
MFSK demodulation in terms of bit error ratio (BER) versus received signal-to-
noise ratio (SNR) per bit, as illustrated in Fig. 7.

TABLE II: Comparative advantages of the proposed TMA approach. FH-MFSK & BS Scheme | # Oscillators | # SPDT switches |  $\eta$ (dB)
---|---|---|---
Conventional (Fig. 6) | $M\in O(M)$ | $12N\in O(N)$ | 11.2
This work: TMA approach (Fig. 2) | $1\in O(1)$ | $6N\in O(N)$ | 3.4
Figure 7: Performance of MFSK demodulation for the proposed TMA approach (see
Fig. 2) and the conventional one (see Fig. 6) in terms of BER versus received
SNR per bit in the conventional scheme.

##  IV Conclusions

We have introduced an innovative TMA approach that seamlessly combines DAFH-
MFSK modulation and beamsteering (BS), making it particularly well suited for
low-power and low-data-rate applications. This approach offers several key
advantages over comparable existing architectures, including better energy
efficiency, simplified design, and the ability to achieve continuous phase-
sensitivity beamsteering.

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  *[IoT]: internet of things
  *[FSK]: frequency-shift keying
  *[MFSK]: M-ary FSK
  *[FH]: frequency-hopping
  *[TMA]: time-modulated array
  *[DAFH-MFSK]: direct antenna frequency hopped M-ary frequency shift keying
  *[TM-FN]: time-modulating feeding network
  *[LP-TM]: linear-phase time-modulating
  *[SP3T]: single-pole triple-throw
  *[SPDT]: single-pole dual-throw
  *[DAM]: direct antenna modulation
  *[HW]: hardware
  *[PAs]: power amplifier
  *[SSB]: single sideband
  *[TMAs]: time-modulated array
  *[TM-FNs]: time-modulating feeding network
  *[VPS]: variable phase shifter
  *[VPSs]: variable phase shifter
  *[MUX]: multiplexer
  *[BPF]: bandpass filter
  *[PA]: power amplifier
  *[SP4T]: single-pole four-throw
  *[BER]: bit error ratio
  *[SNR]: signal-to-noise ratio
  *[BS]: beamsteering