We construct the Picard-Fuchs equations of the $N=2$ supersymmetric $SU(2)$
gauge theories with $N_f=0,1,3$ matter multiplets. For the $N_f=0$ theory from
the solutions of the Picard-Fuchs equation the monodromy matrices on the
quantum moduli space are determined. We analyze the Seiberg-Witten solutions to
compute monodromies exactly and present the instanton expansion of the periods
for the $N_f=0,3$ theories.Comment: 8pages, latex, no figure

Argyres

Cadavid

Ceresole

Klemm

Lerche

Seiberg

Shijong Ryang

Physics Letters B

English

arXiv

We construct the Picard-Fuchs equations of the N=2 supersymmetric SU(2) gauge theories with N_f=0, 1, 3 matter multiplets. For the N_f=0 theory from the solutions of the Picard-Fuchs equation the monodromy matrices on the quantum moduli space are determined. We analyze the Seiberg-Witten solutions to compute monodromies exactly and present the instanton expansion of the periods for the N_f=0, 3 theories

Ryang, S

CERN Document Server

August 1995
The Picard-Fuchs Equations, Monodromies and Instantons
in the N = 2 Susy Gauge Theories
Shijong Ryang
Department of Physics
Kyoto Prefectural University of Medicine
Taishogun, Kyoto 603 Japan
Abstract
We construct the Picard-Fuchs equations of the N = 2 supersymmetric SU(2) gauge
theories with Nf = 0; 1; 3 matter multiplets. For the Nf = 0 theory from the solutions
of the Picard-Fuchs equation the monodromy matrices on the quantum moduli space are
determined. We analyze the Seiberg-Witten solutions to compute monodromies exactly
and present the instanton expansion of the periods for the Nf = 0; 3 theories.
Recently, Seiberg and Witten investigated the low energy properties of N = 2 super-
symmetric SU(2) gauge theories without or with matter multiplets [1, 2] and obtained the
non-perturbative results such as the exact expression for the metric on moduli space and
the monopole and dyon spectrums using a version of Olive-Montonen electric-magnetic
duality. The quntum moduli space of the gauge theory with no matter multiplet was
identied with the moduli space of torus and further for the gauge theries withNf matter
multiplets the families of curves controlling the low energy behavior were constructed.
This approach was extended to the SU(Nc) gauge theory with Nf = 0 and the mod-
uli space of quantum vacua was shown to be provided by that of a special set of genus
Nc − 1 hyperelliptic Riemann surfaces [3]. The generalization of it to include Nf matter
multiplets was performed and weak coupling monodromies were computed [4]. The met-
ric on moduli space was determined by a single holomorphic function, whose structures
are analogue of the special coordinates in the context of Calabi-Yau moduli space. The
Picard-Fuchs (PF) equation indicating the existence of a flat holomorphic connection on
a certain holomorphic bundle has been found to be a second order equation for theN = 2
SU(2) gauge theory with no matter multiplet [5]. With respect to the SU(3) gauge the-
ory with no matter multiplet the PF equation has been solved and the leading instanton
corrections in the low energy quantum eective action have been studied [6].
In this letter by using the technique to transform a dening curve into the Weierstrass
form we construct the PF equations for N = 2 SU(2) gauge theories without matter or
with massless Nf = 1; 3 matter multiplets. As for the Nf = 0 theory the solutions of the
PF equation will be related with the Seiberg-Witten (SW) solutions. The monodromy
matrices on the quantum moduli space determined from the former solutions will be
associated with those determined from the latter solutions. By studying the properties
of the SW solutions for the Nf = 0 as well as Nf = 3 theories thoroughly we will extract
the non-perturbative instanton corrections to the metric of the quantum moduli.
We start to consider the PF equation satised by the periods in the moduli space of
quantum vacua in N = 2 supersymmetric gauge theories. So far the PF equations were
studied for the Landau-Ginzburg models whose marginal deformations are associated with
the complex structure moduli of the torus, K3 and Calabi-Yau manifolds [7, 8, 9]. Here
following the approach in Ref. [8] we write down the periods for a curve of torus in the
Weierstrass form y2 = 4x3 − g2(u)x− g3(u) as
!i =
I
γi
dx
y
; (1)
where γi, i = 1; 2 are the two homology generators of the torus. The PF equation is
derived as
d2!i
du2
−
a02
a2
d!i
ds
+
 
a02a1
a2
− a3
!
!i = 0; (2)
where
a1 = −
0
12
; a2 =
3
2
(2g2g
0
3 − 3g
0
2g3);
1
a3 =
1
16
(g2g
02
2 − 12g
02
3) + a
0
1 (3)
with  = g32 − 27g
2
3 which is the discriminant of the polynomial 4x
3 − g2x − g3. The
low energy quantum eective theory for the N = 2 supersymmetric SU(2) gauge theory
has a single N = 2 vector supermultiplet which can be decomposed into a N = 1 vector
multiplet and a N = 1 chiral multiplet whose scalar component has the VEV denoted by
a. The quantum moduli space of it is given by the moduli space of the family of genus one
Riemann surface y2 = (x− 1)(x+ 1)(x−u) parametrized by the gauge invariant quantity
u which becomes a2=2 for large u. A change of variables x0 = (x − u=3)=2 , y0 = y=
p
2
makes it’s curve in the Weierstrass form with g2 = 1 + u2=3, g3 = u3=27− u=3. From (2)
we obtain
d2!i
du2
+
2u
u2 − 1
d!i
du
+
!i
4(u2 − 1)
= 0; (4)
which shows the Z2 symmetry u! −u and contains two singular points u = 1;−1 which
are associated with two massless states, a monopole and a dyon. This PF equation is
rewritten in terms of z = u2 as
[(1− z)z
d2
dz2
+

1
2
−
3z
2

d
dz
−
1
16
]!i = 0; (5)
whose two independent solutions around z = 0, that is, u = 0 with suitable normalizations
are expressed in terms of the hypergeometric function as
f1 =
Γ2(1
4
)
Γ(1
2
)
F

1
4
;
1
4
;
1
2
; u2

;
f2 =
Γ2(3
4
)
Γ(3
2
)
uF

3
4
;
3
4
;
3
2
; u2

; (6)
which indicate logarithmic singularities at u = 1. Each singularity at u = 1 and u = −1
leads to the monodromy matrices in the basis (f1; f2)t
T1 =
 
1− i i
−i 1 + i
!
; T−1 =
 
1− i −i
i 1 + i
!
(7)
around u = 1 and u = −1 respectively, whose entries are complex numbers [9]. The PF
equation (4) is in the same form as the Landau-Ginzburg model with superpotential of
modality-one singularity type X9 [10]. We turn to the N = 2 SU(2) gauge theories with
Nf matter multiplets1. As for the Nf = 1 theory with zero bare mass , whose curve
is given by y2 = x2(x − u) + 1
6 we make the same change of variables as Nf = 0 to
1As for the Nf > 0 theories we use the notation of Ref.[2].
2
have g2 = u2=3, g3 = u3=27 − 1
6=2. The PF equation can be expressed in terms of
t = (4=27)1=3u=1
2 as
d2!i
dt2
+
2t3 + 1
t(t3 − 1)
d!i
dt
+
t
4(t3 − 1)
!i = 0; (8)
which shows Z3 symmetry under t! exp(2in=3)t (n = 0; 1; 2). It includes three singular
points u = 31
2=41=3 exp(2in=3) (n = 0; 1; 2) permuted by the Z3 symmetry, which
correspond to three massless states, a monopole and two dyons whose magnetic and
electric charges (nm; ne) are given by (1; 0), (1; 1), (1; 2). A change of variable t3 = z
yields
[(1− z)z
d2
dz2
+
1− 4z
3
−
1
36
]!i = 0 (9)
with two independent solutions F (1=6; 1=6; 1=3; z), z2=3F (5=6; 5=6; 5=3; z), which indicate
logarithmic singularities at z = 1 , that is, u = 31
2=41=3 exp(2in=3) (n = 0; 1; 2). In
the massless Nf = 3 th the associated curve is given by y
2 = x2(x − u) − (x − u)2,
which can be transformed into the Weierstrass form with g2 = (u2 − 4u + 1)=3, g3 =
(2u3 +15u2−12u+2)=54 through a change of variables x0 = (x− (u+1)=3)=2, y0 = y=
p
2.
The PF equation is obtained in a compact form as
d2!i
du2
+
8u − 1
u(4u− 1)
d!i
du
+
1
u(4u− 1)
!i = 0; (10)
which has no global symmetry. The regular singularities at u = 0; 1=4 correspond to
two massless states with charges (nm; ne) = (1; 0); (2; 1). The two independent so-
lutions are provided by F (1=2; 1=2; 1; 4u), F1(1=2; 1=2; 1; 4u) + F (1=2; 1=2; 1; 4u) log 4u,
where F1(1=2; 1=2; 1; z) = 2
P1
n=1((1=2)n=n!)
2Pn−1
r=0 (2=(2r+ 1)− 1=(r+ 1))z
n with (j)n =
Γ(j + n)=Γ(j). They include logarithmic singularities at u = 0; 1=4. In this way we have
observed that the PF equations of the Nf = 0; 1; 3 theories are obtained as the hypergeo-
metric dierential ones with appropriate singularities and symmetries, and the solutions
for the Nf = 3 theory show a structure quite dierent from those for the Nf = 0; 1
theories.
From now on we shall return to the Nf = 0 theory. The monodromy matrix acts on
the vector (aD; a)t where aD is the magnetic dual of a. The pair (aD; a) is considered as
a holomorphic section of an SL(2; Z) bundle over the punctured complex u plane, which
is dened by contour integrals aD =
H
γ1
, a =
H
γ2
 with  = dx
q
2(x− u)=2
p
x2 − 1,
where the one cycle γ1 loops around the branch points x = u; 1 and the other cycle γ2
around x = 1;−1 . Explicitly, these are expressed by a =
q
2(u+ 1)F (−1=2; 1=2; 1; 2=(u+
1)) with regular behavior at u = 1 and aD = i(u − 1)=2 F (1=2; 1=2; 2; (1 − u)=2) =
i(u−1)=
q
2(u+ 1)F (1=2; 3=2; 2; (u−1)=(u+1)) , which contains a logarithmic singularity
at u =1 as shown by
aD =
i

a log
u+ 1
2
+
i

(u− 1)
1X
n=0
(1
2
)n(
3
2
)n
(n!)2
kn

2
u+ 1
n+1=2
(11)
3
with kn = 2 (n+ 1) −  (n+ 1=2) −  (n+ 3=2) where  is the digamma function. This
expression yields directly a transformation aD ! −aD + 2a under a circuit of the u plane
at large u , being accompanied with a! −a . This result conrms the derivation in the
weak coupling limit using only the asymptotic expression a 
p
2u, aD  i
p
2u= log u[1].
The regular behavior of aD at u = 1 is compared with the logarithmic singularity of a =q
(u+ 1)=2(F (1=2; 1=2; 1; 2=(u+ 1)) +F (−1=2; 3=2; 1; 2=(u+ 1)) at u = 1. It is described
by a  iaD= log ((u− 1)=(u + 1)) which is in the dual form of (11). Therefore denitely
we obtain aD ! aD, a ! a − 2aD when u loops around 1. This conrmative derivation
of the strong coupling monodromy is compared with the leading order prescription aD 
i=2 (u−1), a  4=− (u−1)=2 log(u−1), where the coecient (u−1) of the logarithm
is identied with aD . Here we focus our attention on the periods along the two nontrivial
cycles γ1; γ2 , that is , the derivative of the electric and magnetic coordinates (aD; a) with
respect to u, which are expressed by
da
du
= −
p
2
4
I
γ1
dx
y
=
1q
2(u+ 1)
F

1
2
;
1
2
; 1;
2
u+ 1

; (12)
daD
du
= −
p
2
4
I
γ2
dx
y
=
i
2
F

1
2
;
1
2
; 1;
1− u
2

: (13)
In deriving them we represent each contour integral in terms of the hypergeometric func-
tion. Alternatively direct derivative of the above obtained expressions of (aD; a) about u
together with a recursion relation of the hypergeometric functions yields them. Our task is
to associate them with the solutions of the PF equation. Since F (1=2; 1=2; 1; (1−u)=2) =
(Γ(1=2)=Γ2(3=4)) F (1=4; 1=4; 1=2; u2) + (Γ(−1=2)=Γ2(1=4)) uF (3=4; 3=4; 3=2; u2) and a0 is
rewritten as 1=
p
2u F (1=4; 3=4; 1; 1=u2), the basis vector (aD0; a0)t is transformed into
(f1; f2)t of (6) as  
aD
0
a0
!
=
1
4
 
i −i
1 + i 1− i
! 
f1
f2
!
; (14)
which is denoted by  = P! . Then we can derive the monodromy matrices around
u = 1 and u = −1 as M1 =
 
1 0
−2 1
!
, M−1 =
 
−1 2
−2 3
!
through M1 = PT1P−1,
M−1 = PT−1P−1 with T1; T−1 in (7). We note that the monodromy matrices in the
basis (aD 0; a0) are the same as in the basis (aD; a) . It should be emphasized that the
quantum moduli space of N = 2 SU(2) gauge theory is related with the moduli space of
complex deformation of the torus , in particular, of type X9. From aD 0 = i=4 (f1 − f2)
and F (j; j; 2j; u2) = (1− u2)−jF (j; j; 2j; u2=(u2 − 1)) with j = 1=4; 3=4 ,which contains a
logarithmic singularity at u =1, we obtain
daD
du
=
i
2
da
du
log(1− u2) +
i
4

G(
1
4
)− uG(
3
4
)

; (15)
4
whereG(j) =
P1
n=0 2((j)n=n!)
2( (n+1)− (n+j))(1−u2)−(n+j). This expression , which is
compared with (11), and a0 = 1=
p
2u
P1
n=0(
1
4
)n(
3
4
)n
1
(n!)2
u−2n have the instanton contribu-
tions u−2n (n > 0) from terms with an odd number of instantons as well as an even number
and reproduce the monodromy matrixM1 =
0BBBBBBBBBB@
−1
0 −1 about innity in the u plane. This structure is in
daD
du
=
i
2
s
i

da
du
log
u+ 1
2
+
i
2
1X
n=0
 
(1
2
)n
n!
!2
kn

2
u+ 1
n+1=2
;
2( (n+ 1)−  (n+ 1=2) = 2(
Pn
r=1(1=r − 2=(2r − 1)) + log 4).
Moreover, we consider the massless Nf = 3 curve y2 = (x − u)(x2 − x + u),whose
right-hand side has zeroes at x = u and x = x = (1i
p
4u− 1)=2 for the region
u > 1=4 which includes the semiclassical regime of u !1 . The periods are dened by
a0 =
H
γ1
!, a0D =
H
γ2
! with ! = (
p
2=8)dx=y. The contours γ1, γ2 loop around the pair
points (x+; x−), (x+; u) respectively. Contrary to the Nf = 1 theory we can get analytic
expressions for a0; a0D
da
du
=
1
2(2u− 1 + i
p
4u− 1)1=2
F

1
2
;
1
2
; 1;
2
1− iX

; (16)
daD
du
=
p
2i
4(i
p
4u− 1)1=2
F

1
2
;
1
2
; 1;
1 + iX
2

(17)
with X = (2u − 1)=
p
4u − 1. The above expressions have some similarity to (12), (13)
only diering in u replaced by −iX. The period da=du can be shown to be equated with
da=du = 2−1(2u − 1− i
p
4u − 1)−1=2F (1=2; 1=2; 1; 2=(1 + iX)) through a relation for the
hypergeometric functions and further be transformed into a manifestly real expression
da
du
=
1
2
p
2u− 1
F

1
4
;
3
4
; 1;−
1
X2

=
1
2
p
2u− 1
1X
n=0
(1
4
)n(
3
4
)n
(n!)2
 
1−
u
1− 1
4u
!−n
; (18)
which shows the expected asymptotic behavior 1=2
p
2u for large u in the semiclassical
region. In the general Nf > 0 theory the section (a; aD) is expected to be
a =
1
2
p
2u
0@1 + 1X
n=1
an(Nf )
 
2Nf
u
!n(4−Nf )1A ;
aD = i
4−Nf
2
a log
u
2Nf
+
p
u
1X
n=0
aDn(Nf)
 
2Nf
u
!n(4−Nf )
(19)
with dynamically generated scale 2Nf , where even number instantons contribute [2]. The
a in (19) for Nf = 3 can be compared with the above result (18). On the other hand in
5
order to make (17) related with aD in (19) we need to transform (17) into
daD
du
=
p
2i
4(i
p
4u− 1)1=2
 
Γ(1
2
)
Γ2(3
4
)
F (
1
4
;
1
4
;
1
2
;−X2)− iX
Γ(−1
2
)
Γ2(1
4
)
F (
3
4
;
3
4
;
3
2
;−X2)
!
; (20)
whose hypergeometric functions are further changed into (1+X2)−1=4F (1=4; 1=4; 1=2; X2=
(X2 + 1)), (1 + X2)−3=4F (3=4; 3=4; 3=2; X2=(X2 + 1)), which include a logarithmic sin-
gularity at u = 1. This logarithmic singularity is expressed in a form similar to (15)
as
daD
du
=
i
2
da
du
log
 
u
1− 1
4u
!
+
1
8
s
i
u

H(
1
4
) + i(1−
1
2u
)H(
3
4
)

(21)
with H(j) =
P1
n=0 2((j)n=n!)
2( (n+ 1) −  (n+ j))(u=(1− 1=4u))−n, whose behavior in
the u variable is the wanted one compared with the instanton expansion of the aD in (19)
for Nf = 3. The relation (20) corresponds to aD 0 = i=4(f1 − f2) which has a role to
produce (15). From (21) together with (16) the monodromy matrix around u = 1 can
be exactly extracted as M1 =
 
−1 1
0 −1
!
.
Finaly we would like to present the alternative compact expression of a0; aD 0 as well
as a; aD for the Nf = 0 theory.
da
du
=
k

K(k) ;
daD
du
=
ik

K 0(k);
a =
4
k
E(k) ; aD =
4i
k
(K 0(k)− E0(k)); (22)
where k =
q
2=(u + 1) and K(k), E(k) are complete elliptic integrals of the rst and
second kinds. There is an interesting identity EK 0 + E0K − KK 0 = =2 , which turns
out to be adaD=du − aDda=du = 2i=. The usual modular parameter 0 = iK 0=K of the
torus straightfowardly becomes aD 0=a0 which is the eective couplings  = =2 + i4=g2
combined by the dimensionless gauge coupling constant and the theta parameter in the
quantum vacuum . Through the expansion of q1=4 in powers of 1=(u+1) for q = exp (i0),
the metric on the u plane given by (ds)2 =Im jdaj2 is expressed in terms of u as
Im = −
4

log
 
(
k
4
)1=2(1 + 2(
k
4
)2 + 15(
k
4
)4 + : : :)
!
; (23)
which resembles (). Conversely it is possible to describe u in terms ratio a0D=a0 by using
k = (2(0)=3(0))2 as
u = 2
 
1 + 2
P1
n=1 q
n2
2
P1
n=0 q
(n+1=2)2
!4
− 1: (24)
In conclusion we have derived the monodromy matrices denitely for the N = 2
SU(2) gauge theory with Nf = 0 by taking advantage of the interrelations between
6
the hypergeometric functions. Our exact derivation provides support for the leading
order prescription using the asymptotic expression. They have been also obtained by
studying the solutions of the PF equation. Concerning theNf = 1; 3 theories the obtained
second order dierential equations for the periods yield solutions expressed in terms of
the hypergeometric function and it’s analogue. By making appropriate transformations
of the hypergeometric functions, which is suggested from analyzing the relation between
the explicit expression ot the SW solution for the Nf = 0 theory and the two independent
solutions of the PF equation , we have gotten the desired power of the moduli parameter
in the instanton expansion of the periods for the Nf = 0; 3 theories. Unlike the Nf = 0
theory it is however unclear how the periods of the Nf = 3 theory are connected with
the two independent solutions of the PF equation. We hope the other representation of
periods expressed in terms of the complete elliptic integrals for the Nf = 0 theory sheds
new light on further understanding of the quantum moduli space.
7
References
[1] N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19.
[2] N. Seiberg and E. Witten, Nucl. Phys. B431 (1994) 484.
[3] A. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, Phys. Lett. B344 (1995) 169;
P. Argyres and A. Faraggi, Phys. Rev. Lett. 74 (1995) 3931.
[4] A. Hanany and Y. Oz, On the Quantum Moduli Space of N=2 Supersymmetric
SU(Nc) Gauge Theories, TAUP-2248-95, WIS-95/19/May-PH, hep-th/9505075; P.
Argyres, M. Plesser and A. Shapere, The Coulomb Phase of N=2 Supersymmetric
QCD, IASSNS-HEP-95/32, UK-HEP/95-06, hep-th9505100.
[5] A. Ceresole, R. D’Auria and S. Ferrara, Phys. Lett. B339 (1994) 71.
[6] A. Klemm, W. Lerche and S. Theisen, Nonperturbative Eective Actions of N=2 Su-
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[7] A.C. Cadavid and S. Ferrara, Phys. Lett. B267 (1991) 193; W. Lerche, D. Smit and
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8


The Picard-Fuchs equations, monodromies and instantons in the N=2 SUSY gauge theories

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