A connection between the conifold locus of the type II string on the
$W\:P_{11226}^4$ Calabi-Yau manifold and the geometry of the quantum moduli of
$N = 2$ $SU(2)$ super Yang-Mills is presented. This relation is obtained from
the anomalous behaviour of the $SU(2)$ super Yang-Mills special coordinates
under $S$-duality transformation in $Sl(2;Z) / \Gamma_2$.Comment: 7 pages, Late

Gomez, C.

Lopez, E.

Physics Letters B

English

arXiv

A connection between the conifold locus of the type II string on the W\:P_{11226}^4 Calabi-Yau manifold and the geometry of the quantum moduli of N = 2 SU(2) super Yang-Mills is presented. This relation is obtained from the anomalous behaviour of the SU(2) super Yang-Mills special coordinates under S-duality transformation in Sl(2;Z) / \Gamma_2

Gómez, C

López, E

CERN Document Server

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hep-th/9506024
A NOTE ON THE STRING ANALOG OF N=2 SUPER-SYMMETRIC
YANG-MILLS
Cesar Gomez
1
and Esperanza Lopez
1
Departement de Physique Theorique, Universite de Geneve,
Geneve 6, Switzerland
Abstract
A connection between the conifold locus of the type II string on the W P
4
11226
Calabi-Yau
manifold and the geometry of the quantum moduli of N =2 SU(2) super Yang-Mills is
presented. This relation is obtained from the anomalous behaviour of the SU(2) super
Yang-Mills special coordinates under S-duality transformation in Sl(2;Z)= 
2
.
June 1995
1
Permanent address: Instituto de Matematicas y F

isica Fundamental (C.S.I.C.), Serrano 123, 28006
Madrid, Spain.
1. A string analog of N=2 supersymmetric Yang-Mills (SYM) is obtained in reference
[1] by means of a "rank three" compactication of the heterotic string on T
2
K
3
. The
corresponding d = 4 N = 2 theory contains two vector multiplets, one of them to be
identied with the dilaton S, and an extra U(1) vector associated with the graviphoton.
The basic claim in [1] is the existence of a dual type II string [2, 3, 4, 5] on the Calabi-Yau
manifold W P
4
11226
=M [6, 7] such that the quantum moduli of the heterotic string is
given by the moduli of the vector multiplets of the type II string on M
2
, which receives
no quantum corrections. Accepting the previous framework, it should be expected that
the geometry of the quantum moduli of N=2 SU(2) SYM, solved in [9], can be recovered
from the structure of the moduli of vector multiplets onM . In reference [1] some evidence
supporting this picture is presented.
The moduli of complex structures for the mirror of M can be parametrized by two
complex variables ,  , dened by
P
11226
= z
12
1
+ z
12
2
+ z
6
3
+ z
6
4
+ z
2
5
  12 z
1
z
2
z
3
z
4
z
5
  2z
6
1
z
6
2
(1)
In terms of the coordinates x =
 
864 
6
, y =
1

2
introduced in [7], the locus of conifold
singularities for (1) is given by
(1  x)
2
  x
2
y = 0 (2)
where j is the Jacobi elliptic function. For xed y 6=0, equation (2) provides two singular
points x

. Identifying, at leading order in the coupling, y = e
 S
, it was observed in [1]
that for S!1 the conifold locus merges to the point x=1. Using the relation found in
[6]
x =
1728
j( )
(j(i) = 1728); (3)
the point x= 1 corresponds to  = i, which, for  the elliptic parameter of the internal
torus used in the heterotic compactication [1], coincides precisely with the SU(2) point
of enhanced gauge symmetry. The picture emerging from this comment being that, for
nite but small y, we should be able to derive the way in which the classical singularity
of N =2 SU(2) SYM, where the SU(2) symmetry is restored perturbatively, splits into
the two singular points of the quantum moduli [9]. It is important to keep in mind that,
in the case of N = 2 SU(2) SYM, the quantum moduli is not the fundamental domain
with respect to the full modular (duality) Sl(2;Z) group, while for the string analog, the
elliptic parameter  in (3) is, as a consequence of T-duality, in a Sl(2;Z) fundamental
domain.
In this note we will approach the problem of connecting the quantum moduli for
SU(2) SYM and the moduli of vector multiplets on M . Our approach will consist in
implementing the strong-weak coupling duality transformations of N=2 SYM as elements
2
It was also suggested in reference [8] that the quantum moduli for a heterotic string of rank r should
look as the complex moduli of a Calabi-Yau manifold with h
21
=r.
1
in Sp(6;Z). This will imply a deep connection between the dilaton and the geometry of
the quantum moduli of SYM, from which we will try to derive the conifold locus of the
Calabi-Yau manifold (1).
2. In [10] we have shown that the obstruction to strong-weak coupling duality invari-
ance in N=2 SYM, comes from the fact that
3
a((u)) 6= a
D
(u) (4)
where the transformation  maps a strong-coupling singular point into the asymptotically
free point at innity. However the eective gauge coupling parameter  = i
4
g
2
eff
+

eff
2
behaves in a dual-like way
 ((u)) =  
1
 (u)
(5)
For the curve [9] which parametrized the quantum moduli of N=2 SU(2) SYM
y
2
= (x+ 
2
)(x 
2
)(x  u) (6)
with  the dynamically generated scale, the transformation  is given by
(u) = 
2
u+ 3
2
u 
2
(7)
which maps the singular point u = 
2
into 1. This transformation is an element of
 
W
=Sl(2;Z)= 
2
, which consists in transformations of the coordinates (x; y) of the curve
that can be compensated by a change in the moduli parameter u.
After coupling to gravity, i.e. in non-rigid special geometry, we can improve (4) to [10]
a
gr
((u)) =
~
f
;
(u) a
gr
D
(u) (8)
if at the same time we perform the transformation
a
0
(u)!
~
f
;
(u) a
0
(u) (9)
with
~
f
;
(u) =
p
2
4
 
2
2

2
  u
!
3=2
(10)
Let us explain briey this result. In the context of rigid special geometry, the 1-form
(x; u), whose periods dene the holomorphic sections (a; a
D
), satises
d
du
 
1
(11)
3
From now on we will use the notation of reference [9].
2
with the proportionality factor determined by imposing the physically correct asymptotic
behaviour at the singularities, and where 
1
(x; u) is the everywhere non-zero holomorphic
1-form of the curve (6). The 1-form 

(x; u) which denes (a((u); a
D
((u)), also veries
equation (11) with
d

du
=
~
f
;

1
(12)
In order to pass from (4), where the relation between a((u)) and a
D
(u) is not a symplectic
transformation, to (8) which, up to the holomorphic function
~
f
;
, is symplectic, we
can replace the derivative in (12) by a covariant derivative relative to the graviphoton
(Hodge) U(1)-connection [10]. From this, equation (9) immediately follows. However, the
gauge transformation
~
f
;
is singular and generates a topological obstruction to implement
strong-weak coupling duality. Thus, the more natural way to interpret (8) and (9) would
be to add an extra dilaton eld S.
Namely, for the "rank three" heterotic string compactication on T
2
K
3
we can
parametrize the two vector multiplets by the special coordinates (X
0
;X
1
;X
2
), with S=
 iX
1
=X
0
). As it was shown in [11, 12], in order to have all the couplings proportional to
S, it is necessary to do the following symplectic change of variables
^
X
I 6=1
= X
I
;
^
F
I 6=1
= F
I
^
X
1
= F
1
;
^
F
1
=  X
1
=  iSX
0
(13)
In these new variables, we can interpret transformation (9) as
^
X
0
! iS
^
X
0
=  
^
F
1
(14)
Performing twice the transformation  given in (7), we get the identity (up to a sign
ambiguity). This implies that under  satisfying (9), the dilaton changes as
S ! S
0
=
1
S
(15)
At the level of the special coordinates, this equation implies that
^
F
1
!X
0
, which very
likely, together with (14), will be part of a symplectic transformation in Sp(6;Z).
Interpreting now equation (9) as equation (14) boils down to the following formal
identication
S 
~
f
;
(u) (16)
3. Our next task will be to give a natural physical meaning to relation (16). Before
doing that it would be worth to come back to our original problem. Once we have added
the dilaton eld S, we can parametrize the moduli of the two vector multiplets by u,
the coordinate of the N =2 SYM quantum moduli, and S. Naively this looks like a one
3
parameter family of u-planes labeled by S. The natural physical picture, according with
the identication of the conifold locus for S!1 with the classical singularity of enhanced
SU(2) symmetry, will be to have, for S!1, a classical moduli with one singularity at
u=0 and, for nite S, the quantum moduli with two singularities at u=
2
, where 
will now depend on the expectation value of the dilaton. However, the previous picture
is too crude since it simple bers the N = 2 SYM quantum moduli on the dilaton line.
This reduces the eect of the dilaton to simply changing the scale , and in this way, the
location of the two singularities. Our approach will be to parametrize the moduli space
of the two vector multiplets by (u; S) variables, and to try to recover the conifold locus
(2) directly from relation (10), which we have derived from the geometry of the quantum
moduli of reference [9], by interpreting the transformations (8) and (9) as the symplectic
transformation (14) of the heterotic string.
In order to use (16) for dening a locus in the (u; S) variables, we need rst to obtain
some explicit relation between the dynamical scale  and the value of the dilaton eld
S. Using that the unrenormalized gauge coupling constant g
0
of the eective eld theory
derived from an heterotic string is proportional to the dilaton expectation value and that
changes in g
0
can be absorbed in changing the scale , we will impose, based on the
renormalization group equation for N=2 SU(2) SYM, the relation

2
= e
 S=2
(17)
Combining this with (10), equation (16) becomes
 
4
p
2
S
!
2=3
=
2e
 S=2
e
 S=2
  u
(18)
which can now be used to dene a locus S(u) in the (u; S) space. We claim that the
so-dened locus in (u; S) variables should coincide with the conifold locus (2).
From (18) we get
e
 S=2
=
u
1  2

p
2
4S

2=3
(19)
For S!1 we can use, at leading order, the identication y=e
 S
[1]. Solving the conifold
locus (2), we obtain
x

=
1
1 
p
y
(20)
which, using (19), can be writen as
x

=
1  2

p
2
4S(u)

2=3
1 u  2

p
2
4S(u)

2=3
(21)
4
Notice that this equation is equivalent to postulate that the locus (19) is equivalent to
the conifold locus (2). However, motivated by this equation we can propose a map from
the (u; S) variables into the (x; y) variables parametrizing the complex moduli of (1)
x(u; S) =
1   2

p
2
4S

2=3
1 + u  2

p
2
4S

2=3
(22)
We need now to check the consistency of (22). First we observe that it reects correctly
the global Z
2
symmetry of the N=2 SU(2) SYM gauge theory. The action of this global
symmetry in the SYM quantum moduli is given by u! u, therefore mapping the two
nite singular points between themselves. We observe that the change u! u in (22) also
maps x
+
into x
 
and vice-versa.
Second, in the string weak-coupling limit S!1, we get
x =
1
u+ 1
+O(S
 2=3
) (23)
which implies that the asymptotically free regime of the N=2 SYM theory, which corre-
sponds to u!1, is associated with x=0
4
. Using now the relation x=
 
864 
6
, we obtain
that
1
u
=
 
864 
6
in a neighborhood of the asymptotically free regime (u!1). In order to
compare the instanton expansion of SU(2) SYM with the expansion in the (; ) variables
of the periods of the holomorphic top form for the Calabi-Yau (1) [6], we will consider 
and e
 S
as independent parameters dening the following doble limit
5
lim
;S!1

4
e
 S
= 1 (24)
The instanton expansion of SU(2) SYM goes like

1
a(u)

4k
, k > 0. Using that a=
p
2u
when u!1 [9], in the u variable, we get

1
u

2k
. On the other hand, the leading term as
!1 in the expansion of the periods of the top form behaves as


 
6

k
. The presence
in one case of powers of 2k and in the other of powers of k can be explained by the fact
that, for the Calabi-Yau case, the two singular points of the locus (2) are related by the
transformation !  which is not a symmetry, while the Z
2
transformation u! u, that
corresponds by (19) to ! , is a global symmetry of N = 2 SU(2) SYM. Therefore,
in order to reproduce the SYM quantum moduli inside the Calabi-Yau moduli space, we
should mod by the transformation ! , or equivalently, consider only even powers of

 
6
.
4
In reference [1] is is also suggested that x=0 can be interpreted as a weak coupling region related to
the degeneration of the compactication torus.
5
In reference [1] it has been already proposed to use a suitable "doble scaling" limit in order to recover
the N =2 SU (2) SYM quantum moduli from the moduli space of complex deformations of the W P
4
11226
Calabi-Yau.
5
Summarizing, we have shown that by identifying the conifold locus with equation
(16) we get, at leading order in S, a consistent physical picture. The important thing
for us is that equation (16) was obtained by promoting, introducing a dilaton eld, the
anomalous strong-weak coupling duality transformations of the N =2 SYM eld theory,
to good symplectic transformations. In other words the quantum moduli in reference [9]
describes more than the complex structure of the curve (6), something which is related to
the existence of a dynamically generated scale . When we include the dilaton and work
in the string framework, what was geometrical (no moduli) information for SU(2) SYM
becomes now pure moduli information of the Calabi-Yau manifold (1).
This work was partially supported by european community grant ERBCHRXCT920069,
by PB 92-1092 and by OFES contract number 93.0083. The work of E.L. is supported by
a M.E.C. fellowship AP9134090983.
6
References
[1] S. Kachru and C. Vafa, "Exact Results for N=2 Compactications of Heterotic
Strings", hep-th/9505105
[2] C. Hull and P. Townsend, "Unity of Superstring Dualities", hep-th/9410167
[3] E. Witten, "String Theory Dynamics in Various Dimensions", hep-th/9503124
[4] A. Sen, "String-String Duality Conjecture in Six Dimensions and Charged Solitonic
Strings", hep-th/9504027
[5] J. Harvey and A. Strominger, "The Heterotic String is a Soliton", hep-th/9504047
[6] P. Candelas, X. De la Ossa, A. Font, S. Katz, and D. Morrison, "Mirror Symmetry
for Two Parameter Models - I", Nucl. Phys. B416 (1994) 481, hep-th/9308083
[7] S. Hosono, A. Klemm, S. Theisen, and S.T. Yau, "Mirror Symmetry, Mirror Map
and Applications to Calabi-Yau Hypersurfaces", Comm. Math. Phys. 167 (1995) 301,
hep-th/9308122
[8] A. Ceresole, R. D'Auria and S. Ferrara, Phys. Lett. B339 (1994) 71 hep-th/9408036;
A. Ceresole, R. D'Auria, S. Ferrara and A. Van Proyen, "Duality Transformations in
Supersymmetric Yang-Mills Theories Coupled to Gravity", hep-th/9502072
[9] N. Seiberg and E. Witten, "Electric-magnetic Duality, Monopole Condensation, and
Connement in N=2 Supersymmetric Yang-Mills Theory", Nucl. Phys. B426 (1994)
19, hep-th/9407087
[10] C. Gomez and E. Lopez, "From Quantum Monodromy to Duality", hep-th/9505135
[11] B. de Wit, V. Kaplunovsky, J. Louis, and D. Lust, "Perturbative Couplings of Vector
Multiplets in N=2 Heterotic String Vacua", hep-th/9504006
[12] I. Antoniadis, S. Ferrara, E. Gava, K.S. Narain, and T.R. Taylor, "Perturbative
Prepotential and Monodromies in N=2 Heterotic String", hep-th/9504034
7


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