Modular properties of 6d (DELL) systems
HJE
Modular properties of 6d (DELL) systems
G. Aminov 1 4
A. Mironov 0 1 2 3 4
A. Morozov 0 1 3 4
0 Institute for Information Transmission Problems
1 Moscow 127994 , Russia
2 Lebedev Physics Institute , Moscow 119991 , Russia
3 National Research Nuclear University MEPhI
4 ITEP , Moscow 117218 , Russia
If superYangMills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge 1 . The lowenergy SeibergWitten prepotential F (a), however, is not explicitly invariant, because the at moduli also change a a modular form and depends also on the anomalous Eisenstein series E2. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N ) theory with two independent modular parameters modular anomaly equation changes, because the modular transform of by an (N dependent!) shift of ^ and vice versa. This is a new peculiarity of doubleelliptic
Anomalies in Field and String Theories; Integrable Hierarchies; Supersym

= 2 + 4g2{
!
systems, which deserves further investigation.
1 Introduction
Doubleelliptic SeibergWitten prepotential
First modular anomaly equation
Second modular anomaly equation
The limit
( j ^) ! 1
Modular anomaly for N = 2
6.1
Evaluating functions c^n ( ; ; ^)
Modular anomaly at N = 3
2
3
4
5
6
7
8
Conclusion A function
1
Introduction
where the main unknown ingredient is the doubleelliptic (DELL) generalization [38{43]
of the CalogeroRuijsenaars model [44{51]. The both approaches are currently technically
involved and not yet very well related. In this paper, we demonstrate that, despite the
complexity of the subject, one can already formulate very clear and elegant statements
extracted from a series of pretty sophisticated and tedious calculations. This is a sign that
the whole 6d/DIM/DELL story will nally acquire a simple and transparent form suitable
for a textbook level presentation.
N = 2 supersymmetric gauge theories can be studied in the string theory framework,
which provides a transparent description for the Coulomb branch of such models. Since
we are interested in the low energy e ective actions and the corresponding integrable
systems, it is useful to formulate the gauge theories under consideration as the quantum
{ 1 {
eld theories derived from various con gurations of branes in the superstring and M theory.
Let us start with the gauge theories in four dimensions and recall their description via M
theory introduced by E. Witten in [52], which was a continuation of a series of previous
studies in [53{56]. According to [52], a wide class of 4d gauge theories can be obtained by
considering D4 branes extended between NS5 branes in Type IIA superstring theory on
R10 with coordinates x0; x1; : : : ; x9. The worldvolumes of NS5 branes are six dimensional
with coordinates x0; x1; : : : ; x5 and the worldvolumes of D4 branes are
ve dimensional
with coordinates x0; x1; x2; x3; x6. One can locate the NS5 branes at x7 = x
8 = x
9 = 0
and, in the classical approximation, at some xed values of x6, while the D4 branes are
nite in the x6 direction and terminate on the NS5 branes. Following [52], we introduce a
complex variable v = x4 + {x5 and, classically, every D4 brane is located at a de nite value
of v. Such brane con gurations can be illustrated by the following picture with vertical
and horizontal directions being v and x6 correspondingly:
If one has n + 1 vebranes labeled by
= 0; : : : ; n, and k fourbranes attached to
the (
Qn=1 SU (k ).
group is given by
1)th and
th
vebranes, the gauge group of the fourdimensional theory is
The positions of the fourbranes ai; , i = 1; : : : ; k
correspond to the
Coulomb moduli of the gauge theory. The coupling constant g of the SU (k ) gauge
1
g2 =
x
6
x
where x6 is the position of the th vebrane in the x6 direction and
is the string coupling
constant. In fact, the vebranes do not really have any de nite values of x6 as the classical
brane picture suggests. The position x6 is determined as a function of v by minimizing the
total vebrane worldvolume. Thus, g is also a function of v and g (v) can be interpreted
as the e ective coupling of the SU (k ) theory at mass jvj. To include the e ective theta
angle
of the SU (k ) gauge theory, one has to lift Type IIA superstring theory to the M
theory on the R10
2 R10. The theta angle
the (
1)th and th vebranes:
S1. The eleventh dimension x10 in M theory is periodic with period
is determined by the separation in the x10 direction between
=
x10
R10
, where R4 is parameterized by the rst four coordinates x0; x1; x2; x3
is a twodimensional surface in R
3
S1 parameterized by x4; x5; x6; x10. If we
S1 with the complex structure with holomorphic variables v = x4 + {x5
and s = x6 + {x10, then, due to the N = 2 supersymmetry,
is a complex Riemann
surface. This surface plays a great role in connecting M theory with the theory of integrable
systems [51, 57{59]. In particular, the low energy e ective action of the N = 2 gauge theory
can be determined by an integrable Hamiltonian system [3, 4, 60] with the spectral curve
given by , which is usually called the SeibergWitten curve.
In this paper, we use methods from the theory of integrable systems to study some
particular curves
and the corresponding low energy e ective actions. We focus on a
special case of systems with x6 direction compacti ed onto a circle. This case describes
theories with adjoint matter hypermultiplets, their bare masses m
being given by di
erences between the average positions in the v plane of the fourbranes to the left and right
of the th
vebranes:
m =
k
1 X ai;
i
1
k +1 j
X aj; +1:
Besides, the numbers of D4 branes k
are all coincide and the gauge group is U(1)
SU(k)n. Various brane con gurations provide us with gauge theories of this type in di erent
dimensions. From the M theory point of view, there is a natural set of gauge theories in
dimensions 4, 5 and 6. First, consider the 4d case and the following brane con guration in
Type IIA theory on R
9
S1:
(1.4)
(1.5)
where there is one NS5 brane and N D4 branes wrapped around a circle in the x6 direction,
i.e. the gauge group is U (1)
SU (N ). In fact, the particular con guration depicted
{ 3 {
in
gure (1.5) corresponds to the N
= 4 theory with gauge group U (k), because the
hypermultiplet bare mass is zero. This is due to the simple choice of the spacetime, which,
in coordinates x
6 and v = x4 + {x5, is just S1
C. Thus, each D4 brane is ending at
the same point to the left and right of the NS5 brane, resulting in zero di erence between
the average positions of the fourbranes on two sides of the vebrane. To introduce a
nonzero hypermultiplet bare mass and to break the N = 4 supersymmetry down to N = 2,
one needs to replace S1
C part of the spacetime by a certain C bundle over S1. The
procedure introduced in [52] is to start with x6 and v as coordinates on R
C and divide
by the following symmetry:
where an arbitrary complex constant m de nes the hypermultiplet bare mass and the
corresponding type IIA brane con guration is
x
6 ! x6 + 2 L;
v ! v + m;
(1.6)
Now, upon going around the x6 circle, one comes back with a shifted value of v. The M
theory uplift of this model also requires some particular choice of the spacetime. To get a
nonzero theta angle, one divides R
x10, and v by the combined symmetry
S1
C part of the spacetime with coordinates x6,
x
6
x10
! x6 + 2 L;
v ! v + m;
{ 4 {
where
de nes the e ective theta angle and x10 is still periodic with period 2 R10. The
quotient of the s plane by these equivalences, i.e. of the R
S1 part of the space is a complex
Riemann surface
of genus one with modulus
giving the complexi ed coupling constant
of the theory. The resulting quotient of the whole R
C by (1.8) is a complex manifold
Xm, which can be regarded as a C bundle over . The type IIA brane con guration (1.7)
in terms of M theory is described by a single M5 brane, which propagates in Xm. The
S1
worldvolume of this vebrane is given by R
4
, where
is a twodimensional Riemann
surface in Xm. An important part of the Xm structure is the map Xm !
provided by
forgetting C. Under this map, the curve
Xm maps to , thus giving an interpretation of
(1.7)
(1.8)
as an N sheeted covering of the base torus . From the viewpoint of integrable systems,
corresponds to the spectral curve
CM of the elliptic CalogeroMoser model [4, 61, 62]
known to have the same geometrical description [63] (generalization to the case of more
than two NS5 branes leads to the spin Calogero model, see [17]). To avoid uncertainties
in the notation, from now on, we denote the curve
of the 4d theory under consideration
by
Before going to the 5d and 6d cases, we brie y review some basic properties of the
curve
CM and the corresponding low energy e ective action. Theories resulting from the
brane con gurations described above, with the x6 direction compacti ed onto a circle, are
known to be conformally invariant [52]. The duality group of the fourdimensional model
is SL (2; Z). In other words, the curve CM is invariant under the modular transformations
!
1= . The low energy e ective action is not invariant, but has
very distinctive properties under the action of the duality group. These properties can be
understood by describing the low energy e ective action in terms of the SeibergWitten
CM, whose second derivatives with respect to the Coulomb moduli ai give the
period matrix T CM of the complex Riemann surface
CM. Using this connection between
CM and the curve
CM, the modular anomaly equation describing the
CM on the second Eisenstein series E2 ( ) was derived by J. Minahan, D.
which is associated with the classical part of the prepotential. To obtain the pure gauge
limit of the N = 2 theory, one should bring the value of m and { to in nity in a consistent
way (double scaling limit) [61, 63, 64, 66]:
{ 5 {
Nemeschansky and N. Warner in [64]. This equation has an elegant form
=
1 XN
2
i=1
and is equivalent to the holomorphic anomaly equation [65] in the limit of 1; 2 ! 0. Brane
con guration also provides valuable insights into the dependence of the low energy e ective
action on the Coulomb moduli. Since the U (1) factor decouples from the SU (N ) part of
the theory, the period matrix T CM depends only on the di erences (ai
aj ). In Type IIA
theory, the Coulomb moduli ai describe the positions of the fourbranes in the v plane,
and these fourbranes are all identical. Therefore, the curve
CM in M theory is invariant
under permutations of the moduli ai, and the period matrix is a symmetric function of
aj ). The same is true for the perturbative and instanton parts of
CM. Another basic aspect of the theory is its behavior at particular
values of the bare mass m. As it was mentioned earlier, N = 2 theory with gauge group
SU (N ) becomes N = 4 theory with gauge group U (k) at m = 0. Thus, the
(1.10)
(1.11)
so that the resulting cuto
is nite. From the M theory point of view, this limit of
in nite mass in the fourdimensional theory is accompanied with the decompacti cation of
the x6 direction.
Roughly speaking, the 5d and 6d theories can be obtained by successively compactifying
the x4 and x5 directions in M theory. To get the proper gauge theory description, one
should start with Type IIA superstring theory and perform the T duality transformation
that turns Type IIA theory into Type IIB. In this way, the vedimensional gauge theory
can be described in terms of the Type IIB D5 and NS5 branes, which form a Type IIB (p;
q)brane web [59, 67{71]. For our purposes of studying the SeibergWitten curves and the low
energy e ective actions of the 5d and 6d theories, it is su cient to use the earlier described
con guration of the single M5 brane and further compactify the x4 and x5 directions. In
particular, 5d SYM theory with one compacti ed KaluzaKlein dimension and the adjoint
matter hypermultiplet [72{74] corresponds to the brane con guration with x
4 direction
compacti ed onto a circle of radius R4 =
x6, x10, and v = x4 + {x5 is divided by the symmetry
1=2. The part of the spacetime with coordinates
of x4 direction a ects the low energy e ective action and the curve
RS in a very manifest
way. Since the Coulomb moduli ai take values in the v plane with the periodic real
coordinate x4, the curve should be invariant under the shifts ai ! ai +
the period matrix T RS can be represented as a symmetric function of sin ( aij)2 with
1
. Thus,
aij
ai
aj. According to (1.12), the mass parameter
describes the shift in the v plane,
and there should be another symmetry of the curve
RS, that is,
!
+ . The 5d
theory under consideration is conformally invariant and the duality group is SL (2; Z). As
it was established in several works [43, 75], the SeibergWitten prepotential F
the same modular anomaly equation (1.9) as in the 4d case. Also, at
RS admits
= 0, the N = 2
supersymmetry becomes N = 4 and
F
RS
=0 =
N
X a2:
i
2 i=1
(1.12)
(1.13)
!
The pure gauge limit of the 5d theory, however, is di erent. The curve is invariant under
+ , and T RS depends on
only through (sin ) . This results in the following
2
de nition of the 5d cuto e
:
Im
! +1; (sin )2N exp (2 { ) ! ( 1)N e2N :
Again, the limit of in nite mass in the vedimensional theory is accompanied with the
decompacti cation of the x6 direction.
{ 6 {
The most general system that can be obtained in the present setup is the 6d SYM
theory with two compacti ed KaluzaKlein dimensions and the adjoint matter
hypermultiplet. The corresponding brane con guration is a single M5 brane in a spacetime, where
the v plane is compacti ed to a torus S1
the radius of the x5 direction. The R
S1
S1 = T
2 with modulus ^ = { R5=R4, and R5 is
T2 part of the spacetime with coordinates x6,
x10, and v is divided by the symmetry (1.12), and the resulting quotient is a complex
manifold X( ;^), which can be regarded as a T2 bundle over . The twodimensional Riemann
surface
Dell
X( ;^), which is a part of the M5 brane worldvolume R
4
Dell, corresponds
to the spectral curve of the doubleelliptic integrable system [38{40, 76] of N interacting
particles. The term doubleelliptic re ects the fact that there are two elliptic curves,
T2 with moduli
maps to , we consider
and ^ correspondingly. Since under the map X( ;^) !
Dell as an N sheeted covering of the base torus
the curve
. This system
and
Dell
can be also described with the help of Type IIB theory, and the relevant (p; q)brane web
was introduced recently in [17]. Similar to the 5d case, the compactness of the forth and
fth spacetime dimensions can be used to describe some basic properties of the low energy
e ective action. The Coulomb moduli ai now take values in the torus T2, which means
that there is an additional symmetry ai ! ai +
matrix T Dell should depend on the di erences (ai
1 ^ of the curve
Dell. Thus, the period
aj) through an elliptic function. The
most common way to obtain such functions is to consider the second logarithmic derivatives
of the Riemann theta function. In this paper, we use the function
(zj ^) de ned as
1zj ^ ;
where 11
1zj ^ is the usual notation for the Riemann theta function with characteristics
11
1zj ^ = X exp
n2Z
{ (n + 1=2)2 ^ + 2 { (n + 1=2) (z + 1=2) :
For small z, (1.15) can be rewritten with the help of the Eisenstein series fE2kg and of the
Riemann zeta function (k):
1
One could expect that the dependence of the period matrix on the mass parameter
is
also through an elliptic function. However, the curve
Dell is not invariant under the shift
+
^ alone. It turns out that the shift of the mass parameter is accompanied with
the shift of the rst elliptic parameter , and the actual symmetry of Dell is
!
+
^;
!
+ N ^ + 2
1 :
Since this symmetry is observed in the low energy limit of the theory, it probably has
more involved structure in the superstring and M theory. Nonetheless, the following
elementary interpretation can be suggested. In Type IIA theory,
describes the distance
{ 7 {
(1.15)
(1.16)
(1.17)
(1.18)
on T2 between the two ends of a D4 brane. The brane con guration in S1
the spacetime with one D4 brane can be represented by the following embedding into the
threedimensional space:
(1.19)
HJEP1(207)3
The A and B cycles on T2 correspond to the compacti ed x4 and x5 directions respectively.
Upon moving one end of the D4 brane all the way around the x4 direction, the line
representing this D4 brane in the 3d embedding goes around the A cycle, and we get the same
con guration we started with. This describes the symmetry
+ . When we move
one end of the fourbrane all the way around the x5 direction, the line in the 3d embedding
!
wraps around the B cycle. This could be interpreted as some e ective extension of the
fourbrane length or the radius of the x6 direction. In M theory, this D4 brane becomes
a part of a single M5 brane and its wrapping around x5 direction could be interpreted as
some e ective shift of the rst elliptic parameter . The above interpretation is based on
the particular form of the 3d embedding and does not explain the exact value of the shift
in . As a result of the symmetry (1.18), the period matrix T Dell depends on the mass
parameter
not only through the elliptic function
( j ^), but also through the Riemann
theta function 11
After compactifying the x
1 j ^ , which will be seen later in formulas (2.7), (6.47), and (7.6).
5 direction, the theory remains conformally invariant, but the
duality group changes. The obvious reason is that now one has two elliptic curves and two
duality groups describing the modular transformations of two elliptic parameters
is accompanied by the shift of the other and, for generic values of the parameters of the
theory, this shift is not even an element of the group SL (2; Z) . The four generators of
this duality group are
1)
2)
3)
4)
!
!
+ 1;
1= ;
1=^;
The actions of the second and the forth generators from (1.20) on the SeibergWitten
Dell can be described by two modular anomaly equations. The rst
equation is a generalization of the fourdimensional MNW modular anomaly equation and has
one additional term, the derivative of the prepotential with respect to the second elliptic
{ 8 {
parameter ^:
The second modular anomaly equation is
2 2
N
X ai
i=1
!2
with the notation E^2
E2 (^). At this point, one can see that, in the present setup, the
low energy e ective action is not invariant under the simple permutation of the two elliptic
and ^. This is because we started with Type IIA theory and, within the
obtained formulation of M theory, the two tori
and T2 are not exactly equivalent. These
tori could become equivalent after a series of T dualities and appropriate changes of the
spacetime. We expect that the 6d modular anomaly equations can be lifted to the level of
Nekrasov functions, as it was done for the 4d case in [77{80] and to the level of 2d conformal
eld theories in [81, 82]. Note that, in the recent paper by S. Kim and J. Nahmgoong, [75],
the Sduality in 6d (2; 0) theory was studied. From the point of view of SYM theories,
the partition function considered in [75] corresponds to the Nekrasov instanton partition
function of the 5d SYM theory with the adjoint matter hypermultiplet. One of the results
described in [75] is that the 5d prepotential admits the same modular anomaly equation
as the 4d one, in accordance with what was stated in [43].
One more topic we are going to discuss in this paper is the behavior of the theory at
particular values of the bare mass . At
theory, and the prepotential is
= 0, the theory becomes N = 4 supersymmetric
F
Dell
=0
=
2
N
describes the shift in two compact dimensions, there is neither the limit of in nite
mass nor the pure gauge limit in the 6d case. Yet there is a special point
=
1, at
which the elliptic function
( j ^) goes to in nity. In fact, an elliptic function must have
at least two poles in a fundamental parallelogram, but we will use the single notation 1
keeping in mind that 1 can take several values. The exact value of 1 depends on the
particular choice of elliptic function, and, in our case, it can be described as a solution to
the following equation:
} ( 1j ^) =
1
3 E2 (^) ;
where relation (A.1) between the
function and the Weierstrass } function was used. By
analogy with the 4d and 5d cases, we consider the limit:
!
1;
2N 101 (0j ^)2N exp (2 { ) ! ( 1)N ^ 2N ;
where the new parameter ^ plays the role of the e ective cuto in the prepotential. Since
the Riemann theta function 11
1 j ^ has no poles and is nite at
=
1, there is
(1.21)
(1.22)
(1.23)
(1.24)
(1.25)
no need to decompactify the x6 direction and bring the rst elliptic parameter
to the
imaginary in nity. In what follows, we refer to (1.25) as the limit
( j ^) ! 1. Despite
all the di erences, one can still recover (1.14) and (1.11) from (1.25) by considering the
limit of Im ^ ! +1. Since Im 1 is proportional to Im ^, and (1.24) implies that Im 1 is
degenerate into the (sin )2N and to get the nite cuto one restores the limit Im
nonzero, 1 goes to imaginary in nity in the limit Im ^ ! +1. Theta functions in (1.25)
The rest of the paper is organized as follows. In section 2, we introduce the
doubleelliptic SeibergWitten prepotential for N
2. In section 3, we discuss the curve
and its properties under the modular transformations of the rst elliptic parameter ,
which leads to the
rst modular anomaly equation (1.21). In section 4, the modular
transformations of the second elliptic parameter ^ are studied, and the second modular
anomaly equation (1.22) is derived. The limit
( j ^) ! 1 is described in section 5, and
the convergency condition is formulated as some nontrivial restriction on the coe cients
in the series expansion of the doubleelliptic prepotential (2.1). In section 6, the N = 2
doubleelliptic prepotential is considered. We demonstrate that the rst modular anomaly
equation along with the convergency condition for the limit
to calculate this prepotential as a series in the mass parameter . The second modular
anomaly equation also proves to be very e cient in the N = 2 case, because it reduces the
problem of computation of the prepotential to the problem of nding of one single function
c^1 ( ; ; ^). In a similar way, we use the rst modular anomaly equation and the limit of
( j ^) ! 1 to compute the N = 3 prepotential in section 7. The results for the N = 3
case are in complete agreement with the calculations from [43], where the involutivity
conditions for the doubleelliptic Hamiltonians were used to compute the prepotential. For
3, the second modular anomaly equation is not that e cient as in the N = 2 case.
However, it provides nontrivial relations between the coe cients in the series expansion of
the doubleelliptic prepotential. In both N = 2 and N = 3 cases, we evaluate the rst few
orders in the qexpansions, q
exp (2 { ), of the rst nontrivial coe cient Cbi1;:::;in ( ; ; ^)
in the expansion (2.1) with i1 = 1 and i2 =
= in = 0. The results given in (6.47)
and (7.6) clearly manifest the symmetry (1.18) and are consistent with the limit (1.25).
Moreover, due to the properties described by (1.18), (1.23), and (1.25), we conclude that
the structure of the qexpansions is uniform for all the coe cients Cbi1;:::;in ( ; ; ^): each
power of q is multiplied by the Riemann theta functions to the power of 2N as in (1.25) and
( j ^) ! 1 can be used
by the
nite linear combination of nonpositive powers of
( j ^) with coe cients being
quasimodular forms in ^ with some particular weights. Thus, the exact expression for any
given order in q of any given function Cbi1;:::;in ( ; ; ^) can be computed.
! +1.
Dell
HJEP1(207)3
2
Doubleelliptic SeibergWitten prepotential
According to [43], there exist nonlinear equations for the SeibergWitten prepotential,
which have exactly the N particle doubleelliptic system as its generic solution.
With
the help of these equations, the expression for the N = 3 doubleelliptic SeibergWitten
prepotential was derived. After some minor simpli cations, the obtained result can be
generalized to the case of N
2 as
F
root system. The coe cients Cbi1;:::;in are fully symmetric under the permutation of indices
i1; : : : ; in and depend on the both elliptic parameters only through the Eisenstein series.
For example, Cbi1;:::;in can be decomposed in powers of in the following way:
where (m) stands for the multiindex (m1; m2; m3),
^
E2
E2 (^) ;
^
E4
E4 (^) ;
^
E6
E6 (^) ;
and Cbi1;:::;in;k;(m) ( ) are quasimodular forms of weight 2 i1 +
+ 2 in + 2k. Also, one
should impose some additional restrictions on the summation over the indices i1; : : : ; in
in (2.1), since otherwise not all the coe cients Cbi1;:::;in ( ; ; ^) are independent: there are
some relations between the functions ( ~ k ~a j ^).
Since the functions Cbi1;:::;in;k;(m) ( ) are quasimodular forms, they can be realized as
polynomials in the Eisenstein series E2, E4, and E6:
E2 ( ) = 1
E4 ( ) = 1 + 240 X
E6 ( ) = 1
24 X
n2N
1
The constant terms in the expansions of Cbi1;:::;in;k;(m) ( ) in powers of q = exp (2 { )
correspond to the perturbative part of the prepotential F
perturbative part is known and can be written in terms of the second derivatives as follows:
Dell. The exact answer for the
i 6= j :
2 Dell
2 Dell
N
+X q
k2N
2 (k)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
where aij
ai
aj and the functions F
(k) = F
(k) (a; ; ; ^) describing the instanton
corrections do not depend on the rst elliptic parameter . As one can note, at the
righthand sides of (2.7) there are some speci c aindependent terms that are essential for the
computation of the limit
First modular anomaly equation
From the M theory point of view, the curve
Dell is a twodimensional Riemann surface in
a compact fourdimensional manifold X( ;^) de ned earlier in the introduction. X( ;^) can
be thought of as a T2 bundle over , where
and T2 are two di erent tori with moduli
and ^. Under the projection X( ;^) !
, the curve
Dell maps to
, and this gives
rise to the interpretation of
Dell as an N sheeted covering of the base torus . To get a
proper geometrical description of this covering, one needs to determine the corresponding
multivalued function from
to T2. In the 4d case, when T2 is decompacti ed to a complex
plane C, the homology basis (Ai; Bi) for the curve
cycles A; B on the base
to each sheet:
CM is given by the lifts Ai; Bi of the
To draw a similar picture for the curve
Dell, one needs to compactify each copy of C to
a torus, which can be done, for example, by adding two cuts on each sheet. However, the
placement of the resulting four cuts is crucial and a ects the basic properties of the curve,
since some of the cuts might be coincident. Thus, instead of guessing the right geometrical
interpretation, we use the explicit expression for the doubleelliptic prepotential (2.1) and
de ne the N
N period matrix of Dell by
This implies that the homology basis for
Dell is still given by (Ai; Bi) and properties
of the curve are described by the picture (3.1). In particular, (3.1) is very useful for
understanding the properties of Tij with respect to the modular transformations of the rst
Tij =
2 Dell
(3.2)
elliptic parameter . On the other hand, the picture (3.1) is not applicable to description
of the behavior of the second elliptic parameter ^ under the modular transformations of .
The same is true for the properties of Tij with respect to the modular transformations of
^. At this point, the explicit expression (2.1) comes into play.
Let us start with the modular transformations of the rst elliptic parameter
!
The rst transformation is trivial and results in the following shift of the period matrix:
The second transformation from (3.3) interchanges the cycles A and B on the base torus:
Taking into the account the de nitions of the at moduli ai and their duals aiD
! 1=
! Bi;
Bi
! 1=
!
Ai:
ai =
1 I
2 { Ai
kdz;
a
iD =
1 I
2 { Bi
kdz =
one gets
functions 0, 0, ^0:
ai
! 1=
! aiD;
a
D
i
! 1=
!
ai;
T
! 1=
!
The transformations for the other parameters can be written in terms of yet unknown
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
Since the cycles Ai and Bi on each sheet of the covering are situated exactly above the
cycles A and B, (3.5) results in
0 =
;
;
^0 = ^
2 :
In what follows, we treat 0, 0, ^0 as series expansions in powers of and require, that the
coe cients in these expansions do not depend on the at moduli a.
The modular transformations (3.8) and the quasimodular properties of the coe cients
in the series expansion (2.1) allow one to determine 0, 0, ^0. To this end, we reformulate
the transformation of the period matrix T (a; ; ; ; ^) as
T
aD; 0; 0;
1
; ^0
=
N . Evaluation for N = 2; 3; 4 demonstrates that the solution is very simple and, in terms
of 0, 0, and ^0, can be represented as:
There are di erent ways to con rm that (3.11) is a proper general solution. A
straightforward way is to solve (3.10) for higher values of N . An easier way is to consider the rst
modular anomaly equation, which is introduced below in (3.20), and solve it perturbatively
in . Then, in the rst nonzero order, the appearance of N in the function ^0 is necessary
to ensure the consistency of the equation. Summarizing the results, we describe the action
of the second modular transformation from (3.3) as
!
1
;
2 ;
!
;
!
;
^ ! ^
ai ! aiD;
T !
:
(3.12)
!
at moduli
An interesting feature of (3.12) to pay attention is the transformation law of the parameter
. As we explained in the introduction, the
inverse is proportional to the radius of the
forth spacetime dimension:
the modular transformation ^ !
1 = 2R4. Therefore, the natural transformation for
under
1=^ of the second elliptic parameter ^ = { R5=R4 is
=^ and this will be the case in the next section. The fact that we have
under the modular transformation
1= of the rst elliptic parameter could mean
that one of the cycles of
cuts on each sheet of
is mapped onto one of the cycles of T
2 and some of the four
Dell coincide in accordance with our earlier assumptions.
To derive the 6d modular anomaly equation, consider the linear combination of the
a
D
i
ai =
Fe
F
Dell
2
N
i=1
depends on the elliptic parameter
following modular properties:
E2
1
=
2E2 ( ) +
6
{
;
E4
Fe aD; ; ; 2E2 +
; 4E4; 6E6; ^
This allows us to simplify (3.15):
6
{
a +
1
As it can be seen from (2.1) and (2.2), Fe possesses some type of scaling invariance
ra Fe; E2 +
6
{
; ^
=
With the help of (3.8), we obtain
or
a
D
i
aD; ; ;
ai
1
; ^
! 1=
1
!
D
i
ai
Function Fe is the sum of the perturbative and instanton parts of the prepotential and
only through the Eisenstein series E2, E4, E6 with the
!
1
N 2
2
1
6
{
!
=
(3.13)
(3.14)
(3.15)
2
:
(3.17)
(3.18)
=
4E4 ( ) ;
E6
=
6E6 ( ) :
(3.16)
= Fe
aD
; ; ; E2 +
; E4; E6; ^
where ra = (@=@a1; : : : ; @=@aN ) and the dependence on the other arguments is implied on
the both sides of the equality. This equation manifest the new symmetry of the function
Fe and describes the dependence of the prepotential on the second Eisenstein series E2 ( ).
Consider the rst order in the expansion of (3.18) in powers of 1= :
N
j=1
= 0:
Integrating with respect to ai and omitting the constant of integration, we obtain the 6d
generalization of the MNW [64] modular anomaly equation:
N 2 @F Dell
2
=
2
i=1
Second modular anomaly equation
We learned in the previous section that the two tori
and T2 play di erent roles in the
geometrical description of the curve
Dell. In particular, the de nitions of the moduli a
and aD are essentially connected with the cycles A and B on the base torus , and the
period matrix T has the U(1)decoupling property:
N
i=1
X Tij = ;
8j = 1; : : : ; N:
This indicates that the theory should behave di erently under the modular transformations
of the rst and the second elliptic parameters.
In order to understand the behavior of the period matrix T under the modular
transformations of the second elliptic parameter ^, we
rst consider it at the classical and
perturbative levels. With the help of the exact expressions (2.7) and of the expansion
log
11
1
zj ^
1z 101 (0j ^)
=
+1
X
k=1
(2k)
k 2k E2k (^) z2k;
we establish that the sum of the classical and perturbative parts of the period matrix is
invariant under the following transformations of the moduli:
and
^ !
1
^
;
2 ^ ;
ai ! ai;
Tij ! Tij
2
2 ^ :
In fact, the second modular transformation (4.4) shifts the period matrix. However, this
shift can be removed by adding to the classical part of the prepotential a term proportional
to (Pi ai)2. This term is also relevant for the computation of the limit ( j ^) ! 1, which
we will discuss in the next section.
Tij (a; ; ; ; ^ + 1) = Tij (a; ; ; ; ^) ;
Tij a; ; ;
^ ^
2 ^ ;
1
^
= Tij (a; ; ; ; ^)
Since the period matrix depends on the second elliptic parameter only through the
Eisenstein series, the rst equation from (4.5) is trivial. The second equation from (4.5) gives
Dell a; ; ;
^ ^
2 ^ ;
1
^
F
Dell (a; ; ; ; ^)
=
Taking into the account the scaling properties with respect to the second elliptic parameter
F
Dell a; ; ;
^ ^
= F
Dell a; ; ;
We notice that the transformations (4.3) and (4.4) do not mix the instanton part of
the period matrix with the classical and perturbative parts. Thus, it is natural to assume
that the instanton part is also invariant under the modular transformations of the second
elliptic parameter (4.3) and (4.4). This assumption provides us with a nontrivial equation
on the prepotential, which can be reformulated in terms of the linear relations between
the functions Cbi1;:::;in ( ; ; ^) and their derivatives. We derive later the exact expressions
for some rst functions Cbi1;:::;in;k;(m) ( ) in the cases when N = 2; 3 and the relations will
be valid for all the computed expressions. We consider this as a strong evidence in favor
of the assumption being made. A less direct evidence is provided by the fact that the
transformation laws for the parameters , , , and ^ are covariant under the permutation
The invariance of the period matrix under the modular transformations of the second
elliptic parameter imply that the following equations on the period matrix hold:
we rewrite (4.6) as
Dell a; ; ;
N 2
2 ^ ; E^2 +
6
{ ^
F
Dell a; ; ; ; E^2
=
2
2 ^ j=1
N
X aj :
This leads us to the second modular anomaly equation:
N 2 @F Dell
2
2
2 2
N
X ai
i=1
!2
;
where the integration constant coming from (4.8) was omitted.
Expanding (4.9) in powers of , in the rst nonzero order we get the equation
X log 11
;
2
2 ^ :
2
2 ^
N
X aj :
j=1
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
which can be checked with the help of (4.2), and the particular value of the Riemann
zeta function
(2) =
2=6. As it was mentioned earlier, the second modular anomaly
equation (4.9) can be solved in each order in , and the solution is given by the linear
relations between the functions Cbi1;:::;in ( ; ; ^) and their derivatives. In the simplest case
of N = 2, the exact solution is given by the recurrence relations (6.9).
5
Before we start solving the rst modular anomaly equation (3.20), it is useful to understand
the limit of
( j ^) ! 1. In the 4d case, the limit of in nite mass m (or the pure gauge
limit) was very well studied [63, 64]. At the same time, as m goes to in nity, one should
simultaneously bring the elliptic parameter
to imaginary in nity, so that the resulting
HJEP1(207)3
cuto
is nite:
(5.1)
and
(5.2)
(5.3)
(5.4)
!
matrix is required:
The limit of the 4d prepotential can be described as an in nite series in powers of
corresponds to the periodic Toda integrable system [3, 60]. Analogous considerations in
the 6d case could result in the elliptic generalization of the Toda system [83, 84].
The pure gauge limit is replaced in the 6d theory by the special point
= 1, which
is de ned by (1.24). The fact that there is no limit of in nite mass in the 6d case, is clear
from both the M theory and the pure analytical point of view. The M theory viewpoint
was described in the introduction, and it relies on the interpretation of the mass parameter
as a shift in two compact dimensions. In the pure analytical approach, one would consider
the exact expression (2.7) for the perturbative part of the period matrix and notice that it
depends on the elliptic function of , which does not have a limit, when
tends to a real or
imaginary in nity. Since 1 is just some nite value of , one might argue that there is no
need to approach this value in the special limit of
1. However, the elliptic function
!
= 1 and it is a nontrivial requirement for the prepotential to have
1. We treat this limit by analogy with the pure gage limits of
the 4d and 5d theories. The complexi ed coupling constant
and the mass parameter
should be replaced by one new parameter ^ . To this end, the following shift of the period
This shift corresponds to an additional classical term in the prepotential:
Then, a counterpart of the pure gauge limit in the 6d theory is
8i; j :
Ti1j = Tij
N
:
F 1 = F
Dell
N
X ai
i=1
!2
:
2N
1;
2N 101 (0j ^)2N exp (2 { ) ! ( 1)N ^ 2N :
For the period matrix one gets
!lim1 Ti1i =
i 6= j : !lim1 Ti1j =
1
log
1
2 (k)
2 (k)
:
As in the 4d case, the convergency condition for the instanton part of (5.5) imposes
additional restrictions on the coe cients in the series expansion of the prepotential (2.1). In
order to satisfy this condition, we rewrite the instanton part as a power series in the new
( j ^) and qe
exp (2 { ), where
Then, the coe cients in the expansion (2.1) transform as
e
=
+
N
{
e
log
e is
11
1 j ^
:
2(i1+ +in+k+1)E^m1 E^m2 E^m3 Cbi1;:::;in;k;(m) ( )
2 4 6
2(i1+ +in+k+1)E^m1 E^m2 E^m3 Cbi11;:::;in;k;(m) (e)
2 4 6
(5.5)
(5.6)
(5.7)
(5.8)
(6.1)
+1
X
X
k=0 m1m+12;mm22+;m33m30=k
=
+1
X
X
k=0 m1m+12;mm22+;m33m30=k
+1
l=0
F
and the functions Cbi11;:::;in;k;(m) are linear combinations of Cbi1;:::;in;k;(m) and their derivatives.
In terms of these new parameters, the limit (5.4) can be described as 2N qe ! ( 1)N ^ 2N .
Since there should be no divergent terms 2Nm+k q m with m; k 2 N at the r.h.s. of (5.7),
we get the following restrictions on the functions Cbi11;:::;in;k;(m):
e
Cbi11;:::;in;k;(m) (e) = X Cbi11;:::;in;k;(m);l qel = Cbi11;:::;in;k;(m);0 +
X
l (i1+ +in+k+1)=N
l
Cbi11;:::;in;k;(m);l qe ;
where the rst d i1+ +in+k+1
2
e
1 coe cients Cbi11;:::;in;k;(m);l, l 2 N should vanish, and
the constant term Cbi11;:::;in;k;(m);0 was already taken into account in the limit (5.5) of the
perturbative part of the prepotential.
6
Modular anomaly for N = 2
In this section, we consider the twoparticle doubleelliptic integrable system in the center
of mass frame (a1 + a2 = 0). This case is the simplest one from the computational point
of view, and, at the same time, it re ects all the relevant phenomena arising in the general
N particle case. The corresponding prepotential can be written as
T (a; ; ;
+ 1; ^) = T (a; ; ; ; ^) + 1;
T
=
The rst modular anomaly equation is
2
=
1
2
2
:
It can be reformulated in terms of the recurrence relations for the functions ^cn ( ; ; ^).
Restoring the integration constant in (6.5) and using the standard di erential equations
for the function
3
2 : 3
(zj ^) (see A), we get the set of relations
+ 2E^2 c^1
E^22
^
E4
c^21 + 2 c^2 +
E^23
3E^2E^4 + 2E^6 c^1c^2 = 0;
(n
1) c^n 1
m (n
m
1) c^m c^n m 1
+ n 2E^2 c^n
+
4
6
6
54
E^22
E^23
4
6
6
n
X
m=1
n + 1 4 E^22
n 2
2
m=1
^
E4 c^n+1 +
n+1
X
m=1
X
m=1
m (n
m + 1) c^m c^n m+1
These relations are somewhat similar to the AMM/EO topological recursion [85{89] and
can be solved exactly, if the proper boundary conditions are imposed. For example, one
can use the convergency condition for the limit described in section 5.
The second modular anomaly equation is
where
c^n ( ; ; ^) =
X
k=0
2k
X
The second derivative of the prepotential (6.1) de nes the period matrix:
T =
2 Dell
log
2 (k)
The modular transformations of the rst elliptic parameter
act on the period matrix as
Solving (6.8) for all orders in , we get the following recurrence relations
N = 2 :
= 0:
n
1 :
3
m (n
m) c^m c^n m
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
n
1 :
c^n =
(6.10)
Thus, the second modular anomaly equation reduces the problem of evaluation of the N = 2
doubleelliptic prepotential to the problem of nding of one single function ^c1 ( ; ; ^).
Equations (6.4) and (6.5) along with the convergency condition for the limit of ( j ^) ! 1
can be used to de ne the instanton part of the N = 2 prepotential completely, without
making any additional assumptions about the functions ^cn ( ; ; ^). The equations for the
period matrix (6.4) describe the quasimodular properties of the coe cients ^cnk(m) ( ) in
the expansion (6.2). Then, the modular anomaly equation (6.5) allows one to compute the
dependence of c^n ( ; ; ^) on the second Eisenstein series E2 ( ). Finally, the convergency
condition from section 5 provides us with E2independent part of the N = 2 prepotential.
To simplify the notation, we replace the multiindex (m) in (6.2) by an ordinary index m
for some rst orders in :
c^n ( ; ; ^) = c^n00 + 2E^2 c^n10 + 4 E^22 c^n20 + E^4 c^n21
+ 6 E^23 c^n30 + E^2E^4 c^n31 + E^6 c^n32 + O
Quasimodular properties of c^nk(m) ( ). Expanding the second equation from (6.4)
in powers of , we obtain in the rst nonzero order:
The rst equation from (6.4) is equivalent to the periodicity condition ^cnkm ( + 1) =
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
and so on. In general, (6.4) describes an important property that the functions ^cnk(m) ( )
are quasimodular forms of weight 2n + 2k, which was assumed in (6.1) from the outset.
Moreover, these equations de ne the dependence of each ^cnk(m) ( ) on E2. The only
problem is that the second equation of (6.4) is quite complicated, and there is no simple way
to reformulate it in terms of the recurrence relations for general ^cn ( ; ; ^), as it was done
in (6.7) for the modular anomaly equation (6.5). Thus, we are going to use (6.7) to de ne
the dependence of c^n ( ; ; ^) on E2 ( ). Since the recurrence relations in the 6d case involve
an additional partial derivative with respect to the second elliptic parameter ^, it is useful
to start with the simpler 4d and 5d cases.
c^nkm ( ), which gives
In the next order, we get
c^100
= 0;
2 : 3
1) cn 1
m (n
m
1) cm cn m 1 = 0;
where 4d functions cn ( ) are related to the 6d functions through
Taking into account that the functions cn ( ) are quasimodular forms of weight 2n, we
realize them as polynomials of three generators E2, E4 and E6:
8n 2 N :
cn =
n E2n + n E2n 2E4 + nE2n 3E6 + : : : ;
where 1 = 0 and 1 =
2 = 0. The rst coe cient 1 is de ned by the rst equation
Then, the recurrence relations for n
2 provide us with the general expression for n:
+ 2 c1 = 0;
e
m (n
m) ecm ecn m = 0;
n 2
c1 ( ) =
n
2 n 1
X
2 m=1
(6.16)
(6.17)
of q in its expansion:
which gives
and
8n
1 :
n =
2 n
5
3n
1 (2n
1)!!
(n + 2)!
:
To obtain the general expression for n, we use the convergency condition from section 5.
According to the constraints (5.8), the function c^200 ( ) should not contain the rst power
c2 =
2 E22 + 2 E4 =
E22 + 2 E4 =
+ 2 +
In the same way, the general expressions for all the coe cients in (6.18) can be obtained.
Evaluating n, one gets
cn =
1 (2n 1)!! 1
3n (n+1)!
E2n +
Recurrence relations in 5d. Taking Im ^ ! +1, we get the 5d limit of (6.6) and (6.7):
n
2 : 3 e
1) ecn 1 + n 2 cn
e
m (n
m
1) ecm ecn m 1
(6.24)
where
cn = ecn ( ; ) =
e
X
k=0
2k
X
The rst equation from (6.24) describes an exponential dependence of ec1 ( ; ) on E2:
and the unknown function
1 ( ; E4; E6) can be xed by the convergency condition for the
limit
which, along with the quasimodular properties of c^nk0 ( ), provides us with the following
( j ^) ! 1. First of all, the expansion of ec1 ( ; ) should not contain any poles in ,
n;m 0
1nm 4n+6mE4n E6m;
To de ne other coe cients 1nm, we use the de nition of the 5d cuto
e
:
Then, as we established in the previous section, the prepotential should be convergent as
a power series in a new parameter
order by order all the coe cients
e = sin , and this requirement allows one to evaluate
1nm. However, the exact answer can be obtained, if we
notice that a perfect candidate for the function convergent in the limit (6.28) would be the
Riemann theta function. To establish the connection between the Eisenstein series and the
theta functions, we consider the expansion (4.2) with z = :
log
11
1
j
+1
X
k=1
(2k)
k 2k E2k ( ) 2k:
11
1
2
4 ec1 ( ; +1) =
2
2
(sin ) ;
2
The theta functions at the l.h.s. of (6.29) are automatically convergent in the limit (6.28), if
we appropriately rescale the rst few terms in their expansion in powers of q. At the same
time, the r.h.s. of (6.29) tells us how to apply these theta functions to (6.26). The rst term
in the expansion (6.29) is 2 E2=6, which gives the exact answer for the function ec1 ( ; ):
ec1 ( ; ) =
1
j
2
the rst few terms in the qexpansion are
The combination 4 ec1 ( ; ) is just the theta function up to some qindependent shift, and
4 ec1 ( ; ) =
2
2
1
2
An additional check of (6.30) is provided by the perturbative limit Im
! +1:
which matches the exact answer for the perturbative part of the prepotential (6.3).
2
3
1
c2 ( ; ) =
3
+
which can be rewritten with the help of (6.30) as follows:
2
2
+
4
8 6
11
1
4
Due to the quasimodular properties, the unknown function 1 has the following expansion:
1
2
X
n;m 0
n+m>0
1 ( ; E4; E6) =
1nm 4n+6mE4n E6m:
This function is nontrivial, because in (6.35) there is the term
The equation for the second function ec2 ( ; ) is
ec1 +
2
Substituting ec1 in the form (6.26), we get the following solution:
(6.33)
(6.35)
(6.36)
(6.38)
(6.39)
(6.40)
(6.41)
4 11
j
4
4
= (sin )4 + 16 q (sin )6 + 48 q2 1 + 2 (sin )
2 (sin )6 + O q3 ; (6.37)
which leads at generic 1 to the divergence of 6 c2 when the imaginary part of
in nity. In fact, there are two divergent terms: the perturbative one (sin )4 and
nongoes to
e
perturbative q (sin )
6
(sin )2. The former one is separately dealt with in (6.3), and the
perturbative limit of ec2 is given by
This xes only the initial condition for 1
:
6 ec2 ( ; +1) =
1
16
This gives a hint that 1 can be expressed in terms of the theta functions, which should
be checked by the requirement of cancelling the nonperturbative q (sin )
gency. Indeed, consider the second partial derivative of (6.29) with respect to :
6
(sin )2
diver1
j
1
2 2 + X
+1
k=2
(2k)
(2k)
1
j
:
4 E4
E4
1360800
4
1
2
22680
E2
4
136080
+
8981280
The r.h.s. of this latter equation perfectly ts the perturbative limit (6.39) and the l.h.s.
provides the series expansion for 1
. Making sure that the rst few terms in the expansion
of 6 ec2 are convergent, we get the exact expression
4
1
( j )
In the same way, the recurrence relations (6.24) allow one to evaluate the functions ecn at
Recurrence relations in 6d. Finally, we come to the most general 6d case and to
the recurrence relations (6.6) and (6.7). The rst equation (6.6) includes both functions
c^1 ( ; ; ^) and c^2 ( ; ; ^). However, the second function appears in terms of higher orders
in . The same is true for other equations (6.7), where the function c^n+1 ( ; ; ^) appears
in 4 and 6terms. This allows one to calculate the coe cients c^nk(m) ( ) in the series
expansions (6.2) order by order. Using the convergency condition for the limit ( j ^) ! 1,
we compute some rst coe cients in the expansion of the functions c^n ( ; ; ^):
(6.43)
(6.44)
(6.45)
(6.46)
180
245E23 +42E2E4 17E6
455E24 +91E22E4 46E2E6 +4E42
1225E24 +700E22E4 +60E2E6 +31E42 +O
6 ;
175E23 +84E2E4 +11E6
665E24 +357E22E4 2E2E6 12E42
E^22 19250E25 +12089E23E4 759E22E6 319E2E42 21E4E6
E^4 3927E25 +3773E23E4 +737E22E6 +572E2E42 +63E4E6
+O
6 ;
where the notation is usual: E2k
E2k ( ) and E^2k
E2k (^). It can be easily checked
that the above expressions are in complete agreement with the recurrence relations (6.9)
coming from the second modular anomaly equation (6.8). In other words, each function
c^n ( ; ; ^) with n
2 can be obtained from c^1 ( ; ; ^) with the help of relation (6.10):
n
2 : c^n =
c^1:
This claim is supported by the rst 22 orders in the expansion of c^1 ( ; ; ^). Thus,
provided the second modular anomaly equation is correct, the computation of the N = 2
doubleelliptic prepotential reduces to nding just the rst function ^c1. To this end, the two
modular anomaly equations can be combined. Using (6.10) with n = 2 and equation (6.6),
one gets the rstorder partial di erential equation for the function ^c1, and the boundary
condition for this equation is given by the limit
To conclude the N = 2 part of the paper, let us discuss some important properties of
the function c^1 ( ; ; ^), in particular, look at the series expansions in powers of the other
two parameters q = exp (2 { ) and q^ = exp (2 { ^). Relying on the computed orders in
the expansion of c^1, we establish the rst few orders in the qexpansion:
c^1 ( ; ; ^) = c^1pert ( ; ^)
2 2 8 11
4
4 4 11
4
following equation:
lim
Im !+1
c^n ( ; ; ^)
The expansion (6.47) is consistent with the symmetry (1.18) and justi es the choice of the
in section 5. The function c^1pert ( ; ^) contributes to the exact expression
for the perturbative part of the period matrix (6.3). With the help of (6.3), we derive the
E^23
3 E^2 E^4 + 2E^6
3
2
E^22
E4 c^pert +
^ 1
9
4 log
11
1 j ^
which can be used to calculate the function c^pert ( ; ^) up to any given order in .
1
The rst few orders in the q^expansion of the function c^1 are
1
2 11
4 11
1
1
j
2
4
2
4
2q^
2 +2 2 11
1
( j )
2
+
9
1
j
2
j
6
6
2E22
3E2
( j )
2
(6.50)
(6.47)
(6.48)
As expected, the zeroth order is the 5d function ec1 ( ; ) de ned in (6.30). The structures
of the both q and q^expansions are similar: the coe cients are speci c theta functions of
with moduli ^ and
correspondingly. Thus, the exact expressions at any nite order in
these expansions can be computed.
7
In this section, we use the modular anomaly equations in order to compute a few rst orders
in the expansion of the functions Cbi1;:::;in ( ; ; ^) in the case of N = 3. Since Cbi1;:::;in are
X log 11
X X X 2(n+ni+j) Cbn;ni;j ( ; ; ^) s^n;ni;j ( a; ^) ;
where again the summation over the index i was restricted, since otherwise not all the coe
cients Cbi1;i2;i3 ( ; ; ^) would be independent because of the relations between ( ~ k ~a j ^)
and, hence, between s^i1;i2;i3 ( a; ^).
With the help of the rst modular anomaly equation (3.20) and of the convergency
condition (5.8), we compute the coe cients in the prepotential (7.2) up to the order of 14.
This corresponds to computing quasimodular forms Cbi1;i2;i3;k;(m) ( ) in the expansions (2.2)
up to the weight 12. For a few rst functions Cbi1;i2;i3 ( ; ; ^), the expansions are
E2
6
2 E2 5 E22 E4 + 4 E^2
^ 2
7 560
35 E23 7 E2 E4 +2 E6
35 E23 +21 E2 E4 2E6 +O
6 ;
E22 E4 + 2 E2
6 480
25 E23 33 E2 E4 +8 E6
s^i1;i2;i3 ( a; ^) =
with
+ = f~e1
fully symmetric under the permutation of indices i1; : : : ; in, we introduce the variables
(7.2)
~e3g and ~ei ~a = ai. Then, the prepotential (2.1) can be
180
4 E4
^
22 680
1
144
4
E4
2 177 280
E4
2 177 280
10 886 400
+
1
27
2 E^2
( j ^)6 +
1
2
Cb100 ( ; ; ^) =
Cb101 ( ; ; ^) =
Cb200 ( ; ; ^) =
3535 E24 5082 E22 E4 +1592 E2 E6 45 E42
385 E24 294 E22 E4 88 E2 E6 3 E42 +O
6 ;
2 E2
22 680
245 E23 +42 E2 E4 17 E6
6965 E24 +1722 E22 E4 632 E2 E6 +9 E42
8225 E24 +6930 E22 E4 +1000 E2 E6 27 E42 +O
6 ;
and so on. Then, the rst few orders in the qexpansion of the function Cb100 can be
estabCb100 ( ; ; ^) = Cb1p0e0rt ( ; ^) q 4
6 11
E^22
E4
1 j ^ 6
101 (0j ^)6
1
( j ^)4
1
q
E^23
3 E^2 E^4 + 2 E^6
18 E^22 E^4 + 8 E^2 E^6 + 3 E^42
+ O q3 :
We checked that the second modular anomaly equation (4.9) works in all computed
orders. For example, (4.9) provides us with the relation
= O
6 ;
(7.7)
where the r.h.s. is nonzero due to the presence of other functions Cbi1;i2;i3 ( ; ; ^) at higher
orders in .
The results of this section are in complete agreement with the calculations from [43],
where the involutivity conditions for the doubleelliptic Hamiltonians were used to
compute the corresponding SeibergWitten prepotential. To compare the prepotential in the
form (7.2) and the prepotential from [43], one should use the following transition formula:
where the function sn (zj ^) is just the rescaled Jacobi elliptic function:
1
and the function # (^) is
Moreover, the expansion of the prepotential in [43] is written in terms of the theta
conseries E2 (^), E4 (^), and E^6 (^), we use the thetaconstant identities of the form
stants # (^), 00 (0j ^)4, and 10 (0j ^)4. To rewrite this expansion in terms of the Eisenstein
E2 (^) = 00 (0j ^)4 + 10 (0j ^)4
E4 (^) = 00 (0j ^)8 + 10 (0j ^)8
E6 (^) = 00 (0j ^)12 + 10 (0j ^)12
3 # (^) ;
00 (0j ^)4 10 (0j ^)4 ;
3
2 00 (0j ^)4 10 (0j ^)4
00 (0j ^)4 + 10 (0j ^)4 :
8
We discussed a number of questions concerning the low energy e ective action of the
6d SYM theory with two compacti ed KaluzaKlein dimensions and the adjoint matter
hypermultiplet. The main focus of our study was on the properties of the theory under
the modular transformations of the two elliptic parameters
and ^. As a result, two
modular anomaly equations were derived, and the corresponding duality group of the
theory was described by four generators (1.20). We demonstrated that the rst modular
anomaly equation (3.20) provides a new method to compute the doubleelliptic
SeibergWitten prepotential, if proper boundary conditions are imposed. For small enough values
of N , such boundary conditions are given by the limit
( j ^) ! 1. This method of
calculating the SeibergWitten prepotentials is rather simple and could help to achieve
further advance in study of the doubleelliptic integrable systems.
(7.8)
(7.9)
(7.10)
(7.11)
(7.12)
(7.13)
Of course, there are still many problems to investigate. In particular, the curve
Dell
lacks any clear geometrical description that would manifest all the basic properties
discussed in this paper. There were di erent attempts in this direction, including an
interpretation of
Dell as a complex Riemann surface of genus N + 1 in [76] and the thetaconstant
representation for the SeibergWitten curves in [42]. In section 3, we also gained some
clues on how the curve should be described. Nonetheless, all these pieces of data are quite
unrelated, and a separate e ort needs to be done to put it all together. An interpretation of
the obtained results at the level of Nekrasov partition functions could also be an interesting
research direction. As it was mentioned in the Introduction, we are almost certain that
there is an uplift of the 6d modular anomaly equations to the level of Nekrasov functions.
What would happen to other properties such as the symmetry (1.18) is not that clear.
Acknowledgments
We are grateful to Y. Zenkevich for helpful discussions. The work was partly supported
by the grant of the Foundation for the Advancement of Theoretical Physics \BASIS"
(A.Mor.), by grants molaved153120484 and 163200920mola (G.A.), by RFBR grants
150204175 (G.A.), 160100291 (A.Mir.) and 160201021 (A.Mor.), by joint grants
175150051YaF (A.Mir. and A.Mor.), 155152031NSCa, 165153034GFEN,
165145029INDa.
A
function
The
function is directly connected with the Weierstrass } function:
1
1
zj ^ =
1
3 E2 (^) + } (zj ^) :
Some standard di erential equations for (zj ^) are
0 (zj ^)2 = 1
00 (zj ^) =
2
E2
2 +
1
E2
E2
E^22
E^22
E^22
E4
E^23
E^23
zj ^ ;
where we use the notation
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
Open Access.
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