Modular properties of 6d (DELL) systems

Journal of High Energy Physics, Nov 2017

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge . The low-energy Seiberg-Witten prepotential ℱ(a), however, is not explicitly invariant, because the flat moduli also change a − → a D  = ∂ℱ/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2. 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 τ and \( \widehat{\tau} \), the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift of \( \widehat{\tau} \) and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

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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 super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge 1 . The low-energy Seiberg-Witten 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 double-elliptic Anomalies in Field and String Theories; Integrable Hierarchies; Supersym- - = 2 + 4g2{ ! systems, which deserves further investigation. 1 Introduction Double-elliptic Seiberg-Witten 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 double-elliptic (DELL) generalization [38{43] of the Calogero-Ruijsenaars 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 text-book 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 four-dimensional 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 two-dimensional 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 Seiberg-Witten 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 non-zero 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 two-dimensional 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 Calogero-Moser 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 four-dimensional 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 Seiberg-Witten 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 four-dimensional 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 ve-dimensional 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 Seiberg-Witten 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 Kaluza-Klein 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 Seiberg-Witten 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 ve-dimensional 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 Kaluza-Klein 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 two-dimensional Riemann surface Dell X( ;^), which is a part of the M5 brane worldvolume R 4 Dell, corresponds to the spectral curve of the double-elliptic integrable system [38{40, 76] of N interacting particles. The term double-elliptic 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 three-dimensional 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 Seiberg-Witten Dell can be described by two modular anomaly equations. The rst equation is a generalization of the four-dimensional 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 S-duality 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 non-zero, 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 Seiberg-Witten 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 double-elliptic prepotential (2.1). In section 6, the N = 2 double-elliptic 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 double-elliptic 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 double-elliptic prepotential. In both N = 2 and N = 3 cases, we evaluate the rst few orders in the q-expansions, 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 q-expansions 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 non-positive 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 Double-elliptic Seiberg-Witten prepotential According to [43], there exist non-linear equations for the Seiberg-Witten prepotential, which have exactly the N -particle double-elliptic system as its generic solution. With the help of these equations, the expression for the N = 3 double-elliptic Seiberg-Witten 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 multi-index (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 a-independent 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 two-dimensional Riemann surface in a compact four-dimensional 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 double-elliptic 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 non-zero 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 non-trivial 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 non-trivial 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 two-particle double-elliptic 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 double-elliptic 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 E2-independent part of the N = 2 prepotential. To simplify the notation, we replace the multi-index (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 q-expansion are The combination 4 ec1 ( ; ) is just the theta function up to some q-independent 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 non-perturbative 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 6-terms. 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 double-elliptic 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 rst-order 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 q-expansion: 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 q-expansion 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 non-zero 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 double-elliptic Hamiltonians were used to compute the corresponding Seiberg-Witten 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 theta-constant 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 Kaluza-Klein 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 double-elliptic 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 Seiberg-Witten prepotentials is rather simple and could help to achieve further advance in study of the double-elliptic 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 theta-constant representation for the Seiberg-Witten 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 mol-a-ved-15-31-20484 and 16-32-00920-mol-a (G.A.), by RFBR grants 15-02-04175 (G.A.), 16-01-00291 (A.Mir.) and 16-02-01021 (A.Mor.), by joint grants 1751-50051-YaF (A.Mir. and A.Mor.), 15-51-52031-NSC-a, 16-51-53034-GFEN, 16-51-45029IND-a. 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G. Aminov, A. Mironov, A. Morozov. Modular properties of 6d (DELL) systems, Journal of High Energy Physics, 2017, 23, DOI: 10.1007/JHEP11(2017)023