E-strings, F 4 , and D 4 triality

We study the E-string theory on R 4 × T 2 with Wilson lines. We consider two examples where interesting automorphisms arise. In the ﬁrst example, the spectrum is invariant under the F 4 Weyl group acting on the Wilson line parameters. We obtain the Seiberg–Witten curve expressed in terms of Weyl invariant F 4 Jacobi forms. We also clarify how it is related to the thermodynamic limit of the Nekrasov-type formula. In the second example, the spectrum is invariant under the D 4 triality combined with modular transformations, the automorphism originally found in the 4d N = 2 supersymmetric SU(2) gauge theory with four massive ﬂavors. We introduce the notion of triality invariant Jacobi forms and present the Seiberg–Witten curve in terms of them. We show that this Seiberg–Witten curve reduces precisely to that of the 4d theory with four ﬂavors in the limit of T 2 shrinking to zero size.


Introduction
The E-string theory is a fundamental superconformal field theory (SCFT) in six dimensions [1,2].It is commonly defined as the low energy theory of a coincident M5-M9 brane system.The theory preserves eight supercharges and possesses a global E 8 symmetry.When toroidally compactified down to four dimensions, it admits the Seiberg-Witten description [3,4].Upon compactification one can introduce Wilson lines for the global E 8 symmetry.The Wilson lines are specified by eight parameters m 1 , . . ., m 8 corresponding to the Cartan part of the E 8 , which are viewed as the mass parameters of fundamental matters in the low-energy gauge theory [5,6].By tuning these parameters m i together with the modulus τ and the size L of the torus, the E-string theory can flow to almost all rank-one gauge theories and SCFTs preserving eight supercharges in five and four dimensions [5][6][7][8][9].Among others, of particular interest is the flow to the 4d N = 2 supersymmetric SU (2) gauge theory with four massive flavors (N f = 4 theory).This theory is special in many respects [4], especially it retains the infrared gauge coupling τ IR as a parameter.
A natural identification of the parameters in the above flow is that the modulus τ of the torus turns into τ IR while four out of eight Wilson line parameters m i give the masses of 4d flavor matters.
However, there is a puzzle concerning this reduction.There are known various ways to obtain the 4d N f = 4 theory from the E-string theory, but none of them realizes the above parameter identification.For instance, one can first reduce the E-string theory to the (mass-deformed) 4d Minahan-Nemeschansky SCFT of E 8 type [10] and then further reduce it to obtain the N f = 4 theory.In this case, m i are trivially identified, but τ is sent to a special value while τ IR is pulled out from the extra mass parameters of the 4d E 8 theory.In another reduction [11], τ is identified with τ IR as desired, but the identification of m i and the 4d masses is quite non-trivial.
In this paper we study in detail the E-string theory on R 4 × T 2 with four Wilson line parameters and resolve this puzzle.One may think that a natural candidate of the Wilson line configuration that leads to the 4d N f = 4 theory is m = m h := (m 1 , m 2 , m 3 , m 4 , 0, 0, 0, 0). (1.1) Clearly, this configuration admits the invariance of the spectrum under the D 4 Weyl group W (D 4 ) acting on m i , the symmetry that the massive 4d N f = 4 theory evidently possesses.As pointed out in [11], this configuration is equivalent to m = m d := (µ 1 , µ 2 , µ 3 , µ 4 , µ 1 , µ 2 , µ 3 , µ 4 ), (1.2) where m i and µ j are related in a simple manner.The E-string theory with the latter Wilson line configuration has been studied, e.g. in [11][12][13].Indeed, there is a way to identify the 6d theory with this configuration and the 4d N f = 4 theory [11], but this is at the cost of conceding nontrivial identification of mass parameters, as mentioned above.What makes the identification so intricate?The answer we find is that the symmetry acting non-trivially on the above configuration is in fact larger than W (D 4 ): We point out that the 6d theory admits a W (F 4 ) automorphism.
Since the 4d N f = 4 theory does not possess the full W (F 4 ) symmetry, 1 one has to break it down to W (D 4 ) at some point of the reduction.At the same time, it is well known that the N f = 4 theory has an interesting D 4 triality symmetry: the theory is invariant under the outer automorphism of the D 4 Dynkin diagram combined with modular transformations [4].How and when does this symmetry arise from the 6d theory?We find that there exists a purely 6d setup where just enough symmetry is already present before reduction.This can be achieved by adding a further twist to the Wilson line parameters.There are several equivalent variations, but one of such twisted configurations is We show that the E-string theory with this configuration admits the above D 4 triality symmetry.
Another aim of this paper is to investigate a fully symmetric description of physical quantities in the E-string theory with the above configurations.We concentrate on the Seiberg-Witten curve, since it determines various important supersymmetric indices directly or indirectly.Apart from technical complication, a fully symmetric description can be obtained straightforwardly for the F 4 configurations m = m h and m = m d : We present the Seiberg-Witten curve expressed in terms of the W (F 4 )invariant Jacobi forms in both cases.
The case of m = m * is more interesting.As expected, one can write the Seiberg-Witten curve in terms of the W (D 4 )-invariant Jacobi forms, but this is not the end of the story.As mentioned above, the theory in addition exhibits the D 4 triality symmetry.To take this symmetry into account, we introduce the notion of the D 4 triality invariant Jacobi forms and present the Seiberg-Witten curve in terms of them.As far as we know, Jacobi forms of this kind have never been considered in 1 Note, however, that in the N f = 4 theory physical quantities on which the modular group trivially acts can have the W (F 4 ) invariance: For instance, the superconformal index is known to be W (F 4 ) invariant [14,15].This can also be seen from the viewpoint of the E-string theory [16].
the literature before.Further investigation of them may provide us with new insights into the study of Jacobi forms.
We also study the reductions of these theories down to five and four dimensions.For the W (F 4 )-invariant theory in five and four dimensions, we write the Seiberg-Witten curve in terms of F 4 Weyl orbit characters and F 4 Casimir invariants respectively.For the D 4 triality invariant theory in the 5d limit, we find that the Seiberg-Witten curve degenerates everywhere in the Coulomb branch moduli space, meaning that the theory becomes trivial.In the 4d limit, the theory is expected to flow to the N f = 4 theory.This is indeed the case.We explicitly show that the Seiberg-Witten curve written in terms of triality invariant Jacobi forms reduces precisely to that of the 4d theory in the limit of T 2 shrinking to zero size.
We also clarify how the Seiberg-Witten curve for the W (F 4 )-invariant model is related to the thermodynamic limit of the Nekrasov-type formula for the prepotential proposed in [11].The Nekrasov-type formula is a special case of the 6d generalization [17] of the Nekrasov partition function [18,19], whose thermodynamic limit has been well studied [17].However, the resulting Seiberg-Witten curve is commonly expressed in a non-elliptic form.We describe in detail how to transform it into a quartic elliptic curve and subsequently into a cubic Weierstrass form, which precisely reproduces our F 4 curve.This completes the proof of the Nekrasov-type formula for the E-string theory, which has been done only in certain special cases [20].
The paper is organized as follows.In section 2 we study the W (F 4 )-invariant model.We first show the emergence of the W (F 4 ) symmetry and then present the Seiberg-Witten curve in terms of W (F 4 )-invariant Jacobi forms.We also study the 5d and 4d limits.In section 3 we study the thermodynamic limit of the Nekrasovtype formula for the E-string theory and show its equivalence with the F 4 curve.
Along the way, we present two simple expressions for the F 4 curve.Section 4 is devoted to the D 4 triality invariant model.We first show that the E-string theory with the twisted Wilson line configuration admits the D 4 triality symmetry.We then clarify the automorphism group that includes the triality symmetry and introduce the notion of triality invariant Jacobi forms.We present the Seiberg-Witten curve written in terms of them and consider its 5d and 4d limits.In particular, we explicitly show that it reproduces the known Seiberg-Witten curve for the N f = 4 theory.
In section 5 we summarize our results and discuss possible directions for further studies.The definitions of special functions and some useful formulas are presented in Appendix A.

W (F
Here p i and q i are E 8 Cartan charges along the fifth and sixth directions respectively and τ is the complex structure modulus of the torus.We fix the size of the torus in such a way that m satisfies the periodicity condition described below.Thus the E-string theory on R 4 × T 2 with Wilson lines is specified by the coordinate in the parameter space H × C 8 , where H = {τ ∈ C|Im τ > 0} is the upper half plane.
However, the above parametrization is of course redundant and different points on the parameter space are in fact identified with one another.In other words, the parameter space has non-trivial automorphisms.For general m the theory exhibits three kinds of automorphisms [5]: (ii) Double periodicity: (iii) SL(2, Z) automorphism: Here W (E 8 ) and L E 8 denote the Weyl group and the root lattice of E 8 respectively.
In the rest of this subsection let us look into the Weyl group automorphism in more detail.
Let e i (i = 1, . . ., 8) be the standard basis of R 8 .The roots of E 8 are given by ) with even number of '+'. (2.6) The Weyl group of E 8 is generated by the reflections with respect to these roots.Let s α denote the reflection with respect to the root α.Specifically, s α acts on m as For instance, using the expression m = 8 i=1 m i e i we have (2.8) From this we see that s e i −e j interchanges the ith and jth components of m.Let us express this reflection as (2.9) Similarly, we have Note that the combination of the above two reflections yields the simultaneous change of two signs s e i +e j s e i −e j : m i → −m i , m j → −m j . (2.11) In a general context, reflections (2.9) and (2.10) correspond to the D n roots e i ± e j (1 ≤ i = j ≤ n) and thus generate the Weyl group W (D n ).In the present case they generate W (D 8 ).In addition, for the root vector we have where

Weyl invariant Jacobi forms
Jacobi forms are functions which have characteristics of both elliptic functions and modular forms.In particular, Weyl invariant Jacobi forms provide us with a natural language in describing physical quantities when the system possesses the three kinds of symmetries (2.3)- (2.5).In this subsection we recall the definition of the Weyl invariant Jacobi forms and present the concrete generators that we will use.
Let R be an irreducible root system of rank r and W (R) the Weyl group of R. Let L R be the root lattice of R and L * R the dual lattice of L R .When L R is odd, we rescale its bilinear form by 2 so that it becomes an even lattice.A holomorphic function satisfies the following properties [21,22]: (ii) Quasi-periodicity: (iii) Modular transformation law: (iv) ϕ k,m (τ, z) admits a Fourier expansion of the form If ϕ k,m (τ, z) further satisfies the condition that the coefficients c(n, w) of the Fourier expansion (2.23) vanish unless w 2 ≤ 2mn, it is called a W (R)-invariant holomorphic Jacobi form.In this paper a Jacobi form means a weak Jacobi form unless otherwise specified. Let denote the vector space of W (R)-invariant Jacobi forms of weight k and index m.
The space of all W (R)-invariant Jacobi forms is a bigraded algebra over the ring of modular forms.It is known that for any irreducible root system R not of type E 8 , J R * , * is a polynomial algebra generated by r +1 generators [22].This means that every W (R)-invariant Jacobi form is expressed uniquely as a polynomial of r + 1 generators and the Eisenstein series E 4 (τ ), E 6 (τ ).
The generators have been explicitly constructed for all types (except E 8 ) of root systems [22][23][24][25][26][27][28].On the other hand, the ring J E 8 * , * is not a polynomial algebra [29] and has a complicated structure [30,31].Nevertheless, one can systematically construct these Jacobi forms using nine "generators" A i , B j or a i , b j [30,31].In the rest of this subsection we explicitly present the generators for R = D 4 , F 4 , E 8 , which we will use later.
D 4 The generators of the ring of W (D 4 )-invariant Jacobi forms were constructed in [22,24].In particular, fully explicit forms were given by Bertola [24].In this paper we adopt the notation of the reference [32], where Bertola's generators are expressed more explicitly (with slight change of normalization) in terms of wellknown functions.The five generators are of the following weights and indices and are given by (2.28) Here η = η(τ ), ϑ k (z, τ ) and ℘(z) are the Dedekind eta function, the Jacobi theta functions and the Weierstrass elliptic function respectively (see Appendix A for our convention).In this paper we often abbreviate ϑ k (0, τ ) as ϑ k .
2 By abuse of notation we let m denote 8-vector for E 8 and 4-vector for D 4 and F 4 .
F 4 The generators of the ring of W (F 4 )-invariant Jacobi forms were constructed in [22,28].In particular, fully explicit forms were given by Adler [28].Here we follow the convention of [33].The five generators are of the following weights and indices and are expressed in terms of the D 4 generators (2.27) as (2.31) We remind the reader that these W (F 4 )-invariant Jacobi forms are constructed for the rescaled root lattice Correspondingly, the rescaled F 4 roots are given by long roots: ± 2e i (i = 1, 2, 3, 4), ±e 1 ± e 2 ± e 3 ± e 4 , short roots: ± e i ± e j (1 ≤ i < j ≤ 4), (2.33) rather than (2.15).Note that this change of the basis does not alter how the whole W (F 4 ) acts on (m 1 , . . ., m 4 ), nor does it harm the discussion in the last subsection.
The study of W (E 8 )-invariant Jacobi forms was initiated in [34].Although W (E 8 )-invariant Jacobi forms do not form a polynomial algebra over the ring of modular forms, they are still expressed as polynomials of nine independent generators if one allows modular functions, rather than modular forms, as coefficients [26,29] (see [30,31] for systematic constructions of such polynomials).A natural choice of the nine generators is to take 3 There is a small difference between the definition of the D 4 generators in [32], which we use in this paper, and that in [33]: generators in the former and latter are related as (2.30) in [35].They are W (E 8 )-invariant holomorphic Jacobi forms of the following weights and indices and are given by Here, Θ E 8 is the theta function of the E 8 root lattice ϑ k (τ ) = ϑ k (0, τ ) and the functions e j (τ ), h 0 (τ ) are defined as (2.37) The normalizations of A n , B n are determined so that In this subsection we determine the Seiberg-Witten curve for the E-string theory with the partially fixed Wilson line configuration m = m h given in (2.14).As we saw in section (2.1), the theory in this case admits the W (F 4 ) automorphism.We call this Seiberg-Witten curve the F 4 curve hereafter.
We start with the manifestly W (E 8 )-invariant Seiberg-Witten curve for general m constructed in [8].It takes the form where In [35], a i , b j were expressed in terms of A k , B l given in (2.35).For instance, (2.40) given in (2.31).The results for A i , B i with i = 1, 2, 3 are as follows: symmetry.This means that the elliptic fibration described by the Seiberg-Witten curve must have a singular fiber of Kodaira type I * 0 .In other words, the F 4 curve must take the form (2.42) Comparing the x 1 -part of (2.42) with that of (2.39), one obtains the relations (2.43) By using (2.40)-(2.41) the third equation is rewritten as (2.44) From this one obtains In principle, (2.44) is a cubic equation in u 0 and there are two other solutions; but one can check that they do not satisfy the fourth equation in (2.43) and are therefore ruled out.
Given the explicit form (2.45) of u 0 , the data (2.41) of A i , B i for i ≤ 3 suffice to determine the entire form of the F 4 curve.It turns out that the final result can be expressed in the following surprisingly simple form:

49)
Therefore m h and m d specify the same physical system under the identification In [11] it was clarified how the Seiberg-Witten curve at the value m d is related to the original curve of Seiberg and Witten for the 4d N f = 4 theory [4] (the curve will also be given in (4.43)-(4.44)).Here we present a different point of view.It was already pointed out in [11] that the fundamental W (E 8 )-actions (2.9), ( (2.13), which generate the full W (E 8 ), are expressed for m = m d as • µ i ↔ −µ i for any i, µ j for all i = 1, . . ., 4. (2.51) As we learned in section 2.1, these are the very actions that generate W (F 4 ) acting on (µ 1 , µ 2 , µ 3 , µ 4 )!In other words, the system exhibits two different W (F 4 ) automorphisms, one acts on m j and the other acts on µ j .Consequently, the F 4 curve (2.46) written in terms of φ k = φ k (τ, m j ) can also be expressed in terms of φk = φ k (τ, µ j ). (2.52) The result is as follows:  to rewrite φk = φ k (τ, µ j ) in terms of the W (D 4 )-invariant Jacobi forms ϕ l (τ, µ j ), ψ 4 (τ, µ j ).The relation between the D(d 4 )-invariant subring and the ring J F 4 * , * of W (F 4 )-invariant Jacobi forms has been discussed in [33].Our result is a good exemplification of this relation.
The fact that two different W (F 4 ) automorphisms arise is due to the isomorphism between the root lattice L D 4 and its dual lattice L * D 4 .Specifically, they are given by (2.56) These are in fact identical with the rescaled and unrescaled F 4 root lattices respectively (see (2.32)).The isomorphism between these two lattices is encoded as the identity between the lattice theta functions: (2.57) From this it is clear that the two lattices differ only by their scale; the minimal norms of L D 4 and L * D 4 are 2 and 1 respectively.The above relation is also expressed as where

5d limit
In the limit of T 2 shrinking to S 1 , one obtains a 5d theory on R 4 × S 1 .The corresponding Seiberg-Witten curve is obtained by merely taking the limit of q → 0. In this limit, the fundamental W (F 4 )-invariant Jacobi forms (including E 4 , E 6 ) become (2.59) Here, v j (j = 1, . . ., 4) denote the Weyl orbit characters associated with the fundamental weights Λ F 4 j of F 4 (see Figure 1 for our labeling of the fundamental weights) (2.61) Then v j are expressed in terms of w k as (2.62) Furthermore, w j are explicitly written as w 1 = even e πi(±m 1 ±m 2 ±m 3 ±m 4 ) , w 2 = odd e πi(±m 1 ±m 2 ±m 3 ±m 4 ) , where the first (second) sum is taken for all terms with even (odd) number of '+' in the exponent.Substituting these into (2.46), one obtains the 5d F 4 curve.Of course, the same curve is obtained by evaluating the 5d E 8 curve given in [7,8] at the value (2.14).
In the same manner, the 5d F 4 curve expressed in terms of µ j can be obtained either from (2.53)-(2.54)or from the 5d E 8 curve with (2.48).As is clear from this construction, the 5d F 4 curve merely describes certain restricted cases of the 5d E 8 theory, namely the 5d N = 1 SU(2) gauge theory on R 4 × S 1 with N f = 7 flavors.

4d limit
In the limit of T 2 shrinking to zero size, one obtains a 4d theory on R 4 .It is well known that the Seiberg-Witten curve for a 4d theory is expressed in terms of Casimir invariants.For a Lie algebra g, Casimir invariants form a polynomial algebra which is the center of the universal enveloping algebra of g.For F 4 the polynomial algebra is generated by four generators p 2 , p 6 , p 8 , p 12 , where each p k is a homogeneous polynomial in m 1 , m 2 , m 3 , m 4 of degree k.The choice of p k is not unique.We define where the product is taken over all long roots α of F 4 : (2.65) In [8] it was clarified how to obtain the 4d E 8 curve [10] from the 5d E 8 curve.
Following the method of [8], let us start with the F 4 curve (2.46) with the 5d coefficients (2.59).We reinstate the circumference L of the S 1 in the fifth dimension by rescaling the variables as (2.66) Expanding the curve in L, we see that all the terms up to the order of L 29 cancel as in [8].At the order of L 30 we obtain (2.67) We verified that this curve can also be directly obtained from the original 4d E 8 curve of Minahan and Nemeschansky [10] by setting the mass parameters as in (2.14) and rescaling the variables as Similarly, we can take the 4d limit of the curve (2.54) expressed in terms of µ.
However, the result is simply obtained from (2.67) by the change of variables (2.50).

Equivalence with thermodynamic limit of Nekrasov-type formula
In this section we take the thermodynamic limit of the Nekrasov-type formula [11] for the E-string theory with four Wilson line parameters, generalizing the result of [20]. 4  In particular, we derive the Seiberg-Witten curve of the following simple form where φ −2,1 , φ 0,1 are the classic weak Jacobi forms of Eichler and Zagier [21] Moreover, we show that this curve is equivalent to the F 4 curve studied in the last section. 4An attempt in this direction was made earlier in [36].

Nekrasov-type formula for prepotential of E-string theory
Let us first briefly recall the Nekrasov-type formula proposed in [11,37].The formula is obtained as a special case of the 6d generalization [17] of the original Nekrasov partition function for the U(N) gauge theory with 2N fundamental matters [18].
Let R = (R 1 , . . ., R N ) denote an N-tuple of partitions.Each partition R k is a nonincreasing sequence of nonnegative integers Let |R k | denote the size of R k , i.e. the number of boxes in the Young diagram of R k : Similarly, the size of R is denoted by We also introduce the notation which represents the relative hook-length of a box at (i, j) between the Young diagrams of R k and R l .
In [17], Hollowood, Iqbal and Vafa proposed the 6d (or elliptic) generalization of the original Nekrasov partition function for the U(N) gauge theory with 2N fundamental matters [18].It takes the simple form: Here the sum is taken over all possible partitions R, including the empty partition.
The indices (i, j) run over the coordinates of all boxes in the Young diagram of R k .
Note that the consistency condition 2 is required, where the equality is understood modulo periods of the torus C/(Z+τ Z).
Let ω k (k = 1, 2, 3, 4) be half periods of the torus: The Nekrasov-type formula [11,37] for the E-string theory with Wilson line parameters (2.48) is obtained by simply setting The Seiberg-Witten prepotential for the E-string theory is then given by An important remark is that the above Nekrasov-type formula is limited to the genus zero prepotential and does not work for higher genus parts [37].This is in contrast to the ordinary Nekrasov partition functions.However, this does not cause any problem in deriving the Seiberg-Witten curve since only the genus zero part concerns.

Thermodynamic limit of Nekrasov-type formula
Nekrasov and Okounkov proved that the Seiberg-Witten curve is obtained from the Nekrasov partition function by taking the thermodynamic limit [19].In [20] the thermodynamic limit of the above Nekrasov-type formula was studied for the massless case µ j = 0 and also for some other special cases.Since the derivation is rather technical and all the fine points were explained in full detail in [20], here we only describe how the derivation in [20] is modified for the case of general values of µ j .
In the rank-one Seiberg-Witten theory, the expectation value ϕ of the Higgs field is expressed through the relation [3,4] Here α is a fundamental one-cycle of the torus and we have introduced the rescaled variables X = u −2 x, Y = u −3 y.On the other hand, by taking the thermodynamic limit of the Nekrasov-type formula, the same quantity is expressed as [20] Here function H(z) for the case of Wilson line configuration (2.48) is given by [20] H ω j and m n are given in (3.9) and (3.10) respectively.By using the identities H(z) is rewritten as .
(3.17) (3.13) is then written as This is identified with (3.12) by the map Substituting the identity and eliminating ℘(z), one obtains Finally using the well-known relation one obtains (3.1).
We close this subsection with a few remarks in connection with some related works.First, in the limit of µ 4 → 0, the quartic curve (3.1) reduces to the cubic one Essentially the same curve (expressed in terms of ℘(µ j ) as in (3.21)) was considered in [38].Our curve (3.1) can be viewed as its generalization to the case of four Wilson line parameters.Second, instead of using the variables X and Y , one can write (3.13) as and regard the second equation as the Seiberg-Witten curve expressed in the variables z and w.By using the identity .25)this curve can be rewritten as The curve of this form has been discussed in [12,13].

Equivalence
We are now in a position to show that the curve (3.1) is equivalent to the F 4 curve studied in the last section.To do this, we first rescale the variables by the replacement5 X → u −2 X, Y → u −3 Y and write the curve (3.1) as This is a quartic curve of the form It is well known that any quartic curve of this form with nonzero c 0 can be transformed into the Weierstrass form y 2 = 4x 3 − f x − g: By the change of variables which is converted to the Weierstrass form by the map By further rescaling the variables as y, the quartic curve (3.28) finally becomes  symmetry (acting on four 0's in (2.14)).However, this is not the case: 6 In the limit of m i → 0, the 6d theory specified by (2.14) recovers the global E 8 symmetry and correspondingly the elliptic fibration described by the Seiberg-Witten curve exhibits a singular fiber of Kodaira type II * , which corresponds to an E 8 singularity.On the other hand, the elliptic fibration of the N f = 4 theory in the massless limit exhibits two singular fibers of Kodaira type I * 0 , which correspond to two D 4 singularities.In the massless case, a natural 6d generalization of the N f = 4 theory was studied in detail in [39].The corresponding Wilson line parameters are given by Note that by using the W (E 8 ) action (2.49) this is mapped to The Wilson line setup for any natural 6d generalization of the N f = 4 theory should reduce to a configuration equivalent to these in the limit of m i → 0.
A natural candidate is This is not equivalent to m * , meaning that the theory is no longer W (F 4 )-invariant.
However, if one combines s e 4 with the modular transformation τ → τ +1, one obtains By the W (E 8 ) action s e 4 −e 5 s e 4 +e 5 , this is mapped to By the periodicity condition (2.4), this is equivalent to m * .
Next, consider This is again not equivalent to m * .However, further applying the modular trans- By the S-transformation (which is a part of the modular invariance (2.5)) of the original E-string theory this is equivalent to Again, by the periodicity condition (2.4) this is equivalent to m * .Note that τ is also mapped to itself by the whole sequence of transformations.
Thus we have shown that the system specified by m = m * is invariant under not only W (D 4 ), but also As we will see in the next subsection, W (D 4 ) and the above transformations generate an automorphism group, which was originally found as a symmetry of the N f = 4 theory [4].
The configuration m = m * in (4.3) is not the only one that exhibits the D 4 triality, but there are several equivalent expressions.For instance, the configuration is equivalent to m * .This can be seen immediately by rewriting it as The equivalence follows from the periodicity condition (2.4).

D 4 triality invariant Jacobi forms
We have seen that the E-string theory with the configuration (4.3) exhibits a set of automorphisms which involves both W (D 4 ) and the D 4 triality combined with modular transformations.The automorphism of this kind was first found by Seiberg and Witten in the study of 4d N f = 4 theory [4].The whole automorphism forms a group, which we call the modular triality group.In this subsection we clarify the structure of this group and introduce Jacobi forms which respect this automorphism.
Let us first define the modular triality group.Let (g, w) denote an element of the We call the subgroup G generated by the following elements the modular triality group: (iii) T = (g T , w T ) with (iv) S = (g S , w S ) with Here the principal congruence subgroup of level N is defined as As clearly seen, T and S represent the transformations in (4.12) under the identification z i = m i .It is also clear that G satisfies the following inclusions: G can also be described more abstractly as follows: Let f 1 , f 2 be group homomorphisms given by the quotient maps We identify the two S 3 's in such a way that the correspondence of the generators is given by g T ↔ w T , g S ↔ w S .Then we can define G as the fiber product of SL(2, Z) and W (F 4 ) over the common S 3 : (ii) Quasi-periodicity: By the inclusions (4.20) we have This in particular means that W (F 4 )-invariant Jacobi forms are automatically triality invariant Jacobi forms.
Moreover, there exist triality invariant Jacobi forms that are not W (F 4 )-invariant.
For instance, one can verify that the following combinations of W (D 4 )-invariant Jacobi forms ϕ 4 , ψ 4 give such triality invariant Jacobi forms: 2 )ϕ 4 − 3(ϑ 8 3 − ϑ 8 4 )ψ 4 . (4.29) Investigating the structure of the ring J G * , * of triality invariant Jacobi forms is an interesting problem.It turns out that J G * , * is not a polynomial algebra, as in the case of J E 8 * , * .The details will be studied separately in a forthcoming paper [40].Here we only mention that any triality invariant Jacobi form is expressed in terms of the following five fundamental Jacobi forms where φ k are the W (F 4 )-invariant Jacobi forms given in (2.31).To be more exact, one can prove that any triality invariant Jacobi form of index m is expressed uniquely as [40] P where P denotes a polynomial of the given variables and [m/2] is the integer part of m/2.Indeed, W (F 4 )-invariant Jacobi forms φ 8 , φ 12 are expressed as

D 4 triality invariant curve
In this subsection we determine the Seiberg-Witten curve for the E-string theory at the value (4.3), which we call the triality invariant curve.
First, let us present the reduction of W (E 8 )-invariant Jacobi forms to the triality invariant Jacobi forms.We set where m * is given in (4.3).Then c j A j (τ, m * ), c j B j (τ, m * ) are expressed in terms of Substituting these data into the E 8 curve, one obtains the triality invariant curve.
Due to the equivalence of (4.3) and (4.13), we know that the triality invariant curve can be viewed as the F 4 curve with slightly altered Wilson line parameters.
This means that the triality invariant curve also takes the factorized form (2.42).
Therefore, as in the case of the F 4 curve, the data (4.34) of A j , B j (j ≤ 3) suffice to determine the triality invariant curve.By the change of variables we finally obtain where

5d limit
Let us now consider the limit of q → 0, namely the limit of T 2 shrinking to S 1 .In this limit, the triality invariant Jacobi forms become where w i are the Weyl orbit characters of D 4 defined in (2.61).Substituting these into (4.36)-(4.37), the curve becomes One sees that the discriminant of the curve identically vanishes This means that the Seiberg-Witten curve degenerates everywhere in the Coulomb branch moduli space parametrized by u.Thus the theory becomes trivial in this limit.
Note, however, that if one introduces the Omega background [18], the supersymmetric index exhibits a non-trivial structure.See [39, Section 4.1] for the discussion in the case of m i = 0.It would be interesting to see how the index is deformed for general m i .

4d limit
Recall that the Seiberg-Witten curve for the N f = 4 theory is given by [4] where e j are given in (2.37) and A = (e 1 − e 2 )(e 2 − e 3 )(e 3 − e 1 ), The Weierstrass form of this curve is obtained immediately by performing a translation of x in such a way that the quadratic term in x vanishes.More specifically, this is achieved by replacing x with x + (E 4 /144) j m 2 j .Let us now show that our 6d triality invariant curve (4.36)-(4.37)reduces to this Weierstrass form.The 4d limit corresponds to the small mass limit.Therefore we expand quantities in the 6d curve in m j .The 6d curve depends on m j through j=1 ϑ k (m j , τ ) and 4 j=1 ℘(m j ).By using the formulas in Appendix A, we see that they are expanded as where m j = 2πm j .Using these expansions together with the expressions (2.27) of the fundamental W (D 4 )-invariant Jacobi forms, we obtain the expansion of the 6d curve in m j .Let us then rescale the variables as and expand the equation of the curve (4.36) in L. We see that all the terms of the order of L n (n < 6) cancel out.The first non-zero part appears at the order of L 6 , which is precisely the Weierstrass form of the N f = 4 curve (4.43)!

Conclusions and outlook
In Jacobi forms and thus manifestly symmetric.The last two expressions are not manifestly symmetric, but are very concise as one can see.We have derived these last two expressions from the Nekrasov-type formula proposed in [11].The derivation, together with the equivalence mentioned above, serves as a proof of the Nekrasovtype formula in the presence of four general mass parameters.This generalizes the earlier proof [20] performed in certain special cases.[32,35,[41][42][43] and the elliptic genera [44,45].In particular, in [32,35,[43][44][45] the results are already expressed in terms of the W (E 8 )-invariant Jacobi forms A i , B j given in (2.theory.This should be given as a suitable generalization of the correspondence studied previously in the massless case [39].From a broader perspective, it would also be interesting to consider the D 4 triality invariant deformation of the elliptic genera of the E 8 × E 8 heterotic string theory [46,47].It would be worth understanding how our results are related to recent studies such as the twisted elliptic genera of [48] and the bimodular forms of [49].
Another intriguing subject is the quantum Seiberg-Witten curve.In this paper we have treated the Seiberg-Witten curve mainly in a manifestly symmetric form.
However, when the symmetry is large, a curve of this form inevitably has intricate dependence on the Wilson line parameters and is not suitable for the quantization.
Quantum Seiberg-Witten curves for the E-string theory have been studied in [13,50,51], where different forms of the classical curve, more akin to (3.26), have been employed.Regarding this, it is worth noting that the D 4 triality invariant curve as well as the rescaling (4.35).This form is neither symmetric nor elliptic, but is probably more suitable for quantizing the curve along the lines of [13].
In this paper we have introduced the notion of D 4 triality invariant Jacobi forms.
As far as we know, Jacobi forms of this sort, i.e. those for which the underlying modular group is non-trivially combined with the other automorphism group have never been considered in the literature before.As mentioned in section 4.2, the ring of triality invariant Jacobi forms is not a polynomial algebra, as in the case of W (E 8 )invariant Jacobi forms [29][30][31].The structure of the ring deserves to be investigated further.We will report some results on this problem in a forthcoming paper [40].
One can show (see e. (A.10)

4 )-invariant model 2 . 1 .
Emergence of W (F 4 ) automorphism The E-string theory is a 6d SCFT and has a global E 8 symmetry.When the theory is partially compactified on T 2 , one can introduce Wilson lines for the E 8 symmetry.The Wilson lines are specified by eight Wilson line parameters m = (m 1 , . . ., m 8 ),

12 .( 2
.54)We observe that the above fi , gj are identical with the generators of the D(d 4 )invariant subring[32] of W (D 4 )-invariant Jacobi forms, where D(d 4 ) denotes the symmetry group of the D 4 Dynkin diagram.The precise relations are f1 in[32, eq.(B.33)].This is easily checked by using(2.31) 28).Recall that W (D 4 )-invariant Jacobi forms of index one are certain linear combinations of T k (τ, m) (with some coefficient functions in τ ), while T k (τ /2, µ) constitute W (D 4 )-invariant Jacobi forms of index two.The same holds for F 4 .This explains why the coefficients in the F 4 curve (2.46) are Jacobi forms of indices 0, 1, 2, 3 while the coefficients (2.54) of the curve (2.53) are Jacobi forms of indices 0, 2, 4, 6.

Figure 1 :
Figure 1: Dynkin diagrams for F 4 (left) and D 4 (right): numbers attached to nodes denote the labels of the corresponding fundamental weights.

4. D 4 triality invariant model 4 . 1 .
Emergence of D 4 triality in E-string theory One of the purposes of this paper is to establish a natural flow from 6d E-string theory to the 4d N = 2 supersymmetric SU(2) gauge theory with four massive flavors (N f = 4 theory).It is sometimes believed that the E-string theory with half of the Wilson line parameters turned on as in (2.14) gives a natural 6d generalization of the N f = 4 theory, since the system exhibits a W (D 4 ) automorphism acting on m i (which is actually promoted to W (F 4 ) as we have seen) and keeps a global D 4

. 3 )
By the discussion in section 2.1, it is clear that this configuration exhibits W (D 4 ) automorphism acting on m 1 , . . ., m 4 .In what follows we will show that this configuration in addition exhibits extra automorphisms, namely the D 4 triality combined with modular transformations.First, as in (2.16), consider the action of s e 4 on m * :

(4. 22 )
Let us next define G-invariant Jacobi forms, which we call the triality invariant Jacobi forms.Let k, m be integers (m ≥ 0) and G the modular triality group defined above.We call a holomorphic function ϕ k,m : H × C 4 → C a triality invariant weak Jacobi form of weight k and index m if it satisfies the following properties: (i) Weyl invariance and modular transformation law: ϕ k,m aτ + b cτ + d , w(z) cτ + d = (cτ + d) k exp mπi c cτ + d z 2 ϕ k,m (τ, z), (4.23)

. 24 ) 4 c
(iii) ϕ k,m (τ, z) admits a Fourier expansion of the formϕ k,m (τ, z) = ∞ n=0 w∈L * D (n, w)e 2πiw•z q n/2 .(4.25)It follows immediately that triality invariant Jacobi forms of index 0 space of triality invariant Jacobi forms of weight k and index m.As in the case of W (R)-invariant Jacobi forms, the space of all triality invariant Jacobi forms algebra over the ring of modular forms.Let JD 4 Γ(2) * , *denote the bigraded algebra of W (D 4 )-invariant Jacobi forms over the ring of modular forms for Γ(2).
this paper we have studied two concrete examples of the E-string theory on R 4 ×T 2 with Wilson lines, one exhibits a W (F 4 ) symmetry and the other exhibits a D 4 triality symmetry.We have explicitly shown the emergence of these symmetries by means of the automorphisms (2.3)-(2.5)known for the case of the most general Wilson lines.In the first example, we have constructed four different explicit forms of the F 4 curve, namely (2.46), (2.53)-(2.54),(3.1) and (3.26).We have shown that they are all equivalent.The first two expressions are written in terms of W (F 4 )-invariant The second example is the D 4 triality invariant model specified by the twisted Wilson line configuration (4.3), (4.13) or(4.15).Concerning this model, we have clarified the discrete automorphism of the 4d N f = 4 theory and introduced the notion of triality invariant Jacobi forms.We have then expressed the Seiberg-Witten curve (4.36)-(4.37) in terms of them.We have shown explicitly that this curve reduces precisely to that of the 4d N f = 4 theory in the limit of T 2 shrinking to zero size.There are several directions for further studies.It is definitely interesting to study the supersymmetric indices in these two examples.For the E-string theory with general Wilson line parameters m, there are several systematic studies of the topological string partition functions 35).As explained in the main text, given the Seiberg-Witten curves (2.46) and (4.36) one immediately obtains the data of the forms (2.41) and (4.34) for all A i , B j .The supersymmetric indices are then obtained by simply substituting these data into the above known results expressed in terms of A i , B j .In particular, it is interesting to see how the topological string partition function of the D 4 triality invariant model is related to the Nekrasov partition function of the N f = 4

c
g.[37, Appendix B]) that the Weierstrass elliptic function and the Jacobi theta functions are expanded as℘(z) = 4π 2 e k (k = 1, 2, 3) are defined in (2.37) and the coefficients c n , d n (x) are determined by the recurrence relations k c n−k−1(n ≥ 3),d 1 (x) = x, d 2 (x) = 6x 2 − E 4 24 , d n (x) = d 2 (x)∂ x d n−1 (x) + 4x 3 − 4As long as this constraint is imposed, the parameter space is effectively H × C 4 .What is the nontrivial Weyl group automorphism acting on this space?From the above discussion it is obvious that there is a Weyl group of D 4 acting on m 1 , . . ., m 4 , −e 8 s e 7 +e 8 s e 5 −e 6 s e 5 +e 6 s s )(m h ) = m ′ h .
.41) Substituting relations like these into the curve (2.39) with (2.40), one obtains the F 4 curve.In fact, there is a better way to obtain the F 4 curve, which does not require the full data of the form (2.41) for A 4 , B 4 , A 5 , B 6 : It is clear that the Wilson line configuration (2.14)only partially breaks the global E 8 symmetry and leaves a D 4 [31]j at the value(2.14)interms of φ k .Note that the expressions of A i , B j in terms of a k , b l are explicitly given in[31].Using this one can easily obtain the data of the form (2.41) for A 4 , B 4 , A 5 , B 6 as well.2.4.Another W (F 4 ) automorphismLet us next consider the configurationm = m d := (µ 1 , µ 2 , µ 3 , µ 4 , µ 1 , µ 2 ,µ 3 , µ 4 ).+e 4 s s s e 1 −e 2 s e 1 +e 2 s e 5 −e 6 s e 5 +e 6 s s s e 3 −e 8 s e 1 −e 6 s e 4 −e 6 s e 4 +e 6 .9) Next, applying the W (E 8 ) action s e 5 +e 8 s e 2 −e 3 s e 1 −e 4 s s s e 3 −e 4 s e 3 +e 4 s e 1 −e 2 s e 1 +e 2 s s .37)As in the F 4 case, by comparing the above curve with the original Seiberg-Witten curve (2.39), one immediately obtains the expressions of a i , b j at the value (4.3) in terms of φ k , χ l .More specifically, using the change of variables (4.35) inversely as [31]B j in terms of a k , b l are explicitly given in[31].