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Seiberg-Witten prepotential for E-string theory and global symmetries

We obtain Nekrasov-type expressions for the Seiberg-Witten prepotential for the six-dimensional (1,0) supersymmetric E-string theory compactified on T^2 with nontrivial Wilson lines. We consider compactification with four general Wilson line parameters, which partially break the E_8 global symmetry. In particular, we investigate in detail the cases where the Lie algebra of the unbroken global symmetry is E_n + A_{8-n} with n=8,7,6,5 or D_8. All our Nekrasov-type expressions can be viewed as special cases of the elliptic analogue of the Nekrasov partition function for the SU(N) gauge theory with N_f=2N flavors. We also present a new expression for the Seiberg-Witten curve for the E-string theory with four Wilson line parameters, clarifying the connection between the E-string theory and the SU(2) Seiberg-Witten theory with N_f=4 flavors.


Introduction
The E-string theory is one of the simplest interacting quantum field theories with (1,0) supersymmetry in six dimensions [1][2][3][4][5]. It is obtained as the low energy theory of the heterotic string theory on K3 when an E 8 instanton shrinks to zero size [1,2].
The theory is decoupled from gravity. It is probably a conventional local quantum field theory, though it does not have a Lagrangian description. Another unusual feature is that fundamental excitations are strings, called E-strings, rather than particles. The moduli space of vacua consists of a Coulomb branch with one tensor multiplet and a Higgs branch with 29 hypermultiplets. There are no vector multiplets and E 8 appears as a global symmetry group.
The E-string theory shows extremely rich properties when toroidally compactified down to lower dimensions [3][4][5][6][7][8][9][10][11][12]. In the compactified theories one can break the When the theory is toroidally compactified down to four dimensions, the low energy dynamics in the Coulomb branch is described by Seiberg-Witten theory [13,14]. The Seiberg-Witten curve was constructed with the most general Wilson line parameters [4,15]. Recently, it was found that the Seiberg-Witten prepotential admits a Nekrasov-type expression [16]. The expression is for the case with no Wilson line parameters. In this paper we present a Nekrasov-type expression with four general Wilson line parameters. It is verified up to a sufficiently high order (involving Young diagrams with 10 boxes) that the prepotential given by this expression is in perfect agreement with that computed from the Seiberg-Witten curve.
The Seiberg-Witten curve with full eight Wilson line parameters takes a rather complicated form. In the case of four Wilson line parameters, however, the curve reduces to a much simpler expression. Interestingly, it is expressed in terms of the curve for the SU(2) Seiberg-Witten theory with N f = 4 flavors. It has been known that the low energy theory of the E-string theory on T 2 can flow to that of the four-dimensional SU(2) N f = 4 theory. In particular, it was argued that the SL(2, Z) duality of the latter theory is derived from the SL(2, Z) action on the T 2 [4]. Our expression clarifies how it occurs, in particular how the SO(8) triality emerges from the E-string theory.
By adjusting four Wilson line parameters to special values, one can realize the cases where the Lie algebra of the unbroken global symmetry is E n ⊕ A 8−n with n = 8, 7, 6, 5 or D 8 . We present explicit forms of the Seiberg-Witten curves and the Nekrasov-type expressions for these specific cases. In each of these cases the Seiberg-Witten prepotential counts multiplicities of BPS E-strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta.
The multiplicities are equivalent to Gromov-Witten invariants associated with the E n del Pezzo surface or P 1 × P 1 embedded in a Calabi-Yau threefold [3,9,17,18].
Our Nekrasov-type expressions provide us with the generating functions of these invariants in very simple, closed forms. In particular, these expressions respect the modular properties of the partition functions of wound BPS E-strings [5,11,12].
As in the case with no Wilson line parameters [16], our Nekrasov-type expression can be viewed as a special case of the elliptic analogue of the Nekrasov partition function for the SU(4) gauge theory with N f = 8 flavors [19,20]. Moreover, for particular values of the Wilson line parameters our expression can be embedded in the elliptic analogue of the Nekrasov partition functions for the SU(N) N f = 2N theories with N = 3, 2. In fact, for all the cases with E n ⊕ A 8−n with n = 8, 7, 6, 5 and D 8 the Nekrasov-type expressions can be viewed as special cases of the elliptic Nekrasov partition function for the SU(3) N f = 6 theory. Furthermore, expressions for the cases with E 7 ⊕ A 1 , E 5 ⊕ A 3 and D 8 can also be viewed as special cases of the elliptic Nekrasov partition function for the SU(2) N f = 4 theory.
The organization of this paper is as follows. In section 2, we present our new expression for the Seiberg-Witten curve with four general Wilson line parameters and discuss its properties. In section 3, we present the Nekrasov-type expression with four Wilson line parameters. We then focus on some particular cases in which the general formula reduces to a sum over fewer partitions. In section 4, we investigate in detail the cases with global symmetries E n ⊕ A 8−n with n = 8, 7, 6, 5 and D 8 .
Conventions of special functions are summarized in Appendix A.

Seiberg-Witten curve with four Wilson line parameters
In this section we present a new expression for the Seiberg-Witten curve for the E-string theory compactified on T 2 with four Wilson line parameters. We clarify how it is related to the Seiberg-Witten curve for the four-dimensional SU(2) gauge theory with N f = 4 flavors.
The Seiberg-Witten curve for the E-string theory compactified on T 2 with the most general Wilson line parameters was constructed in [4,15]. (An improved expression in terms of E 8 -invariant Jacobi forms is available in [21].) It takes the following form The coefficients a j , b j depend on nine complex parameters, τ and µ = (µ 1 , . . . , µ 8 ).
τ is the complex modulus of the T 2 and µ are the Wilson line parameters which specify the background E 8 gauge field along the T 2 [6]. In this paper we restrict Here Γ w denotes the weight lattice of D 4 , The group generated by the above transformations is in fact the affine automorphism group of Γ w . There is also an SL(2, Z) automorphism generated by the following In principle, the explicit form of the curve with four Wilson line parameters is obtained by simply substituting (2.3) or (2.4) into the expression in [21]. However, the expression constructed in this way is rather complicated for practical purposes.
In what follows we will express the same Seiberg-Witten curve in a more convenient form by means of the curve for the SU(2) gauge theory with N f = 4 fundamental hypermultiplets. Recall that the Seiberg-Witten curve for the SU(2) N f = 4 theory is given by [14] (2.16) Here E 2k := E 2k (τ ) and η := η(τ ) are the Eisenstein functions and the Dedekind eta function respectively. The curve (2.14)-(2.15) can then be written in the form Hereã j ,b j are some functions in τ , lM i and u 0 . l is a parameter that gives an inverse mass scale. It can be absorbed in the definitions ofũ,x,ỹ and M i , but let us keep it for later use. Next, we identify the parameters as and instead of (2.16). The curve (2.14) is then obtained by taking the limit l → 0. In fact, under this limit the identification (2.16) coincides with (2.20).
In the above reduction the bare gauge coupling τ of the SU(2) N f = 4 theory is identified with the complex modulus τ of T 2 on which the E-string theory is compactified. This means that the SL(2, Z) duality of the N f = 4 theory is identified with the SL(2, Z) action of the T 2 [4]. Recall that the SL(2, Z) duality of the SU(2) N f = 4 theory is mixed with SO(8) triality [14]. That is, the spectrum of the theory is not invariant under the transformations τ → τ + 1 and τ → −1/τ , but is invariant under the combinations overall factor which can be absorbed into l). This peculiar identification was first found in [22], where a different connection between the two theories was considered.
The identification (2.19) admits the following interpretation in connection with the automorphism group of the curve. We saw that the automorphism group of the present Seiberg-Witten curve is governed by the weight lattice of D 4 denoted by Γ w .
This lattice can be viewed as the union of four sublattices In terms of these sublattices, (2.19) can be expressed as In the rest of this section let us sketch out how to calculate the prepotential from the Seiberg-Witten curve. Our Seiberg-Witten curve given by (2.14)-(2.19) is expressed in the Weierstrass form. An elliptic curve in the Weierstrass form can be parametrized as Hereτ is the complex structure modulus and ω (multiplied by 2π) is one of the fundamental periods of the elliptic curve. By comparing this expression with the explicit form of the Seiberg-Witten curve, one can calculate ω(u, τ, m),τ (u, τ, m) as series expansions in 1/u. They are related to the scalar vev ϕ and the prepotential These relations parametrically determine the function F 0 (ϕ, τ, m). The integration constants are determined accordingly. The reader is referred for the details of these calculations to [21].

Nekrasov-type expression with four Wilson line parameters
In this section we present an explicit expression for the Seiberg-Witten prepotential for the E-string theory with four Wilson lines and discuss its properties.

General expression
Let R (N ) = (R 1 , . . . , R N ) denote an N-tuple of partitions. Each partition R k is a nonincreasing sequence of nonnegative integers Here the number of nonzero ν k,i is denoted by ℓ(R k ). R k is represented by a Young diagram. We 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 (N ) is denoted by We let R ∨ k = {ν ∨ k,1 ≥ ν ∨ k,2 ≥ · · · } denote the conjugate partition of R k . 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 our expression we consider a sum over four partitions. For our present purpose, it is convenient to express these partitions as The prepotential is then given by and Q := e 2πiϕ+πiτ . (3.8) Here the sum is taken over all possible partitions R (4) (including the empty partition). If we set m = 0, the expression reduces to the one studied in [16].
We find that the above F 0 coincides with the prepotential computed from the Seiberg-Witten curve in the last section. We verified it by computing the series expansion of F 0 in Q independently by each of the methods and comparing the coefficients up to order Q 10 . In doing this, the following identities turn out to be useful. Using these identities one can rewrite both the Seiberg-Witten curve and the Nekrasov-type expression in such a way that all the dependence on m cd is expressed through ϑ 00 (2m cd , 2τ ) and ϑ 10 (2m cd , 2τ ). The comparison can then be made in the same way as in the case of m = 0 by using the Taylor expansions of the theta functions [16].
As in [16], one can express Z as a special case of the elliptic analogue of the Nekrasov partition function for the SU(N) gauge theory with N f = 2N fundamental hypermultiplets [19,20] Z SU(N ) In terms of this function, (3.7) can be expressed as

(3.13)
In the following sections we will see that this type of notation provides us with an efficient, universal way of expressing various Nekrasov-type formulas for specific cases. The prepotential F 0 for the E-string theory represents the genus zero topological string amplitude for a family of local 1 2 K3 [5]. However, as was observed in the case of m = 0 [16], higher order parts of the expansion ln Z = 1 2 F 0 −2 + · · · do not give higher genus amplitudes [12,21,23]. The disagreement can be clearly seen as the difference of modular anomalies. With the help of (3.11) one immediately sees that Z exhibits the same modular anomaly as in the case of m = 0. This deviates from the modular anomaly of the all-genus topological string partition function for the local 1 2 K3 starting at genus one.

Reductions to sums over fewer partitions
For particular values of the Wilson line parameters the above Nekrasov-type sum over partitions reduces to that over fewer partitions. This structure has already been found in the case of no Wilson line parameters [16].
Furthermore, recall that for any function f (x) the following identity holds: In this setting, the expression for Z reduces to the form where R (2) = (R 1 , R 2 ). In terms of the elliptic Nekrasov partition function (3.12), Z with (3.19) can be expressed as As we will see in the next section, the cases of global symmetries and D 8 are realized as special cases of this setting.
Furthermore, if we set Z vanishes. This is consistent with the fact that the corresponding unbroken global symmetry is D 4 ⊕ D 4 and the Seiberg-Witten curve in this case describes a constant elliptic fibration over the moduli space [22].

Two-parameter families
In this section we consider the cases in which the Lie algebra of the unbroken global symmetry is E 9−N ⊕ A N −1 with N = 1, 2, 3, 4 or D 8 . These cases are of particular interest because the prepotential in each case generates Gromov-Witten invariants associated with the E 9−N del Pezzo surface or P 1 × P 1 embedded in a Calabi-Yau threefold. In particular, we consider two-parameter families of Calabi-Yau, whose prepotentials depend not only on Kähler modulus ϕ but also on another Kähler modulus τ . These two-parameter families have been studied by means of mirror symmetry [3,12]. precisely to the simple Lie algebras constituting the unbroken global symmetry [22].
We will also present explicit Nekrasov-type expressions for each of the cases.
The prepotential is obtained from Z through (3.6). Following [5] we introduce the winding number expansion of the prepotential by Z n for E 9−N ⊕ A N −1 with N = 1, 2, 3, 4 can be expressed in terms of E 2 (τ ) and modular forms of Γ 1 (N) = a b c d ∈ SL(2, Z) a ≡ d ≡ 1, c ≡ 0 mod N . We will introduce generators α N , β N of modular forms of Γ 1 (N) as well as a function λ N and present explicit forms of Z n for small n. In the D 8 case, Z n are expressed in terms of E 2 (τ ) and modular forms of Γ 1 (2).
In each of the above cases, the prepotential can be expressed as Integer N n,k represents the multiplicity of BPS E-strings wound around one of the circles of the toroidal compactification with winding number n and momentum k. Up to an overall normalization the values of N n,n turn out to be equal to the genus zero Gromov-Witten invariants associated with the E 9−N del Pezzo surface or P 1 × P 1 embedded in a Calabi-Yau threefold [3,9,17,18]. Our formulas also generate invariants N n,k with k = n, where k is the degree associated with the homology class of the elliptic fiber. It would be very interesting to see how our combinatorial expressions are related to the geometric computation of these invariants [18].

E 8
Let us first consider the case with an E 8 global symmetry. This is the case originally discussed in [16] and is realized by the trivial Wilson line parameters m = (0, 0, 0, 0) . (4. 3) The corresponding Seiberg-Witten curve is given by The Nekrasov-type expression can be written as Here the set of indices (a, b), (c, d) take values (1, 0), (0, 0), (0, 1) and we let the three partitions be denoted by R (3) = (R 10 , R 00 , R 01 ). The first three coefficients of the expansion (4.1) are where α 1 := E 4 , β 1 := E 6 (4.8) and These Z n agree with the original results [11]. Table 1 shows the values of N n,k for low n and k. These numbers were originally computed by using mirror symmetry [3].  (4.10) The Seiberg-Witten curve is given by The Nekrasov-type expression in this case takes a remarkably simple form (4.13) The first three coefficients of the expansion (4.1) are , (4.14) where and (4.16) As expected, these Z n are in agreement with the results obtained in [12]. Table 2 shows the values of N n,k for low n and k. The values of N n,k multiplied by two agree with the rational instanton numbers of the E 7 model in [12].
The E 6 ⊕ A 2 symmetry is realized by the following Wilson line parameters The Seiberg-Witten curve is given by where α 3 := ϑ 3 (0, 2τ )ϑ 3 (0, 6τ ) + ϑ 2 (0, 2τ )ϑ 2 (0, 6τ ),  Table 3: BPS multiplicities N n,k for the E 6 ⊕ A 2 case. Table 3 shows the values of N n,k for low n and k. The values of N n,k multiplied by three agree with the rational instanton numbers of the E 6 model in [12].

E 5 ⊕ A 3
The E 5 ⊕ A 3 symmetry is realized by the following Wilson line parameters The Seiberg-Witten curve and the Nekrasov-type expression in this case are given respectively by The first three coefficients of the expansion (4.1) are  and (4.30) Table 4 shows the values of N n,k for low n and k. The values of N n,k multiplied by four agree with the rational instanton numbers of the E 5 model in [12].

D 8
The (4.34) The first three coefficients of the expansion (4.1) are   where α 2 , β 2 are defined in (4.15) and (4.36) Table 5 shows the values of N n,k for low n and k. We observe that the values of N n,n are related to the genus zero Gromov-Witten invariants of the local P 1 × P 1 as N n,n = n 1 +n 2 =n N P 1 ×P 1 n 1 ,n 2 . (4.37) Note that the values of N n,k for the D 8 case are very similar to those for the E 5 ⊕ A 3 case. This has been explained by the similarity between the Picard-Fuchs operators for P 1 ×P 1 (which is equal to the quadric surface in P 3 ) and those for the E 5 del Pezzo surface [9]. From the point of view of the E-string theory this may be explained by the similarity between the Weyl orbits of D 8 and those of E 5 ⊕ A 3 ∼ = D 5 ⊕ D 3 .

Acknowledgments
The author would like to thank T. Eguchi