BPS index and 4d N=2 superconformal field theories

We study the BPS index for the four-dimensional rank-one N=2 superconformal field theories H_0, H_1, H_2, E_6, E_7, E_8. We consider compactifications of the E-string theory on T^2 in which these theories arise as low energy limits. Using this realization we clarify the general structure of the BPS index. The index is characterized by two exponents and a sequence of invariants. We determine the exponents and the first few invariants.

In the study of 4d N = 2 supersymmetric theories the BPS index is of crucial importance as it captures the details of the exact quantum spectrum. For U(n) gauge theories the BPS index can be expressed in a very concise, explicit form known as the Nekrasov partition function [13,14]. 1 It is natural to ask how the BPS index looks near the SCFT singularities on the moduli space of supersymmetric theory. The answer is not obvious, even in the case of a gauge theory whose Nekrasov partition function is explicitly known. This is because the Nekrasov partition function is given by the instanton expansion about the classical singularity of the moduli space, but the expansion is no longer valid at the SCFT singularities. Some sort of analytic continuation is required.
In this paper we focus on the 4d rank-one N = 2 SCFTs H n , E 8−n (n = 0, 1,2) and study the BPS index. (The index for the D 4 SCFT can be computed from the Nekrasov partition function by using the modular properties of the index [15].) These SCFTs are realized in many ways. For example, the H n SCFTs can be studied by means of the original realization in SU (2) super Yang-Mills [3]. Analysis of the BPS index along these lines was carried out [16]. In this paper we study the SCFTs by means of toroidal compactifications of the 6d E-string theory [17][18][19][20][21][22][23]. This approach has the following advantages.
Firstly, one can make full use of the known results of the well-studied BPS index of E-strings. It has several entirely different interpretations, providing us with complementary ways to compute it. In particular, the E-string theory has a world-sheet description, which enables us to compute the index as the generating function for 1 In this paper the term 'Nekrasov partition function' means the explicit expression (typically given as a sum over partitions), while the term 'BPS index' abstractly denotes the observable that can, in certain cases, be expressed as the Nekrasov partition function. For 5d gauge theories on R 4 × S 1 the BPS index is defined as a generalized supersymmetric index [13]. For 4d gauge theories it can be defined as the partition function given by the path integral in the Omega background [14]. For the SCFTs studied in this paper we do not know any explicit, direct definition of the BPS index. It can, however, be defined indirectly by means of theories which flow to the SCFTs and one can unambiguously compute it (at least as a series expansion). the sequence of elliptic genera of multiple E-strings [23]. Secondly, in this approach one can study the above SCFTs (including the D 4 case [24]) in a unified manner.
The E-string theory encompasses almost all 5d N = 1 and 4d N = 2 rank-one gauge theories [20,22]. All the above SCFTs arise in the moduli space of this single theory.
We clarify the general structure of the BPS index. For the sake of simplicity we consider the unrefined case, where the chemical potentials ǫ 1 , ǫ 2 for the Lorentz spins are fixed as ǫ 1 = −ǫ 2 =: . Our main results are summarized as follows. The BPS index for the type g = E 8−n , H n (n = 0, 1, 2) SCFTs takes the form where φ is the Higgs expectation value and The differential operator is introduced so that the 'descendants' that are determined by the modular anomaly equation are concealed from view. Put in this form the BPS index is characterized by two exponents β g , γ g and a sequence of invariants c g k . In this paper we determine β g , γ g (as above), invariants c E 8 k , c H 0 k (k = 1, 2, 3, 4) and c E 7 1 , c H 1 1 by using the known results of the BPS index for the E-string theory. The values of γ Hn have been known [16] and our results are in agreement with them.
Our method can easily be generalized to the case with general ǫ 1 , ǫ 2 . Another interesting generalization of this work is to turn on the chemical potentials for the other global symmetry charges. It would also be interesting to clarify how our results are related to the superconformal index for the SCFTs [25][26][27][28][29][30].
The rest of the paper is organized as follows. In section 2 we review the definition and the basic properties of the BPS index of E-strings. In section 3 we consider the setting of the E-string theory on R 4 × T 2 that realizes E 8 ⊕ H 0 singularities on the moduli space. We first study the index of E-strings in this setting and then take limits to obtain the index for the E 8 and H 0 SCFTs. Section 4 and section 5 are devoted to the E 7 ⊕ H 1 case and the E 6 ⊕ H 2 case respectively.

Review of BPS index of E-strings
The BPS index of E-strings is defined as the 5d BPS index [13] for the E-string theory on R 5 × S 1 . It is given by a trace over the space of the BPS particles as follows: and SO (4). Throughout this paper we consider the unrefined case ǫ 1 = −ǫ 2 =: .
The index Z is interpreted in several ways. One interpretation is associated with the expansion where Z n is the elliptic genus of n E-strings. Z n with any n can in principle be computed by using the localization technique [23]. Explicit forms of Z n (n ≤ 4) were obtained in [23]. Another useful interpretation is where F g is the genus-g topological string amplitude for the local 1 2 K3. F g (g ≤ 3) with general m were computed in [32], which we will use mainly in this paper. A third interpretation relates Z with the partition function of a certain five-brane web system [33]. This picture enables us to compute Z as a power series expansion in q.
The index Z satisfies two important constraints. One is known as the modular anomaly equation [34] where ∂ E 2 is the formal derivative in Eisenstein series E 2 (τ ). The other is the gap This follows from the geometric structure of the local 1 2 K3 [21].

Seiberg-Witten curve
Let us first consider the E-string theory on R 4 × T 2 without E 8 Wilson lines, i.e.
The low energy effective theory is characterized by the following elliptic Seiberg-Witten curve [20,22] Here u parametrizes the Coulomb branch of the moduli space of vacua. In this section we focus on the case where the complex structure of the torus is fixed to the special value This is a value for which Eisenstein series E 4 (τ ) vanishes We let subscript * denote that the quantity is evaluated at τ = τ * . For example, Other Eisenstein series take the following values (see, e.g., [35]) The Seiberg-Witten curve (3.2) becomes Because of the absence of the linear term in x, the j-invariant of this elliptic curve is identically 0. Therefore the complex structure of the curve takes the constant value τ * over the moduli space. The discriminant of the curve has zeros at u = 0 and u = 864/E 6 * , where the elliptic fibration has singular fibers of Kodaira type II * and II respectively. This means that near the point at u = 0 the theory looks like the E 8 SCFT while near the point at u = 864/E 6 * the theory looks like the H 0 SCFT [20].
In the following, we will study how the BPS index looks near these SCFT points.

Mirror map
In this subsection we establish the mirror map which connects the global moduli space coordinate u with the scalar expectation value φ on local patches of the moduli space.
We will see that the mirror map reduces to a very simple form in the present case with τ = τ * . This is similar to what was observed in the D 4 ⊕ D 4 case [24].
Since the mirror map described below is merely a special case of the general one [22], we will summarize the main points only. The reader is referred for the details to [32]. Note that the variable φ used in [32] should not be confused with φ in this paper. They are related to each other by In our present case with m = 0 and τ = τ * , one of the periods (divided by 2π) of the Seiberg-Witten curve is written as Here z is a new coordinate of the moduli space defined by The singularities corresponding to the E 8 and H 0 SCFTs are mapped to z = 0 and z = ∞ respectively. The Higgs expectation value is given by Here φ E 8 0 is an integration constant. Near the E 8 singularity at z = 0, φ is expanded as (3.14) Near the H 0 singularity at z = ∞, on the other hand, φ is expanded as (3.15) It is possible to determine the values of the integration constants φ E 8 0 and φ H 0 0 with respect to the convention of [32], but for our purposes their values are not important.

6d amplitudes
Let us now consider the BPS index of E-strings in the present setting (3.16) We will study it mainly in the expansion of the form By using the modular anomaly equation (2.5), the gap condition (2.6), the known forms of F g (g ≤ 3) [32] and elliptic genera Z n (n ≤ 4) [23], we are able to determine The first few amplitudes are as follows:  The rest of the results are rather lengthy and thus we do not present them here.
Instead, in what follows we clarify the structure of F E 8 ⊕H 0 g with general g.
First, F E 8 ⊕H 0 g (g ≥ 3) takes the following form where c g,k,i are some numerical coefficients. Next, one can easily see that F E 8 ⊕H 0 g with g = 3n+2, 3n+3 (n ∈ Z ≥0 ) are completely determined by the modular anomaly equation, given the data of F E 8 ⊕H 0 g (g ≤ 3n + 1). Therefore all the essential data are provided by F E 8 ⊕H 0 g with g = 3n + 1 (n ∈ Z ≥0 ). Furthermore, apart from the n = 0 case, they are written as with unknowns c 3n+1,k,i , but except for k = n they are again determined by the modular anomaly equation, given the data of F E 8 ⊕H 0 3m+1 (m < n). To sum up, the data of F E 8 ⊕H 0 1 and c 3n+1,n,i (n ∈ Z >0 , i = 0, . . . , 6n − 2) completely determine the BPS index.
Before closing this subsection let us sketch out a proof of the general form (3.23).
First, it is well-known that topological string amplitudes for Calabi-Yau threefolds are polynomials in a finite number of generators [36,37]. The generators for the most general BPS index of E-strings were identified [32]. By reducing the results to the present case, one sees that F E 8 ⊕H 0 g (g ≥ 2) are polynomials in ∂ k φ ln ω (k = 1, 2, 3) and in E 2 * , E 6 * . Moreover, F E 8 ⊕H 0 g (g ≥ 3) is of degree 2g − 2 and weight 2g − 2, where we assign degree k to ∂ k φ ln ω and weight 2n to E 2n * . It then follows that F E 8 ⊕H 0 g (g ≥ 3), as a function in z, is written as with f 2g−2 (x) being a degree 2g − 2 polynomial function. Next, let us show that contains the factor (1 − z 6 ) 2 . We first note that the elliptic genus of a single E-string in the present case vanishes In terms of F E 8 ⊕H 0 g this means that in the power series expansion in p = e −φ . Since the variable z is expanded in p as has to contain the factor (1 − z) 2 in order to satisfy (3.27). This, combined with (3.25), means that F E 8 ⊕H 0 g (g ≥ 3) has to contain the factor (1 − z 6 ) 2 .

4d limits
We have so far studied 6d theory on R 4 × T 2 . Let us now consider scaling limits in which the torus T 2 shrinks to zero size. To do this we first recover the length scale R, which is proportional to the radii of the T 2 . We then take the limit of R → 0 while keeping the complex structure of the T 2 to be τ = τ * . In the low energy Seiberg-Witten description, this procedure corresponds to zooming in on a point in the moduli space. In particular, by suitably zooming in on the singularity at z = 0 (z = ∞), one obtains the E 8 (H 0 ) SCFT.
Let us first consider the scaling limit in which only the local structure of the E 8 singularity at z = 0 contributes to the physics. This is achieved by first replacing the variables as and then taking the limit of R → 0. In this limit a function in z is dominated by the leading order term of its expansion in z. The mirror map (3.14) is simplified as The BPS index for the 4d E 8 SCFT is obtained from that for the E-string theory as The normalization constant is fixed as follows. Z E 8 is expanded in as We fix the normalization of Z E 8 so that F E 8 1 takes the above simple form without constant term. Note also that the form of F E 8 0 is not characteristic of the SCFT itself, but rather of how it is embedded in the bulk theory. The explicit forms of F E 8 g (g ≤ 15) are immediately obtained from the results of the last subsection.
Instead of listing them all, we will present the results in a very concise form, taking account of the fact that the BPS index satisfies the modular anomaly equation.
For Z that has the structure (2.3), one can show that the modular anomaly equation (2.5) is formally solved as whereZ is independent of E 2 . After we set τ = τ * , E 2 becomes a numerical constant and the modular anomaly equation does not make sense, but the structure (3.34) remains intact in the index. Note that the first prefactor in (3.34) becomes trivial when we take the 4d limit.
By exploiting this fact, the results of F g (g ≤ 15) are packed into the following concise expression: with (3.36) Similarly, one can compute Z H 0 by zooming in on the H 0 singularity at z = ∞. This is done by first replacing the variables as and then taking the limit of R → 0. The mirror map (3.15) becomes φ = 6 5z 5 . (3.38) The BPS index for the H 0 SCFT is obtained as The results are given as follows: with In this section we consider the compactification of the E-string theory that realizes the E 7 and H 1 SCFTs and compute the BPS index. Let us first recall the E 7 ⊕ A 1 reduction of the global symmetry E 8 of the E-string theory. This is achieved by setting the Wilson line parameters as [38] m = (0, 0, 0, 0, 0, 0, π, π) . (4.1) In this case the general Seiberg-Witten curve [22,32] reduces to [39] Here ϑ j ≡ ϑ j (0, τ ). The E 7 singularity has already realized in this setting. To obtain the H 1 singularity and make the complex structure of the curve constant, we set the complex structure of T 2 as This value is connected with τ = i by an SL(2, Z) transformation. 2 As in the last section, subscript ♯ denotes that the quantity is evaluated at τ = τ ♯ . We see that The curve (4.2) then becomes (4.7) The mirror map is given as follows. The period is where this time we have defined z as The Higgs expectation value is given by Near the E 7 singularity at z = 0, φ is expanded as (4.12) Near the H 1 singularity at z = ∞, on the other hand, φ is expanded as (4.13) Using the explicit forms of F g (g ≤ 3) [32], one can immediately compute . The results are F E 7 ⊕H 1 0 = 0, (4.14) By taking limits as in the last section, one obtains the BPS index for the E 7 and H 1

SCFTs as
with To determine the invariants c E 7 k , c H 1 k for k ≥ 2, we need more data of F g or Z n .
We expect that c E 7 k , c H 1 k for general k are given by E k 4♯ multiplied by some rational numbers, as in the case of the E 8 and H 0 SCFTs.

E 6 ⊕ H 2 case
In this section we consider the compactification of the E-string theory that realizes the E 6 and H 2 SCFTs and compute the BPS index. Let us first recall the E 6 ⊕ A 2 reduction of the global symmetry E 8 of the E-string theory. This is achieved by setting the Wilson line parameters as [38] The general Seiberg-Witten curve [22,32] reduces to [39] where α 3 := (m,n)∈Z 2 q m 2 +n 2 −mn = ϑ 3 (0, 2τ )ϑ 3 (0, 6τ ) + ϑ 2 (0, 2τ )ϑ 2 (0, 6τ ), The E 6 singularity has already realized in this setting. To obtain the H 2 singularity and make the complex structure of the curve constant, we set the complex structure of T 2 as This value is connected with τ = e 2πi/3 by an SL(2, Z) transformation. We let subscript ♭ denote that the quantity is evaluated at τ = τ ♭ . We see that Near the E 6 singularity at z = 0, φ is expanded as z 3k+1 3k + 1 = 3 z + z 4 4 + z 7 7 + · · · . (5.13) Near the H 2 singularity at z = ∞, on the other hand, φ is expanded as (5.14) Using the explicit forms of F g (g ≤ 3) [32], one can immediately compute . The results are In contrast to the E 8 ⊕H 0 case, the holomorphic anomaly equation, the gap condition and the data of elliptic genera Z n (n ≤ 4) are not enough to determine the invariants c E 6 k , c H 2 k . To determine the first invariants c E 6 1 , c H 2 1 , one needs slightly more data, e.g., the explicit form(s) of F E 6 ⊕H 2