Exactly calculable quantities in supersymmetric field theories play important roles in recent progress in quantum field theories. The Schur index of superconformal field theories is one such quantity. There are different ways to calculate the index.

The Schur index is a specialization of the superconformal index of superconformal field theories [1,2]. (See Ref. [3] for various limits of the superconformal index.) The superconformal index can be regarded as the supersymmetric partition function in . For Lagrangian theories it is defined as the path integral in the background, and we can reduce it as a finite-dimensional integral by using the localization method. This is also available for a non-Lagrangian theory if we know a UV Lagrangian theory that flows to the theory.

Cordova and Shao [4] proposed an interesting formula for the Schur index which gives the index as the trace of a so-called quantum monodromy operator. With this formula we can calculate the index from the information of the BPS spectrum of the theory in the Coulomb branch. Although the BPS spectrum depends on the Coulomb moduli parameters and jumps on walls of marginal stability the quantum monodromy operator is wall-crossing invariant, and so is the index. The formula is generalized to decorated indices by introducing defect operators [5–7].

For class S theories, which are realized on M5-branes wrapped on Riemann surfaces, the Schur index is expressed as a correlation function of a two-dimensional topological field theory on the associated Riemann surface [8–14].

The Schur index receives contributions of a special class of gauge-invariant operators, which are called Schur operators. Beem et al. [15] show that the set of Schur operators form a chiral algebra of two-dimensional conformal field theory (CFT). Once we identify the chiral algebra associated with a four-dimensional theory the Schur index is obtained as the vacuum character of the chiral algebra. The characters of other modules are also important and are related to line and surface operator insertions [7,16]. For theories of class S there is a prescription to obtain the corresponding chiral algebra [17].

The purpose of this paper is investigate a generalization of the Schur index defined by replacing the background manifold with its orbifold . Such an orbifold generalization for the superconformal index of and theories has already been investigated in Refs. [18–22]. The index is defined in Ref. [18] by using , which preserves a supercharge of a specific chirality. Because we need only one supercharge (and its Hermitian conjugate) for the definition of the superconformal index, this orbifolding is consistent with the definition of the index. However, in the case of the Schur index, we need to preserve two supercharges with opposite chirality (and their Hermitian conjugates). Therefore, we need to modify the definition of the orbifold index by introducing an extra twist. The application of the localization technique for the orbifold Schur index is straightforward and will be shown in Sect. 2. Then, we discuss a generalization of the IR formula to the orbifold case. We suggest a natural generalization of Cordova and Shao’s formula based on a physical interpretation of the formula, and apply it to some simple examples. For systems consisting of free hypermultiplets we find agreement of the UV and IR formulae. Unfortunately, however, for more general systems including dynamical vector multiplets the suggested formula does not give the desired results.

Let us consider an superconformal field theory defined in . Let , , , , and be the Hamiltonian, the third component of the left-handed spin, that of the right-handed spin, the Cartan generator, and the charge, respectively. The superconformal index is defined by (1) where the trace is taken over all gauge-invariant states in . We denote the set of Cartan generators of the flavor group by . Other Cartan generators are defined by (2) (3) (4) (5) where and are supercharges with the following quantum numbers: (6)

The definition in Eq. (1) respects . Namely, all the Cartan generators used in Eq. (1) commute with . Because is -exact, the index in Eq. (1) is independent of .¹

The superconformal index can be regarded as the supersymmetric partition function in . If the theory has a Lagrangian description with gauge group we can define the index as the path integral. By using the localization technique we can reduce the path integral to the finite-dimensional integral (7) where is the plethystic exponential defined by (8)

is the one-particle index defined by (9) where the trace is taken over all one-particle states including gauge-non-invariant states. includes both flavor and gauge Cartan generators, and is the set of the corresponding fugacities. Namely, . is the integral over the gauge fugacities, (10) where is the size of the Weyl group of .

The Schur index is obtained from the superconformal index by the specialization . Then disappears from Eq. (5), and all the remaining Cartan charges commute with not only but also . Therefore, the Schur index receives contributions of operators that carry . Such operators are called Schur operators. As a result, the Schur index is a function of the superconformal fugacity and the flavor fugacities .

There is another index that receives from the same set of operators as the Schur index. It is the Macdonald index defined from the superconformal index by taking the limit . We do not consider the Macdonald index here because its definition does not respect the supercharges and that are important when we discuss the BPS configuration associated with the IR formula. Even so, the Macdonald index is closely related to the Schur index, and can be reproduced from the chiral algebra up to some ambiguity that can be fixed by using information from four-dimensional SCFT [23]. In the recent paper [24] an orbifold version of the Macdonald index is studied, and a close relation to the chiral algebra is observed. It will be important to study the relation between the orbifold Schur index defined below and the orbifold Macdonald index in Ref. [24].

The one-particle Schur index is given by (11) where , , and represent the contributions of a single vector multiplet, charged vector multiplets, and half-hypermultiplets, respectively. They are explicitly given as (12) where is the character of the adjoint representation of with the Cartan contribution subtracted, and is the character of the representation of the half-hypermultiplets.² Because does not depend on the gauge fugacities we can factor out the Cartan contribution in Eq. (7) as , with defined by (13) where is the -Pochhammer symbol defined by (14)

A generalization of the superconformal index to the orbifold background was first investigated in Ref. [18]. They defined the orbifold with the discrete group generated by (15)

This is consistent with the definition in Eq. (1) of the superconformal index in the sense that commutes with the supercharge . If the theory has gauge and/or flavor symmetry we can turn on holonomies by using instead of as the generator. Once we choose the generator, it is straightforward to generalize the formula in Eq. (7) to the orbifold case. What we have to do first is project away the contribution of non-invariant states from the one-particle index in Eq. (9). This projection is carried out by inserting the projection (16) into the trace in Eq. (9). Because can be expressed in terms of the Cartan generators appearing in Eq. (1) as , insertion of is equivalent to the replacement of the fugacities (17) and the one-particle index for the orbifold is (18)

By using this orbifold one-particle index, the orbifold index is given by (19) where the factor is the contribution of the zero-point energy. For the ordinary index this factor just gives an overall factor and is usually neglected. However, in the orbifold case this factor is important because it depends on the gauge holonomy . The zero-point factor is easily obtained by taking the product of all contributions of one-particle states. For example, if the one-particle index is expanded as (we explicitly show and dependence for simplicity, and the dots include other fugacities) then the corresponding zero-point factor is given by . Although this infinite product is usually divergent, we can obtain a finite result by using an appropriate regularization such as -function regularization.

Now, let us apply the same orbifolding prescription to the Schur index. We cannot obtain the orbifold Schur index by the specialization from the orbifold superconformal index above because this contradicts the orbifold action in Eq. (17). For consistency with we use (20) instead of . Namely, we combine with the additional twist to keep both and invariant. Again, we can turn on holonomies , and the insertion of is equivalent to the replacement (21)

Therefore, the orbifold one-particle index is (22)

With this one-particle index we can calculate the orbifold Schur index of Lagrangian theories by using Eq. (19), which we call the UV formula in the following.

Before discussing the IR formula for the orbifold Schur index, let us apply the UV formula to some simple systems. In the following examples the zero-point factors give overall factors independent of the holonomies and we omit them.

The IR formula proposed by Cordova and Shao [4] gives the Schur index by using the information of the BPS spectrum in the Coulomb branch.

Let be the charge lattice of flavor and gauge charges, and be the associated Dirac pairing. The central charge of a particle is determined by its charge , and we denote it by . The flavor sublattice is defined as the set of charges which have vanishing pairing with arbitrary .

Let be the set of primitive charges such that the charge of an arbitrary BPS particle is given by with and . The BPS spectrum at a point of the Coulomb branch is encoded in a set of functions defined for each . These functions are called quantum Kontsevich–Soibelman factors. The functional form of depends on the helicity of the BPS particle . (By an abuse of notation we use as a label for sorts of particles.) In the following we deal with only BPS particles belonging to a half-hypermultiplet, and the function for such a particle is given by (29) where is the -factorial defined by .

Cordova and Shao’s IR formula [4] is (30) where is the phase-ordered product according to ; is the rank of the theory, which is the number of massless gauge fields at a generic point of the Coulomb branch. are operators satisfying the quantum torus algebra³ (31)

The trace is defined as an isomorphic map from to . If , .

To physically interpret this formula, we should understand the structure of BPS configurations in the Coulomb branch [5]. In general, a BPS configuration contains massive BPS particles with different charges. For a configuration in to preserve the supercharges and , which are necessary to define the Schur index, the particles must be aligned along a large circle in , and the position on the circle is determined by the central charge by . Thus we can specify the particle distribution in a BPS configuration by giving a set of occupation numbers for . We denote this set by .

If we expand the function as (32) then the index is given as the summation over the occupation numbers: (33)

The summand gives the contribution of a specific set of occupation numbers, and consists of the following three factors:

∘ The factor is the one-loop contribution of the massless vector multiplets.

Now we suggest an IR formula for the orbifold as a natural generalization of the original one. acts on the large circle as a shift by and the orbifolding makes it a circle with circumference . In other words, the large circle in the covering space consists of fundamental regions, and only BPS particles in one of the fundamental regions are independent. Let be the set of primitive charges associated with a specific fundamental region. The charge distribution of a BPS configuration in the orbifold is specified by . Therefore, the index should be given as the summation over .

The first factor in the summand should be replaced by the factor , with given in Eq. (23). The second factor associated with the internal degrees of freedom of BPS particles should be the product of for . The third factor, as we mentioned above, can be regarded as the classical contribution to the angular momentum and the flavor charges. The angular momentum can be obtained by integrating the contribution of the Poynting vector over the orbifold. The same result is obtained by first calculating the angular momentum for the covering space , and dividing the result by . This is also the case for the flavor charges. Correspondingly, the trace factor should be replaced by the th root of the trace for the configuration in the covering space. Combining these, we obtain (34)

This is the IR formula we want to discuss in the next section.

An additional comment on the trace factor would be in order. As we mentioned above, only charge distribution in a specific fundamental region is independent, and the distribution in other regions should be determined by the symmetry. The phase-ordered trance in Eq. (34) must be taken over all charges in the covering space. It is important that the acts on charges non-trivially, and the product is not the simple th power of the product in the fundamental region. In the case, for example, the action flips the signs of charges, as we will see explicitly in the next section, and the trace factor takes the form .

Let us apply the IR formula in Eq. (34) to a few simple examples and compare the results with those obtained by the UV formula.

We first consider the system of a free hypermultiplet . Before computing the index by the UV and IR formulae it is important to check the consistency between BPS configurations in the Coulomb branch and the orbifold action.

In the system of a single hypermultiplet, we have only two sorts of particles with flavor charge . In a BPS configuration there are particles with charge at and anti-particles with charge at . This cannot be invariant under action with . Even for , exchanges particles and anti-particles, and the orbifolding seems to contradict the BPS configuration in the Coulomb branch. When , actually, we can take it back to the original configuration by performing additional charge conjugation. This additional transformation is also necessary to keep the mass term in the Lagrangian invariant. To make the hypermultiplet massive we need to turn on the vacuum expectation value (VEV) of the scalar component of the non-dynamical background vector multiplet coupling to the current. Because carries a charge of , the action changes its sign. This is compensated by the additional charge conjugation, which flips the sign of the background vector multiplet.

The charge conjugation exchanging and is realized by the transformation with (35)

This anti-commutes with the generator , and works as the charge conjugation. For consistency, we should set the Wilson line to be . (A generic Wilson line is not allowed because the operator does not commute with , and is incompatible with the orbifold action.)

Now we have chosen a generator consistent with the Coulomb branch VEV. Let us calculate the index by using the UV and IR formulae. When , the orbifold with the charge conjugation twist is in fact equivalent to the orbifold with fractional holonomies studied in the previous section up to rotation, because the fractional holonomy is conjugate to the charge conjugation . Therefore, the index is still given by Eq. (27) with . By setting , the one-particle index becomes (36) and the Schur index is (37)

On the IR side, we have only one charge in . The suggested formula in Eq. (34) gives (38) and this agrees with Eq. (37).

By introducing multiple hypermultiplets we can realize more general orbifolds. Let us consider a system of free hypermultiplets. There are sorts of particles. For invariance, we need to tune the Coulomb branch VEV and introduce an appropriate twist . Let be the Coulomb branch VEV (we use the basis in which the invariant tensor is given by ). We tune so that the central charges are symmetric and given by (39)

The action of shifts to for and to . To keep the background unchanged, we need a twist such that the transformation acts on as (40)

The following realize this transformation: (41)

(This is a generalization of the charge conjugation in Eq. (35) in the case.) For the same reason as the case we set the Wilson line to vanish. The diagonalization of gives the eigenvalues with (42)

For each pair of eigenvalues this is the same as the action of the fractional holonomy on a single hypermultiplet, and the one-particle index of the whole system is the sum of Eq. (26) over : (43)

The Schur index is , and this is the same as what we obtain from the IR formula. Each fundamental region of the large circle contains one sort of particle and the IR formula in Eq. (34) gives .

We can consider a more general case in which is not but its divisor, . Such an orbifold is defined by using as the generator, and the one-particle index becomes (44) the Schur index is . Again, this is reproduced by the IR formula of Eq. (34) by taking account of the sorts of particles in a fundamental region.

Up to here we have found good agreement between the UV and IR results. Let us move on to a more complicated system, QED with hypermultiplets. Although this system is not conformal, it is known that the two formulae give the same answer for the ordinary Schur index [4], and it is natural to expect this to hold for orbifolds, too. However, disappointingly, we find a discrepancy for the orbifold index in this case.

For example, let us consider the orbifold of two-flavored QED. The UV formula gives the index in Eq. (28). When we use the IR formula we need to take account of the existence of two sorts of particles in the fundamental region. Let and be their charges and and be the corresponding occupation numbers. In the covering space the charges of particles in a fundamental region are always canceled by the mirror images in the other fundamental region, and the trace factor in Eq. (34) does not impose any constraints on and . If we assume the trivial flavor holonomy, the trace factor simply gives . This means that the index is essentially the same as that for two half-hypermultiplets in , and given by (45)

This is obviously different from the result of the UV formula in Eq. (28).

In this paper we generalized the Schur index to the orbifold in such a way that the action preserves the two supersymmetries respected by the definition of the Schur index. We also naively generalized Cordova and Shao’s IR formula for the Schur index to the orbifold case. This reproduces the correct index for a system of free hypermultiplets when the fugacities and holonomies are chosen in a -invariant way. However, it is far from satisfactory, with the following deficiencies:

∘ Although any SCFT admits a orbifold with an arbitrary in the UV description, the IR formula works only for special values of depending on the theory.

The second and third deficiencies may be related to each other. The incompatibility of the generic Wilson lines seems to prevent us from performing path integrals over all gauge field configurations. At present, we cannot claim anything definite for this point, and more investigation is desired. It is important to study more general systems to clarify to what extent our formula works and (if possible) how we should improve it to make it applicable to general systems. In particular, it would be interesting and important to study the orbifold Schur index of non-Lagrangian theories such as Argyres–Douglas theories. Lagrangian theories that flow to a class of Argyres–Douglas theories have been proposed in Refs. [25–31], and this enables us to apply the UV formula of the orbifold index to the non-Lagrangian theories.

It is also interesting to analyze the relation between the orbifold index and chiral algebra. In the unorbifolded case, the chiral algebra is realized in a complex plane, which is identified by Weyl rescaling with , where is the large circle in on which BPS particles are aligned. The Virasoro generators and on the plane are related to the superconformal generators by (46)

For Schur operators with , and the generator is given by (47)

Namely, this is the orbifold acting on the worldsheet. Therefore, in the context of the chiral algebra, the orbifolding should be regarded as the insertion of a twist operator at the origin. Such a relation is studied for an orbifold version of the Macdonald index in Ref. [24]. It would be interesting to do a similar analysis for the orbifold Schur index.