Indices for 6 dimensional superconformal field theories
Seok Kim and Kimyeong Lee
Department of Physics and Astronomy & Center for Theoretical Physics,
Seoul National University, Seoul 151-747, Korea
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
We review some recent developments in the 6 dimensional superconformal field theories, focusing on their BPS spectra in the Coulomb and symmetric phases computed by various Witten indices. We shall discuss the instanton partition function of 5d maximal super-Yang-Mills theory, and the 6d superconformal index.
This is a contribution to the review volume “Localization techniques in quantum field theories” (eds. V. Pestun and M. Zabzine) which contains 17 Chapters available at 
- 1 Introduction
- 2 Coulomb branch indices for the self-dual strings
- 3 Superconformal indices of the 6d theories
- 4 Discussions
- 5 Appendix. Off-shell supergravity analysis on
With various string dualities found in mid 90’s, interacting quantum field theories in spacetime dimensions larger than were discovered from string theory [3, 4, 5]. Many aspects of these QFTs are counterintuitive from the conventional viewpoint and have enriched our notion on what quantum field theory is. The higher dimensional QFTs are also the key to understanding the strong-coupling aspects of string and M theories. Multiple M5-branes and 6d theories are such examples.
However, we still do not know their intrinsic definitions. For instance, they are strongly interacting CFTs, and no Lagrangian descriptions are known. Despite this situation, in the last few years there has been interesting progress in our understanding on the 5 and 6 dimensional superconformal field theories, based on various effective descriptions of these theories. In particular, we shall focus on the advances in supersymmetric observables of these higher dimensional field theories.
There have been many works on the BPS observables of 5d and 6d SCFTs, especially from 2012 when the techniques of curved space SUSY QFT were applied to higher dimensions. For instance, in 5d SCFTs, there have been extensive studies on the partition functions on [6, 7, 8, 9] and [10, 11, 12]. There have also been many studies on 6d SCFTs. Their partition functions were studied on [13, 14, 15, 16, 17, 18, 19], , [21, 22], and , where and are 2 and 3 dimensional manifolds. Various 6d defect partition functions on curved manifolds were also studied, such as the dimension surfaces [17, 18, 24, 25] and dimension surfaces [26, 25]. The progress was made possible largely due to the technical advances in 5d super-Yang-Mills theories on curved manifolds. See [27, 28, 29, 13, 30] and references therein for some early developments,  for some systematic formulations on 5d maximal SYM on curved backgrounds, [6, 16, 17, 32, 33, 34] for the factorizations of 5d partition functions on and ,  for the saddle point structures of the supersymmetric path integral of 5d SYM on Sasaki-Einstein spaces. Often, via factorization, some curved space observables are related to those of the same QFT on flat spacetime, such as or , in the Coulomb phase. The last Coulomb phase observables have been studied from relatively long time ago, after the pioneering works by Nekrasov et al. [36, 37]. There have been continuing developments in these observables [38, 39, 40, 41, 42, 43, 44, 45, 46], especially in the recent few years after the realization of their relations to the conformal phase observables.
Especially in this review paper, we shall discuss the BPS spectra of these theories captured by Witten index partition functions. The main objects will be the partition functions of 6d SCFTs on the Omega deformed in the Coulomb phase, and also the superconformal index partition function on . We shall mostly discuss the CFTs, since major progress has been made only for these theories so far. We shall however comment on possible generalizations to a wider class of CFTs at various places. It will mostly be reviews of some papers cited above, but contains some unpublished materials as well. In the rest of the introduction, we shall briefly motivate the objects that we study in this paper and also our methods and approaches.
One observable discussed in this paper is the superconformal index of the 6d SCFT . This is a Witten index which counts BPS local operators of the CFT on . Or equivalently, it counts BPS states of the radially quantized CFT on , weighted by various chemical potentials. Being a supersymmetric version of the thermal partition function, we can regard it as the partition function on with supersymmetric boundary conditions of fields along . Schematically, we shall be considering expressions for this index of the form
where denotes the ‘scalar VEV in the Coulomb branch’ which is integrated over in the above expression, and is the so-called ‘classical action’ which shall be explained later. The three ingredients are the Coulomb branch Witten index of the circle comapactified 6d theory on flat space, which we shall explain in detail in section 2. collectively denotes the chemical potentials. In particular, it will contain the (dimensionless) ‘inverse temperature’ like variable , where are the radii of the and factor, respectively. Other chemical potentials, in our parametrization, will be the three rotation chemical potentials on , and those for the flavor symmetries.
The expression above is just one of the many occasions in which the SUSY QFT partition functions on compact manifolds are related to the Coulomb branch partition functions. A canonical example can be found for gauge theories on , related to the Coulomb phase partition function on . The Coulomb branch partition function has been an extremely useful observable by itself, for many reasons, and has been extensively studied since [36, 37]. In the context of 6d CFTs, it provides useful information on the BPS spectrum of wrapped self-dual strings [49, 50]. Also, understanding its properties better has been (and will be) the key to the developments in the conformal phase observables, such as (1.1). So our section 2 will review the old and new developments on the Coulomb branch partition function on . Somewhat interestingly, the recent demand on refined understanding of this observable triggered a technically clearer derivation of this rather old observable, especially for many subtle QFTs for which this partition function could not be computed before.
Coming back to the superconformal index (1.1), we do not have a self-contained formulation to justify it. However, considering the regime with small circle, , we can try to understand the structure of (1.1) using a 5 dimensional effective description. When , the expression (1.1) admits a ‘weak coupling’ expansion in , either perturbative one in power series of , or nonperturbative one in a series of , where . The last ‘weak coupling’ expansion acquires a more precise sense when the 6d SCFT compactified on a small circle admits a weakly coupled 5d Yang-Mills theory description. For instance, when we compactify the 6d CFT of ADE type on small with radius , then at low energy we would have a 5 dimensional maximal super-Yang-Mills description on .111‘Maximal SYM’ will often mean a QFT with the field content of maximal SYM, subject to deformations due to curvature and chemical potential parameters. So the number of preserved SUSY could be less than . For instance, mass-deformed maximal SYM, the theory, will often be called just maximal SYM. Such a 5d SYM limit exists for some other SCFTs.222We shall comment on cases in which no 5d SYM limits exist, in which case the expression (1.1) could still make sense. The radius of the circle gets mapped to the 5d gauge groupling via
in our convention for . So here, the small expansion is indeed the weak coupling expansion.
The partition function at thus reduces to 5d SYM partition functions on , which has been studied in great detail since . This decomposes into the perturbative part and instanton corrections,
where is the 1-loop contribution which is independent of , and
with acquires contributions from Yang-Mills instantons localized on and extended along . These instanton solitons in 5d SYM are interpreted as Kaluza-Klein modes of the 6d CFT compactified on circle, so captures nontrivial dependence even after compactification on small circle. in (1.1) can also be computed from 5d SYM. So pragmatically, we shall be able to understand all the ingredients of (1.1) from 5d SYM. Having obtaining the weakly coupled expression (1.1) for the 6d index, one may sum over the series and re-expand the result at if one has a good technical control over . The strong coupling result is useful because the spectral information can be obtained only after the expansion in the small fugacity . We explain in section 3.2 how to explicitly do this in some special cases.
At this point, we also note that there is another version of the 6d index formula taking the form (1.1), which is obtained from 5d SYM on . This expression takes a manifest form of the index, given as an expansion in at . We shall explain it for the theory in section 3.3, emphasizing its virtue and new physics visible from this setting.
Conceptually, it will be interesting to understand whether the formulae of the type (1.1) are correct for all 6d SCFTs without relying on 5d SYM descriptions. Also, it would be nice to understand whether it is the unexpected feature of 5d SYM or our specific choice of SUSY observables which made 5d SYM useful here. For the Coulomb phase index explained in section 2, we can completely bypass the 5d SYM description logically (although it is still useful), and directly compute the index from string/M-theory by taking decoupling limits and starting from UV complete 1d or 2d gauge theories. We do not know whether we can bypass the 5d SYM description for the superconformal index.
The remaining part of this paper is organized as follows. In section 2, we explain the computation and physics of the Coulomb branch indices of 6d CFTs on Omega deformed , mainly from 1 dimensional gauge theories (also with detailed comments on studies from 2d gauge theories). In section 3, we explain the 6d superconformal index and the physics contained in it. Section 4 concludes with open questions and comments. Appendix A elaborates on the SUSY gauge theory on , including background supergravity construction for the vector multiplets.
2 Coulomb branch indices for the self-dual strings
In this section, we study the spectrum of self-dual strings in the Coulomb phase of the 6d SCFTs. On one hand, this will be interesting data of the theory by itself. On the other hand, these Coulomb phase observables play important roles in understanding supersymmetric partition functions at the conformal point, such as the 6d superconformal index .
In the ‘Coulomb phase,’ scalars in the 6d tensor multiplets assume nonzero expectation values . In such a phase, there appear tensionful self-dual strings whose tension is proportional to the Coulomb VEV. Let us first explain the SUSY preserved by these strings, when they are extended along a straight line. The 6d theory in the Coulomb phase preserves or Poincare supersymmetry. We only use the part of the SUSY to define our BPS self-dual strings. For our purpose, we write the supercharges as , . is the doublet index for the R-symmetry. , are the doublet indices of the spatial rotation on the 6d field theory direction , transverse to the string. These supercharges are subject to the reality condition
The supersymmetry algebra contains the following anti-commutatiaon relations:
where is energy, is momentum along the string, and is the momenta along . Here we have compactified one direction of the 6d theory on with radius , and wrapped the strings on that circle. We shall study the self-dual strings whose 5d masses saturate the BPS bound , so preserve supercharges . The other half-BPS states preserving would have similar spectrum.
2.1 Elliptic Genus Method
In particular, we shall be interested in the Witten index which counts the BPS degeneracies of these strings wrapping the circle. Namely, the 6d CFT is put on , and there are real scalar VEVs (). The index is defined by
where is the energy over the string rest mass , , , , and collectively denotes all the other conserved global charges which commute with the supercharges. The charges appearing inside the trace is chosen so that they commute with the two supercharges , , among . From the algebra (2), the most general states preserving these two supercharges will be the -BPS states preserving all . So with this index we are counting the states in the -BPS multiplet, with a further refinement given by (which does not commute with all four ). We also define the partition function by summing over the winding numbers of the self-dual strings,
where . Here, we introduce the (dimensionless) chemical potentials conjugate to the winding numbers . These are just scaled version of the scalar VEVs that we used above but should not be confused with them.
is computed in various ways. Currently, in most nontrivial theories, it is only computable in series expansions. One series expansion takes the form of (2.4), and the coefficients are computed from the elliptic genera of suitable 2 dimensional supersymmetric quantum field theories living on the worldsheets of these strings [39, 40, 43, 46]. A different kind of series expansion can be made in , when :
The momentum charge on is given a weight . These Kaluza-Klein momentum states are regarded as massive particles in 5d. can be computed from the quantum mechanics of the ‘instanton solitons’ of 5 dimensional gauge theory, if one has a 5d weakly coupled SYM description at small radius. In this section, we shall mostly focus on the latter quantum mechanical index. The usefulness of these two approaches will be commented later.
We first explain the general ideas of computing the two types of coefficients and , before studying an example. Both computations essentially rely on the string theory completion of the 6d SCFT, and suitable decoupling limits when the contribution of some charges to the BPS mass become large.
Let us first explain the strategy of computating . Firstly, as the 6d SCFT lacks intrinsic definition, we rely on its string theory or M-theory engineering. In all such constructions, one engineers suitable string/M-theory backgrounds, and takes suitable low energy decoupling limits in which the 6 dimensional states decouple from the bulk states (e.g. 10/11 dimensional gravity, stringy states, so on). After this limit, certain 6 dimensional decoupled sector of 6d SCFT exists. Furthermore, we are interested in the 1+1 dimensional strings in the Coulomb phase, with nonzero VEV for the 6d scalar whose mass dimension is . The tension of the self-dual strings is proportional to . At energy scale much below , the 6d system will again exhibit a decoupling, between the 2d QFT on the strings and the rest of the 6d system. is computed by studying the last 2d QFT living on the strings’ worldsheet. We generally expect the 2d QFT to be an interacting conformal field theory. The computation of the observables is generally very difficult with strongly interacting QFT. Here, the crucial step is to engineer a 2d gauge theory which is weakly coupled in UV, and flows to the desired interacting CFT in the IR. The construction of the UV gauge theory will often be easy with brane construction engineering of the 6d SCFT and the associated self-dual strings. Such UV gauge theories are constructed for the self-dual strings of a few interesting 6d CFTs, such as ‘M-strings’ , ‘E-strings’ , and some others . The UV gauge theories for many interesting self-dual strings are still unknown at the moment and are under active studies. With a weakly-coupled UV gauge theory which flows to the desired CFT, the elliptic genus can in principle be easily computed from the UV theory, as the elliptic genus is independent of the continuous coupling parameters of the theory. In fact the general elliptic genus formula for 2d SUSY Yang-Mills theories was recently derived in [52, 53].
can also be computed in a similar manner, for some classes of self-dual strings. This approach is applicable to the cases in which the circle compactification of the 6d theory yields weakly coupled 5d Yang-Mills theories at low energy. Then, the momentum is given by the topological charge
carried by the Yang-Mills instanton solitons of the 5d gauge theory. The dynamics of these solitons are often described by a quantum mechanical gauge theory. is essentially computed by the quantum mechanical index for the instantons. More precisely, one finds
where is computed from the perturbative degrees of freedom in 5d SYM, and is given by the instanton quantum mechanics. The last instanton partition function has been first computed in [36, 37], and has been intensively studied since then for various reasons. Although we used the notion of 5d SYM to explain the strategy, we can often get the quantum mechanical gauge theory description from the full string theory set up by taking a suitable decoupling limit, bypassing the UV incomplete 5d SYM description at all. For instance, for the theory compactified on circle, one just obtains the quantum mechanics from the D0-D4 system by taking a low energy decoupling limit, without relying on 5d SYM description at all.
The two quantities and are supersymmetric indices of the 2d and 1d gauge theories on and , respectively. Although both types of indices have been extensively studied in the literature from long time ago, their general structures for gauge theories have been fully clarified only recently. See [51, 52, 53] for the developments in the 2d elliptic genus, and [42, 54, 55] for the 1d Witten index.
Before proceeding with concrete examples, we also comment that the quantity can often be computed from topological string amplitudes on suitable Calabi-Yau 3-folds. This happens when the 6d SCFTs are engineered from F-theory on singular elliptic Calabi-Yau 3-folds [56, 57, 58]. Changing the moduli of CY in a way that specific 2-cycles shrink to zero volume, one obtains a 6 dimensional singularity which supports decoupled degrees of freedom at low energy, defining 6d SCFTs. One important ingredient of these theories is D3-branes wrapping these collapsing 2-cycles, which yield self-dual strings that become tensionless in the singular limit. Therefore, the volume moduli of these 2-cycles are the Coulomb branch VEVs in the 6d tensor supermultiplets.
So in this setting, we consider the F-theory on in the Coulomb phase. We wrap D3-branes along times the 2-cycles in CY. This system can be T-dualized on to the dual circle of the type IIA theory. The D3-branes map to D2-branes transverse to . Consider the regime with large , or equivalently small , and make an M-theory uplift on an extra circle . Then and combine to a torus and fiber the 4d base of the original CY we started from, meaning that we get M-theory on the same CY. The self-dual string winding numbers over the 2-cycles maps to the M2-brane winding numbers on the same cycles. The momentum on maps to M2-brane winding number on the fiber. So the counting of the self-dual string states maps to counting the wrapped M2-branes on CY in M-theory. The last BPS spectrm is computed by the topological string partition function on CY [59, 60]. In particular, consider an expansion of in the rotation paramters given by
The coefficients of the expansion are computed by the topological string amplitudes on CY. The series in (2.8) is the genus expansion of refined topological string. So from this viewpoint, the elliptic genus we study in this section is the all genus sum of the topological string amplitudes. A few low genus expansions are known for many interesting 6d self-dual strings. This provides an alternative method of computing some data of the full elliptic genus when neither 2d nor 1d gauge theories are known. For instance, see  for the results 6d strings engineered by F-theory on Hirzebruch surfaces, where many such strings do not have known gauge theory descriptions yet.
2.2 Instanton Partition Method
With the above comments in mind, we shall now explain the studies of the Coulomb branch indices from 1d gauge theories. We shall specifically explain the 6d SCFT of type, to be concrete. Although this quantity has been studied in the context of ‘instanton counting’ of 5d SYM , one does not have to rely on 5d SYM description at all, as everything can be directly understood from the full string theory setting. For applications of the similar techniques to the Coulomb branch CFTs with SUSY, see [42, 43].
The maximal superconformal field theory in 6d of type is engineered by taking M5-branes on top of another, in the flat M-theory background. In the low energy limit, the system contains a 6d SCFT on M5-branes’ worldvolume which is decoupled from the bulk. In the Coulomb branch, we take M5-branes separated along one of the five transverse directions of . The self-dual strings are suspended between separated M5-branes along this direction, and also wrap of the 5-brane worldvolume.
We are interested in the index of the circle compactified self-dual strings. The index is invariant under the change of continuous parameters of the theory, and also of the background parameters as long as they do not appear in the supercharges that are associated with the definition of the Witten index. So we can take the circle radius to be very small, and use the type IIA string theory description for the computation. Let us denote by (with ) the scalar VEVs, or positions of M5-branes along a line in . These are related to the 5d VEVs by a multiplication of . Let denote the number of self-dual strings ending on a given M5-branes, with orientations taken into account. If the strings have units of Kaluza-Klein momentum, one obtains in the small limit a system of D0-branes bound to fundamental strings with charges stretched between the D4-branes. In particular, the energy of the compactified self-dual strings is bounded as
In the regime with very small , where we plan to compute the index, we can use the effective description with fixed , as the particles with large rest mass become non-relativistic. So the quantum mechanics of D0-branes bound to D4-branes would capture the exact index . The quantum numbers will be realized as Noether charges of this mechanical system. This is simply the decoupling limit of the D0-branes bound to D4-branes, and could also be regarded as the discrete lightcone quantization (DLCQ) of M5-branes .
The quantum mechanics of D0-branes on D4-branes (in the Coulomb phase) preserves SUSY, since the D0-D4 system preserves of the type IIA SUSY. The system has rotation symmetry on D4 worldvolume transverse to D0, and rotation transverse to the D4’s. When D4’s are displaced along one of the five directions of , with VEV , is broken to . We denote by the doublet indices of the four ’s, respectively, in the order presented above. The supercharges can be written by , with reality conditions similar to (2.1). The degrees of freedom are:
with . The D4-D4 strings move along transverse to the D0’s, and decouple at low energy. (These will be perturbative 5d SYM degrees.) This system can be formally obtained by a dimensional reduction of a 2 dimensional SUSY gauge theory, in which and respectively define left-moving and right-moving supercharges. The first line of the above field content is called the vector multiplet. The second and third line separately form a hypermultiplet. The action of this system is very standard, and could be found e.g. in , whose notations we followed here.
The index is defined in this quantum mechanics by
where denotes the charge, , and , are the Cartans of , , respectively. is the usual regulator parameter which does not appear in the index. The trace is over the Hilbert space of the quantum mechanics. Note that the measure in the trace commutes with two supercharges , among , as we explained at the beginning of this section. This index was computed by Nekrasov  in 2002. We shall briefly review it with adding more recent clarifications on the computational step, for which Nekrasov wrote down a prescription for computation. These clarification of the prescriptions is somewhat crucial to compute the indices for more general 6d SCFTs [42, 43].
The 1d gauge theory for D0-D4 system that we explained above is strongly coupled at low energy, since the quantum mechanical gauge coupling has dimension . We can however compute the index in the limit, as the Witten index is generically expected to be insensitive to the changes of continuous parameters of the theory. This is what Nekrasov has done in , and also in more recent studies of [42, 54, 55]. The computation of the index is done by going to the path integral representation of the index with Euclidean quantum mechanics, put on a circle with circumference , and computing it in the limit. The computation consists of (1) identifying the zero modes of the quantum mechanical path integral on , in the limit (carefully defined in [52, 53, 42, 55]); (2) Gaussian path integral over the non-zero modes; (3) finally making an exact integration over the zero modes.
We briefly explain the results of these three steps, within our example for simplicity. Firstly, the zero modes in the limit consist of the constant modes of and which commute with each other. Here, is the Wick-rotated variable in the Euclidean quantum mechanics on . More precisely, defines a holonomy of the gauge group along the circle. For the gauge group, one can take to be
using rotation, locally labeled by complex parameters. Each parameter satisfies , so lives on a cylinder. These variables are subject to further identification given by permuting the variables. This is the permutation subgroup of which acts within (2.12). For gauge groups other than , especially for disconnected groups, the zero mode structure could be more complicated. See  for examples. There are also some fermionic zero modes in the strict limit, which we shall not explain here, but plays important roles in the final step (3) above.
Secondly, in the above background, the 1-loop determinants over non-zero modes yield the following factor:
where , and the expressions with in the arguments mean multiplying the factors with all possible signs. The factor on the second line comes from the integral over the fundamental hypermultiplet , and the second factor on the right hand side of the first line comes from the adjoint hypermultiplet . Finally, the first factor on the first line comes from the vector multiplet nonzero modes .
The final task is to integrate over the complex, or real, variables . Naively, it appears that one has to do a dimensional integral on copies of cylincers, with a meromorphic measure given by (2.13). This naive prescription will not work because the measure will diverge at various poles, implying that the integration over non-zero modes becomes subtle near the poles even in the limit. In , Nekrasov gave a dimensional contour integral prescription, rather than a dimensional real integral, with the measure (2.13). The result is the sum over residues for a subset of poles in the integrand (2.13). The relevant poles are labeled by all possible -tuple of Young diagrams with total number of boxes. These are sometimes called -colored Young diagrams with boxes. The summation of residues from these poles is given by [62, 63, 38]
Here, labels the boxes in the ’th Young diagram . is the distance from the box to the edge on the right side of that one reaches by moving horizontally to the right. is the distance from to the edge on the bottom side of that one reaches by moving down (and may be negative if one has to move up to the bottom of ). See [62, 63, 38] for more detailed explanations on notation. This result can be obtained from the following rule for the contour. First of all, the contours will be a closed curve on the plane. The rules of the contour choices, or equivalently the residues to be kept by the contour integral, are as follows: (1) exclude all the poles in (2.13) coming from the factors whose arguments include (from 5d SYM, this amounts to ignoring all the poles coming from 5d adjoint hypermultiplet); (2) exclude all poles at or ; (3) as for the remaining poles, take , and include all poles within the unit circles .
Although well known and used, this prescription was given a satisfactory derivation only rather recently , using and generalizing the methods of [52, 53]. The strategy of [52, 53] is to carefully re-do the supersymmetric path integral computation when is near its pole location, and also carefully considering the lift of some gaugino zero modes . After some analysis, the final contour integral reduces to a set of residue sum, which is called the Jeffrey-Kirwan residue . The Jeffrey-Kirwan residue rules are slightly different from the above (1), (2), (3) in general, but it was shown for the above theory that the two rules yield the same result .
The result (2.14) is useful to understand various aspects of the theory in Coulomb phase, and its self-dual strings comapactified on a circle . It is also useful to understand the conformal phase (with zero Coulomb VEV) of the theory. An early finding of this sort was that (2.14) could be used to study the index of the DLCQ theory, which is the 6d CFT compactified on a light-like circle. Namely, one takes (2.14) and suitably integrates over the Coulomb VEV with Haar measure inserted, to extract out the gauge invariant spectrum . More recently, and this will be reviewed in our section 3, (2.14) was used as the building block of more sophisticated CFT observable, the superconformal index on . Again several factors of the form (2.14) are multiplied (with other factors that we shall call the ‘classical measure,’ see section 3), and we suitably integrate over the Coulomb VEV parameter .
Here, we find one virtue of the index (2.5) obtained by 1d gauge theory, over (2.4) which is obtained by 2d gauge theory. Namely, in many recent applications, is kept as a fixed fugacity, while the Coulomb VEV is introduced temporarily and should be integrated over to obtain CFT observables. The computations explained in this subsection keeps the dependence exact, at a given order in . So in this sense, knowing the coefficients of (2.5) exactly could be more useful, rather than knowing those of (2.4).
On the other hand, the elliptic genus (2.4) has the virtue of making the modular property under the transformation clear, with the modular parameter . So when one has to make a strong coupling re-expansion of the partition function, as explained in the introduction and section 3.2, this could potentially be very useful. Also, the elliptic genus (2.4) can often be computed when the circle reduction of 6d CFT does not flow to weakly coupled SYM, so that the 1d approach of this section becomes difficult to apply [43, 46]. However, (2.4) takes the form of the Coulomb VEV expansion when acquires large expectation values (compared to other parameters such as ). So apparently it is unclear how to integrate over them in the curved space partition functions.
3 Superconformal indices of the 6d theories
In this section, we explain the current status of our understanding on the superconformal index of the 6d theories. Possible extensions to the 6d theories have not yet been developed in detail, on which we shall just make general statements and brief comments.
6d SCFT has superconformal symmetry, as well as possible global symmetries whose charges we collectively call . The bosonic part of the superconformal symmetry is . We are interested in the radially quantized CFT, living on . Then the maximal commuting set of charges of the bosonic subgroup are taken to be in the R-symmetry, which are rotations on , and for the translational symmetry along the time direction . Often, we make dimensionless by multiplying the radius of . We normalize to have eigenvalues for spinors. The superconformal index of a general 6 dimensional SCFT is defined by 
where , and is the trace over the Hilbert space of the CFT on . Note that the 6d SCFT has Poincare supercharges , with for the R-symmetry, and for the symmetry, where the last three signs are constrained by . These supercharges have energy ( scale dimension) . The conformal supercharges are given by , where this time the three signs for are constrained by . They have scale dimension . Among these supercharges, the measure of (3.1) commutes with , . So the index counts BPS states (with minus sign for fermions) which are annihilated by at least these two supercharges. Equivalently, by the operator-state map, the index counts BPS local operators of the CFT on . The energies (dimensions) of the BPS states (operators) are given by , from the vanishing of acting on these BPS states.
Specifying to the 6d SCFTs, the superalgebra is , and there are no extra flavor symmetries. The supercharges are now given by , , where , are the two spinor charges. are given and constrained in the same way as the previous paragraph. We can pick and and define the index
Even without a microscopic formulation of the 6d SCFTs, we have fairly well-motivated expressions for these partition functions (3.1), (3.2). We shall write down two such expressions, one in section 3.1 and another in section 3.3. Both of them are inspired by 5 dimensional super-Yang-Mills theories, obtained by circle reductions of 6d SCFTs on down to 5d.
3.1 The partition function on
The first expression for the partition function , is given as follows. It uses the Coulomb branch partition function that we explained in section 2, and is given by
where , . ( appearing in the arguments may be replaced by , as was more commonly used in [16, 33], using the period shifts of the arguments.) Here, is the gauge group of the low energy 5d SYM that one obtains by reducing the 6d SCFT, and is the Weyl group of . More abstractly, in the 6d CFT, acquires meaning as the Weyl group acting on the Coulomb branch as , and parametrizes the Coulomb branch . is a term which depends only on the background parameters , which we shall explain further below. This expression has been proposed with two different motivations. See  for discussions involving topological strings. Here, we explain how (3.1) was proposed from the viewpoint of 5 dimensional supersymmetric Yang-Mills theory.
First consider the 6d theory on . The partition function (3.1) would be computed by a Euclidean 6d theory path integral on , where the has circumference and various fields satisfy twisted boundary conditions due to the extra insertion .333For the convenience of arguments, we formally assume the existence of a 6d Lagrangian description and the path integral representation of (3.1). This is true for the free Abelian theory. For interacting theories, concrete arguments will only rely on the Lagrangian formulation of the 5d SYM at low energy, which exists. The twisted boundary conditions given by can be represented by deforming the background metric of in a ‘complex’ manner as follows :
where and . Here ’s satisfy , and , periodicities are assumed. If one is uncomfortable about the complex metric, one can simply take the chemical potentials ’s to be imaginary first, and later continue to real ’s in the partition function (3.1) or 5d SYM. (It will be deforming the action to be complex.) We would like to understand the partition function (3.1) first in the regime , in which case one can make the Kaluza-Klein reduction of the 6d theory on a small circle to a 5d SYM on . in the dimensionless convention is the ratio of the radii of and . In particular, when , this is identified with the 5d SYM gauge coupling as . All terms in appearing in the right hand side can be understood as non-perturbative instanton corrections for small even from the 5d viewpoint, as we saw in the section 2.444Sometimes, (3.1) makes sense even if the small circle reduction does not yield weakly-coupled 5d SYM. For instance, some 6d SCFTs on a circle flow to strongly interacting 5d SCFTs rather than 5d SYMs. However, viewing , as the 6d partition functions and 6d scalars, (3.1) still makes sense, although we do not know how to derive it.
If we Kaluza-Klein reduce the 6d metric on circle, one would naturally expect to have a supersymmetric Yang-Mills theory on a ‘squashed’ whose metric is given by above, also with a background ‘dilaton’ field and the background ‘gravi-photon’ field . The last statement can be made more precise by finding (3.4) as a 5 dimensional off-shell supergravity background . We find that this is the case. More precisely, we divide the construction of 5d SYM on with metric into two steps. We first obtaining the vector multiplet part of the action using off-shell supergravity methods, which is more cumbersome to achieve in a more conventional method. We then construct the hypermultiplet part of the action in a more brutal manner. The former can be easily done by using the 5d off-shell supergravity of , which realizes off-shell SUSY of the background gravity and the dynamical vector multiplets. Construction of the hypermultiplet part of the action with one off-shell SUSY closely follows . The results are summarized in appendix A.
At this point, let us comment that the metric of (3.4) may be just one special way of geometrizing the chemical potentials . In the literature, alternative geometric realizations are also discussed, which lead to the same supersymmetric partition function (3.1) [16, 33].
With the action, SUSY and notations on the squashed summarized in appendix A, we can understand the partition function (3.1) in more detail from 5d SYM. We first study the classical action at the possible saddle points. Expanding three factors in the series of , with , we find the following factor at each value of and given ,
The exponent can be understood as the action of the following supersymmetric configurations. The SUSY transformation of the gaugino in the vector multiplet is given by
where . Some off-shell supersymmetric configurations are given by taking and
with constant , as explained in appendix A. can be taken to be in the Cartan subalgebra, using the global part of the gauge transformation. This is not the most general supersymmetric configurations. To understand more general possibilities, we consider
with , . The vector