Superconformal Index Research Articles

Exact Schur index in closed form

The Schur limit of the superconformal index of a four-dimensional N = 2 superconformal field theory encodes rich physical information about the protected spectrum of the theory. For a Lagrangian model, this limit of the index can be computed by a contour integral of a multivariate elliptic function. However, surprisingly, so far it has eluded exact evaluation in closed, analytical form. In this paper we propose an elementary approach to bring to heel a large class of these integrals by exploiting the ellipticity of their integrand. Our results take the form of a finite sum of (products of) the well-studied flavored Eisenstein series. In particular, we derive a compact formula for the fully flavored Schur index of all theories of class S of type a1, we put forward a conjecture for the unflavored Schur indices of all N=4 super Yang-Mills theories with gauge group SU(N), and we present closed-form expressions for the index of various other gauge theories of low ranks. We also discuss applications to non-Lagrangian theories, modular properties, and defect indices.

The Hilbert space of large N Chern-Simons matter theories

We demonstrate that the known expressions for the thermal partition function of large N Chern-Simons matter theories admit a simple Hilbert space interpretation as the partition function of an associated ungauged large N matter theory with one additional condition: the Fock space of this associated theory is projected down to the subspace of its quantum singlets i.e. singlets under the Gauss law for Chern-Simons gauge theory. Via the Chern-Simons / WZW correspondence, the space of quantum singlets are equivalent to the space of WZW conformal blocks. One step in our demonstration involves recasting the Verlinde formula for the dimension of the space of conformal blocks in SU(N)k and U(N)k,k′ WZW theories into a simple and physically transparent form, which we also rederive by evaluating the partition function and superconformal index of pure Chern-Simons theory in the presence of Wilson lines. A particular consequence of the projection of the Fock space of Chern-Simons matter theories to quantum (or WZW) singlets is the ‘Bosonic Exclusion Principle’: the number of bosons occupying any single particle state is bounded above by the Chern-Simons level. The quantum singlet condition (unlike its Yang-Mills Gauss Law counterpart) has a nontrivial impact on thermodynamics even in the infinite volume limit. In this limit the projected Fock space partition function reduces to a product of partition functions, one for each single particle state. These single particle state partition functions are q-deformations of their free boson and free fermion counterparts and interpolate between these two special cases. We also propose a formula for the large N partition function that is valid for arbitrary finite volume of the spatial S2 and not only at large volume.

Superconformal index for [formula omitted][formula omitted] Chern-Simons quiver gauge theories

We compute superconformal indices for N=3ADEˆ Chern-Simons quiver gauge theories with a product gauge group ∏iU(N)i, using the method of supersymmetric localization. We also perform a large N analysis of the index. This index includes contribution from non zero magnetic flux sector. The fact that these theories have a weakly coupled UV completion in terms of N=3 supersymmetric Chern-Simons Yang-Mills theories enables us to apply the localization technique. Such theories have dual M-theory description on AdS4×M7, where M7 is a tri-Sasaki Einstein manifold.

Global form of flavor symmetry groups in 4d N=2 theories of class S

We provide a systematic method to deduce the global form of flavor symmetry groups in 4d N=2 theories obtained by compactifying 6d N=(2,0) superconformal field theories (SCFTs) on a Riemann surface carrying regular punctures and possibly outer-automorphism twist lines. Apriori, this method only determines the group associated to the manifest part of the flavor symmetry algebra, but often this information is enough to determine the group associated to the full enhanced flavor symmetry algebra. Such cases include some interesting and well-studied 4d N=2 SCFTs like the Minahan-Nemeschansky theories. The symmetry groups obtained via this method match with the symmetry groups obtained using a Lagrangian description if such a description arises in some duality frame. Moreover, we check that the proposed symmetry groups are consistent with the superconformal indices available in the literature. As another application, our method finds distinct global forms of flavor symmetry group for pairs of interacting 4d N=2 SCFTs (recently pointed out in the literature) whose Coulomb branch dimensions, flavor algebras and levels coincide (along with other invariants), but nonetheless are distinct SCFTs.

Logarithmic corrections to the entropy of rotating black holes and black strings in AdS5

We investigate logarithmic corrections to the entropy of supersymmetric, rotating, asymptotically AdS5 black holes and black strings. Within the framework of the AdS/CFT correspondence, the entropy of these black objects is determined, on the field theory side, by the superconformal index and the refined topologically twisted index of mathcal{N} = 4 supersymmetric Yang-Mills theory, respectively. We read off the logarithmic correction from those field-theoretic partition functions. On the gravity side, we take the near-horizon limit and apply the Kerr/CFT correspondence whose associated charged Cardy formula describes the degeneracy of states at subleading order and determines the logarithmic correction to the entropy. We find perfect agreement between these two approaches. Our results provide a window into precision microstate counting and demonstrate the efficacy of low-energy, symmetry-based approaches such as the Kerr/CFT correspondence for asymptotically AdS black objects under certain conditions.

Zero-form and one-form symmetries of the ABJ and related theories

The zero-form and one-form global symmetries of the Aharony-Bergman-Jafferis (ABJ) and related theories, with at least mathcal{N} = 6 supersymmetry in three dimensions, are examined in detail. Starting from well-known dualities between theories with orthogonal and symplectic gauge groups and those with unitary gauge groups, we gauge their one-form symmetries or their subgroups and obtain new dualities. One side of the latter involves theories with special orthogonal and symplectic gauge groups, and the other side involves theories with unitary gauge groups; there is a discrete quotient on one or both sides of the duality. We study the refined superconformal indices of such theories and map the symmetries across the dualities, with particular attention to their discrete part. As a generalisation, we also find a new duality between a circular quiver with a discrete quotient of alternating special orthogonal and symplectic gauge groups and a three-dimensional mathcal{N} = 4 circular (Kronheimer-Nakajima) quiver with unitary gauge groups, whose Higgs or Coulomb branch describes an instanton on a singular orbifold.

3d mathcal{N} = 3 generalized Giveon-Kutasov duality

We generalize the Giveon-Kutasov duality for the 3d mathcal{N} = 3 U(N)k,k+nN Chern-Simons matter gauge theory with F fundamental hypermultiplets by introducing SU(N) and U(1) Chern-Simons levels differently. We study the supersymmetric partition functions and the superconformal indices of the duality, which supports the validity of the duality proposal. From the duality, we can map out the low-energy phases: for example, confinement appears for F + k − N = −n = 1 or N = 2F = k = −n = 2. For F + k − N < 0, supersymmetry is spontaneously broken, which is in accord with the fact that the partition function vanishes. In some cases, the theory shows supersymmetry enhancement to 3d mathcal{N} = 4. For k = 0, we comment on the magnetic description dual to the so-called “ugly” theory, where the usual decoupled sector is still interacting with others for n ≠ 0. We argue that the SU(N)0 “ugly-good” duality (which corresponds to the n → ∞ limit in our setup) is closely related to the S-duality of the 4d mathcal{N} = 2 SU(N) superconformal gauge theory with 2N fundamental hypermultiplets. By reducing the number of flavors via real masses, we suggest possible ways to flow to the “bad” theories.

OPE of line defects in 5d En SCFT

5d ray index is the 5d superconformal index in the presence of a ray-like defect. An elementary ray defect flows to a fundamental Wilson ray in IR gauge theory. In 5d mathcal{N} = 1 superconformal field theory(SCFT) with En global symmetry, we compute the ray index associated with an adjoint Wilson ray. We show that the operators that appear in the index are non-trivially charged under the center of the global symmetry En. The charge under the center being twice compared with the center charge carried by the elementary ray operator indicates a non-trivial OPE between the ray-like defects in the UV SCFT.

Deconfining mathcal{N} = 2 SCFTs or the art of brane bending

We introduce a systematic approach to constructing mathcal{N} = 1 Lagrangians for a class of interacting mathcal{N} = 2 SCFTs. We analyse in detail the simplest case of the construction, arising from placing branes at an orientifolded ℂ2/ℤ2 singularity. In this way we obtain Lagrangian descriptions for all the R2,k theories. The rank one theories in this class are the E6 Minahan-Nemeschansky theory and the C2 × U(1) Argyres-Wittig theory. The Lagrangians that arise from our brane construction manifestly exhibit either the entire expected flavour symmetry group of the SCFT (for even k) or a full-rank subgroup thereof (for odd k), so we can compute the full superconformal index of the mathcal{N} = 2 SCFTs, and also systematically identify the Higgsings associated to partial closing of punctures.

Growth of the 116 -BPS index in 4d N=4 supersymmetric Yang-Mills theory

We study the microcanonical superconformal index of $\frac{1}{16}$-supersymmetric operators in 4d $\mathcal{N}=4$ $U(N)$ super Yang-Mills (SYM) theory. We show, numerically for $N\ensuremath{\le}10$, that the large-charge asymptotics are consistent with the entropy of supersymmetric black holes in the dual anti de Sitter space. We then prove, using representation theory, that the index agrees precisely with the multigraviton index, when the charge is less than $2(N+1)$, and begins to deviate for larger values of charge. Thus the $U(N)$ SYM index interpolates between multigraviton values at small charges and black hole growth at large charges.

Conformal S-dualities from O-planes

We study 4d SCFTs obtained by orientifold projections on necklace quivers with fractional branes. The models obtained by this procedure are \U0001d4a9 = 1 linear quivers with unitary, symplectic and orthogonal gauge groups, bifundamental and tensorial matter. Remarkably, models that are not dual in the unoriented case can have the same central charges and superconformal index after the projection. The reason for this behavior rests upon the ubiquitous presence of adjoint fields with R-charge one. We claim that the presence of such fields is at the origin of the notion of inherited S-duality on the models’ conformal manifold.

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

Calculating Superconformal Index of N = 4 Super-Yang-Mills Theory for Finite N

The examination of AdS/CFT correspondence for finite N, where quantum gravitational effects become important, reveals that finite N corrections for the superconformal index can be reproduced as D-3 brane contributions.

SL(3, \u2124) Modularity and New Cardy limits of the mathcal{N} = 4 superconformal index

The entropy of 1/16-th BPS AdS5 black holes can be microscopically accounted for by the superconformal index of the mathcal{N} = 4 super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the “S-transformation” of the elliptic Γ function. In this paper, we derive more general SL(3, ℤ) modular properties of the elliptic Γ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the mathcal{N} = 4 superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS5 black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions.

Finite-N superconformal index via the AdS/CFT correspondence

Abstract We propose a prescription to calculate the superconformal index of the ${\cal N}=4$U(N) supersymmetric Yang–Mills theory with finite N on the anti-de Sitter side. The finite-N corrections are included as contributions of D3-branes wrapped around three-cycles in ${\boldsymbol S}^5$, which are calculated as the index of the gauge theories realized on the wrapped branes. The single-wrapping contribution has been studied in a previous work, and we further confirm that the inclusion of multiple-wrapping contributions correctly reproduces the higher-order terms as far as we have checked numerically.

Gravity interpretation for the Bethe Ansatz expansion of the N=4 SYM index

The superconformal index of the maximally supersymmetric SU(N) Yang-Mills theory can be expressed as a sum over solutions to Bethe Ansatz Equations. Within the AdS/CFT framework, the authors find a one-to-one mapping of such solutions, which have good large N limit, to (complex) Euclidean black hole solutions on the gravity side. This mapping captures both the leading contribution from the classical gravity action and non-perturbative corrections.

Entropy function from toric geometry

It has recently been claimed that a Cardy-like limit of the superconformal index of 4d N=4 SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS5 rotating black hole. Here we study this Cardy-like limit for N=1 toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.

The 4d superconformal index near roots of unity and 3d Chern-Simons theory

We consider the S3×S1 superconformal index ℐ(τ) of 4d mathcal{N} = 1 gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential τ coupled to a combination of angular momenta on S3 and the U(1) R-charge. We develop the all-order asymptotic expansion of the index as q = e2πiτ approaches a root of unity, i.e. as overset{sim }{tau } ≡ mτ+n → 0, with m, n relatively prime integers. The asymptotic expansion of log ℐ(τ) has terms of the form overset{sim }{tau } k, k = −2, −1, 0, 1. We determine the coefficients of the k = −2, −1, 1 terms from the gauge theory data, and provide evidence that the k = 0 term is determined by the Chern-Simons partition function on S3/ℤm. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the S1. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the overset{sim }{tau } → 0 expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index.

On the superconformal index of Chern-Simons theories and their KK spectrometry

We study the large-N limit of superconformal index for two strongly interacting Chern-Simons theories in three dimensions with mathcal{N} = 2 supersymmetry, and compare the result against the AdS/CFT dual, namely the data of full Kaluza-Klein reduction spectra obtained using the exceptional field theory technique. The two theories of interest are mABJM and GJV theories, which are obtained as IR fixed point of RG whose UV description is ABJM and maximally supersymmetric Yang-Mills theory respectively. We confirm agreement and the duality persists. For the case of mABJM it turns out that we need a refinement of the gravity index which reflects the fact that the UV description on the field theory side has enhanced global symmetry.

Integral representations of rank two false theta functions and their modularity properties

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type A_2 and B_2. This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze {hat{Z}}-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing mathtt{H}-graphs. Along the way, our method clarifies previous results on depth two quantum modularity.