Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a $c=24$ two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a $c=12$ CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group $M_{24}$ has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving $(4,4)$ superconformal symmetry was studied in beautiful work by Gaberdiel, Taormina, Volpato, and Wendland. The present paper shows that in both the GTVW and $c=12$ theories the construction of superconformal generators can be understood via the theory of quantum error correcting codes. The automorphism groups of these codes lift to symmetry groups in the CFT preserving the superconformal generators. In the case of the $N=1$ supercurrent of the GTVW model our result, combined with a result of T. Johnson-Freyd implies the symmetry group is the maximal subgroup of $M_{24}$ known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on \cite{gtvw} the Ramond-Ramond sector of the GTVW model is related to the Miracle Octad Generator which in turn leads to a role for the Golay code as a group of symmetries of RR states. Moreover, $(4,1)$ superconformal symmetry suffices to define and decompose the elliptic genus of a K3 sigma model into characters of the $N=4$ superconformal algebra. The symmetry group preserving $(4,1)$ is larger than that preserving $(4,4)$.
We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4 superconformal algebra. Similarly, any subgroup of Co_0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2 superconformal algebra. Through the decomposition of the corre…
We construct a model of moonshine phenomenon based on the use of N=2 superconformal algebra. We consider an extremal Jacobi form of weight 0 and index 2, and expand it in terms of N=2 massless and massive representations. We find the multiplicities of massive representations are decomposed into a sum of dimensions of irreducible representations of the group L2(11).
We compute the $\mathcal{N}=1$ superconformal blocks from the networks of open Wilson lines in the $\text{osp}(1|2)$ Chern-Simons theory in the expansion of large central charge $c$. We first reproduce the $1/c$ correction of conformal weight from an open Wilson line by adopting the regularization prescription developed in our previous works. We then obtain the closed form expressions of superconformal blocks including $1/c$ corrections, which were not available before. We also examine heavy operators corresponding to supersymmetric conical spaces, and the geometry is quantized by utilizing th…
We discuss higher-order corrections to superconformal invariance for a class of N=2 supersymmetric Chern-Simons theories including the ABJM model. We argue that corrections are inevitable for general theories in this class; but that it is probable that any corrections are of a particular "maximally transcendental" form.
We study the superconformal index of S-fold theories by using AdS/CFT correspondence. It has been known that the index in the large $N$ limit is reproduced as the contribution of bulk Kaluza-Klein modes. For finite $N$ D3-branes wrapped around the non-trivial cycle in $\boldsymbol{S}^5/\mathbb{Z}_k$, which corresponds to Pfaffian-like operators, give the corrections of order $q^N$ to the index. We calculate the finite $N$ corrections by analyzing the fluctuations of wrapped D3-branes. Comparisons to known results show that our formula correctly reproduces the corrections up to errors of order …
The superconformal index of quiver gauge theories realized on D3-branes in toric Calabi-Yau cones is investigated. We use the AdS/CFT correspondence and study D3-branes wrapped on supersymmetric cycles. We focus on brane configurations in which a single D3-brane is wrapped on a cycle, and we do not take account of branes with multiple wrapping. We propose a formula that gives finite $N$ corrections to the index caused by such brane configurations. We compare the predictions of the formula for several examples with the results on the gauge theory side obtained by using localization for small si…
Recently it has been conjectured that N=6, U(N)_{k} \times U(N)_{-k} Chern-Simons theory is dual to M-theory on AdS_4\times S^7/Z_{k}. By studying one-loop correction to the M-theory effective action, we calculate the correction to the entropy of thermal field theory at strong coupling. For large k level, we have also found the alpha' correction to the entropy from the string correction of the type IIA effective action. The structure of these two corrections at strong t'Hooft coupling are different.
We investigate the AdS/CFT correspondence for quiver gauge theories realized on D3-branes put on abelian orbifolds by using the superconformal index. We assume that on the gravity side the finite $N$ corrections of the index are reproduced by D3-branes wrapped on three particular three-cycles in the internal space ${\cal Y}$, the abelian orbifold of $\boldsymbol{S}^5$. We first establish the relation between baryonic charges on the gauge theory side and the D3-brane wrapping numbers and holonomies on D3-branes. Then we confirm our proposal by comparing the results of localization for gauge the…
Using the duality conjecture between N=6 supersymmetric U(N)_k \times U(N)_{-k} Chern-Simons theory and M-theory on AdS_4 \times S^7/Z_{k}, we calculate the corrections to the shear viscosity of the field theory at temperature T. At strong 't Hooft coupling and at small k level, we have considered one-loop correction to the M-theory effective action. At large k level, we have considered the alpha' correction to the type IIA effective action. In both cases the correction to the ratio of shear viscosity to the entropy density is positive.
We study the Argyres-Douglas theories realized at the superconformal point in the Coulomb moduli space of $\mathcal{N}=2$ supersymmetric $SU(2)$ QCD with $N_f=1,2,3$ hypermultiplets in the Nekrasov-Shatashvili limit of the Omega-background. The Seiberg-Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg-Witten curve, we calculate the quantum corrections to the Seiberg-Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.
In \cN = 2, 4 superconformal field theories in four space-time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in \cN = 1 superspace, we test the Dine-Seiberg proposal by comparing the two-loop F^4 quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) \cN = 4 SYM (i.e. \cN = 2 SYM …
We consider the superconformal index of three-dimensional ${\cal N}=2$ supersymmetric field theories computed via localization on $S^1\times S^2$. We systematically develop an expansion where the ratio of the radius of $S^1$ to the radius of $S^2$ is taken very small -- a Cardy-like expansion. We emphasize the sub-leading structures in this Cardy-like expansion as well as their interplay with the large-$N$ limit for theories with gauge group of the form product of $U(N)$ factors. We demonstrate that taking the large-$N$ limit first leads to an expression for the index that only includes terms …
We investigate the large $N$ instanton effects of partition functions in a class of $\mathcal{N}=4$ circular quiver Chern-Simons theories on a three-sphere. Our analysis is based on the supersymmetry localization and the Fermi-gas formalism. The resulting matrix model can be regarded as a two-parameter deformation of the ABJM matrix model, and has richer non-perturbative structures. Based on a systematic semi-classical analysis, we find analytic expressions of membrane instanton corrections. We also exactly compute the partition function for various cases and find some exact forms of worldshee…
In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau--Zaslow (capturing the Euler characters of the moduli spaces of D2-branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz--Klemm--Vafa (KKV), and Katz--Klemm--Pandharipande (KKP). We show that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently bee…
In this note we consider Coulomb-branch chiral primary correlation functions in ${\cal N} = 2$ superconformal QCD with gauge group $SU(2)$, in the limit of large R-charge ${\cal J} = 2n$ for the chiral primary operators $[{\cal O}(x)]^ n$ with the inverse gauge coupling $τ$ held fixed. In previous work, these correlation functions were determined to all orders in $n$, up to unknown exponentially small corrections. In this paper we determine the first several orders of the asymptotic expansion of the exponentially small correction itself. To do this we use: the physical interpretation of the ex…
We develop the method of the hamiltonian reduction of affine Lie superalgebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which connects the diagonal gauge with the Drinfeld-Sokolov gauge and considering the quantum corrections, we get generic expressions for the classical and quantum free field realizations of the algebras.
We study thermodynamics of N=6 superconformal Chern-Simons theory by computing quantum corrections to the free energy. We find that in weakly coupled ABJM theory on R(2) x S(1), the leading correction is non-analytic in the 't Hooft coupling lambda, and is approximately of order lambda^2 log(lambda)^3. The free energy is expressed in terms of the scalar thermal mass m, which is generated by screening effects. We show that this mass vanishes to 1-loop order. We then go on to 2-loop order where we find a finite and positive mass squared m^2. We discuss differences in the calculation between Coul…
The quantized form of the soft N=8 superconformal algebra is investigated. Its operator product expansions are shown to exhibit a one-parameter-class of (soft) anomalies, which may be arbitrarily shifted by certain suitable quantum corrections of the generators. In particular, the BRST operator can be constructed and made nilpotent in the quantum version of all known realizations of the algebra. This generalizes the results of Cederwall and Preitschopf, who studied the $\widehat{S^7}$-algebra, that is contained as a soft Kac-Moody part in the superconformal algebra. A Fock-space representation…
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 AdS 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.
We consider strong 't Hooft coupling expansion in special four-dimensional $\mathcal N=2$ superconformal models that are planar-equivalent to $\mathcal N=4$ super Yang-Mills theory. Various observables in these models that admit localization matrix model representation can be expressed at large $N$ in terms of a Fredholm determinant of a Bessel operator. The latter previously appeared in the study of level spacing distributions in matrix models and, more recently, in four-point correlation functions of infinitely heavy half-BPS operators in planar $\mathcal N=4$ SYM. We use this relation and a…
In arXiv:0908.2463 we computed the four-loop correction to a function depending on the 't Hooft coupling(s) that appears in the magnon dispersion relation of the spin chains derived from single trace operators in N=6 superconformal Chern-Simons theories. In this paper we give detailed descriptions of this calculation and the computation of the four-loop wrapping corrections for a length four operator in the 20 of SU(4), the R-symmetry group for these theories. Here, we give all relevant Feynman diagrams and loop integrals explicitly, and also demonstrate the cancellation of double poles in the…
We review recent progress in the understanding of symmetries for scattering amplitudes in N=4 superconformal Yang-Mills theory. It is summarized how the superficial breaking of superconformal symmetry by collinear anomalies and the renormalization process can be cured at tree and loop level. This is achieved by correcting the representation of the superconformal group on amplitudes. Moreover, we comment on the Yangian symmetry of scattering amplitudes and how it inherits these correction terms from the ordinary Lie algebra symmetry. Invariants under this algebra and their relation to the Grass…
We study $\mathcal{N}=2$ supersymmetric Jackiw-Teitelboim (JT) gravity at finite temperature coupled to matter. The matter fields are related to superconformal primaries by AdS/CFT duality. Due to broken super reparametrisation invariance in the SCFT dual, there are corrections to superconformal correlators. These are generated by the exchange of super-Schwarzian modes which is dual to the exchange of 2D supergravity modes. We compute corrections to four-point functions for superconformal primaries and analyse the behaviour of out-of-time-ordered correlators. In particular, four-point function…
We propose and test a novel conjectural relation satisfied by the superconformal index of maximally supersymmetric $U(N)$ gauge theory in four dimensions. Analogous relations appear to be also valid for the superconformal indices of a large collection of other gauge theories, as well as for a broad class of index-like generating functions. The relation expresses the finite $N$ index as a systematic series of corrections to a large $N$ answer. Individual corrections have an holographic interpretation as the analytic continuation of contributions from "giant graviton" branes fixed by a specific …
We calculate the superconformal indices of the $\mathcal N=2$ superconformal field theories realized on $N$ coincident D3-branes in 7-brane backgrounds with constant axiodilaton via the AdS/CFT correspondence. We include the finite-$N$ corrections as the contribution of D3-branes wrapped around 3-cycles in the internal space. We take only single-wrapping contributions into account for simplicity. We also determine the orders of the next-to-leading corrections which we do not calculate. The orders are relatively high, and we obtain many trustable terms. We give the results for $N=1,2,3$ explici…
We show that the reduction of a planar free spin-1/2 particle system by the constraint fixing its total angular momentum produces the one-dimensional Akulov-Pashnev-Fubini-Rabinovici superconformal mechanics model with the nontrivially coupled boson and fermion degrees of freedom. The modification of the constraint by including the particle's spin with the relative weight $n\in \N$, $n>1$, and subsequent application of the Dirac reduction procedure (`first quantize and then reduce') give rise to the anomaly free quantum system with the order $n$ nonlinear superconformal symmetry constructed re…
Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When $\N=4$, results are obtained for the three point function of various descendant operators, including the energy momentum tensor and SU(4) current. For both $\N=2$ or 4 superconformal identities are derived for the functions of the two conformal invariants appearing in the four point function for the chiral primary operator. These are solved in terms of a single arbitrary function of the two …
We establish a theorem about non-trivial 11D supergravity fluctuations that are conformally related to flat superspace geometry. Under the assumption that a theory of conformal 11D supergravity exists, similar in form to that of previously constructed theories in lower dimensions, this theorem demands the appearance of non-vanishing dimension 1/2 torsion tensors in order to accommodate a non-trivial 11D conformal compensator and thus M-theory corrections that break super-conformal symmetry. At the complete non-linear level, a presentation of a conventional minimal superspace realization of Wey…
We compute classical and first quantum finite-size corrections to the recently found giant magnon solutions in two different subspaces of $\mathbb{CP}^3$. We use the Lüscher approach on the recently proposed exact S-matrix for $\mathcal{N}=6$ superconformal Chern-Simons theory. We compare our results with the string and algebraic curve computations and find agreement, thus providing a non-trivial test for the new $AdS_4/CFT_3$ correspondence within an integrability framework.
We calculate the superconformal indices of a class of six-dimensional ${\cal N}=(1,0)$ superconformal field theories realized on M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularity by using the method developed in previous works of the authors and collaborators. We use the AdS/CFT correspondence, and finite $N$ corrections are included as the contribution of M2-branes wrapped on two-cycles in $S^4/\mathbb{Z}_k$. We confirm that the indices are consistent with the expected flavor symmetries.
We present the basic $\mathcal{N} =1$ superconformal field theories in four-dimensional de Sitter space-time, namely the non-abelian super Yang-Mills theory and the chiral multiplet theory with gauge interactions or cubic superpotential. These theories have eight supercharges and are invariant under the full $SO(4,2)$ group of conformal symmetries, which includes the de Sitter isometry group $SO(4,1)$ as a subgroup. The theories are ghost-free and the anti-commutator $\sum_α\{Q_α, Q^{α\dagger}\}$ is positive. SUSY Ward identities uniquely select the Bunch-Davies vacuum state. This vacuum state…
Quantization of the nonlinear supersymmetry faces a problem of a quantum anomaly. For some classes of superpotentials, the integrals of motion admit the corrections guaranteeing the preservation of the nonlinear supersymmetry at the quantum level. With an example of the system realizing the nonlinear superconformal symmetry, we discuss the nature of such corrections and speculate on their possible general origin.
The Konishi superconformal multiplet is an important theoretical laboratory where one can test AdS/CFT methods to compute strong coupling corrections to the spectrum of superstrings in AdS_5 x S^5. In particular, one can exploit integrability for finite charge states/operators. The multiplet ground state is a singlet operator with two simple descendants in the rank-1 sectors sl(2) and su(2) of N=4 super Yang-Mills theory. Recently, the next-to-leading quantum correction to the sl(2) state has been computed. Here, we use the algebraic curve approach to determine the correction to the other stat…
We bootstrap loop corrections to AdS${}_5$ supergravity amplitudes by enforcing the consistency of the known classical results with the operator product expansion of $\mathcal{N}=4$ super Yang-Mills theory. In particular this yields much new information on the spectrum of double-trace operators which can then be used, in combination with superconformal symmetry and crossing symmetry, to obtain a prediction for the one-loop amplitude for four graviton multiplets in AdS. This in turn yields further new results on subleading $O(1/N^4)$ corrections to certain double-trace anomalous dimensions.
In addition to the superconformal symmetry of the underlying Lagrangian, the scattering amplitudes in planar N=4 super-Yang-Mills theory exhibit a new, dual superconformal symmetry. We address the question of how powerful these symmetries are to completely determine the scattering amplitudes. We use the example of the NMHV superamplitudes to show that the combined action of conventional and dual superconformal symmetries is not sufficient to fix all the freedom in the tree-level amplitudes. We argue that the additional information needed comes from the study of the analytic properties of the a…
A new class of models of dynamical supersymmetry breaking is proposed. The models are based on SU(N_C) gauge theories with N_F(<N_C) flavors of quarks and singlets. Dynamically generated superpotential exibits runaway behavior. By embedding the models into conformal field theories at high energies, the runaway potential is stabilized by strong quantum corrections to the Kahler potential. The quantum corrections are large but nevertheless can be controlled due to superconformal symmetry of the theories.
The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving e…
We compute a certain class of corrections to (specific) screening lengths in strongly coupled nonabelian plasmas using the AdS/CFT correspondence. In this holographic framework, these corrections arise from various higher curvature interactions modifying the leading Einstein gravity action. The changes in the screening lengths are perturbative in inverse powers of the 't Hooft coupling or of the number of colours, as can be made precise in the context where the dual gauge theory is superconformal. We also compare the results of these holographic calculations to lattice results for the analogou…
We investigate logarithmic corrections to the entropy of supersymmetric, rotating, asymptotically AdS$_5$ 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 ch…
We first discuss how the longstanding confusion in the literature concerning one-loop quantum corrections to 1+1 dimensional solitons has finally been resolved. Then we use 't Hooft and Veltman's dimensional regularization to compute the kink mass, and find that chiral domain wall fermions, induced by fermionic zero modes, lead to spontaneous parity violation and an anomalous contribution to the central charge such that the BPS bound becomes saturated. On the other hand, Siegel's dimensional reduction shifts this anomaly to the counter terms in the renormalized current multiplet. The superconf…
We determine the one-loop deformation of the conformal symmetry of a general N}=2 superconformally invariant Yang-Mills theory. The deformation is computed for several explicit examples which have a realization as world-volume theories on a stack of D3 branes. These include (i) N=4 SYM with gauge groups SU(N), USp(2N) and SO(N); (ii) USp(2N) gauge theory with one hypermultiplet in the traceless antisymmetric representation and four hypermultiplets in the fundamental; (iii) quiver gauge theory with gauge group SU(N)xSU(N) and two hypermultiplets in the bifundamental representations (N,\bar N) a…
