Chiral Primary Operators Research Articles

A limit for large R-charge correlators in mathcal{N} = 2 theories

Using supersymmetric localization, we study the sector of chiral primary operators (Tr ϕ2)n with large R-charge 4n in mathcal{N} = 2 four-dimensional superconformal theories in the weak coupling regime g → 0, where λ ≡ g2n is kept fixed as n → ∞, g representing the gauge theory coupling(s). In this limit, correlation functions G2n of these operators behave in a simple way, with an asymptotic behavior of the form {G}_{2n}approx {F}_{infty}left(uplambda right){left(frac{uplambda}{2pi e}right)}^{2n} nα, modulo O(1/n) corrections, with alpha =frac{1}{2} dim left(mathfrak{g}right) for a gauge algebra mathfrak{g} and a universal function F∞(λ). As a by-product we find several new formulas both for the partition function as well as for perturbative correlators in mathcal{N}=2kern0.5em mathfrak{s}mathfrak{u}(N) gauge theory with 2N fundamental hypermultiplets.

Wilson loops and its correlators with chiral operators in mathcal{N} = 2, 4 SCFT at large N

In this paper we compute the vacuum expectation value of the Wilson loop and its correlators with chiral primary operators in mathcal{N} = 2, 4 superconformal U(N ) gauge theories at large N . After localization these quantities can be computed in terms of a deformed U(N ) matrix model. The Wilson loops we deal with are in the fundamental and symmetric representations.

Exact Holography of Massive M2-brane Theories and Entanglement Entropy

We test the gauge/gravity duality between the N = 6 mass-deformed ABJM theory with Uk(N) × U-k(N) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)=ℤk × SO(4)=ℤk isometry. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein (KK) holography to the LLM geometries. We focus on the chiral primary operator (CPO) with conformal dimension Δ = 1. The non-vanishing vacuum expectation value (vev) implies the breaking of conformal symmetry. In that case, we show that the variation of the holographic entanglement entropy (HEE) from it’s value in the CFT, is related to the non-vanishing one-point function due to the relevant deformation as well as the source field. Applying Ryu Takayanagi’s HEE conjecture to the 4-dimensional gravity solutions, which are obtained from the KK reduction of the 11-dimensional LLM solutions, we calculate the variation of the HEE. We show how the vev and the value of the source field determine the HEE.

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in {mathcal {N}} = 6 U_k(N)times mathrm{U}_{-k}(N) mass-deformed ABJM theory and the LLM geometries with {mathbb {Z}}_k orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension Delta = 1, which is given by langle {mathcal {O}}^{(Delta =1)}rangle = N^{frac{3}{2}} , f_{(Delta =1)}, for large N and k=1. Here the factor f_{(Delta )} is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of kne 1 for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

Mass-deformed ABJM theory and LLM geometries: exact holography

We present a detailed account and extension of our claim in arXiv:1610.01490. We test the gauge/gravity duality between the mathcal{N} = 6 mass-deformed ABJM theory with Uk(N )×U−k(N ) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/ℤk × SO(4)/ℤk isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with conformal dimension Δ = 1. We show that leftlangle {mathcal{O}}^{left(varDelta =1right)}rightrangle ={N}^{frac{3}{2}}{f}_{left(varDelta =1right)} for all supersymmetric vacuum solutions and LLM geometries with k = 1, where the factor f(Δ) is independent of N. We also confirm that the vacuum expectation value of the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of k ≠ 1 for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the mathcal{N} = 4 super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.

Large-N correlation functions in mathcal{N} = 2 superconformal QCD

We study extremal correlation functions of chiral primary operators in the large-N SU(N ) mathcal{N} = 2 superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large-N matrix model free energy given by the four-sphere partition function. Special emphasis is given to singletrace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak ’t Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.

Exactly marginal deformations from exceptional generalised geometry

We apply exceptional generalised geometry to the study of exactly marginal deformations of mathcal{N} = 1 SCFTs that are dual to generic AdS5 flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries.Our analysis holds for any mathcal{N} = 2 AdS5 flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.

Operator mixing in large N superconformal field theories on S4 and correlators with Wilson loops

We find a general formula for the operator mixing on the $\mathbb{S}^4$ of chiral primary operators (CPO) for the ${\cal N}=4$ theory at large $N$ in terms of Chebyshev polynomials. As an application, we compute the correlator of a CPO and a Wilson loop, reproducing an earlier result by Giombi and Pestun obtained from a two-matrix model proposal. Finally, we discuss a simple method to obtain correlators in general ${\cal N}=2$ superconformal field theories in perturbation theory in terms of correlators of the ${\cal N}=4$ theory.

Search for a Minimal N=1 Superconformal Field Theory in 4D.

We discuss a candidate for a minimal interacting four-dimensional N=1 superconformal field theory. The model contains a chiral primary operator u satisfying the chiral ring relation u^{2}=0, and its scaling dimension is Δ(u)=1.5. The model is derived by turning on a N=1 preserving deformation of N=2 A_{2} Argyres-Douglas theory. The central charges are given by (a,c)=(263/768,271/768)≃(0.342,0.353). There is no moduli space of vacua, no flavor symmetry, and the chiral ring is finite.

Large N correlation functions in superconformal field theories

We compute correlation functions of chiral primary operators in N=2 superconformal theories at large N using a construction based on supersymmetric localization recently developed by Gerchkovitz et al. We focus on N=4 SYM as well as on superconformal QCD. In the case of N=4 we recover the free field theory results as expected due to non-renormalization theorems. In the case of superconformal QCD we study the planar expansion in the large N limit. The final correlators admit a simple generalization to a finite N formula which exactly matches the various small $N$ results in the literature.

On superconformal anyons

In d=2+1 dimensions, there exist field theories which are non-relativistic and superconformal. These theories describe two species of anyons, whose spins differ by 1/2, interacting in a harmonic trap. We compute the dimensions of chiral primary operators. These operators receive large anomalous dimensions which are related to the unusual angular momentum properties of anyons. Surprisingly, we find that the dimensions of some chiral primary operators violate the unitarity bound and we trace this to the fact that the associated wavefunctions become non-normalisable. We also study BPS non-perturbative states in this theory: these are Jackiw-Pi vortices. We show that these emerge at exactly the point where perturbative operators hit the unitarity bound. To describe the low-energy dynamics of these vortices, we construct a novel type of supersymmetric gauged linear sigma model.

On exact correlation functions in SU(N) N = 2 $$ \mathcal{N}=2 $$ superconformal QCD

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.

Generalized bootstrap equations for N = 4 $$ \mathcal{N}=4 $$ SCFT

We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N=4 superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.

An extremal chiral primary three-point function at two-loops in ABJ(M)

I compute the leading correction to the structure constant for the three-point function of two length-two and one length-four chiral primary operators in planar ABJ(M) theory at weak 't Hooft coupling. The computation is reduced to four-loop propagator type Feynman integrals via a manifestly finite integration over the position of the length-four operator.

Entanglement entropy and D1-D5 geometries

In Conformal Field Theories with a gravitational AdS dual it is possible to calculate the entanglement entropy of a region $A$ holographically by using the Ryu-Takayanagi formula. In this work we consider systems that are in a pure state that is not the vacuum. We study in particular the 2D Conformal Field Theory dual to type IIB string theory on AdS$_3 \times S^3 \times T^4$ and focus on the $1/4$-BPS states described holographically by the 2-charge microstate geometries. We discuss a general prescription for the calculation of the entanglement entropy in these geometries that are asymptotically AdS$_3 \times S^3$. In particular we study analytically the perturbative expansion for a single, short interval: we show that the first non-trivial terms in this expansion are consistent with the expected CFT structure and with previous results on the vevs of chiral primary operators for the $1/4$-BPS configurations.

Semiclassical strings and non-Abelian T-duality

We study semiclassical strings in the Klebanov–Witten and in the non-Abelian T-dual Klebanov–Witten backgrounds. We show that both backgrounds share a subsector of equivalent states up to conditions on the T-dual coordinates. We also analyse string configurations where the strings are stretched along the T-dual coordinates. This semiclassical analysis predicts the existence of (almost) chiral primary operators for the dual superconformal field theory whose (anomalous) bare dimensions depend on the T-dual coordinates. We briefly discuss the Penrose limit of the dualised background.

ABJ(M) chiral primary three-point function at two-loops

We compute the leading correction to the structure constant for the three-point function of length-two chiral primary operators in planar ABJ(M) theory at weak 't Hooft coupling.

Bounds on operator dimensions in 2D conformal field theories

We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $\Delta_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary operators. The bound we find is of the same form as found for $\Delta_1$: $\Delta_2 \leq c_{tot}/12 + O(1)$. We find a similar bound on the conformal dimension $\Delta_3$, and present a method for deriving bounds on $\Delta_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that $\Delta_n \leq \frac{c_{tot}}{12}+O(1)$. We conclude with a brief discussion of the gravitational implications of these results.

The role of leading twist operators in the Regge and Lorentzian OPE limits

We study two kinematical limits, the Regge limit and the Lorentzian OPE limit, of the four-point function of the stress-tensor multiplet in Super Yang-Mills at weak coupling. We explain how both kinematical limits are controlled by the leading twist operators. We use the known expression of the four-point function up to three loops, to extract the pomeron residue at next-to-leading order. Using this data and the known form of pomeron spin up to next-to-leading order, we predict the behaviour of the four-point function in the Regge limit at higher loops. Specifically, we determine the leading log behaviour at any loop order and the next-to-leading log at four loops. Finally, we check the consistency of our results with conformal Regge theory. This leads us to predict the behaviour around $J=1$ of the OPE coefficient of the spin $J$ leading twist operator in the OPE of two chiral primary operators.

Simplifying instanton corrections to $ \mathcal{N} $ = 4 SYM correlators

We compute instanton corrections to non-minimal correlation functions of chiral primary operators in the N=4 super Yang-Mills super-current multiplet. Using a representation in terms of Mellin integrals, we find that these corrections can be written as conformal integrals in AdS_5 times certain "kinematic" prefactors that are independent of the instanton moduli. We then consider the consecutive, pairwise light-like limit x_{i, i+1}^2 -> 0 of such correlators, and prove that the ratio between the instanton contribution and the corresponding tree-level expression vanishes in this limit. We also speculate on the extension to the non-perturbative level of the proposed duality with polygonal Wilson loops and briefly discuss its potential implications for scattering amplitudes.