Gauge Theory Operators Research Articles

Integrability in dipole-deformed super Yang–Mills**Dedicated to the memory of Petr Petrovich Kulish.

We study the null dipole deformation of super Yang–Mills theory, which is an example of a potentially solvable ‘dipole CFT’: a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrödinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld–Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrödinger holography.

Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identified with refined topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.

Decay constants in soft wall AdS/QCD revisited

Phenomenological AdS/QCD models, like hard wall and soft wall, provide hadronic mass spectra in reasonable consistency with experimental and (or) lattice results. These simple models are inspired in the AdS/CFT correspondence and assume that gauge/ gravity duality holds in a scenario where conformal invariance is broken through the introduction of an energy scale. Another important property of hadrons: the decay constant, can also be obtained from these models. However, a consistent formulation of an AdS/QCD model that reproduces the observed behavior of decay constants of vector meson excited states is still lacking. In particular: for radially excited states of heavy vector mesons, the experimental data lead to decay constants that decrease with the radial excitation level. We show here that a modified framework of soft wall AdS/QCD involving an additional dimensionfull parameter, associated with an ultraviolet energy scale, provides decay constants decreasing with radial excitation level. In this version of the soft wall model the two point function of gauge theory operators is calculated at a finite position of the anti-de Sitter space radial coordinate.

A note on instanton operators, instanton particles, and supersymmetry

We clarify certain aspects of instanton operators in five-dimensional super-symmetric gauge theories. In particular, we show how, in the pointlike limit, they become supersymmetric and provide the natural bridge with the instantonic states contributing to the index as well as with the zero mode counting leading to broken current multiplets.

Line defects and 5d instanton partition functions

We consider certain line defect operators in five-dimensional SUSY gauge theories, whose interaction with the self-dual instantons is described by 1d ADHM-like gauged quantum mechanics constructed by Tong and Wong. The partition function in the presence of these operators is known to be a generating function of BPS Wilson loops in skew symmetric tensor representations of the gauge group. We calculate the partition function and explicitly prove that it is a finite polynomial of the defect mass parameter $x$, which is an essential property of the defect operator and the Wilson loop generating function. The relation between the line defect partition function and the qq-character defined by N. Nekrasov is briefly discussed.

Instanton operators in five-dimensional gauge theories

We discuss instanton operators in five-dimensional gauge theories. These are defined as disorder operators which create a non-vanishing second Chern class on a foursphere surrounding their insertion point. As such they may be thought of as higherdimensional analogues of three-dimensional monopole (or ‘t Hooft) operators. We argue that they play an important role in the enhancement of the Lorentz symmetry for maximally supersymmetric Yang-Mills to SO(1, 5) at strong coupling.

An operator product expansion for the mutual information in AdS/CFT

We investigate the behavior of the mutual information IAB between two “small” and wide separated spherical regions A and B in the N=4 SYM gauge theory dual to Type IIB string theory in AdS5×S5. To this end, the mutual information is recasted in terms of correlators of surface operators W(Σ) defined along a surface Σ within the boundary gauge theory. This construction relies on the strong analogies between the twist-field operators appearing in the replica trick method used for the computation of the entanglement entropy, and the disorder-like surface operators in gauge theories. In the AdS/CFT correspondence, a surface operator W(Σ) corresponds to having a D3-brane in AdS5×S5 ending on the boundary along the prescribed surface Σ. Then, a long distance expansion for IAB is provided. The coefficients of the expansion appear as a byproduct of the operator product expansion for the correlators of the operators W(Σ) with the chiral primaries of the theory. We find that, while undergoing a phase transition at a critical distance, the holographic mutual information, instead of strictly vanishing, decays with a power law whose leading contributions of order O(N0), originate from the exchange of pairs of the lightest bulk particles between A and B. These particles correspond to operators in the boundary field theory with the smallest scaling dimensions.

Large J expansion in ABJM theory revisited.

Recently there has been progress in the computation of the anomalous dimensions of gauge theory operators at strong coupling by making use of the AdS/CFT correspondence. On the string theory side they are given by dispersion relations in the semiclassical regime. We revisit the problem of a large-charge expansion of the dispersion relations for simple semiclassical strings in an mathrm{AdS}_4times {mathbb {CP}}^3 background. We present the calculation of the corresponding anomalous dimensions of the gauge theory operators to an arbitrary order using three different methods. Although the results of the three methods look different, power series expansions show their consistency.

Anomalous dimensions from rotating open strings in AdS/CFT

We propose a new entry within the dictionary of the AdS/CFT duality at strong coupling: in the limit of a large spin or a large R-charge, the anomalous dimension of the gauge theory operator dual to a semiclassical rotating string is proportional to the string proper length. This conjecture is motivated by a generalization to strings of the rule for computing anomalous dimensions of massive particles and supergravity fields in the anti-de Sitter space. We show that this proportionality holds for a rotating closed string in global AdS space, representing a high spin operator made of fields in the adjoint representation. It is also valid for closed strings rotating in $S^5$ (representing operators with large R-charge), for closed strings with multiple AdS spin, and for giant magnons. Based on this conjecture, we calculate the anomalous dimension $\delta$ of operators made of fields in the fundamental representation, associated with high spin mesons, and which are represented by rotating open strings attached to probe D7-branes. The result is a logarithmic dependence upon the spin, $\delta\sim \sqrt{\lambda}\ln S$, similar to the closed string case. We show that the operator properties --- anomalous dimension and spin --- are obtained from measurements made by a local observer in the anti-de Sitter space. For the open string case, this ensures that these quantities are independent of the mass scale introduced by the D7-branes (the quark mass), as expected on physical grounds. In contrast, properties of the gauge theory states, like the energy, correspond to measurements by a gauge theory observer and depend upon the mass scale --- once again, as expected.

Anomalous dimensions and scalar glueball spectroscopy in AdS/QCD

An extended version of the AdS/QCD Soft-Wall model that incorporates QCD-like anomalous contributions to the dimensions of gauge theory operators is proposed. This exploratory approach leads to a relation between scalar glueball masses and beta functions. Using this relation, properties of the glueball mass spectroscopy that emerge from phenomenological beta functions proposed in the literature are investigated. The reverse problem is also considered: starting from a linear Regge trajectory which fits the lattice glueball masses, beta functions with different asymptotic infrared behaviours are found. Remarkably, some of them present a fixed point at finite coupling.

Quivers as calculators: counting, correlators and Riemann surfaces

The spectrum of chiral operators in supersymmetric quiver gauge theories is typically much larger in the free limit, where the superpotential terms vanish. We find that the finite N counting of operators in any free quiver theory, with a product of unitary gauge groups, can be described by associating Young diagrams and Littlewood-Richardson multiplicities to a simple modification of the quiver, which we call the split-node quiver. The large N limit leads to a surprisingly simple infinite product formula for counting gauge invariant operators, valid for any quiver with bifundamental fields. An orthogonal basis for the operators, in the finite N CFT inner product, is given in terms of quiver characters. These are constructed by inserting permutations in the split-node quivers and interpreting the resulting diagrams in terms of symmetric group matrix elements and branching coefficients. The fusion coefficients in the chiral ring - valid both in the UV and in the IR - are computed at finite N. The derivation follows simple diagrammatic moves on the quiver. The large N counting and correlators are expressed in terms of topological field theories on Riemann surfaces obtained by thickening the quiver. The TFTs are based on symmetric groups and defect observables associated with subgroups play an important role. We outline the application of the free field results to the construction of BPS operators in the case of non-zero super-potential.

Holographic Kondo model in various dimensions

We study the addition of localised impurities to U(N) Supersymmetric Yang-Mills theories in (p + 1)-dimensions by using the gauge/gravity correspondence. From the gravity side, the impurities are introduced by considering probe D(8 − p)-branes extending along the time and radial directions and wrapping an (7 − p)-dimensional submanifold of the internal (8 − p)-sphere, so that the degrees of freedom are point-like from the gauge theory perspective. We analyse both the configuration in which the branes generate straight flux tubes — corresponding to actual single impurities — and the one in which connected flux tubes are created — corresponding to dimers. We discuss the thermodynamics of both the configurations and the related phase transition. In particular, the specific heat of the straight flux-tube configuration is negative for p < 3, while it is never the case for the connected one. We study the stability of the system by looking at the impurity fluctuations. Finally, we characterise the theory by computing one- and two-point correlators of the gauge theory operators dual to the impurity fluctuations. Because of the underlying generalised conformal structure, such correlators can be expressed in terms of an effective coupling constant (which runs because of its dimensionality) and a generalised conformal dimension.

Fine structure of the string spectrum in AdS 5 × S 5

The spectrum of an infinite spinning string in AdS(5) does not precisely match the spectrum of dual gauge theory operators, interpolated to the strong coupling regime with the help of Bethe-ansatz equations. We show that the mismatch is due to interactions in the string sigma-model which cannot be neglected even at asymptotically large 't Hooft coupling.

’t Hooft operators in gauge theory from Toda CFT

We construct loop operators in two dimensional Toda CFT and calculate with them the exact expectation value of certain supersymmetric 't Hooft and dyonic loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and \Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly speculate on the Toda CFT realization of arbitrary loop operators in these gauge theories in terms of topological web operators in Toda CFT.

More three-point correlators of giant magnons with finite size

In the framework of the semiclassical approach, we compute the normalized structure constants in three-point correlation functions, when two of the vertex operators correspond to heavy string states, while the third vertex corresponds to a light state. This is done for the case when the heavy string states are finite-size giant magnons with one or two angular momenta, and for two different choices of the light state, corresponding to dilaton operator and primary scalar operator. The relevant operators in the dual gauge theory are Tr(F_{\mu\nu}^2 Z^j+...) and Tr(Z^j). We first consider the case of AdS_5 x S^5 and N = 4 super Yang-Mills. Then we extend the obtained results to the gamma-deformed AdS_5 x S^5_\gamma, dual to N = 1 super Yang-Mills theory, arising as an exactly marginal deformation of N = 4 super Yang-Mills.

Semiclassical short strings in AdS5 × S 5

We present results for the one-loop correction to the energy of a class of string solutions in AdS_5 x S^5 in the short string limit. The computation is based on the observation that, as for rigid spinning string elliptic solutions, the fluctuation operators can be put into the single-gap Lame' form. Our computation reveals a remarkable universality of the form of the energy of short semiclassical strings. This may help to understand better the structure of the strong coupling expansion of the anomalous dimensions of dual gauge theory operators.

-algebras and surface operators in gauge theories

A general class of -algebras can be constructed from the affine sl(N) algebra by (quantum) Drinfeld–Sokolov reduction and are classified by partitions of N. Surface operators in an SU(N) 4D gauge theory are also classified by partitions of N. We argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding -algebra. We test this proposal by analysing the Polyakov–Bershadsky algebra obtaining results that are in agreement with the known partition functions for SU(3) gauge theories with a so-called simple surface operator. As a byproduct, our proposal implies relations between and algebras.

Exact computation of one-loop correction to the energy of pulsating strings in AdS5 × S5

In this paper, which is a sequel to Beccaria et al (2010 J. Phys. A: Math. Theor. 43 165402), we compute the one-loop correction to the energy of pulsating string solutions in AdS5 × S5. We show that, as for rigid spinning string elliptic solutions, the fluctuation operators for pulsating solutions can also be put into the single-gap Lamé form. A novel aspect of pulsating solutions is that the one-loop correction to their energy is expressed in terms of the stability angles of the quadratic fluctuation operators. We explicitly study the ‘short-string’ limit of the corresponding one-loop energies, demonstrating a certain universality of the form of the energy of ‘small’ semiclassical strings. Our results may help to shed light on the structure of strong-coupling expansion of anomalous dimensions of dual gauge theory operators.

On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT

We consider the 2-point function of string vertex operators representing string state with large spin in AdS_5. We compute this correlator in the semiclassical approximation and show that it has the expected (on the basis of state-operator correspondence) form of the strong-coupling limit of the 2-point function of single trace minimal twist operators in gauge theory. The semiclassical solution representing the stationary point of the path integral with two vertex operator insertions is found to be related to the large spin limit of the folded spinning string solution by a euclidean continuation, transformation to Poincare coordinates and conformal map from cylinder to complex plane. The role of the source terms coming from the vertex operator insertions is to specify the parameters of the solution in terms of quantum numbers (dimension and spin) of the corresponding string state. Understanding further how similar semiclassical methods may work for 3-point functions may shed light on strong-coupling limit of the corresponding correlators in gauge theory as was recently suggested by Janik et al in arXiv:1002.4613.

Strings on bubbling geometries

We study gauge theory operators which take the form of a product of a trace with a Schur polynomial, and their string theory duals. These states represent strings excited on bubbling AdS geometries which are dual to the Schur polynomials. These geometries generically take the form of multiple annuli in the phase space plane. We study the coherent state wavefunction of the lattice, which labels the trace part of the operator, for a general Young tableau and their dual description on the droplet plane with a general concentric ring pattern. In addition we identify a density matrix over the coherent states on all the geometries within a fixed constraint. This density matrix may be used to calculate the entropy of a given ensemble of operators. We finally recover the BMN string spectrum along the geodesic near any circle from the ansatz of the coherent state wavefunction.