Hooft Coupling Research Articles

Fermi seas from Bose condensates in Chern-Simons matter theories and a bosonic exclusion principle

We generalize previously obtained results for the (all orders in the ’t Hooft coupling) thermal free energy of bosonic and fermionic large N Chern-Simons theories with fundamental matter, to values of the chemical potential larger than quasiparticle thermal masses. Building on an analysis by Geracie, Goykhman and Son, we present a simple explicit formula for the occupation number for a quasiparticle state of any given energy and charge as a function of the temperature and chemical potential. This formula is a generalization to finite ’t Hooft coupling of the famous occupation number formula of Bose-Einstein statistics, and implies an exclusion principle for Chern-Simons coupled bosons: the total number of bosons occupying any particular state cannot exceed the Chern-Simons level. Specializing our results to zero temperature we construct the phase diagrams of these theories as a function of chemical potential and the UV parameters. At large enough chemical potential, all the bosonic theories we study transit into a compressible Bose condensed phase in which the runaway instability of free Bose condensates is stabilized by the bosonic exclusion principle. This novel Bose condensate is dual to — and reproduces the thermodynamics of — the fermionic Fermi sea.

Toverline{T} -deformation of q-Yang-Mills theory

We derive the Toverline{T} -perturbed version of two-dimensional q-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the Toverline{T} -deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large N factorization into chiral and anti-chiral sectors. For the U(N) gauge theory on the sphere, we show that the large N phase transition persists, and that it is of third order and induced by instantons. The effect of the Toverline{T} -deformation is to decrease the critical value of the ’t Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for (q, t)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large N limit of Yang-Mills theory, showing that the Toverline{T} -deformation decreases the contribution of the Boltzmann entropy.

Unitarity Entropy Bound: Solitons and Instantons

AbstractWe show that non‐perturbative entities such as solitons and instantons saturate bounds on entropy when the theory saturates unitarity. Simultaneously, the entropy becomes equal to the area of the soliton/instanton. This is strikingly similar to black hole entropy despite absence of gravity. We explain why this similarity is not an accident. We present a formulation that allows to apply the entropy bound to instantons. The new formulation also eliminates apparent violations of the Bekenstein entropy bound by some otherwise‐consistent unitary systems. We observe that in QCD, an isolated instanton of fixed size and position violates the entropy bound for strong 't Hooft coupling. At critical 't Hooft coupling the instanton entropy is equal to its area.

Bootstrapping conformal four-point correlators with slightly broken higher spin symmetry and 3D bosonization

Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the planar four-point functions at arbitrary ’t Hooft coupling λ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the λ expansions of the correlation functions. The λ expansions terminate at order λ2 and the solutions are exact in λ. Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the 3D Chern-Simons vector models and their holographic duals in AdS4. Besides, we show that the same results can also be obtained rather simply from bosonization duality of 3D Chern-Simons vector models. The odd term at order O(λ) in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the 3D bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.

Mathcal{N} = 2 Conformal SYM theories at large mathcal{N}

We consider a class of mathcal{N} = 2 conformal SU(N) SYM theories in four dimensions with matter in the fundamental, two-index symmetric and anti-symmetric representations, and study the corresponding matrix model provided by localization on a sphere S4, which also encodes information on flat-space observables involving chiral operators and circular BPS Wilson loops. We review and improve known techniques for studying the matrix model in the large-N limit, deriving explicit expressions in perturbation theory for these observables. We exploit both recursive methods in the so-called full Lie algebra approach and the more standard Cartan sub-algebra approach based on the eigenvalue distribution. The sub-class of conformal theories for which the number of fundamental hypermultiplets does not scale with N differs in the planar limit from the mathcal{N} = 4 SYM theory only in observables involving chiral operators of odd dimension. In this case we are able to derive compact expressions which allow to push the small ’t Hooft coupling expansion to very high orders. We argue that the perturbative series have a finite radius of convergence and extrapolate them numerically to intermediate couplings. This is preliminary to an analytic investigation of the strong coupling behavior, which would be very interesting given that for such theories holographic duals have been proposed.

Multi-particle finite-volume effects for hexagon tessellations

Correlation functions of gauge-invariant composite operators in mathcal{N} = 4 super Yang-Mills theory can be computed by integrability using triangulations. The elementary tile in this process is the hexagon, which should be glued by appropriately inserting resolutions of the identity involving virtual (“mirror”) magnons. We consider this problem for five-point functions of protected operators. At one-loop in the ’t Hooft coupling, it is necessary to glue three adjacent tiles which involves two virtual magnons scattering among each other. We show that the result can be simplified by using an adapted mirror rotation and employing appropriate summation techniques. The mirror-particle contributions then yield hyperlogarithms of weight two. Finally, we use these results to investigate braiding prescriptions introduced in earlier work on the problem.

Non-perturbative defect one-point functions in planar [formula omitted] super-Yang-Mills

The four dimensional N=4 super-Yang-Mills (SYM) theory exhibits rich dynamics in the presence of codimension-one conformal defects. The new structure constants of the extended operator algebra consist of one-point functions of local operators which are nonvanishing due to the defect insertion and carry nontrivial coupling dependence. We study an important class of half-BPS superconformal defects engineered by D5 branes that share three common directions with the D3 branes and involve Nahm pole configurations for the SYM fields on the D3 brane worldvolume. In the planar large N limit, we obtain non-perturbative results in the 't Hooft coupling λ for the defect one-point functions of both BPS and non-BPS operators, building upon recent progress in localization [1] and integrability methods [2,3].For BPS operator insertions in the SYM with D5-brane type boundary or interface, we derive an effective two dimensional defect-Yang-Mills (dYM) theory from supersymmetric localization, which gives an efficient way to extract defect observables and generates a novel matrix model for the defect one-point function. By solving the matrix model in the large N limit, we obtain exact results in λ which interpolate between perturbative Feynman diagram contributions in the weak coupling limit and IIB string theory predictions on AdS5×S5 in the strong coupling regime, providing a precision test of AdS/CFT with interface defects. For general non-BPS operators, we develop a non-perturbative bootstrap-type program for integrable boundary states on the worldsheet of the IIB string theory, corresponding to the interface defects in the planar SYM. Such integrable boundary states are constrained by a set of general consistency conditions for which we present explicit solutions that reproduce and extend the known results at weak coupling from integrable spin-chain methods.

The QCD trace anomaly at strong coupling from M-theory

Obtaining a lattice-consistent result for the temperature dependence of the QCD conformal anomaly from a top-down M-theory dual (valid) for all temperatures – both, T<T_c and T>T_c – of thermal QCD at intermediate gauge coupling, has been missing in the literature. We fill this gap by addressing this issue from the M-theory uplift of the SYZ type IIA mirror at intermediate gauge/string coupling [both obtained in Dhuria et al. (JHEP 1311:001, 2013)] of the UV-complete type IIB holographic dual of large-N thermal QCD of Mia et al. (Nucl Phys B 839:187, 2010), and comparing with the very recent lattice results of Bazavov et al. (Phys Rev D 97(1):014510, 2018). Estimates of the mathcal{O}(R^4) higher derivative corrections in the D=11 supergravity action relevant to considering the aforementioned M theory uplift in the intermediate ’t Hooft coupling (in addition to gauge coupling) limit, are also presented. We also show that after a tuning of the (small) Ouyang embedding parameter and radius of a blown-up S^2 when expressed in terms of the horizon radius, a QCD deconfinement temperature T_c=150 MeV from a Hawking–Page phase transition at vanishing baryon chemical potential consistent with lattice QCD in the heavy-quark limit, can be obtained.

Octagon at finite coupling

We study a special class of four-point correlation functions of infinitely heavy half-BPS operators in planar mathcal{N} = 4 SYM which admit factorization into a product of two octagon form factors. We demonstrate that these functions satisfy a system of nonlinear integro-differential equations which are powerful enough to fully determine their dependence on the ’t Hooft coupling and two cross ratios. At weak coupling, solution to these equations yields a known series representation of the octagon in terms of ladder integrals. At strong coupling, we develop a systematic expansion of the octagon in the inverse powers of the coupling constant and calculate accompanying expansion coefficients analytically. We examine the strong coupling expansion of the correlation function in various kinematical regions and observe a perfect agreement both with the expected asymptotic behavior dictated by the OPE and with results of numerical evaluation. We find that, surprisingly enough, the strong coupling expansion is Borel summable. Applying the Borel-Padé summation method, we show that the strong coupling expansion correctly describes the correlation function over a wide region of the ’t Hooft coupling.

All order exact result for the anomalous dimension of the scalar primary in Chern-Simons vector models

We present a conjecture for the leading $1/N$ anomalous dimension of the scalar primary operator in $U(N{)}_{k}$ Chern-Simons theories coupled to a single fundamental field, to all orders in the t'Hooft coupling $\ensuremath{\lambda}=\frac{N}{k}$. Following this we compute the anomalous dimension of the scalar in a Regular Bosonic theory perturbatively at two-loop order and demonstrate that matches exactly with the result predicted by our conjecture. We also show that our proposed expression for the anomalous dimension is consistent with all other existing two-loop perturbative results, which constrain its form at both weak and strong coupling thanks to the bosonization duality. Furthermore, our conjecture passes a novel nontrivial all loop test which provides a strong evidence for its consistency.

On bulk viscosity at weak and strong ’t Hooft couplings

Bulk viscosity is an important transport coefficient that exists in the hydrodynamical limit only when the underlying theory is non-conformal. One example being thermal QCD with large number of colors. We study bulk viscosity in such a theory at low energies and at weak and strong ’t Hooft couplings when the temperature is above the deconfinement temperature. The weak coupling analysis is based on Boltzmann equation from kinetic theory whereas the strong coupling analysis uses non-conformal holographic techniques from string and M-theories. Using these, many properties associated with bulk viscosity may be explicitly derived.

Vacuum energy of the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$ in the $1/N$ expansion

Abstract By employing the $1/N$ expansion, we compute the vacuum energy $E(\delta\epsilon)$ of the two-dimensional supersymmetric (SUSY) $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions to the second order in a SUSY-breaking parameter $\delta\epsilon$. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter $\delta\epsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the $S^1$, $R$, $RE(\delta\epsilon)$ behaves as inverse powers of $\Lambda R$ for $\Lambda R$ small, where $\Lambda$ is the dynamical scale. Since $\Lambda$ is related to the renormalized ’t Hooft coupling $\lambda_R$ as $\Lambda\sim e^{-2\pi/\lambda_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in $\lambda_R$.

Quiver CFT at strong coupling

The circular Wilson loop in the two-node quiver CFT is computed at large-N and strong ’t Hooft coupling by solving the localization matrix model.

The planar limit of mathcal{N} = 2 superconformal field theories

We obtain the perturbative expansion of the free energy on S4 for four dimensional Lagrangian mathcal{N} = 2 superconformal field theories, to all orders in the ’t Hooft coupling, in the planar limit. We do so by using supersymmetric localization, after rewriting the 1-loop factor as an effective action involving an infinite number of single and double trace terms. The answer we obtain is purely combinatorial, and involves a sum over tree graphs. We also apply these methods to the perturbative expansion of the free energy at finite N , and to the computation of the vacuum expectation value of the 1/2 BPS circular Wilson loop, which in the planar limit involves a sum over rooted tree graphs.

Exact null octagon

We consider the so-called simplest correlation function of four infinitely heavy half-BPS operators in planar mathcal{N} = 4 SYM in the limit when the operators are light-like separated in a sequential manner. We find a closed-form expression for the correlation function in this limit as a function of the ’t Hooft coupling and residual cross ratios. Our analysis heavily relies on the factorization of the correlation function into the product of null octagons and on the recently established determinant representation for the latter. We show that the null octagon is given by a Fredholm determinant of a certain integral operator which has a striking similarity to those previously encountered in the study of two-point correlation functions in exactly solvable models at finite temperature and of level spacing distributions in random matrices. This allows us to compute the null octagon exactly by employing a method of differential equations.

Deuteron electric dipole moment from holographic QCD

We compute the electric dipole moment (EDM) of the deuteron in the holographic QCD model of Witten-Sakai-Sugimoto. Previously, the leading contribution to the EDM of nucleons was computed, finding opposite values for the proton and the neutron which then cancel each other in the deuteron state. Here we compute the next-to-leading order contribution which provides a splitting between their absolute value. At large $N_{\rm c}$ and large 't Hooft coupling $\lambda$, nuclei are bound states of almost isolated nucleons. In particular, we find that in this limit the deuteron EDM is given by the splitting between proton and neutron EDMs. Our estimate for the deuteron EDM extrapolated to the physical values of $N_{\rm c}$, $\lambda$, $M_{\rm KK}$ and $m_q$ is $\mathcal{D}_D = -0.92\times 10^{-16} \theta\ e \cdot {cm}$. This is consistent, in sign and magnitude, with results found previously in the literature and obtained using completely different methods.

Correlation functions in mathcal{N} = 2 Supersymmetric vector matter Chern-Simons theory

Correlation functions of the higher-spin current operators in large N Chern-Simons theories are important to understand approximate higher-spin symmetries in these theories. Moreover, they also provide stronger checks for conjectured dualities in these theories. In this paper, we compute the two, three and four-point functions of the operators in the spin zero multiplet of mathcal{N} = 2 Supersymmetric vector matter Chern-Simons theory at large N to all orders of ’t Hooft coupling. While the two- and three-point functions are computed by solving the Schwinger-Dyson equation, this method becomes intractable for the computation of the four-point functions. Thereby, we use bootstrap method to evaluate four-point function of scalar operator {J}_0^f=overline{psi}psi and {J}_0^b=overline{phi}phi . Interestingly, because leftlangle {J}_0^f{J}_0^f{J}_0^brightrangle is a contact term, the four point function of {J}_0^f operator resembles the corresponding correlation function in the free theory, up to overall coupling constant dependent factors and up to some ‘bulk AdS’ contact terms. On the other hand the {J}_0^b four-point function receives an additional contribution compared to the free theory expression due to the {J}_0^f exchange. We find that the double discontinuity of this single trace operator {J}_0^f vanishes and hence it only contributes to AdS-contact term.

Scattering amplitudes in mathcal{N} = 3 supersymmetric SU(N) Chern-Simons-matter theory at large N

We study mathcal{N} = 3 supersymmetric Chern-Simons-matter theory coupled to matter in the fundamental representation of SU(N). In the ’t Hooft large N limit, we compute the exact 2 → 2 scattering amplitudes of the fundamental scalar superfields to all orders in the ’t Hooft coupling λ. Our computations are presented in mathcal{N} = 1 superspace and make significant use of the residual SO(2)R symmetry in order to solve for the exact four-point correlator of the scalar superfields. By taking the on-shell limit, we are able to extract the exact 2 → 2 scattering amplitudes of bosons/fermions in the symmetric, anti-symmetric and adjoint channels of scattering. We find that the scattering amplitude of the mathcal{N} = 3 theory in the planar limit is tree-level exact to all orders in the ’t Hooft coupling λ. The result is consistent with the conjectured bosonization duality and is expected to have enhanced symmetry structures such as dual superconformal symmetry and Yangian symmetry.

Genus-2 holographic correlator on AdS5\xd7 S5 from localization

We consider the four-point function of the stress tensor multiplet superprimary in mathcal{N} = 4 super-Yang-Mills (SYM) with gauge group SU(N ) in the large N and large ’t Hooft coupling lambda equiv {g}_{mathrm{YM}}^2N limit, which is holographically dual to the genus expansion of IIB string theory on AdS5× S5. In [1] it was shown that the integral of this correlator is related to derivatives of the mass deformed mathcal{N} = 2∗ sphere free energy, which was computed using supersymmetric localization to leading order in 1/N2 for finite λ. We generalize this computation to any order in 1/N2 for finite λ using topological recursion, and use this any order constraint to fix the R4 correction to the holographic correlator to any order in the genus expansion. We also use it to complete the derivation of the 1-loop supergravity correction, and show that analyticity in spin fails at zero spin in the large N expansion as predicted from the Lorentzian inversion formula. In the flat space limit, the R4 term in the holographic correlator matches that of the IIB S-matrix in 10d, which is a precise check of AdS5/CFT4 for local operators at genus-one. Using the flat space limit and localization we then fix D4R4 in the holographic correlator to any order in the genus expansion, which is nontrivial at genus-two, i.e. 1/N6. This is the first result at two orders beyond the planar limit at strong coupling for a holographic correlator.

Mathcal{N} = 2 conformal gauge theories at large R-charge: the SU(N) case

Conformal theories with a global symmetry may be studied in the double scaling regime where the interaction strength is reduced while the global charge increases. Here, we study generic 4d mathcal{N} = 2 SU(N ) gauge theories with conformal matter content at large R-charge QR → ∞ with fixed ’t Hooft-like coupling kappa ={Q}_{mathrm{R}}{g}_{mathrm{YM}}^2. Our analysis concerns two distinct classes of natural scaling functions. The first is built in terms of chiral/anti-chiral two-point functions. The second involves one-point functions of chiral operators in presence of frac{1}{2} -BPS Wilson-Maldacena loops. In the rank-1 SU (2) case, the two-point sector has been recently shown to be captured by an auxiliary chiral random matrix model. We extend the analysis to SU(N) theories and provide an algorithm that computes arbitrarily long perturbative expansions for all considered models, parametric in the rank. The leading and next-to-leading contributions are cross-checked by a three- loops computation in mathcal{N} = 1 superspace. This perturbative analysis identifies maximally non-planar Feynman diagrams as the relevant ones in the double scaling limit. In the Wilson-Maldacena sector, we obtain closed expressions for the scaling functions, valid for any rank and κ. As an application, we analyze quantitatively the large ’t Hooft coupling limit κ ≫ 1 where we identify all perturbative and non-perturbative contributions. The latter are associated with heavy electric BPS states and the precise correspondence with their mass spectrum is clarified.