We study four-dimensional Chern-Simons theory on $D \times \mathbb{C}$ (where $D$ is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an "analytically-continued" toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.
Recent progress in holographic correspondence uncovered remarkable relations between key characteristics of the theories on both sides of duality and certain integrable models. In this note we revisit the problem of the role of certain invariants in low-dimensional holography. As motivating example we consider first the entanglement entropy in 2d CFT and projective invariants. Next we consider higher projective invariants and suggest generalization to higher spin theories. Quadratic in invariants deformations is considered and conjectured to play role in low-dimensional higher spin holography.
Classical world-sheet string theory has recently been shown to be nonintegrable and chaotic in various confining string theory backgrounds -- the AdS soliton background in particular. In this paper we study a minisuperspace quantization of the theory and look at properties of the spectrum like the distribution of level spacing, which are indicative of quantum order or chaos. In the quantum spectrum we find a gradual transition from chaotic (Wigner GOE) to integrable (Poisson) regime as we look at higher energies. This is expected since our system is integrable asymptotically, and at higher ene…
We consider operators in ${\cal N}=4$ super Yang-Mills theory dual to closed string states propagating on a class of LLM geometries. The LLM geometries we consider are specified by a boundary condition that is a set of black rings on the LLM plane. When projected to the LLM plane, the closed strings are polygons with all corners lying on the outer edge of a single ring. The large $N$ limit of correlators of these operators receives contributions from non-planar diagrams even for the leading large $N$ dynamics. Our interest in these fluctuations is because a previous weak coupling analysis argu…
Over the last twenty years, work based on lattice supersymmetry has generated many new results and insights into the non-perturbative nature of string theory, quantum black holes, and gravity. This endeavor is a broad research program encompassing lattice field theory, supersymmetry, string theory, and quantum gravity. In this volume, we look at a selected subset of the topics covering recent progress in lattice supersymmetry and holography.
These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)
We review the study, on the lattice, of the Green-Schwarz gauge-fixed string action describing worldsheet fluctuations about the minimal surface holographically dual to the null cusp Wilson loop, useful to evaluate the cusp anomaly of N = 4 super Yang-Mills (sYM). We comment on discretization, numerical explorations and challenges for the non-perturbative study of this benchmark model of gauge-fixed worldsheet actions.
In this paper we use lattice simulation to study four dimensional $N=4$ super Yang-Mills (SYM) theory. We have focused on the three color theory on lattices of size $12^4$ and for 't Hooft couplings up to $λ=40.0$. Our lattice action is based on a discretization of the Marcus or GL twist of $N=4$ SYM and retains one exact supersymmetry for non-zero lattice spacing. We show that lattice theory exists in a single non-Abelian Coulomb phase for all 't Hooft couplings. Furthermore the static potential we obtain from correlators of Polyakov lines is in good agreement with that obtained from holograp…
Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in in…
We elucidate how integrable lattice models described by Costello's 4d Chern-Simons theory can be realized via a stack of D4-branes ending on an NS5-brane in type IIA string theory, with D0-branes on the D4-brane worldvolume sourcing a meromorphic RR 1-form, and fundamental strings forming the lattice. This provides us with a nonperturbative integration cycle for the 4d Chern-Simons theory, and by applying T- and S-duality, we show how the R-matrix, the Yang-Baxter equation and the Yangian can be categorified, that is, obtained via the Hilbert space of a 6d gauge theory.
This is a brief review of my work on the correspondence between four-dimensional $\mathcal{N} = 1$ supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction.
We investigate the behavior of energy momentum tensor correlators in strongly coupled large-N_c Yang-Mills theory at nonzero temperature, working within the Improved Holographic QCD model. In particular, we determine the spectral functions and corresponding imaginary time correlators in the bulk and shear channels, and compare the results to recent perturbative and lattice calculations where available. For the bulk channel imaginary time correlator, for which all three results exist, lattice data is seen to favor the holographic prediction over the perturbative one over a wide range of tempera…
We derive an explicit mapping between the spectra of conserved local operators of integrable quantum lattice models and the density distributions of their thermodynamic particle content. This is presented explicitly for the Heisenberg XXZ spin chain. As an application we discuss a quantum quench scenario, in both the gapped and critical regimes. We outline an exact technique which allows for an efficient implementation on periodic matrix product states. In addition, for certain simple product states we obtain analytic closed-form expressions in terms of solutions to Hirota functional relations…
It is argued that the supersymmetric index of a certain system of branes in M-theory is equal to the partition function of an integrable three-dimensional lattice model. The local Boltzmann weights of the lattice model satisfy a generalization of Zamolodchikov's tetrahedron equation. In a special case the model is described by a solution of the tetrahedron equation discovered by Kapranov and Voevodsky and by Bazhanov and Sergeev.
We consider maximally supersymmetric SU(N) Yang--Mills theory in Euclidean signature compactified on a flat two-dimensional torus with anti-periodic (`thermal') fermion boundary conditions imposed on one cycle. At large N, holography predicts that this theory describes certain black hole solutions in Type IIA and IIB supergravity, and we use lattice gauge theory to test this. Unlike the one-dimensional quantum mechanics case where there is only the dimensionless temperature to vary, here we emphasize there are two more parameters which determine the shape of the flat torus. While a rectangular…
Finite-density QCD is difficult to study numerically because of the sign problem. We prove that, in a certain region of the phase diagram, the phase quenched approximation is exact to O(Nf/Nc). It is true for any physical observables. We also consider the implications for the lattice simulations and find a quantitative evidence for the validity of the phase quenching from existing lattice QCD results at Nc=3. Our results show that the phase-quench approximation is rather good already at Nc=3, and the 1/Nc correction can be incorporated by the phase reweighting method without suffering from the…
Recently there has been a notable progress in the study of glueball states in lattice gauge theories, in particular extrapolating their spectrum to the limit of large number of colors $N$. In this note we compare the large $N$ lattice results with the holographic predictions, focusing on the Klebanov-Strassler model. We note that glueball spectrum demonstrates approximate universality across a range of gauge theory models. Because of this universality the holographic models can give reliable predictions for the spectrum of pure $SU(N)$ Yang-Mills theories with and without supersymmetry. This i…
A new class of $A^{(1)}_n$ integrable lattice models is presented. These are interaction-round-a-face models based on fundamental nimrep graphs associated with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each value of the rank and level. The Boltzmann weights are parameterized by elliptic theta functions and satisfy the Yang-Baxter equation for any fixed value of the elliptic nome q. At q=0, the models provide representations of the Hecke algebra and are expected to lead in the continuum limit to coset conformal field theories related to the $A^{(1)}_n$ conjugate modu…
We study the local lattice integrable regularization of the Sine-Gordon model written down in terms of the lattice Bose-operators. We show that the local spin Hamiltonian obtained from the six-vertex model with alternating inhomogeneities in fact leads to the Sine-Gordon model in the low-energy limit. We show that the Bethe Ansatz results for this model lead to the correct general relations for different critical exponents of the coupling constant.
We study the holographic field theory dual of a probe SU(2) Yang-Mills field in a background $(4+1)$-dimensional asymptotically Anti-de Sitter space. We find a new ground state when a magnetic component of the gauge field is larger than a critical value. The ground state forms a triangular Abrikosov lattice in the spatial directions perpendicular to the magnetic field. The lattice is composed of superconducting vortices induced by the condensation of a charged vector operator. We perform this calculation both at finite temperature and at zero temperature with a hard wall cutoff dual to a confi…
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N=3 the $τ$-function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.
Our goal is to develop a more general scheme for constructing integrable lattice regularisations of integrable quantum field theories. Considering the affine Toda theories as examples, we show how to construct such lattice regularisations using the representation theory of quantum affine algebras. This requires us to clarify in particular the relations between the light-cone approach to integrable lattice models and the representation theory of quantum affine algebras. Both are found to be related in a very natural way, suggesting a general scheme for the construction of generalised Baxter Q-o…
Recently a remarkable agreement was found between lattice simulations of long Wilson lines and behavior of the Nambu Goto string in flat space-time. However, the latter fails to fit the short distance behavior since it admits a tachyonic mode for a string shorter than a critical length. In this paper we examine the question of whether a classical holographic Wilson line can reproduce the lattice results for Wilson lines of any length. We determine the condition on the the gravitational background to admit a Coulombic potential at short distances. We analyze the system using three different ren…
We construct new trigonometric solutions of the Yang-Baxter equation, using the Fuss-Catalan algebras, a set of multi-colored versions of the Temperley-Lieb algebra, recently introduced by Bisch and Jones. These lead to new two-dimensional integrable lattice models, describing dense gases of colored loops.
We numerically construct asymptotically AdS black brane solutions of $D=4$ Einstein-Maxwell theory coupled to a pseudoscalar. The solutions are holographically dual to $d=3$ CFTs at finite chemical potential and in a constant magnetic field, which spontaneously break translation invariance leading to the spontaneous formation of abelian and momentum magnetisation currents flowing around the plaquettes of a periodic Bravais lattice. We analyse the three-dimensional moduli space of lattice solutions, which are generically oblique, and show, for a specific value of the magnetic field, that the fr…
We propose Lobachevsky boundary conditions that lead to asymptotically H^2xR solutions. As an example we check their consistency in conformal Chern-Simons gravity. The canonical charges are quadratic in the fields, but nonetheless integrable, conserved and finite. The asymptotic symmetry algebra consists of one copy of the Virasoro algebra with central charge c=24k, where k is the Chern-Simons level, and an affine u(1). We find also regular non-perturbative states and show that none of them corresponds to black hole solutions. We attempt to calculate the one-loop partition function, find a rem…
We compute boundary correlation functions for scalar fields on tessellations of two- and three-dimensional hyperbolic geometries. We present evidence that the continuum relation between the scalar bulk mass and the scaling dimension associated with boundary-to-boundary correlation functions survives the truncation of approximating the continuum hyperbolic space with a lattice.
The effect of the complex phase of the fermion determinant is a key question related to the sign problem in finite-density QCD. Recently it has been shown that ignoring the complex phase -- the phase quenching -- does not change physics in a certain region of the phase diagram when a number of colors N_c is large. In this paper we study the effect of the phase quenching within the frameworks of effective models and holographic models. We show, in a unified manner, that the phase quenching gives exact results for any fermionic observables (e.g., chiral condensate) in the mean-field approximatio…
We construct a holographic model in the framework of Q-lattices whose dual exhibits metal-insulator transitions. By introducing an interacting term between the Q-lattice and the electromagnetic field in bulk geometry, we find such kind of transition can be Mott-like. The evidences are presented as follows. i) The transition from a metallic phase to an insulating phase occurs when the lattice constant becomes larger. ii) A hard gap in the insulating phase can be manifestly observed in the optical conductivity. Nevertheless, in the zero temperature limit this model exhibits novel metallic behavi…
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudodifferential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld--Sokolov realization.
We show that integrable vertex and RSOS models with trigonometric Boltzmann weights and appropriate inhomogeneities provide a convenient lattice regularization for massive field theories and for the recently studied massless field theories that interpolate between two non trivial conformal field theories. Massive and massless S matrices are computed from the lattice Bethe ansatz.
Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that provides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete sub…
Recent lattice simulation has uncovered many interesting properties of SU(N) gauge theory at finite temperature. Especially, above the deconfinement phase transition all the thermodynamics quantities acquire significant quadratic contributions in inverse temperature. Such a term is also found to dominate the logarithmic of the renormalized Polyakov loop. Using the Hawking-Page transition in Anti-de Sitter space as an example, we show how such contributions can be naturally generated in the holographic approach.
We consider Einstein gravity in AdS in the presence of a deformed conformal boundary metric, in the limit of large spacetime dimension. At leading order we find a new set of effective near-horizon equations. These can be understood as covariant generalisations of the undeformed equations with new source terms due to the curvature. We show that these equations are given by the conservation of the exact second-order Landau-frame hydrodynamic stress tensor. No derivative expansions are invoked in this identification. We use the new equations to study CFTs with 2d lattice deformations, computing t…
We consider large Wilson loops with quarks in higher representations in SU(N) Yang-Mills theories. We consider representations with common N-ality and check whether the expectation value of the Wilson loop depends on the specific representation or only on the N-ality. In the framework of AdS/CFT we show that <W_R> = dim R exp -sigma_k A, namely that the string tension depends only on the N-ality k but the pre-exponent factor is representation dependent. The lattice strong coupling expansion yields an identical result at infinite N, but shows a representation dependence of the string tension at…
We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical integrable models. Thus new as well as known lattice systems of different classes are generated including new types of collective integrable models and canonical models with nonstandard $r$ matrices.
New boundary conditions for integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank 1 ansatz for the 2x2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be the one of dynamical symmetry for the A1 and BC2 Calogero-Moser problems. Other physical realizations of our quadratic…
We consider the integrable XXZ model with the special open boundary conditions. We perform Quantum Group reduction of this model in roots of unity and use it for the definition Minimal Models of Interable lattice theory. It is shown that after this Quantum Group reduction Sklyanin's transfer-matrices satisfy the closed system of the truncated functional relations. We solve these equations for the simplest case.
We propose a regular way to construct lattice versions of $W$-algebras, both for quantum and classical cases. In the classical case we write the algebra explicitly and derive the lattice analogue of Boussinesq equation from the Hamiltonian equations of motion. Connection between the lattice Faddeev-Takhtadjan-Volkov algebra [1] and q-deformed Virasoro is also discussed.
We give a rational form of a generic two-dimensional "quad" map, containing the so-called $Q_4$ case, but whose coefficients are free. Its integrability is proved using the calculation of algebraic entropy.
We investigate instantons in finite temperature QCD via Witten's holographic QCD. To study the deconfinement phase, we use the setup proposed in [1] (arXiv:1107.4048). We find that the sizes of the instantons are stabilized at certain values both in the confinement and deconfinement phases. This agrees with the numerical result in the lattice gauge theory. Besides we find that the gravity duals of the large and small instantons in the deconfinement phase have different topologies. We also argue that the fluctuation of the topological charges is large in confinement phase while it is exponentia…
This note gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holomorphic. Spin-chain models arise in this way from a twisted, deformed version of N=1 gauge theory.
We consider lattice analogues of some conformal theories, including WZW and Toda models. We describe discrete versions of Drinfeld-Sokolov reduction and Sugawara construction for the WZW model. We formulate perturbation theory in chiral sector. We describe the spaces of Integrals of Motion in the perturbed theories. We interpret the perturbed WZW model in terms of NLS hierarchy and obtain of this model into the lattice KP-hierarchy.
We consider the case of an integrable quantum spin chain with ``soliton non-preserving'' boundary conditions. This is the first time that such boundary conditions have been considered in the spin chain framework. We construct the transfer matrix of the model, we study its symmetry and we find explicit expressions for its eigenvalues. Moreover, we derive a new set of Bethe ansatz equations by means of the analytical Bethe ansatz method.
We obtain off-critical (elliptic) Boltzmann weights for lattice models whose continuum limits correspond to massive, $N=2$ supersymmetric, quantum integrable field theories. We also compute the free energies of these models and show that they are analytic in the region of parameter space where we believe that the supersymmetry is unbroken. While the supersymmetry is not directly realized on the lattice, there is still a very close connection between the models described here and topological lattice models. A simple example is discussed in detail and some corner transfer matrix computations are…
A pedagogical presentation of integrable models with special reference to the Toda lattice hierarchy has been attempted. The example of the KdV equation has been studied in detail, beginning with the infinite conserved quantities and going on to the Lax formalism for the same. We then go on to symplectic manifolds for which we construct the Lax operator. This formalism is applied to Toda Lattice systems. The Zakharov Shabat formalism aimed at encompassing all integrable models is also covered after which the zero curvature condition and its fallout are discussed. We then take up Toda Field The…
The light-cone approach is reviewed. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, taking the massive Thirring model (sine-Gordon) as an example. They are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras $U_q({\hat\G})$. Based on le…
The light--cone lattice approach to the massive Thirring model is reformulated using a local and integrable lattice Hamiltonian written in terms of discrete fermi fields. Several subtle points concerning boundary conditions, normal--ordering, continuum limit, finite renormalizations and decoupling of fermion doublers are elucidated. The relations connecting the six--vertex anisotropy and the various coupling constants of the continuum are analyzed in detail.
We derive the expressions for $ψ$-functions and generic solutions of lattice principal chiral equations, lattice KP hierarchy and hierarchy including lattice N-wave type equations. $τ$-function of $n$ free fermions plays fundamental role in this context. Miwa's coordinates in our case appear as the lattice parameters.
The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomeogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop mode…
