A D3-D5 intersection gives rise to a defect CFT, wherein the rank of the gauge group jumps by k units across a domain wall. The one-point functions of local operators in this set-up map to overlaps between on-shell Bethe states in the underlying spin chain and a boundary state representing the D5 brane. Focussing on the k=1 case, we extend the construction to gluonic and fermionic sectors, which was prohibitively difficult to achieve for k>1. As a byproduct, we test an all-loop proposal for the one-point functions in the su(2) sector at the half-wrapping order of perturbation theory.
The encoding of all possible sets of Bethe equations for a spin chain with SU(N|M) symmetry into a QQ-system calls for an expression of spin chain overlaps entirely in terms of Q-functions. We take a significant step towards deriving such a universal formula in the case of overlaps between Bethe eigenstates and integrable boundary states, of relevance for AdS/dCFT, by determining the transformation properties of the overlaps under fermionic as well as bosonic dualities which allows us to move between any two descriptions of the spin chain encoded in the QQ-system. An important part of our anal…
We analyze the defect scaling Lee-Yang model from the perturbed defect conformal field theory (DCFT) point of view. First the defect Lee-Yang model is solved by calculating its structure constants from the sewing relations. Integrable defect perturbations are identified in conformal defect perturbation theory. Then pure defect flows connecting integrable conformal defects are described. We develop a defect truncated conformal space approach (DTCSA) to analyze the one parameter family of integrable massive perturbations in finite volume numerically. Fusing the integrable defect to an integrable…
We consider the expectation value of Wilson lines in two defect versions of N = 4 SYM, both with supersymmetry completely broken, where one is described in terms of an integrable boundary state, the other one not. For both cases, imposing a certain double scaling limit, we find agreement to two leading orders between the expectation values calculated from respectively the field theory and the string theory side of the AdS/dCFT correspondence.
We study the matrix product state which appears as the boundary state of the AdS/dCFT set-up where a probe D7 brane wraps two two-spheres stabilized by fluxes. The matrix product state plays a dual role, on one hand acting as a tool for computing one-point functions in a domain wall version of N=4 SYM and on the other hand acting as the initial state in the study of quantum quenches of the Heisenberg spin chain. We derive a number of selection rules for the overlaps between the matrix product state and the eigenstates of the Heisenberg spin chain and in particular demonstrate that the matrix p…
We determine in a closed form all scalar one-point functions of the defect CFT dual to the D3-D5 probe brane system with k units of flux which amounts to calculating the overlap between a Bethe eigenstate of the integrable SO(6) spin chain and a certain matrix product state of bond dimension k. In particular, we show that the matrix product state is annihilated by all the parity odd charges of the spin chain which has recently been suggested as the criterion for such a state to correspond to an integrable initial state. Finally, we discuss the properties of the analogous matrix product state f…
We study tree level one-point functions of non-protected scalar operators in the defect CFT, based on N=4 SYM, which is dual to the SO(5) symmetric D3-D7 probe brane system with non-vanishing instanton number. Whereas symmetries prevent operators from the SU(2) and SU(3) sub-sectors from having non-vanishing one-point functions, more general scalar conformal operators, which in particular constitute Bethe eigenstates of the integrable SO(6) spin chain, are allowed to have non-trivial one-point functions. For a series of operators with a small number of excitations we find closed expressions in…
Integrable boundary states can be built up from pair annihilation amplitudes called $K$-matrices. These amplitudes are related to mirror reflections and they both satisfy Yang Baxter equations, which can be twisted or untwisted. We relate these two notions to each other and show how they are fixed by the unbroken symmetries, which, together with the full symmetry, must form symmetric pairs. We show that the twisted nature of the $K$-matrix implies specific selection rules for the overlaps. If the Bethe roots of the same type are paired the overlap is called chiral, otherwise it is achiral and …
We consider a particular set of two-point functions in the setting of N = 4 SYM with a defect, dual to the fuzzy-funnel solution for the probe D5-D3-brane system. The two-point functions in focus involve a single trace operator in the SU(2)-subsector of arbitrary length and a length-two operator built out of any scalars. By interpreting the contractions as a spin-chain operator, simple expressions were found for the leading contribution to the two-point functions, mapping them to earlier known formulas for the one-point functions in this setting.
We consider different supersymmetric mixed boundary conditions for scalar and fermionic fields in $AdS_2$, searching for the dual description of a family of interpolating Wilson Loops in ABJM theory. The family, which interpolates between the bosonic 1/6 BPS loop and the 1/2 BPS loop, can be thought of as an exact marginal deformation in a defect CFT$_1$. Confronting this property against holographic correlators and vacuum energy corrections singles out a particular boundary condition which we propose as dual to the interpolating family of Wilson loops.
We consider defect composite operators in a defect superconformal field theory obtained by inserting an AdS_4 x S^2-brane in the AdS_5 x S^5 background. The one-loop dilatation operator for the scalar sector is represented by an integrable open spin chain. We give a description to construct coherent states for the open spin chain. Then, by evaluating the expectation value of the Hamiltonian with the coherent states in a long operator limit, a Landau-Lifshitz type of sigma model action is obtained. This action is also derived from the string action and hence we find a complete agreement in both…
In this review we discuss recent advances in the computation of one-point functions in defect conformal field theories with holographic duals. We briefly review the appearance of integrable spin chains in N=4 super Yang--Mills theory and reformulate the problem of computing one-point functions to determining overlaps between Bethe states and a Matrix Product State. We will then demonstrate how these overlaps can be computed by determinant formulas. This work is based on lectures given at the Young Researchers Integrability School and Workshop 2018. To appear in a special issue of J. Phys. A.
Via a challenging field-theory computation, we confirm a supergravity prediction for the non-supersymmetric D3-D7 probe-brane system with probe geometry AdS_4 x S^2 x S^2, stabilized by fluxes. Supergravity predicts, in a certain double-scaling limit, the value of the one-point functions of chiral primaries of the dual defect version of N=4 SYM theory, where the fluxes translate into SO(3) x SO(3)-symmetric, Lie-algebra-valued vacuum expectation values for all six scalar fields. Using a generalization of the technique based on fuzzy spherical harmonics developed for the related D3-D5 probe-bra…
Defect conformal field theories (dCFTs) have been attracting increased attention recently, mainly because they enable us to bridge the gap between idealistic, highly symmetric models of our world (such as the particle/string duality) and real-world systems. This talk is about the AdS/defect CFT correspondence, an exciting new proposal that joins the forces of holography, integrability, supersymmetric localization and the conformal bootstrap program in a framework that is appropriate for the study of defects in real-world systems. After introducing dCFTs and some of their holographic realizatio…
We build the framework for performing loop computations in the defect version of N=4 super Yang-Mills theory which is dual to the probe D5-D3 brane system with background gauge-field flux. In this dCFT, a codimension-one defect separates two regions of space-time with different ranks of the gauge group and three of the scalar fields acquire non-vanishing and space-time-dependent vacuum expectation values. The latter leads to a highly non-trivial mass mixing problem between different colour and flavour components, which we solve using fuzzy-sphere coordinates. Furthermore, the resulting space-t…
We consider open spinning string solutions on an AdS_4 x S^2-brane (D5-brane) in the bulk AdS_5 x S^5 background. By taking account of the breaking of SO(6)_R to SO(3)_H x SO(3)_V due to the presence of the AdS-brane, the open rotating string ansatz is discussed. We construct the elliptic folded/circular open string solutions in the SU(2) and the SL(2) sectors, so that they satisfy the appropriate boundary conditions. On the other hand, in the SU(2) sector of the gauge theory, we compute the matrix of anomalous dimension of the defect operator, which turns out to be the Hamiltonian of an open …
We demonstrate that the one-loop dilatation generator for the scalar sector of a certain perturbation of N=4 Super Yang-Mills with fundamentals is the Hamiltonian of an integrable spin chain with open boundary conditions. The theory is a supersymmetric defect conformal field theory (dCFT) with the fundamentals in hypermultiplets confined to a codimension one defect. We obtain a K-matrix satisfying a suitably generalized form of the boundary Yang-Baxter equation, study the Bethe ansatz equations and demonstrate how Dirichlet and Neumann boundary conditions arise in field theory, and match to ex…
We consider a non-supersymmetric domain-wall version of $\mathcal{N} = 4$ SYM theory where five out of the six scalar fields have non-zero classical values on one side of a wall of codimension one. The classical fields have commutators which constitute an irreducible representation of the Lie algebra $\mathfrak{so}(5)$ leading to a highly non-trivial mixing between color and flavor components of the quantum fields. Making use of fuzzy spherical harmonics on $S^4$, we explicitly solve the mixing problem and derive not only the spectrum of excitations at the quantum level but also the propagator…
We calculate the leading contributions to the connected two-point functions of protected scalar operators in the defect version of N=4 SYM theory which is dual to the D5-D3 probe-brane system with k units of background gauge field flux. This involves several types of two-point functions which are vanishing in the theory without the defect, such as two-point functions of operators of unequal conformal dimension. We furthermore exploit the operator product expansion (OPE) and the boundary operator expansion (BOE), which form the basis of the boundary conformal bootstrap equations, to extract con…
The 1/2-BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with $AdS_2 \times S^2$ and $AdS_2 \times S^4$ worldvolume geometries, ending at the $AdS_5$ boundary along a one-dimensional contour. We first compute the…
We propose a prescription for describing correlation functions in higher-dimensional defect conformal field theories (DCFTs) by those in ancillary conformal field theories (CFTs) without defects, which is a vast generalization of the image method in two-dimensional boundary CFTs. A correlation function of $n$ operators inserted away from a defect in a DCFT is represented by a correlation function of $2n$ operators in the ancillary CFT, each pair of which is placed symmetrically with respect to the defect. For scalar operators, we establish the correspondence by matching the constraints on corr…
One-point functions of certain non-protected scalar operators in the defect CFT dual to the D3-D5 probe brane system with k units of world volume flux can be expressed as overlaps between Bethe eigenstates of the Heisenberg spin chain and a matrix product state. We present a closed expression of determinant form for these one-point functions, valid for any value of k. The determinant formula factorizes into the k=2 result times a k-dependent prefactor. Making use of the transfer matrix of the Heisenberg spin chain we recursively relate the matrix product state for higher even and odd k to the …
We consider four dimensional $U(N)$ $\mathcal N=4$ SYM theory interacting with a 3d $\mathcal N=4$ theory living on a codimension-one interface and holographically dual to the D3-D5 system without flux. Localization captures several observables in this dCFT, including its free energy, related to the defect expectation value, and single trace $\frac{1}{2}$-BPS composite scalars. These quantities may be computed in a hermitian one-matrix model with non-polynomial single-trace potential. We exploit the integrable Volterra hierarchy underlying the matrix model and systematically study its $1/N$ ex…
The non-integrable mass is studied explicitly in this paper. We study Einstein-scalar gravities with weakened boundary conditions, and calculate the mass with the Hamiltonian formula and Wald's formula respectively. We find the masses calculated by these two formulas are non-integrable. One way to solve this non-integrability problem is to impose boundary conditions; however, we find the mass calculated in this way has many other problems. This implies the macroscopic thermodynamic properties of the scalar hairy black holes should be described by one more charge beside the mass, which we call …
A class of explicitly integrable models of 1+1 dimensional dilaton gravity coupled to scalar fields is described in some detail. The equations of motion of these models reduce to systems of the Liouville equations endowed with energy and momentum constraints. The general solution of the equations and constraints in terms of chiral moduli fields is explicitly constructed and some extensions of the basic integrable model are briefly discussed. These models may be related to high dimensional supergravity theories but here they are mostly considered independently of such interpretations. A brief r…
Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension $d\geq 5$ with a conformally-invariant spatial boundary (BCFTs) or $4$-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT's Weyl tensor. F…
We study the ground state energy of integrable $1+1$ quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new, ``R-channel TBA'', where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scal…
We compute the two-loop anomalous dimension matrix in the scalar sector of planar ${\cal N}=3$ flavored ABJM theory. Using coordinate Bethe ansatz, we obtain the reflection matrix and confirm that the boundary Yang-Baxter equations are satisfied. This establishes the integrability of this theory in the scalar sector at the two-loop order.
We study deformations of 2D Integrable Quantum Field Theories (IQFT) which preserve integrability (the existence of infinitely many local integrals of motion). The IQFT are understood as "effective field theories", with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields $X_s$, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars $X_s$ are built from the components of the associated conserved currents in a universal way. The first of these scalars, $X_1$, coincid…
Integrable models of 1+1 dimensional gravity coupled to scalar and vector fields are briefly reviewed. A new class of integrable models with nonminimal coupling to scalar fields is constructed and discussed.
A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to twisted affine Kac-Moody algebras. Explicit examples of the $N=1,2$ super sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets $sl(p,1)/sl(p)\otimes u(1)$ when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when $p=2$ leads to the constrained Bukhvostov-Lipatov, Thirring, …
In integrable field theories the S-matrix is usually a product of a relatively simple matrix and a complicated scalar factor. We make an observation that in many relativistic integrable field theories the scalar factor can be expressed as a convolution of simple kernels appearing in the nested levels of the nested Bethe ansatz. We formulate a proposal, up to some discrete ambiguities, how to reconstruct the scalar factor from the nested Bethe equations and check it for several relativistic integrable field theories. We then apply this proposal to the AdS asymptotic Bethe ansatz and recover the…
We consider a theory of scalar superfields in two dimensions with arbitrary superpotential. By imposing no particle production in tree level scattering, we constrain the form of the admissible interactions, recovering a supersymmetric extension of the sinh-Gordon model.
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 address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr, Caetano:2016ydc}. We argue that their 3-dimensional $ϕ^{6}$ fishnet model becomes perturbatively stable under renormalization in the large $N$ limit, in contrast to what happens in their 4-dimensional $ϕ^{4}$ fishnet model, in which double trace terms are known to be generated by the RG flow. We point out that there is a direct way to modify this second theory that protects it from such corrections. Additionally, we observe that the 6-dimensional $ϕ…
We compute scalar products and norms of Bethe vectors in the massless sector of AdS_3 integrable superstring theories, by exploiting the general difference form of the S-matrix of massless excitations in the pure Ramond-Ramond case, and the difference form valid only in the BMN limit in the mixed-flux case. We obtain determinant-like formulas for the scalar products, generalising a procedure developed in previous literature for standard R-matrices to the present non-conventional situation. We verify our expressions against explicit calculations using Bethe vectors for chains of small length, a…
Chiral Integrable Boundary States in the SU(4) Alternating Spin Chain
Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院量子联合研究中心及物理系）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）
We study dynamics of scalar fields on a large class of geometries described by integrable sigma models. Although equations of motion are not separable due to absence of isometries and Killing tensors, we completely determine the spectra using algebraic and group-theoretic methods.
We consider a special double scaling limit, recently introduced by two of the authors, combining weak coupling and large imaginary twist, for the $γ$-twisted $\mathcal{N}=4$ SYM theory. We also establish the analogous limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in the planar limit. In spite of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions defined by specific, integrable Feynman diagrams. …
A class of integrable models of 1+1 dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to 0+1 dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real space-time dimension are derived. A generalization of the `no-hair' theorem is pointed out and the Hamiltonian formulation that enables quantizing the s-integrable syste…
Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type.
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU(3)-invariant Heisenberg chain.
We consider the $γ$-deformed $\mathcal{N}=4$ SYM in the double scaling limit of large imaginary twists and small coupling, which discards the gauge fields and retains only certain Yukawa and scalar interactions with three arbitrary effective couplings. In the 't~Hooft limit, these 4D theories are integrable, with most of the correlators being conformal such that the whole arsenal of AdS/CFT integrability remains applicable. In particular, for one non-zero effective coupling, we obtain a QFT of two complex scalars with a chiral, quartic interaction. The BMN vacuum anomalous dimension is dominat…
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of a bilinear combination of their highest coefficients. We obtain various different representations for the highest coefficient in terms of sums over partitions. We also obtain multiple integral representations for the highest coefficient.
In this work, we study the $\frac{1}{8}$-BPS heavy-heavy-light-light correlators in the D1D5 CFT and its holographic dual. On the field theory side, we compute the fermionic four-point correlators at the free orbifold point. On the dual gravity side, we compute the correlators of the scalar operators in the supergravity limit of the D1D5 CFT. Following the strategy of \cite{Galliani:2017jlg}, the four-point function is converted into a two-point function in non-trivial geometries known as superstrata which are supergravity solutions preserving $1/8$ supersymmetries. We focus on a family of int…
We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang-Mills and Wess-Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain of SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integr…
We calculate the scalar product of Bethe states of the XXZ spin-$\frac{1}{2}$ chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.
We show that several integrable (i.e., exactly solvable) scalar cosmologies considered by Fré, Sagnotti and Sorin (Nuclear Physics \textbf{B 877}(3) (2013), 1028--1106) can be generalized to include cases where the spatial curvature is not zero and, besides a scalar field, matter or radiation are present with an equation of state $p^{(m)} = w\, ρ^{(m)}$; depending on the specific form of the self-interaction potential for the field, the constant $w$ can be arbitrary or must be fixed suitably.
Global canonical transformations to free chiral fields are constructed for DG models minimally coupled to scalar fields. The boundary terms for such canonical transformations are shown to vanish in asymptotically static coordinates if there is no scalar field.
We compute the scalar curvature of 7-dimensional ${G}_2$-manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable ${G}_2$-structure into a cocalibrated one of pure type $W_3$.
The representation of the Bethe wave functions of certain integrable models via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The algebraic relations arising in the calculation of the scalar products and the correlation functions are based on the Binet-Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of the Bethe state-vectors in terms of nests of the self-avoiding lattice paths constituting the so-called watermelon configur…
We propose a $D$-dimensional generalization of $4D$ bi-scalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a strong-twist double scaling limit of $γ$-deformed $\mathcal{N}=4$ SYM theory. Similarly to the $4D$ case, this D-dimensional CFT is also dominated by "fishnet" Feynman graphs and is integrable in the planar limit. The dynamics of these graphs is described by the integrable conformal $SO(D+1,1)$ spin chain. In $2D$ it is the analogue of L. Lipatov's $SL(2,\mathbb{C})$ spin chain for the Regge limit of $QCD$, but with the spins $s=1/4$ instead o…
We study the integrable model with minimally and non-minimally coupled scalar fields and the correspondence of their general solutions. Using the model with a minimally coupled scalar field and a the constant potential as an example we demonstrate the difference between the general solutions of the corresponding models in the Jordan and the Einstein frames.
In arXiv:1704.05807, it was shown that the planar flavored ABJM theory is integrable in the scalar sector at two-loop order using coordinate Bethe ansatz. A salient feature of this case is that the boundary reflection matrices are anti-diagonal with respect to the chosen basis. In this paper, we relax the coefficients of the boundary terms to be general constants in order to search for integrable systems among this class. We found that at each end of the spin chain, the only integrable boundary interaction, besides the one in arXiv:1704.05807, is the one with vanishing boundary interactions le…
