We define a perturbatively calculable quantity--the on-shell correlator--which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. Afterwards, we compute tree-level on-shell correlators for biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn that color-kinematics duality is manifested at the level of fields under a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold. Last but not least, we present a field theoretic derivation of the fundamental BCJ relations for on-shell correlators following from the existence of certain conserved currents in BAS theory and the NLSM.
We use gauge/string duality to analytically evaluate correlation lengths of the renormalized field strength correlators in pure Yang-Mills theories at zero and finite temperature.
In this note we compare planar correlators such as $< \prod_i^n Tr M^{2k_i} >$ in the Gaussian matrix model with corresponding genus zero correlators of the A-model topological string theory on $P^1$. We find a simple relation between them which provides additional evidence for the duality between the two theories, as proposed in arXiv:1104.2386.
We calculate 3-point correlation functions of Delta-BMN operators with 3 scalar impurites in N=4 supersymmetric gauge theory. We use these results to test the pp-wave/SYM duality correspondence of the vertex--correlator type. This correspondence relates the coefficients of 3-point correlators of Delta-BMN operators in gauge theory to the 3-string vertex in lightcone string field theory in the pp-wave background. We verify the vertex--correlator duality equation of hep-th/0301036 at the 3 scalar impurites level for supergravity and for string modes.
We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic non-MHV amplitudes in planar N=4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. Our correlators are computed at Born level. If all of their points form a light-like polygon, the correlator is dual to the tree-level ampli…
We continue the study of the duality between super-correlators and scattering super-amplitudes in planar N=4 SYM. We provide a number of further examples supporting the conjectured duality relation between these two seemingly different objects. We consider the five- and six-point one-loop NMHV and the six-point tree-level NNMHV amplitudes, obtaining them from the appropriate correlators of strength tensor multiplets in N=4 SYM. In particular, we find exact agreement between the rather non-trivial parity-odd sector of the integrand of the six-point one-loop NMHV amplitude, as obtained from the …
In these notes we describe how to formulate the Lagrangian insertion technique in a way that mimics generalized unitarity. We introduce a notion of cuts in real space and show that the cuts of the correlators in the super-correlators/super-amplitudes duality correspond to generalized unitarity cuts of the equivalent amplitudes. The cuts consist of correlation functions of operators in the chiral part of the stress-tensor multiplet as well as other half-BPS operators. We will also discuss the application of the method to other correlators as well as non-planar contributions.
The AdS boundary correlators and their dual correlation functions of boundary operators have been the main dynamic observables of the holographic duality relating a bulk AdS theory and a boundary conformal field theory. We show that tree-level AdS boundary correlators for generic states can be expressed as nonlocal differential operators of a certain structure acting on contact Witten diagrams. We further write the boundary correlators in a form that is very similar to flat space amplitudes, with Mandelstam variables replaced by certain combinations of single-state conformal generators, prove …
We continue the study of n-point correlation functions of half-BPS protected operators in N=4 super-Yang-Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the l-loop corrections by making l Lagrangian insertions. We argue that there exists a simple relation between the (n+l)-point tree-level correlator with l Lagrangian insertions and the integrand of the n-particle l-loop MHV scattering amplitude, as obtained by the recent momentum twistor construction of Arkani-Hamed et al. We present several examples of this new duality, at one …
Motivated by recent works in which the FZZ duality plays an important role, we revisit the computation of correlation functions in the sine-Liouville field theory. We present a direct computation of the three-point function, the simplest to the best of our knowledge, and give expressions for the $N$-point functions in terms of integrated Liouville theory correlators. This leads us to discuss the relation to the $H^+_3$ WZW-Liouville correspondence, especially in the case in which spectral flow is taken into account. We explain how these results can be used to study scattering amplitudes of win…
We consider Chern-Simons theory coupled to massive fundamental matter in three spacetime dimensions at finite temperature, in the large $N$ limit. We compute several thermal correlators in this theory for both fermionic and bosonic matter separately. The results are computed in the large $N$ 't Hooft limit but for arbitrary values of the 't Hooft coupling. Furthermore, we generalize the computations of the four-point function of fundamental scalars in the bosonic theory to finite temperature. As a consistency check, we see that the results obtained here agree with the existing previous results…
We show that there exists a duality between the local coordinates and the solutions of the Klein-Gordon equation in curved spacetime in the same sense as in the Minkowski spacetime. However, the duality in curved spacetime does not have the same generality as in flat spacetime and it holds only if the system satisfies certain constraints. We derive these constraints and the basic equations of duality and discuss the implications in the quantum theory.
We obtain the T-duality transformations of space-time spinors (the supersymmetry transformation parameters, gravitinos and dilatinos) of type-II theories in curved backgrounds with an isometry. The transformation of the spinor index is shown to be a consequence of the twist that T-duality introduces between the left and right-moving local Lorentz frames. The result is then used to derive the T-duality action on Ramond-Ramond field strengths and potentials in a simple way. We also discuss the massive IIA theory and, using duality, give a short derivation of ``mass''-dependent terms in the Wess-…
We examine a strong/weak duality between a Heisenberg coset of a theory with $\mathfrak{sl}_n$ subregular $\mathcal{W}$-algebra symmetry and a theory with a $\mathfrak{sl}_{n|1}$-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras propos…
In the paper [1] we showed that in double space, where all initial coordinates $x^μ$ are doubled $x^μ\to y_μ$, the T-duality transformations can be performed by exchanging places of some coordinates $x^a$ and corresponding dual coordinates $y_a$. Here we generalize this result to the case of weakly curved background where in addition to the extended coordinate we will also transform extended argument of background fields with the same operator $\hat {\cal T}^a$. So, in the weakly curved background T-duality leads to the physically equivalent theory and complete set of T-duality transformations…
We consider a gauge/gravity dual model of thermalization which consists of a collapsing thin matter shell in asymptotically Anti-de Sitter space. A central aspect of our model is to consider a shell moving at finite velocity as determined by its equation of motion, rather than a quasi-static approximation as considered previously in the literature. By applying a divergence matching method, we obtain the evolution of singularities in the retarded unequal time correlator $G^R(t,t')$, which probes different stages of the thermalization. We find that the number of singularities decreases from a fi…
The elements of $O(d,d,\Z)$ are shown to be discrete symmetries of the space of curved string backgrounds that are independent of $d$ coordinates. The explicit action of the symmetries on the backgrounds is described. Particular attention is paid to the dilaton transformation. Such symmetries identify different cosmological solutions and other (possibly) singular backgrounds; for example, it is shown that a compact black string is dual to a charged black hole. The extension to the heterotic string is discussed.
We generalize the color/kinematics duality of flat-space scattering amplitudes to the embedding space formulation of AdS boundary correlators. Kinematic numerators and propagators are replaced with differential operators acting on a scalar contact diagram that is the AdS generalization of the momentum conserving delta function of flat space scattering amplitudes. We show that color/kinematics duality implies differential relations among AdS boundary correlators that naturally generalize the flat space BCJ amplitude relations and verify them for the correlators of Yang-Mills theory and of the N…
Braneworld effective action is constructed by two different methods based respectively on the Dirichlet and Neumann boundary value problems. The equivalence of these methods is shown due to nontrivial duality relations between special boundary operators of these two problems. Previously known braneworld action algorithms in two-brane Randall-Sundrum model are generalized to curved branes with deSitter and Anti-deSitter geometries.
Previous studies of high-energy jet stopping in strongly-coupled plasmas have lacked a clear gauge-theory specification of the initial state. We show how to set up a well-defined gauge theory problem to study jet stopping in pure {\cal N}=4 super Yang Mills theory (somewhat analogous to Hofman and Maldacena's studies at zero temperature) and solve it by using gauge-gravity duality for real-time, finite-temperature 3-point correlators. Previous studies have found that the stopping distance scales with energy as E^{1/3} (with disagreement on the gauge coupling dependence). We do find that none o…
With two typical parent actions we have two kinds of dual worlds: i) one of which contains an electric as well as magnetic current, and ii) the other contains (generalized) Chern-Simons terms. All these fields are defined on a curved spacetime of arbitrary (odd) dimensions. A new form of gauge transformations is introduced and plays an essential role in defining the interaction with a magnetic monopole or in defining the generalized Chern-Simons terms.
We consider the closed string propagating in the weakly curved background which consists of constant metric and Kalb-Ramond field with infinitesimally small coordinate dependent part. We propose the procedure for constructing the T-dual theory, performing T-duality transformations along coordinates on which the Kalb-Ramond field depends. The obtained theory is defined in the non-geometric double space, described by the Lagrange multiplier $y_μ$ and its $T$-dual $\tilde{y}_μ$. We apply the proposed T-duality procedure to the T-dual theory and obtain the initial one. We discuss the standard rela…
We point out that the duality symmetry of free electromagnetism does not hold in the quantum theory if an arbitrary classical gravitational background is present. The symmetry breaks in the process of renormalization, as also happens with conformal invariance. We show that a similar duality-anomaly appears for a massless scalar field in $1+1$ dimensions.
We study non-linear electrodynamics in curved space from the viewpoint of dualities. After establishing the existence of a topological bound for self-dual configurations of Born-Infeld field in curved space, we check that the energy-momentum tensor vanishes. These properties are shown to hold for general duality-invariant non-linear electrodynamics. We give the dimensional reduction of Born-Infeld action to three dimensions in a general curved background admitting a Killing vector. The SO(2) duality symmetry becomes manifest but other symmetries present in flat space are broken, as is U-dualit…
The PP-wave/SYM proposal in its original form emphasizes a duality relation between the masses of the string states and the anomalous dimensions of the corresponding BMN operators in gauge theory, the mass--dimension type duality. In this paper, we give evidence in favour of another duality relation of the vertex--correlator type, which relates the coefficients of 3-point correlators of BMN operators in gauge theory to 3-string vertices in lightcone string field theory in the pp-wave background. We verify that all the available field theory results in the literature, as well as the newly obtai…
We initiate a program to study the relationship between the target space, the spectrum and the scattering amplitudes in string theory. We consider scattering amplitudes following from string theory and quantum field theory on a curved target space, which is taken to be the $SU(2)$ group manifold, with special attention given to the duality between contributions from different channels. We give a simple example of the equivalence between amplitudes coming from string theory and quantum field theory, and compute the general form of a four-scalar field theoretical amplitude. The corresponding str…
We consider a theory of an open string moving in a weakly curved background, composed of a constant metric and a linearly coordinate dependent Kalb-Ramond field with an infinitesimal field strength. We find its T-dual using the generalized Buscher procedure developed for the closed string moving in a weakly curved background, and the fact that solving the boundary conditions, the open string theory transforms to the effective closed string theory. So, T-dualizing the effective theory along all effective directions we obtain its T-dual theory and resume the open sting theory which has such an e…
We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formall…
We perform a systematic study of S-duality for ${\cal N}=2$ supersymmetric non-linear abelian theories on a curved manifold. Localization can be used to compute certain supersymmetric observables in these theories. We point out that localization and S-duality acting as a Legendre transform are not compatible. For these theories S-duality should be interpreted as Fourier transform and we provide some evidence for this. We also suggest the notion of a coholomological prepotential for an abelian theory that gives the same partition function as a given non-abelian supersymmetric theory.
Bosonization in curved spacetime maps massive Thirring model (self-interacting Dirac fermions) to a generalized sine-Gordon model, both living in $1+1$-dimensional curved spacetime. Applying this duality we have shown that the Thirring model fermion, in non-relativistic limit, gets identified with the soliton of non-linear Scrodinger model with Kerr form of non-linearity. We discuss one particular optical soliton in the latter model and relate it with the Thirring model fermion.
We use AdS/CFT duality to compute in N=4 Yang-Mills theory the finite temperature spatial correlator G(r) of the scalar operator F^2, integrated over imaginary time. The computation is carried out both at zero frequency and integrating the spectral function over frequencies. The result is compared with a perturbative computation in finite T SU(N_c) Yang-Mills theory.
We investigate the relation between 3d N=2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S^2xS^1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recentl…
In one of our previous papers we generalized the Buscher T-dualization procedure. Here we will investigate the application of this procedure to the theory of a bosonic string moving in the weakly curved background. We obtain the complete T-dualization diagram, connecting the theories which are the result of the T-dualizations over all possible choices of the coordinates. We distinguish three forms of the T-dual theories: the initial theory, the theory obtained T-dualizing some of the coordinates of the initial theory and the theory obtained T-dualizing all of the initial coordinates. While the…
A class of exactly solvable string models can be obtained by starting with flat space and combining T-duality and shifts of angular coordinates of several polar planes. The models are the analog of the Lunin-Maldacena β-deformation of the AdS_5 x S^5 type IIB string background, which is dual to a Leigh-Strassler deformation of \N=4 Super Yang-Mills Theory. We determine the complete physical string spectrum for two string models obtained in this way, by explicitly solving the string equations and quantizing in terms of free creation and annihilation operators. We also show that the 3-parameter …
We have solved a sigma-model in curved background using the fact that the Poisson-Lie T-duality can transform the curved background into the flat one. For finding solution of the flat model we have used transformation of coordinates that makes the metric constant. The T-duality transform was then explicitly performed.
It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS$_2$ background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS$_2$ are directly related to the correlation functions of the chiral $\mathcal W$-symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite…
We carry out a systematic study of correlation functions of momentum modes in the Euclidean c=1 string, as a function of the radius and to all orders in perturbation theory. We obtain simple explicit expressions for several classes of correlators in terms of special functions. The Normal Matrix Model is found to be a powerful calculational tool that computes c=1 string correlators even at finite N. This enables us to obtain a simple combinatoric formula for the 2n-point function of unit momentum modes, which after T-duality determines the vortex condensate. We comment on possible applications …
The AdS/CFT duality maps supersymmetric heavy operators with conformal dimension of the order of the central charge to asymptotically AdS supergravity solutions. We show that by studying the quadratic fluctuations around such backgrounds it is possible to derive the 4-point correlators of two light and two heavy states in the supergravity approximation. We provide an explicit example in the AdS$_3$ setup relevant for the duality with the D1-D5 CFT. Contrary to previously studied examples, the supergravity correlator derived in this work differs from the result obtained at the CFT orbifold poin…
The NSF AI Institute for Artificial Intelligence and Fundamental Interactions（美国国家科学基金会人工智能与基础交互AI研究所）；Northeastern University（东北大学）
Parameter-space and function-space provide two different duality frames in which to study neural networks. We demonstrate that symmetries of network densities may be determined via dual computations of network correlation functions, even when the density is unknown and the network is not equivariant. Symmetry-via-duality relies on invariance properties of the correlation functions, which stem from the choice of network parameter distributions. Input and output symmetries of neural network densities are determined, which recover known Gaussian process results in the infinite width limit. The me…
We explore color-kinematic duality for tree-level AdS/CFT correlators in momentum space. We start by studying the bi-adjoint scalar in AdS at tree-level as an illustrative example. We follow this by investigating two forms of color-kinematic duality in Yang-Mills theory, the first for the integrated correlator in AdS$_4$ and the second for the integrand in general AdS$_{d+1}$. For the integrated correlator, we find color-kinematics does not yield additional relations among $n$-point, color-ordered correlators. To study color-kinematics for the AdS$_{d+1}$ Yang-Mills integrand, we use a spectra…
The source-free Maxwell action is invariant under electric-magnetic duality rotations in arbitrary spacetimes. This leads to a conserved classical Noether charge. We show that this conservation law is broken at the quantum level in presence of a background classical gravitational field with a non-trivial Chern-Pontryagin invariant, in a parallel way to the chiral anomaly for massless Dirac fermions. Among the physical consequences, the net polarization of the quantum electromagnetic field is not conserved.
The equations of motion of a super non-Abelian T-dual sigma model on the Lie supergroup $(C^1_1+A)$ in the curved background are explicitly solved by the super Poisson-Lie T-duality. To find the solution of the flat model we use the transformation of supercoordinates, transforming the metric into a constant one, which is shown to be a supercanonical transformation. Then, using the super Poisson-Lie T-duality transformations and the dual decomposition of elements of Drinfel'd superdouble, the solution of the equations of motion for the dual sigma model is obtained. The general form of the dilat…
We propose various new 3d N=2 dualities exploiting their recently discovered connection to the duality relations for 2d free field CFT correlators. Most of the dualities involve, as the main building block, a quiver theory with monopole superpotential which enjoys various interesting properties such as being self-dual and reducing, in a suitable real mass deformation, to the familiar T[SU(N)] theory. In particular we propose a duality for the U(N) theory with one adjoint and k+1 fundamental flavors. By iterating some basic dualities we can bring the theory to a stable form which, in turns, all…
We use AdS/QCD duality to compute the finite temperature Green's function G(omega,k;T) of the shear operator T_12 for all omega,k in hot Yang-Mills theory. The goal is to assess how the existence of scales like the transition temperature and glueball masses affects the correlator computed in the scalefree conformal N=4 supersymmetric Yang-Mills theory. We observe sizeable effects for T close to T_c which rapidly disappear with increasing T. Quantitative agreement of these predictions with future lattice Monte Carlo data would suggest that QCD matter in this temperature range is strongly intera…
We calculate, using holographic duality, the thermal two-point function in finite temperature little string theory. The analysis of those correlators reveals possible instabilities of the thermal ensemble, as in previous discussions of the thermodynamics of little string theory. We comment on the dependence of the instability on the spatial volume of the system.
We describe a combinatorial approach to the analysis of the shape and orientation dependence of Wilson loop observables on two-dimensional noncommutative tori. Morita equivalence is used to map the computation of loop correlators onto the combinatorics of non-planar graphs. Several nonperturbative examples of symmetry breaking under area-preserving diffeomorphisms are thereby presented. Analytic expressions for correlators of Wilson loops with infinite winding number are also derived and shown to agree with results from ordinary Yang-Mills theory.
By making use of the duality transformation, gauge field correlators of the Abelian Higgs Model are studied in the London limit. The obtained results are in a good agreement with the dual Meissner scenario of confinement and with the Stochastic Model of QCD vacuum. The nontrivial contribution to the quartic correlator arising due to accounting for the finiteness of the coupling constant is discussed.
The holographic principle in the pp-wave limit proposed in our previous works is further confirmed by studying impurity non-preserving processes which contain a fermionic BMN operator with one scalar and one fermion impurities. We show that the previously proposed duality relation between the matrix elements of the three point interaction Hamiltonian in the holographic string field theory and the OPE coefficients in super Yang-Mills theory holds to the leading order in the large $μ$ limit. Operator mixing is required to obtain the BMN operator of definite conformal dimension which corresponds …
We outline the nonlinear transformation in the path integral representation for partition function of O(N) symmetric oscillator systems bringing their duality to certain one-dimensional oscillators with unstable potential shapes. This duality transformation realizes the equivalence between spectra of a Hermitian and a non-Hermitian Hamiltonians. It is exploited to calculate exactly the spectra of quantum rotators . The zero-temperature limit is considered and the dual representation of n-point correlators of coordinate operators is obtained.
Color-Kinematics Duality and the Double Copy in Curved Spacetimes
Abdus Salam Centre for Theoretical Physics, Imperial College（阿卜杜斯·萨拉姆理论物理中心，帝国理工学院）；School of Physics and Astronomy, University of Southampton（南安普顿大学物理与天文学学院）
