Checkerboard CFT

The Checkerboard conformal field theory is an interesting representative of a large class of non-unitary, logarithmic Fishnet CFTs (FCFT) in arbitrary dimension which have been intensively studied in the last years. Its planar Feynman graphs have the structure of a regular square lattice with checkerboard colouring. Such graphs are integrable since each coloured cell of the lattice is equal to an R-matrix in the principal series representations of the conformal group. We compute perturbatively and numerically the anomalous dimension of the shortest single-trace operator in two reductions of the Checkerboard CFT: the first one corresponds to the fishnet limit of the twisted ABJM theory in 3D, whereas the spectrum in the second, 2D reduction contains the energy of the BFKL Pomeron. We derive an analytic expression for the Checkerboard analogues of Basso--Dixon 4-point functions, as well as for the class of Diamond-type 4-point graphs with disc topology. The properties of the latter are studied in terms of OPE for operators with open indices. We prove that the spectrum of the theory receives corrections only at even orders in the loop expansion and we conjecture such a modification of Checkerboard CFT where quantum corrections occur only with a given periodicity in the loop order.Comment: 66 pages, 24 figure

Bethe ansatz inside Calogero-Sutherland models

We study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura and Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest.Comment: 42 pages, 3 figure

Non-compact integrable spin chain for the conformal fishnet theory

Avant d’être étendue à toute dimension, la théorie fishnet a d’abord été obtenue en quatre dimensions comme une limite de fort twist et de faible couplage de la théorie N = 4 super Yang–Mills. C’est une théorie non-unitaire de deux champs scalaires matriciels complexes interagissant d’une manière si restrictive que, dans la limite planaire, très peu de graphes de Feynman survivent. Il est alors possible de montrer que la théorie est conforme. En outre, l’intégrabilité apparaît naturellement à travers une relation avec une chaîne non-compacte de spins dans une représentation de la série principale du groupe conforme. Certaines classes de graphes de Feynman peuvent en effet être obtenues par l’application répétée d’opérateurs coïncidant avec des charges conservées de la chaîne de spins. La théorie fishnet constitue ainsi un exemple simple et rare d’une théorie conforme des champs intégrable sans supersymétrie en dimension arbitraire. Nous présentons, dans cette thèse, la diagonalisation exacte d’opérateurs associés à la chaîne de spins ouverte. Les vecteurs propres ont une interprétation en tant que fonctions d’onde d’états à plusieurs particules dans une théorie unidimensionnelle miroir. La détermination de la relation de dispersion et de la matrice de diffusion de ces particules miroir nous permet de formuler les équations de l’ansatz de Bethe thermodynamique pour les dimensions conformes d’une famille d’opérateurs dans la théorie fishnet. Dans l’intention de simplifier davantage ce problème spectral nous développons le système Q pour des modèles intégrables avec une symétrie SO(2r). Nous obtenons aussi, pour de tels modèles, de nouvelles expressions pour les matrices de transfert en termes des fonctions Q, quantifiant ainsi les formules classiques de Weyl pour les caractères.The fishnet theory was first obtained in four dimensions as a strongly twisted, weakly coupled limit of N = 4 super Yang– Mills before being extended to arbitrary dimension. It is a non-unitary theory of two complex matrix scalar fields interacting in such a manner that, in the planar limit, only very few Feynman graphs are allowed and, moreover, the bulk of these graphs must be a piece of a square lattice. As a consequence, the theory can be shown to be conformal and integrability naturally appears through a relation with a non-compact chain of spins in principal series representations of the conformal group. Certain classes of Feynman graphs can indeed be built from the repeated application of operators coinciding with conserved charges of the chain. The fishnet theory thus constitutes a rare and simple example of an integrable non-supersymmetric conformal field theory in arbitrary dimension. We present, in this thesis, the exact diagonalisation of the graph-building operators associated with the open spin chain. The eigenvectors have an interpretation as wave functions of multi-particle states in a mirror one-dimensional theory. Extracting the dispersion relation and the scattering matrix of these mirror particles allow us to formulate the thermodynamic Bethe ansatz equations for the conformal dimensions of a whole class of operators in the fishnet theory. As a first step towards a further simplification of this spectral problem, we develop the Q-system for integrable models with SO(2r) symmetry. We also obtain, for such models, new expressions for the transfer matrices, or T-functions, in terms of the Q-functions, thus quantising the classical Weyl formulae for characters

Chaîne de spins intégrable non compacte pour la théorie fishnet conforme

The fishnet theory was first obtained in four dimensions as a strongly twisted, weakly coupled limit of N = 4 super Yang– Mills before being extended to arbitrary dimension. It is a non-unitary theory of two complex matrix scalar fields interacting in such a manner that, in the planar limit, only very few Feynman graphs are allowed and, moreover, the bulk of these graphs must be a piece of a square lattice. As a consequence, the theory can be shown to be conformal and integrability naturally appears through a relation with a non-compact chain of spins in principal series representations of the conformal group. Certain classes of Feynman graphs can indeed be built from the repeated application of operators coinciding with conserved charges of the chain. The fishnet theory thus constitutes a rare and simple example of an integrable non-supersymmetric conformal field theory in arbitrary dimension. We present, in this thesis, the exact diagonalisation of the graph-building operators associated with the open spin chain. The eigenvectors have an interpretation as wave functions of multi-particle states in a mirror one-dimensional theory. Extracting the dispersion relation and the scattering matrix of these mirror particles allow us to formulate the thermodynamic Bethe ansatz equations for the conformal dimensions of a whole class of operators in the fishnet theory. As a first step towards a further simplification of this spectral problem, we develop the Q-system for integrable models with SO(2r) symmetry. We also obtain, for such models, new expressions for the transfer matrices, or T-functions, in terms of the Q-functions, thus quantising the classical Weyl formulae for characters.Avant d’être étendue à toute dimension, la théorie fishnet a d’abord été obtenue en quatre dimensions comme une limite de fort twist et de faible couplage de la théorie N = 4 super Yang–Mills. C’est une théorie non-unitaire de deux champs scalaires matriciels complexes interagissant d’une manière si restrictive que, dans la limite planaire, très peu de graphes de Feynman survivent. Il est alors possible de montrer que la théorie est conforme. En outre, l’intégrabilité apparaît naturellement à travers une relation avec une chaîne non-compacte de spins dans une représentation de la série principale du groupe conforme. Certaines classes de graphes de Feynman peuvent en effet être obtenues par l’application répétée d’opérateurs coïncidant avec des charges conservées de la chaîne de spins. La théorie fishnet constitue ainsi un exemple simple et rare d’une théorie conforme des champs intégrable sans supersymétrie en dimension arbitraire. Nous présentons, dans cette thèse, la diagonalisation exacte d’opérateurs associés à la chaîne de spins ouverte. Les vecteurs propres ont une interprétation en tant que fonctions d’onde d’états à plusieurs particules dans une théorie unidimensionnelle miroir. La détermination de la relation de dispersion et de la matrice de diffusion de ces particules miroir nous permet de formuler les équations de l’ansatz de Bethe thermodynamique pour les dimensions conformes d’une famille d’opérateurs dans la théorie fishnet. Dans l’intention de simplifier davantage ce problème spectral nous développons le système Q pour des modèles intégrables avec une symétrie SO(2r). Nous obtenons aussi, pour de tels modèles, de nouvelles expressions pour les matrices de transfert en termes des fonctions Q, quantifiant ainsi les formules classiques de Weyl pour les caractères

Mirror channel eigenvectors of the d-dimensional fishnets

International audienceWe present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d-dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers (u$_{k}$, l$_{k}$ ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in (1 + 1)-dimensions with O(d) internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation, new to our knowledge, of the matrix elements of the fused R-matrix with O(d) symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions

QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

International audienceWe propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the D$_{r}$ Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length

A large twist limit for any operator

Abstract We argue that for any single-trace operator in N $$\mathcal{N}$$ = 4 SYM theory there is a large twist double-scaling limit in which the Feynman graphs have an iterative structure. Such structure can be recast using a graph-building operator. Generically, this operator mixes between single trace operators with different scaling limits. The mixing captures both the finite coupling spectrum and corrections away from the large twist limit. We first consider a class of short operators with gluons and fermions for which such mixing problems do not arise, and derive their finite coupling spectra. We then focus on a class of long operators with gluons that do mix. We invert their graph-building operator and prove its integrability. The picture that emerges from this work opens the door to a systematic expansion of N $$\mathcal{N}$$ = 4 SYM theory around the large twist limit

Bethe ansatz inside Calogero-Sutherland models

International audienceWe study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura and Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest

Thermodynamic Bethe Ansatz for Biscalar Conformal Field Theories in Any Dimension

International audienceWe present the TBA equations for the exact spectrum of multi-magnon local operators in the D-dimensional bi-scalar fishnet CFT. The mixing matrix of such operators is given in terms of fishnet planar graphs of multiwheel and multispiral type. These graphs probe the two key building blocks of the TBA approach, the magnon dispersion relation and scattering matrix, which we obtain by diagonalizing suitable graph-building operators. We also obtain the dual version of the TBA equations, which relates, in the continuum limit, D-dimensional graphs to two-dimensional sigma models in AdS Dþ1

RF endoluminal coil with NMR electro-optical conversion and transmission for colon wall imaging: Initial finding

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