13 research outputs found
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 dâĂȘtre Ă©tendue Ă toute dimension, la thĂ©orie fishnet a dâabord Ă©tĂ© obtenue en quatre dimensions comme une limite de fort twist et de faible couplage de la thĂ©orie N = 4 super YangâMills. Câest une thĂ©orie non-unitaire de deux champs scalaires matriciels complexes interagissant dâune maniĂšre si restrictive que, dans la limite planaire, trĂšs peu de graphes de Feynman survivent. Il est alors possible de montrer que la thĂ©orie est conforme. En outre, lâintĂ©grabilitĂ© apparaĂźt naturellement Ă travers une relation avec une chaĂźne non-compacte de spins dans une reprĂ©sentation de la sĂ©rie principale du groupe conforme. Certaines classes de graphes de Feynman peuvent en effet ĂȘtre obtenues par lâapplication rĂ©pĂ©tĂ©e dâopĂ©rateurs coĂŻncidant avec des charges conservĂ©es de la chaĂźne de spins. La thĂ©orie fishnet constitue ainsi un exemple simple et rare dâune thĂ©orie conforme des champs intĂ©grable sans supersymĂ©trie en dimension arbitraire. Nous prĂ©sentons, dans cette thĂšse, la diagonalisation exacte dâopĂ©rateurs associĂ©s Ă la chaĂźne de spins ouverte. Les vecteurs propres ont une interprĂ©tation en tant que fonctions dâonde dâĂ©tats Ă plusieurs particules dans une thĂ©orie unidimensionnelle miroir. La dĂ©termination de la relation de dispersion et de la matrice de diffusion de ces particules miroir nous permet de formuler les Ă©quations de lâansatz de Bethe thermodynamique pour les dimensions conformes dâune famille dâopĂ©rateurs dans la thĂ©orie fishnet. Dans lâintention de simplifier davantage ce problĂšme spectral nous dĂ©veloppons le systĂšme Q pour des modĂšles intĂ©grables avec une symĂ©trie SO(2r). Nous obtenons aussi, pour de tels modĂšles, de nouvelles expressions pour les matrices de transfert en termes des fonctions Q, quantifiant ainsi les formules classiques de Weyl pour les caractĂšres.The fishnet theory was first obtained in four dimensions as a strongly twisted, weakly coupled limit of N = 4 super Yangâ 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
Chaßne 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 Yangâ 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 dâĂȘtre Ă©tendue Ă toute dimension, la thĂ©orie fishnet a dâabord Ă©tĂ© obtenue en quatre dimensions comme une limite de fort twist et de faible couplage de la thĂ©orie N = 4 super YangâMills. Câest une thĂ©orie non-unitaire de deux champs scalaires matriciels complexes interagissant dâune maniĂšre si restrictive que, dans la limite planaire, trĂšs peu de graphes de Feynman survivent. Il est alors possible de montrer que la thĂ©orie est conforme. En outre, lâintĂ©grabilitĂ© apparaĂźt naturellement Ă travers une relation avec une chaĂźne non-compacte de spins dans une reprĂ©sentation de la sĂ©rie principale du groupe conforme. Certaines classes de graphes de Feynman peuvent en effet ĂȘtre obtenues par lâapplication rĂ©pĂ©tĂ©e dâopĂ©rateurs coĂŻncidant avec des charges conservĂ©es de la chaĂźne de spins. La thĂ©orie fishnet constitue ainsi un exemple simple et rare dâune thĂ©orie conforme des champs intĂ©grable sans supersymĂ©trie en dimension arbitraire. Nous prĂ©sentons, dans cette thĂšse, la diagonalisation exacte dâopĂ©rateurs associĂ©s Ă la chaĂźne de spins ouverte. Les vecteurs propres ont une interprĂ©tation en tant que fonctions dâonde dâĂ©tats Ă plusieurs particules dans une thĂ©orie unidimensionnelle miroir. La dĂ©termination de la relation de dispersion et de la matrice de diffusion de ces particules miroir nous permet de formuler les Ă©quations de lâansatz de Bethe thermodynamique pour les dimensions conformes dâune famille dâopĂ©rateurs dans la thĂ©orie fishnet. Dans lâintention de simplifier davantage ce problĂšme spectral nous dĂ©veloppons le systĂšme Q pour des modĂšles intĂ©grables avec une symĂ©trie SO(2r). Nous obtenons aussi, pour de tels modĂšles, de nouvelles expressions pour les matrices de transfert en termes des fonctions Q, quantifiant ainsi les formules classiques de Weyl pour les caractĂšres
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, l ), 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 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 = 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 = 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 DĂŸ1
RF endoluminal coil with NMR electro-optical conversion and transmission for colon wall imaging: Initial finding
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