scholarly journals Bootstrapping mixed correlators in three-dimensional cubic theories II

2020 ◽  
Vol 8 (6) ◽  
Author(s):  
Stefanos R. Kousvos ◽  
Andreas Stergiou
Keyword(s):  
Condensed Matter ◽  
Three Dimensional ◽  
Global Symmetry ◽  
Energy Tensor ◽  
Conformal Field Theories ◽  
Cubic Symmetry ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Perturbative Methods ◽  
Stress Energy

Conformal field theories (CFTs) with cubic global symmetry in 3D are relevant in a variety of condensed matter systems and have been studied extensively with the use of perturbative methods like the \varepsilonε expansion. In an earlier work, we used the nonperturbative numerical conformal bootstrap to provide evidence for the existence of a previously unknown 3D CFT with cubic symmetry, dubbed “Platonic CFT”. In this work, we make further use of the numerical conformal bootstrap to perform a three-dimensional scan in the space of scaling dimensions of three low-lying operators. We find a three-dimensional isolated allowed region in parameter space, which includes both the 3D (decoupled) Ising model and the Platonic CFT. The essential assumptions on the spectrum of operators used to provide the isolated allowed region include the existence of a stress-energy tensor and the irrelevance of certain operators (in the renormalization group sense).

scholarly journals Bootstrapping mixed correlators in three-dimensional cubic theories

2019 ◽  
Vol 6 (3) ◽  
Author(s):  
Stefanos R. Kousvos ◽  
Andreas Stergiou
Keyword(s):  
Phase Transitions ◽  
Parameter Space ◽  
Structural Phase ◽  
Three Dimensional ◽  
Ferromagnetic Phase ◽  
Structural Phase Transitions ◽  
Cubic Symmetry ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Crossing Symmetry

Three-dimensional theories with cubic symmetry are studied using the machinery of the numerical conformal bootstrap. Crossing symmetry and unitarity are imposed on a set of mixed correlators, and various aspects of the parameter space are probed for consistency. An isolated allowed region in parameter space is found under certain assumptions involving pushing operator dimensions above marginality, indicating the existence of a conformal field theory in this region. The obtained results have possible applications for ferromagnetic phase transitions as well as structural phase transitions in crystals. They are in tension with previous \epsilonϵ expansion results, as noticed already in earlier work.

scholarly journals Perturbative and nonperturbative studies of CFTs with MN global symmetry

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Johan Henriksson ◽  
Andreas Stergiou
Keyword(s):  
Fixed Points ◽  
Numerical Data ◽  
Three Dimensions ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
State Of Affairs ◽  
Perturbative Limit ◽  
Good Agreement

Fixed points in three dimensions described by conformal field theories with \ensuremath{M N}_{m,n} = O(m)^n\rtimes S_nMNm,n=O(m)n⋊Sn global symmetry have extensive applications in critical phenomena. Associated experimental data for m=n=2m=n=2 suggest the existence of two non-trivial fixed points, while the \varepsilonε expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters mm and nn, with critical exponents in good agreement with experimental determinations in the m=n=2m=n=2 case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters mm and nn. We find that one family of kinks approaches a perturbative limit as mm increases, and using large spin perturbation theory we construct a large mm expansion that fits well with the numerical data. This new expansion, akin to the large NN expansion of critical O(N)O(N) models, is compatible with the fixed point found in the \varepsilonε expansion. For the other family of kinks, we find that it persists only for n=2n=2, where for large mm it approaches a non-perturbative limit with \Delta_\phi\approx 0.75Δϕ≈0.75. We investigate the spectrum in the case \ensuremath{M N}_{100,2}MN100,2 and find consistency with expectations from the lightcone bootstrap.

scholarly journals QUANTUM ENERGY INEQUALITIES IN TWO-DIMENSIONAL CONFORMAL FIELD THEORY

2005 ◽  
Vol 17 (05) ◽  
pp. 577-612 ◽  
Author(s):  
CHRISTOPHER J. FEWSTER ◽  
STEFAN HOLLANDS
Keyword(s):  
Field Theories ◽  
Positive Energy ◽  
Energy Tensor ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Stress Energy Tensor ◽  
Quantum Energy ◽  
Conformal Field ◽  
Quantum Energy Inequalities ◽  
Stress Energy

Quantum energy inequalities (QEIs) are state-independent lower bounds on weighted averages of the stress-energy tensor, and have been established for several free quantum field models. We present rigorous QEI bounds for a class of interacting quantum fields, namely the unitary, positive energy conformal field theories (with stress-energy tensor) on two-dimensional Minkowski space. The QEI bound depends on the weight used to average the stress-energy tensor and the central charge(s) of the theory, but not on the quantum state. We give bounds for various situations: averaging along timelike, null and spacelike curves, as well as over a space-time volume. In addition, we consider boundary conformal field theories and more general "moving mirror" models. Our results hold for all theories obeying a minimal set of axioms which — as we show — are satisfied by all models built from unitary highest-weight representations of the Virasoro algebra. In particular, this includes all (unitary, positive energy) minimal models and rational conformal field theories. Our discussion of this issue collects together (and, in places, corrects) various results from the literature which do not appear to have been assembled in this form elsewhere.

scholarly journals Analytic and numerical bootstrap of CFTs with $O(m)\times O(n)$ global symmetry in 3D

2020 ◽  
Vol 9 (3) ◽  
Author(s):  
Johan Henriksson ◽  
Stefanos R. Kousvos ◽  
Andreas Stergiou
Keyword(s):  
Power Series ◽  
Fixed Points ◽  
Parameter Space ◽  
Field Theories ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Bootstrap Techniques ◽  
Perturbative Methods ◽  
Anomalous Dimensions

Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with O(m)\times O(n)O(m)×O(n) global symmetry in d=3d=3 spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in \varepsilon=4-dε=4−d and in 1/n1/n, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for O(2)\times O(n)O(2)×O(n) theories for various values of nn. Some of these islands can be attributed to fixed points predicted by perturbative methods like the \varepsilonε and large-nn expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.

scholarly journals STRESS ENERGY TENSOR IN LCFT AND LOGARITHMIC SUGAWARA CONSTRUCTION

2003 ◽  
Vol 18 (25) ◽  
pp. 4771-4788 ◽  
Author(s):  
IAN I. KOGAN ◽  
ALEXANDER NICHOLS
Keyword(s):  
Central Charge ◽  
Energy Tensor ◽  
Conformal Field Theories ◽  
Stress Energy Tensor ◽  
Moody Algebra ◽  
Conformal Field ◽  
Stress Energy ◽  
Dimension One ◽  
Independent Parameters ◽  
Construction Works

We discuss the partners of the stress energy tensor and their structure in Logarithmic conformal field theories. In particular we draw attention to the fundamental differences between theories with zero and non-zero central charge. However they are both characterised by at least two independent parameters. We show how, by using a generalised Sugawara construction, one can calculate the logarithmic partner of T. We show that such a construction works in the c=-2 theory using the conformal dimension one primary currents which generate a logarithmic extension of the Kac-Moody algebra. This is an expanded version of a talk presented by A. Nichols at the conference on Logarithmic Conformal Field Theory and its Applications in Tehran Iran, 2001.

scholarly journals Bootstrap bounds on closed Einstein manifolds

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
James Bonifacio ◽  
Kurt Hinterbichler
Keyword(s):  
Compact Riemannian Manifold ◽  
Tree Level ◽  
Conformal Field Theories ◽  
Einstein Manifolds ◽  
Consistency Conditions ◽  
Conformal Field ◽  
Geometric Data ◽  
Conformal Bootstrap ◽  
Laplacian Operators ◽  
Kaluza Klein

Abstract A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

scholarly journals Bounds in 4D conformal field theories with global symmetry

2010 ◽  
Vol 44 (3) ◽  
pp. 035402 ◽  
Author(s):  
Riccardo Rattazzi ◽  
Slava Rychkov ◽  
Alessandro Vichi
Keyword(s):  
Field Theories ◽  
Global Symmetry ◽  
Conformal Field Theories ◽  
Conformal Field

scholarly journals The conformal anomaly action to fourth order (4T) in $$d=4$$ in momentum space

2021 ◽  
Vol 81 (8) ◽  
Author(s):  
Claudio Corianò ◽  
Matteo Maria Maglio ◽  
Dimosthenis Theofilopoulos
Keyword(s):  
Momentum Space ◽  
Form Factors ◽  
Conformal Anomaly ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Independent Form ◽  
Stress Energy ◽  
Complete Determination ◽  
Non Local

AbstractWe elaborate on the structure of the conformal anomaly effective action up to 4-th order, in an expansion in the gravitational fluctuations (h) of the background metric, in the flat spacetime limit. For this purpose we discuss the renormalization of 4-point functions containing insertions of stress-energy tensors (4T), in conformal field theories in four spacetime dimensions with the goal of identifying the structure of the anomaly action. We focus on a separation of the correlator into its transverse/traceless and longitudinal components, applied to the trace and conservation Ward identities (WI) in momentum space. These are sufficient to identify, from their hierarchical structure, the anomaly contribution, without the need to proceed with a complete determination of all of its independent form factors. Renormalization induces sequential bilinear graviton-scalar mixings on single, double and multiple trace terms, corresponding to $$R\square ^{-1}$$ R □ - 1 interactions of the scalar curvature, with intermediate virtual massless exchanges. These dilaton-like terms couple to the conformal anomaly, as for the chiral anomalous WIs. We show that at 4T level a new traceless component appears after renormalization. We comment on future extensions of this result to more general backgrounds, with possible applications to non local cosmologies.

CONFORMAL THEORIES, CURVED PHASE SPACES, RELATIVISTIC WAVELETS AND THE GEOMETRY OF COMPLEX DOMAINS

1990 ◽  
Vol 02 (01) ◽  
pp. 1-44 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. JADCZYK
Keyword(s):  
General Relativity ◽  
Complex Geometry ◽  
Three Dimensional ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Relativistic Phase

We investigate some aspects of complex geometry in relation with possible applications to quantization, relativistic phase spaces, conformal field theories, general relativity and the music of two and three-dimensional spheres.

Sign in / Sign up

Export Citation Format

Share Document