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 CFT unitarity and the AdS Cutkosky rules

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
David Meltzer ◽  
Allic Sivaramakrishnan
Keyword(s):  
Flat Space ◽  
Tree Level ◽  
Optical Theorem ◽  
Conformal Field Theories ◽  
Strongly Coupled ◽  
Conformal Field ◽  
Weakly Coupled ◽  
S Matrix ◽  
Space Limit ◽  
Diagrammatic Method

Abstract We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

scholarly journals Bootstrap experiments on higher dimensional CFTs

2018 ◽  
Vol 33 (07) ◽  
pp. 1850036 ◽  
Author(s):  
Yu Nakayama
Keyword(s):  
Empirical Relationship ◽  
Adjoint Representation ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Time Dimension ◽  
Unitarity Bound ◽  
Conformal Field ◽  
Dimension Formula ◽  
Conformal Bootstrap ◽  
Higher Dimensional

Recent programs on conformal bootstrap suggest an empirical relationship between the existence of nontrivial conformal field theories and nontrivial features such as a kink in the unitarity bound of conformal dimensions in the conformal bootstrap equations. We report the existence of nontrivial kink-like behaviors in the unitarity bound of scalar operators in the adjoint representation of the [Formula: see text] symmetric conformal field theories. They have interesting properties: (1) the kink-like behaviors exist in [Formula: see text] dimensions; (2) the location of kink-like behaviors are when the unitarity bound hits the space–time dimension [Formula: see text]; (3) there exists a “conformal window” of [Formula: see text], where [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text].

scholarly journals A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions

2021 ◽  
Vol 11 (6) ◽  
Author(s):  
Yin-Chen He ◽  
Junchen Rong ◽  
Ning Su
Keyword(s):  
Gauge Group ◽  
Gauge Theories ◽  
Higher Dimensions ◽  
Equation Of Motion ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Gauge Groups ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Null Operator

We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions d>2d>2. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of decoupling operator, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of 2d2d Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a U(1)U(1) gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both d=3d=3 and d=2+\epsilond=2+ϵ dimensions.

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 Two-photon processes in conformal QCD: resummation of the descendants of leading-twist operators

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
V.M. Braun ◽  
Yao Ji ◽  
A.N. Manashov
Keyword(s):  
Deeply Virtual Compton Scattering ◽  
Virtual Compton Scattering ◽  
Field Theories ◽  
Tree Level ◽  
Operator Product ◽  
Conformal Field Theories ◽  
Closed Expression ◽  
Conformal Field ◽  
Two Photon ◽  
Twist Operators

Abstract Using some techniques of conformal field theories, we find a closed expression for the contribution of leading twist operators and their descendants, obtained by adding total derivatives, to the operator product expansion (OPE) of two electromagnetic currents in QCD. Our expression resums contributions of all twists and to all orders in perturbation theory up to corrections proportional to the QCD β-function. At tree level and to twist-four accuracy, our result agrees with the expression derived earlier by a different method. The results are directly applicable to deeply-virtual Compton scattering and, e.g., γγ∗ annihilation in two mesons. As a byproduct, we derive a simple representation for the OPE of two scalar currents that is convenient for applications.

scholarly journals Applications of dispersive sum rules: $ε$-expansion and holography

2021 ◽  
Vol 10 (6) ◽  
Author(s):  
Dean Carmi ◽  
Joao Penedones ◽  
Joao A. Silva ◽  
Alexander Zhiboedov
Keyword(s):  
Sum Rules ◽  
Dispersion Relations ◽  
Field Theories ◽  
Tree Level ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Anomalous Dimensions ◽  
Fisher Model ◽  
Twist Operators ◽  
Space Dispersion

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in d=4-\epsilond=4−ϵ dimensions. We re-derive many of the known results to order \epsilon^4ϵ4 and we make new predictions. No assumption of analyticity down to spin 00 was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.

scholarly journals Double copy structure of parity-violating CFT correlators

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Sachin Jain ◽  
Renjan Rajan John ◽  
Abhishek Mehta ◽  
Amin A. Nizami ◽  
Adithya Suresh
Keyword(s):  
Flat Space ◽  
Field Theories ◽  
Tree Level ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Higher Spin ◽  
Space Limit ◽  
Double Copy ◽  
Conserved Currents ◽  
Massless Fields

Abstract We show that general parity-violating 3d conformal field theories show a double copy structure for momentum space 3-point functions of conserved currents, stress tensor and marginal scalar operators. Splitting up the CFT correlator into two parts — called homogeneous and non-homogeneous — we show that double copy relations exist for each part separately. We arrive at similar conclusions regarding double copy structures using tree-level correlators of massless fields in dS4. We also discuss the flat space limit of these correlators. We further extend the double copy analysis to correlators involving higher-spin conserved currents, which suggests that the spin-s current correlator can be thought of as s copies of the spin one current correlator.

scholarly journals Exclusion inside or at the border of conformal bootstrap continent

2020 ◽  
Vol 35 (06) ◽  
pp. 2050036
Author(s):  
Yu Nakayama
Keyword(s):  
Bound States ◽  
Pauli Exclusion Principle ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Field Theories ◽  
Exclusion Principle ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Anomalous Dimensions

How large can anomalous dimensions be in conformal field theories? What can we do to attain larger values? One attempt to obtain large anomalous dimensions efficiently is to use the Pauli exclusion principle. Certain operators constructed out of constituent fermions cannot form bound states without introducing nontrivial excitations. To assess the efficiency of this mechanism, we compare them with the numerical conformal bootstrap bound as well as with other interacting field theory examples. In two dimensions, it turns out to be the most efficient: it saturates the bound and is located at the (second) kink. In higher dimensions, it more or less saturates the bound but it may be slightly inside.

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