26 research outputs found
Analytic Bounds and Emergence of Physics from the Conformal Bootstrap
We study analytically the constraints of the conformal bootstrap on the
low-lying spectrum of operators in field theories with global conformal
symmetry in one and two spacetime dimensions. We introduce a new class of
linear functionals acting on the conformal bootstrap equation. In 1D, we use
the new basis to construct extremal functionals leading to the optimal upper
bound on the gap above identity in the OPE of two identical primary operators
of integer or half-integer scaling dimension. We also prove an upper bound on
the twist gap in 2D theories with global conformal symmetry. When the external
scaling dimensions are large, our functionals provide a direct point of contact
between crossing in a 1D CFT and scattering of massive particles in large
. In particular, CFT crossing can be shown to imply that
appropriate OPE coefficients exhibit an exponential suppression characteristic
of massive bound states, and that the 2D flat-space S-matrix should be analytic
away from the real axis.Comment: 46+2 pages, 12 figure
Bootstrapping the 3d Ising twist defect
Recent numerical results point to the existence of a conformally invariant
twist defect in the critical 3d Ising model. In this note we show that this
fact is supported by both epsilon expansion and conformal bootstrap
calculations. We find that our results are in good agreement with the numerical
data. We also make new predictions for operator dimensions and OPE coefficients
from the bootstrap approach. In the process we derive universal bounds on
one-dimensional conformal field theories and conformal line defects.Comment: 24+8 pages, 12 figures, references adde
Explorations in the Conformal Bootstrap
We investigate properties of various conformally invariant quantum systems, especially from the point of view of the conformal bootstrap.
First, we study twist line defects in three-dimensional conformal field theories. Numerical results from lattice simulations point to the existence of such conformal defect in the critical 3D Ising model. We show that this fact is supported by both epsilon expansion and the conformal bootstrap calculations. We find that our results are in a good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects.
Second, we analyze the constraints imposed by the conformal bootstrap for theories with four supercharges in spacetime dimension between 2 and 4. We show how superconformal algebras with four Poincaré supercharges can be treated in a formalism applicable to any, in principle continuous, value of d and use this to construct the superconformal blocks for any dimension between 2 and 4. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the d=2, N=(2, 2) minimal model with central charge c=1 and the theory of a free chiral multiplet in d=4, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions.
Finally, we turn to the question of the analytic origin of the conformal bootstrap bounds. To this end, we introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large AdS. In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis
The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
We study a general class of functionals providing an analytic handle on the
conformal bootstrap equations in one dimension. We explicitly identify the
extremal functionals, corresponding to theories saturating conformal bootstrap
bounds, in two regimes. The first corresponds to functionals that annihilate
the generalized free fermion spectrum. In this case, we analytically find both
OPE and gap maximization functionals proving the extremality of the generalized
free fermion solution to crossing. Secondly, we consider a scaling limit where
all conformal dimensions become large, equivalent to the large radius
limit of gapped theories in . In this regime we demonstrate analytically
that optimal bounds on OPE coefficients lead to extremal solutions to crossing
arising from integrable field theories placed in large . In the process,
we uncover a close connection between asymptotic extremal functionals and
S-matrices of integrable field theories in flat space and explain how 2D
S-matrix bootstrap results can be derived from the 1D conformal bootstrap
equations. These points illustrate that our formalism is capable of capturing
non-trivial solutions of CFT crossing.Comment: 39 pages + appendices, version 2: references adde
Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula
We constrain the low-energy spectra of Laplace operators on closed hyperbolic
manifolds and orbifolds in three dimensions, including the standard
Laplace-Beltrami operator on functions and the Laplacian on powers of the
cotangent bundle. Our approach employs linear programming techniques to derive
rigorous bounds by leveraging two types of spectral identities. The first type,
inspired by the conformal bootstrap, arises from the consistency of the
spectral decomposition of the product of Laplace eigensections, and involves
the Laplacian spectra as well as integrals of triple products of eigensections.
We formulate these conditions in the language of representation theory of
and use them to prove upper bounds on the first
and second Laplacian eigenvalues. The second type of spectral identities
follows from the Selberg trace formula. We use them to find upper bounds on the
spectral gap of the Laplace-Beltrami operator on hyperbolic 3-orbifolds, as
well as on the systole length of hyperbolic 3-manifolds, as a function of the
volume. Further, we prove that the spectral gap of the
Laplace-Beltrami operator on all closed hyperbolic 3-manifolds satisfies
. Along the way, we use the trace formula to estimate the
low-energy spectra of a large set of example orbifolds and compare them with
our general bounds, finding that the bounds are nearly sharp in several cases.Comment: v1: 61 pages + appendices and reference
No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit
We introduce high-energy limits which allow us to derive recursion relations
fixing the various couplings of Lagrangians of two-dimensional relativistic
quantum field theories with no tree-level particle production in a very
straightforward way. The sine-Gordon model, the Bullough-Dodd theory, Toda
theories of various kinds and the U(N) non-linear sigma model can all be
rediscovered in this way. The results here were the outcome of our explorations
at the 2017 Perimeter Institute Winter School.Comment: 20 page
Automorphic Spectra and the Conformal Bootstrap
We point out that the spectral geometry of hyperbolic manifolds provides a
remarkably precise model of the modern conformal bootstrap. As an application,
we use conformal bootstrap techniques to derive rigorous computer-assisted
upper bounds on the lowest positive eigenvalue of the
Laplace-Beltrami operator on closed hyperbolic surfaces and 2-orbifolds . In
a number of notable cases, our bounds are nearly saturated by known surfaces
and orbifolds. For instance, our bound on all genus-2 surfaces is
, while the Bolza surface has
. Our methods can be generalized to
higher-dimensional hyperbolic manifolds and to yield stronger bounds in the
two-dimensional case.Comment: 53+9 pages, 5 figure
Dispersive CFT Sum Rules
We give a unified treatment of dispersive sum rules for four-point
correlators in conformal field theory. We call a sum rule dispersive if it has
double zeros at all double-twist operators above a fixed twist gap. Dispersive
sum rules have their conceptual origin in Lorentzian kinematics and absorptive
physics (the notion of double discontinuity). They have been discussed using
three seemingly different methods: analytic functionals dual to double-twist
operators, dispersion relations in position space, and dispersion relations in
Mellin space. We show that these three approaches can be mapped into one
another and lead to completely equivalent sum rules. A central idea of our
discussion is a fully nonperturbative expansion of the correlator as a sum over
Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion
utilizes the data of two separate channels, while having (term by term) good
Regge behavior in the third channel. We construct sum rules which are
non-negative above the double-twist gap; they have the physical interpretation
of a subtracted version of superconvergence sum rules. We expect dispersive sum
rules to be a very useful tool to study expansions around mean-field theory,
and to constrain the low-energy description of holographic CFTs with a large
gap. We give examples of the first kind of applications, notably, we exhibit a
candidate extremal functional for the spin-two gap problem.Comment: 87 pages + appendices, 16 figures; v2: references adde
Dispersive CFT Sum Rules
We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule dispersive if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of superconvergence sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably, we exhibit a candidate extremal functional for the spin-two gap problem