140 research outputs found
Carving Out the Space of 4D CFTs
We introduce a new numerical algorithm based on semidefinite programming to
efficiently compute bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. Using our
algorithm, we dramatically improve previous bounds on a number of CFT
quantities, particularly for theories with global symmetries. In the case of
SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal
technicolor. In N=1 superconformal theories, we place strong bounds on
dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the
line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive
anomalous dimensions in this region. We also place novel upper and lower bounds
on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we
find examples of lower bounds on central charges and flavor current two-point
functions that scale with the size of global symmetry representations. In the
case of N=1 theories with an SU(N) flavor symmetry, our bounds on current
two-point functions lie within an O(1) factor of the values realized in
supersymmetric QCD in the conformal window.Comment: 60 pages, 22 figure
Bootstrapping Mixed Correlators in the 3D Ising Model
We study the conformal bootstrap for systems of correlators involving
non-identical operators. The constraints of crossing symmetry and unitarity for
such mixed correlators can be phrased in the language of semidefinite
programming. We apply this formalism to the simplest system of mixed
correlators in 3D CFTs with a global symmetry. For the leading
-odd operator and -even operator
, we obtain numerical constraints on the allowed dimensions
assuming that and are
the only relevant scalars in the theory. These constraints yield a small closed
region in space compatible with the known
values in the 3D Ising CFT.Comment: 39 pages, 6 figure
Bootstrapping the O(N) Vector Models
We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We
obtain rigorous upper bounds on the scaling dimensions of the first O(N)
singlet and symmetric tensor operators appearing in the
OPE, where is a fundamental of O(N). Comparing these bounds to
previous determinations of critical exponents in the O(N) vector models, we
find strong numerical evidence that the O(N) vector models saturate the
bootstrap constraints at all values of N. We also compute general lower bounds
on the central charge, giving numerical predictions for the values realized in
the O(N) vector models. We compare our predictions to previous computations in
the 1/N expansion, finding precise agreement at large values of N.Comment: 26 pages, 5 figures; V2: typos correcte
A spacetime derivation of the Lorentzian OPE inversion formula
Caron-Huot has recently given an interesting formula that determines OPE data
in a conformal field theory in terms of a weighted integral of the four-point
function over a Lorentzian region of cross-ratio space. We give a new
derivation of this formula based on Wick rotation in spacetime rather than
cross-ratio space. The derivation is simple in two dimensions but more involved
in higher dimensions. We also derive a Lorentzian inversion formula in one
dimension that sheds light on previous observations about the chaos regime in
the SYK model.Comment: 26 pages plus appendice
Non-gaussianity of the critical 3d Ising model
We discuss the 4pt function of the critical 3d Ising model, extracted from
recent conformal bootstrap results. We focus on the non-gaussianity Q - the
ratio of the 4pt function to its gaussian part given by three Wick
contractions. This ratio reveals significant non-gaussianity of the critical
fluctuations. The bootstrap results are consistent with a rigorous inequality
due to Lebowitz and Aizenman, which limits Q to lie between 1/3 and 1.Comment: 10 pages, 6 figures; v2: refs added; v3: refs updated, published
version; v4: acknowledgement adde
Weight Shifting Operators and Conformal Blocks
We introduce a large class of conformally-covariant differential operators
and a crossing equation that they obey. Together, these tools dramatically
simplify calculations involving operators with spin in conformal field
theories. As an application, we derive a formula for a general conformal block
(with arbitrary internal and external representations) in terms of derivatives
of blocks for external scalars. In particular, our formula gives new
expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple
derivations of identities between external-scalar blocks with different
dimensions and internal spins. We comment on additional applications, including
derivation of recursion relations for general conformal blocks, reducing
inversion formulae for spinning operators to inversion formulae for scalars,
and deriving identities between general 6j symbols (Racah-Wigner
coefficients/"crossing kernels") of the conformal group.Comment: 84 page
Bootstrapping the O(N) Archipelago
We study 3d CFTs with an global symmetry using the conformal bootstrap
for a system of mixed correlators. Specifically, we consider all nonvanishing
scalar four-point functions containing the lowest dimension vector
and the lowest dimension singlet , assumed to be the only
relevant operators in their symmetry representations. The constraints of
crossing symmetry and unitarity for these four-point functions force the
scaling dimensions to lie inside small islands. We
also make rigorous determinations of current two-point functions in the
and models, with applications to transport in condensed matter systems.Comment: 32 pages, 13 figures; updated Fig.2, added references and minor
corrections in Sec.3.
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