The effective bootstrap
Received: June
The e ective bootstrap
Alejandro Castedo Echeverri 0 1 2
Benedict von Harling 0 1 2
Marco Serone 0 1 2
SISSA 0 1 2
0 Strada Costiera 11 , I-34151 Trieste , Italy
1 Notkestrasse 85 , 22607 Hamburg , Germany
2 Via Bonomea 265, I-34136 Trieste , Italy
We study the numerical bounds obtained using a conformal-bootstrap method | advocated in ref. [1] but never implemented so far | where di erent points in the plane of conformal cross ratios z and z are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point z = z = 1=2, we can consistently \integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this \e ective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature.
Conformal and W Symmetry; Conformal Field Theory
Convergence of the OPE
Bootstrapping with multiple points
3.1
Choice of points
Comparison with generalized free theories and asymptotics for z ! 1
Remainder for CFTs with O(n) symmetry
3D Ising and O(n) models
A closer look at the spectrum of 3D O(n) models
1 Introduction
2.1
2.2
4.1
4.3
Results
4.2 4D CFTs
2
3
4
5
6
1
Details of the implementation
Conclusions
Introduction
There has recently been a great revival of interest in the conformal bootstrap program [2, 3]
after ref. [4] observed that its applicability extends to Conformal Field Theories (CFTs) in
d > 2 dimensions. Since ref. [4], considerable progress has been achieved in understanding
CFTs in d 2 dimensions, both numerically and analytically. Probably the most striking progress has been made in the numerical study of the 3D Ising model, where amazingly precise operator dimensions and OPE coe cients have been determined [5{7].
Essentially all numerical bootstrap studies so far have used the constraints imposed
by crossing symmetry on 4-point correlators evaluated at a speci c value of the conformal
cross-ratios, u = v = 1=4, or equivalently in z-coordinates at z = z = 1=2 [8]. This is
the point of best convergence for the combined conformal block expansions in the s and t
channels. Taking higher and higher derivatives of the bootstrap equations evaluated at this
point has proven to be very e ective and successful in obtaining increasingly better bounds.
We will denote this method in the following as the \derivative method". A drawback of
the derivative method | both in its linear [4, 6, 9] or semi-de nite [10, 11]
programming incarnations | is the need to include a large number of operators in the bootstrap
equations. This makes any, even limited, analytical understanding of the obtained results
quite di cult.
A possible approximation scheme is in fact available: ref. [12] has determined the rate of
convergence of the Operator Product Expansion (OPE), on which the bootstrap equations
are based. This allows us to extract the maximal error from neglecting operators with
dimensions larger than some cuto
in the bootstrap equations and thus to consistently
truncate them. These truncated bootstrap equations can then be evaluated at di erent
points in the z-plane. This method, which we denote as the \multipoint method", has
{ 1 {
and study the resulting bounds. It is important to emphasize that the method of ref. [
1
]
combines what are in principle two independent ideas: i) multipoint bootstrap and ii)
truncation of the bootstrap equations. One could study i) without ii), or try to analyze ii)
without i). We will not consider these other possibilities here.
We begin in section 2 with a brief review of the results of refs. [
1, 12, 13
] on the
convergence of the OPE. We use generalized free theories as a toy laboratory to test some
of the results obtained in ref. [12]. We then generalize the results of ref. [12] for CFTs with
an O(n) global symmetry.
We write the bootstrap equations and set the stage for our numerical computations in
section 3. Our results are then presented in section 4. For concreteness, we study bounds
on operator dimensions and the central charge in 3D and 4D CFTs, with and without an
O(n) global symmetry (with no supersymmetry). For these bounds, extensive results are
already available in the literature (see e.g. refs. [5{7, 10, 14{22]). In particular, we focus our
attention on the regions where the 3D Ising and O(n) vector models have been identi ed.
We show how the results depend on the number N of points in the z-plane at which we
evaluate the bootstrap equations and the cut-o
on the dimension of operators in the
bootstrap equations. Using values for the dimension of the operator
in O(n) vector
models available in the literature and a t extrapolation procedure, we then determine the
dimensions of the second-lowest O(n) singlet and symmetric-traceless operators S0 and T 0
for n = 2; 3; 4. To our knowledge, these have not been obtained before using bootstrap
techniques. Our (...truncated)