The conformal bootstrap: Theory, numerical techniques, and applications

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible due to both significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world-record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.

Figure 1

The crossing relation for the 4pt function $⟨{\mathcal{O}}_1{\mathcal{O}}_2{\mathcal{O}}_3{\mathcal{O}}_4⟩$.

Figure 2

A conformal frame defining the $z$ coordinate. From [281].

Figure 3

The positivity of the 6pt function implies reality of the 3pt function coefficient ${\lambda }_{123}$; see the text.

Figure 4

A conformal frame defining the radial coordinate. From [281].

Figure 5

Left: A case where vectors can sum to zero with positive coefficients. Right: A case where vectors cannot sum to zero with positive coefficients and there exists a separating plane $\alpha$ such that all vectors point on one side of the plane. From [433].

Figure 6

The upper bound on the dimension of the second singlet scalar ${{\mathcal{O}}_0}^{\prime }$ as a function of the dimension of the first ${\mathcal{O}}_0$. From [400], Supplementary Material.

Figure 7

The upper bound on ${\mathrm{\Delta }}_{\epsilon }$ as a function of ${\mathrm{\Delta }}_{\sigma }$ in 3D CFTs. From [173].

Figure 8

The allowed region in the $\{{\mathrm{\Delta }}_{\sigma },{\mathrm{\Delta }}_{\epsilon }\}$ plane under the assumption that $\epsilon$ is the only relevant scalar. From [173].

Figure 9

The lower bound on the central charge as a function of ${\mathrm{\Delta }}_{\sigma }$. From [173].

Figure 10

The upper bound on the dimension ${\mathrm{\Delta }}_{T^{\prime }}$ of the first ${\mathbb{Z}}_2$-even spin-2 operator after the stress tensor as a function of ${\mathrm{\Delta }}_{\sigma }$. From [173].

Figure 11

The upper bound on the dimension of the leading ${\mathbb{Z}}_2$-even spin-4 operator. From [173].

Figure 12

An allowed region following from the analysis of three 4pt functions assuming ${\mathrm{\Delta }}_{{\sigma }^{\prime }}\ge 3$ with no assumption on ${\mathrm{\Delta }}_{{\epsilon }^{\prime }}$. From [325].

Figure 13

This plot assumes ${\mathrm{\Delta }}_{{\epsilon }^{\prime }}\ge 3$ (light blue), ${\mathrm{\Delta }}_{{\sigma }^{\prime }}\ge 3$ (medium blue), or both gaps simultaneously (dark blue). From [325].

Figure 14

An allowed region in the $\{{\mathrm{\Delta }}_{\sigma },{\mathrm{\Delta }}_{\epsilon },{\lambda }_{\epsilon \epsilon \epsilon }/{\lambda }_{\sigma \sigma \epsilon }\}$ space obtained by [329].

Figure 15

Projection of the 3D region in Fig. 14 on the $\{{\mathrm{\Delta }}_{\sigma },{\mathrm{\Delta }}_{\epsilon }\}$ plane and its comparison with a Monte Carlo prediction for the same quantities. From [329].

Figure 16

Variation of ${\lambda }_{\epsilon \epsilon \epsilon }$ and ${\lambda }_{\sigma \sigma \epsilon }$ within the allowed region in Fig. 14. From [329].

Figure 17

The spectrum of ${\mathbb{Z}}_2$-even scalar operators appearing in the solution to crossing minimizing $C_T$ near ${\mathrm{\Delta }}_{\sigma }$ corresponding to the 3D Ising model. Line 1 corresponds to the $\epsilon$ operator and shows little variation on the scale of this plot. All other lines exhibit the spectrum rearrangement phenomenon. From [175].

Figure 18

Comparison of the extremal functional spectrum with the analytic bootstrap; see the text. From [484].

Figure 19

The non-Gaussianity ratio $Q$ in the critical 3D Ising model. From [461].

Figure 20

The upper bound on the dimension of $s$. From [326].

Figure 21

The upper bound on the dimension of $t$. From [326].

Figure 22

The lower bound on $C_T$ computed under the assumption ${\mathrm{\Delta }}_s,{\mathrm{\Delta }}_t\ge 1$. From [326].

Figure 23

The Lower bound on $C_J$. From [398].

Figure 24

The $O(N)$ archipelago. From [328].

Figure 25

Allowed regions in the parameter space of $O(2)$ or $U(1)$ symmetric CFTs. The strongest constraint (dark blue) has been obtained from the analysis of three correlators $\{⟨\varphi \varphi \varphi \varphi ⟩,⟨\varphi \varphi ss⟩,⟨ssss⟩\}$, assuming that $s$ and $\varphi$ are the only two relevant scalars of charge 0 and 1, and imposing the OPE coefficient relation ${\lambda }_{\varphi \varphi s}={\lambda }_{\varphi s\varphi }$. From [328].

Figure 26

Allowed islands in the mixed correlator analysis of $O(2)$ or $U(1)$ symmetric CFTs after performing a scan over the OPE coefficient ratio ${\lambda }_{sss}/{\lambda }_{\varphi \varphi s}$. From [329].

Figure 27

The $O(3)$ analog of Fig. 26. From [329].

Figure 28

The upper bounds on the dimension of (a) the first parity-odd scalar $\sigma$ and (b) the first parity-even scalar $\epsilon$ in the OPE $\psi \times \psi$ as a function of ${\mathrm{\Delta }}_{\psi }$. Here $\psi$ is a Majorana fermion primary operator in a 3D parity-invariant unitary CFT. From [291].

Figure 29

The lower bounds on $C_T$ as a function of ${\mathrm{\Delta }}_{\psi }$, where $\psi$ is (a) a Majorana fermion or (b) a multiplet of Majorana fermions in the fundamental representation of an $O(N)$ global symmetry group. From [291] and [290].

Figure 30

The effect of imposing a gap until the second pseudoscalar ${\sigma }^{\prime }$ on the parameter space of 3D parity-invariant CFTs. From [291].

Figure 31

The effect of imposing that there is only one relevant pseudoscalar ${\mathrm{\Delta }}_{{\sigma }^{\prime }}\ge 3$ in 3D parity-invariant CFTs. From [291].

Figure 32

The effect of imposing a gap ${\mathrm{\Delta }}_{{\sigma }^{\prime }}\ge 3$ in the singlet pseudoscalar sector of $O(N)$-symmetric fermionic CFTs. The kinks at low $N$ may perhaps be identified with the GNY and GNY* CFTs. From [290].

Figure 33

The upper bounds on the dimension of the symmetric traceless pseudoscalar ${\sigma }_T$ in the OPE ${\psi }_i\times {\psi }_j$ in $O(N)$-symmetric fermionic CFTs. Notice the mysterious jumps in the wide view of the bounds (a) when they cross marginality. (b) A zoom on the small ${\mathrm{\Delta }}_{\psi }$ region, where the bounds exhibit kinks, in agreement with the GNY dimensions at large $N$. From [290].

Figure 34

Bounds on (a) ${\mathrm{\Delta }}_{M_1}$ in terms of ${\mathrm{\Delta }}_{M_{1/2}}$ in $d=3$ for $N_f=2$ and 6, and (b) with various assumptions on the gaps in the uncharged sector in the same $SU(N_f)$ representation as $M_1$. From [106].

Figure 35

(a) The analog of Fig. 34 for $N_f=4$. (b) Starting from the ${\mathrm{\Delta }}_2\ge 3$ case of (a) and showing that placing an additional gap ${\mathrm{\Delta }}_{M_1^{\prime }}$ above ${\mathrm{\Delta }}_{M_1}$ turns the kink into a peninsula. From [106].

Figure 36

The 3D upper bound on ${\mathrm{\Delta }}_2$ as a function of ${\mathrm{\Delta }}_1$. It may be possible to improve this bound if ${\mathrm{\Delta }}_0$ is known. The same bound applies to $M_2\times M_2\sim M_4$. From [400].

Figure 37

An upper bound on ${\mathrm{\Delta }}_3$ as a function of $\{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2\}$ under the assumptions that ${\mathrm{\Delta }}_0>1.044$, ${\mathrm{\Delta }}_4>3$. It follows from Fig. 36 that the range of ${\mathrm{\Delta }}_2$ is restricted by the latter assumption from below, and, for fixed ${\mathrm{\Delta }}_1$, from above. From [400].

Figure 38

Bounds on $C_T$ as a function of the $⟨TTT⟩$ 3pt function parameter $\theta$. From [165].

Figure 39

Bounds on $C_T$ as a function of the $⟨JJT⟩$ 3pt function parameter $\gamma$. From [166].

Figure 40

Bounds on $C_T$ as a function of the $⟨TTT⟩$ 3pt function parameter $\theta$ for different values of the parity-odd scalar gap ${\mathrm{\Delta }}_{\mathrm{odd}}$. From [165].

Figure 41

Bounds on $C_T$ as a function of the $⟨TT⟩$ 3pt function parameter $\theta$ assuming that the leading parity-even scalar is irrelevant. From [165].

Figure 42

Bounds on $C_T$ as a function of the $⟨JJT⟩$ 3pt function parameter $\gamma$ with no assumptions (lower solid curve), parity-odd scalars irrelevant (lower dashed curve), parity-even scalars irrelevant (upper dashed curve), and all scalars irrelevant (upper solid curve). From [166].

Figure 43

Bounds on $C_T$ as a function of the $⟨TTT⟩$ 3pt function parameter $\theta$ with gap assumptions plausible for the Ising model. From [165].

Figure 44

Bounds on $C_T$ as a function of the $⟨JJT⟩$ 3pt function parameter $\gamma$ with gap assumptions plausible for the critical $O(2)$ model. From [166].

Figure 45

Allowed region for parity-even and parity-odd scalar gaps from (a) the $⟨TTTT⟩$ bootstrap. From [165]. (b) The $⟨JJJJ⟩$ bootstrap. From [166].

Figure 46

(a) The upper bound on the dimension of the first scalar in the $\varphi \times \varphi$ OPE as a function of ${\mathrm{\Delta }}_{\varphi }$ in 4D unitary CFTs; (b) the lower bound on the central charge $C_T$, computed by maximizing the OPE coefficient ${\lambda }_{\varphi \varphi T}$. From [434].

Figure 47

(a) The upper bounds on the singlet scalar dimension in $SO(N)$ and $SU(N)$ symmetric 4D CFTs, as a function of ${\mathrm{\Delta }}_{\varphi }$ in the fundamental; (b) the lower bounds on $C_T$ in the same theories. From [434].

Figure 48

The lower bounds on $C_J$ in $SO(N)$-symmetric unitary 4D CFTs as a function of the dimension of a scalar in the fundamental ([434]). $SU(N/2)$ adjoint currents satisfy the same bound ([79]).

Figure 49

The lower bounds on the inverse-square OPE coefficient of a singlet current in $SU(N)$-symmetric unitary 4D CFTs as a function of dimension of a scalar in the fundamental. From [434].

Figure 50

Viable regions in the $\{{\mathrm{\Delta }}_H,{\mathrm{\Delta }}_S\}$ plane for conformal technicolor models in the flavor-generic (red) and flavor-optimistic (cross-hatched green) cases, superimposed with the $SO(4)$ bound. Regions for no tuning, 10%, and 1% tuning are shown in successively lighter shades of each color, with the largest region corresponding to 1% tuning in each case. Flavor-generic models are ruled out. From [434].

Figure 51

The upper bound on the dimension of the first singlet operator appearing in the OPE $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ for $N=8$ as a function of ${\mathrm{\Delta }}_{\mathrm{\Phi }}$. From [395].

Figure 52

The upper bounds on the dimensions of the first $TT$ and $AA$ operators appearing in the OPE $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ for $N=8$ as a function of ${\mathrm{\Delta }}_{\mathrm{\Phi }}$. From [395].

Figure 53

(a) The upper bound on the dimension of the operator $\mathcal{R}$ as a function of ${\mathrm{\Delta }}_{\mathrm{\Phi }}$. The shaded area is excluded. The dashed line at ${\mathrm{\Delta }}_{\mathcal{R}}=2{\mathrm{\Delta }}_{\mathrm{\Phi }}$ corresponds to generalized free theories. From [350] and [434]. (b) The lower and upper bounds on the OPE coefficient of the chiral operator ${\mathrm{\Phi }}^2$ entering the $\mathrm{\Phi }\times \mathrm{\Phi }$ OPE. The vertical dotted line is at ${\mathrm{\Delta }}_{\mathrm{\Phi }}=1.407$ and the horizontal dashed line is at the free theory value ${\lambda }_{\mathrm{\Phi }\mathrm{\Phi }{\mathrm{\Phi }}^2}\equiv {\lambda }_{\varphi }^2=\sqrt{2}$. From [435].

Figure 54

(a) The lower bound on the central charge as a function of ${\mathrm{\Delta }}_{\mathrm{\Phi }}$ assuming that ${\mathrm{\Delta }}_{\mathcal{R}}$ is consistent with the unitarity bound (thin line) or it saturates the upper bound in Fig. 53 (thick line). The shaded area is excluded. From [350]. (b) The lower and upper bounds on the central charge as a function of ${\mathrm{\Delta }}_{\mathrm{\Phi }}$, with the assumption that there is no ${\mathrm{\Phi }}^2$ operator. The upper bounds correspond to different gaps until the second spin-1 superconformal primary ${\mathrm{\Delta }}_{\ell =1}\ge 3.1,3.3,3.5,3.7,3.9,4,4.1$ (from left to right). The shaded area is excluded. From [435].

Figure 55

The upper bounds on the OPE coefficients $c_{\mathcal{O}}\equiv 2^{-\ell /2}{\lambda }_{JJ\mathcal{O}}$ appearing in the $J\times J$ OPE arising from (a) $J$ itself or (b) the stress-tensor supermultiplet $V$, as a function of the dimension of the first unprotected scalar $\mathcal{O}$ in the $J\times J$ OPE. The region to the right of the dotted vertical line at ${\mathrm{\Delta }}_{\mathcal{O}}=5.246$ is not allowed. From [350].

Figure 56

The lower bounds on the effective 2pt function coefficient ${\kappa }_{\mathrm{eff}}=1/{\lambda }_{\mathrm{\Phi }\mathrm{\Phi }J}^2$ of $SU(N)$ singlet currents appearing in $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$, where $\mathrm{\Phi }$ is a chiral scalar of dimension $d$ in the fundamental of $SU(N)$, for $N=2,\dots ,14$. The bounds are normalized to the value ${\kappa }_{\text{chiral}}$ corresponding to a free chiral superfield. Each dot connected to a bound corresponds to the exact value in an SQCD theory with the same symmetry. From [434].

Figure 57

The bound on the dimension of the first unprotected scalar $\mathrm{\Phi }\overline{\mathrm{\Phi }}$ in the $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ OPE in 3D SCFTs with $\mathcal{N}=2$ supersymmetry. From [71].

Figure 58

The light cone in the embedding space; light rays are in one-to-one correspondence with points of ${\mathbb{R}}^d$. The Poincaré section of the cone is also shown. From [132].