Recent Progress inConformal Bootstrap
＠JICFuS seminar, 2015/4/15

Tomoki Ohtsuki (Kavli IPMU)

Part 1 : Bootstrap Minimal

Relevance of CFTs
 Why are conformal field theories (CFTs) so important?
 UV & IR limit of RG flows must be scale invariant.
 In most cases scale invariance together with unitarity &Lorentz invariance implies an enhanced symmetry, that is,
conformal symmetry (see e.g. arXiv:1302.0884).

 Thus, CFTs are ubiquitous in theoretical (especially highenergy & condensed matter) physics.

Philosophy of the conformalbootstrap program
 “Solve the CFTs from consistency
conditions, without assuming Lagrangian”(Ferrara-Gatto-Grillo ‘73, Polyakov ‘74)

 … Seems quite unwieldy because there is an
infinite # of unknown parameters in CFTs and theconsistency conditions are also infinitedimensional.

 Is it just an empty dream?

Numerical conformal bootstrap（Rattazzi, Rychkov, Tonni, Vichi, `08)
 It is realized that the CFT consistency condition alone allows us
to delineate the following curve.
Δ

Δ
Figure from (Poland et. al., ‘11)

 Meaning: for general 4d CFTs with dimension Δ scalar , theOPE must contain a scalar with dimension smaller than
∆ Δ .

the 3d Ising “solution”
 Subsequently the analysis forin:

3 was made, resultingΔ

Δ It developed a “kink” at the position where the 3d Isingmodel parameters sit. (El-showk et. al., ‘12)
 It is quite mysterious why such a phenomenon take place.

Goal of Part I Tell you how we can obtain such non-trivial curves solely
from CFT consistency conditions, without relying on
Lagrangians!
Outline1. Conformal kinematics
2. Conformal block decomposition of 4pt functions
3. The bootstrap equation and linear functional argument

1.1 Some conformal kinematics

The conformal invariance Conformal transformation means diffeomorphism which
preserves the metric up to position dependent Weyl
rescaling:= e It is easy to classify the solutions for the flat background.
They are spanned by the usual Poincare transformation,
dilatation, and special conformal transformation (SCT):
⇒

1

2 ⋅

The state-operator
correspondence We require that the states in the Hilbert space are in oneto-one with the local operators:
0 |0|(Note : If there’s a Lagrangian, this can be derived.)

 Operators (states) which can be written as the derivatives( -action) of other local operators is called descendant
operators (states). Otherwise they are called primary.

 The CFT Hilbert space are spanned by
0 |0 ,
…, |
,

|

0 |0
,
|

| ,
| ,
,…, and so on.

|,Kinematical constraints
from unitarity
 The scaling dimensions of operators cannot be arbitrary inunitary CFTs.

 In particular, primary operators (except for identity op. )must have its scaling dimension
2
0
Δ
2
2
0
, where is the spin of . Otherwise some descendantstates acquire negative norm!

 But no upper bound in kinematical level.

Conformal 2 and 3pt functions
 Thanks to dilation, 2pt function of scale invariant QFT is determined
to be

1

Δ : scaling dim. of Thanks to the special conformal transf., in CFTs the 3pt functions is
also determined by kinematics:(First taketo origin by translation, thento infinity by SCT andto (1,0,…,0) by rotation+ dilatation.)
finally Hererefers the OPE coefficient,
0 ~04pt function
 4pt functions cannot be determined from kinematics alone.
 All we can do for
1. First takeis toto the origin.

2. Thento the infinity by special conformal transf.

3. Then

to 1,0, ⋯ , 0 by dilation + rotation.

4. Finallyto, , 0, ⋯ , 0 by rotation (which fixes Thus the 4pt function is recovered from 2d-like“standard” configuration.

).

1.2 Conformal block
decomposition of 4pt function

A lesson from elementary
quantum mechanics
 4pt correlators are encoded in some functions on 2dplane, which is not fixed kinematically.

 But to some extent we can pursue, if one remembers thelogic in elementary quantum mechanics, that is,
1

|

|

:

 In CFTs the sum over states reads:
|
:

|

|
:

:

|

“Conformal partial wave”
 Insert the complete set of states in the identical scalar
4pt function||:

 Then organize them as a double-summation:
|
:

:

|

Matrix elements as 3pt function
 Fixing the primary , evaluate the matrix elements like0.

 Recall that |and |is of the form⋯
0 |0 hence
⋯

|

|
⋯

0 |0 .

 Thus matrix elements are computed from the 3pt function,
lim
→

⋯

0

0

Recall that 3pt function is determined kinematically up to
OPE coefficients!

The conformal block Thus the partial sum can be rewritten,
∑ :

|

|

, ̅; Δ , Hereis theconfiguration.

 The function,Δcoordinate in the “standard”

is fixed by CFT kinematics and called the“conformal block”. (In old literature it is called “conformal
partial wave”)

The “s-channel” decomposition, ̅; Δ , Δ It turns out

is independent of Δ .

 The final form for 4pt function is1, ̅; Δ ,: This expression is rapidly converging. (Pappadpulo, Espin,
Rychkov, Rattazzi, ‘12)

1.3 The Bootstrap Equation andLinear functional Argument

Crossing symmetry
 Actually our conformal correlators behave properly only
when they are “radial ordered”, i.e., operator ordering
means

| |.

 We can equally compute the 4pt function from anotherordering, say
conformal block decomposition.

, and perform the These two expression must agree when both of themconverge!

Conformal block in t -channel
 Correlator in the ordering

can bedecomposed
1

1

,1

̅; Δ ,:

 Note that,
1. Conformal block argument has been changed to 1
2. OPE factor and the summation range are the same!

The bootstrap equation
 Thus we have, (adding prefactor0,), ̅ ;Δ ,,Δ:, where, ̅: Δ ,,Δ, ̅: Δ ,1,1

̅: Δ ,

 Thus in CFT, infinitely many unknowns are constrained byinfinite-dimensional constraint.

Linear functional argument -1
 Fix Δ and assume there is a linear functionalΛ ∶ functions onplane →and some “hypothetical gap” Δ , with the following property:Λ, ̅, Δ, , ΔΛ, ̅, Δ, , ΔΛ

0 for Δ0 for Δ, ̅, 0,0, Δ

2, if
Δ , if000

 Then apply this functional to the bootstrap equation,, ̅ ;Δ ,0:,ΔLinear functional argument -2
 0

Λ 0

∑

Λ

Λ

, ̅; Δ ,, ̅; Δ ,
,Δ

:

,

Λ

, ̅; Δ ,Λ, ̅; Δ ,Λ, ̅; 0,0, Δ

,Δ

:

,Δ

:
,

 The first, second and the last term is positive by the assumption!
  Contribution of the third term must be present!

 There must be an operator with!!

Difficulties
 Thus finding such Λ and Δ has quite important meaning.
 The smaller the value Δ , the stronger the bound.
 How can we find such a linear functional? There are two
difficulties.

1. The space of all the linear functionals is ∞-dimensional.
2. Also ∞ly many inequalities must be checked.

Truncating the difficulties 1.
 Though not ideal, we can restrict our attentions within
finite dimensional subspace.

 Adopt the ansatz
Λ ∗

,
̅

∗

,

, whereis some cutoff and search for,.

→ ∞, we expect we can approach theideal choice for Δ .

 As you takeTruncating the difficulties 2.
 The difficulty 2. is actually harder to over come.
 An obvious way is to discretize: i.e., check the inequalitiesΛ, ̅, Δ, , Δ

0

for Δ
unitarity bound∗ , starting from0, 1, ⋯ until ∗ becomes sufficiently large.

 Since conformal block take universal form ascheck this for→ ∞,30 is enough.

 Note: now there is a sophisticated way of avoiding thisugly discretization.

Reduction to thelinear programming Now the problem is to find real numbers
satisfying inequality,,

for with finite # of Δ,̅, ̅, Δ, , Δ

,

,

0

pairs.

 Linear programming! The computers can address the
problem much better.

 Repeating the analysis for each Δ we obtain Δ Δ .

Numerical output in 4d(Rattazzi-Rychkov-Tonni-Vichi, ‘08)
 The output of the above procedure with variousΔ

Δ
Figure from (Poland et.al., ‘11)

 The convergence w.r.t.

is not so bad.

Numerical output in 2d（Rychkov-Vichi, ‘09） For

2, the same procedure gives us
ΔΔ This encouraged Slava Rychkov and collaborators toanalyze the same problem in
3, though there was nogood algorithm to compute conformal block for
2,4 atthat time. (They had to develop!)3d Ising “solution”(El-showk, et. al., ‘12)
 The output for 3d then turned out to be:ΔΔ Although still far from the “solution”, the 3d Ising model iscornered by the bootstrap method.

Previously found kinks
 Wilson-Fisher fixed points in the 2

4 interval

3-vector models (to be explained below)

5-universality class

3

1,2 super-Ising model



3

2

1

- ABJ model (with

8 SUSY)For a much more exciting example,
stay here for Part 2!

Summary
 CFTs are much more tightly constrained than the ordinary
QFTs due to conformal invariance & unitarity& crossing
relations.

 The bootstrap equation together with the linear functional
method gives a powerful bound, sometimes showing “kink”
at the actual CFT(e.g. Ising) location, but the fundamental
reason is mysterious…

Part 2 : Applications to
CFTs
Based on, arXiv:1404.0489, 1407.6195 with Yu Nakayama

The

2.1 Introduction:
wonderland

Pisarski-Wilczek Argument for
the Chiral Phase Transition
 2-flavor QCD classically has the symmetry,2but the last factor

1

2

1
1
is explicitly broken by anomaly.

 Consider the chiral symmetry breaking transition. There
the effective DOF comprises of mesons, neutral under1 .

 After the thermal compactification the effective action isthat of 3d22 ≃4 -symmetric sigma model.

 The transition is either 1st order or 2nd order with
universality class…

4 --controversies
 But the anomaly is less visible at higher temperature.
 It is quite controversial to what extent the anomaly effectis weaken around the chiral phase transition.

 (Aoki-Fukaya-Taniguchi, ’12) argued it is invisible (at thelevel of effective-model) above the critical temperature.

 If it is the case, the relevant -model is2
2
1 ≃-symmetric Landau-Ginzburg model.

42

Controversies overControversies
 Even if the anomaly-restoration scenario is true, we still
have controversies: the presence of IR-stable fixed points
in the 4
2 -LG model is so hard to investigate!
Reference

MethodResultPisarski-Wilczek ‘81

1-loopNo

Wetterich, ‘97

Functional RG(LPA)

No

Calabrese et. al. , ’03(A cond-mat paper!)5-loop + resummation

Yes

Calabrese-Paruccini, ‘04

5-loop + resummation

No

Fukushima et .al., ‘10

Functional RG (∞ dim LPA)

No

Grahl ‘14

Functional RG (LPA’)

Depends heavily on how you
truncate. No critical exponent
agreed with the 5-loop result.

Another Realization
-LG model offers a big business also forcondensed-matter theorists: imagine –component antiferromagnetic spin systems on triangular lattice. When2, two ground states are possible: The effective field theory at criticality is described by
2 LGW model (Kawamura `85).

 Presence of IR fixed points is again controversial!

What is problematic in theperturbative RG methods?
 The fixed points are quite unusual: they exist only in theregion ~3 (and not in d4

) and small .

 Perturbative results can be altered when one considers

even higher loop series. As an example, consider 3 -LG
model, with an IR (Heisenberg) fixed point. At 3-loop orderthe fixed point vanishes! Only at 4-loop order is it
restored…

 Dependence on the resummation parameters is also
criticized.

What is problematic in thefunctional RG method?
 Functional RG equation is 1-loop exact!
 But is formulated in the infinite-dimensional space of all
possible interaction term.

 We have to truncate this space to some convenient
subspace. (E.g. LPA) The truncation procedure cannot be justified. Indeed it isreported that the results (existence of fixed point) depend
heavily on the order of truncation (Grahl, ‘14.)

Milestone Ifhave an

LG model has an IR FP, there we must
symmetric CFT there.

 We can derive the bootstrap constraints in these cases aswell (by a generalized method to be explained.)

 Can we observe the kink as in the 3d Ising (no continuous
global symmetry) case?

Outline
1. Introduction : The

wonderland

2. Mouse :3. Monkey :3 with≫3

4. Human :2 with3, 4

2.2 Mouse:-LG model Before applying the conformal bootstrap tomodels, we briefly review how it proceeds formodels.

 Recall that-symmetric Landau-Ginzburg model orvector model, we have a scalar field transforming ina vector (v) rep. of, .

 Consider a 4pt correlation function : Then we perform conformal block decomposition as well.

Structured conformal block
decomposition
 In order for 0|| to survive, has totransform in some irreducible rep contained in v ⊗ v.

 Thusis either scalar (S) or traceless-symmetric(T) or
anti-symmetric (A).

 The final form of the decomposition contains Kronecker
deltas because
0
S ∝
0

T:

∝

0

A:

∝

2

Vectorial Bootstrap equation
 Here we require the crossing symmetry w.r.t simultaneous
exchange↔

, ↔ .

 We have to match each coefficient of independent .
 Three constraints. The bootstrap equation looks like
0
:

1
:

1

2
2

:

0
0
0

Linear functional argument
 Thus to generalize the linear functional argument wesearch forΛ: 3 component vector valued function →satisfying positivity condition for S,T,A sectors separately.

 According to in which sector (S,T,A) we assume

hypothetical gap Δ , we can independently bound thelowest dimension of operators.

bounds for
scalar(Kos, Poland, Simons-Duffin, `13)

 The precise meaning： a scalar operator in the -rep
OPE with dimension below , ,.
inbounds for
scalarLessons
 Kinks corresponding to-LG models show up.

 The operator dimension bound can be derived equally wellfor every global symmetry sector.
-case, the S and T bounds point out the single
model (but there’s no guarantee for this).

 In this2.3 Monkey:
withBased on arXiv.1404.0489

More serious experiment with
 The situation in

2 with3,4 looked likeswampland…

 We decided to work in the case where the RG-theoreticalstudies are solid.
3 -model with ≫ 3.
In the → ∞ limit the model is solvable! We have 1/ –expansion.

 Thus we started with the Still dynamically rich: the presence of conformal window(more precisely “conformal half-line”) is predicted.

Lagrangian description ofthe-LG model
 Consider a Lagrangian formed from scalar field,

Here the indices run over ,1, . . . , ,transforms as a bifundamental of The term proportional to
When

0,

→

is actually,

1, … ,.

, i.e.,- invariant.
explicitly.

RG prediction
 Whenis sufficiently large, the RG flow looks like

 2 more fixed points. The (un)stable one is called the(anti-)chiral fixed point.

The bootstrap equation:
too lengthy to repeat Can we observe these additional fixed points??
 To obtain the bootstrap constraints we consider => There are 9 global symmetry sectors, denoted by
SS, ST, SA, TS, TT, TA, AS, AT, AA

 The bootstrap equation is huge. Consequently thecomputation is ~100 times heavier than the Ising case.

Bounds for SS spin 0 operator
in
model
15
3 model, where thepresence of non-Heisenberg FPs (called “chiral” and“anti-chiral”) is undoubtable.

 Our first sample is The bound for SS, spin 0 operator is shown by red dots:

Symmetry enhancement
 Within the precision the bound is identical to that of O(45).
 Such “symmetry enhancement” has been reported for the
4d SU(N)/SO(2N). Is it a general mathematical statement?

 largeprediction for the additional FPs are well-below
the bounds. There are two aspects:

1. 😊 The upper bounds are satisfied and consistent!
2. 😢 We cannot observe any symptom of these fixed points
from this computation. Can’t we “solve” them??

Salvation : Bounds for spin 1
operator in TA sector
 Then we computed the dimension bounds for spin 1
operator in TA representation. Note that such operator
has dimension exactly 2 at
Heisenberg fixed points
but not whenis broken to.

“Kink” in the bound When differentiated, it becomes apparent that the slopechanges around≅ 0.515.

Spectral study (El-Showk, Paulos `12) has shown that once a CFT saturate
this kind of bounds, spectrum contained inuniquely reproduced from the bootstrap output.

can be Our result:Note: although this CFT saturate ∆ ,the bound in the other sector like ∆ ,, it may not do so for!

 The 1/ – prediction for “anti-chiral” fixed point :=>anti-chiral fixed point is observed!

family Varying , the bounds ∆ ,changes its form like

Slope change disappearance Around6~7, the kink in ∆ , According to large
7.3 .

disappears.

analysis, such a fixed point disappears at

Summary for
15
3 model in various global symmetry sector and foundthat the one in TA sector is saturated by the anti-chiral
fixed point. => It is “solvable” as in the
3 Ising!

 We examined operator dimension bounds for For smaller values of , the kink present in spin 1 TA
sector bounds of3 model disappears.
＝＞Might be a reflection of the conformal window.

 This is the first example where we can observe multiple
interacting CFTs in single bootstrap eq.

 Conclusion: Everything is consistent with the bootstrap!

2.4 Human :Based on: arXiv:1407.6195

: a signal of
frustrated magnet transitions
 For

3

2 , the bound for ST sector look like:The spectra agree!

 The spectra read off around the kink and the higher orderresults agree within systematic errors.

 Most natural explanation:

the fixed point actually exists!
 According to the perturbative analysis, this is IR-stable.

: Signal of thechiral phase transition CFTThe spectral agreement

 Again they agree and we conjecture that the FP exists.
 IR stable according to the perturbative results.

Summary & Discussions
 Despite various criticism, resumed perturbative RG seems
to be robust from the comparison with the bootstrap.

 In particular certain frustrated Heisenberg models, i.e.,
3

2 LGW model can transit continuously.

1 is restored, 2-flavor QCD chiral phasetransition could be of second order!! We predicted thecritical exponents most precisely.

 Even when

Theoretical backup needed?
 Our working hypothesis “kink => CFT” hasnot been rigorously proven even for thesimplest cases. At this stage our results arephenomenological.

 The deeper understanding of the bootstrap
program would provide the complete answer.

Thank you!
 The legend of bootstrap, applied to modern controversy...

Part 3: Future Prospects

Mixed Correlator study
 So far we have been considering only single correlator,.

 Of course there is another opearator

and we are able

to consider the bootstrap equation for,,simultaneously.

 The linear functional argument can be equally applied, but
the machinery there is semi-definite programming (ageneralization of linear programming).

Universality hypothesis
given proof? (Kos et. al.,1406.4858)
 Assume that a CFT has only two relevant operator, , .
 Then the allowed region for Δ , Δ

isBootstrap state of the art(Simmons-Duffin, 1502.02033)

 Note that the error estimate is rigorous here!!

What’s the next?
 Getting more precision?
 The study of correlators with non-scalar operator, likeEM-tensor 4pt function
This is now possible, if we have the expression (goodapproximation algorithm) for the conformal block.

 Now it has become fairly easy to start the bootstrap study,
thanks to a user-friendly package, “SDPB” in 1502.02033!
Anyway all we have to do is to implement the conformal
blocks.