Conformal Field Theory on d-Dimensional Real
Projective Space: Fundamentals and Applications
Chika Hasegawa

Abstract
In this thesis, I study fundamentals and applications of conformal field theory on a ddimensional real projective space. We investigate whether an established method for solving
conformal field theory on a d-dimensional flat Euclidean space Rd is also useful or not in
conformal field theory on a d-dimensional real projective space RPd , which is a curved space anda locally conformal flat space. By examining concrete critical models as application examples,
we confirm that there are no conflicts with known results. First of all, we use a compatibility
between the conformal symmetry and the equations of motion to solve the one-point functionof the lowest dimensional scalar primary operator in the critical ϕ3 theory (a.k.a. the Yang-Leeedge singularity) on the d = 6 − ϵ dimensional real projective space to the first non-trivial
order in the ϵ-expansion. It reproduces the conventional perturbation theory and agree with
the numerical conformal bootstrap results. Secondly, we study the critical O(N ) model on thed = 6 − ϵ dimensional real projective space and we solve the one-point functions of the scalar
primary operators to the first non-trivial order in the ϵ-expansion based on the compatibility
between the conformal invariance and the classical equations of motion. We show that theobtained results are consistent with the known results. Thirdly, we solve a conformal cross-cap
bootstrap equation in the critical ϕ4 theory (a.k.a. the critical Ising model) on the d = 4 − ϵdimensional real projective space by ϵ-expansion and to evaluate the two-point function of thelowest dimensional scalar primary operator with itself to the first non-trivial order in ϵ. We will
also argue that our results are consistent with the results of the ϵ-expansion from conformal
field theory.

Contents
1 Introduction

3

2 Conformal field theory on d-dimensional flat Euclidean space: fundamentals

6

3 Conformal field theory on d-dimensional real projective space: fundamentals 13
3.1 One-point functions of scalar primary . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Two-point functions of scalar primary . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Conformal cross-cap bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Universality classes of critical phenomena
4.1 Critical ϕ4 theory: critical Ising model . . . . . . . . . . . . . . . . . . . . . . .
4.2 Critical ϕ3 theory: Yang-Lee edge singularity . . . . . . . . . . . . . . . . . . . .
4.3 Critical O(N ) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
16
17
18

5 Methods for solving conformal field theory
5.1 ϵ-expansion from conformal field theory . . . . . . . . . . . . . . . . . . . . . . .
5.2 Conformal bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Conventional perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
21
23

6 Conformal field theory on d-dimensional flat Euclidean space: applications

25

7 Conformal field theory on d-dimensional real projective space: applications
7.1 Critical ϕ3 theory on 6 − ϵ dimensional real projective space . . . . . . . . . . .
7.1.1 ϵ-expansion from conformal field theory . . . . . . . . . . . . . . . . . . .
7.1.2 Conventional perturbation theory . . . . . . . . . . . . . . . . . . . . . .
7.1.3 Conformal cross-cap bootstrap . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Critical O(N ) model on 6 − ϵ dimensional real projective space . . . . . . . . . .
7.2.1 ϵ-expansion from conformal field theory . . . . . . . . . . . . . . . . . . .
7.2.2 Conventional perturbation theory . . . . . . . . . . . . . . . . . . . . . .
7.3 Critical ϕ4 theory on 4 − ϵ dimensional real projective space . . . . . . . . . . .
7.3.1 Conformal cross-cap bootstrap . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 ϵ-expansion from conformal field theory . . . . . . . . . . . . . . . . . . .

30
30
30
34
35
37
39
49
49
50
52

8 Conclusion

55

1

A Conformal field theory on real projective space in projective null cone formalism
59
A.1 Projective null cone formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.2 One-point functions and two-point functions in conformal field theory on realprojective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.3 Conformal blocks in conformal field theory on real projective space . . . . . . . 62
B Laplacian acting twice two-point functions

264

Chapter 1
Introduction
The conformal invariant quantum field theories, so-called conformal field theories play a significant role in theoretical physics. Quantum field theory is a theory to describe quantum
systems with infinite degrees of freedom, which is not mathematically well defined except for
free-field theory but it is established as the general framework for the description of the fundamental processes in physics. For example, the Standard Model in elementary particle physics
is a quantum field theory with local gauge symmetry, and it has been accurately verified byhigh energy accelerator experiments. In addition, quantum field theory is useful not only inhigh energy physics such as elementary particle physics, nuclear physics, and cosmology but
also in condensed matter physics such as to explain second order phase transitions. Usually,
quantum field theory is formulated by “path integral” which is physically intuitive but lacks arigor beyond perturbation theories. The idea of limiting the theory by symmetry is useful inorder to understand the quantum field theory more mathematically or non-perturbatively, so
in particular here we would like to pay attention to conformal symmetry. Roughly speaking,conformal symmetry is a transformation that preserves angles between two arcs or lines that arein contact with the same point, and is a position dependent scale transformation. Conformal
symmetry is an extended space-time symmetry, which consists of Poincaré symmetry (consisting of rotation and translation) required by special relativity, scale symmetry and special
conformal transformation symmetry.
Conformal symmetry is realized at fixed points of the local renormalization group. Renormalization group transformation is an operation that performs coarse graining and scale transformation without changing the essence of the system (i.e. keeping the Hamiltonian or thepartition function unchanged). Therefore, on the fixed point of the renormalization group,
scale invariance is realized. Scale invariance can explain the power law which characterizes thecritical phenomenon (i.e. the scaling hypothesis). So the fixed point of the renormalizationgroup is considered to correspond to the critical point. According to the the renormalizationgroup flow, the theory eventually reaches a stable infrared fixed point. The renormalizationgroup with position-dependent coupling is called local renormalization group transformation.
Solving the conformal field theory means to determine the spectrum (that is, the scaling
dimension and the spin) of the operators appearing in the theory and all the operator productexpansion coeﬃcients. From the insights of the renormalization group, the scaling dimension
is related to the eigenvalues of the renormalization group transformation, and the critical exponents are determined by the eigenvalues of the renormalization group transformation andspace dimension, through the scaling relations. Therefore, if we can determine the scaling di-3

mensions of the operators appearing in the conformal field theory, through the scaling relation,
the critical exponent can be estimated and its value can be compared with the experimental
results.
From a viewpoint of algebra and its representation, conformal invariant quantum field theory
can be well-defined mathematically. In fact, the two-dimensional conformal field theory has
a mathematical structure of Virasoro algebra and its representation [1] [2]. Fortunately it
succeeded in classify the universality classes of critical phenomena in statistical physics. In
general, many exactly solvable models are known in the low dimensions.
Moreover, conformal invariant quantum field theories can be solved non-perturbatively by
conformal bootstrap method [3] [4] [5]. Conformal bootstrap method is an idea of evaluating
physical quantity such as correlation functions of the theory from consistency conditions suchas conformal symmetry, crossing symmetry and unitarity. Indeed, in the 1980’s it was applied
to two-dimensional conformal field theory and succeeded [6] [7]. More modernly, since 2008 in
breakthrough paper [8], conformal bootstrap approach has succeeded in numerically solving a
conformal field theory in higher than two dimensions beyond the known facts in 1970’s paper[9] and analytical understanding has also progressed [10] [11]1 . Conformal bootstrap is also
applied to solve quantum chromodynamics and frustrated magnets [15] [16] [17] [18] [19] [20].
If we add supersymmetry to the assumption, there is a possibility that we can solve not only
the critical phenomena [21] [22] but also the eﬀective theory of M/string theory [23] [24] [25]. In
this way, non-perturbative research to solve the conformal field theory in the dimension higher
than two has been progressing.
So far, we have discussed conformal field theories on flat Minkowski space-time (or Euclidean
space). The main theme of this thesis is to solve conformal field theories on curved space-time.
Solving quantum field theories in curved background has a long history in its applications tocosmology, black hole physics, string theory compactification as well as condensed matter with
topological orders or boundaries, but the available tools to solve them is even morelimited.
Again, even the definition is unclear beyond the perturbation theories and most of the “exact”
results are limited to supersymmetric field theories in which the perturbative computation canbe shown to be exact.
It is therefore an interesting question to address if we can use the conformal symmetry andnon-perturbative techniques developed there to solve conformal field theories on non-trivial
curved background as in the flat space-time. Obviously, we may trivially solve conformal field
theories on conformal flat manifold, in which all the conformal symmetry is preserved, by justrescaling all the correlation functions up to possible conformal anomaly. Our target in thisthesis, however, is real projective space, which is locally conformal flat, but not globally. It
preserves half of the original conformal symmetries on flat space-time. The central question
is if the methods useful in solving conformal field theories in flat space-time are sill powerfulenough to solve them on real projective space-time. If so, such a method may be worthwhile
studying further in other more non-trivial space-time.
In this thesis, based on the above facts and background, since we interested in both solving
conformal field theory in higher than two dimensions and solving the conformal field theory on
the real projective space. Therefore the purpose of this thesis is to verify whether old and new
methods (the renormalization group, the bootstrap, etc.) for solving conformal field theory on
a flat space are also useful as a method for solving conformal field theory on real projective
1

The conformal blocks obtained analytically in the 2000s [12] [13] [14] contributed to development of theconformal bootstrap method.

4

space.
One of our motivations of study for conformal field theory is to answer a profound question of “Can the conformal hypothesis explain the universality of the critical phenomena? ” [9](see also [27]). As we have already mentioned, there are the following two well-known succeessful results as positive facts to support the conformal hypothesis. The first is the factthat Belavin-Polyakov-Zamolodchikov succeeded in classifying of universality classes of twodimensional critical phenomena with constructing two-dimensional conformal field theory in
1980’s [1] [2]. And the second is the fact that in recent years the three dimensional critical
Ising model has been solved by numerical conformal bootstrap program [21] [22]. It is known
that the power law, which is the characteristic of the critical phenomena, found in the physical quantity such as the correlation function at the critical point can be explained enough byassuming the scale invariance. In other words, the scaling hypothesis can explain the criticalphenomena. As above two successful facts imply, we may reveal that we explain the criticalphenomena by “the conformal hypothesis” rather than the scaling hypothesis. It is, therefore,of our great interest to understand how and why the conformal symmetry, alone or with some
additional assumptions, determines the universal nature of critical phenomena.
Our motivation in this investigation is to answer a mysterious and an interesting question“How useful is the conformal field theory on the d-dimensional real projective space for solving
fundamental problems in theoretical physics? ” [28]. In particular, can conformal field theory
on the d-dimensional real projective space be useful for research on d + 1 dimensional quantum
gravity theory based on the holographic principle [29] [30]? In [31] [32] [33] [34] [35] [36], theyrealize that the symmetry of bulk local fields in the context of anti-de Sitter/conformal field
theory correspondence may be related to the cross-cap Ishibashi states in dual conformal field
theories. In another viewpoint, can we apply such a theory to condensed matter physics? Sincea real projective space in even dimensions is not orientable, it seems, at first sight, diﬃcult oreven impossible to realize critical systems on such space in our real world and therefore it may
appear to be only of academic interest2 . However, the recent classification of topological phaseof matter reveals putting a system on non-orientable manifolds including a real projective space
gives us a crucial hint to understand the parity anomaly in the condensed matter physics [41].
The organization of this thesis is as follows. In chapter 2, we summarize the well-known facts
about conformal field theory on a d-dimensional flat Euclidean space. In chapter 3, we define
conformal field theory on a d-dimensional real projective space. In chapter 4, we introduce thetypical universality classes of critical phenomena, which can be interpreted as conformal field
theory. In chapter 5, we explain three possible and consistent methods for solving conformal
field theory for determining conformal field theory data of local operators appearing in thetheory. In chapter 6, we apply these methods to the Yang-Lee edge singularity as the simplest
example on the d-dimensional real projective space. In chapter 7, we also apply the methods tothree famous models belonging to diﬀerent universality classes describing critical phenomena:
the first is the Yang-Lee edge singularity, the second is the critical O(N ) model, and the thirdis so-called the critical Ising model on the d-dimensional real projective space. In chapter 8, we
will conclude this thesis and discuss for future directions. In appendix A, we derivate propertiesof conformal field theories on the real projective space in the projective null cone formalism.
In appendix B, we put some results on the calculation of Laplacian acting twice two-point
functions.
2

In fact, the conformal field theory on the two-dimensional real projective space has been investigated as theunoriented string world sheet theory [37] [38] [39] [40].

5Chapter 2
Conformal field theory on
d-dimensional flat Euclidean space:fundamentals
In this chapter, we summarize the well-known facts about conformal field theory on a ddimensional flat Euclidean space, discussed for the first time in [9]. We will see the following:by the finite number of conformal symmetries in the higher than two dimensions, the functional
form of the two-point functions and the three-point functions are completely determined, andthe four-point functions are fixed up to the ambiguity of the arbitrary function of the conformal
invariant parameter, and the n-point functions are reduced to n − 1 point functions by operator product expansions. Finally, we will also see that solving the conformal field theory is todetermine the spectrum (i.e. set of the scaling dimension and the spin) of the local operators
appearing in the theory and operator product expansion coeﬃcients.
A conformal transformation is a transformation that keeping an angle between a vector
toward one point and the other vector starting from the same point. The conformal transformation is expressed as
ds2 → e2σ(x) ds2 ,(2.0.1)ds2 = gµν (x)dxµ dxν ,(2.0.2)where the line element isand gµν (x) is Riemann metric. The Greek indices µ, ν run over from 0 to d − 1 in the case of
Minkowski space-time, while they run over from 1 to d in the case of Euclidean space.
For a d-dimensional Cartesian coordinate vectorxµ = (x0 , x1 , · · · , xd−1 ),(2.0.3)

we consider the general space-time coordinate transformations
xµ → x′µ .

(2.0.4)The line element transforms under the general space-time coordinate transformations (2.0.4)
asds2 → ds′2 = gµν (x′ )dx′µ dx′ν∂x′µ ∂x′ν ρ σ
dx dx .
= gµν (x′ ) ρ∂x ∂xσ6(2.0.5)(2.0.6)

In order to interpret the transformation (2.0.6) as the conformal transformation (2.0.1), we
need the following conditiongµν (x′ )∂x′µ ∂x′ν
= e2σ(x) gρσ (x),∂xρ ∂xσ(2.0.7)that means the metric change as followsgµν (x) → e2σ(x) gµν (x).

(2.0.8)

Now, let us consider the infinitesimal transformation. We consider the infinitesimal coordinatetransformation isxµ → x′µ = xµ − ϵµ (x),ϵµ ≪ 1,(2.0.9)and the infinitesimal conformal transformation (σ(x) ≪ 1)′gµν (x) → gµν(x′ ) ≃ (1 + 2σ(x))gµν (x).

(2.0.10)

Under this infinitesimal coordinate transformation, since the metric transforms as′gµν(x′ ) =∂xρ ∂xσgρσ (x),∂x′µ ∂x′ν

(2.0.11)

we obtain′gµν(x′ ) = gµν (x) + ∂µ ϵν + ∂ν ϵµ .

(2.0.12)Thus, in order for that this infinitesimal transformation (2.0.12) is an infinitesimal conformal transformation (2.0.10), the coordinate dependent parameter ϵµ must satisfy the following
equation∂µ ϵν + ∂ν ϵµ = 2σ(x)gµν (x).

(2.0.13)This equation is so-called a conformal Killing equation.
Taking the trace on both sides and∂ ρ ϵρ
solving for the function σ(x), we obtain σ(x) = d . So, the conformal Killing equation can be
rewritten
2∂µ ϵν + ∂ν ϵµ = ∂ ρ ϵρ gµν (x).
d(2.0.14)The solution of this conformal Killing equation (2.0.14) for gµν (x) = ηµν is ϵµ (x) which producestranslation, rotation, dilatation and special conformal transformation, that are summarized asfollows
ϵµ (x) = aµ + bAµ ν xν +A µ 1
x + (−B µ x2 + 2Bν xν xµ ),24bAµ ν = −bAν µ .

(2.0.15)

The first term generates translation, the second term generates rotation, the third term generates dilatation, and the last term generates special conformal transformation respectively.
µAfter replacing infinitesimal parameters as ω µν := bAµ ν , λ := A2 , bµ := − B4 , we obtain
xµ → x′µ = xµ − aµ − ω µν xν − λxµ − (bµ x2 − 2bν xν xµ )7(2.0.16)

where ω µν is the antisymmetric tensor (i.e. ω µν = −ωνµ ). By integrating this infinitesimaltransformation, finite conformal transformations can be obtained
xµ → x′µ = xµ − aµ ,xµ → x′µ = Λµν xν ,xµ → x′µ = λxµ ,xµ − bµ x2
xµ → x′µ =.
1 − 2b · x + b2 x2(2.0.17)(2.0.18)(2.0.19)(2.0.20)

Note that the finite form of the special conformal transformation can be obtained by inversionµ(i.e. xµ → xx2 ) → translation → inversion. Also note that the inversion is a discrete conformaltransformation that is not connected to the identity element of the conformal group.
The generators are expressed as follows
Pµ = −i∂µ ,Mµν = i(xµ ∂ν − xν ∂µ ),D = −ixµ ∂µ ,Kµ = −i[2xµ (xν ∂ν ) − x2 ∂µ ],(2.0.21)(2.0.22)(2.0.23)(2.0.24)

where Pµ , Mµν , D and Kµ generate translation, rotation, dilatation (i.e. scaling transformation), and special conformal transformation respectively. These generators satisfy following
commutation relations[Mµν , Mρσ ] = i(gνρ Mµσ − gµρ Mνσ + gνσ Mρµ − gµσ Mρν ),[Mµν , Pρ ] = i(gνρ Pµ − gµρ Pν ),[Mµν , Kρ ] = i(gνρ Kµ − gµρ Kν ),[D, Pµ ] = iPµ ,[D, Kµ ] = −iKµ ,[Kµ , Pν ] = i(2gµν D − 2Mµν ).

(2.0.25)(2.0.26)(2.0.27)(2.0.28)(2.0.29)(2.0.30)This algebra is called the d-dimensional conformal algebra1 . Note that, if we consider in thecase of Minkowski space-time, we use gµν = ηµν = diag(−1, +1, · · · , +1), while if we consider
in the case of Euclid space, we take gµν = δµν = diag(+1, +1, · · · , +1).
The conformal algebra in d-dimensional frat Euclidean space Rd can be interpreted as
Lorentz algebra so(d+1, 1) in d+2 dimensional Minkowski space Rd+1,1 . In fact, if we define theantisymmetric generators acting on d + 2 dimensional Minkowski space JAB = −JBA (A, B =−1, 0, 1, · · · , d) as follows
Jµν = Mµν ,1
J−1µ = (Pµ − Kµ ),2
J−10 = D,
1J0µ = (Pµ + Kµ ),2
1

Mµν makes Lorentz algebra, and Pµ and Mµν make Poincaré algebra

8(2.0.31)(2.0.32)(2.0.33)(2.0.34)

where {Pµ , Mµν , D, Kµ } are the generators of d-dimensional Euclidean conformal algebra, andwe can show that the generators JAB satisfy the Lorentz algebra[JAB , JCD ] = i(gAD JBC − gBC JAD + gAC JBD − gBD JAC ),(2.0.35)

where gAB is the Minkowski space-time metric gAB = ηAB = diag(−1, +1, +1, · · · , +1). Note
that, in d-dimensional Minkowski space-time Rd−1,1 , the above d-dimensional conformal algebra
can be embedded into so(d, 2) algebra in a (d+2)-dimensional space Rd,2 , while in d-dimensional
Euclid space Rd , the above d-dimensional conformal algebra can be embedded into so(d + 1, 1)
algebra in a (d + 2)-dimensional space Rd+1,1 . The number of generators of the conformal
algebra is (d + 2)(d + 1)/2, which is consistent with the fact that d-dimensional conformal
algebra consists of d translations, d(d − 1)/2 rotations, 1 dilatation and d special conformaltransformations2 . Remember that, we will see that the restricted symmetry group SO(d + 1)
which is a subgroup of the full Euclidean conformal group SO(d + 1, 1) remains in the theory
on a d-dimensional real projective space.
From now on, let us consider unitary conformal field theory on d-dimensional flat Euclidean
space Rd . The Euclidean conformal field theory has the conformal symmetry with Euclidean
conformal group SO(d + 1, 1). As we have already introduced, the number of generators are
finite and the generators consist of translation Pµ , rotation Mµν , dilatation D, and special
conformal translation Kµ (µ, ν = 1, · · · d). Operators (or fields) O∆,ℓ appearing in conformal
field theory are classified by the eigenvalues of dilatation D and rotation Mµν (i.e. scaling
dimension ∆ and spin ℓ) from the representation theory. For the operator O∆,ℓ inserting at theorigin, we have
[D, O∆,ℓ (0)] = i∆O∆,ℓ (0),[Mµν , O∆,ℓ (0)] = Sµν O∆,ℓ (0),(2.0.36)(2.0.37)

where Sµν is a spin matrix. In conformal field theory, the operators can be divided two diﬀerent
types, one is primary and the other is descendant. Primary O∆,ℓ (0) is defined by the highest
weight state of SO(d + 1, 1), which means the operator vanishing under special conformaltransformation Kµ[Kµ , O∆,ℓ (0)] = 0.

(2.0.38)

On the other hand, descendant Õ∆,ℓ is defined by the operator which is constructed by applying
diﬀerentiation to the primaryÕ∆,ℓ = [Pµ , · · · , [Pµ , O∆,ℓ (0)]],(2.0.39)

2

In the case of d = 2 dimensions, if we consider complex coordinate z = x0 + ix1 , then arbitrary holomorphic
function z → z ′ = f (z) gives a conformal mapping. Therefore the number of generators of conformal symmetry
enhances infinite and the generators satisfy the following algebra[Lm , Ln ] = (m − n)Lm+n +cm(m2 − 1)δm+n,0 ,12n∈Z

where c is central charge. This algebra is well-known infinite dimensional Lie algebra called Virasoro algebra.
This algebra is mostly studied in the context of string world sheet theory or two-dimensional critical phenomena
in statistical physics.

9

where[Pµ , O∆,ℓ (0)] = −i∂µ O(0).

(2.0.40)

Recalling that Taylor expansion of the function yields higher order diﬀerential terms, and basedon the idea of operator product expansion, we can see that the higher order diﬀerential term
corresponds to the descendant generated from a certain primary. By using the commutation
relations (2.0.28) and (2.0.29), we can see the following property respectively[D, [Pµ , O∆,ℓ (x)]] = i(∆ + 1)[Pµ , O∆,ℓ (x)],(2.0.41)[D, [Kµ , O∆,ℓ (x)]] = i(∆ − 1)[Kµ , O∆,ℓ (x)],(2.0.42)

andwhere O∆,ℓ (x) denotes the operator O∆,ℓ inserting at the arbitrary point x. Therefore, we caninterpret Kµ as lowering operator which decreases the scaling dimension by 1, while we caninterpret Pµ as raising operator which increases the scaling dimension by 1 respectively.
Note that, from SO(d + 1, 1) symmetry, we can obtain unitarity bound for primary withscale dimension ∆, which is diﬀerent for the spin ℓ as followsd− 1, for ℓ = 0,2
∆ ≥ ℓ + d − 2, for ℓ ≥ 1.
∆≥

(2.0.43)(2.0.44)

Equality holds in the case of the free theory for the scalar primary (ℓ = 0). On the other hand,equal signs hold for the conserved currents such as Jµ and the energy-momentum tensor Tµν(ℓ ≥ 1).
Next, we take operator product expansion between a scalar primary ϕ with scaling dimension∆ϕ and ϕ∑O|x1 − x2 |∆ Cϕϕ ∆,ℓ C(x1 − x2 , ∂2 )O∆,ℓ  , (2.0.45)
ϕ(x1 )ϕ(x2 ) = |x1 − x2 |−2∆ϕ 1 +O∆,ℓ=even

where 1 comes from exchanging the identity operator I and O∆,ℓ is an intermediate stateO
primary operator3 and Cϕϕ ∆,ℓ related to three-point function coeﬃcient. Note that the function
C(x1 − x2 , ∂2 ) containing descendant is determined by conformal symmetry. Since there areinfinite number of primaries, the above sum is taken over infinite number of O∆,ℓ . It is known
that operator product expansion is convergent series in conformal field theory [42]. By operator
product expansion∑(2.0.46)
O1 (x1 )O2 (x2 ) =C12k (x12 , ∂2 )Ok (x2 ),k:primarythe n-point function can be reduced to the n − 1 point function in the conformal field theory
as follows∑⟨O1 (x1 )O2 (x2 )O3 (x3 ) · · · On (xn )⟩ =C12k (x12 , ∂2 )⟨Ok (x2 )O3 (x3 ) · · · On (xn )⟩ (2.0.47)
k:primary3

For operator product expansions of the same operators, only even spins will appear as intermediate states.

10where x12 := |x1 − x2 |. So, if two-point functions and three-point functions are known, allcorrelation functions can be determined in principle. In order to prove this fact, it suﬃces toassume that the associativity of operator product expansion for four-point functions hold [3][4] [5].
From the conformal Ward-Takahashi identity4 , we can determine the functional form ofcorrelation functions. One-point function of a scalar primary Oi is vanishes except identity
operator I⟨Oi (x)⟩ = δiI .

(2.0.48)

Two-point correlation functions between a scalar primary Oi with the scaling dimension ∆iand a scalar primary Oj with the scaling dimension ∆j , we have obtained⟨Oi (x1 )Oj (x2 )⟩ =cij δ∆i ∆j1(x212 ) 2 (∆i +∆j ),(2.0.49)

where cij is a normalization factor. Note that although the functional form of the two-point
functions are determined only from the conformal symmetry, the scaling dimensions can not
be determined.
Three-point correlation functions among scalar primaries, we can find⟨Oi (x1 )Oj (x2 )Ok (x3 )⟩ =

Cijk
1
1
1
(x212 ) 2 (∆i +∆j −∆k ) (x223 ) 2 (∆j +∆k −∆i ) (x231 ) 2 (∆k +∆i −∆j ),(2.0.50)

where Cijk is a three-point function coeﬃcient. Note that Cij k is a operator product expansion
coeﬃcient, which is obtained by raising and lowering index by the normalization constant of
the two-point function (i.e. Cijk = ckl δ∆k ∆l Cij l ). Again, we note that although the functional
form of the three-point functions are determined only from the conformal symmetry, not only
the scaling dimensions but also the three-point function coeﬃcients (or the operator productexpansion coeﬃcients) can not be determined.
Four-point correlation functions between scalar primaries whose scaling dimensions are generally diﬀerent are obtained as( 2 )a ( 2 )bx14
x14
Gijkl (u, v)⟨Oi (x1 )Oj (x2 )Ok (x3 )Ol (x4 )⟩ =,(2.0.51)122x24x13 (x212 ) 2 (∆i +∆j ) (x234 ) 12 (∆k +∆l )where ∆ij := ∆i − ∆j ,a := −

∆ij
,
2

b :=

∆kl
.
2

(2.0.52)

Note that, there is a function Gijkl (u, v), which is an arbitrary function of two conformal
invariant parameters u and v so-called cross-ratios
u :=x212 x234,
x213 x224v :=4x214 x223.
x213 x224(2.0.53)

The conformal Wrad-Takahashi identity is an important relation that holds for the correlation function, which holds when assuming that the action and integral measure are invariant under conformal transformation in terms of path integral formalism (in other words, the consequence of symmetry and its conservation law). It is expressed by the following equation for scalar primaries: ⟨ϕ1 (x1 )ϕ2 (x2 ) · · · ϕn (xn )⟩ =∏n ∂x′ ∆di′′′′i=1 | ∂x |x=xi ⟨ϕ1 (x1 )ϕ2 (x2 ) · · · ϕn (xn )⟩, where x is a coordinate transformed by conformal transformation.

11

We would like to introduce the famous useful coordinates (a.k.a. Dolan-Osborn coordinates[12][13][14])v = (1 − z)(1 − z̄).

u = z z̄,(2.0.54)In the case of flat Euclidean space, z and z̄ are complex conjugate each other, while in thecase of Minkowski space-time these are independent real parameters. Crossing symmetry (i.e.
associativity of operator product expansion) requires⟨Oi (x1 )Oj (x2 )Ok (x3 )Ol (x4 )⟩ = ⟨Oi (x1 )Ol (x4 )Ok (x3 )Oj (x2 )⟩,(2.0.55)

and this leads following non-trivial infinite number of constraints
1

Gijkl (u, v) =u 2 (∆k +∆l )1v 2 (∆j +∆k )Gilkj (v, u).

(2.0.56)

This non-trivial relation is called conformal bootstrap equation. We can decompose an arbitrary
function of cross-ratios Gijkl (u, v) in terms of eigenfunctions of a quadratic conformal Casimir
equation (this decomposition is called the Conformal partial wave decomposition) as follows∑
Gijkl (u, v) =Cij O Ckl O GO (u, v),(2.0.57)
O:primarywhere GO (u, v) are called conformal blocks, which are determined by conformal symmetry.
Based on the facts we have seen so far, solving the conformal field theory is to determine
the spectrum (i.e. the set of both scaling dimension and spin of the local operator appearing
in the theory) and all operator product expansion coeﬃcients. We call the set of spectrum andoperator product expansion coeﬃcients conformal field theory data.

12

Chapter 3
Conformal field theory on
d-dimensional real projective space:fundamentals
In this chapter, we define conformal field theory on a d-dimensional real projective space basedon mainly [28]. For details on derivation of properties in conformal field theory on the ddimensional real projective space, see the appendix A.
A d-dimensional real projective space RPd is defined by involution ⃗x → − |⃗x⃗x|2 for d-dimensional
Cartesian coordinate vector ⃗x = (x1 , x2 , · · · , xd ) on a d-dimensional Euclid space Rd . The fundamental region of RPd is either 1 ≤ |⃗x| ≤ ∞ or 0 ≤ |⃗x| ≤ 1. Identification of each antipodal
points breaks down the Euclidean conformal symmetry SO(d+1, 1) into its subgroup SO(d+1)1 .
In radial quantization the dilatation D leads “time-evolution”2 . Note that, a real projective
space can not be oriented in even dimensions.

3.1.

One-point functions of scalar primary

We can fix the functional form of one-point functions of a scalar primary Oi with scaling
dimension ∆i up to a constant in the conformal field theory on the d-dimensional real projective
space by using the restricted conformal symmetry SO(d + 1) as follows⟨Oi (⃗x)⟩RP =dAi.
(1 + |⃗x|2 )∆i(3.1.1)

Note that Ai is additional conformal field theory data on the real projective space comparedwith conformal field theory data on the flat Euclidean space, so that solving conformal field
theories on the real projective space is equivalent to specifying all Ai . This fact that there are
non-vanishing one-point functions of a scalar primary is a significant feature of conformal field1

If we consider the case of Lorentzian signature manifold instead of the case of Euclidean signature manifold,
we have to replace the original Euclidean conformal symmetry group SO(d+1, 1) with the Lorentzian conformal
symmetry group SO(d, 2). And also the restricted conformal symmetry replace SO(d+1) in the case of Euclidean
signature with SO(d, 1) in the case of Lorentzian signature.
2
Strictly speaking, the radial direction in the flat Euclidean space is not the time direction but the spatial
direction.

13

theory on the real projective space. We note that one-point functions of spinning operators
vanish under the invariance of d-dimensional rotation group transformation.

3.2.

Two-point functions of scalar primary

We can also determine the functional form of two-point functions of each scalar primary up to
an arbitrary unknown function of a single conformal invariant parameter in the conformal field
theory on d-dimensional real projective space as follows⟨O1 (⃗x1 )O2 (⃗x2 )⟩RPd=(1 + |⃗x1 |2 )−∆1 +∆2
2(1 + |⃗x2 |2 )∆1 +∆2|⃗x1 − ⃗x2 |2( 2 )−∆2 +∆1
2

G12 (η),(3.2.1)x2 |where η := (1+|⃗x|⃗x1 |12−⃗
is invariant under the restricted conformal symmetry SO(d + 1) on)(1+|⃗
x2 |2 )
the real projective space, which is called the cross-cap cross-ratio. Two-point functions are
fixed by conformal symmetry up to the ambiguity that G12 (η) which is the arbitrary functionof the conformal invariant parameter η still remains as the unknown function depending on thetheory. The function G12 (η) can be decomposed by conformal blocks which satisfy conformal
quadratic Casimir equation as follows)(∑∆i∆1 − ∆2 + ∆i ∆2 − ∆1 + ∆idi
G12 (η) =C12 Ai η 2 2 F1,
; ∆i + 1 − ; η ,(3.2.2)222i2where C12i are the operator product expansion coeﬃcients3 and Ai are the one-point function
coeﬃcients. We note that the sum is taken only over the scalar primary appearing in the theory.
This manipulation is called the conformal partial wave decomposition 4 .

3.3.

Conformal cross-cap bootstrap

We will see that there is a consistency condition for the two-point functions on the real projective
space. For operator identification between a point ⃗x and its antipodal point ⃗x˜ = − |⃗x⃗x|2 i.e. ⃗x ∼ ⃗x˜,
we can evaluate essentially the same two-point functions by two diﬀerent ways of the operator
product expansion. In other words, on one hand we take the operator product expansion as⃗x1 to ⃗x2 , and on the other hand we take the operator product expansion as ⃗x1 to ⃗x˜2 = − |⃗x⃗x22|2 ,because of similarity ⃗x2 ∼ ⃗x˜2 we obtain
(

1−η
η2

) ∆1 +∆
2
6(
G12 (η) =η
(1 − η)2() ∆1 +∆
2
2) ∆1 +∆
2
6

G12 (1 − η)(3.3.1)or equivalently
G12 (η) =η
1−η

G12 (1 − η).

(3.3.2)This equation is called the conformal cross-cap bootstrap equation.
3

If the superscript i is lowered by two-point function’s normalization factor, the operator product expansion
coeﬃcients (or operator product expansion structure constants) become three-point function coeﬃcients.
4
Sometimes one can see the term which is the conformal block decomposition in the similar context.

14

Chapter 4
Universality classes of criticalphenomena
In this chapter, we introduce the typical universality classes of critical phenomena, which canbe interpreted as conformal field theory. We will see concretely three critical models such as
the critical Ising model, the Yang-Lee edge singularity, and the critical O(N ) vector model.
Before introducing a concrete critical model, we will review the basic knowledge of thecritical phenomenon. The critical phenomena in statistical physics is a singular phenomenathat appears in the second order phase transition like the gas-liquid phase transition of wateror the paramagnet-ferromagnet phase transition. The physical quantities such as correlation
functions obey the power law near the critical point, and they diverge at the critical point. Since
the power appearing in the power law which characterizes the critical phenomena is called acritical exponent, which is a universal quantity determined for each critical phenomena, thecritical phenomena can be classified by the value of the critical exponents. The correlation
length ξ diverge at the critical point (i.e. ξ ∝ |t|−ν → ∞ as T → Tc , where t := (T − Tc )/Tc ),so that scale invariance is realized at the critical point. It is therefore believed that the powerlaw is occurred from the consequence of scale invariance, that means the eﬀect of fluctuationof any energy scale can not be ignored.
The renormalization group transformation consists of coarse-graining and scale transformation, so that the fixed points under this transformation, which is called the renormalizationgroup fixed point, has scale invariance. We assume that the free energy satisfies the scalinglaw (this is called the scaling hypothesis) and the power law of the thermodynamic quantity
can be explained and scaling relations that holds among the critical exponents can be derived.
However, with the scaling hypothesis alone, the each value of the critical exponents themselves
can not be determined. Therefore, we assume that a conformal invariance, which is a larger
symmetry including scale invariance, is realized (this is called the conformal hypothesis) at thecritical point, and we expect that the critical phenomena is classified with conformal invariant
fixed points1 . Indeed, as we have already mentioned in introduction, two-dimensional criticalphenomena were completely classified by two-dimensional conformal invariant quantum field
theory [1] [2]. In addition, recently, the critical exponents of the three-dimensional critical
Ising model were obtained with conformal symmetry and some physically reasonable assumptions (e.g. crossing symmetry or unitarity) i.e. by conformal bootstrap approach, and it is
known that the values agree well with the experimental values [21] [22].
1

The conformal invariant fixed point is realized at the fixed point of the local renormalization group.

15

4.1.

Critical ϕ4 theory: critical Ising model

The Ising model is a model of (anti-)ferromagnet in statistical physics. The dynamical variablesin the Ising model are discrete spin variables taking two values, upward and downward, on theeach lattice sites. Each spin variables interact among with the nearest neighbor lattice sites.
If we suppose the spin variables are continuous variables rather than discrete variables and its
absolute values are fixed by constant, the Ising model can be rewritten as the ϕ4 theory. The
Landau-Ginzburg type eﬀective Euclidean action of the Ising model in d dimensions is given by[]∫
m2 2 λ 41d2S[ϕ] = d x (∂ϕ) +ϕ + ϕ + hϕ .
(4.1.1)
2
2
4!
The scaling operator σ := ϕ is called the spin operator and the scaling operator ε := ϕ2 is called
the energy density operator. Then the relevant parameters in the Ising model are temperatureand external magnetic field. More strictly speaking, a reduced temperature t = (T − Tc )/Tc instatistical physics is related to the mass square of the scalar field m22 , which is a coeﬃcient of
the quadratic term of the scalar field ϕ. The critical model belonging to the same universality
class as the Ising model are, for example, a lattice gas model of a fluid, a binary alloy model.
The critical exponents are determined by the scaling dimensions of the scaling operatorsand the space(-time) dimensions of the system, based on the method of the renormalizationgroup. For example, the critical exponents η and ν is defined by respectively
G(r) ∝ r−(d−2+η) ,ξ ∝ |t|−ν ,(4.1.2)(4.1.3)where G(r) is the two-point correlation function and ξ is the correlation length. According to
the scaling hypothesis based on the idea of the renormalization group, the following scaling
relations are obtained
η = d + 2 − 2yh ,
1
ν= ,
yt

(4.1.4)(4.1.5)where d is space(-time) dimension, yh is the renormalization group eigenvalue for the external
magnetic field h and yt is the renormalization group eigenvalue for the reduced temperaturet. Since we require that the action dose not change under the renormalization transformation(i.e. length x → x′ = b−1 x, couplings gi → gi′ = byi gi , and scaling operators Oi (x) → Oi′ (x′ ) =b∆i Oi (x). b is a rescaling factor),∫S[gi , Oi ] = dd x gi Oi (x)(4.1.6)∫∫′′d ′ ′ ′
(4.1.7)→ S = d x gi Oi (x ) = dd x b−d+yi +∆i gi Oi (x) = S,
The correlation length ξ ∝ |t|−ν diverges at the second order phase transition point (naively, that cord−2
responds to critical point), and the correlation function obey the power law G(r) ∝ r−2( 2 +γ ) that means
r
d−1
d−1
massless. (c.f. near the critical point, G(r) is exponentially damping: G(r) ∝ r− 2 e− ξ = r− 2 e−mr , r ≫ ξ.)216

we obtain the following relation∆i = d − yi .

(4.1.8)Therefore, given the scaling dimensions ∆i of the scaling operators appearing in the critical ϕ4theory and space(-time) dimension d, the critical exponents of the same universality class as
the Ising model are obtained as follows
η = 2∆ϕ − d + 2,
1ν=.
d − ∆ϕ2(4.1.9)(4.1.10)Note that the scaling dimension of the spin operator ∆ϕ = d−2+ γϕ and the scaling dimension
2of the energy density operator ∆ϕ2 = d − 2 + γϕ2 can be written by anomalous dimension of
the spin operator γϕ , anomalous dimension of the energy density operator γϕ2 , and space-time
dimensions d.
In the Ising model, on the lower temperature side than the critical temperature, a spin
configuration with the spin orientation aligned to minimize the free energy is realized, and thesystem is in the (anti-)ferromagnetic phase. In the order phase in which such spin directions
are aligned, because one vacuum (e.g. upward or downward) has been chosen, symmetry of
the system is spontaneously broken. On the other hand, on the higher temperature side thanthe critical temperature, a spin configuration with the spin orientation disordered is realized tominimizes the free energy, and the system is in the paramagnetic phase (disordered phase). At
the critical point which is just the boundary where the phase transition from the paramagnetic
phase to the ferromagnetic phase occurs when the external magnetic field switched to h = 0,
the correlation length diverges and the theory is scale invariant3 . The one-point function of
the spin variable, that is, the magnetization plays the role of the order parameter, and whether
the system is in the ordered phase or the disordered phase can be determined depending onwhether the value is zero or non zero. Then it is also possible to judge whether the symmetryof the theory is broken or not.

4.2.

Critical ϕ3 theory: Yang-Lee edge singularity

The Yang-Lee edge singularity is a critical phenomenon which appears when applying an pure
imaginary external magnetic field to the Ising model. It is an important well-known fact that
the Yang-Lee edge singularity can be described by the critical ϕ3 theory in d = 6 − ϵ dimension[43]. The Landau-Ginzburg type eﬀective Euclidean action of the Yang-Lee edge singularity isgiven by][∫m2 2 λ 4
12d
ϕ + ϕ + ih̃ϕ .
(4.2.1)
S[ϕ] = d x (∂ϕ) +2
2
4!
By suitable shifting of scaling operator ϕ, above action can be rewritten the following critical
ϕ3 theory in d = 6 − ϵ dimensions
]
[
∫
g̃ 3
1
2
d
(4.2.2)
S[ϕ] = d x (∂ϕ) + i(h − hc )ϕ + i ϕ ,23!
3This fact is explained in the context of the renormalization group, as the critical point is realized as a fixed
point of the renormalization group.

17

where hc is the critical external magnetic field. Here, it is assumed that the ϕ4 interacting
term has already been neglected because of including irrelevant parameter in context of powercounting. Since the three-point coupling g := ig̃ is pure imaginary, this model is not reflection
positive. In the case of d = 2, the Yang-Lee edge singularity is described by a non-unitary
minimal model with the negative central charge c = − 22
< 0 and the operator product expan5
sion between the scalar primary ϕ and ϕ contains only two Virasoro primaries, which are theidentity operator and ϕ i.e. [ϕ] × [ϕ] = I + [ϕ]. Note that the composite operator ϕ2 behave as
redundant operator, that means we can eliminate the ϕ2 term by an appropriate variable shift
ϕ → ϕ + const.

4.3.

Critical O(N ) model

The O(N ) vector model is an extended model, which has a dynamical spin variable as N components vector i.e. ϕi = (ϕ1 , ϕ2 , · · · , ϕN ), i = 1, 2, · · · , N . This model has a global symmetryof rotation group O(N ). In particular, this model corresponds to just the Ising model when
N = 1, an XY model describing superfluid when N = 2, and when N = 3 a Heisenberg model,that is a ferromagnetic model with three components spin variable, which have the x direction,
the y direction and the z direction. The Landau-Ginzburg type eﬀective Euclidean action of
the O(N ) vector model is given by[]∫1
m2 i i λ ( i i ) 2di 2
S[ϕ] = d x (∂ϕ ) +
ϕϕ +
ϕϕ
,
(4.3.1)2
2
4
where m is a mass of each ϕi and λ is a four-point interacting coupling.
First, let us consider the case of N real massless scalars ϕi with O(N ) global symmetry in
2 < d(= 4 − ϵ) < 4 dimensions (ϵ > 0),[]∫1
λ ( i i )2di 2
S[ϕ] = d x (∂ϕ ) +ϕϕ.
(4.3.2)
2
4
From the following one-loop beta function for the coupling λβλ = −ϵλ + (N + 8)

λ2
+ O(λ3 ),8π 2(4.3.3)

we found that this theory has the Wilson-Fisher type fixed point, which is a weekly interacting
nontrivial infrared renormalization group flow fixed point, at
λ+
∗ =

8π 2
ϵ + O(ϵ2 ).
N +8(4.3.4)

The anomalous dimensions of the scaling operator ϕi and its composite operator ϕi ϕi are
obtained respectively
N +2 2ϵ + O(ϵ3 ),4(N + 8)N +2
ϵ + O(ϵ2 ).
γϕi ϕi =
N +8
γϕ =

18

(4.3.5)(4.3.6)

Note that γϕ starts from order ϵ2 . If we consider the sign flip ϵ → −ϵ,
λ−
∗ = −

8π 2
ϵ + O(ϵ2 ).
N +8(4.3.7)

Under the sign flip ϵ → −ϵ, the anomalous dimension of ϕ starts from the order of ϵ2 (i.e.
γϕ = O(ϵ2 )), so it is unchanged4 , but since the critical coupling λ∗ becomes negative, one
worries about whether this fixed point is stable or not.
Next, we will see another eﬀective action describing the same critical O(N ) fixed point
which is a negative critical coupling λ−∗ (4.3.7) in d > 4 dimensions. By introducing the
Hubbard-Stratonovich auxiliary field σ, we can rewrite it as
[
]
∫
1
1 i i
1 2
d
i 2
S[ϕ] = d x (∂ϕ ) + σϕ ϕ − σ .
(4.3.8)22
4λThis above action return to the original critical O(N ) vector model (4.3.2) if we integrate out
the auxiliary field σ by using its equation of motion σ = λϕi ϕi . At the critical point, we may
drop the third term in (4.3.8) and we obtain
[
]
∫
1
1 i i
d
i 2
S[ϕ] = d x (∂ϕ ) + σϕ ϕ .
(4.3.9)2
2
In order to reduce to the usual critical ϕ3 theory when N = 0, we add the kinetic terms of σand cubic interaction term σ 3 to the previous action (4.3.9). Thus, we obtain another eﬀective
action describing the same fixed point which is a negative critical coupling λ−∗ (4.3.7) in d > 4
dimensions as follows [44].
[]∫1 ( i )2 1
g1 i i g23d2
S[ϕ] = d x∂ϕ + (∂σ) + σϕ ϕ + (σ) .
(4.3.10)22
2!
3!
Starting from the critical ϕ3 theory in the 6 − ϵ dimension, one expect that if one searches
for a nontrivial fixed point that exists at 4 < d < 65 , it will flow to the previous fixed point3
λ−∗ (4.3.7). Therefore, the above action (4.3.10), which is the critical ϕ theory consisting theN + 1 real scalar fields in d = 6 − ϵ dimensions, can be interpreted as the model describing thecritical O(N ) model as large N .

4This fact guarantees the unitarity of the infrared fixed point.
The ultraviolet fixed points were found in higher dimensions (d > 4) in the large-N O(N ) model, using
non-perturbative renormalization group approach [45], and the higher dimensional ultraviolet fixed points arefound using a polynomial ansatz for the eﬀective potential in the same case [46].
519Chapter 5
Methods for solving conformal field
theoryIn this chapter, we explain three possible and consistent methods for solving conformal field
theory i.e. for determining conformal field theory data of local operators appearing in thetheory. Recent doctoral thesis are good references [47] [48] [49] [50].

5.1.

ϵ-expansion from conformal field theory

In this section, we review the ϵ-expansion from conformal field theory based on [51]. For related
research using this method, see [52] [53] [54] [55] [56] [57] [58].
The ϵ-expansion is one of the useful methods in the perturbative renormalization group for
calculating critical exponents at the renormalization group flow fixed point which is near theGaussian fixed point [59] [60]. We assume that ϵ is positive small parameter ϵ > 0, ϵ ≪ 1.
In [51], Rychkov-Tan proposed three axioms in order to solve mainly higher than two dimensional interacting theory at the critical point from the perspective of conformal field theory
not numerically but analytically:
Axiom I
A Wilson-Fisher type fixed point, which is one of the non-trivial fixed points and stands
for a renormalization group flow fixed point interpreted as a weakly coupled interacting
theory, is conformal invariant.
Axiom II
Correlation functions evaluated in the interacting theory (the Wilson-Fisher type fixed
point) approach correlation functions evaluated in the free theory (the Gaussian fixed
point) if we take as ϵ → 0.
Axiom IIIA particular operator which is a primary in the free theory (the Gaussian fixed point)
behaves a descendant in the interacting theory (the Wilson-Fisher type fixed point), whichis called multiplet recombination phenomenon.
These axioms are given as reproduce the known results evaluated by using Feynman diagrams
in perturbation theory. If we accept these axioms as starting point once, we can use the20techniques having been developed in the context of investigations of conformal field theory
without referencing not only perturbative information but also Lagrangian descriptions in orderto calculate anomalous dimensions of the relevant operators. In keeping with above facts,
Rychkov and Tan named this alternative ϵ-expansion approach the ϵ-expansion from conformal
field theory [51].
We have some comments on the above alternative ϵ-expansion approach. In [61], one-loop
anomalous dimensions of scalar primaries, which are the quantity to the order in ϵ, were calculated based on conformal field theoretic structures without using Feynman diagrams. In [62][63], there is pointed out that the mechanism of the recombination of conformal multiplets canbe directly read from the analytic properties of the conformal blocks without further assumptions. Whether including axiom III or not, the approach of the ϵ-expansion from conformal field
theory may be a useful method for solving conformal field theory even in the case of non-unitary
theory, unlike numerical conformal bootstrap program.
In the section 7, we will see that the ϵ-expansion from conformal field theory is applied tothe actual critical models on the real projective space concretely in order to solve the one-point
function of the lowest dimensional primary operator at least to the first non-trivial order inϵ. Especially, we will use the classical equations of motion derived from the classical action asaxiom III, like studies in [64].

5.2.

Conformal bootstrap

In this section, we will summarize basic ideas of conformal bootstrap. For more details see
pedagogical lecture-notes presented by pioneers e.g. [65] [66].
The conformal bootstrap is an old idea, which was born in 1970’s [3] [4] [5], that is nonperturbative method for solving the theory i.e. for determining the correlation functions in thetheory by the physically relevant consistency conditions such as conformal symmetry, crossing
symmetry from associativity of the operator product expansion, unitarity and so on1 . The
modern conformal bootstrap starting from [8]. This breakthrough leads to important resultsin the history of theoretical physics that the three-dimensional critical Ising model was solved
non-perturbatively by numerical conformal bootstrap program [21] [22] [67] [68].
Form now on, we would like to derive the conformal bootstrap equation based on crossing symmetry from associativity of the operator product expansion for four-point correlation
function. Now, let us begin to introduce conformal bootstrap, which gives infinite number of
constraints from the consistency condition based on the crossing symmetry for four-point function. If we consider a four-point function among the same four scalar primaries ϕ with scaling
dimension ∆ϕ⟨ϕ(x1 )ϕ(x2 )ϕ(x3 )ϕ(x4 )⟩ =g(u, v)(x212 )∆ϕ (x234 )∆ϕ,(5.2.1)

where x12 := |x1 − x2 | is a distance between two points and g(u, v) is an arbitrary function ofx2 x223x2 x234
and v := x14
two cross-ratios (i.e. u := x12
2 x2
2 x2 ).
13 24
13 24
First, the four-point function (5.2.1) is unchanded under the position exchanging x1 ↔ x2
⟨ϕ(x2 )ϕ(x1 )ϕ(x3 )ϕ(x4 )⟩ =1

g(u/v, 1/v),
(x212 )∆ϕ (x234 )∆ϕ(5.2.2)

Unitarity in the Lorentzian space-time case will be replaced as reflection positivity in the Eucliean space.

21

so that we have obtain a following non-trivial consistency condition
g(u, v) = g(u/v, 1/v),(5.2.3)that is one of crossing equations. Here, g(u, v) can be decomposed by conformal blocks g∆,ℓ (u, v)
as follows∑
g(u, v) = 1 +p∆,ℓ g∆,ℓ (u, v),(5.2.4)∆,ℓ=evenO

where p∆,ℓ := (Cϕϕ ∆,ℓ )2 is positive definite from unitarity. Remind that the first term 1, is anindication that the identity operator I is always present in the theory. Therefore, the crossing
equation gives nontrivial constraints on conformal blocks labeled with scale dimensions andspins g∆,ℓ (u, v).
Second, similarly the four-point function (5.2.1) is also unchanged under the position exchanging x1 ↔ x3v ∆ϕ g(u, v) = u∆ϕ g(v, u).

(5.2.5)This equation is also one of crossing equation and so-called conformal bootstrap equation, which
gives nontrivial infinite number of constraints on conformal blocks g∆,ℓ (u, v). This means that
operator product expansion is associative for the four-point function.
As shown in [12] [13] (see also [14]), conformal blocks g∆,ℓ (u, v) have a explicit formula insome special cases. If we focus on the case of scalar exchange ℓ = 0, and conformal block havea double power series representation∞∑[( ∆2 )m ( ∆2 )m+n ]2g∆,ℓ=0 (u, v) =um (1 − v)n ,∆∆m!n!(∆ + 1 − 2 )m ( 2 )2m+nm,n(5.2.6)∏
Γ(x+m)
where (x)m := m
a=1 (x + a − 1) = Γ(x) is the Pochhammer symbol. Especially, in the case of
d = 4, it is known that the conformal blocks are written by Gaussian hypergeometric functions
d=4g∆,ℓ(u, v) =where(−1)ℓ z z̄[K∆+ℓ (z)K∆−ℓ−2 (z̄) − K∆+ℓ (z̄)K∆−ℓ−2 (z)] ,2ℓ z − z̄)β β, , β, x ,Kβ (x) := x 2 F1
2 2
u = z z̄, v = (1 − z)(1 − z̄).
β2(5.2.7)((5.2.8)(5.2.9)Remark that z̄ is the complex conjugate of z in the case of Euclidean space, while in the case ofLorentzian signature space-time two parameters (i.e. z and z̄) are independent parameters eachother. Then, it is also known that the conformal blocks are written by Gaussian hypergeometricfunctions in the case of d = 2(−1)
d=2(u, v) =
g∆,ℓ
ℓ
2

ℓ

[K∆+ℓ (z)K∆−ℓ (z̄) + K∆+ℓ (z̄)K∆−ℓ (z)] .

22(5.2.10)

We emphasize that the fact that a conformal block can be written with a function of Kβ (x)proportional to the Gaussian hypergeometric function is similar to the functional form of aconformal block in conformal field theory on the real projective space. For d = 3, although
there are less-known explicit formula, we have analytic formula of conformal blocks when we
consider them at a symmetric point z = z̄.
After studying the upper bound of the lowest dimensional primary appearing in the operator
product expansion for d = 4 unitary conformal field theory in [8], for d = 2 unitary conformal
field theory, it was shown that the well-known unitary minimal sequences are on the boundaryof the region allowed by consistent solutions satisfying the conformal bootstrap equation [69].
As mentioned at the beginning, the three-dimensional critical Ising model was solved nonperturbatively by modern numerical conformal bootstrap. The result of conformal bootstrap
method succeeded in explaining the experimental value with higher accuracy than other methods (the Monte-Carlo simulation, the high temperature expansion, the functional renormalization group method etc.) [21] [22] [67] [68]. And the results of the ϵ-expansion of five-loops
based on the method of the perturbative renormalization group are consistent with the conformal bootstrap result, and the ϵ-expansion will be reevaluated based on this fact [70].
In recent years, inspired by the success of numerical conformal bootstrap, the method ofanalytical conformal bootstrap, which had been studied since the 1970’s, has been revisited andreclaimed. For example, analytical conformal bootstrap results with the Mellin space approach
have been shown to reproduce epsilon’s third order results in the ϵ-expansion [71] [72]. Here,let’s recall that the ϵ-expansion from conformal field theory introduced in the previous section
was a computational technique based on analytic conformal bootstrap philosophy in order forsolving the conformal field theory non-perturbatively without using a Hamiltonian or Feynman
diagrams. As you can see, from the past through now, through various research motivations, themethods for solving the conformal invariant quantum field theory has been highly developed,
both numerically and analytically, and also both perturbatively and non-perturbedly.

5.3.

Conventional perturbation theory

In this section, we recall the conventional perturbation theory in ordinary quantum field theory.
The perturbation theory in quantum field theory is one of the standard approximate methodsof evaluating physical quantities such as scattering amplitudes and correlation functions by
the formal power-series expansion in the coupling constant. This expansion is known to be
an asymptotic expansion. In principle, the correlation functions in the interacting theory canbe formally evaluated from the correlation functions in the free theory by the perturbation
expansion in the coupling constant.
For example, we consider the interacting theory of the ϕ3 theory on the d = 6−ϵ dimensional
flat Euclidean space Rd . First, let us consider the following Euclidean path integral∫
Z = [Dϕ]e−S[ϕ](5.3.1)
where the action S[ϕ] is set to the ϕ3 theory
[
]
∫
1
g 3
d
2
S[ϕ] = d x (∂ϕ) + ϕ .
2
3!
23(5.3.2)

Then the n-point function can be written as∫⟨ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )⟩ = [Dϕ]ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )e−S[ϕ] .

(5.3.3)

So, when the coupling g can be regarded as small value (i.e. g ≪ 1), we can expand
g
− 3!

e

∫

dd xϕ3

g
=1−
3!

∫

1
d xϕ (x) +
2
d

3

(

g
3!

∫

)2
d xϕ (x) + O(g 3 ),d

3(5.3.4)and by using this above result we can evaluate the n-point function in conventional perturbation
theory as follows⟨ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )⟩ = ⟨ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )⟩free∫g−
dd x⟨ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )ϕ3 (x)⟩free + O(g 2 ).
3!

(5.3.5)Note that ⟨· · · ⟩free denotes the expected value evaluated in the free theory with g = 0. In
this way, the interacting theory can be approximately evaluated in the sense of asymptotic
expansion by using the Feynman rule of the free-field theory in most cases. Note that then-point correlation function in the free-field theory can be written by all possible combinations
by two-point functions of free-field theory, and this fact is called Wick’s theorem.

24

Chapter 6
Conformal field theory on
d-dimensional flat Euclidean space:
applications
In this chapter, based on the conformal hypothesis, we will solve a critical model on the higher
than two dimensional flat Euclidean space by using the appropriate methods for solving conformal field theory that we have introduced in previous chapter 5. We emphasize the fact thatthe logic to solve the conformal field theory on a flat Euclidean space is the same even if themodel is diﬀerent (e.g. the critical ϕ3 theory in 6 − ϵ dimension, the critical O(N ) model in
6 − ϵ dimension, and the critical ϕ4 theory in 4 − ϵ dimension), and that the obtained results
are shown to be consistent with each other. Therefore, for simplicity, we focus on to solve thecritical ϕ3 theory (the Yang-Lee edge singularity) on 6 − ϵ dimensional flat Euclidean space. A
good reference similar to this chapter is [64].
The critical ϕ3 theory is a model that explains the critical phenomenon called the Yang-Leeedge singularity that appears when applying an pure imaginary external magnetic field to the
Ising model. Note that this theory is a non-unitary (or no reflection positivity in the case of
Euclidean space), so it is not trivial whether conformal bootstrap method is useful or not.
The action of the critical ϕ3 theory is given by
[
]
∫
1
g 3
2
d
S [ϕ, g] = d x(∂ϕ(x)) + ϕ (x) , d = 6 − ϵ, (ϵ > 0).
(6.0.1)
2
3!
From the principle of the least action, we obtain the classical equation of motion obeying thelowest dimensional scalar primary as followsg□x ϕ(x) = ϕ2 (x),2(6.0.2)where □x := (∂)2 is d-dimensional Laplacian. The scaling dimension of the scalar primary ϕ+ γϕ = 2 − 2ϵ + γϕ 1 , where the anomalous dimension γϕ is interpreted
is written as ∆ϕ = d−22
as one of critical exponents η, appearing in the correlation function at the critical point (i.e.
G(r) ∝ r−(d−2+η) ). This relation (the equation of motion (6.0.2)) can be interpreted as just1

If we consider the free-field theory, the anomalous dimension of a single scalar primary γϕ can be set to
zero.

25

multiplet recombination of axiom III, which is mentioned in previous chapter. This observation
is important when we determine the anomalous dimension.
Let us consider solving the critical ϕ3 theory (6.0.1) in terms of conventional perturbation
theory. This theory has a non-trivial fixed point, in other words, the zero point of the one-loop
beta function in the perturbation theory. The one-loop beta function was obtained by3
g 3 + O(g 5 )332(4π )

β(g) = −ϵg −(6.0.3)in the calculation based on Feynman diagrams, and this result implies its zero pointβ(g∗ ) = 0,

(6.0.4)whereg∗2 = −(4π 3 )32
ϵ + O(ϵ2 ).
3(6.0.5)Thus, g∗ is the critical coupling giving the Wilson-Fisher type non-trivial fixed point at the
one-loop order. This result tells us that g∗ is pure imaginary, so that the theory has no reflection
positivity, and note that it is proportional to not ϵ but ϵ1/2 . We can also calculate the anomalous
dimension of operator ϕ in the perturbation theory from the wavefunction renormalization of
ϕ − ϕ two-point correlation function, so we obtain
γϕ =

1 g2
.
4π 3 192(6.0.6)

Substituting the critical coupling giving the Wilson-Fisher type non-trivial fixed point at the
one-loop order g∗ for the anomalous dimension γϕ (6.0.6), we found
γϕ = −

1
ϵ + O(ϵ2 ).
18(6.0.7)

These results are well-known results in the conventional perturbation theory to the first orderin ϵ (see, e.g. [73]).
This model was also solved in the literature [64] from the standpoint of explicitly using thedescription by Lagrangian (or Hamiltonian) by using classical equations of motion as axiom III
in the ϵ-expansion from conformal field theory. In [64], the anomalous dimension of a single
scalar primary γϕ is reproduced by the ϵ-expansion from conformal field theory and the critical
coupling g∗ is determined without Feynman diagrams.
The two-point function of the lowest dimensional scalar primary ϕ becomes⟨ϕ(x)ϕ(y)⟩ = gϕϕ |x − y|−2∆ϕ ,(6.0.8)where gϕϕ is a constant that matches the normalization constant of the ϕ-ϕ two-point functionin the free-field theory when we take the limit as ϵ → 0 (i.e. ⟨ϕ(x)ϕ(y)⟩free = 4π1 3 |x − y|−4 ).
Then, the ϕ-ϕ-ϕ three-point function becomes⟨ϕ(x)ϕ(y)ϕ(z)⟩ = Cϕϕϕ |x − y|−∆ϕ |y − z|−∆ϕ |z − x|−∆ϕ ,(6.0.9)where Cϕϕϕ is the three-point function coeﬃcient related to the operator product expansion
coeﬃcient Cϕϕϕ (i.e. Cϕϕϕ = gϕϕ Cϕϕϕ ) in the critical ϕ3 theory.
26

From now on, we apply the ϵ-expansion from conformal field theory for solving the critical
ϕ3 theory on d = 6 − ϵ dimensional flat Euclidean space Rd . First, we take the free theory limit
to the conformal invariant two-point function
lim⟨ϕ(x)ϕ(y)⟩ = ⟨ϕ(x)ϕ(y)⟩free .
ϵ→0(6.0.10)Comparing the both side in O(1), we foundgϕϕ =1+ O(ϵ).
4π 3(6.0.11)

Next, we focus on the two-point function. For the Laplacian acting twice on the twopoint function between the lowest dimensional scalar primary ϕ and ϕ, satisfying axiom I (i.e.
conformally invariant), we apply axiom III (i.e. using classical equation of motion as multiplet
recombination phenomenon) to the correlation function2 ,( g )2⟨□x ϕ(x)□y ϕ(y)⟩ =⟨ϕ2 (x)ϕ2 (y)⟩.
(6.0.12)2
The left-hand side is evaluated as(LHS(6.0.12)) = gϕϕ □x □y |x − y|−2∆ϕ∼ 2gϕϕ ∆ϕ (2∆ϕ + 2)(2∆ϕ + 2 − d)(2∆ϕ + 4 − d)|x − y|−2∆ϕ −4
1∼ 42 · 6 · 3 γϕ |x − y|−8 .
(6.0.13)4π
On the other hand, the right-hand side is calculated by using ordinary Wick’s theorem at thefirst non-trivial order in ϵ (i.e. g 2 = O(ϵ))
g2 2⟨ϕ (x)ϕ2 (y)⟩free4
g2=[⟨ϕ(x)ϕ(y)⟩free ]2 · 2
4
(
)2
g2
1
=2·
|x − y|−8 ,4 4π 3(RHS(6.0.12)) ∼(6.0.14)

where factor 2 comes from the number of combinations of Wick contraction. Thus, after
applying axiom II (i.e. taking the free theory limit, we can find that the correlation functionin the interacting theory approaches the one in the free theory) to above results, we obtain
γϕ =

1 g2
.
4π 3 192(6.0.15)

At this stage, it seems that the coupling g is unknown, while next we will see that the numberof equations and the number of unknown parameters can be matched and solved by applying
the same analysis to the three-point functions.
To go further, we pay attention to the three-point functions. For the Laplacian acting once
on a three-point function among the lowest dimensional scalar primary ϕ, satisfying axiom I
Note that if the Laplacian anting once, ⟨□x ϕ(x)ϕ(y)⟩ = g2 ⟨ϕ2 (x)ϕ(y)⟩ will vanish when we take the freetheory limit because this becomes the quantity which is proportional to the one point function.
227

(i.e. conformally invariant), we apply axiom III (i.e. using classical equation of motion as
multiplet recombination phenomenon) to the correlation functiong⟨□x ϕ(x)ϕ(y)ϕ(z)⟩ = ⟨ϕ2 (x)ϕ(y)ϕ(z)⟩.
(6.0.16)2
The left-hand side is evaluated as(LHS(6.0.16)) = Cϕϕϕ ∆ϕ (2∆ϕ + 2 − d)|x − y|−∆ϕ −2 |y − z|−∆ϕ |z − x|−∆ϕ+ Cϕϕϕ ∆ϕ (2∆ϕ + 2 − d)|x − y|−∆ϕ |y − z|−∆ϕ |z − x|−∆ϕ −2− Cϕϕϕ (∆ϕ )2 |x − y|−∆ϕ −2 |y − z|−∆ϕ +2 |z − x|−∆ϕ −2∼ −4Cϕϕϕ |x − y|−4 |z − x|−4(6.0.17)(6.0.18)

On the other hand, the right-hand side is calculated by using ordinary Wick’s theorem at thefirst non-trivial order in ϵ (i.e. g = O(ϵ1/2 ))g(RHS(6.0.16)) ∼ ⟨ϕ2 (x)ϕ(y)ϕ(z)⟩free2g= ⟨ϕ(x)ϕ(y)⟩free ⟨ϕ(x)ϕ(z)⟩free · 2
2
(
)2
1
g
=2·
|x − y|−4 |z − x|−4(6.0.19)32 4π
where factor 2 comes from the number of combinations of Wick contraction. Thus, after
applying axiom II (i.e. if take the free theory limit and we found that the correlation functionin the interacting theory terns out the one in the free theory) to above results, we obtain()2
1
g
Cϕϕϕ = −+ O(g 2 ).
(6.0.20)34π
4Note that the operator product expansion coeﬃcient is Cϕϕϕ = − 4π1 3 g4 + O(g 2 ).
Then, we consider that for the Laplacian acting twice on the ϕ-ϕ-ϕ2 three-point function,satisfying axiom I, we apply axiom III to the correlation function( g )2
2⟨□x ϕ(x)□y ϕ(y)ϕ (z)⟩ =⟨ϕ2 (x)ϕ2 (y)ϕ2 (z)⟩.
(6.0.21)2
The left-hand side is evaluated by using the equation of motion □ϕ = g2 ϕ2 (i.e. ϕ2 (z) =2□ ϕ(z)), the result of (6.0.17) and the three-point function coeﬃcient (6.0.20) as followsg z
2(LHS(6.0.21)) = ⟨□x ϕ(x)□y ϕ(y)□z ϕ(z)⟩g()
4γϕ 2γϕ − ϵ∼+|x − y|−4 |y − z|−4 |z − x|−4 .
π6π6(6.0.22)On the other hand, the right-hand side is calculated by using ordinary Wick’s theorem as atfirst non-trivial order in ϵ (i.e. g 2 = O(ϵ))
g2 2(RHS(6.0.21)) = ⟨ϕ (x)ϕ2 (y)ϕ2 (z)⟩free4g2= ⟨ϕ(x)ϕ(y)⟩free ⟨ϕ(y)ϕ(z)⟩free ⟨ϕ(z)ϕ(x)⟩free · 23
4
)3
(
2
1
3 g
|x − y|−4 |y − z|−4 |z − x|−4 .
∼2 ·4 4π 328(6.0.23)

where factor 23 comes from the number of combinations of Wick contraction. Thus, after
applying axiom II to above results, we obtainϵ1 g2
γϕ = + 3 .
6 4π 48(6.0.24)

Therefore, these results about anomalous dimension γϕ , that are both (6.0.24) and (6.0.15),
lead to the following critical coupling and anomalous dimension at the first nontrivial order inϵg∗2 = −(4π 3 )γϕ = −

32
ϵ + O(ϵ2 ),31
ϵ + O(ϵ2 ).
18(6.0.25)

These results are consistent with known results in the conventional perturbation theory (see,e.g. [73]). In this way, we have been seen that the ϵ-expansion from conformal field theory
works well in the concrete case of conformal field theory on the flat Euclidean space at least to
the first non-trivial order in ϵ.

29

Chapter 7
Conformal field theory on
d-dimensional real projective space:
applications
In this chapter, based on the conformal hypothesis, we will solve the three critical models on
the higher than two dimensional real projective space by using the appropriate methods forsolving conformal field theory that we have introduced in chapter 5. For simplicity, we focus on
the well-known three critical models such as the Yang-Lee edge singularity, the critical O(N )vector model, and the critical Ising model.

7.1.

Critical ϕ3 theory on 6−ϵ dimensional real projective
space

In this section, based on our work that has been published in [74], as the main topic of thisthesis, we explain that we solve the one-point function of a lowest dimensional scalar primaryin the critical cubic scalar theory on the d = 6 − ϵ dimensional real projective space by using
a compatibility between the conformal symmetry and the equation of motion. Remark that
we solve the one-point function of the lowest dimensional scalar primary to the first non-trivial
order in ϵ with analytic methods. Main analytical calculations is based on the modern method
for solving conformal field theory proposed in [51] and developed in [64].

7.1.1.

ϵ-expansion from conformal field theory

In this subsection, we apply the ϵ-expansion from conformal field theory to solve the critical ϕ3theory with conformal invariance, and determine the conformal field theory data including theone-point function on the real projective space to the first nontrivial order in ϵ.
The action of the critical cubic scalar ϕ3 theory (a.k.a. the Yang-Lee edge singularity) is
given by
[
]
∫
1
g 3
2
d
S [ϕ, g] = d x(∂ϕ(x)) + ϕ (x) , d = 6 − ϵ, (ϵ > 0),(7.1.1)2
3!

30

and from the principle of the least action the classical equation of motion for the scalar primaryisg□x ϕ(x) = ϕ2 (x),(7.1.2)2where the scalar primary field ϕ has a scaling dimension
∆ϕ =

d−2
ϵ
+ γϕ = 2 − + γϕ .
22(7.1.3)Here, □x := (∂)2 denotes Laplacian in the d-dimensional Cartesian coordinate ⃗x. Since a single
derivative operator corresponds to the generator of translation, the above equation of motion(7.1.2) implies that the composite operator ϕ2 behaves as a descendant of the operator ϕ in
the ϕ3 interacting theory. This observation is interpreted as axiom III of the ϵ-expansion from
conformal field theory. Recall that, since there is no reflection positivity in this theory, it is nottrivial whether bootstrap is useful to solve this theory.
As we have mentioned in the previous chapter, this theory has a non-trivial fixed point, in3
3other words, the zero point of the one-loop beta function β(g∗ ) = 0, β(g) = −ϵg − 32(4π3) g +O(g 5 ) in the conventional perturbation theory. Recall that the critical coupling g∗ giving the
Wilson-Fisher type fixed point at the one-loop order is as follows
g∗2 = −(4π 3 )32
ϵ + O(ϵ2 ).
3(7.1.4)This result tells us that g∗ is pure imaginary, so that the theory has no reflection positivity. Andnote that it is proportional to not ϵ but ϵ1/2 , so we assume that the coupling g can be expanded
in terms of the formal power series of ϵ1/2 at the critical point which is near the Gaussian fixed
point. Let us also recall that the anomalous dimension of operator ϕ in perturbation theory is
g2
known as γϕ = 3·41 3 4π
3 , so substituting above g∗ for γϕ , we found
γϕ = −

1
ϵ + O(ϵ2 ).
18(7.1.5)

Again, these results are well-known results in the conventional perturbation theory to the firstorder in ϵ.
In critical ϕ3 theory, which is a interacting theory, operator product expansion between alowest dimensional scalar primary ϕ and ϕ is obtained by[ϕ] × [ϕ] = I + [ϕ] + [ϕ3 ] + · · · .

(7.1.6)On the other hand, in a free theory, we know that the operator product expansion is[ϕ]free × [ϕ]free = Ifree + [ϕ2 ]free + · · · .

(7.1.7)Compare with above two operator product expansions, we see the operator ϕ2 does not appear
in the interacting theory, while the operator ϕ2 behaves as a primary in the free theory. Thisdiﬀerence comes from the fact that a particular primary operator in the free theory (i.e. ϕ2in this case) behaves as a descendant in the interacting theory from the classical equation of
motion (7.1.2). So this phenomenon is just multiplet recombination pointed out in [51] as axiom
III of the ϵ-expansion from conformal field theory.
31

The one-point function of the lowest dimensional scalar primary ϕ becomes⟨ϕ(x)⟩RP =dAϕ,(1 + x2 )∆ϕ(7.1.8)and the ϕ-ϕ two-point function becomes⟨ϕ(x)ϕ(y)⟩RP = |x − y|−2∆ϕ Gϕϕ (η)
d(7.1.9)in the critical ϕ3 theory. Here, η = (1+x|x−y|2 )(1+y 2 ) is invariant under the restricted conformal group
SO(d + 1) transformation. We are able to decompose the arbitrary function of the cross-cap
cross-ratio G(η) into conformal partial waves as follows
(
)
∑
∆i
∆i ∆i
d
i
Gϕϕ (η) =Cϕϕ Ai η 2 2 F1
, ; ∆i + 1 − ; η .
(7.1.10)2 2232i=I,ϕ,ϕ ,···

Note that the sum is taken over only the scalar primaries appearing in the operator productexpansion in the critical ϕ3 theory (7.1.6).
Before studying to determine conformal field theory data in the critical ϕ3 theory (theinteracting theory at the Wilson-Fisher type non-trivial fixed point) on 6 − ϵ dimensional realprojective space, we need to fix the normalization of some correlation functions in the freetheory at the Gaussian fixed point as follows:⟨ϕ(x)⟩RP
free = 0,11d⟨ϕ2 (x)⟩RP,free =3
4π (1 + x2 )4
[
(
)2 ]
η
1
1
d
⟨ϕ(x)ϕ(y)⟩RP
1+,free =
4π 3 |x − y|4
1−η

 [
)2
(
)2 ]2
(
1
η
1
d
2 · 1 +⟨ϕ2 (x)ϕ2 (y)⟩RP+ η4 .
free =
384π|x − y|1−ηd(7.1.11)(7.1.12)(7.1.13)(7.1.14)

From now on, we apply the ϵ-expansion from conformal field theory for solving the critical
ϕ theory on 6 − ϵ dimensional real projective space. First, we take the free theory limit to theconformal invariant two-point function
3

lim⟨ϕ(x)ϕ(y)⟩RP = ⟨ϕ(x)ϕ(y)⟩RPfree .
dϵ→0

d(7.1.15)Comparing both sides at O(η 0 ), we found1+ O(ϵ),
4π 3(7.1.16)

1
1= 3 + O(ϵ).
γϕ
4π(7.1.17)

CϕϕI AI =and comparing both sides at O(η 2 ), we foundCϕϕϕ Aϕ32

These are our first main results. After applying axiom III to the correlation function thatapplied the Laplacian once to the one-point function of the lowest dimensional scalar primaryϕ, satisfying axiom I, we obtaingdd⟨□x ϕ(x)⟩RP = ⟨ϕ2 (x)⟩RP .
2(7.1.18)

Then we apply axiom II to both sides of 7.1.18, and we evaluate them to the first non-trivial
order in ϵ respectively
24Aϕ,(1 + x2 )4
g
g 1
1
d
(RHS(7.1.18)) ∼ ⟨ϕ2 (x)⟩RP.
free =
2
2 4π 3 (1 + x2 )4(LHS(7.1.18)) = □x ⟨ϕ(x)⟩RP ∼ −d(7.1.19)(7.1.20)1Comparing both sides in O(ϵ 2 ) when ϵ is suﬃciently smaller than 1, we obtainAϕ = −1 g+ O(g 3 ).
4π 3 48(7.1.21)

This result is the most important our result. We can also derive the same result based on theconventional perturbation theory in the next subsection.
After applying the Laplacian twice to the two-point function that satisfies axiom I andapplying axiom III to the correlation function, we find⟨□x ϕ(x)□y ϕ(y)⟩RP =
d

g2 2
d
⟨ϕ (x)ϕ2 (y)⟩RP .
4(7.1.22)

For the left-hand side, we know the concrete form of the two-point function decomposed into
conformal partial waves, so we can also just diﬀerentiate it as follows
(
)
∑
∆O
∆O ∆O
d
O
−2∆ϕ
2
Cϕϕ AO η 2 F1
(LHS(7.1.22)) = □x □y |x − y|,; ∆O + 1 − ; η222
O=I,ϕ,···
[
(
)
]
∆ϕ
∆ϕ ∆ϕ
ϕ
−2∆ϕ
I
= □x □y |x − y|
Cϕϕ AI + Cϕϕ Aϕ η 2 2 F1,
; γϕ ; η + · · ·
2 2
[
(
)
)
]
∆ϕ (
1
ϵ
∼ 3 □x □y |x − y|−2∆ϕ 1 + η 2 γϕ + 1 − + γϕ η + O(η 2 ) + · · · ,4π2(7.1.23)
where we expanded around η = 0 and we used (7.1.16) and (7.1.17) in the last line. For
the right-hand side, since the prefactor g 2 ∼ O(ϵ) is multiplied, the two-point function on the
Wilson-Fisher type fixed point may be approximated by the correlation function of the free-field
theory, we can calculate it using the ordinary Wick’s theorem as follows
g2 2d⟨ϕ (x)ϕ2 (y)⟩RP
free
4 [
]
[
]2
g2
RPd
2
RPd
2
RPd
=
2 · ⟨ϕ(x)ϕ(y)⟩free + ⟨ϕ (x)⟩free ⟨ϕ (y)⟩free
4
)2
(
[
]
g2
1
1
2
3
=
2
+
4η
+
O(η
)
,
4 4π 3|x − y|8(RHS(7.1.22)) ∼33(7.1.24)

where we expanded around η = 0 in the last line. Then we apply axiom II to the correlation
function (7.1.22) and compare both sides of them in O(ϵη 0 ), we find
γϕ =

1 g2
,
4π 3 192(7.1.25)and comparing both sides of (7.1.22) in O(ϵη 2 ), we see
γϕ =

ϵ
1 g2
+ 3 .
6 4π 48(7.1.26)

1

Therefore, we know g = O(ϵ 2 ) and γϕ = O(ϵ), so that we can express g 2 and γϕ in terms of ϵg 2 = −(4π 3 )γϕ = −

32
ϵ + O(ϵ2 ),31
ϵ + O(ϵ2 ).
18(7.1.27)

With all these constraints from axiom I, II and III, we can completely specify the conformal
field theory data including the one-point function of the lowest dimensional scalar primary inthe critical ϕ3 theory on 6 − ϵ dimensional real projective space as follows
√
3
2 1
1 g
1
3
√ ϵ 2 + O(ϵ 2 ),Aϕ = − 3+ O(g ) = −i √(7.1.28)34π 48
4π 12 3
√
3
2 1
1 g
1
ϕ
3
√ ϵ 2 + O(ϵ 2 ),Cϕϕ = − 3 + O(g ) = −i √(7.1.29)4π 4
4π 3 3d−2ϵ1∆ϕ =+ γϕ = 2 − + γϕ , γϕ = − ϵ + O(ϵ2 ).
(7.1.30)
2
2
18
We note that Aϕ is additional conformal field theory data in conformal field theory on realprojective space. Other conformal field theory data are consistent with in the case of conformal
field theory on a flat Euclidean space [64].
The quantity Cϕϕϕ Aϕ appearing in conformal partial wave decomposition is obtained by
ϵ-expansion as
Cϕϕϕ Aϕ = −1 1
ϵ + O(ϵ2 ).
3
4π 18(7.1.31)

Comparing the result with the result by numerical truncated conformal cross-cap bootstrap (inthe case of the Yang-Lee edge singularity on the flat Euclidean space, see [75] [76]), we find theabove result is good agreement within 10 percent error when ϵ = 0.05.

7.1.2.

Conventional perturbation theory

In this subsection, we derive the one-point function on the real projective space from theconventional perturbation theory in the weak coupling region. The classical action of thecritical ϕ3 theory in d = 6 − ϵ dimensions is (7.1.1) and the model is defined by the Euclidean
path integral∫
Z[g] = [Dϕ] e−S[ϕ,g] ,(7.1.32)
34

with the perturbative expansions in the weak coupling g ≪ 1. At g = 0, the free-field correlation
functions on the real projective space are given by (7.1.11) (7.1.12) (7.1.13) (7.1.14). Using the
perturbative expansions, we obtain
∫
g
d
RPd
RPd
2
⟨ϕ(x)⟩= ⟨ϕ(x)⟩free −dd y ⟨ϕ(x)ϕ3 (y)⟩RP(7.1.33)
free + O(g ),
3!
∫
g
d
2
RPd
2
=−
dd y ⟨ϕ(x)ϕ(y)⟩RP(7.1.34)
free ⟨ϕ (y)⟩free + O(g ),2where the integral range is 0 ≤ |y| ≤ 1 and d = 6 as ϵ → 0. In the second line, we use
the standard Wick contraction. After setting ⃗x to ⃗0 and plugging in (7.1.12) and (7.1.13) for(7.1.34), we obtaing 1d⟨ϕ(0)⟩RP = −+ O(g 2 ).
(7.1.35)
48 4π 3Since ⟨ϕ(0)⟩RP equals to Aϕ in conformal filed theory on the real projective space (see (7.1.8)),
this perturbative result agrees with (7.1.21) which is obtained by using the axioms in the critical
ϕ3 theory with conformal symmetry discussed in the previous subsection.
d7.1.3.

Conformal cross-cap bootstrap

In this subsection, we will solve the conformal cross-cap bootstrap equation.
j() ∆i +∆2ηGij (η) =Gij (1 − η)1−η(7.1.36)

Rewriting the conformal cross-cap bootstrap equation using conformal partial wave decomposition, the following equation giving infinite number of constraints is obtained()∑∆k∆i − ∆j + ∆k ∆j − ∆i + ∆kdk2
Cij Ak η 2 F1,
; ∆k + 1 − ; η
2
2
2
k
(

=

η
1−η

j
) ∆i +∆∑2k

Cij Ak (1 − η)
k

∆k
2

(

2 F1

)
∆i − ∆j + ∆k ∆j − ∆i + ∆kd,; ∆k + 1 − ; 1 − η .
2
2
2(7.1.37)

Note that the sum is taken only over the scalar primaries appearing in the theory.
First, for practice, let us study the conformal cross-cap bootstrap equation for a ϕ-ϕ twopoint function in the case of the free theory in d-dimensions. The operator product expansion
between the scalar primary ϕ and ϕ in the free theory is given by[ϕ]free × [ϕ]free = I + [ϕ2 ]free .
So, the cross-cap bootstrap equation can be written as
)
(
∆ 2
ϕ
∆ϕ2 ∆ϕ2d
ϕ2
I,; ∆ϕ2 + 1 − ; η
Cϕϕ AI + Cϕϕ Aϕ2 η 2 2 F1
2
2
2
(
)∆ϕ
[ I
η
=
Cϕϕ AI
1−η
)]
(
∆ 2
ϕ
∆ϕ2 ∆ϕ2d
ϕ2
2+ Cϕϕ Aϕ2 (1 − η),; ∆ϕ2 + 1 − ; 1 − η .
2 F1
2
2
2
35

(7.1.38)(7.1.39)

Consistent solutions of the (7.1.39) are as follows:
CϕϕI AI = 1 : normalization,(7.1.40)

ϕ2

Cϕϕ Aϕ2 = 1(7.1.41)

with the scaling dimensions in the free theoryd−2,2∆ϕ2 = 2∆ϕ .
∆ϕ =(7.1.42)Thus, the solution of the conformal cross-cap bootstrap equation in the case of the free theory
becomes as follows()∆free
[(
ϕ
)free
η
free
Gϕϕ (η) =CϕϕI AI
1−η
)]
(
free
free
(
)free
∆free
2
∆
∆
ϕ
2
2
2
d
ϕ
ϕ
+ Cϕϕϕ Aϕ2(1 − η) 2 2 F1,
; ∆free. (7.1.43)
ϕ2 + 1 − ; 1 − η2
22
Since the hypergeometric function appearing in (7.1.43) is
(
)
∞
∑
∆free
∆free
(∆free
d
free
free
ϕ2
ϕ2
ϕ )n
free
,
; ∆ϕ2 + 1 − ; 1 − η =(1 − η)n = [1 − (1 − η)]−∆ϕ = η −∆ϕ ,
2 F1
2
2
2
n!
n=0
(7.1.44)
we can rewrite as(
Gfree
ϕϕ (η) = 1 +η
1−η)∆freeϕ.

(7.1.45)

In this way, we write down the function in the case of the free theory Gfree
ϕϕ (η) that satisfies the
cross-cap bootstrap equation and derive a consistent two-point function in the free-field theory:
[
]
(
) d−2
2
η
d
−∆ϕ
⟨ϕ(x)ϕ(y)⟩RP1+.
(7.1.46)
free = |x − y|
1−η
d/2

1
, Sd := 2π
instead of 1 in (7.1.40), we
If we set the normalization factor as CϕϕI AI = (d−2)SΓ( d )d2

need multiply this above result (7.1.46) by the normalization factor.
Next, let us see in the case of the critical ϕ3 theory in d = 6 − ϵ dimensions. We reconsider
the cross-cap bootstrap equation in the case of the ϕ-ϕ two-point function is()∆ϕη
Gϕϕ (1 − η).
(7.1.47)Gϕϕ (η) =1−η
In the case of the critical ϕ3 theory in d = 6 − ϵ dimensions, the operator product expansion
between a scalar primary ϕ and ϕ is[ϕ] × [ϕ] = I + [ϕ] + [ϕ3 ] + · · · .
36(7.1.48)

The scaling dimension of ϕ is ∆ϕ = d−2+ γϕ = 2 − 2ϵ + γϕ . From now on, we consider expanding
2
conformal field theory data to the lowest order in ϵ as follows
γϕ = (γϕ )(1) ϵ + O(ϵ2 ),()(1)
Cϕϕϕ Aϕ = Cϕϕϕ Aϕϵ + O(ϵ2 ).

(7.1.49)(7.1.50)

As in the case of the free-field theory, comparing both sides of the cross-cap bootstrap equation
after using the connection formula of Gaussian hypergeometric function (i.e, expansion aroundη ∼ 0), we obtain the following non-trivial relation among conformal field theory dataCϕϕϕ Aϕ

1= CϕϕI AI .
γϕ(7.1.51)

This non-trivial relation is a necessary condition for establishing the cross-cap bootstrap equation. Note that the value of the anomalous dimension γϕ expanded in ϵ is not determined.
We note the following two things. First of all, the reason why we cannot determine γϕ is dueto the fact that the same equation is satisfied by the O(N ) symmetric version of ϕ3 theories.
Therefore, the crosscap bootstrap equations alone should not determine Cϕϕϕ Aϕ completely
without specifying γϕ . Secondly, only with a finite operators in the conformal block expansions,the cross-cap bootstrap equation can be solved only at O(1) and not at O(ϵ). Solving crosscap bootstrap equations at O(ϵ) requires the infinite number of primary operators. This is incontrast with the ϕ4 theory case, in which we may solve it at O(ϵ) with only finite number ofprimary operators. Technically this is due to the fact that γϕ is O(ϵ) in the critical ϕ3 theory
while it is O(ϵ2 ) in the critical ϕ4 theory.

7.2.

Critical O(N ) model on 6 − ϵ dimensional real projective space

In this section, we study the critical O(N ) model on the d = 6 − ϵ dimensional real projective
space, which is proposed in [44], by using a compatibility between the conformal invariance andthe classical equations of morion. This section is based on our unpublished note.
First, we define the critical O(N ) model in d = 6 − ϵ dimensions from the viewpoint of aconformal field theory. We are going to solve this model in formal power series of ϵ1/2 . The
classical action of the critical O(N ) model in d = 6 − ϵ dimensions is[]∫
1 ( i )2 1
g1 i i g2 3
2idS[ϕ , σ, g1 , g2 ] = d x∂ϕ + (∂σ) + σϕ ϕ + σ ,(7.2.1)2
2
2
3!
where the global O(N ) symmetry vector index i runs from 1 to N in a natural number. The
classical equations of motion are□x ϕ = g1 σϕi ,g1
g2□x σ = ϕ i ϕ i + σ 2 ,2
2(7.2.2)(7.2.3)where □x := (∂)2 is Laplacian in d-dimensions. The classical equation of motion (7.2.3) means
the mixed scalar operator O+ := g21 ϕi ϕi + g22 σ 2 is the descendant operator of the scalar primary37

operator σ, and this relation leads the fact of ∆σ + 2 = ∆O+ , where ∆O+ := 4 − ϵ + γO+ is thescaling dimension of O+ (γO denotes the anomalous dimension of the local operator O).
Given this perturbative picture, we postulate the following three axioms: I: The non-trivial
fixed point has conformal symmetry. II: If we take the ϵ → 0 limit, correlation functions in
the interacting theory will approach the ones in the free theory. III: From the equations of
motion, a particular primary operator in the free theory (i.e. σϕi , O+ ) behaves as a descendant
operator at the non-trivial fixed point (i.e. σϕi is a descendant of ϕi , and O+ is a descendantof σ by acting Laplacian as in (7.2.2) and (7.2.3) respectively).
We have a comment on axiom III. From the purely conformal field theory viewpoint, we,a priori, do not know the magnitude of couplings g1 and g2 at the fixed point nor if these
are related to the operator product expansion coeﬃcients such as Cϕi ϕσj , Cσσσ and so on, butfrom the expectation in the conventional perturbation theory, it is consistent to assume that
couplings g1 and g2 are of order ϵ1/2 and so will we in the following.
As we mentioned in the previous section, our main interest is to determine the one-point
functions on the real projective space. In particular, we would like to focus on the one-point
function of the lowest dimensional scalar primary operator σ with the scaling dimension ∆σ :=d−2+γσ = 2− 2ϵ +γσ , and the next-lowest dimensional scaler primary operator O− := − Ngg21 ϕi ϕi +2
σ 2 with the scaling dimension ∆O− := 4 − ϵ + γO− :Aσ,(1 + x2 )∆σAO−d⟨O− (x)⟩RP =.
(1 + x2 )∆O−
⟨σ(x)⟩RP =d(7.2.4)(7.2.5)Since ϕi with the scaling dimension ∆ϕi := d−2+ γϕi = 2 − 2ϵ + γϕi has the vector index, the2done-point function for ϕi vanishes under the global O(N ) symmetry (i.e. ⟨ϕi (x)⟩RP = 0).
For this purpose, we are going to study their two-point functions:δ ij
Gϕi ϕj (η),|x − y|2∆ϕi
1d⟨σ(x)σ(y)⟩RP =Gσσ (η),|x − y|2∆σ1d⟨O− (x)O− (y)⟩RP =GO− O− (η),|x − y|2∆O−⟨ϕi (x)ϕj (y)⟩RP =dwith the conformal partial wave decomposition [34]:
)
(
∑
∆O
d
∆O ∆O
O
2
,
; ∆O + 1 − ; η ,Gϕi ϕj (η) =Cϕi ϕj AO η 2 F1
2
2
2
O

(7.2.6)(7.2.7)(7.2.8)(7.2.9)2

where η := (1+x(x−y)
2 )(1+y 2 ) is a cross-cap cross-ratio. For later purposes, we expand (7.2.9) to thefirst few terms in η:
]
[
( )2
∆σ
1
∆
σ
η + O(η 2 )Gϕi ϕj (η) = Cϕi ϕI j AI + Cϕi ϕσj Aσ η 2 γσ +γσ2−

+ Cϕi ϕOj AO− η

∆ −
O
2

[1 + O(η)] · · · ,38(7.2.10)where the operator I denotes the identity operator with the scaling dimension ∆I = 0. Notethat although the scalar operator product expansion is given by [ϕi ] × [ϕj ] = I + [σ] + [O− ] + · · ·in the critical O(N ) model.

7.2.1.

ϵ-expansion from conformal field theory

In this subsection, we apply the axioms of the critical O(N ) model with conformal invarianceto determine the critical exponents and the one-point function on the real projective space tothe first non-trivial order in the ϵ-expansion.
First of all, let us fix the normalization of the correlation functions and use axiom II of thecontinuity of the correlation functions to the free-field theory in the ϵ → 0 limit. We fix thenormalization of the two-point functions for the lowest dimensional primary operator in thed/2
1free theory as (d−2)S= 4π1 3 , where Sd := 2π
is the surface area of a unit d-sphere. With thisΓ( d2 )dnormalization, the free-field correlation functions on the real projective space are given by⟨ϕi (x)⟩RPfree = 0,N1d⟨ϕi ϕi (x)⟩RP,free =3
4π (1 + x2 )4
[
(
)2 ]
ij
η
1
δ
d
1+
,
⟨ϕi (x)ϕj (y)⟩RPfree =
4π 3 |x − y|41−η

(7.2.11)⟨σ(x)⟩RP
free = 0,11d,⟨σ 2 (x)⟩RPfree =3
4π (1 + x2 )4
[
(
)2 ]
η
1
1
RPd
1+
,
⟨σ(x)σ(y)⟩free = 3
4π |x − y|41−η

(7.2.14)dd(7.2.12)(7.2.13)(7.2.15)(7.2.16)

where we mean by ⟨· · · ⟩RP
free that the expectation values are evaluated in the free theory with
ϵ = 0. The above correlation functions are obtained by using the method of image under the
involution ⃗x → − |⃗x⃗x|2 .
We now demand that (7.2.6) approaches (7.2.13) in the ϵ → 0 limit. For this to be possible,
as more explicitly seen in (7.2.10), we needd

δ ij
+ O(ϵ),
4π 3
1
δ ij
−
Cϕi ϕσj Aσ + Cϕi ϕOj AO− = 3 + O(ϵ).
γσ
4πCϕi ϕI j AI =(7.2.17)(7.2.18)

Similarly, we also demand that (7.2.7) approaches (7.2.16) as ϵ → 0, we require1+ O(ϵ),
4π 3
1
1−
Cσσσ Aσ + CσσO AO− = 3 + O(ϵ).
γσ
4π
CσσI AI =(7.2.19)(7.2.20)

The next goal is to determine each piece of the left-hand side of (7.2.18) and (7.2.20)
separately. For this purpose, we combine axiom II and III in the correlation functions and takethe ϵ → 0 limit.
39

Let us begin with the one-point function. Axiom III means that we can use the classicalequations of motion (7.2.3)⟨□x σ(x)⟩RP =d

g1 i ig2dd⟨ϕ ϕ (x)⟩RP + ⟨σ 2 (x)⟩RP ,2
2(7.2.21)inside the one-point function to derive the consistency condition to the first non-trivial orderin the ϵ-expansion. By acting Laplacian on (7.2.4) and comparing it with (7.2.15) and (7.2.21)
to the first non-trivial order in the ϵ-expansion
24Aσ,(1 + x2 )4
g1
g2 2
d
d
(RHS(7.2.21)) ∼ ⟨ϕi ϕi (x)⟩RP⟨σ (x)⟩RP
free +
free
2 [] 2
1 g1 N
g2
1
= 3
,
+
4π
2
2 (1 + x2 )4(LHS(7.2.21)) = □x ⟨σ(x)⟩RP ∼ −d(7.2.22)(7.2.23)

we obtain
Aσ = −

11[g1 N + g2 ] 3 + O(ϵ).
48
4π(7.2.24)

This result may be also derived from the conventional perturbation theory by evaluating a
Feynman diagram on the real projective space (see next subsection).
To go further, we study the two-point functions with axiom II and III. We apply the classicalequations of motion twice in the two-point functions:⟨□x ϕi (x)□y ϕj (y)⟩RP = g12 ⟨σϕi (x)σϕj (y)⟩RP ,
g2
g2
d
d
d
⟨□x σ(x)□y σ(y)⟩RP = 1 ⟨ϕi ϕi (x)ϕj ϕj (y)⟩RP + 2 ⟨σ 2 (x)σ 2 (y)⟩RP
4
4]
g1 g2 [ i idd⟨ϕ ϕ (x)σ 2 (y)⟩RP + ⟨σ 2 (x)ϕj ϕj (y)⟩RP .
+
4
d

d

(7.2.25)(7.2.26)

We now evaluate the left-hand side and the right-hand side separately to the first non-trivial
order in the ϵ-expansion to derive additional necessary conditions in order for the criticalexponents to be compatible with the conformal symmetry. We focus on the limit when x
approaches y (i.e. η → 0) by using the operator product expansion. Expanding with respectto η, in the case of the two-point function for ϕi -ϕj , the left-hand side of (7.2.25) becomes(LHS(7.2.25)) = □x □y ⟨ϕi (x)ϕj (y)⟩RP
[

d

1 ∆σ
∼ □x □y δ ij |x − y|−2∆ϕi Cϕi ϕ1j A1 + Cϕi ϕσj Aσ η 2
γσ]−+Cϕi ϕOj AO− (1 + O(η)) + · · · ,

(

[

∆σ
γσ +
2

)

]2

η + O(η 2 )(7.2.27)

while the right-hand side of (7.2.25) becomesijRP(RHS(7.2.25)) ∼ g12 ⟨σ(x)σ(y)⟩RPfree ⟨ϕ (x)ϕ (y)⟩free()2]
1
δ ij [
2
2
3
= g1
1
+
2η
+
O(η
)
,
4π 3
|x − y|8d40

d(7.2.28)

in the first non-trivial order in the ϵ-expansion.
Similarly, in the case of the two-point function for σ-σ, the left-hand side of (7.2.26) becomes(LHS(7.2.26)) = □x □y ⟨σ(x)σ(y)⟩RP[−2∆σ∼ □x □y |x − y|d

1 ∆σ
CσσI AI + Cσσσ Aσ η 2γσ]−+CσσO AO− (1 + O(η)) + · · · ,

(

[

∆σ
γσ +
2

)

]2
2

η + O(η )(7.2.29)

while the right-hand side of (7.2.25) becomes
g12 i i
g22 2
d
j j
RPd
(RHS(7.2.26)) ∼ ⟨ϕ ϕ (x)ϕ ϕ (y)⟩free + ⟨σ (x)σ 2 (y)⟩RP
free
4
4
]
g1 g2 [ i i
d
2
RPd
+
⟨ϕ ϕ (x)σ (y)⟩free + ⟨σ 2 (x)ϕj ϕj (y)⟩RP
free
4
(
)
2
[
]
g 2 N + g22
1
1= 1
2 + 4η 2 + O(η 3 )384
4π|x − y|()2
1
1
g1 g2 N
+
η4,
3
8
2
4π
|x − y|(7.2.30)in the first non-trivial order in the ϵ-expansion.
The equality must be satisfied as a power series expansion with respect to η. We will pay
attention to the terms of order η 0 , η, and η 2 because the O(η 3 ) term has a contribution fromhigher dimensional primary operators on the left-hand side, which we are not interested in.
First of all, we will see in the case of the two-point function for ϕi -ϕj . At order η 0 , directly
acting the Laplacian twice on the left-hand side of (7.2.25) (see also (7.2.27)), we obtain(i) (LHS(7.2.25)) ⊃ Cϕi ϕ1j A1 δ ij □x □y |x − y|−2∆ϕi= Cϕi ϕ1j A1 δ ij (2∆ϕi )(2∆ϕi + 2 − d)(2∆ϕi + 2)(2∆ϕi + 4 − d)|x − y|−2∆ϕi −41(7.2.31)∼ 3 δ ij 2 · 3 · 42 γϕi |x − y|−8 ,4πwhere we take ϵ → 0 in the last line. On the other hand, the coeﬃcient of this term must agreewith the right-hand side to the first non-trivial order in ϵ, i.e. g12 δ ij |x − y|−8 /(4π 3 )2 at order η 0 .
Thus, we obtain
g12 1.
(7.2.32)
6 · 42 4π 3
The computation here is essentially same as in the case of flat Euclidean space obtained in[61][64].
The comparison at order η and η 2 is more involved. With the results caused by twice
Laplacian acting the two-point functions (see appendix B), let us first compare the term oforder η. Actually, this term vanishes on the right-hand side of (7.2.25) to the first non-trivial
order in the ϵ-expansion (see (7.2.28)), and so must be on the left-hand side. Indeed, this isithe case because the aϕ(0)σ term is the only contribution at order η, but it is of order ϵ:
γϕi =

i

aϕ(0)σ = (∆σ − 2∆ϕi − 2)(2∆ϕi − ∆σ )(2∆ϕi − ∆σ + 2 − d)(∆σ − 2∆ϕi − 4 + d) = O(ϵ),(7.2.33)
41

and it is further multiplied by γσ = O(ϵ) from the expansion coeﬃcient that appeared in(7.2.27). Thus, the term of order η does not appear as expected.
Now we compare the term of order η 2 to the first non-trivial order in the ϵ-expansion. As
we shown the contribution to the order η term behaves order ϵ2 , we concentrate on the terms
at order η 2 which start just order ϵ. There are three contributions to this order in the left-handiside. (i) From the bϕ(0)σ term: since there is an O(ϵ) prefactor in γσ , we have to focus on the O(1)iterm that appeared in bϕ(0)σ to compare with the right-hand side which is of order g12 = O(ϵ).
[
][ ]
i
2
(ii) From the aϕ(1)σ term: since there is a (O(1) + O(ϵ)) prefactor of Cϕi ϕσj Aσ γ1σ ∆2σ , we haveii−

to focus on the O(1) and O(ϵ) terms that appeared in aϕ(1)σ . (iii) From the aϕ(0)O term: since][−
there is a (O(1) + O(ϵ)) prefactor of Cϕi ϕOj AO− , we have to focus on the O(1) and O(ϵ) termsi−that appeared in aϕ(0)O . Thus, we approximateibϕ(0)σ = (∆σ )(2∆σ − 4∆ϕi )(∆σ − 2∆ϕi − 2)(2∆ϕi − ∆σ + 2 − d)
− 2d(∆σ )(2∆ϕi − ∆σ )(2∆ϕi − ∆σ + 2 − d) + O(x2 )∼ 2 · 42 = O(1),(7.2.34)

ϕi σa(1) = (∆σ − 2∆ϕi )(2∆ϕi − ∆σ − 2)(2∆ϕi − ∆σ − d)(∆σ − 2∆ϕi − 2 + d)(ϵ)
= O(ϵ),(7.2.35)∼ 42 2γϕi − γσ −2i −aϕ(0)O = (∆O− − 2∆ϕi − 2)(2∆ϕi − ∆O− )(2∆ϕi − ∆O− + 2 − d)(∆O− − 2∆ϕi − 4 + d)()
∼ 42 2γϕi − γO− = O(ϵ).
(7.2.36)
Therefore, the second term in (7.2.27) can be evaluated as
[
]
[ ]2
∆σ
∆
1
σ
(ii) (LHS(7.2.25)) ⊃ Cϕi ϕσj Aσ δ ij □x □y |x − y|−2∆ϕi η 2 γσ +
η + O(η 2 )γσ
2
1 ij [ 2 (ϵ )]σ|x − y|−8 η 2(7.2.37)∼ Cϕi ϕj Aσ δ 4 2γϕi + γσ −γσ2and the third term in (7.2.27) turns out−(iii) (LHS(7.2.25)) ⊃ Cϕi ϕOj AO− δ ij □x □y |x − y|−2∆ϕi η[ ()]−∼ Cϕi ϕOj AO− δ ij 42 2γϕi − γO− |x − y|−8 η 2 .

∆ −
O
2

[1 + O(η)](7.2.38)

Combining the above two contributions, the coeﬃcient of order η 2 which is proportional
to )]
[ 2(
1
ij
−8
σ
δ |x−y| to first non-trivial order in the left-hand side is obtained as Cϕi ϕj Aσ γσ 4 2γϕi + γσ − 2ϵ +
( )2
)]
[ (
−
Cϕi ϕOj AO− 42 2γϕi − γO− , and this should agree with the right-hand side (i.e. 2g12 4π1 3 ) atorder η 2 . As a result, by noticing that the relation (7.2.18), we obtain
(
1 2(
ϵ)
ϵ)
O−
2
−
−
i
4
2γ
+
γ
−
−
C
A
4
γ
+
γ
−
= 2g12
i
j
σ
σ
O
O
ϕ
ϕϕ
4π 3
2
2
comparing both sides of (7.2.25) at order η 2 .
42

(

1
4π 3

)2
,

(7.2.39)

Secondly, we will see the case of the two-point function for σ-σ similarly. At order η 0 ,directly acting the Laplacian twice on the left-hand side of (7.2.26) (see also (7.2.29)), we
obtain(i) (LHS(7.2.26)) ⊃ CσσI AI □x □y |x − y|−2∆σ= CσσI AI (2∆σ )(2∆σ + 2 − d)(2∆σ + 2)(2∆σ + 4 − d)|x − y|−2∆σ −41∼ 3 2 · 3 · 42 γσ |x − y|−8 ,4π(7.2.40)

where we take ϵ → 0 in the last line. On the other hand, the coeﬃcient of this term must agreewith the right-hand side to the first non-trivial order in ϵ, i.e. [g12 N + g22 ]|x − y|−8 /[2 · (4π 3 )2 ]at order η 0 . Thus, we obtain
γσ =

1
1
[g12 N + g22 ] 3 .
3
3·4
4π

(7.2.41)

The computation here is essentially same as in the case of flat Euclidean space obtained in [64].
The comparison at order η and η 2 is more involved. With the results caused by twice
Laplacian acting the two-point functions (see appendix B), let us first compare the term oforder η. Actually, this term vanishes on the right-hand side of (7.2.25) to the first non-trivial
order in the ϵ-expansion (see (7.2.28)), and so must be on the left-hand side. Indeed, this isithe case because the aϕ(0)σ term is the only contribution at order η, but it is of order ϵ:aσσ(0) = (−∆σ − 2)(∆σ )(∆σ + 2 − d)(−∆σ − 4 + d) = O(ϵ),
and it is further multiplied by γσ = O(ϵ) from the expansion coeﬃcient that appeared in(7.2.27). Thus, the term of order η does not appear as expected.
Now we compare the term of order η 2 to the first non-trivial order in the ϵ-expansion. Since
the contribution to the order η term behaves order ϵ2 for the same reason in the case of theϕi -ϕj correlation function, we concentrate the terms at order η 2 which start just order ϵ. There
are three contributions to this order in the left-hand side. (i) From the bσσ(0) term: since there isan O(ϵ) prefactor in γσ , we have to focus on the O(1) term that appeared in bσσ(0) to compare
22σσwith the right-hand side which is of order [g1 N + g2 ] = O(ϵ). (ii) From the a(1) term: since[][ ]2there is a (O(1) + O(ϵ)) prefactor of Cσσσ Aσ γ1σ ∆2σ , we have to focus on the O(1) and O(ϵ)−σOterms that appeared in aσσ(1) . (iii) From the a(0) term: since there is a (O(1) + O(ϵ)) prefactor[]−−of CσσO AO− , we have to focus on the O(1) and O(ϵ) terms that appeared in aσO(0) . Thus,
we approximate
22bσσ(0) = (∆σ )(−2∆σ )(−∆σ − 2)(∆σ + 2 − d) − 2d(∆σ ) (∆σ + 2 − d) + O(x )∼ 2 · 42 = O(1),(7.2.42)σσa(1) = (−∆σ )(∆σ − 2)(∆σ − d)(−∆σ − 2 + d)(ϵ)
= O(ϵ),(7.2.43)∼ 42 γσ −2−aσO(0) = (∆O− − 2∆σ − 2)(2∆σ − ∆O− )(2∆σ − ∆O− + 2 − d)(∆O− − 2∆σ − 4 + d)
∼ 42 (2γσ − γO− ) = O(ϵ),(7.2.44)43

Thus, the second term in (7.2.29) can be evaluated as
[
]
[ ]2
∆σ
1
∆
σ
(ii) (LHS(7.2.26)) ⊃ Cσσσ Aσ □x □y |x − y|−2∆σ η 2 γσ +
η + O(η 2 )
γσ
2
[
(
)]
ϵ
1 2
4 3γσ −|x − y|−8 η 2(7.2.45)∼ Cσσσ Aσγσ2and the third term in (7.2.29) turns out−(iii) (LHS(7.2.26)) ⊃ CσσO AO− δ ij □x □y |x − y|−2∆σ η[]−∼ CσσO AO− 42 (2γσ − γO− ) |x − y|−8 η 2 .

∆ −
O
2

[1 + O(η)](7.2.46)

Combining the above two contributions, the coeﬃcient of order η 2 which is [proportional
(
)]to
1
ϵ
−8
σ
2
|x − y| to first non-trivial order in the left-hand side is obtained as Cσσ Aσ γσ 4 3γσ − 2 +( )2−CσσO AO− [42 (2γσ − γO− )], and this should agree with the right-hand side (i.e. [g12 N +g22 ] 4π1 3 )at order η 2 . As a result, by noticing that the relation (7.2.20), we obtain
(
)
(
] 1 2
1 2(
ϵ)
ϵ) [ 2O−
2
2
4 3γσ −− Cσσ AO− 4 γO− + γσ −= g1 N + g2,
(7.2.47)
4π 3
2
2
4π 3
comparing both sides of (7.2.26) at order η 2 .
From now on, we will check the consistency of the obtained results (7.2.39) and (7.2.47)
with known results based on both [44] and [64]. For this purpose, we have to (re-)derive theconformal filed theory data to the first non-trivial order in ϵ.
As a first step, we solve the one-point function of the next-lowest dimensional scaler primary
operator O− := − Ngg21 ϕi ϕi + σ 2 with the scaling dimension ∆O− := 4 − ϵ + γO− (γO− denotes
the anomalous dimension of the local operator O− ) to the leading order in ϵ. The one-point
function of the next-lowest dimensional scaler primary operator O− is given by⟨O− (x)⟩RP ∼ ⟨O− (x)⟩RPfree ,
dd(7.2.48)and this can be evaluated at the tree-level approximately:AO −,(1 + x2 )4
g2
d
2
RPd
(RHS(7.2.48)) = − ⟨ϕi ϕi (x)⟩RPfree + ⟨σ (x)⟩free
g
( 1
)
g2
1
1
= 1−
.
3
g1 4π (1 + x2 )4(7.2.50)()
g2
1
AO − = 1 −+ O(ϵ1/2 ).
g1 4π 3(7.2.52)(LHS(7.2.48)) ∼Thus we obtain(7.2.49)(7.2.51)

Next, we compute the two-point function of the next-lowest dimensional scaler primary
operator O− is given by⟨O− (x)O− (y)⟩RP ∼ ⟨O− (x)O− (y)⟩RPfree ,dd

44(7.2.53)

and this can be evaluated at the tree-level approximately:
(
)
[ I
∆σ
∆σ ∆σ
d
σ
2
(LHS(7.2.53)) = |x − y|
CO− O− AI + CO− O− Aσ η 2 F1,; ∆σ + 1 − ; η
2 2
2
(
)]
∆ −
∆O− ∆O−d−O+ CO−OO− AO− η 2 2 F1,
; ∆O − + 1 − ; η
2
2
2
[
[
]
[ ]2
∆σ
∆
1
σ
= |x − y|−2∆O− CO−I O− AI + CO−σO− Aσ η 2 γσ +
η + O(η 2 )
γσ
2
]
∆ −
O
O−
+CO− O− AO− η 2 [1 + O(η)] ,(7.2.54)
−2∆O−

g22
i i
j j
RPd
2
2
RPd
⟨ϕ
ϕ
(x)ϕ
ϕ
(y)⟩
+
⟨σ
(x)σ
(y)⟩
free
free
2
N 2 g1
]
[
g2
i i
2
RPd
2
j j
RPd
−
⟨ϕ ϕ (x)σ (y)⟩free + ⟨σ (x)ϕ ϕ (y)⟩free
N g1
( 2
)(
)2
[
]
g2
1
1
=
+1
2 + 4η 2 + O(η 3 )
2
3
8
N g1
4π
|x − y|
( 2
)(
)2
g2
1
1
g2
+
−2 +1
η4.
2
3
g1
g1
4π
|x − y|8(RHS(7.2.53)) =(7.2.55)

Thus we obtain()2
1
CO− O− AI = 2+ O(ϵ1/2 ),
4π 3
( 2
)(
)2
g2
1
1
O−
σ
+1
.
CO− O− Aσ + CO− O− AO− = 4
γσ
N g12
4π 3
I

g22
+1
N g12

3 2

)(

(7.2.56)(7.2.57)

N g2

1
Note that the inverse of (7.2.56) is (4π2 ) N g2 +gMoreover, since CO−σO− = O(ϵ1/2 ) and2.
1
2
Aσ /γσ = O(1) at the tree-level approximation, we can calculate the operator product expansion−
coeﬃcient CO−OO− through the non-trivial relation (7.2.57) as follow:−

CO−OO− =4(g12 N + g22 ) 1+ O(ϵ1/2 ).
N g1 (g1 + g2 ) 4π 3(7.2.58)

Now, we will consider the three-point functions and let us read the coeﬃcients to the treelevel approximately. First of all, we will focus on the ϕ-ϕ-O− three-point function:⟨ϕi (x)ϕj (y)O− (z)⟩RP ∼ ⟨ϕi (x)ϕj (y)O− (z)⟩RP
free
g2
d
i
j
2
RPd
⟨ϕi (x)ϕj (y)ϕk ϕk (z)⟩RP=−free + ⟨ϕ (x)ϕ (y)σ (z)⟩free
N g1
[
[
(
)2
(
)2 ]
(
)2 ]
g2
1
δ ij
ηxz
1
ηyz
= −2
1+
1+
N g1 4π 3|x − z|4
1 − ηxz|y − z|4
1 − ηyz
[
]
(
)(
)2
(
)2
g2
1
δ ij
ηxy
1
+ 1−
1+
,
(7.2.59)34g1
4π|x − y|1 − ηxy(1 + z 2 )4
d

d

45

2

2

2

(y−z)
(z−x)
where ηxy := (1+x(x−y)
2 )(1+y 2 ) , ηyz := (1+y 2 )(1+z 2 ) , and ηzx := (1+z 2 )(1+x2 ) are cross-cap cross-ratios.
Thus we obtain the three-point function coeﬃcient Cϕi ϕj O− as follows
(
)2
1
ij g2
Cϕi ϕj O− = −2δ
+ O(ϵ1/2 ).
(7.2.60)
N g1 4π 3
3 2

N g2

1
Therefore, by using (7.2.60) and the inverse of (7.2.56) (4π2 ) N g2 +g
2 , we can obtain the operator
1−

2

product expansion coeﬃcient Cϕ−Oϕ− as−Cϕi ϕOj = −g1 g2+ O(ϵ1/2 ),N g12 + g22(7.2.61)

and then by using (7.2.52) additionally we can see−

Cϕi ϕOj AO− =g2 (g2 − g1 ) 1+ O(ϵ1/2 ).
N g12 + g22 4π 3(7.2.62)

Then, substituting the one-point function coeﬃcient for σ i.e. Aσ (7.2.24), the anomalous
dimension of σ i.e. γσ (7.2.41) and the product of Cϕi ϕσj multiplied by Aσ (7.2.62) for the nontrivial relation (7.2.20) obeyed from axiom II, we can re-derive the operator product expansion
coeﬃcient of Cϕi ϕσj as follows
Cϕi ϕσj = −δ ij

g1 1+ O(ϵ).
4 4π 3(7.2.63)2

Note that the three-point function coeﬃcient Cϕi ϕj σ = −δ ij g41 4π1 3 to the order ϵ1/2 , since thenormalization factor of the σ-σ two-point function is 4π1 3 , so Cϕi ϕj σ can be evaluated as equal
to Cϕi ϕσj multiplied by 4π1 3 .
Secondly, we will focus on the σ-σ-O− three-point function similarly:RP⟨σ(x)σ(y)O− (z)⟩RP ∼ ⟨σ(x)σ(y)O− (z)⟩free
g2
d
2
RPd
=−
⟨σ(x)σ(y)ϕk ϕk (z)⟩RPfree + ⟨σ(x)σ(y)σ (z)⟩free
N g1
[
[
)2
(
)2 ]
(
)2 ]
(
1
ηxz
1
ηyz
1
1+
1+
=2
4π 3
|x − z|4
1 − ηxz|y − z|4
1 − ηyz
]
[
)(
)2
(
)2
(
1
1
ηxy
1
g2
1
+
.
+ 1−
g1
4π 3
|x − y|41 − ηxy(1 + z 2 )4dd

Thus we obtain the three-point function coeﬃcient CσσO− as follows()2
1+ O(ϵ1/2 ).
CσσO− = 24π 3(7.2.64)(7.2.65)

Therefore, by using (7.2.65) and the inverse of (7.2.56), we can obtain the operator product−expansion coeﬃcient CσσO as−

CσσO =N g12+ O(ϵ1/2 ),N g12 + g22
46(7.2.66)

and then by using (7.2.52) additionally we can see−

CσσO AO− = −N g1 (g2 − g1 ) 1+ O(ϵ1/2 ).
2
2
3
N g1 + g2 4π(7.2.67)

Then, substituting the one-point function coeﬃcient for σ i.e. Aσ (7.2.24), the anomalous−
dimension of σ i.e. γσ (7.2.41) and the product of CσσO multiplied by AO− (7.2.66) for the nontrivial relation (7.2.20) obeyed from axiom II, we can re-derive the operator product expansion
coeﬃcient of Cσσσ as follows
g2 1+ O(ϵ).
(7.2.68)
4 4π 3( )2Note that three-point function coeﬃcient Cσσσ = − g42 4π1 3 to the order ϵ1/2 , since the normalization factor of the σ-σ two-point function is 4π1 3 , so Cσσσ can be evaluated as equal toCσσσ multiplied by 4π1 3 .
Let us check the consistency of above results with the known results, which describe thecertain non-trivial fixed point theory based on the ϵ-expansion. There are three unknown
parameters: g1 , g2 and γO− , and we recognize that there are two necessary conditions (7.2.39)
and (7.2.47). Therefore one parameter should be remaining unknown (note that these threeparameters must be rewritten in terms of ϵ and N ). So, we will show that the obtained results
are consistent with the known results when we give the known results as input data.
Two non-trivial relations (7.2.39) (7.2.47) which are obtained by comparing at order η 2 inthe correlation functions both the interacting theory and the free theory as ϵ → 0:
(
)2
(
1 2(
1
ϵ)
ϵ)
O−
2
2
4 2γϕi + γσ −− Cϕi ϕj AO− 4 γO− + γσ −= 2g1,
(7.2.69)
4π 3
2
2
4π 3
(
)
(
] 1 2
1 2(
ϵ)
ϵ) [ 2O−
2
2− Cσσ AO− 4 γO− + γσ −= g1 N + g2
4 3γσ −,
(7.2.70)
4π 3
2
2
4π 3
Cσσσ = −can be rewritten by using the following relation, based on (7.2.61) and (7.2.66),−

CσσO
ij N g1+ O(ϵ1/2 ),
=
−δ
−
O
g
Cϕi ϕj2
therefore we obtain
(
ϵ)
ϵ)
ϵ
1
1 2(
O−
2
− + γσ −−4i + γσ −
2γ
γ
4
−
C
A
= 3 · 43 x2 3 ,
i
j
O
O
ϕ
ϕϕ
3
4π
2
2
[N 4π2 ]()
1 2(ϵ ) Nxϵy1−O
4 3γσ −+
Cϕi ϕjAO− 42 γO− + γσ −= 6 · 42 ϵ x2 +
,
3
4π
2
y
2
N 4π 3(7.2.71)(7.2.72)(7.2.73)

where we define x and y by respectively
√
x :=
√
y :=

N
g1 ,
6ϵ(4π 3 )3(7.2.74)N
g2 ,6ϵ(4π 3 )3(7.2.75)47

as the same notation in [44].
Let us solve the anomalous dimension of the next-lowest primary operator O− i.e. γO−
appearing in (7.2.72) and (7.2.73) as a function of x and y, which is related to both ϵ and N ,[] 2 y2x +N
ϵ 2x2 y 2
ϵ
ϵ
ϵ 2
γO− =
x − 20 +− 1 y2 xy− − x2 −
y ,
2
N
N
2 2
N2
−
N
N
[ (
)
]
2
2
2
y
y
ϵ
y x2 + yN
ϵ
ϵ
ϵ 2
2
2
12 x +− 3x −
γO− =+1− − x2 −
y ,
2
y
xy
2
N
N
Nx −
2 2
N2
N

(7.2.76)(7.2.77)

Nby noticing the anomalous dimension of σ i.e. γσ (7.2.41) is written as γσ = 2ϵ x2 + Nϵ2 y 2 and2−
1
Cϕi ϕOj AO− = δ ij xy2 N−xy
(we consider the case i = j just now) in terms of x and y. Then we+y 2 4π 3
notice that the following nontrivial condition[ ()]
x2 y 2
y2
y2
y
2
2
2
− 3x −
x − 20 +− 1 = 12 x ++1(7.2.78)
N
N
N
N
Nx
must be satisfied at the non-trivial fixed point to the first non-trivial order in ϵ in the context
of the ϵ-expansion.
The nontrivial condition (7.2.78) for the interacting nontrivial fixed point theory can be
rewritten as
[
]
9 2
9
20
y2
xy +
y + 1 + 1 + x2 − x2 −= 0.
(7.2.79)NN
N
N
This equation is true if the known results in [44] [28], that means each beta functions β1 and β2
of g1 (∼ x) and g2 (∼ y) respectively vanish in the context of ϵ-expansion to the leading order:(N − 8)g13 − 12g12 g2 + g1 g22ϵ= 0,β1 = − g 1 +212(4π 3 )ϵ
−4N g13 + N g12 g2 − 3g23
β2 = − g 2 += 0,
24(4π 3 )(7.2.80)(7.2.81)or equivalently,
8 2 12
1x − xy + y 2 ,
N
N
N
x3
9
1 = −12 + 3x2 − y 2yN
1 = x2 −(7.2.82)(7.2.83)

are given. So, at least, we have checked that the zero of beta functions for g1 (∼ x) (7.2.82)
and g2 (∼ y) (7.2.83) are suﬃcient conditions for the equation (7.2.78) from the consistency ofmatching γO− which was obtained from two ways in our calculation based on the ϵ-expansion
form conformal filed theory on the real projective space. However, as we have already mentioned, there are three unknown parameters: g1 (∼ x), g2 (∼ y) and γO− , and we recognize that
there are two necessary conditions (7.2.39) and (7.2.47). Therefore one parameter remains unknown, but the obtained results are consistent with known results which describe the non-trivial
fixed point as the interacting theory.
48

7.2.2.

Conventional perturbation theory

In this subsection, we derive the one-point function on the real projective space from theconventional perturbation theory in the weak coupling region. The classical action of the criticalO(N ) model in d = 6 − ϵ dimensions is (7.2.1) and the model is defined by the Euclidean path
integral∫iZ[g1 , g2 ] = [Dϕi ][Dσ] e−S[ϕ ,σ,g1 ,g2 ] ,(7.2.84)
with the perturbative expansions in weak couplings g1 ≪ 1 and g2 ≪ 1. At g1 = 0 and g2 = 0,
the free-field correlation functions on the real projective space are given by (7.2.11) (7.2.12)(7.2.13) (7.2.14) (7.2.15) (7.2.16) Using the perturbative expansions, we obtain
∫
g1
d
RPd
RPd
⟨σ(x)⟩= ⟨σ(x)⟩free −dd y ⟨σϕi ϕi (x)σ(y)⟩RP
free
2
∫
g2
d
2
−
dd y ⟨σ(x)σ 3 (y)⟩RPfree + O(g ),
3!
∫
g1
d
i 2
RPd
=−
dd y ⟨σ(x)σ(y)⟩RPfree ⟨(ϕ ) (y)⟩free
2
∫
g2
d
2
RPd
2
dd y ⟨ϕ(x)ϕ(y)⟩RP−(7.2.85)
free ⟨σ (y)⟩free + O(g ),2where the integral range is 0 ≤ |y| ≤ 1 and d = 6 as ϵ → 0. In the second line, we use thestandard Wick contraction. After setting ⃗x to ⃗0 and plugging in (7.2.12), (7.2.13), (7.2.12) and(7.2.16) for (7.2.85), we obtain⟨σ(0)⟩RP = −d1
1[g1 N + g2 ] 3 + O(g 2 ).
48
4π(7.2.86)

Since ⟨σ(0)⟩RP equals to Aσ in conformal filed theory on the real projective space (see (7.2.4)),
this perturbative result agrees with (7.2.24) which is obtained by using the axioms in the criticalO(N ) model with conformal symmetry discussed in the previous subsection.
d

7.3.

Critical ϕ4 theory on 4−ϵ dimensional real projective
space

In this section, we would like to solve the conformal cross-cap bootstrap equation to limit theconformal field theory data from the viewpoint which is slightly diﬀerent from the approachof solving a one-point function using the consistency between the equation of motion andconformal invariance described in the previous section [77]. Since conformal field theory data is
still determined by using the ϵ-expansion to the first non-trivial order in ϵ, it is expected that
results obtained using any method will be consistent each other. Note that we can obtain only
the quantity of the operator product expansion coeﬃcient multiplied by the one-point function
coeﬃcient in this approach. In other words, we cannot determine the one-point function by
itself with solving such a cross-cap bootstrap equation.

49

7.3.1.

Conformal cross-cap bootstrap

In order to solve the cross-cap bootstrap equation(
Gij (η) =η1−ηj) ∆i +∆
2

Gij (1 − η),(7.3.1)

which gives infinite number of constraint conditions, we will take the conformal block decomposition. We know the arbitrary function of a single cross-cap cross-ratio Gij (η) can be
decomposed by conformal partial waves
)
(
∑
∆k
d
∆i − ∆j + ∆k ∆j − ∆i + ∆kk
Gij (η) =,; ∆k + 1 − ; η ,(7.3.2)
Cij Ak η 2 2 F1
2
2
2
kwhere Ak is the non-vanishing one point function coeﬃcient (3.1.1), which is also one of important properties of conformal field theory on the d-dimensional real projective space RPd . Notethat this quantity Ak appears the product with operator product expansion coeﬃcient Cij k in
the conformal partial wave decomposition (i.e. Cij k Ak ). Again, the sum is taken only over thescalar primaries appearing in the theory.
From now on, let us consider the case of the critical ϕ4 theory as a concrete, simple andthe most suggestive example, that is the universality class of critical Ising model in the context
of critical phenomena. In the conformal bootstrap approach, we do not necessarily require the
Hamiltonian (or Lagrangian) description. However, as will be mentioned later, for example,
when confirming the consistency with the calculation result based on the perturbation theory,
we will concretely write Lagrangian of the critical ϕ4 theory. So let us write down an action of
the critical ϕ4 theory
[
]
∫
1
g 4
d
2
S[ϕ, g] = d x (∂ϕ) + ϕ , d = 4 − ϵ, (ϵ > 0)
(7.3.3)2
4!
and a equation of motion of the lowest dimensional scalar primary ϕ is obtained□x ϕ(x) =g 3
ϕ (x).
3!

(7.3.4)Note that g is a quartic coupling and according to the known result of the zero of the one-loop
beta function in the conventional perturbation theory, g is evaluated as order ϵ.
For the ϕ-ϕ two-point function⟨ϕ(x)ϕ(y)⟩RP = |x − y|−2∆ϕ Gϕϕ (η),
dwe have obtained the following cross-cap bootstrap equation()∆ϕη
Gϕϕ (η) =Gϕϕ (1 − η).
1−η

(7.3.5)(7.3.6)

Here, in this interacting theory, since the operator product expansion for the lowest dimensional scalar primary is[ϕ] × [ϕ] = I + [ϕ2 ] + [ϕ4 ] + · · · ,50(7.3.7)

we sum over the three scalar primaries (i.e. I, ϕ2 , and ϕ4 ) in the arbitrary function of thecross-cap cross-ratio decomposed by conformal partial waves (i.e. Gϕϕ (η)).
(
)
∑
∆O
∆O ∆O
d
O
2
Gϕϕ (η) =Cϕϕ AO η 2 F1,
; ∆O + 1 − ; η .
(7.3.8)222
2 4
O=I,ϕ ,ϕ ,···

As far as the lowest order of ϵ is concerned, it is enough to consider the contribution of finite summation from the three scalar primaries i.e. the identity operator I, the next-lowest
dimensional scalar primary operator ϕ2 , and the next-next lowest dimensional scalar primary
operator ϕ4 , which is quite nontrivial (the same situation is found in the boundary conformalbootstrap [82]).
As a first step, we expand the scaling dimensions ∆i and the operator product expansion
coeﬃcients Cij k multiplied by the one-point function coeﬃcients Ak (i.e. Cij k Ak ) by ϵ. Eachof the scaling dimension of ϕ, ϕ2 and ϕ4 can be expanded in ϵ as follows
d−2
ϵ+ γϕ = 1 − + (γϕ )(1) ϵ + O(ϵ2 ),22∆ϕ2 = 2 − ϵ + (γϕ2 )(1) ϵ + O(ϵ2 ),(7.3.10)∆ϕ4 = 4 − 2ϵ + (γϕ4 )(1) ϵ + O(ϵ2 ),(7.3.11)∆ϕ =(7.3.9)and the products of the operator product expansion coeﬃcient and the one-point function
2
4
coeﬃcient (i.e. CϕϕI AI , Cϕϕϕ Aϕ2 and Cϕϕϕ Aϕ4 ) can be written as follows
1: normalization,
4π 2
2
2
2
Cϕϕϕ Aϕ2 = (Cϕϕϕ Aϕ2 )(0) + (Cϕϕϕ Aϕ2 )(1) ϵ + O(ϵ2 ),CϕϕI AI =ϕ4

ϕ4

Cϕϕ Aϕ4 = (Cϕϕ Aϕ4 )(1) ϵ + O(ϵ2 ).

(7.3.12)(7.3.13)(7.3.14)

We will see that not only the anomalous dimension (γϕ )(1) and (γϕ2 )(1) but also the quan2
4
tity (Cϕϕϕ Aϕ2 )(1) and (Cϕϕϕ Aϕ4 )(1) have been determined by the cross-cap bootstrap equation
2(7.3.6). Of cause, (Cϕϕϕ Aϕ2 )(0) is also determined by the cross-cap bootstrap equation (7.3.6).
So let us solve the cross-cap bootstrap equation analytically in the context of ϵ-expanded
conformal field theory data to the first nontrivial order in ϵ. Substitute ϵ-expanded conformal
field theory data into the cross-cap bootstrap equation (7.3.6) and compare both sides order
by order in ϵ after using the analytic connection formula of Gaussian hypergeometric function:
2 F1 (a, b, c; 1 − η) =Γ(c)Γ(c − a − b)
2 F1 (a, b; 1 + a + b − c; η)Γ(c − a)Γ(c − b)Γ(c)Γ(a + b − c) c−a−bη+
2 F1 (c − a, c − b; 1 + c − a − b; η).
Γ(a)Γ(b)(7.3.15)

As a result, comparing the terms of O(ϵ0 ) on both sides of the cross-cap bootstrap equation
gives:
2(Cϕϕϕ Aϕ2 )(0) = CϕϕI AI ,51

(7.3.16)

on the other hand, when comparing the terms of both sides O(ϵ), the following relationshipamong conformal field theory data are obtained
2

4(Cϕϕϕ Aϕ2 )(1) = −2(Cϕϕϕ Aϕ4 )(1) ,(7.3.17)ϕ4(1)(γϕ2 )

4(Cϕϕ Aϕ4 )(1)=,CϕϕI AI(γϕ )(1) = 0.

(7.3.18)(7.3.19)

As we have already mentioned, the relationship (7.3.16) is trivial because of the normalization
when we take the free theory limit. While, the other relationships (i.e. (7.3.17), (7.3.18) and(7.3.19) ) are non-trivial. Since it is known that the anomalous dimension γϕ starts from order
ϵ2 according to the perturbation theory, we can find that immediately one of the relationship(7.3.19) is consistent with known result. In this way, we have obtained the solution of thecross-cap bootstrap equation. So, next our task is to check the consistency of the obtained
results with the another calculation for solving the same ϕ-ϕ two-point function.

7.3.2.

ϵ-expansion from conformal field theory

In this subsection, we will solve the critical ϕ4 theory by using a compatibility between theconformal symmetry and the equation of motion as the consistency check of the obtained
results from the cross-cap bootstrap equation in the previous subsection.
Let us recall the action of the critical ϕ4 theory
[
]
∫
1
g 4
d
2
S[ϕ, g] = d x (∂ϕ) + ϕ , d = 4 − ϵ, (ϵ > 0)
(7.3.20)2
4!
and the equation of motion of the lowest dimensional scalar primary ϕ with the scaling dimension ∆ϕ = d−2+ γϕ = 1 − 2ϵ + γϕ is obtained2□x ϕ(x) =g 3
ϕ (x),
3!

(7.3.21)

where □ := (∂)2 is d-dimensional Laplacian. Note that g is a quartic coupling and according
to the known result of the zero of the one-loop beta function in the conventional perturbation
theory, g is evaluated as order ϵ. This equation of motion (7.3.21) implies that the operator ϕ3 behaves as a descendant of the scalar primary ϕ, that is the multiplet recombination
phenomenon known as the character of the interacting theory, which diﬀerent form the freetheory.
Given this perturbative picture, we also recall three axioms in the ϵ-expansion from conformal field theory as follows I: The non-trivial fixed point has conformal symmetry. II: If wetake the ϵ → 0 limit, correlation functions in the interacting theory will approach the ones in
the free theory. III: From the equations of motion, a particular primary operator in the freetheory (i.e. ϕ3 ) behaves as the descendant operator at the non-trivial fixed point (i.e. ϕ3 is thedescendant of ϕ by acting Laplacian as in (7.3.21)).
Before studying to determine conformal field theory data in the critical ϕ4 theory (a interacting theory at the Wilson-Fisher type non-trivial fixed point) on 4 − ϵ dimensional real52

projective space, we need to fix the normalization of some correlation functions in the freetheory at the Gaussian fixed point as follows:⟨ϕ(x)⟩RP
free = 0,11d⟨ϕ2 (x)⟩RP,free =2
4π (1 + x2 )2
[
(
)]
1
1
η
RPd
1+
.
⟨ϕ(x)ϕ(y)⟩free = 2
4π |x − y|21−ηd(7.3.22)(7.3.23)(7.3.24)

These correlation functions are needed when we take the free theory limit for the correlation
function in the interacting theory (i.e. the Wilson-Fisher type fixed point) based on axiom II
in the ϵ-expansion from conformal field theory.
From now on, we apply the ϵ-expansion from conformal field theory for solving the critical4
ϕ theory on 4−ϵ dimensional real projective space. First, we consider that for Laplacian acting
once the ϕ-ϕ two-point function, which is satisfied with the axiom I (i.e. conformal invariance),
we apply the equation of motion (7.3.21) as axiom III (i.e. a particular operator ϕ3 behaves as
the descendant of a primary ϕ, that is the multiplet recombination phenomenon)g 3d⟨ϕ (x)ϕ(y)⟩RP .
3!

⟨□x ϕ(x)ϕ(y)⟩RP =d(7.3.25)

Then based on the axiom II (i.e. correlation functions in the interacting theory will approach
the ones in the free theory if we take the ϵ → 0 limit), we would like to compare both sides of(7.3.25) to the first non-trivial order in ϵ.
For the left-hand side, we know the concrete form of the two-point function, so we can justdiﬀerentiate it
)
(
∑
∆O
∆O ∆O
d
−2∆ϕ
O
(LHS(7.3.25)) = □x |x − y|
Cϕϕ AO η 2 2 F1,
; ∆O + 1 − ; η .
2
2
2
2 4
O=I,ϕ ,ϕ ,···

(7.3.26)
For the right-hand side, since the prefactor g ∼ O(ϵ) is multiplied, the two-point function on
the Wilson-Fisher type fixed point may be approximated by the correlation function of thefree-field theory, we can calculate it using the ordinary Wick’s theoremg 3d⟨ϕ (x)ϕ(y)⟩RP
free
3!
g
d
2
RPd
= · 3⟨ϕ(x)ϕ(y)⟩RPfree ⟨ϕ (x)⟩free .
3!

(RHS(7.3.25)) ∼(7.3.27)

Comparing both sides, we found the anomalous dimension of the next-lowest dimensional scalar
primary ϕ2γϕ2 =g+ O(ϵ2 ).
16π 2(7.3.28)

Next, for Laplacian acting twice the ϕ-ϕ two-point function, which satisfies the axiom I, weapply the equation of motion (7.3.21) as axiom III⟨□x ϕ(x)□y ϕ(y)⟩RP =dg2d⟨ϕ3 (x)ϕ3 (y)⟩RP .
2
(3!)

53

(7.3.29)For the left-hand side, we know the concrete form of the two-point function decomposed into
conformal partial waves, so we can also just diﬀerentiate it
(
)
∑
∆O
∆O ∆O
d
−2∆ϕ
O
(LHS(7.3.29)) = □x □y |x − y|
Cϕϕ AO η 2 2 F1,
; ∆O + 1 − ; η .
2
2
2
2 4
O=I,ϕ ,ϕ ,···

(7.3.30)
For the right-hand side, since the prefactor g 2 ∼ O(ϵ2 ) is multiplied, the two-point function
on the Wilson-Fisher type fixed point may be approximated by the correlation function of thefree-field theory, we can calculate it using the ordinary Wick’s theorem
g2 3d⟨ϕ (x)ϕ3 (y)⟩RP
free
36 [
]
[
]3
g2
2
RPd
RPd
RPd
2
RPd
=
3! ⟨ϕ(x)ϕ(y)⟩free + 9⟨ϕ(x)ϕ(y)⟩free ⟨ϕ (x)⟩free ⟨ϕ (y)⟩free .
36(RHS(7.3.29)) ∼(7.3.31)

Comparing both sides, we found the anomalous dimension of the lowest dimensional scalar
primary ϕγϕ =g2+ O(ϵ3 )3 · 43 (4π 2 )2(7.3.32)

and the critical coupling
16π 2
g∗ =
ϵ + O(ϵ2 ).
3(7.3.33)

Note that, in the process of obtaining the above latter result (7.3.33), the previous result(7.3.28) was used. These results are consistent with known results in perturbation theory andin the case of flat Euclidean space [64].

54Chapter 8
Conclusion
We have studied conformal field theories on the d-dimensional real projective space. We havefound out that the method for solving conformal field theory useful not only for conformal field
theory on the flat d-dimensional Euclidean space Rd but also for conformal field theory on thed-dimensional real projective space RPd . We have examined concrete three critical models asapplication examples. As a result, we have confirmed that there are no conflicts with known
results.
First of all, we use a compatibility between the conformal symmetry and the equations of
motion to solve the one-point function of the lowest dimensional scalar primary operator in
the critical ϕ3 theory (a.k.a. the Yang-Lee edge singularity) on the d = 6 − ϵ dimensional realprojective space to the first non-trivial order in the ϵ-expansion. It reproduces the conventional
perturbation theory and agree with the numerical conformal bootstrap results. Secondly, we
study the critical O(N ) model on the d = 6 − ϵ dimensional real projective space and wesolve the one-point functions of the scalar primary operators to the first non-trivial order inthe ϵ-expansion based on the compatibility between the conformal invariance and the classicalequations of motion. We show that the obtained results are consistent with the known results.
Thirdly, we solve a conformal cross-cap bootstrap equation in the critical ϕ4 theory (a.k.a. thecritical Ising model) on the d = 4 − ϵ dimensional real projective space by ϵ-expansion and toevaluate the two-point function of the lowest dimensional scalar primary operator with itself
to the first non-trivial order in ϵ. We will also argue that our results are consistent with theresults of the ϵ-expansion from conformal field theory.
We conclude that the methods useful in solving conformal field theories in flat space-time
are sill powerful enough to solve them on real projective space-time. So, methods for solving
the conformal field theory on the real projective space we used, such as the ϵ-expansion from
conformal field theory, the conformal cross-cap bootstrap, the conventional perturbation theory,
may be worthwhile studying further in other more non-trivial space-time. The other methods
developed for solving conformal field theories on the d-dimensional flat Euclidean space may
be also useful for the purpose. Solving conformal field theories which can be interpreted asconcrete models belonging to a universality classes of critical phenomena on the real projection
space, we confirmed that the conformal field theory data other than the one-point functions
added in the case of the conformal field theory on the real projective space are consistent withthe conformal field theory data on the flat real Euclid space. Therefore, we conclude that theconformal field theory on the real projective space can be applied to the universality classes ofcritical phenomenon same as in the case of conformal field theory on the flat Euclidean space.

55

The future directions in order to solve conformal field theory on the d-dimensional realprojective space are as follows.
• To determine conformal field theory data (especially, one-point functions) in various conformal field theories on the d-dimensional real projective space analytically or numericallywith high precision using a perturbative or non-perturbative approach. Once the conformal field theory data are obtained, the critical exponents can be calculated from theinsights of the renormalization group and it may be possible to compare with the someknown experimental values.
• To solve conformal bootstrap equations on the d-dimensional real projective space (i.e.
the cross-cap bootstrap equation) analytically or numerically with further more highaccuracy than the first non-trivial order in ϵ. This direction also leads to verifying theconsistency between the results of epsilon deployment, which is a perturbative approach,and the conformal bootstrap conformal bootstrap approach, which is a non-perturbative
approach.
• To apply conformal field theory on the real projective space to fundamental problems
in theoretical physics such as d + 1 dimensional quantum gravity theory based on theholographic principle from string theory, two-dimensional unoriented string world sheet
theory, the classification of topological phase in condensed matter physics. These areapplications other than the application to the universality class of the critical phenomena
we have shown.
It is important to determine the conformal field theory data on the real projective spacecompletely. In order to do that, we need to compute one-point functions of all the scalar
primary operators beyond the only lowest one which has been studied in [74]. Since it becomes
harder and harder to determine the one-point functions of higher dimensional operators from thenumerical conformal bootstrap [64], it is important to develop alternative method such as theone pursued in [74]. At the same time, the ϵ-expansion must break down for suﬃciently higher
dimensional operators because of the non-perturbative operator mixing, and it is interesting tosee how this breakdown becomes manifest in this approach based on conformal field theory.
We would like to point out the possibility that conformal field theory on real projective
space can be applied other than application examples we have shown. The first is the expected
application through the comparison of conformal field theory on the real projective space andthe boundary conformal field theory which is similar in that symmetry is restricted. The second
is the application of d+1 dimensional quantum gravity theory through d-dimensional conformal
field theory based on the holographic principle.
First, it is interesting to pursue the diﬀerence between the conformal field theory on the realprojective space and conformal field theory on a flat space with a boundary (or boundaries),
which is called boundary conformal field theory [78] [79]. Because there is a common pointthat the conformal symmetry is not fully preserved but partially preserved (see also [80] [81][82]). In the former case the restricted conformal group is SO(d + 1), while in the lattercase the restricted conformal group is SO(d, 1). Since we consider the case of the Euclidean
signature in both cases, the each group (i.e. SO(d + 1) and SO(d, 1)) is a subgroup of the
Euclidean conformal group SO(d + 1, 1). In addition, it is known that boundary conformal
field theory has two important applications (see also [83]). The first is the application to thephysics of open strings and D-branes in string theory. The second is the application to the56boundary critical behavior and quantum impurity models in condensed matter physics. Sinceboth theoretical structures are similar, we expect that the methods for solving will be useful
in both cases. However, the restricted conformal symmetry themselves are strictly diﬀerent,
so it is considered that conformal field theory on the real projective space can be applied toa physical situation diﬀerent from boundary conformal field theory. Therefore, understanding
the physical diﬀerence between boundary conformal field theory and conformal field theory on
the real projective space will be useful in considering the applications of conformal field theory
on real projective space.
Second, from the viewpoint of holography, it is important to investigate the bulk reconstruction from conformal field theory on the real projective space. At the present twenty years
since anti-de Sitter/conformal field theory correspondence [30] was proposed, its verification
and application continue, but there is no proof yet. For a while after advocating for anti-deSitter/conformal field theory correspondence, “strong-weak duality” under the large N limitthat the strongly coupled region of the SU (N ) gauge theory, which is the quantum field theory,can be solved by the holographic dual classical gravity theory attracts attention, and analysisof the quantum theory of the d-dimensional strongly coupled gauge field has been carried out
exclusively from the d + 1 dimensional classical gravity theory. At the same time, as the opposite direction, attempts to construct d + 1 dimensional quantum gravity theory higher than
1 dimensional from the d-dimensional conformal invariant quantum field theory based on theoriginal holographic principle have also been discussed. Due to the development of the conformal bootstrap method, attempts to solve the conformal field theory not only in two dimensions
but higher than two dimensions have succeeded, at present, as a study related to AdSd+1 /CFTd
correspondence.
It is important to solve conformal field theory and to propose applications of conformal
field theory. Elucidation of critical phenomena based on the conformal hypothesis is not yetachieved, and construction of d+1 dimensional quantum gravity theory based on d-dimensional
conformal field theory from holography is also incomplete. Therefore our challenge to nature
is still going on.

57

Acknowledgments
First of all, I am sincerely grateful to my first research collaborator i.e. Yu Nakayama for giving
me part of his research work as a joint research project for my doctoral degree of Ph.D in physics
application, and also I would like to deeply thank my supervisors for giving me a lot of essential
and critical advices, i.e. Yu Nakayama and Thoru Eguchi, who are my supervisors in the terms
when I was a doctoral course student at Rikkyo University from April 2013 to March 2018 (i.e.
the former is from April 2017 to March 2018, and the latter is from April 2013 to March 2017).
Secondly, I would really like to thank all the members of Institute for Theoretical Physics,
Research Center for Mathematical Physics, and Research Center for Measurement in Advanced
Science, in Rikkyo University, those who were overlapped with me during the terms that I wasa student at Rikkyo University. Thirdly, I would like to thank all researchers of the high energy
physics or mathematical physics theorist (in particular the researchers of strings and fields)that have been active in domestic and overseas those who interacted with me actually througha lot of my research activities (such as workshops, symposiums, conferences, seminars, journal
clubs, and intensive lectures) about theoretical physics. Finally, I would also like to thank all
the members of faculty staﬀs, students and graduates, Rikkyo University (especially those who
supported my research activities and my educational activities of the department of physics,
Rikkyo University).
These research activities were supported in part by Rikkyo University graduate school salaryscholarship in each academic year 2013 and 2015. In addition, our works were supported in partby Rikkyo University Special Fund for Research (for short, Rikkyo SFR) in academic year 2017
(Research category: graduate student package type study. In more details, this is individual
investigation under the guidance of the applicant’s supervisor and the adoption number is 29).

58

Appendix A
Conformal field theory on realprojective space in projective null cone
formalism
In this appendix, we derivate properties of conformal field theories on real projective space inprojective null cone formalism. Sometimes this formalism is called the embedded formalism.

A.1.

Projective null cone formalism

In projective null cone formalism, we write down a d-dimensional physical Euclidean spacecoordinate vector as embedded coordinate in a d + 2 dimensional Minkowski space Rd+1,1⃗
X A = (X + , X − , X)

(A.1.1)

where the superscript A is an index of embedded space which runs d+2 directions which consistof null cone directions (i.e. + and −) and the other space directions (i.e. from 1 to d). Notethat the terminology “Projective ” means
X A ∼ λX A ,(A.1.2)where λ is a real constant and the other terminology “null cone” says
X 2 = 0,(A.1.3)⃗ ·X⃗ = 01 . While, if we consider a line
where X 2 := X · X = ηAB X A X B = −X + X − + X
element ds2 in the embedded Minkowski space Rd+1,1 like ds2 = ηAB dX A dX B where ηAB =diag(−1, +1, +1, · · · , +1) and A, B = −1, 0, 1, · · · , d, the light cone coordinates become X + :=1
(X −1 + X 0 ) and X − := 12 (X −1 − X 0 ). Note that the time-like coordinate X −1 is the invariant
2
coordinate direction under the restricted conformal group SO(d + 1) transformation. We canNote that the metric components of the light cone directions η+− = η−+ = − 12 have minus sign. This causesthat the dependence of the cross-cap cross-ratio η for conformal blocks in real projective space is diﬀerent fromthe dependence of the cross-ratio ξ for conformal blocks in the case of boundary conformal field theory multipliedby minus.
1

59

fix the section as
XA
XA ∼ + =
X

(

⃗
X− X
1, + , +X X)=: (1, x2 , ⃗x),

(A.1.4)⃗

where x2 := ⃗x · ⃗x = |⃗x|2 , and the vector ⃗x := XX+ is a d-dimensional coordinate vector in original
flat Euclidean space Rd , so that this section is called the Euclidean section.
In d-dimensional real projective RPd space which is defined by involution ⃗x → − |⃗x⃗x|2 ona frat Euclidean space Rd , the restricted conformal symmetry SO(d + 1) is preserved. If weconsider the case of Lorentzian signature space like Minkowski space-time, SO(d, 1) which isa subgroup of the Lorentzian conformal group SO(d, 2) is preserved. In both the Euclideancase and the Lorentzian case, dilatation D and one of the linear combinations of translation Pµ
and special conformal translation Kµ i.e. Pµ − Kµ are violated and the other one Pµ + Kµ androtation Mµν are preserved in d-dimensional real projective space.

A.2.

One-point functions and two-point functions in conformal field theory on real projective spaceFirst, we derive one-point functions which are fixed by the restricted conformal symmetrySO(d+1). Since the scalar scaling operators Oi with scaling dimensions ∆i behave homogeneousfunctions of coordinate like Oi (λX) = λ−∆i Oi (X), we can fix the one-point functions by using
the conformally invariant time-like coordinate X −1 = X + +X − under appropriate normalization⟨Oi (X)⟩RP =
d

Ai
.
+
(X + X − )∆i(A.2.1)⃗ = (1, x2 , ⃗x), i.e. λ = 1+ ,We assume that we take the Euclidean section λX A = X1+ (X + , X − , X)
X
by using the following scaling relation⟨Oi (λX)⟩RP = (λ)−∆i ⟨Oi (X)⟩,d(A.2.2)

we obtain one-point functions in physical coordinate space⟨Oi (⃗x)⟩RP =dAi.
(1 + |⃗x|2 )∆i(A.2.3)

Similarly, we can also fix the functional form of two-point functions for the scalar scaling
operators respected to Oi (λX) = λ−∆i Oi (X)⟨Oi (X1 )Oj (X2 )⟩RP =d

1

gij (η)(X1+ + X1− )∆i (X2+ + X2− )∆j(A.2.4)up to gij (η), which is an arbitrary function of the conformally invariant dimensionless parameterη so-called corss-cap cross-ratioη=−2X1 · X2.
+(X1 + X1− )(X2+ + X2− )60(A.2.5)

The conformally invariant parameter η consists of a Lorentz scalar X1 · X2 = ηAB X1A X2B =X1+ X2+ [− 12 |⃗x1 − ⃗x2 |2 ], which is proportional to the distance between two points in physicalcoordinate space, and time-like directions of each points X + +X − = X + (1+|⃗x|2 ), each of whichis invariant direction under the transformation by the restricted conformal group SO(d + 1).
In projective null cone formalism, the cross-cap cross-ratio η is invariant under an involution⃗ → −X.
⃗ If we choose thetransformation i.e. inversion X + ↔ X − and parity transformation X
1
1
A
+
− ⃗
2
Euclidean section λX = X + (X , X , X) = (1, x , ⃗x) (i.e. λ = X + ), we obtain the cross-cap
cross-ratio η in physical coordinate space as follows
η=|⃗x1 − ⃗x2 |2.
(1 + |⃗x1 |2 )(1 + |⃗x2 |2 )(A.2.6)We also assume that we choose the Euclidean section, by using the following scaling relation⟨Oi (λX1 )Oj (λX2 )⟩RP = (λ)−∆i −∆j ⟨Oi (X1 )Oj (X2 )⟩,dand we obtain two-point functions in physical coordinate space1d⟨Oi (⃗x1 )Oj (⃗x2 )⟩RP =gij (η).
(1 + |⃗x1 |2 )∆i (1 + |⃗x2 |2 )∆j(A.2.7)(A.2.8)Note that the asymptotic form of the arbitrary function gij (η) is determined by the following
operator product expansion as ⃗x1 → ⃗x2 , (i.e. η → 0)∑
Cij k C[|⃗x1 − ⃗x2 |, ∂2 ]Ok (⃗x2 )(A.2.9)Oi (⃗x1 )Oj (⃗x2 ) =kwhere k runs all scalar primaries appearing in the theory. After taking the expected value inconformal field theory on the real projective space for both sides, we obtain∑dd⟨Oi (⃗x1 )Oj (⃗x2 )⟩RP =Cij k C[|⃗x1 − ⃗x2 |, ∂2 ]⟨Ok (⃗x2 )⟩RP .
(A.2.10)kThis equation turns out∑
1
1
1
Cij k Ak |⃗x1 − ⃗x2 |−2(− 2 (∆i +∆j −∆k ))g(η)∼. (A.2.11)
ij
∆
2
∆
2
2 ∆
j
i
(1 + |⃗x1 | ) (1 + |⃗x2 | )
(1
+
|⃗
x
2| ) kkx2 |is proportional to a distance between ⃗x1 andSince the cross-cap cross-ratio η = (1+|⃗x|⃗x1 |12−⃗)(1+|⃗
x2 |2 )
2⃗x2 (i.e. |⃗x1 −⃗x2 | ) divided by remained conformally invariant directions (1 + |⃗x1 |2 )(1 + |⃗x2 |2 ), we1
found that at least gij (η) needs a factor Cij k Ak η − 2 (∆i +∆j −∆k ) . Therefore we may write down
1
gij (η) as gij (η) = η − 2 (∆i +∆j ) Gij (η).
For later purposes, we rewrite the two-point functions in terms of Gij (η) instead of gij (η)
1through the relation gij (η) = η − 2 (∆i +∆j ) Gij (η) as follows
2⟨Oi (⃗x1 )Oj (⃗x2 )⟩RP =d(1 + |⃗x1 |2 )−∆i +∆j2(1 + |⃗x2 |2 )2|⃗x1 − ⃗x2 |−∆j +∆i2( ∆ +∆ )ij

Gij (η)(A.2.12)2

in physical coordinate space, or equivalently
RPd⟨Oi (X1 )Oj (X2 )⟩=(X1+ + X1− )−∆i +∆j2(X2+ + X2− )( ∆ +∆ )(−2X1 · X2 )in embedded coordinate space.
61

i

j

2

−∆j +∆i2

Gij (η)(A.2.13)

A.3.

Conformal blocks in conformal field theory on realprojective space

In this subsection, we derivate a single conformal block of two-point functions, which is aneigenfunction of a conformal quadratic Casimir equation by the modern method which is basedon projective null cone formalism (see Appendix A in [82]).
According to the result of the previous subsection, we can rewrite two-point functions ofscalar primary in the projective null cone formalism as follows⟨Oi (X1 )Oj (X2 )⟩RPd=(X1+ + X1− )−∆i +∆j2(X2+ + X2− )−∆j +∆i2( ∆ +∆ )(−2X1 · X2 )ijGij (η).

(A.3.1)2

Here, for later purposes, we introduce a pre-factor function as
F (X) :=

(X1+ + X1− )−∆i +∆j2(X2+ + X2− )−∆j +∆i2( ∆ +∆ )(−2X1 · X2 )

i

j

.

(A.3.2)

2

A particular single conformal block Gij∆,ℓ with scaling dimension ∆ and spin ℓ satisfy thefollowing conformal Casimir equation
L2 (F (X)Gij∆,ℓ ) = C∆,ℓ (F (X)Gij∆,ℓ ) ,(A.3.3)

where
1 (1)
(1)
(2)
(2)
L2 = − (LAB L(1)AB + LAB L(2)AB + LAB L(1)AB + LAB L(2)AB ),2()∂∂
LAB = i XA
− XB
.
∂XB
∂XA

(A.3.4)(A.3.5)

L2 is a quadratic Casimir operator of the rotation generator LAB in the embedded space Rd+1,1 ,and C∆,ℓ = ∆(∆ − d) + ℓ(ℓ + d − 2) is an eigenvalue of the quadratic Casimir operator for thecorresponding eigenfuntion.
The conformal Casimir equation A.3.3 becomes
[
]
∂2
1
d−2 ∂
1
2
2 ∂
η (1 − η) 2 − η− (∆i − ∆j )(∆j − ∆i )η −η
Gij∆,ℓ (η) = C∆,ℓ Gij∆,ℓ (η),
∂η
∂η 4
2
∂η
4
(A.3.6)
up to a common pre-factor function F (X). So if we consider in the case of the scalar primaryℓ = 0 i.e. Gij∆,0 , its asymptotic functional form is fixed with a situation exchanging O∆,ℓ , thatis∆Gij∆,0 (η) = η 2 gij∆,0 (η).

(A.3.7)The conformal quadratic Casimir equation turns out
η(1 − η)∂2∂
gij∆,0 (η) + [c − (a + b + 1)η] gij∆,0 (η) − ab gij∆,0 (η) = 0,
2
∂η
∂η
62

(A.3.8)

where1a = (∆i − ∆j + ∆),2
1
b = (∆j − ∆i + ∆),2
dc=∆+1− .
2(A.3.9)(A.3.10)(A.3.11)This diﬀerential equation A.3.8 is so-called the Gaussian hypergeometric diﬀerential equation,and the solution is known asgij∆,0 (η) = 2 F1 (a, b; c; η) :=∞∑(a)n (b)nn=0(c)n n!

(η)n ,(A.3.12)

where the symbol (a)n is called the Pochhammer symbol, which is defined by(a)0 := 1(a)n := (a)(a + 1) · · · (a + n − 1),n ≥ 1.

(A.3.13)(A.3.14)

Note that the another solution η 1−c 2 F1 (a − c + 1, b − c + 1; 2 − c; η) is dropped because the factor∆∆
η 1−c = η − 2 will cancel the factor η 2 which is determined by the operator product expansion
in Gij∆,0 (η). As a result, we can see a single conformal block with scaling dimension ∆ andspin ℓ = 0 as follows
)
(
∆
1
1
d
Gij∆,0 (η) = η 2 2 F1(∆i − ∆j + ∆), (∆j − ∆i + ∆); ∆ + 1 − ; η .
(A.3.15)2
2
2
In this way, we have found conformal blocks for the conformal partial wave decomposition in
conformal field theory on the d-dimensional real projective space.

63

Appendix B
Laplacian acting twice two-point
functions
In this appendix, we compute the following terms appearing in the Laplacian acting twice
the O1 -O2 two-point correlation functions, which are expanded by the conformal partial wave
decomposition:
) [
(
]
∆O
∆O
2
2 +n
O1 O2
O1 O2
2
−2∆O1
2
□x □y |x − y|= a(n) + b(n) η + O(η ) |x − y|−2∆O1 −4 η 2 +n , (B.0.1)
η
1 O2
aO
:= (∆O2 − 2∆O1 − 2 + 2n)(2∆O1 − ∆O2 − 2n)(n)× (2∆O1 − ∆O2 + 2 − d − 2n)(∆O2 − 2∆O1 − 4 + d + 2n),(B.0.2)O1 O2b(n):= (∆O2 + 2n)(2∆O2 − 4∆O1 + 4n)

× (∆O2 − 2∆O1 − 2 + 2n)(2∆O1 − ∆O2 + 2 − d − 2n)− 2d(∆O2 + 2n)(2∆O1 − ∆O2 − 2n)(2∆O1 − ∆O2 + 2 − d − 2n)+ O(x2 ),(B.0.3)where ∆O1 and ∆O2 are the scaling dimension of the local operator O1 and O2 respectively.
Since the above terms have appeared in the expansion (7.2.27) and (7.2.29), we need to set a
suitable integer number n in order to evaluate the order η terms and the order η 2 terms. In
fact, in the case of O2 = σ, which has the canonical scaling dimension 2, we need to calculate
n = 0 and n = 1, then in the case of O2 = O− , which has the canonical scaling dimension 4,
we need to calculate n = 0.

64

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