Luis F Alday

Gregory P Korchemsky

Instanton corrections to twist-two operators

We present the calculation of the leading instanton contribution to the scaling dimensions of twist-two operators with arbitrary spin and to their structure constants in the OPE of two half-BPS operators in N = 4 SYM. For spin-two operators we verify that, in agreement with N = 4 superconformal Ward identities, the obtained expressions coincide with those for the Konishi operator. For operators with high spin we find that the leading instanton correction vanishes. This arises as the result of a rather involved calculation and requires a better understanding.

Introduction

In this paper we continue the study of instanton corrections to correlation functions in maximally supersymmetric N = 4 Yang-Mills theory. Although these corrections are exponentially small in the planar limit, they are expected to play an important role in restoring the S-duality of the theory. At weak coupling, the leading instanton contribution can be found in the semiclassical approximation by neglecting quantum fluctuation of fields. In this approximation, the calculation amounts to evaluating the product of operators in the background of instantons and integrating the resulting expression over the collective coordinates. For a review see [START_REF] Belitsky | Yang-Mills and D instantons[END_REF][START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Bianchi | Instantons and Supersymmetry[END_REF].

Previous studies revealed [START_REF] Bianchi | On the logarithmic behavior in N=4 SYM theory[END_REF] that the leading instanton contribution to four-point correlation function of half-BPS operators in N = 4 SYM scales at weak coupling as e -8π 2 /g 2 . An OPE analysis showed, however, that this correction does not affect twist-two operators [START_REF] Arutyunov | Perturbative and instanton corrections to the OPE of CPOs in N=4 SYM(4)[END_REF] and, therefore, does not modify the leading asymptotic behaviour of correlation functions in the light-cone limit. This led to the conclusion [START_REF] Bianchi | Properties of the Konishi multiplet in N=4 SYM theory[END_REF][START_REF] Kovacs | On instanton contributions to anomalous dimensions in N=4 supersymmetric Yang-Mills theory[END_REF] that the leading instanton contribution to the conformal data of twist-two operators (scaling dimensions ∆ S and OPE coefficients C S ) should be suppressed by a power of the coupling constant and scale as g 2n e -8π 2 /g 2 with some n ≥ 1. The calculation of such corrections within the conventional approach is way more complicated as it requires going beyond the semiclassical approximation.

In [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF] we argued that, by virtue of N = 4 superconformal symmetry, the above mentioned instanton effects can be determined from the semiclassical computation of two-and threepoint correlation functions for another operator in the same supermultiplet. Following this approach, we computed the leading non-vanishing correction to the scaling dimension of the Konishi operator, ∆ (inst) K = O(g 4 e -8π 2 /g 2 ) and to its structure constant in the OPE of two half-BPS operators, C (inst) K = O(g 2 e -8π 2 /g 2 ) (see [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF] for explicit expressions). In this paper we extend the analysis to twist-two operators O S with arbitrary even Lorentz spin S. For spin zero, the operator O S=0 coincides with the half-BPS operator and is protected from quantum corrections. For spin-two, the operator O S=2 belongs to the same supermultiplet as the Konishi operator and, therefore, has the same conformal data. For S ≥ 4, quite surprisingly, our calculation yields a vanishing result for the instanton contribution. This implies that the leading instanton corrections to the conformal data of twist-two operators O S with S ≥ 4 are suppressed at least by a power of g 2 as compared with those for the Konishi operator

∆ (inst) S = δ S,2 ∆ (inst) K + O(g 6 e -8π 2 /g 2 ) , C (inst) S = δ S,2 C (inst) K
+ O(g 4 e -8π 2 /g 2 ) .

(1.1)

Notice that these two expressions differ by a power of the coupling constant, ∆

(inst) S

/C

(inst) S = O(g 2 ), whereas the leading perturbative corrections to both quantities have the same scaling in g 2 at weak coupling.

The paper is organized as follows. In Section 2 we define operators of twist two and discuss their relation to light-ray operators. In Section 3 we construct the one-instanton solution to the equations of motion in N = 4 SYM for the SU (2) gauge group. In Section 4 we present the calculation of correlation functions involving half-BPS and twist-two operators in the semiclassical approximation and discuss its generalization to the SU (N ) gauge group. Section 5 contains concluding remarks. Some details of the calculation are summarized in four Appendices.

Twist-two operators

All twist-two operators in N = 4 SYM belong to the same supermultiplet and share the same conformal data. This allows us to restrict our consideration to the simplest twist-two operator, of the form

O S (x) = tr Z D S + Z(x) + . . . , (2.1)
where Z(x) is a complex scalar field and D + = n µ D µ (with n 2 = 0) is a light-cone component of the covariant derivative

D µ = ∂ µ + i[A µ , ].
All fields take values in the SU (N ) algebra, e.g. Z(x) = Z a (x)T a with the generators normalized as tr(T a T b ) = δ ab /2. The dots on the right-hand side of (2.1) denote a linear combination of operators with total derivatives of the form ∂ ℓ + tr ZD S-ℓ + Z(x) with 0 ≤ ℓ ≤ S. The corresponding expansion coefficients are fixed by the condition for O S (x) to be a conformal primary operator and depend, in general, on the coupling constant. To lowest order in the coupling, they are related to those of the Gegenbauer polynomials (see Eq. (2.9) below).

In this paper, we compute the leading instanton corrections to correlation functions of twist-two operators (2.1) and half-BPS scalar operators of the form

O 20 ′ (x) = Y AB Y CD tr[φ AB φ CD (x)] , (2.2)
where the complex scalar fields φ AB = -φ BA (with A, B = 1, . . . , 4) satisfy reality condition φAB = 1 2 ǫ ABCD φ CD . The auxiliary antisymmetric tensor Y AB is introduced to project the product of two scalar fields onto the representation 20 ′ of the SU (4) R-symmetry group. It satisfies ǫ ABCD Y AB Y CD = 0 and plays the role of the coordinate of the operator in the isotopic SU (4) space. The scalar field Z entering (2.1) is a special component of φ AB

Z = φ 14 = (Y Z ) AB φ AB , (2.3)
where (Y Z ) AB has the same properties as the Y -tensor in (2.2) and has the only nonvanishing components (Y Z ) 14 = -(Y Z ) 41 = 1/2. Conformal symmetry fixes the form of two-and three-point correlation functions of the operators (2.1) and (2.2)

O S (x) ŌS ′ (0) = δ SS ′ N S [2(xn)] 2S (x 2 ) ∆ S +S , O 20 ′ (x 1 )O 20 ′ (x 2 )O S (0) = C S (x 2 12 ) 2 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 S x 2 12 x 2 1 x 2 2 (∆ S -S)/2 . (2.4)
Here the scaling dimension of twist-two operator ∆ S , the normalization factor N S and threepoint coefficient function C S depend on the coupling constant whereas the scaling dimension of the half-BPS operator is protected from quantum corrections.

Light-ray operators

To compute the correlation functions (2.4), it is convenient to introduce a generating function for the twist-two operators (2.1), the so-called light-ray operator,

O(z 1 , z 2 ) = tr Z(nz 1 )E(z 1 , z 2 )Z(nz 2 )E(z 2 , z 1 ) . (2.5)
In distinction with (2.1), it is a nonlocal operator -the two scalar fields are separated along the light-ray direction n µ and two light-like Wilson lines are inserted to restore gauge invariance,

E(z 1 , z 2 ) = P exp i z 2 z 1 dt n µ A µ (nt) , (2.6)
with E(z 1 , z 2 )E(z 2 , z 1 ) = 1. The scalar variables z 1 and z 2 define the position of the fields along the null ray. Making use of gauge invariance of (2.5), we can fix the gauge n µ A µ (x) = 0 in which Wilson lines (2.6) reduce to 1. Then, the expansion of (2.5) in powers of z 1 and z 2 takes the form

O(z 1 , z 2 ) = k,n≥0 z k 1 k! z n 2 n! tr [∂ k + Z(0)∂ n + Z(0)] , (2.7)
where ∂ + = (n∂). To restore gauge invariance, it suffices to replace ∂ + → D + in this relation.

The local operators on the right-hand side of (2.7) are not conformal primaries but, for given S = k + n, they can be expanded over the conformal primary operators O S (0) and their descendants ∂ ℓ + O S-ℓ (0). As mentioned at the beginning of this section, the conformal operators have the following general form

O S (0) = k+n=S c kn tr [D k + Z(0)D n + Z(0)] , (2.8)
with the expansion coefficients c kn depending on the coupling constant. To lowest order in the coupling, these coefficients coincide (up to an overall normalization) with those of (

x 1 + x 2 ) S C 1/2 S ((x 1 -x 2 )/(x 1 + x 2 )) = c kn x k 1
x n 2 involving the Gegenbauer polynomial [START_REF] Makeenko | Conformal Operators In Quantum Chromodynamics[END_REF][START_REF] Ohrndorf | Constraints From Conformal Covariance on the Mixing of Operators of Lowest Twist[END_REF]

c kn = (-1) n S! (k! n! ) 2 + O(g 2 ) .
(2.9)

Note that the sum in (2.8) vanishes for odd S and the conformal operators are defined for even nonnegative S. Inverting (2.8) we can expand the light-ray operator (2.7) over the conformal twist-two operators and their descendants. In this way, we find that the operators O S (0) appear as the coefficients in the expansion of the light-ray operator (2.5) in powers of

z 12 ≡ z 1 -z 2 [11] 1 O(z 1 , z 2 ) = S≥0 z S 12 S! [O S (0) + . . . ] . (2.10)
Here the dots denote the contribution from descendant operators of the form ∂ ℓ + O S (0). This contribution is fixed by conformal symmetry, see [START_REF] Braun | The Uses of conformal symmetry in QCD[END_REF].

From light-ray to twist-two operators

The rationale for introducing (2.5) is that finding instanton corrections to light-ray operators proves to be simpler as compared to that for twist-two operators. Then, having computed the correlation function 2

G(x 1 , x 2 , nz 1 , nz 2 ) = O 20 ′ (x 1 )O 20 ′ (x 2 )O(z 1 , z 2 ) , (2.11)
we can then apply (2.10) to obtain the three-point correlation function

O 20 ′ (x 1 )O 20 ′ (x 2 )O S (0) .
To lowest order in the coupling, we can simplify the calculation by making use of the following relation between the operators O S (0) and O(z 1 , z 2 )

O S (0) = ‹ dz 1 dz 2 (2πi) 2 (z 1 -z 2 ) S (z 1 z 2 ) S+1 O(z 1 , z 2 ) , (2.12)
where the integration contour in both integrals encircles the origin. Indeed, replacing O(z 1 , z 2 ) on the right-hand side with (2.7) and computing the residue at z 1 = 0 and z 2 = 0 we obtain (2.8) with c kn given by (2.9). Using the operator identity (2.12) inside corrrelation functions we arrive at

O S (0)O 20 ′ (x 1 )O 20 ′ (x 2 ) = ‹ dz 1 dz 2 (2πi) 2 (z 1 -z 2 ) S (z 1 z 2 ) S+1 O(z 1 , z 2 )O 20 ′ (x 1 )O 20 ′ (x 2 ) . (2.13)
We would like to emphasize that relations (2.12) and (2.13) hold to the lowest order in the coupling constant. Beyond this order, we have to take into account O(g 2 ) corrections to (2.9). Relation (2.13) offers an efficient way of computing the correlation functions of twist-two operators. As an example, we show in Appendix C how to use (2.13) to obtain the correlation functions (2.4) in the Born approximation (see Eqs. (C.4) and (C.6)).

Instantons in N = 4 SYM

The general one-instanton solution to the equations of motion in N = 4 SYM with the SU (N ) gauge group depends on 8N fermion collective coordinates, see [START_REF] Belitsky | Yang-Mills and D instantons[END_REF][START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Bianchi | Instantons and Supersymmetry[END_REF]. Among them 16 modes are related to N = 4 superconformal symmetry. The remaining 8(N -2) fermion modes are not related to symmetries in an obvious way and are usually called 'nonexact modes'. This makes the construction of the instanton solution more involved.

For the SU (2) gauge group the general one-instanton solution to the equations of motion of N = 4 SYM can be obtained by applying superconformal transformations to the special solution corresponding to vanishing scalar and gaugino fields and gauge field given by the celebrated BPST instanton [START_REF] Belavin | Pseudoparticle Solutions of the Yang-Mills Equations[END_REF] 2 Such correlation functions appeared in the study of the asymptotic behaviour of four-point correlation functions

O 20 ′ (x 1 )O 20 ′ (x 2 )O 20 ′ (x 3 )O 20 ′ (x 4
) in the light-cone limit x 2 34 → 0, see [START_REF] Alday | From correlation functions to Wilson loops[END_REF].

φ (0),AB = λ (0),A α = λ(0) α,A = 0 , A (0) µ (x -x 0 ) = 2 η a µν (x -x 0 ) ν T a (x -x 0 ) 2 + ρ 2 , (3.1)
where η a µν are the 't Hooft symbols and the SU (2) generators are related to Pauli matrices T a = σ a /2. It depends on the collective coordinates ρ and x 0 defining the size and the position of the instanton, respectively.

In this section, we present explicit expressions for the one-instanton solution in N = 4 SYM for the SU (2) gauge group. Then, in the next section, we explain how to generalize the expressions for the correlation functions (2.4) to the case of the SU (N ) gauge group.

The field configuration (3.1) is annihilated by (the antichiral) half of the N = 4 superconformal generators. Applying to (3.1) the remaining (chiral) N = 4 superconformal transformations (see (A.11) in Appendix A), we obtain a solution to the N = 4 equations of motion that depends on 16 fermionic collective coordinates, ζ A α and η αA . The resulting expressions for gauge and scalar fields can be expanded in powers of fermion modes

A µ = A (0) µ + A (4) µ + • • • + A (16) µ , φ AB = φ AB,(2) + φ AB,(6) + • • • + φ AB,(14) , (3.2)
where

A (n) µ
denotes the component of the gauge field that is homogenous in ζ A α and η αA of degree n and similar for scalars. Gaugino fields admit similar expansions (see (B.4) in Appendix B) but we will not need them for our purposes. Explicit expressions for various components of (3.2) are given below.

By virtue of superconformal invariance, the action of N = 4 SYM evaluated on the instanton configuration (3.2) does not depend on the fermionic modes and coincides with the one for pure Yang-Mills theory3

S inst = d 4 x L(x) = -2πiτ , (3.3)
where τ is the complex coupling constant

τ = θ 2π + 4πi g 2 . (3.4)
Notice that, due to our definition of the Lagrangian (see (A.9) in Appendix A), the instanton solution (3.1) and (3.2) does not depend on the coupling constant.

It is straightforward to work out the leading term of the expansion (3.2) by subsequently applying N = 4 superconformal transformations to (3.1), see [START_REF] Belitsky | Yang-Mills and D instantons[END_REF]. The direct calculation of the subleading terms becomes very involved due to the complicated form of these transformations (see (A.11) in Appendix A). There is, however, a more efficient approach to computing higher components in (3.2) which is presented in Appendix B. It makes use of the known properties of fields with respect to conformal symmetry, R-symmetry and gauge transformations and allows us to work out the expansion (3.2) recursively with little efforts.

To present the resulting expressions for the instanton configuration (3.2) for the SU (2) gauge group it is convenient to switch to spinor notation and use a matrix representation for (3.1) (see Appendix A for our conventions)

(A α α) i j = iA a µ (x)(T a ) i j (σ µ ) α α , (3.5)
where the four-dimensional vector of 2 × 2 matrices σ µ = (1, iσ) and the SU (2) generators T a = σ a /2 are built from Pauli matrices. This field carries two pairs of indices, Lorentz indices (α, α = 1, 2) and SU (2) indices (i, j = 1, 2). Here we distinguish lower and upper SU (2) indices and define the product of two matrices by

(A α α) i j (A β β ) j k ≡ (A α αA β β ) i k . (3.6)
All indices are raised and lowered with the help of the antisymmetric tensor, e.g.

(A α α) i k = (A α α) ij ǫ jk , (A (0)
α α) ij = ǫ iα x j α + ǫ jα x i α x 2 + ρ 2 , (3.7)
where the second relation follows from (3.1) and (3.5). The advantage of (A α α) ij as compared to (3.5) is that it is symmetric with respect to the SU (2) indices. The instanton (3.1) and (3.7) is a self-dual solution to the equations of motion in pure Yang-Mills theory, F (0) α β = 0. The corresponding (chiral) strength tensor is given by (F

(0) αβ ) ij = ǫ β γ ∂ (α β A (0) β) γ + A (0) (α β A (0) β) γ ij = - 1 2 f (x)(ǫ iα ǫ jβ + ǫ iβ ǫ jα ) , (3.8)
where ∂ α β = (σ µ ) α β ∂ µ and angular brackets denote symmetrization with respect to indices, A (αβ) = A αβ + A βα . Here a shorthand notation was introduced for the instanton profile function

f (x) = 16ρ 2 (x 2 + ρ 2 ) 2 .
(3.9)

It is easy to verify that (F αβ ) ij is symmetric with respect to both pair of indices and satisfies the equations of motion D αα F αβ = [∂ αα + A αα , F αβ ] = 0. Notice that (3.7) and (3.8) do not depend on the position of the instanton x 0 . To restore this dependence it suffices to apply the shift x → xx 0 . We can further simplify (3.7) and (3.8) by contracting all Lorentz indices with auxiliary (commuting) spinors |n ≡ λ α and |n] ≡ λ α depending on their chirality, e.g.

n|A ij |n] ≡ λ α λ α(A α α) ij , n|F ij |n ≡ λ α λ β (F αβ ) ij .
The resulting expressions only have SU (2) indices and are homogenous polynomials in λ and λ. In particular,

n|A (0) ij |n] = - λ i x j α + λ j x i α x 2 + ρ 2 λ α , n|F (0)
ij |n = -f (x)λ i λ j , (3.10)
where the superscript '(0)' indicates that these expressions correspond to the lowest term in the expansion (3.2). An unusual feature of the expressions on the right-hand side of (3.10) is that the chiral Lorentz indices are identified with the SU (2) indices.

To obtain the subleading corrections in the instanton solution (3.2), depending on fermion modes, we apply the method described in Appendix B. Namely, we make use of (B.9) and replace the gauge field by its expression (3.10). Going through the calculation we get

n|A (4) ij |n] = 8ρ 2 (ρ 2 + x 2 ) 3 ǫ ABCD nζ A ρ 2 [η B n] -ξ B |x|n] ζ C i ζ D j , n|F (4)
ij |n = 1 4 f 2 (x)ǫ ABCD nζ A nζ B ζ C i ζ D j , (3.11)
where ζ stands for a linear x-dependent combination of fermion modes

ζ A α (x) = ξ A α + x α α η αA , (3.12)
and a shorthand notation is used for various contractions of Lorentz indices

nζ A = λ α ζ A α , [η B n] = ηB α λ α , ξ B |x|n] = ξ αB x α α λ α . (3.13)
Note that the dependence on the fermion modes enters into n|F (4) ij |n through the linear combination (3.12). As explained in Appendix B, this property can be understood using conformal symmetry.

We recall that, to leading order, the instanton solution (3.10) satisfies the self-duality condition F (0) α β = 0. Using the obtained expression for A (4) α α, we find that the anti-self-dual component of the gauge field strength tensor

[n| Fij |n] ≡ λα λ β F α β,ij receives a nonvanishing correction [n| F (4) ij |n] = f 2 (x) 2ρ 2 ǫ ABCD ρ 2 [η A n] -ξ A |x|n] ρ 2 [η B n] -ξ B |x|n] ζ C i ζ D j , (3.14)
which contains four fermion modes. This relation illustrates that higher components of fields depend on fermion modes in a nontrivial manner.

Repeating the same analysis we can evaluate subleading corrections to all fields in (3.2). In particular, using the relations (B.7) and (B.9) we get the following expressions for the scalar field on the instanton background

φ AB,(2) ij = - f (x) 2 √ 2 ζ [A i ζ B] j , φ AB,(6) ij = - f 2 (x) 20 √ 2 ǫ CDEF (ζ 2 ) AC (ζ 2 ) BD ζ E i ζ F j , (3.15)
where brackets in the first relation denote antisymmetrization with respect to the SU (4) indices, the variable ζ is defined in (3.12) and (

ζ 2 ) AB = (ζ 2 ) BA = ζ βA ǫ βγ ζ γB .
The expressions (3.11) and (3.15) depend on the size of the instanton, ρ, as well as on 16 fermion modes, ξ A α and η αA . To restore the dependence on the position of the instanton, we apply the shift x → xx 0 .

Correlation functions in the semiclassical approximation

In the semiclassical approximation, the calculation of correlation functions reduces to averaging the classical profile of the operators over the collective coordinates of instantons [START_REF] Belitsky | Yang-Mills and D instantons[END_REF][START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Bianchi | Instantons and Supersymmetry[END_REF]

O 1 . . . O n inst = dµ phys e -S inst O 1 . . . O n , (4.1)
where the gauge invariant operators O i on the right-hand side are evaluated on the instanton background (3.2). For the SU (2) gauge group, the collective coordinates of the one-instanton solution are the size of the instanton ρ, its localtion x µ 0 and 16 fermion modes, ξ A α and ηA α . The corresponding integration measure is [START_REF] Bianchi | Instantons in supersymmetric Yang-Mills and D instantons in IIB superstring theory[END_REF] dµ phys e -S inst = g 8 2 34 π 10 e 2πiτ d4 x 0

dρ ρ 5 d 8 ξ d 8 η , (4.2)
where the complex coupling constant τ is defined in (3.4). We recall that, due to our normalization of the Lagrangian (A.9), the instanton background (3.2) does not depend on the coupling constant. The same is true for the operators (2.2), (2.5) and (2.8) built from scalar and gauge fields. As a consequence, the instanton correction to the correlation function of these operators scales in the semiclassical approximation as O(g 8 e 2πiτ ) independently on the number of operators n. At the same time, the same correlation functions in the Born approximation scale as O(g 2n ) with one power of g coming from each scalar field (see Eqs. (C.4) and (C.6)). 4 Thus, the ratio of the two contributions scales as O(g 8-2n e 2πiτ ) with n being the number of operators. As we show in a moment, the instanton correction vanishes in the semiclassical approximation for n > 5. Later in the paper we shall compute this correction explicitly for n = 2 and n = 3.

For the correlation function (4.1) to be different from zero upon integration of fermion modes, the product of operators O 1 . . . O n should soak up the product of all fermion modes (ξ) 8

(η) 8 ≡ A ξ A 1 ξ A 2 ηA 1 ηA 2 .
Let us denote by k i the minimal number of fermion modes in O i . Then, by virtue of the SU (4) invariance, the total number of modes in the product O 1 . . . O n is given by k min + 4p with k min = k 1 + • • • + k n and p = 0, 1, . . . . For k min > 16 the product O 1 . . . O n is necessarily proportional to the square of a fermion mode and, therefore, the correlation function (4.1) vanishes. To obtain a nontrivial result for the correlation function, we have to go beyond the semiclassical approximation and take into account quantum fluctuations of fields. For k tot ≤ 16 and k min multiple of 4, the integral over fermion modes in (4.1) does not vanish a priori. This case corresponds to the so-called minimal correlation functions [START_REF] Bianchi | Instantons and Supersymmetry[END_REF][START_REF] Green | Instanton induced Yang-Mills correlation functions at large N and their AdS(5) x S**5 duals[END_REF]. Another interesting feature of these correlation functions is that the results obtained in the semiclassical approximation for the SU (2) gauge group can be extended to the general case of SU (N ) gauge group for the one-instanton solution [START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Dorey | Yang-Mills instantons in the large N limit and the AdS/CFT correspondence[END_REF] and for multi-instanton solutions at large N [START_REF] Dorey | Multi-instanton calculus and the AdS / CFT correspondence in N=4 superconformal field theory[END_REF].

Since the operators (2.2), (2.5) and (2.8) involve two scalar fields, we deduce from (3.2) that each of them has at least four fermion modes, k = 4. Then, for the product of these operators O 1 . . . O n we have k min = 4n and, as a consequence, the correlation function O 1 . . . O n inst is minimal for n ≤ 4.

Instanton profile of twist-two operators

To compute the correlation functions (2.4) in the semiclassical approximation, we have to evaluate the operators O 20 ′ and O S on the instanton background by replacing all fields by their explicit expressions (3.2) and, then, expand their product in powers of 16 fermion modes.

For the half-BPS operators (2.2) we find [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF] O (4)

20 ′ (x) = 1 2 f 2 (x)Y AB Y CD (ζ 2 ) AC (ζ 2 ) BD , (4.3)
where f (x) is the instanton profile function (3.9) and the superscript on the left-hand side indicates the number of fermion modes. Since O 20 ′ (x) is annihilated by half of the N = 4 supercharges, its expansion involves only four fermion modes.

As argued above, in order to study correlators involving twist-two operators it is convenient to introduce light-ray operators (2.5). In order to evaluate those in the instanton background we replace scalar and gauge fields by their expressions Z = Z (2) + Z (6) + . . . and A = A (0) + A (4) + . . . , which leads to O(z 1 , z 2 ) = O (4) + O (8) + O (12) + O (16) , (

where the leading term contains four fermion modes and each subsequent term has four modes more. The last term is proportional to the product of all 16 modes (ξ) 8 (η) 8 . The first two terms on the right-hand side of (4.4) are given by

O (4) (z 1 , z 2 ) = tr Z (2) (nz 1 )E (0) (z 1 , z 2 )Z (2) (nz 2 )E (0) (z 2 , z 1 ) , O (8) (z 1 , z 2 ) = tr Z (6) (nz 1 )E (0) (z 1 , z 2 )Z (2) (nz 2 )E (0) (z 2 , z 1 ) + tr Z (2) (nz 1 )E (4) (z 1 , z 2 )Z (2) (nz 2 )E (0) (z 2 , z 1 ) + (z 1 ↔ z 2 ) , (4.5)
where E (n) denote specific terms in the expansion of the light-like Wilson line (2.6) on the instanton background

E(z 1 , z 2 ) = E (0) + E (4) + . . . . (4.6)
Explicit expressions for E (0) and E (4) can be found in Appendix D. The remaining terms O (12) and O (16) are given by a priori lengthy expressions but their evaluation leads to a surprisingly simple result

O (12) (z 1 , z 2 ) = O (16) (z 1 , z 2 ) = 0 . (4.7)
As we show in Appendix D, these terms are necessarily proportional to the square of a fermion mode and, therefore, have to vanish. Let us consider the correlation function (2.11). As was explained at the beginning of the section, this correlation function is minimal and scales as O(g 8 e 2πiτ ) in the semiclassical approximation. Applying (4.1), we have to find the instanton profile of

O(z 1 , z 2 )O 20 ′ (x 1 )O 20 ′ (x 2 )
and identify the contribution containing 16 fermion modes. Making use of (4.3) and (4.4) we find

O(z 1 , z 2 )O 20 ′ (x 1 )O 20 ′ (x 2 ) inst = dµ phys e -S inst O (8) (z 1 , z 2 )O (4) 20 ′ (x 1 )O (4) 20 ′ (x 2 ) . (4.8)
Later we shall use this relation to find the instanton correction to the three-point correlation function

O S (0)O 20 ′ (x 1 )O 20 ′ (x 2 ) inst .
In a similar manner, in order to find O S (0) ŌS ′ (x) inst we have to consider the correlation function of two light-ray operators separated by a distance x and conjugated to each other. For this purpose, we have to generalize the definition (2.5) by allowing the light-ray to pass through an arbitrary point x

O(z 1 , z 2 |x) = e iP •x O(z 1 , z 2 ) e -iP •x , (4.9)
where P µ is the operator of total momentum. The operator (4.9) is given by the same expression (2.5) with scalar and gauge fields shifted by x. Obviously, for x = 0 we have

O(z 1 , z 2 |0) = O(z 1 , z 2 )
. Substituting (4.4) into (4.9), we find that O(z 1 , z 2 |x) has a similar expansion in powers of fermion modes. Then, we take into account (4.7) to get

O(z 1 , z 2 |0) Ō(z 3 , z 4 |x) inst = dµ phys e -S inst O (8) (z 1 , z 2 |0) Ō(8) (z 3 , z 4 |x) , (4.10)
where Ō(z 3 , z 4 |x) is a conjugated operator. Notice that the lowest O (4) term of the expansion (4.4) does not contribute to (4.8) and (4.10).

We are now ready to compute the correlation functions (4.8) and (4.10). Using the expression for the integration measure (4.2), we first perform the integration over fermion modes. Replacing O (4) 20 ′ and O (8) by their explicit expressions, Eqs. (4.3) and (4.5), respectively, and going through a lengthy calculation we find (see Appendix D for details)

d 8 ξd 8 η O (8) (z 1 , z 2 )O (4) 20 ′ (x 1 )O (4) 20 ′ (x 2 ) = -2 29 × 3 3 × (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × (z 1 -z 2 ) 2 [(nx 2 )x 2 1 -(nx 1 )x 2 2 ] 2 ρ 14 [(ρ 2 + (nz 1 -x 0 ) 2 )(ρ 2 + (nz 2 -x 0 ) 2 )] 3 [(ρ 2 + x 2 10 )(ρ 2 + x 2 20 )] 4 , (4.11)
where

(Y i Y j ) = ǫ ABCD Y i,AB Y i,
d 8 ξd 8 η O (8) (z 1 , z 2 |0) Ō(8) (z 3 , z 4 |x) = 2 30 × 3 2 × (z 1 -z 2 ) 2 (z 3 -z 4 ) 2 (xn) 4 ρ 12 [(ρ 2 + (nz 1 -x 0 ) 2 )(ρ 2 + (nz 2 -x 0 ) 2 )(ρ 2 + (x + nz 3 -x 0 ) 2 )(ρ 2 + (x + nz 4 -x 0 ) 2 )] 3 (4.12)
Notice that (4.11) and (4.12) vanish quadratically for z 1 → z 2 (as well as for z 3 → z 4 ). This property can be understood as follows. For z 2 → z 1 we can use the definition of the light-ray operator (2.5) to show that

O(z 1 , z 2 ) = tr [Z 2 (nz 1 )] + 1 2 (z 2 -z 1 )∂ + tr [Z 2 (nz 1 )] + O (z 2 -z 1 ) 2 . (4.13)
The first two terms of the expansion involve the half-BPS operator tr Z 2 , it can be obtained from the general expression (2.2) for Y = Y Z . According to (4.3), the half-BPS operators have exactly four fermion modes and, therefore, the first two terms on the right-hand side of (4.13) do not contribute to

O (8) (z 1 , z 2 ) leading to O (8) (z 1 , z 2 ) = O((z 2 -z 1 ) 2 ).
To obtain the correlation functions, we have to integrate (4.11) and (4.12) over the bosonic coordinates ρ and x 0 with the measure (4.2). The resulting integrals can be expressed in terms of the so-called D-functions

D ∆ 1 ...∆n (x 1 , . . . , x n ) = d 4 x 0 dρ ρ 5 n i=1 ρ ρ 2 + x 2 i0 ∆ i , (4.14)
where x i0 = x ix 0 . Multiplying (4.11) and (4.12) by the additional factors coming from the integration measure (4.2), we finally find the leading instanton correction to the correlation functions

O(z 1 , z 2 )O 20 ′ (1)O 20 ′ (2) inst = - 27 128π 10 g 8 e 2πiτ (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × (z 1 -z 2 ) 2 (x 2 1 x 2 2 ) 2 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 2 D 3344 (nz 1 , nz 2 , x 1 , x 2 ) , (4.15)
O(z 1 , z 2 |0) Ō(z 3 , z 4 |x) inst = 9 16π 10 g 8 e 2πiτ × (z 1 -z 2 ) 2 (z 3 -z 4 ) 2 (xn) 4 D 3333 (nz 1 , nz 2 , x + nz 3 , x + nz 4 ) . (4.16)
The same correlation functions in the Born approximation are given by (C.2) and (C.5), respectively, evaluated for N = 2. Dividing (4.15) and (4.16) by the Born level expressions, we find that the instanton corrections to the two correlation functions scale as O(g 2 e 2πiτ ) and O(g 4 e 2πiτ ), respectively, in agreement with the analysis at the beginning of this section.

Instanton corrections to correlation functions

Let us apply (4.15) and (4.16) to derive the correlation functions of twist-two operators. We recall that to the lowest order in the coupling constant, the correlation functions involving light-ray and twist-two operators are related to each other through the relations (2.12) and (2.13). It is not obvious however whether the same relations should work for the instanton corrections in the semiclassical approximation. To show that this is the case, we apply below (2.12) and ( 2

O S (0)O 20 ′ (1)O 20 ′ (2) inst = - 27 128π 10 g 8 e 2πiτ (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × (x 2 1 x 2 2 ) 2 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 2 I S (x 1 , x 2 ) , (4.17)
where I S (x 1 , x 2 ) denotes the following integral

I S (x 1 , x 2 ) = ‹ dz 1 dz 2 (2πi) 2 (z 1 -z 2 ) S (z 1 z 2 ) S+1 (z 1 -z 2 ) 2 D 33∆∆ (nz 1 , nz 2 , x 1 , x 2 ) , (4.18)
with ∆ = 4. In what follows we relax this condition and treat ∆ as an arbitrary parameter.

The reason for this is that the same integral with ∆ = 3 enters the calculation of (4.16).

Replacing the D-function in (4.18) by its integral representation (4.14) and exchanging the order of integration, we find that the integrand has triple poles at z 1 , z 2 = (ρ 2 + x 2 0 )/(2(x 0 n)). Blowing up the integration contour in (4.18) and picking up the residues at these poles, we find that the integral over z 1 and z 2 vanishes for all nonnegative integer S except for S = 2, leading to

I S (x 1 , x 2 ) = 6 δ S,2 D 6∆∆ (0, x 1 , x 2 ) = 6 δ S,2 d 4 x 0 dρ ρ 9 (ρ 2 + x 2 0 ) 6 (ρ 2 + x 2 10 ) ∆ (ρ 2 + x 2 20 ) ∆ . (4.19)
For ∆ = 4 the calculation of this integral yields

D 644 (0, x 1 , x 2 ) = π 2 90 1 x 2 12 (x 2 1 x 2 2 ) 3 . ( 4
O S (0)O 20 ′ (1)O 20 ′ (2) inst = -δ S,2 9 640π 8 g 8 e 2πiτ (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × 1 x 2 12 x 2 1 x 2 2 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 2 . (4.21)
Surprisingly enough, this expression vanishes for all spins except S = 2. For S = 0 the corresponding twist-two operator O S=0 = tr Z 2 is half-BPS. In this case, the three-point correlation function is protected from quantum corrections and, therefore, the instanton correction (4.21) should vanish for S = 0.

To obtain the two-point correlation function O S (0)O S ′ (x) from (4.16) we apply (2.12) to both operators to get

O S (0)O S ′ (x) inst = 9 16π 10 g 8 e 2πiτ (xn) 4 I SS ′ (x) , (4.22)
where I SS ′ (x) is given by a folded contour integral over four z-variables. Using (4.18) this integral can be rewritten as

I SS ′ (x) = ‹ dz 3 dz 4 (2πi) 2 (z 3 -z 4 ) S ′ +2 (z 3 z 4 ) S ′ +1 I S (x + nz 3 , x + nz 4 ) , (4.23)
where I S is evaluated for ∆ = 3. Replacing I S with its integral representation (4.19), we evaluate the integral by picking up the residue at the poles z 3 , z 4 = (ρ 2 +(x 0 -x) 2 )/(2(x 0 -x)n) in the same manner as (4.19). In this way, we arrive at

O S (0) ŌS ′ (x) inst = δ S,2 δ S,S ′ 81 4π 10 g 8 e 2πiτ (xn) 4 D 66 (0, x) , (4.24)
where the D-function is defined in (4.14). Similar to (4.21), this expression vanishes for all spins except S = S ′ = 2. For S = S ′ = 0 this property is in agreement with protectiveness of two-point correlation functions of half-BPS operators. A close examination of (4.14) shows that D 66 (0, x) develops a logarithmic divergence. It comes from integration over small size instantons, ρ → 0, located close to one of the operators, x 2 0 → 0 and (xx 0 ) 2 → 0. This divergence produces a logarithmically enhanced contribution ∼ ln x 2 which modifies the scaling dimensions of twist-two operators. To identify this contribution, we regularize the integral by modifying the integration measure over x 0 5

D 66 (0, x) → d 4-2ǫ x 0 dρ ρ 5 ρ 12 (ρ 2 + x 2 0 ) 6 (ρ 2 + (x -x 0 ) 2 ) 6 = π 2 20(x 2 ) 6 - 1 ǫ + ln x 2 + . . . , (4.25)
where dots denote terms independent of x and/or vanishing for ǫ → 0. In the standard manner, the pole 4.26) define one-instanton corrections to the correlation functions. Antiinstanton corrections are given by the complex conjugated expressions. Finally, we combine (4.21) and (4.26) with analogous expressions in the Born approximation (given by (C.4) and (C.6) for N = 2) and add the anti-instanton contribution to get the following expressions for the correlation functions in N = 4 SYM for the SU (2) gauge group

O S (0)O 20 ′ (1)O 20 ′ (2) = 3g 6 (4π 2 ) 3 (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) x 2 12 x 2 1 x 2 2 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 S × 1 -δ S,2 3g 2 10π 2 (e 2πiτ + e -2πiτ ) , (4.27)
O S (0) ŌS ′ (x) = δ SS ′ 3g 4 c S 2(4π 2 ) 2 [2(nx)] 2S (x 2 ) 2+2S × 1 + δ S,2 9 5 g 2 4π 2
2 (e 2πiτ + e -2πiτ ) ln x 2 , (4.28)

where the normalization factor c S is defined in (C.6). We recall that these relations were derived in the one (anti)instanton sector in the semiclassical approximation and are valid up to corrections suppressed by powers of g 2 . Comparing (4.28) with the general expression for a two-point correlation function (2.4), we obtain the leading instanton correction to the scaling dimension of twist-two operators As follows from (4.28), the instanton correction to the normalization factor N S entering the first relation in (2.4) has the same dependence on the coupling constant as (4.29) and does not affect the leading correction to the structure constants in the OPE of two half-BPS operators.

∆ S = 2 + S + γ S γ (inst) S = -δ S,2 9
As was already mentioned, for S = 0 the correlation functions (4.27) and (4.28) are protected from quantum corrections and, therefore, γ S=0 = 0 and C S=0 = 1 for arbitrary coupling constant. For S > 2, there are no reasons for the same relations to hold beyond the semiclassical approximation. For S = 2, the corresponding conformal operator O S=2 ∼ tr(ZD 2 + Z) -2 tr(D + ZD + Z) belongs to the same N = 4 supermultiplet as Konishi operator K = tr(φ AB φAB ). As a consequence, the two operators should have the same anomalous dimension, γ S=2 = γ K , as well as OPE coefficients, C S=2 = C K . As a nontrivial check of our calculation we use the results of [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF] to verify that both relations are indeed satisfied in the semiclassical approximation.

As was already mentioned in Section 4, the correlation functions (4.27) and (4.28) belong to the class of minimal correlation functions. Following [START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Dorey | Yang-Mills instantons in the large N limit and the AdS/CFT correspondence[END_REF][START_REF] Dorey | Multi-instanton calculus and the AdS / CFT correspondence in N=4 superconformal field theory[END_REF], we can then generalize the relations (4.27) and (4.28) to the SU (N ) gauge group and, in addition, include the contribution of an arbitrary number of (anti)instantons at large N . As before, for S = 2 the resulting expressions for γ S and C S do not receive corrections in the semiclassical approximation, whereas for S = 2 they coincide with those for the Konishi operator and can be found in [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF].

Conclusions

In this work we have presented the explicit calculation of the instanton contribution to twoand three-point correlation functions involving half-BPS and twist-two operators. A somewhat surprising outcome of our analysis is the vanishing of the leading instanton corrections to the scaling dimensions and the OPE coefficients of twist-two operators with spin S > 2. This result comes from a rather involved calculation and requires a better understanding. Note that the situation here is very different from that for twist-four operators. As it was shown in [START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF], crossing symmetry implies that twist-four operators with arbitrarily high spin acquire instanton corrections already at order O(g 2 e -8π 2 /g 2 ).

According to (1.1), the instanton corrections to twist-two operators with S > 2 are pushed to higher order in g 2 . Their calculation remains a challenge as it requires going beyond the semiclassical approximation. There is however an interesting high spin limit, S ≫ 1, in which we can get additional insight on the instanton effects. In this limit, the scaling dimensions of twist-two operators scale logarithmically with the spin [START_REF] Korchemsky | Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions[END_REF][START_REF] Alday | Comments on operators with large spin[END_REF] ∆ S = S + 2Γ cusp (g 2 ) ln S + O(S 0 ) ,

where Γ cusp (g 2 ) is the cusp anomalous dimension. The instanton contribution to ∆ S should have the same asymptotic behavior and produce a correction to Γ cusp (g 2 ). According to the first relation in (1.1), it should scale at least as O(g 6 e -8π 2 /g 2 ). We can find however the same correction using the fact that the cusp anomalous dimension governs the leading UV divergences of light-like polygon Wilson loops. The light-like polygon Wilson loop W L is given by the product of gauge links (2.6) defined for L different light-like vectors n i . In the semiclassical approximation, the instanton contribution to W L can be found using (4.1). Since W L does not depend on the coupling constant on the instanton background, the dependence on g 2 only comes from the integration measure (4.2) leading to W L inst = O(g 8 e -8π 2 /g 2 ). At the same time, in the Born approximation, we have W L Born = 1. As a consequence, the leading instanton correction to the cusp anomalous dimension scales at least as Γ (inst) cusp (g 2 ) = O(g 8 e -8π 2 /g 2 ) .

(5.2)

A N = 4 SYM in spinor notations

Performing the calculation of instanton corrections in N = 4 SYM it is convenient to employ spinor notations. We use Pauli matrices σ µ = (1, iσ) to map an arbitrary four-dimensional Euclidean vector x µ into a 2 × 2 matrix

x α β = x µ (σ µ ) α β , (A.1)
and use the completely antisymmetric tensor to raise and lower its indices

x α β = ǫ αβ x β β , x β α = x α αǫ α β , x αβ = ǫ βα x α β ǫ β α , (A.2)
with ǫ αβ ǫ αγ = δ γ β , ǫ α β ǫ α γ = δ γ β and ǫ 12 = ǫ 12 = 1. Then,

x 2 µ = 1 2 x α αx α α = - 1 2 x α αx αα = 1 2 x α αx β β ǫ αβ ǫ α β . (A.3)
For derivatives we have similarly 6 Similar considerations should apply to the structure constants C

∂ α β = ∂ µ (σ µ ) α β , ∂ α αx β β = 2ǫ αβ ǫ α β , ∂ α αx 2 = 2x α α . (A.
(inst) S in the large spin limit. Indeed, from the analysis of [START_REF] Alday | Higher-spin correlators[END_REF] it follows that the structure constants for S ≫ 1 are given in terms of the cusp anomalous dimension.

Using these definitions we obtain for the gauge field A µ and the stress tensor

F µν = -i[D µ , D ν ] A α α = iA µ (σ µ ) α α , F αβ = -iF µν (σ µ σ ν ) αβ , F α β = -iF µν (σ µ σ ν ) α β , (A.6)
where the additional factor of i is introduced for convenience. The symmetric matrices F αβ and F α β describe (anti)self-dual parts of the strength tensor

F αβ = ǫ α β D α αD β β + D β αD α β ≡ -D 2 (αβ) , F α β = ǫ αβ D α αD β β + D β αD α β ≡ -D 2 ( α β) , (A.7)
where angular brackets denote symmetrization with respect to indices and the covariant derivative D α α = D µ (σ µ ) α α is defined as

D α β X = [∂ α β + A α β , X] . (A.8)
The Lorentz indices are raised and lowered according to (A.2). The Lagrangian of N = 4 super Yang-Mills theory takes the following form in spinor notations

L = 1 g 2 tr - 1 16 F 2 αβ - 1 16 F 2 α β - 1 4 D α α φ AB D α α φAB -2i λ αA D αβ λ A β + √ 2λ αA [ φAB , λ B α ] - √ 2 λ αA [φ AB , λ α B ] + 1 8 [φ AB , φ CD ][ φAB , φCD ] + i θ 8π 2 tr 1 16 F 2 αβ - 1 16 F 2 α β , (A.9)
where gaugino fields λ A α and λ α A , and scalar fields, φ AB and φAB , carry the SU (4) indices (A, B = 1, . . . , 4) and satisfy the reality condition φAB = 1 2 ǫ ABCD φ CD . All fields are in the adjoint representation of the SU (N ) gauge group, e.g. A α α = A a α α T a , with the generators satisfying [T a , T b ] = if abc T c and normalized as tr(T a T b ) = 1 2 δ ab . In the special case of the SU (2) gauge group the generators are expressed in terms of Pauli matrices T a = σ a /2.

As follows from (A.9), gauge fields, gaugino and scalars satisfy equations of motion

D α α λ A α -i √ 2[φ AB , λ αB ] = 0 D α α λ α A + i √ 2[ φAB , λ B α ] = 0 D α α F αβ + 4i{λ A β , λ αA } + [ φAB , D β αφ AB ] = 0 D 2 φAB + √ 2{ λ αA , λ α B } - 1 √ 2 ǫ ABCD {λ αC , λ D α } + 1 2 [φ CD , [ φAB , φCD ]] = 0 (A.10)
where

D 2 ≡ D 2 µ = 1 2 D α α D α α .
The advantage of the normalization of the Lagrangian (A.9) is that the equations of motion and their solutions do not depend on the coupling constant.

The relations (A.10) are invariant under (on-shell) N = 4 superconformal transformations. In particular, the transformations generated by chiral Poincare supercharges Q α A and corresponding special superconformal generators S α A look as

δA α β = -2ζ A α λ βA , δφ AB = -i √ 2ζ α[A λ B] α , δ φCD = -i √ 2ǫ ABCD ζ α A λ B α , δλ A α = i 2 F αβ ζ βA -i[φ AB , φBC ]ζ C α , δ λ α A = - √ 2(D αβ φAB )ζ B β + 2 √ 2 φAB η αB , (A.11)
where brackets in the second relation denote antisymmetrization of the SU (4) indices. These relations describe transformations of fields under combined Q-and S-transformations with the corresponding parameters being ξ A α and ηA α , respectively. All relations in (A.11) except the last one depend on the linear (x-dependent) combination

ζ A α (x) = ξ A α + x α α η αA . (A.12)
This property plays an important role in our analysis and it can be understood as follows. We recall that S-transformations can be realized as composition of the inversion and Q-transformations

(η S) = I • (ξ Q) • I , I(ξ A α ) = ηA α , I(η A α ) = ξ A α , (A.13)
where (η S) = ηA α S α A and (ξ Q) = ξ Aα Q αA . Let us consider the last relation in (A.11). We first apply inversions and take into account that I changes the chirality of Lorentz indices

I(x α β ) = x β α x 2 , I( λ α A ) = λ βA x βα x 2 , I( φAB ) = x 2 φAB . (A.14)
Then, we obtain from (A.11) and (A.13)

(ξ Q) λ α A = - √ 2(D αβ φAB )ξ B β , (η S) λ α A = I • (ξ Q) • I = η βB I • Q βA • I( λ α A ) = √ 2 ηB β x α β (x 2 ) 2 I(D ββ φAB ) , (A.15)
where

I(D ββ φAB ) = x β γ x β γ D γγ (x 2 φAB ) = x 2 (x β γ x β γ D γγ φAB -2x ββ φAB ).
Combining the relations (A.15) together, we find that δ λ

α A = (ξ • Q + η • S) λ α
A agrees with the last relation in (A.11). The additional correction to δ λ proportional to η αB comes from the inhomogenous term in I(D ββ φAB ). In other words, the appearance of O(η) term in the expression for δ λ in (A.11) is ultimately related to the fact that D αβ φAB does not transform covariantly under the inversion, or equivalently, that δ λ involves the operator D αβ φAB which is not conformal primary.

The question arises whether the relations (A.11) are consistent with conformal symmetry. Supplementing (A.11) with the relation

I(ζ A α ) = ζ βA (x -1 ) β α , (A.16)
that follows from (A.13) and (A.14), we verify that Q + S variations of all fields, δF αβ , δF α β , δφ AB , δλ A α and δ λ α A , transform under the conformal transformations in the same manner as the fields themselves. The same property should hold for higher order variation of fields, e.g. for δ n φ AB with n = 2, 3, . . . . The explicit calculation of δ n φ AB from (A.11) is very cumbersome and is not efficient for higher n due to proliferation of terms. Instead, we can use conformal symmetry to simply the task. Namely, conformal symmetry restricts the possible form of δ n φ AB and allows us to write its general expression in terms of a few arbitrary coefficients. The latter can be fixed by requiring the fields to satisfy the N = 4 SYM equations of motion (A.10).

B Iterative solution to the equations of motion

The equations of motion (A.10) are invariant under N = 4 superconformal transformations (A.11). Following [START_REF] Belitsky | Yang-Mills and D instantons[END_REF], we can exploit this property to construct solutions to (A.10) starting from the special solution The field configuration (B.1) is invariant under the antichiral Q + S transformations. Then, we use the remaining chiral Q + S generators to get Φ(x; ξ, η) = e i(ξQ)+i(η S) Φ (0) (x) e -i(ξQ)-i(η S) = Φ (0) + Φ (1) + Φ (2)

F (0) α β = φ AB,(0) = λ A,(0) α = λ α, (0)
+ . . . , (B.2)
where Φ stands for one of the fields (scalar, gaugino and gauge field). Here Φ (0) is given by (B.1) and the notation was introduced for

Φ (1) = i[(ξQ) + (η S), Φ (0) ] ≡ δΦ (0) , Φ (n) = 1 n δΦ (n-1) = 1 n! δ n Φ (0) , (B.3)
where the variation of fields δΦ is given by (A.11). It is important to emphasise that applying (B.3) we first perform superconformal transformations (A.11) and, then, replace Φ (0) by their explicit expressions (B.1).

By construction, the field Φ(x; ξ, η) depends on 16 fermion modes ξ A α and η αA (with α, α = 1, 2 and A = 1, . . . , 4). The second relation in (B.2) defines the expansion of Φ(x; ξ, η) in powers of fermion modes, so that Φ (n) is a homogenous polynomial in ξ A α and η αA of degree n. Its maximal degree cannot exceed the total number of fermion modes leading to Φ (n) = 0 for n > 16. An additional condition comes from the requirement for Φ (n) to have the same R-charge. This leads to the following relations for different fields

A α α = A (0) α α + A (4) α α + • • • + A (16) α α , λ A α = λ A, (1)
α + λ A, (5) α

+ • • • + λ A, (13) α
, (14) ,

φ AB = φ AB,(2) + φ AB,(6) + • • • + φ AB,
λ αA = λ(3) αA + λ(7) αA + • • • + λ(15) αA , (B.4)
where φAB = 1 2 ǫ ABCD φ CD and each subsequent term of the expansion has four fermion modes more.

Substituting (B.4) into (A.10) and matching the number of fermion modes on both sides of the relations, we obtain the system of coupled equations for various components of fields. To the leading order, we have from (A.10)

D α α F (0) αβ = D 2 φ(2) AB - 1 √ 2 ǫ ABCD {λ (1),αC , λ (1),D α
} = 0 , D α α λ (1),A α = D α α λ(3), α A + i √ 2[ φ(2) AB , λ (1),B α ] = 0 , (B.5)
where the covariant derivative D α α is given by (A.8) with the gauge field replaced by A (0) α α.

To the next-to-leading order we find in a similar manner

D β αF (4) βα + [A (4),β α , F (0)
βα ] + 4i{λ (1),A α , λ (3)
αA } + [φ (2),BC , D α α φ(2) BC ] = 0 , D αβ λ (5),A β + [A (4), αβ , λ (1),A β
] -i √ 2[φ (2),AB , λ (3), α
B ] = 0 , 1 2 D α α D α α φ(6) AB + [A (4),α α , D α α φ(2) AB ] -1 2 [ φ(2) AB , D α α A (4), α α ] + √ 2{ λ(3) αA , λ (3), α
B } + 1 2 [φ (2),CD , [ φ(2) AB , φ(2) CD ]] - √ 2ǫ ABCD {λ (1),αC , λ (5),D α } = 0 . (B.6)

Leading order solutions

Applying relations (B.3) together with (A.11) and (B.1) we obtain the leading order corrections to scalar and gaugino fields

λ (1),A α = i 2 F αβ ζ βA , φ (2),AB = 1 √ 2 ζ αA F αβ ζ βB ≡ 1 √ 2 (ζF ζ) AB , λ (3) αA
= 1 6 ǫ ABCD ζ βB (ζD β αF ζ) CD + ǫ ABCD ηB α (ζF ζ) CD , (B.7)
where

F αβ = F (0)
αβ is nonvanishing self-dual part of the gauge strength tensor and x-dependent variable ζ A α is defined in (A.12). It is straightforward to check that (B.7) satisfy the equations of motion (B.5). We verify that, in agreement with (B.3), the fields (B.7) are related to each other as 2φ (2),AB = δφ (1)

,AB = -i √ 2ζ α[A λ (1),B] α , 3 λ(3), α A = δ λ(2), α A = - √ 2(D αβ φ(2) AB )ζ B β + 2 √ 2 φ(2) AB η αB , (B.8)
where expressions on the right-hand side follow from (A.11). As was mentioned above, the form of (B.7) is restricted by conformal symmetry. For instance, the expression for λ(3)

αA involves (ζD β αF ζ) CD = ζ γC D β αF γδ ζ δD
which is not a conformal primary operator. As was explained in Section A, this leads to the appearance of the second term in λ(3) αA proportional to ηB α which is needed to restore correct conformal properties of the gaugino field. The relative coefficient between the two terms in the expression for λ(3) αA is uniquely fixed by the conformal symmetry whereas the overall normalization coefficient is fixed by the equations of motion (B.5).

Next-to-leading order solutions

Direct calculation of subleading corrections to fields (B.4) based on (B.3) and (A.11) is very cumbersome. We describe here another, more efficient approach.

We start with next-to-leading correction to the gauge field and try to construct the general expression for A

α α which has correct properties with respect to conformal and R symmetries. By construction, A

α α is a homogenous polynomial of degree 4 in fermion modes ξ A α and η αA . To begin with, we look for an expression that depends on their linear combination ζ A α defined in (A.12) and has quantum numbers of the gauge field. Since ζ A α has scaling dimension (-1/2), the product of four ζ's should be accompanied by an operator carrying the scaling dimension 3. It can only be built from the self-dual part of the strength tensor F αβ and covariant derivatives D α α. In virtue of the equations of motion, ǫ αβ D α αF βγ = 0, such operator takes the form D (α αF β)γ . Contracting its Lorentz indices with those of the product of four ζ's we obtain A (4) Repeating the same analysis for scalar and gaugino fields we get

α α ∼ ǫ ABCD ζ A α ζ βB (ζD β αF ζ) CD .
A (4) α α = - 1 12 ǫ ABCD ζ A α ζ βB (ζD β αF ζ) CD - 1 2 ǫ ABCD ζ A α ηB α (ζF ζ) CD , λ (5),αA = 3i 40 ǫ CDEF ζ αC (ζ 2 ) AD (ζF 2 ζ) EF , φ (6),AB = - 1 20 √ 2 ǫ CDEF (ζ 2 ) AC (ζ 2 ) BE (ζF 2 ζ) F D , (B.9)
where the notation was introduced for

(ζ 2 ) AB = (ζ 2 ) BA = ζ βA ǫ βγ ζ γB , (ζF 2 ζ) EF = -(ζF 2 ζ) F E = ζ βF F 2 βγ ζ γE , F 2 βγ = F 2 γβ = F βα ǫ αδ F δγ . (B.10)
It is straightforward to verify that the fields (B.9) satisfy the system of coupled equations (B.6). Notice that the last two relations in (B.9) involve a conformal primary operator F 2 αβ of dimension 4 and, as a consequence, the dependence on fermion modes only enters through the linear combination ζ A α . 7We recall that the subleading corrections to fields have to satisfy (B.3). In application to (B.9) these relations read 4A (4)

α α = δA (3) α α = -2ζ A α λ(3) αA , 5λ (5),A α = δλ (4),A α = i 2 F (4) αβ ζ βA -i[φ (2),AB , φ(2) BC ]ζ C α , 6φ (6)
,AB = δφ (5)

,AB = -i √ 2ζ α[A λ (5),B] α , (B.11)
where F

αβ = D (α αA α, (4)
defines the correction to self-dual part of the gauge strength tensor. Replacing the fields with their explicit expressions (B.7) and (B.9), we verify that the relations (B.11) are indeed satisfied.

We can now turn the logic around and apply the relations (B.8) and (B.11) to compute subleading corrections to the fields. Indeed, we start with the expression for λ λ [START_REF] Kovacs | On instanton contributions to anomalous dimensions in N=4 supersymmetric Yang-Mills theory[END_REF],

α A = δ λ(6), α A = - √ 2 D αβ φ(6) AB + [A αβ,(4) φ(2) AB ] ζ B β + 2 √ 2 φ(6) AB η αB , 8A (8)
α α = δA (7)
α α = -2ζ A α λ(7) αA . (B.12)
Going through calculation of higher components of fields we find that they are proportional to the square of a fermion mode and, therefore, vanish

λ (9),A α = φ (10),AB = λ(11) αA = A (12)
α α = λ (13),A α = φ (14),AB = λ(15) αA = A (16) α α = 0 . (B.13)
This relation implies that the expansion of fields on the instanton background (B.4) is shorter than one might expect. The same result was independently obtained in [22].

C Projection onto twist-two operators

In this Appendix we explain how to use the light-ray operators (2.5) to compute the correlation functions involving twist-two operators in the Born approximation.

To begin with, we consider the correlation function (2.11) of the light-ray operator and two half-BPS operators in N = 4 SYM with the SU (N ) gauge group. In the Born approximation, we can neglect gauge links in the definition (2.5) of O(z 1 , z 2 ) and express (2.11) in terms of free propagators of scalar fields φ AB (x) = φ a,AB (x)T a φ a,AB (x 1 )φ b,CD (x 2 ) = g 2 δ ab ǫ ABCD D(x 1x 2 ) , (C.1)

where D(x) = 1/(4π 2 x 2 ) and the additional factor of g 2 appears due to our normalization of the Lagrangian (A.9). The generators of the SU (N ) gauge group are normalized as tr(T a T b ) = δ ab /2, so that O 20 ′ (x i ) = Y i,AB Y i,CD φ a,AB φ a,CD . We recall that the light-ray operators (2.5) are built out of the complex scalar field Z = φ 14 . It is convenient to represent this field as Z = Y Z,AB φ AB , with Y Z having the only nonvanishing components Y Z,14 = -Y Z,41 = 1/2. Then, we find

O(z 1 , z 2 )O 20 ′ (x 1 )O 20 ′ (x 2 ) Born = 1 2 g 6 (N 2 -1)(Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × D(x 12 )[D(x 1 -nz 1 )D(x 2 -nz 2 ) + D(x 1 -nz 2 )D(x 2 -nz 1 )] , (C.2)
where x 12 = x 1x 2 and the notation was introduced for (Y i Y j ) = ǫ ABCD Y i.AB Y j,CD . Taking into account that (x inz i ) 2 = x 2 i -2z i (nx i ), we can rewrite this expression as

1 2 g 6 (N 2 -1)(Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) D(x 12 )D(x 1 )D(x 2 ) (1 -ǫ 1 z 1 )(1 -ǫ 2 z 2 ) + (z 1 ↔ z 2 ) , (C.3)
with ǫ i = 2(nx i )/x 2 i . To obtain the three-point correlation function of local twist-two operator, we substitute (C.3) into (2.13). Blowing up the integration contour in (2.13) and picking up the residue at

z i = 1/ǫ i , we arrive at 8 O S (0)O 20 ′ (1)O 20 ′ (2) Born = 1 2 g 6 (N 2 -1)(Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) × D(x 12 )D(x 1 )D(x 2 ) 2(nx 1 ) x 2 1 - 2(nx 2 ) x 2 2 S + (x 1 ↔ x 2 ) . (C.4)
This relation coincides with the general expression for the correlation function of twist-two operators (2.4). Notice that (C.4) vanishes for odd S, in agreement with the fact that the twist-two operators carry nonnegative even spin S.

The same technique can be used to compute two-point correlation function of twist-two operators O S (0) ŌS ′ (x) . We start with the correlation function of two light-ray operators separated by distance x. In the Born approximation we have

tr [Z(nz 1 )Z(nz 2 )] tr [ Z(x + nz 3 ) Z(x + nz 4 )] = g 4 4 (N 2 -1)D(x -nz 13 )D(x -nz 24 ) + (z 1 ↔ z 2 ) = g 4 4 (N 2 -1)D 2 (x) (1 -ǫz 13 )(1 -ǫz 24 ) + (z 1 ↔ z 2 ) , (C.5)
with ǫ = 2(nx)/x 2 . Substituting this relation into (2.13) and performing integration over z 1 and z 2 , we can project the light-ray operator tr [Z(nz 1 )Z(nz 2 )] onto the twist-two operator O S (0). Repeating the same procedure with respect to z 3 and z 4 , we obtain the expression for O S (0) ŌS ′ (x) that is different from zero only for even positive S = S ′ and is given by

O S (0) ŌS ′ (x) Born = 1 2 g 4 c S (N 2 -1)D 2 (x) 2(nx) x 2 2S δ SS ′ , (C.6)
with c S = (2S)! /(S! ) 2 and D(x) defined in (C.1).

D Instanton profile of operators Light-like Wilson line

The calculation of the leading term E (0) (z 1 , z 2 ) of the expansion of the light-like Wilson line (2.6) relies on the following identity

i dt n • A (0) ij (nt) = 1 2 n αα [ǫ iα (x 0 ) j α + ǫ jα (x 0 ) i α] dt (nt -x 0 ) 2 + ρ 2 = Σ ij dt (nx 0 ) (nt -x 0 ) 2 + ρ 2 , (D.1)
where we replaced the instanton field with its explicit expression (3.7) for x = ntx 0 and took into account that n 2 = 0. Here in the second relation we introduced the following 2 × 2 matrix Σ ij

Σ = Σ + -Σ -, Σ + = nx 0 2(nx 0 ) , Σ -= x 0 n 2(nx 0 ) , (D.2) with Σ ± being projectors, Σ 2 ± = Σ ± , Σ + Σ -= 0 and Σ + + Σ -= 1. Notice that Σ + n = n Σ -= n and n Σ + = Σ -n = 0.
Since the Σ-matrix in (D.1) does not depend on the integration variable, the pathordered exponential reduces to the conventional exponential leading to

E (0) (z 1 , z 2 ) = exp (ΣI(z 1 , z 2 )) = e I(z 1 ,z 2 ) Σ + + e -I(z 1 ,z 2 ) Σ -, (D.3)
where the 2 × 2 matrices Σ ± are independent on z i and I(z 1 , z 2 ) = -I(z 2 , z 1 ) is given by

I(z 1 , z 2 ) = z 2 z 1 dt (nx 0 ) (nt -x 0 ) 2 + ρ 2 = - 1 2 ln (nz 2 -x 0 ) 2 + ρ 2 (nz 1 -x 0 ) 2 + ρ 2 . (D.4)
We recall that the matrix indices of Σ ± are identified with the SU (2) indices of E (0) (z 1 , z 2 ). The first subleading correction to the Wilson line, E (4) (z 1 , z 2 ), comes from the A (4) term in the expansion of the gauge field. It is given by

E (4) (z 1 , z 2 ) = 1 2 z 2 z 1 dt E (0) (z 1 , t) n|A (4) (nt)|n]E (0) (t, z 2 ) . (D.5)
Replacing n|A (4) |n] with its explicit expression (3.11) (evaluated for x = ntx 0 ) and taking into account (D.3), we obtain

E (4) (z 1 , z 2 ) = z 2 z 1 dt 4ρ 2 [ρ 2 + (nt -x 0 ) 2 ] 3 ǫ ABCD ρ 2 [η A n] -ξ A |x 0 |n] nζ B t × e I(z 1 ,t) Σ + + e -I(z 1 ,t) Σ -|ζ C t ζ D t | e I(t,z 2 ) Σ + + e -I(t,z 2 ) Σ -, (
) i = (Σ ± ) i j ζ A j .

Light-ray operators

The expansion of the light-ray operator on the instanton background in powers of fermion modes takes the form (4.4). We show below that the last two terms of the expansion vanish, Eq. (4.7). The underlying reason for this is that, by virtue of N = 4 superconformal symmetry, the light-ray operator O(z 1 , z 2 ) only depends on 12 fermion modes, ξ A α and [nη A ]. According to its definition (2.5), the operator O(z 1 , z 2 ) depends on scalar and gauge fields, Z(x) and (nA(x)), evaluated on the light-ray x µ = n µ z. Examining the explicit expressions for the lowest components of these fields, Eqs. (3.11) and (3.15), we observe that the dependence on fermion modes enters either through [nη A ] or through the linear combination ζ A α (x) defined in (A.12). For x = nz the latter simplifies as ζ A α = ξ A α + z|n α [nη A ], so that the above mentioned components of fields depend on the light-ray on ξ A α and [nη A ] only. Then, we can use the recurrence relations (B.3), (B.11) and (B.12) to show that the same is true for all components of fields.

Thus, the light-ray operator O(z 1 , z 2 ) depends on 12 fermionic modes ξ A α and [nη A ] that we shall denote as Θ A i (with i = 1, 2, 3). Then, the top component O (16) (z 1 , z 2 ) is necessarily proportional to the square of a fermion mode and, therefore, vanishes. The next-to-top component O (12) (z 1 , z 2 ) contains the product of all 12 fermion modes. R-symmetry fixes its form to be

Θ 1 Θ 4 Θ 1 Θ 4 ǫ A 1 B 1 C 1 D 1 Θ A 1 Θ B 1 Θ C 1 Θ D 1 ǫ A 2 B 2 C 2 D 2 Θ A 2 Θ B 2 Θ C 2 Θ D 2 , (D.7)
where the first four Θ's carry the SU (4)-charge of two scalar fields Z = φ 14 and the remaining factors are the SU (4) singlets. Here we did not display the lower index of Θ A i . Counting the total number of Grassmann variables in (D.7), we find that it is proportional to Θ

1 i 1 Θ 1 i 2 Θ 1 i 3 Θ 1 i 4 .
Since the lower index can take only three values this product vanishes leading to (4.7).

Derivation of (4.11)

We start by examining the instanton profile of the product of two half-BPS operators. Using (4.3) we find

O (4) 20 ′ (x 1 )O (4) 20 ′ (x 2 ) = 1 4 f 2 (x 1 )f 2 (x 2 )Y 1,A 1 B 1 Y 1,C 1 D 1 Y 2,A 2 B 2 Y 2,C 2 D 2 ×(ζ 2 1 ) A 1 C 1 (ζ 2 1 ) B 1 D 1 (ζ 2 2 ) A 2 C 2 (ζ 2 2 ) B 2 D 2 , (D.8)
where

ζ i ≡ ζ(x i -x 0 ) = ξ + (x i -x 0 )
η and the instanton profile function f (x) is given by (3.9) with x → xx 0 .

It is convenient to rewrite the light-ray operator O (8) (z 1 , z 2 ) defined in (4.5) as O (8) = O (8)

A + O (8) B + (z 1 ↔ z 2 )
. For the first term on the right-hand side we get

O (8) A (z 1 , z 2 ) = tr Z (2) (nz 1 )E (0) (z 1 , z 2 )Z (6) (nz 2 )E (0) (z 2 , z 1 ) = 1 40 f (nz 1 )f 2 (nz 2 )I (8)
A , (D.9)

where we substituted Z (n) = (Y Z ) AB φ (n),AB , replaced φ (n),AB by the explicit expressions (3.15) and introduced

I (8) A = (Y Z ) A 1 B 1 (Y Z ) A 2 B 2 ǫ CDEF (ζ 2 z 2 ) A 2 C (ζ 2 z 2 ) B 2 E ζ B 1 z 1 |E (0) (z 1 , z 2 )|ζ F z 2 ζ D z 2 |E (0) (z 2 , z 1 )|ζ A 1 z 1 (D.10)
with ζ z i = ξ + (z i nx 0 )η. By construction, this expression contains 8 fermion modes. Following (4.11), we multiply expressions on the right-hand side of (D.8) and (D.9) and integrate out 16 fermion modes. In this way, we obtain 1 160 f 2 (x 1 )f 2 (x 2 )f (nz 1 )f 2 (nz 2 )E (0) αγ (z 1 , z 2 )E where the integral over fermion modes I αβγδ is given by the following expression where (Y i Y j ) = ǫ ABCD Y i,AB Y j,CD and chiral Lorentz tensor P αβγδ (x) depends on four points, x 10 , x 20 , nz 1x 0 and nz 2x 0 . The calculation of this tensor can be significantly simplified with a help of conformal symmetry. Namely, denoting the above mentioned four points as y i (with i = 1, . . . , 4) and making use of (A.16), we find from (D.12) and (D.13) that P αβγδ transforms under inversions I(y αβ i ) = y βα i /y where in the second relation we replaced y ij by their expressions and took into account that n 2 = 0. Here angular brackets (αβ) denote symmetrization of Lorentz indices.

I αβγδ (Y, x) = (Y Z ) E 1 F 1 (Y Z ) E 2 F 2 Y 1,A 1 B 1 Y 1,C 1 D 1 Y 2,A 2 B 2 Y 2,C 2 D 2 ǫ B 3 C 3 D 3 E 3 × d 8 ξd 8 η (ζ 2 1 ) A 1 C 1 (ζ 2 1 ) B 1 D 1 (ζ 2 2 ) A 2 C 2 (ζ 2 2 ) B 2 D 2 ζ α,E 1 z 1 ζ β,F 1 z 1 (ζ 2 z 2 ) E 2 B 3 (ζ 2 z 2 ) F 2 C 3 ζ γ,D 3 z 2 ζ δ,E
Finally, we substitute (D.13) and (D.15) into (D.11) and replace F αβ and E (0) with their expressions, Eqs. (3.8) and (D.3), respectively. In this way, we arrive after some algebra at the expression that differs by the factor of 6 from the one on the right-hand side of (4.11).

B .

(D.17)

Here we replaced Z (2) = (Y Z ) AB φ (2),AB using (3.15) and introduced notation for

I (8) B = (Y Z ) A 1 B 1 (Y Z ) A 2 B 2 ζ B 1 z 1 |E (4) (z 1 , z 2 )|ζ A 2 z 2 ζ B 2 z 2 |E (0) (z 2 , z 1 )|ζ A 1 z 1 , (D.18)
where ζ z i = ξ +(z i n-x 0 )η. Replacing E (0) and E (4) with their explicit expressions, Eqs. (D.3) and (D.6), respectively, we get for

I (8) B (Y Z ) A 1 B 1 (Y Z ) A 2 B 2 ǫ ABCD ρ 2 [η A n] -ξ A |x 0 |n] nζ B 0 ζ B 2 z 2 |e I(z 1 ,z 2 ) Σ + + e -I(z 1 ,z 2 ) Σ -|ζ A 1 z 1 × z 2 z 1 4ρ 2 dt [ρ 2 + (nt -x 0 ) 2 ] 3 ζ B 1
z 1 |e I(z 1 ,t) Σ + + e -I(z

B = 8ρ 2 (z 1 -z 2 ) 2 5(ρ 2 + (nz 1 -x 0 ) 2 )(ρ 2 + (nz 2 -x 0 ) 2 ) × (Y Z ) A 1 B 1 (Y Z ) A 2 B 2 ǫ ABCD nζ A 0 nζ B 0 (ζ 2 z 1 ) A 1 C (ζ 2 z 1 ) B 1 D [η A 2 n][η B 2 n] . (D.21)
Then, we substitute this relation into (D.17 Finally, we take the sum of (D. [START_REF] Green | Instanton induced Yang-Mills correlation functions at large N and their AdS(5) x S**5 duals[END_REF]) and (D.22), multiply it by the factor of 2 in order to take into account the contribution of (z 1 ↔ z 2 ) terms and arrive at (4.11).

Derivation of (4.12)

According to (4.9), the instanton profile of O(z 1 , z 2 |x) can be obtained from that of O(z 1 , z 2 ) by shifting the coordinates of all fields by x. As follows from (3.1), this transformation is equivalent to shifting the position of the instanton, x 0 → x 0x. As before we decompose the light-ray operator as O (8) = O

A + O

B + (z 1 ↔ z 2 ). Then, for the product of two operators in (4.12) we have O (8) A is given by (D.9). To get an analogous expression for Ō(8) A , we apply the shift x 0 → x 0 -x to (D.9), change the coordinates, z (D.25)

4 )

 4 Throughout the paper we use the following conventions for contracting Lorentz indices in the product of 2 × 2 matrices (xy) αβ = x α αy α β , (xy) α β = x α αy α β , (xyz) α β = x α αy αβ z β β . (A.5)

A

  dual gauge field in pure Yang-Mills theory with F (0) αβ = 0. Since (B.1) verifies (A.10), we can obtain another solution to (A.10) by applying N = 4 superconformal transformations (A.11) to the fields (B.1).

( 4 )

 4 α α to satisfy the first relation in (B.6). As we will show in a moment, there is much simpler way to fix this coefficient using the second relation in (B.3).

  and use (B.8) and (B.11) recursively to reproduce (B.9). Continuing this procedure we can determine all remaining terms of the expansion (B.4) with little efforts, e.g.

7

 7

D. 6 )

 6 where ζ t ≡ ζ(tnx 0 ) = ξ + (tnx 0 )η depends on the integration variable. Here we used shorthand notations for contraction of the indices, e.g. (Σ ± |ζ A

  δβ (z 2 , z 1 )I αβγδ (Y, x) , (D.11)

3

 3

z 2 . (D. 12 )

 212 Taking into account that the Y -variables satisfy the relation ǫ ABCD Y AB Y CD = 0, we find that the Y -dependence of the integral is uniquely fixed by the SU (4) symmetryI αβγδ (Y, x) = (Y 1 Y 2 )(Y 1 Y Z )(Y 2 Y Z ) P αβγδ (x) , (D.13)

d 8 ξd 8 16 ) 8 )B

 8168 For the second term on the right-hand side of (4.5) we haveO ((z 1 , z 2 ) = tr Z (2) (nz 1 )E (4) (z 1 , z 2 )Z (2) (nz 2 )E (0) (z 2 , z 1 ) = -1 2 f (nz 1 )f (nz 2 )I

3 ×

 3 ), multiply it by (D.8) and integrate over 16 fermion modes to get d 8 ξd 8 η O Eq.(4.11) . (D.22)

(z 1 1 ↔

 11 , z 2 |0) Ō(8) (z 3 , z 4 |x) = O z 2 ) + (z 3 ↔ z 4 ) + (z 1 ↔ z 2 , z 3 ↔ z 4 ) , A (z 3 , z 4 )| x 0 →x 0 -x .Let us consider separately four terms in the first line of (D.23). The instanton profile of O[START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF]

2 ×

 2 1 → z 3 and z 2 → z 4 , and replace Y Z with conjugated Y Z -variables defined as Z = φ14 = φ 23 = (Y Z ) AB φ AB and satisfying (Y Z Y Z ) = 1. In this way we getO 1 )f 2 (nz 2 )f (x + nz 3 )f 2 (x + nz 4 ) × I A isobtained from (D.10) through transformations described above. Integrating out fermion modes we findd 8 ξd 8 η I (8) A Ī(8) A = 1600 (z 1z 2 ) 2 (z 3z 4 ) 2 n|E (0) (z 1 , z 2 )x|n] n|E (0) (z 3 , z 4 )x|n]x 0 →x 0 -x 2 .

  CD and the antisymmetric tensors Y i,AB carry the SU (4) charges of the operators. The tensors Y i,AB (with i = 1, 2) enter into the definition (2.2) of the operators O 20 ′ (x i ) and Y Z,AB defines the complex scalar field Z = φ 14 = Y Z,AB φ AB . In a similar manner, we find from (4.10) (see Appendix D for details)

  .13) to relations (4.15) and (4.16) and verify that the obtained expressions for the correlation functions O S (0)O 20 ′ (1)O 20 ′ (2) and O S (0)O S ′ (x) are indeed consistent with conformal symmetry and have the expected form (2.4).

	Applying (2.13) to (4.15), we get

  1/ǫ can be removed from the correlation function O S (0)O S ′ (x) by multiplying the twist-two operators, O S (x) → Z(1/ǫ)O S (x) by appropriately chosen renormalization Z-factor. Retaining only this term we get from (4.24) O S (0) ŌS ′ (x) inst = δ S,2 δ S,S ′ 81 80π 8 g 8 e 2πiτ (xn) 4 (x 2 ) 6 ln x 2 .

	Relations (4.21) and (
	(4.26)

  Since this operator is not conformal primary, it should receive correction proportional to ǫ ABCD ζ A

	α	ηB α (ζF ζ) CD , the relative coefficient is
	fixed by the conformal symmetry. The overall normalization coefficient can be determined
	by requiring A

  ) 4 P α ′ β ′ γ ′ δ ′ .This relation, combined with the condition for f αβγδ to be a homogenous polynomial in y ij of degree 8, allows us to determine f αβγδ up to an overall normalization factor. Going through calculation we findP αβγδ = -2 4 × 3 2 × 5 × (y 34 y 41 y 12 y 24 ) (αγ (y 34 y 41 y 12 y 24 ) β)δ = -2 4 × 3 2 × 5 × z 2 12 (nx 1 x 12 (x 2nz 2 )) (αγ (nx 1 x 12 (x 2nz 2 )) β)δ , (D.15)

		2 i as
	I P αβγδ =	y αα ′ 3 y ββ ′ 3 y γγ ′ 4 y δδ ′ 4 (y 2 1 y 2 2 y 2 3 ) 2 (y 2 4 (D.14)

  1 ,t) Σ -|ζ C t ζ D t |e I(t,z 2 ) Σ + + e -I(t,z 2 ) Σ -|ζ A 2 z 2 , (D.19)with ζ t = ξ + (ntx 0 )η and ζ 0 = ξx 0 η. This relation can be simplified with a help of identifiesΣ -|ζ A t = Σ -|ζ A 0 , Σ + |ζ A t = Σ + |ζ A 0 + t|η A ] , (D.20)that follow from (D.2). Going through lengthy calculation we arrive at remarkably simple expression

	I	(8)

The relation (2.10) has the following interpretation[START_REF] Belitsky | Superconformal operators in Yang-Mills theories on the light-cone[END_REF]. The light-ray operator (2.5) transforms covariantly under the action of the collinear SL(2; R) subgroup of the conformal group. Then, the relation (2.10) defines the decomposition of the tensor product of two SL(2; R) representations over the irreducible components. The coefficient functions z S 12 are the lowest weights of these components.

For the SU (N ) gauge group the instanton action (3.3) also depends on 8(N -2) nonexact fermion modes, see[START_REF] Belitsky | Yang-Mills and D instantons[END_REF][START_REF] Dorey | The Calculus of many instantons[END_REF][START_REF] Bianchi | Instantons and Supersymmetry[END_REF].

In the previous paper[START_REF] Alday | Revisiting instanton corrections to the Konishi multiplet[END_REF], we used different normalization of operators, O there = O here /g 2 , for which the correlation functions in the Born approximation do not depend on the coupling constant.

We would like to emphasize that this regularization is different from the conventional dimensional regularization. Since the coefficient in front of ln x 2 in (4.25) is independent on the choice of regularization, we can choose it to our convenience.

There is another operator with the same quantum numbers, D 2 F αβ , but it reduces to -F 2 αβ /2 on shell of the equations of motion.

The powers of the coupling constant in (C.4) and (C.6) appear due to our definition of the Lagrangian (A.9). They can be removed by changing the normalization of operators φ AB → φ AB /g, O 20 ′ → O 20 ′ /g 2 and O S → O S /g 2 .

Acknowledgements

We are grateful to Massimo Bianchi, Emeri Sokatchev, Yassen Stanev and Pierre Vanhove for useful discussions. The work of L.F.A. was supported by ERC STG grant 306260. L.F.A. is a Wolfson Royal Society Research Merit Award holder. This work of G.P.K. was supported in part by the French National Agency for Research (ANR) under contract StrongInt (BLANC-SIMI-4-2011).

Combining the last two relations we conclude that ∆ (inst) S = O(g 8 e -8π 2 /g 2 ln S) at large S. Thus, if the leading O(g 6 e -8π 2 /g 2 ) correction to ∆ (inst) S in (1.1) is different from zero, it should approach a finite value for S → ∞. 6 It would be interesting to compute explicitly the instanton correction (5.2).

The matrix elements in this relation can be easily computed with a help of (D.3)

where in the second relation we used the properties of matrices (D.2) and I(z 1 , z 2 ) is given by (D.4). In this way, we obtain from (D.24)

For the second term on the right-hand side of (D.23) we have from (D.17)

where I

B is given by (D.21) and Ī(8) B is obtained from I

B through the same transformation as before. Integration over fermion modes yields

leading to the following relation

× Eq.(4.12) . (D.30)

For the last two terms on the right-hand side of (D.23) we have from (D.9) and (D.17)

Then, we integrate over fermion modes to get

The integral of I (8) B

Ī(8)

A is given by the same expression with variables exchanged, z 1 ↔ z 3 and z 2 ↔ z 4 . Using (D.26) we find

× Eq.(4.12) . (D.33)

Combining together (D.27), (D.30) and (D.33), we find that the sum of four terms in the first line of (D.23) is 1/4 × Eq.(4.12). Since it is invariant under the exchange of points, z 1 ↔ z 2 and z 3 ↔ z 4 , the contribution of terms in the second line of (D.23) is three times larger. As a result, the total contribution of (D.23) is given by (4.12).


David Andriot
email: david.andriot@aei.mpg.de

Johan Blåbäck
email: johan.blaback@cea.fr

Thomas Van Riet
email: thomasvr@itf.fys.kuleuven.be

Minkowski flux vacua of type II supergravities

Keywords: numbers: 11.25.Mj, 11.25.Wx, 04.65.+e, 04.20.Ex I. CONTEXT AND RESULTS

We study flux compactifications of 10d type II supergravities to 4d Minkowski space-time, supported by parallel orientifold Op-planes with 3 ≤ p ≤ 8. With some restrictions, the 4d Ricci scalar can be written as a negative sum of squares involving BPS-like conditions. Setting all squares to zero provides automatically a solution to 10d equations of motion. This way, we characterize a broad class, if not the complete set, of Minkowski flux vacua with parallel orientifolds. We conjecture an extension with non-geometric fluxes. None of our results rely on supersymmetry.

We consider vacua of ten-dimensional (10d) type II supergravities on a 4d maximally symmetric space-time times a 6d compact manifold, with fluxes. Such vacua are a major framework for string phenomenology, given the role of fluxes in moduli stabilization. Having a complete classification of these vacua would thus be a significant achievement. We make here an essential step towards this goal: with few requirements, we reveal and characterize a broad class of Minkowski flux vacua.

The best understood flux compactifications of type II supergravities are those with just fluxes and no sources. Well-known no-go theorems [START_REF] Maldacena | Supergravity description of field theories on curved manifolds and a no go theorem[END_REF] imply that such reductions only lead to anti-de Sitter (AdS) vacua. Even more, those AdS vacua are no genuine compactifications since they lack a tunable hierarchy between the AdS length scale and the Kaluza-Klein scale [2]. Space-time filling orientifold planes are a natural way out of this problem and allow one to find Minkowski vacua [START_REF] Dasgupta | M theory, orientifolds and G-flux[END_REF][START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF] (see [START_REF] Graña | Flux compactifications in string theory: A Comprehensive review[END_REF] for a review) or AdS ones with scale separation (see e.g. [START_REF] Dewolfe | Type IIA moduli stabilization[END_REF]). However such reductions come with an extra level of complication, the backreaction of the orientifolds themselves. In the case of non-intersecting O p -planes (and possibly D p -branes) the backreaction is rather well-understood: it introduces a warp factor e A in the 10d metric ds 2 10 = e 2A (ds 2 4 + ds 2 6|| ) + e -2A ds 2 6⊥ ,

with inverse powers along parallel and transverse directions to the sources; ds 2 6|| and ds 2 6⊥ still depend on all internal coordinates [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF]. So we work here in this context: type II supergravities supplemented by space-time filling, parallel (i.e. non-intersecting), backreacted O p /D p sources, with a fixed size 3 ≤ p ≤ 8, and the standard dilaton value e φ = g s e A(p-3) with a constant g s . In this framework, we study Minkowski flux vacua; note we do not capture F-theory non-perturbative solutions. We follow conventions of a companion paper [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF].

A. O3-planes

Let us recall known results on O 3 compactifications in IIB, following [START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF][START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF][START_REF] Blåbäck | Smeared versus localised sources in flux compactifications[END_REF]. The sourced flux is the 5-form F 5 , the other Ramond-Ramond (RR) fluxes being the 1-and 3-forms F 1 , F 3 , and the NSNS 3-form flux is denoted H. For our purposes, we need two equations: the first one is a combination of the dilaton equation of motion (e.o.m.), the 10d and the 4d traces of the Einstein equation, giving an expression for the Ricci scalar R4 of ds 2 4 ; the second equation is the F 5 Bianchi identity (BI)

e -2A R4 = -1 2 |H| 2 -e 2φ 2 |F 3 | 2 -e 2φ |F 5 | 2 + e φ 4 T 10 + . . . , dF 5 = H ∧ F 3 + ε3 4 T 10 vol 6 , (2)
where the dots stand for terms explicitly dependent on ∂φ and ∂A, and we leave the definitions of the square of forms, the sign ε p , and the sources contributions T 10 to [9]. We see from above that, at least in the smeared limit where ∂φ = ∂A = 0, the flux contributions cannot be canceled to find a Minkowski vacuum without the O 3 contribution in T 10 . We now combine both equations of (2) and rewrite the result as in [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] e -2A R4 = -e 4A * 6 de -4A -ε 3 e φ F 5 2

(3)

-1 2 * 6 H + ε 3 e φ F 3 2 + e -2A ∂(. . . ) ,
where ∂(. . . ) denotes a total derivative over the (unwarped) compact manifold. Upon integration, we deduce that a Minkowski vacuum requires both squares to vanish. One leads to the well-known ISD condition [START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF]: In the present Letter, we derive analogous results for O p -planes with p > 3. Up to little additional structure, R4 can again be written in terms of squares of BPS-like conditions; with few restrictions, a Minkowski vacuum then requires to set these conditions to zero, fixing the sourced flux F 8-p , 3 ≤ p ≤ 8, and relating H to F 6-p . We show that these three fluxes are the only non-trivial ones. Assuming their BI, we verify that all e.o.m. are satisfied, giving us a broad class of Minkowski flux vacua.

H = ε 3 e φ *
Let us point out the main differences when p > 3. For p = 3, all internal directions are transverse to the sources, while for p > 3, some are parallel, as in [START_REF] Maldacena | Supergravity description of field theories on curved manifolds and a no go theorem[END_REF]. A is then only dependent on transverse directions, and * 6 gets replaced by * ⊥ , ∆6 by ∆⊥ , etc. Fluxes may now have different components:

F l = F (0) l + F (1) l + . . . where n in F (n) l
denotes the number of parallel indices of the component. We will as well introduce the projection on the transverse subspace

F l | ⊥ = F (0) l .
A second difference is the possibility of RR fluxes F l with l > 8 -p. Taking these features into account, a lengthy computation [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] leads to an expression for R4 , analogous to (3), given by

e -2A R4 = -e 4A * ⊥ de -4A -ε p e φ F (0) 8-p 2 -(flux) 2 (4) -1 2 * ⊥ H| ⊥ + ε p e φ F 6-p | ⊥ 2 + e -2A ∂(. . . ) -1 2 a || * ⊥ (de a || )| ⊥ -ε p e φ (ι a || F (1) 8-p ) 2 -(R || + R ⊥ || ) + 1 2 |H (2) | 2 + |H (3) | 2 , for 0 ≤ 8 -p ≤ 5
, where we use the internal one-form basis e a = e a m dy m constructed with Vielbeins, and the contraction by a vector ι a e b = δ b a . The 6d metric in flat indices is δ ab . One has de a = -1 2 f a bc e b ∧ e c , where the anholonomicity symbol (from now on "geometric flux") f does not need to be constant. Thus by definition, (de

a || )| ⊥ = -1 2 f a || b ⊥ c ⊥ e b ⊥ ∧ e c ⊥ .
This expression (4) is derived in [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] where on top of the analogue of (2), one uses the trace of the Einstein equation along internal parallel directions. Differences between (4) and (3) are related to those mentioned above: there is a third BPSlike condition involving F

(1) 8-p , the (flux) 2 are squares of

F (n) l>8-p , R || + R ⊥ || are curvature terms given in terms of f a || b || c || , f a || b ⊥ c || , f a ⊥ b || c || , f a ⊥ b ⊥ c || , i.e.
involving parallel directions, similarly to the components H (2) , H (3) ; we refer to [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] for precise definitions.

The last line of ( 4) is the only one with indefinite sign. We thus set it here to zero: first, we restrict to

H (2) = H (3) = 0 , (5)
while for computational convenience, we actually set the following (unwarped) geometric fluxes to zero

f a || b || c || = f a || b ⊥ c || = f a ⊥ b || c || = f a ⊥ b ⊥ c || = 0 , (6)
f a ⊥ b ⊥ c ⊥ = 0 , (7)
where ( 6) implies R || = R ⊥ || = 0; note the latter holds for all known Minkowski flux vacua on twisted tori [10]. The geometric requirements ( 6), ( 7) can be viewed as the analogue of the Ricci flatness for p = 3; they amount to having sources wrap a fiber, over a transverse base. The only restrictions made here on the vacua are ( 5), ( 6), [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF]. Integrating (4) as in [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] over the 6d compact manifold then gives R4 only in terms of BPS-like conditions and squares of fluxes. A Minkowski vacuum requires to set them to zero, making the total derivative vanish, as for p = 3. These conditions are enough to characterize a broad class of Minkowski flux vacua, as we now show.

II. FLUXES

Without the last line of (4), asking for a Minkowski vacuum sets the BPS-like conditions of (4) and the squares of fluxes to zero. This, together with the trace of the Einstein equation along internal parallel directions [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], implies that fluxes (and components) within the BPSlike conditions are the only ones allowed to be non-zero, i.e.

H = H| ⊥ , F 6-p = F 6-p | ⊥ , F 8-p = F (0) 8-p + F (1) 8-p , with F 6-p = -e -φ ε p * ⊥ H , (8)
F (0) 8-p = e -φ ε p e 4A * ⊥ de -4A , F (1)
8-p = e -φ ε p δ ab e a || ∧ * ⊥ (de b || )| ⊥ . Indeed, we get that F l>8-p = 0, and that the lowest RR flux F 4-p , meaning F 0 for p = 4 and F 1 for p = 3, vanishes [11]. The only possible flux content is then (8)!

The BI for these fluxes [START_REF]The fluxes may also enter the F10-p BI through H ∧ F8[END_REF] are given by dH = dF 6-p = 0 , dF 8-p -H ∧ F 6-p = εpT10 p+1 vol ⊥ , (9) and projecting the F 8-p BI on the transverse directions, together with [START_REF] Blåbäck | Smeared versus localised sources in flux compactifications[END_REF], gives

e φ T 10 p + 1 = e 6A ∆⊥ e -4A + e 2φ |F 6-p | 2 + e 2φ |F (1) 8-p | 2 (10) = e 6A ∆⊥ e -4A + |H| 2 + |f a || b ⊥ c ⊥ | 2 , (11)
the last term being defined in [START_REF]= δ ab[END_REF]. Upon integration of the last equation one recovers the RR tadpole condition that expresses the net charge in terms of the fluxes. Both the H-flux and the geometric flux contribute to cancel the O p charge, and the two contributions are schematically viewed as T-dual [START_REF] Blåbäck | Smeared versus localised sources in flux compactifications[END_REF][START_REF] Kachru | New supersymmetric string compactifications[END_REF]. While F

8-p is related to the geometric flux, F (0) 8-p leads to ∆⊥ e -4A generating the δfunctions, and can thus be interpreted as related to the sources backreaction. Explicit vacua with both pieces are given e.g. in [10,[START_REF] Graña | A Scan for new N=1 vacua on twisted tori[END_REF], while a vacuum with all fluxes (8) turned on is given in (6.42) of [START_REF] Graña | A Scan for new N=1 vacua on twisted tori[END_REF].

We now show that the fluxes (8) solve their e.o.m. [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], given here by

e -4A d(e 4A * 6 F 8-p ) = 0 (4 ≤ p ≤ 7) , (12)
e -4A d(e 4A * 6 F 6-p ) + H ∧ * 6 F 8-p = 0 (3 ≤ p ≤ 5) , e -4A d(e 4A-2φ * 6 H) -F 6-p ∧ * 6 F 8-p = 0 ,
provided relevant BI are satisfied. To that end, it is convenient to rewrite F 8-p as follows (details in [START_REF]The internal manifold and transverse subspace having no boundary, one has f a || a || d ⊥ = 0. One then shows that d vol || = e -A(p-3) a || * 6 ([END_REF])

F 8-p = (-1) p ε p e -4A * 6 d e 4A-φ vol || , (13)
with the internal parallel volume form, and use for an l- 13) is the calibration condition that minimizes their energy [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF].

form A l that * 6 A l | ⊥ = (-1) l(p+1) vol || ∧ * ⊥ A l | ⊥ . It

III. SATISFYING ALL E.O.M.

The remaining e.o.m. are the dilaton and the Einstein equation; we split the latter into the 4d one, the transverse internal one, and for p = 3 the internal parallel and off-diagonal ones. Verifying that they are satisfied is purely technical; we give here and in the Appendix only the main insights. For all these e.o.m., we use the 10d Einstein trace [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], leading us to consider in the following the trace reversed Einstein equation. We also use (10) and dilaton and warp factor formulas of Appendix C of [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF]. The dilaton e.o.m. and the 4d Einstein equation are then straightforward to verify. The latter boils down to

R| µν + 2∇| µ ∂| ν φ - gµν 4 (2|∂φ| 2 -∆φ) (14) = gµν 8 e φ 2 7-p p+1 T 10 -|H| 2 + e 2φ 2 6 q=0 (1 -q)|F q | 2 ,
with even/odd RR fluxes for IIA/IIB, and each side can be shown to be equal to (A.1). We have introduced the shorthand | µ ≡ M=µ , and similarly | a ≡ A=a . Internal Einstein equations are treated using flat indices. The Ricci tensor is computed using formulas of Appendix C of [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] and

f a ⊥ BC = δ b B δ c C f a ⊥ bc .
Starting from the general tensor expression, one should pay attention going from 10d to 6d indices, parallel and transverse, and further to warped versus unwarped quantities. We also use the source contribution

T ab = δ a || a δ b || b δ a || b || T 10 /(p+1).
The internal parallel Einstein equation requires the dilaton derivatives (A.2). One should then prove that the Ricci tensor is given by (A.3), which is achieved using assumption [START_REF] Dewolfe | Type IIA moduli stabilization[END_REF]. We also use [START_REF] Dewolfe | Type IIA moduli stabilization[END_REF] for the internal off-diagonal Einstein equation and compute for the dilaton (A.4); this equation eventually becomes

R| a || b ⊥ = (p -2)δ e ⊥ d ⊥ δ a || c || f c || e ⊥ b ⊥ e -A ∂ d ⊥ e A . ( 15
)
The computed Ricci tensor (A.5) is however different. Interestingly, the match is achieved using the F 8-p BI along non-transverse directions, i.e. dF 8-p -(dF 8-p )| ⊥ = 0, which has two components due to F

(1) 8-p . Setting the one along e a || ∧ e b || ∧ e ⊥ , resp. along e a || ∧ e ⊥ , to zero gives identities (A.6), resp. (A.7), using ( 6) and f a ⊥ b ⊥ a ⊥ = 0. Identity (A.7) allows to solve the off-diagonal Einstein equation. Finally, using the expression (A.8) for the dilaton, the internal transverse Einstein equation is given by

R| a ⊥ b ⊥ = -1 2 δ g || h || δ c ⊥ e ⊥ f g || c ⊥ a ⊥ f h || e ⊥ b ⊥ (16)
+ δ a ⊥ b ⊥ (2p -7)| de A | 2 + δ a ⊥ b ⊥ e A ∆⊥ e A -2(p -3)∂ a ⊥ ∂ b⊥ e A -2(p + 1)∂ ã⊥ e A ∂ b⊥ e A + 2ω a ⊥ c ⊥ b ⊥ | (∂A=0) ∂ c ⊥ φ + 1 2 ι a ⊥ H • ι b ⊥ H + e 2φ 2 ι a ⊥ F 6-p • ι b ⊥ F 6-p -δ a ⊥ b ⊥ e 2φ 2 |F 6-p | 2 .
This holds as well for p = 3 where F 5 = F (0) 5 , using [START_REF] Blåbäck | Smeared versus localised sources in flux compactifications[END_REF]. The last three rows of (16) vanish using [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] and [START_REF] Blåbäck | Smeared versus localised sources in flux compactifications[END_REF]. Computing the Ricci tensor with ( 6) and [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], one then obtains a match, up to 1 2 ∂ g || f g || a ⊥ b ⊥ , which is zero due to the identity (A.6). All e.o.m. are thus satisfied.

IV. NON-GEOMETRIC FLUXES EXTENSION

A natural extension of (4) (without its last line) including non-geometric NSNS fluxes would be

2e -2A R4 = -2 e 4A * ⊥ de -4A -ε p e φ F (0) 8-p 2 (17)
-

H a ⊥ b ⊥ c ⊥ * ⊥ e a ⊥ b ⊥ c ⊥ /3 + ε p e φ F (0) 6-p 2 - a || f a || b ⊥ c ⊥ * ⊥ e b ⊥ c ⊥ /2 + ε p e φ (ι a || F (1) 8-p ) 2 - a || b || Q c ⊥ a || b || * ⊥ e c ⊥ + ε p e φ (ι b || ι a || F (2) 10-p ) 2 - a || b || c || R a || b || c || * ⊥ 1 + ε p e φ (ι c || ι b || ι a || F (3) 12-p ) 2 -(flux) 2 + e -2A ∂(. . . ) .
This combination of BPS-like conditions is T-duality invariant from the 4d perspective: there one lifts or lowers T-dualized indices of NSNS fluxes [START_REF] Shelton | Nongeometric flux compactifications[END_REF], while parallel and transverse directions get exchanged when Tdualized. The last line of (4) may also have a T-duality invariant extension with non-geometric fluxes (see [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF]). Finally, one could extend the F 8-p BI on transverse directions (10) by adding

+|Q c ⊥ a || b || | 2 + |R a || b || c || | 2 .
In these extended expressions, all terms are however not present for any p, due to the number of parallel and transverse directions. All fluxes are allowed for p = 6, but not for the others: on top of the first BPS-like condition, only the one in H is present for p = 3, H, f for p = 4, H, f, Q for p = 5, f, Q, R for p = 7 and Q, R for p = 8. This restriction can also be understood schematically by T-dualizing an O 3 vacuum on T 6 with H-flux.

A first setting to derive these conjectured expressions would be 4d gauged supergravities, replacing R4 by the vacuum value of the scalar potential. Another option would be the 10d formalism of β-supergravity [START_REF] Andriot | β-supergravity: a tendimensional theory with non-geometric fluxes, and its geometric framework[END_REF], or extensions thereof, including [START_REF] Geissbühler | Exploring Double Field Theory[END_REF]. But this goes beyond the scope of this Letter.

V. DISCUSSION

In this Letter, we have characterized a broad class of Minkowski flux vacua with parallel localized O p /D p sources; this is achieved thanks to the general rewriting (4) of the 4d Ricci scalar as a negative sum of squares, extending a well-known result for O 3 -planes [START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF] to general O p reductions. The fluxes are (8) (their BI are assumed to be satisfied), the metric is (1) with restrictions ( 6) and [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], and the dilaton is given after (1). Interestingly, requiring only the restrictions (5), ( 6) and ( 7), we have shown that these are the only Minkowski flux vacua! Our analysis does not rely on supersymmetry (SUSY) so our vacua can capture both SUSY and non-SUSY solutions. To get SUSY O 3 vacua, the Ricci flat condition should be supplemented by Calabi-Yau (see [10] for a Ricci flat non-Calabi-Yau solvmanifold), and the ISD 3form flux has to be (1,2) and primitive [START_REF] Giddings | Hierarchies from fluxes in string compactifications[END_REF]. Here, it would be interesting to compare our geometric restrictions to the generalized Calabi-Yau condition and our fluxes to the SUSY ones [10,[START_REF] Graña | A Scan for new N=1 vacua on twisted tori[END_REF], given the integrability result of [START_REF] Koerber | Supersymmetric sources, integrability and generalized-structure compactifications[END_REF]. Already, our sourced flux takes the value (13) of a calibrated brane [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF], which would be the SUSY value in presence of an SU(3)×SU(3) structure.

A natural question is whether there exists other Minkowski flux vacua with parallel sources than the class found here, thus meaning some with the restrictions not satisfied (see a related discussion in [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF]). We may capture all no-scale vacua, but still wonder whether other vacua exist, which do not arise from the BPS-like conditions. We can for instance look at the set of SUSY Minkowski flux vacua on twisted tori: all known solu-tions are reviewed in [10]. All listed vacua with parallel sources verify ( 5), ( 6), [START_REF] Andriot | Refining the boundaries of the classical de Sitter landscape[END_REF] (in particular those of [START_REF] Graña | A Scan for new N=1 vacua on twisted tori[END_REF]), except for the new ones found in [10] that violate [START_REF] Dewolfe | Type IIA moduli stabilization[END_REF]. For the latter however, the Ricci tensor still vanishes and R || = R ⊥ || = 0, indicating a suspected possible refinement of our geometric conditions. Then, up to this detail, we do not find vacua beyond the class presented here.

Appendix: Ricci tensor and dilaton derivatives

Each side of the 4d Einstein equation ( 14) is equal to

+ 1 2 δ b ⊥ e ⊥ δ c || h || ∂ b ⊥ f h || e ⊥ d ⊥ + 1 4 δ c || i || δ d ⊥ g ⊥ δ e ⊥ h ⊥ δ b ⊥ j ⊥ f i || e ⊥ j ⊥ f g ⊥ h ⊥ b ⊥ ,
-2δ l ⊥ d ⊥ f b || c ⊥ l ⊥ ∂ d⊥ e A + δ d ⊥ l ⊥ ∂ d ⊥ f b || c ⊥ l ⊥ (A.7) -1 2 δ k ⊥ c ⊥ δ e ⊥ h ⊥ δ f ⊥ g ⊥ f b || h ⊥ g ⊥ f k ⊥ e ⊥ f ⊥ = 0 .
For the transverse Einstein equation, we compute

∇| a ⊥ ∂| b ⊥ φ = ∂ a ⊥ ∂ b ⊥ φ -ω a ⊥ c ⊥ b ⊥ | (∂A=0) ∂ c ⊥ φ (A.8) + ∂ a ⊥ φ∂ b⊥ e A -δ a ⊥ b ⊥ δ c ⊥ d ⊥ ∂ c ⊥ φ∂ d⊥ e A .

gµν 16 ( 7 2 + 2 f

 16722 -p) e 6A ∆⊥ e -4A -e 8A |de -4A | 2 .(A.1)For the internal parallel Einstein equation, we use∇| a || ∂| b || φ = -δ a || b || 4 e 2φ-2A δ c ⊥ d ⊥ ∂ c ⊥ e 2A ∂ d ⊥ e -2φ (A.2) R| a || b || = δ a || b || 8 2e 6A ∆⊥ e -4A -(p -1)e 8A |de -4A | δa || c || δ b || d || 4 δ e ⊥ f ⊥ δ g ⊥ h ⊥ f c || e ⊥ g ⊥ f d || f ⊥ h ⊥ , (A.3)where the warp factor terms of (A.3) can be rewritten as-δ a || b || δ c ⊥ d ⊥ e A ∂ c⊥ ∂ d⊥ e A + (2p -7)∂ c⊥ e A ∂ d⊥ e A .For the internal off-diagonal Einstein equation, we compute with[START_REF] Dewolfe | Type IIA moduli stabilization[END_REF] and2∂ [a ∂ b] = f c ab ∂ c ∇| a || ∂| b ⊥ φ = ∇| b ⊥ ∂| a || φ = -δa || e || δ c ⊥ d ⊥ e || d ⊥ b ⊥ ∂ c ⊥ φ (A.4) R| c || d ⊥ = (p -3)δ b ⊥ e ⊥ δ c || h || f h || e ⊥ d ⊥ ∂ b⊥ e A (A.5)

  and the F 8-p BI along non-transverse directions gives∂ a || f c || d ⊥ e ⊥ = 0 , (A.6)

  6 F 3 . The combination in the other square is also present in the total derivative, which thus vanishes. This also fixes F 5 , and relates eventually dF 5 to ∆6 e -4A as expected from the BI. With further combinations of e.o.m., one can show that F 1 = 0 and φ is constant (see below). Using the H and F 3 BI (i.e. that those fluxes are closed), one then shows that all e.o.m. are solved, pro-

	vided the (unwarped) compact manifold is Ricci flat. In
	other words, given this geometric requirement, and as-
	suming satisfied BI, one finds all Minkowski flux vacua,
	and those are characterized through the expression (3)
	by setting the squares of BPS-like conditions to zero.
	B. Op-planes with p > 3

ACKNOWLEDGEMENTS

We thank F. F. Gautason and D. Tsimpis for useful discussions. The work of D. A. is part of the Einstein Research Project "Gravitation and High Energy Physics", funded by the Einstein Foundation Berlin. The work of J. B. was supported by the John Templeton Foundation Grant 48222 and the CEA Eurotalents program. The work of T. V. R. is supported by the FWO odysseus grant G.0.E52.14N. We acknowledge support from the European Science Foundation Holograv Network.