The $O(N)$ Model in $4<d<6$: Instantons and Complex CFTs

We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In $d$ dimensions and for large $N$, we find that they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $N$, the $O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional $O(N)$ models may be studied using the techniques of numerical bootstrap.


Introduction and summary
One of the classic models of Quantum Field Theory (QFT) is the O(N )-symmetric theory of N real scalar fields φ i with interaction g 4 (φ i φ i ) 2 . For small values of N there are physical systems in three and two spacetime dimensions, whose critical behavior is described by this QFT. While the infrared (IR) dynamics in these dimensions is strongly coupled, there are various methods for studying it, including dimensional continuation [1,2] and the 1/N expansion (for a review, see [3]). Furthermore, the methods of conformal bootstrap [4][5][6] (for reviews, see [7][8][9][10][11]) have been fruitfully applied to the O(N ) model in various dimensions [12][13][14][15].
There is an interesting range, 4 < d < 6, where the theory appears to be unitary order by order in the 1/N expansion [27][28][29][30][31]. Yet, the fate of the theory with large but finite N is unclear in view of the expectation, supported by rigorous results [32], that the interacting φ 4 theory cannot exhibit true critical behavior in d > 4.
Some light on this issue was shed by the series of papers starting with [31], where a cubic O(N )-symmetric theory with the action for N + 1 scalar fields given by was introduced as a possible UV completion of the O(N ) model in d < 6. Indeed, for N > N crit , where N crit ≈ 1038, an IR stable fixed point of the theory (1.1) was found perturbatively in [31,33,34]. Furthermore, it was found that the 6 − expansions of various observables agree with the results obtained from the formal 1/N expansions. For N = N crit the IR fixed point merges with another fixed point, and for N < N crit these fixed points become complex. This kind of merger of fixed points is a ubiquitous phenomenon in studies of the Renormalization Group [35][36][37][38][39][40][41][42], and theories at the complex fixed points for N < N crit have been recently called "complex CFTs" [40,41].
At the same time, one should expect that, even the model with N > N crit cannot be perfectly stable because of tunneling from the perturbative vacuum at σ = φ i = 0 to large negative values of σ, where the potential is unbounded from below. The instantons mediating this tunneling were found long ago in the 6-d cubic theory of a single scalar field [43], as well as in the 4-d O(N ) model with negative coupling [44,45]. Via application of the instanton methods to the UV fixed point in 4 + dimensions, it was shown [45] that the critical exponents acquire imaginary parts of order exp − N +8 parts that are exponentially suppressed as e −N f (d) at large N . 1 We calculate f (d) and find that it is given by the free energy of a conformal scalar on S d−2 , which has the integral representation [47]: In particular, f (5) = log 2 8 − 3ζ(3) 16π 2 ≈ 0.0638 [48]. The function f (d) is plotted in Figure 1. In agreement with [45], f (4 + ) blows up as 1/(3 ). Similarly, near six dimensions we find f (6 − ) → 1/(90 ); as we show below, this precisely agrees with the contribution of the instanton in the cubic theory (1.1). The relation between the non-perturbative imaginary parts in d dimensions and the sphere free energy of a conformal scalar in d − 2 dimensions is quite striking, and it would be nice to understand its physical origin.
A simple argument for the exponential suppression of the imaginary parts is that on the sphere S d , or cylinder S d−1 × R, the conformal coupling to curvature adds a positive quadratic term to the scalar potential, making the perturbative vacuum metastable. We demonstrate the smallness of imaginary parts explicitly by computing, at large N , the sphere The fact that the imaginary parts are very small for large N , makes the O(N ) models in 4 < d < 6 similar to the robust examples [35][36][37][38][39][40][41][42]  2 Technically similar calculations of instanton corrections to CFT correlation functions and operator scaling dimensions in the N = 4 SYM theory have been performed in [49][50][51][52][53]. In that case there is no instability, and the instanton does not have a negative mode responsible for the imaginary parts.
invariant relevant operators [31,33], one may be able to find the approximately critical behavior by tuning both the nearest-neighbor and next-to-nearest-neighbor couplings on a lattice. 3 Our results may have interesting implications for the AdS/CFT correspondence [54][55][56], in particular its higher spin version (for reviews see [57,58]). The type A Vasiliev higher spin theories [59][60][61] have the minimal field content, which consists of massless higher spin fields of even spin and a massive scalar. It has been conjectured [62] that such a theory in dimension d + 1 is dual to the singlet sector of the d-dimensional O(N ) model, which is either free or interacting depending on the choice of boundary conditions on the bulk scalar field. In d = 3 both choices of boundary conditions produce a stable theory, since both free and interacting O(N ) models are conventional CFTs. In d = 5, however, only the choice of boundary conditions corresponding to the free O(N ) model should be stable. In view of our findings in this paper, the other choice is expected to be metastable, with the decay amplitude ∼ e −const/G , where G ∼ 1/N is the bulk coupling constant. From the bulk point of view, the instability may again be due to instantons, whose existence should depend on the choice of boundary conditions. 4 It would be interesting to search for such instanton solutions of the higher-spin theory in AdS 6 .
The rest of this paper is organized as follows. We begin in Section 2 with a description of the instanton solutions in the d = 6 − and d = 4 + expansions, where such solutions can be found by solving the classical equations of motion of the corresponding theories.
In Section 3 we then describe the instantons at large N , where these instantons extremize the effective action for the Hubbard-Stratonovich field σ that is obtained after integrating out the O(N ) vector fields φ i . These solutions on S d have constant σ = −k(k + 1) (for a special choice of the instanton moduli), where k is a positive integer. In 4 < d < 6 the dominant non-perturbative effects come from the k = 1 saddle point. We continue with calculations of the instanton contribution to the round sphere free free energy in Section 4, to the operator scaling dimensions in Section 5, and to C J and an example of a three-point function coefficient in Section 6. In Section 7 we exhibit the classical solutions on S 4 and S 6 where the fields φ i are not constant, but are rather proportional to spherical harmonics; these solutions correspond to the large N saddle points with k > 1. Several technical details and related calculations are relegated to the Appendices. 3 We are grateful to Slava Rychkov for this suggestion. 4 Similar instanton solutions in AdS d+1 were discussed in [63,64].

Classical instantons in the epsilon expansion 2.1 The instanton near six dimensions
The equations of motion of the O(N ) invariant cubic scalar theory with action (1.1) are given by (2.1) In d = 6, in addition to the trivial solution σ = φ i = 0, these equations admit the O(N )invariant instanton solution Here a and λ are the moduli corresponding to the position and size of the instanton. Since this solution has φ i = 0, it is a simple generalization of the instanton solution in the single scalar cubic theory studied in [43]. 5 Plugging this solution into (1.1), one obtains the finite instanton action The instanton solution (2.3) is responsible for tunneling from the metastable ground state at σ = φ i = 0. As we show explicitly in Appendix B.1, computing the spectrum of quantum fluctuations around this solution, one finds a single negative mode, so that the instanton yields an imaginary contribution ∼ ±ie −S inst to the free energy and to other observables. To leading order in the d = 6 − expansion, for N > N crit ≈ 1048 one finds a fixed point at [31] so that at large N and small the instanton action is Hence, the imaginary contribution to free energy and other observables comes with the exponentially suppressed factor ∼ ±ie − N 90 . Let us point out that the β-function of the theory is also expected to receive such imaginary contributions from the instanton (see [43][44][45]), so that the value of the fixed point couplings (2.5) gets small imaginary parts as well. This would give non-perturbative corrections to the action (2.6), which we neglect in this section as they are further suppressed.
Let us now consider a conformal mapping of the theory (1.1) from flat space to the unit-radius round sphere S d parameterized in stereographic coordinates x, with the metric (2.7) In d = 6, the action (1.1) and the equations of motion (2.1) are conformally invariant, so the instanton solution on S 6 can be simply obtained by a Weyl rescaling, and it is given by where we have used that σ has scaling dimension 2. It is easy to check that the classical action on S 6 evaluated on the solution (2.8) has the same value as (2.4), as expected from conformal invariance. The quadratic terms in the action above come from the conformal coupling to the sphere curvature. 6 We now observe that for a special choice of the moduli, λ = 1 7 and a = 0, the instanton solution on the sphere is just a constant 8 This has a simple interpretation: it is the critical point corresponding to the local maximum 6 In general d, the contribution of the conformal coupling to the Lagrangian for a scalar field φ is given by where R is the Ricci scalar. On S d , we have R = d(d − 1)/R 2 , and we have set R = 1 in (2.9). 7 Or λ = 1/R 2 if we reinstate the radius of the sphere. 8 This was also noticed and used in related models. For example, see [43,44,65].
A plot of this potential is given in Figure 2.
Note that one may also conformally map the theory to the cylinder R t ×S 5 . The constant solution (2.10) is then mapped to the time-dependent configuration This solves the equation where the right-hand side comes from the potential on R t × S 5 , V Rt×S 5 = 2σ 2 + g 2 σ 3 /6.
The solution (2.12) has just the same form as the instanton in quantum mechanics which is responsible for tunneling in a cubic potential. See, for instance, [66,67] for reviews. In the rest of the paper we will focus mainly on the S d description of the instanton.
In section 3, we will describe how an analogous instanton solution arises in the large N treatment of the O(N ) model in 4 < d < 6 (the integer dimension d = 5 is, of course, the most interesting). Essentially, the role of the "fundamental" field σ above will be played by the Hubbard-Stratonovich field, which is used to develop the large N expansion of the model in general d.

The instanton near four dimensions
For completeness, let us also discuss how the instanton solution looks like near the lower end of the range 4 < d < 6, where we can formally use the d = 4 + expansion in the O(N ) invariant quartic scalar theory (2.14) In d = 4 + , the one-loop beta function is given by and thus we see that there is a formal UV fixed point at negative coupling At the level of perturbation theory in , the corresponding fixed point appears to be unitary (all scaling dimensions are real and above the unitarity bound) to all orders in . However, due to the wrong sign quartic potential, we expect the model to be non-perturbatively unstable. Indeed, it is well known that for negative coupling the theory (2.14) in d = 4 has a real instanton solution [44,68] whereû i is a constant unit N -component vector (û iûi = 1), and λ, a are the size and position moduli as in the previous section. Note thatû i are also exact moduli parameterizing an S N −1 . Integration over these moduli restores O(N ) invariance of correlation functions (see for instance [66] for a review).
After a conformal mapping to S 4 , the instanton solution takes the form which solves the equations of motion of the S 4 theory with the action Again, we observe that on the sphere there is a choice of moduli where the solution becomes a constant which can be seen to correspond to the degenerate maxima of the potential As discussed above, the conformal coupling to the sphere curvature makes the perturbative vacuum metastable for g < 0, as shown in Figure 3. The value of the classical action on the ϕ V(ϕ) instanton solution (computed either on R 4 or S 4 ) is Plugging in the value of the coupling at the fixed point in d = 4 + , this yields in agreement with [45]. We will see that this matches with the large N method we introduce in the next section. In Section 7 we show that there are additional classical solutions where φ i is generalized from a constant to any spherical harmonic on S d .

"Hubbard-Stratonovich instanton" at large N
Applying the familiar Hubbard-Stratonovich transformation to the (φ i φ i ) 2 interaction in the O(N ) model, one arrives at the following action describing the critical O(N ) model: where σ is the Hubbard-Stratonovich field, and we have dropped the term ∼ σ 2 /λ which becomes irrelevant in the critical limit. 9 This action can be used to systematically develop the 1/N expansion of the theory. The field σ, which acquires induced dynamics due to φ loops, becomes at large N a conformal scalar operator with scaling dimension ∆ = 2 + O(1/N ).
After a conformal transformation to the sphere metric (2.7), one obtains the action Since this action is quadratic in φ i , we can integrate out these fields exactly, and hence obtain a path integral over σ with action Given the intuition from the classical solution in d = 6 described in the previous section, it is natural to look for configurations of constant σ that solve the equation of motion. Recall that the eigenvalues λ n and degeneracies D n of the scalar laplacian on S d are: (3.4) So the action for constant σ is Following [47], we find Consequently, the action (3.5) is extremized when for some integer k. This gives σ = −k(k + 1) . as the physical interpretation of the additional saddles seems unclear (in particular, as will be shown in Section 7 below, the solutions with k > 1 have several negative modes). For instance, for d < 4 the appropriate contour can be chosen to run along the imaginary σ axis passing only through the k = 0 saddle.
Let us now focus on the k = 1 solution in 4 < d < 6, i.e. that with σ = −2. As in the cubic theory in d = 6, this solution can be seen to correspond to the local maximum of the effective potential for constant σ in Eq. (3.5). A plot of this potential in d = 5 is given in Figure 4 (for general 4 < d < 6, the behavior is analogous); it is qualitatively similar to the classical potential for the cubic theory on S 6 (except that it becomes unbounded from below as σ approaches −15/4 as explained above, while in d = 6 − this happens only asymptotically as σ → −∞). It is then natural to view the σ = −2 solution as the instanton configuration that is the large N counterpart of the classical solution described in the previous section. Indeed, as we will show below, studying the spectrum of fluctuations around then σ = −2 saddle point, one finds a single negative mode and d + 1 zero modes, which we interpret as the size and position moduli of the instanton. Hence, even though we have found the solution specializing to constant configurations of σ, we expect that it belongs to a family of instanton solutions where λ and a are the moduli, as seen explicitly in the d = 6 analysis. We will compute the value of the action (3.5) on the instanton solution, as well as the determinant for quantum fluctuations, in Section 4 below. Before doing that, in the next section we give a useful description of the instanton profile using embedding coordinates in the instanton moduli space.

Instanton profile in embedding coordinates
On the round S d with metric (2.7), the instanton profile in the large N theory is given by (3.9). If we make a conformal transformation to flat R d (parameterized by x and with line element d x 2 ), the profile for σ will then be (3.10) As familiar in instanton calculus, the moduli space is expected to be given by the quotient 11 We can then simplify the formulas by going to a (d + 2)-dimensional embedding space with signature (+, +, · · · , +, −). In this space, H d+1 is given by the hyperboloid The relation between these coordinates and a and λ is For the CFT coordinates, we use null vectors P in the same (d + 2)-dimensional space: For flat space, we have while for the sphere we have These null vectors have the property that the induced line element dP · dP = d+1 i=1 (dP i ) 2 − dP 2 d+2 is precisely equal to the metric on flat space and on the unit sphere, respectively. In terms of these coordinates, the instanton profile (either on flat space or on the sphere) can be written as (3.16) A different coordinate system that makes the symmetries of S d manifest can be found as follows. We parameterize S d by a unit vectorp embedded in R d+1 , and we parameterize the instanton moduli space by a radial coordinate ρ and a unit vectorn in R d+1 . We take P = p , 1 , X = n sinh ρ , cosh ρ .

(3.17)
To find the relation betweenp and x we equate the first equation in (3.17) to (3.15), and to find the relation between (ρ,n) and (λ, a) we should equate the second equation in (3.17) to (3.12). In these new coordinates, the line elements on S d and on the instanton moduli space are dP · dP = dp 2 , dX · dX = dρ 2 + sinh 2 ρ dn 2 .
The instanton profile is Quite nicely, for ρ = 0, an SO(d + 1) rotation of the pointp on S d is equivalent to a rotation of the unit vectorn. If we want to preserve rotational symmetry on S d when integrating over the moduli space of instantons, we should thus impose a cutoff at a fixed value of ρ. We will make such a choice shortly when computing the instanton contribution to the sphere partition function and to various correlation functions.

The S d partition function
In the large N expansion, we have that the S d partition function is (keeping up to order N 0 terms for each saddle): (4.1) The quantities f 0 and f 1 come from the φ determinant, and may be viewed as the "classical action" on the σ saddle point, while A 0 and A 1 come from the σ determinants for quadratic fluctuations around the saddle point. In the contribution of the σ determinant around the instanton saddle, we included an explicit factor of the (divergent) volume Vol(H d+1 ) of the unit curvature radius hyperbolic space, in anticipation of the fact that the integration over the instanton moduli space yields such a factor.
More explicitly, the partition function around a saddle point σ c to quadratic order in fluctuations is given by where G( x, y) denotes the Green's function for φ in the background σ c , which will be computed below.

"Classical action" from φ determinant
Let us now calculate the quantities appearing in (4.1). From (3.5), we find that for the perturbative vacuum at σ = 0 we have For the σ = −2 instanton solution, we instead have (4.4) Subtracting, we obtain It can be easily checked using a useful relation This expression is identical to (4.3) with d shifted down by 2. So we find (4.7) Since the derivation above involved some formal manipulations with divergent sums, in the Appendix we present an alternative derivation of this result-see Eq. (B.6).
In d = 5, we then have where we have used the value of F free scalar S 3 computed in [48].
In general d, the value of F for a free conformal scalar has the integral representation [47] F free scalar This expression can be easily expanded near even integer dimensions, where one finds poles related to the conformal a-anomaly coefficient. For the instanton in d = 6 − dimensions, we find from (4.7) and using the above representation that which is in precise agreement with the result (2.6) obtained from the classical instanton in the 6-d cubic theory. In the case d = 4 + , we find similarly which agrees with the classical instanton in the quartic theory given in Eq. (2.23).

Scalar Green's function on the instanton background
In order to compute the one-loop determinant for the σ fluctuations, we will first need to evaluate the scalar two-point function in the instanton background. We will first perform this calculation on the σ = −2 constant solution (corresponding to moduli λ = 1, a = 0).
In general, for the operator the Green's function is given by The solution of this equation is where C is an m-dependent constant, which we will fix below.
, with s( x 1 , x 2 ) the chordal distance on S d , and the constants a, b, c are given by When m 2 = 0, we have a conformally coupled scalar, whose Green's function is obtained by conformally mapping the flat space Green's function for a massless scalar, gives (4.17) To determine the constant C, we note that G( x 1 , x 2 ) must have the same short-distance singularity as the propagator of a conformally coupled scalar, so setting equal the coefficients (4.18) The general dependence on the moduli may be reinstated by performing a conformal transformation to flat R d , where a and λ are related simply to translations and dilations.
One finds that for the general instanton profile, using the embedding coordinates introduced earlier, we can write where P 1 and P 2 are the embedding space coordinates of the points x 1 and x 2 . Noting that we can also write this formula completely in embedding space as .
This formula is true both on R d and on S d provided we use (3.14) or (3.15) as appropriate.
In the special case d = 5, we have . (4.21)

The σ determinant
As per (4.2), the kernel for the σ fluctuations at any saddle is given by −N G( x 1 , x 2 ) 2 . This kernel is diagonalized by expanding it in spherical harmonics Y n, m ( x), with eigenvalues λ n : Here, the n = 0, 1, 2, . . . index labels the distinct representations of the rotation group appearing in the spherical harmonic decomposition of a scalar function, while the multiindex m labels the various states in a given representation, taking D n values, with D n given in (3.4). The expansion (4.22) can be performed using the general formula . where the Green's function is given in (4.16). Applying (4.23) in this case, we obtain the eigenvalues The same computation at the constant instanton saddle with the Green's function (4.17) gives .
n is negative for n = 0, it vanishes for n = 0, and it is positive for all n > 0. The negative mode at n = 0 (note that the degeneracy is D 0 = 1 in this case) will give an imaginary part to the sphere free energy and other observables. The fact that when n = 1 we have D 1 = d + 1 zero modes is a consequence of the existence of a whole manifold of instanton saddles (3.9) parameterized by the d + 1 parameters a and λ.
The zero modes represent infinitesimal motions along this manifold starting at the a = 0 and λ = 1 saddle. Because these saddles are related by conformal transformations, the spectrum of quadratic fluctuations in δσ must be independent of the instanton moduli a and λ.
If not for the presence of zero modes, the instanton correction to the sphere free energy Due to the presence of the zero modes, however, the factor λ is replaced by an integral over the instanton moduli space. In addition, the divergent integral over the negative mode requires careful treatment [69,70]. Thus, when computing the instanton correction, let us split up the n = 0 and n = 1 modes from the rest, and write where c 0 is a factor arising from the analytically continued integral over the negative mode (to be discussed below), and the zero mode measure µ zero modes (d) will be computed in the next subsection. The factor R d is the contribution from all the modes with n ≥ 2, and we may compute it explicitly as follows: . (4.28) In d = 5, we obtain (see Appendix B) (4.29) We will calculate µ zero modes (d) in the next section.
Let us now discuss the factor c 0 introduced in (4.27). The negative mode arises from the integration over the constant part of σ, so this is essentially a problem of analytically continuing an ordinary integral. Let us consider a concrete example [43] which is qualitatively similar to our situation, namely an Airy-like integral of the form where C is a choice of contour, and we are interested in the behavior of the integral for large N . The function in the exponent has essentially the same form as the classical potential On the other hand, the contours C ± receive contributions from both saddle points and hence either one is a suitable contour for our physical application, where we want the perturbative vacuum to give the dominant contribution. Note that C + and C − can be deformed so that they run along the real axis passing through z = 0, reaching the z = −1 saddle point, and then moving into the complex plane along a direction of steepest descent (either along positive or negative imaginary parts, corresponding to C + and C − respectively). Hence, the imaginary part of the integral at large N receives contribution from the semi-infinite arc of such contours starting at z = −1. In the saddle point approximation this gives half of a Gaussian integral, so that The potential for σ in the large N theory is a more complicated function, but it is qualitatively similar. See Figure 4. So we expect a similar analysis to go through in this case as well (see [70] for a general discussion). In particular we expect the appropriate contour to run along the real axis passing through σ = 0 and reaching σ = −2, and then to move into the complex plane into either positive or negative directions of the imaginary axis. Therefore, for the purpose of extracting the imaginary parts of physical quantities due to the instanton, we should set in (4.27) The choice of sign, corresponding to the ambiguity in the choice of contour, gives two distinct "complex CFTs" that are related to each another by complex conjugation. that we can unit normalize

Measure on the instanton moduli space
Then, the path integral measure is We separate these modes into zero modes indexed by A = 1, . . . , d + 1 and non-zero modes indexed by i: c a = (c A , c i ). If the coordinates v A parameterize the instanton moduli space, then 36) where N A is the norm of dσ/dv A . From c A ψ A = (dσ/dv A )dv A , we then conclude that dc A = √ N A dv A , and so the zero mode measure is Let us now compute N A . Close to ρ = 0, the instanton profile is approximately σ ≈ −2 − 4p · (nρ) + · · · . We can thus take v A = ρn A close to ρ = 0, so So the zero mode measure close to ρ = 0 is The quantity ρ d dρ vol S d is the small ρ approximation to the volume of hyperbolic space of unit radius. Thus, the full zero mode measure is and, lastly, we have the contribution of the modes with n ≥ 2 in (4.29): (4.44) Adding these expression together, exponentiating, and multiplying by e −N (f 1 −f 0 ) we find The expression that appears in the instanton contribution to the S 5 free energy in (4.1) also includes a factor of the volume of the unit radius hyperbolic space, Vol(H 6 ). For general d, one can regularize such a factor using a regulator preserving spherical symmetry. The regularized value is Thus, the regularized value of Vol(H 6 ) in (4.1) should be taken to be Vol(H 6 ) = −8π 3 /15.
Putting everything together, the imaginary part of the S 5 free energy is then approximately given by where we have used c 0 = ±i/2 as explained in Section 4.3.
Note that while an overall factor of Vol(H 6 ) arises in the instanton contribution to the sphere free energy, such a factor will be absent in the computations of other CFT observables presented below (which, in particular, are independent of whether we choose spherical or planar slicing for the hyperbolic space metric).

Imaginary parts of scaling dimensions
A key property of complex CFTs is the presence of some operators whose scaling dimensions are complex. This behaviour has been found in a variety of models, including [35][36][37][38][39][40][41][42]. We will now explore how the instanton effects contribute small imaginary parts to the operator scaling dimensions. While in a number of large N theories the complex scaling dimensions have been found to lie on the principal series d/2 + iα, we find that the instanton effects produce complex dimensions of a more general form, with real parts not necessarily equal to d/2.

Anomalous dimension of φ i
For the perturbative saddle, at leading order in 1/N , the two point function of φ i is where, as before, s( x 1 , x 2 ) is either | x 1 − x 2 | on R 5 or the chordal distance on S 5 . The exponent in the denominator shows that the scaling dimension of φ i equals 3/2 at large N .
The expression (5.1) receives both perturbative and non-perturbative corrections in 1/N .
While the perturbative corrections are real, the first contribution to the imaginary part comes from the instanton saddles discussed in the previous section.
Let us denote the scaling dimension of φ i by ∆ φ = 3/2 + δ φ . Since the instanton contribution to δ φ , which we denote by δ inst φ , is small, its effect is to modify the exponent in (5.1) only slightly. Expanding 1/s( where is the UV cutoff, we find that the instanton contribution to the φ i two-point function must take the form To obtain (5.2), note that when computing the two-point function φ i ( x 1 )φ j ( x 2 ) we should perform the path integral with a φ i ( x 1 )φ j ( x 2 ) insertion by summing over all saddle points, and then divide the answer by the partition function. Thus where for each saddle point we only included the contribution from the φ i and σ determinants.

(5.5)
Comparing this expression with (5.2), we conclude To calculate δ inst φ , all that remains to do is to calculate the logarithmic derivative of I φ with respect to . Note that I φ ( x 1 , x 2 ) is Weyl-invariant, so it is independent on whether the theory is placed on R 5 or on S 5 .

Hard cutoff regularization
Since the logarithmic derivative should not depend on the points x 1 and x 2 , let us set x 1 = 0 and | x 2 | → ∞. On S 5 these two points correspond to the North and South poles of S 5 , respectively. In embedding space, the North pole corresponds to P N = ( 0, 1, 1) and the South pole corresponds to P S = ( 0, −1, 1), so the chordal distance between them is s(N, S) = 2. Then, taking X = (n 4 sin θ sinh ρ, cos θ sinh ρ, cosh ρ), wheren 4 is a unit vector on S 4 , we have = Vol(S 4 ) dρ dθ sinh 5 ρ sin 4 θ 4 1 + sinh 2 ρ sin 2 θ . (5.7) (Here, we used the SO(5) rotational symmetry of this configuration to pull out a volume factor of S 4 .) Both integrals in (5.7) can be done analytically, but for the ρ integral we impose an upper cutoff: where we used Vol(S 4 ) = 8π 2 /3. Taking e ρm = 1/ (if we were to reinstate the radius R of S 5 we would have e ρm = R/ ), we then find From this expression we can extract the logarithmic derivative where we used the fact that the logarithmic derivative should be independent of the choice of points.

Analytic regulator in Poincaré coordinates
Let us now obtain the same result using a different regulator. We could also write I φ ( x 1 , x 2 ) in Poincaré coordinates: If we regularize this integral by multiplying the integrand by (z/ ) s , we can use the formulas from [71] to evaluate (5.11) to The logarithmic derivative of this expression with respect to reproduces (5.10).

Anomalous dimension of σ
We can perform a similar calculation to determine the leading contribution to the imaginary part of the scaling dimension of σ. The leading two-point function of σ can be found from the perturbative saddle by inverting the kernel −N G 0 ( x 1 , x 2 ) 2 multiplying the σ fluctuations.
This is easily done after performing the spherical harmonic decomposition as in (4.23), whereby this kernel becomes −N C 2 0 k n (d − 2), with C 0 being the constant defined in (4.16). Because the inverse of this kernel gives −1/(N C 2 0 k n (d − 2)) = C σ k n (2), with one finds that the leading order two-point function of σ is (5.14) As in the case of φ, the dimension of σ is ∆ σ = 2 + δ σ , so the contribution of the instanton saddle to the two-point function must take the form where is the UV cutoff.
To leading order in N , the contribution from the instanton saddle can be found by simply replacing σ by its classical value σ = −8/(−2X · P ) 2 and computing the functional integral over σ in the presence of the insertion σ 2 . This integral yields the same answer as without the insertion for the non-zero modes. The integration over the instanton moduli space now replaces Vol(H 6 ) in µ zero modes d with the integral of σ 2 over H 6 . Thus, where we included only the leading contributions from the two saddles. Expanding at small A 1 , we identify the instanton contribution to the σ two-point function to be Note that the expansion of the denominator in (5.16) generates a term proportional to σ( x 1 )σ( x 2 ) 0 that is proportional to A 1 /A 0 , but this term is suppressed by one power of 1/N relative to the term written in (5.17). Comparing this expression with (5.15) and noting that the logarithm in (5.15) can only arise from the first term in the square bracket of (5.17), we extract Like I φ ( x 1 , x 2 ), the quantity I σ ( x 1 , x 2 ) is also Weyl-invariant, so it is independent on whether the theory is placed on R 5 or on S 5 .

(5.19)
We can again do both integrals: (5.20) Using e ρm = 1/ , we obtain I σ (N, S) = (power-law divergence) + 64π 3 log + · · · . (5.21) Making use of the fact that the logarithmic derivative of I σ is independent of the choice of points, we conclude that

Analytic regulator in Poincaré coordinates
We can also obtain the same result using an analytic regulator. In Poincaré coordinates, we have (5.23) Let's again regularize the integral by multiplying the integrand by (z/ ) s : A derivative of (5.24) with respect to log then reproduces (5.22).

Numerical values
Since the leading imaginary part of the scaling dimensions comes from the instanton contribution, we have where the quantity A 1 A 0 e −N (f 1 −f 0 ) is given explicitly in (4.45), and the overall sign choice corresponds to c 0 = ±i/2 as in (4.32).
In d = 5, if we require |Im ∆ φ | < 10 −2 or 10 −3 , then we find N > 172 or N > 220, respectively. If we require |Im ∆ σ | < 10 −2 or 10 −3 , we find N > 310 or N > 355, respectively. These constraints are roughly commensurate with the smallest values of N where the "islands" in the bootstrap for the d = 5 O(N ) model were observed [15]. 12 6 Instanton contribution to other quantities where C 0 is the constant defined in (4.16). This definition of c J is such that a free theory of N massless scalar fields, which has an O(N ) global symmetry under which the scalar fields transform as a vector, has c J = 1.
For us (or for a theory of N free massless scalars φ i ), the canonically-normalized O(N ) current is Canonical normalization means that the leading term in the OPE between j µij and an operator O i transforming in the vector representation of O(N ) takes the form as | x| → 0. This equation is such that if we construct the charge operator Q Σ ij = Σ j µ ij n µ associated with a closed surface Σ with outward pointing normal n µ surrounding the origin, then Q Σ ij acts on O k (0) as a generator of the O(N ) symmetry: The equation ( Working around either the perturbative saddle or one of the instanton saddles, we can write It is straightforward to check that with G = G 0 given in (4.16), one reproduces (6.1) with c J = 1. After using (4.19) or (4.20) around a given instanton saddle specified by the moduli (λ, a) and integrating over these moduli, Eq. (6.5) becomes with the quantity Q defined as . (6.7) From (6.6) one can obtain the full instanton contribution (up to one-loop order) after per-forming an integral over the instanton moduli space: where the term in the second line arises precisely in the same way that the G 0 term in the square bracket of (5.4).
Let us now work in d = 5, where The integral over X can be performed as in the previous section. After regularization, we and this integral is independent of weather we use the rotationally-invariant cutoff regulator or the analytic one. From (5.12), we also have which again is independent of which regulator we use. Plugging (6.10) and (6.11) into (6.9) and comparing with the definition of c J in (6.1), we find that the leading approximation for the imaginary part of c J is given in (4.45).

Three-point function coefficients
Another example of a quantity that acquires an imaginary part due to the instanton contribution is the three-point function σσσ . Conformal invariance implies for some numerical coefficient C σσσ . The perturbative contribution to C σσσ is O(1/N 2 ), and since it is real, we are not concerned with it here. The contribution from the instanton saddle gives the leading imaginary part of C σσσ , and can be evaluated simply by plugging in the classical value of σ and integrating over the instanton moduli space. By analogy with the expression (5.17) for the two-point function, we have (6.14) This integral is convergent and was evaluated in [71]: It follows that the leading imaginary contribution to C σσσ is (4.45). While the three-point function σσσ and consequently C σσσ depends on the normalization of the operator σ, one can define the normalization-independent ratio whereê is a unit vector. The leading order imaginary part of r is then 7 Saddle points with k > 1 As shown in Section 3, the large N theory on S d admits a sequence of saddle points with constant σ = −k(k + 1) and k a positive integer. We have identified the k = 1 saddle point as the source of leading non-perturbative effects in the large N limit of the O(N ) model with 4 < d < 6. The corresponding instanton solutions are well-known in the limits where d approaches 4 and 6 so that the theory becomes weakly coupled [43][44][45], as reviewed in Section 2. In this section, we collect for completeness a number of results about the k > 1 solutions, including their "classical actions" and the spectrum of fluctuations around them.
In particular, we present new (as far as we know) classical solutions on S 4 and S 6 where the fields φ i are not constant, but are rather proportional to any spherical harmonic.

Saddle points in the large N theory
The value of the effective action evaluated on the large N saddle point is given by where f k (d) may be given the integral representation in (B.5) or (B.6), for α = k. Using the recursion relation for degeneracies it is not hard to show that From this relation we find and finally we obtain where we used the fact that f 0 (d) = F free scalar S d , with an explicit expression for general d given in (4.9). For example, for the k = 2 saddle, we find that at large N , the effective action evaluated on the saddle point is Let us now consider the spectrum of fluctuations of σ around these saddles. For this we need to diagonalize the "kinetic" operator −N G 2 is the φ's Green's function around the σ = −k(k + 1) solution. Using Eq. (4.14) and hypergeometric function relations we find where we fixed the constant C as before by noting that G k ( x 1 , x 2 ) must have the same short-distance singularity as the propagator of a conformally coupled scalar. As for the k = 1 saddle, the Green's function can be expanded in spherical harmonics Y n, m (x) with eigenvalues λ Using that we find for general k that n is positive for all n, whereas λ (k) n are negative for n = 0, 2, 4, . . . , 2k − 2 and zero for n = 1, 3, 5, . . . , 2k − 1 with degeneracies D n (d). In the case of k = 1, i.e. the physical instanton which was the focus of the paper, we get a single negative mode and d + 1 zero modes. For k > 1 there are several negative modes as well as additional zero modes, making the physical interpretation of these solutions unclear. For instance, for k = 2, one finds (d + 1)(d + 2)/2 negative modes and (d + 1)(d + 2)(d + 3)/6 zero modes.

Expansion around 6 and 4 dimensions
Let us now discuss how these saddles behave close to d = 6 and d = 4. First note that if we take d = 6 − in Eq. (7.5), and expand to leading order in , then only a finite number of terms contribute to the sum, because the general formula (4.9) for the S d free energy implies to leading order in Thus we find that, in the large N theory, the value of the action evaluated on the σ = −k(k + 1) saddle point is Performing a similar computation in d = 4 + dimensions, we find The fact that these expressions are proportional to 1/ suggests that we should be able to find all the k-saddles in perturbation theory in . Indeed, in the following we find analogous classical solutions in the theories (1.1) and (2.14) in d = 6 − and d = 4 + , respectively, at values of N that are not necessarily large.

Instantons close to d = 6
Let us start with the cubic theory (1.1) in d = 6 at arbitrary couplings g 1 and g 2 , conformally mapped to S 6 as in (2.9). Due to conformal symmetry at the classical level, any solution to the classical equations of motion for the S 6 theory (2.9), can be mapped to a solution to the classical equations of motion (2.1) of the R 6 theory (1.1), and vice versa. In (7.14), we relabeled σ → σ when we derived the equations of motion from (2.9) in order not to confuse σ, which appears in the 6d theory with the σ field from the large N theory. The two fields differ by an overall normalization factor.
The feature of the solutions we want to find is that σ is constant on S 6 , 13 so we may ask whether (7.14) admit such solutions. It is easy to see that the first equation in (7.14) then takes the form of an eigenvalue equation for the operator ∇ 2 − 6, with g 1 σ being the eigenvalue. Apart from the trivial solutions with φ i = 0, non-trivial solutions exist only when 6 + g 1 σ = −n(n + 5) for some n = 0, 1, 2, . . .. They are the S 6 spherical harmonics Y n, m . Thus Comparing the coefficients of σφ i φ i in the S 6 action (7.14) and the large N action (3.2), we see that σ = g 1 σ. Therefore, the solution (7.15) precisely matches σ = −k(k + 1) upon the identification k = n + 2. The second equation in (7.14) is solved provided that φ i φ i is a constant equal to 14 It is not hard to arrange for the φ i to be proportional to the spherical harmonics with index n and at the same time for φ i φ i to be constant. Indeed, if D n is the number of linearly independent such spherical harmonics, and {Y n,p }, with p = 1, . . . , D n is an orthonormal 13 There should also be solutions where σ is not a constant. Some of them can be obtained by performing a conformal transformation to flat space, followed by a translation and dilation, followed by a conformal transformation back to the sphere. We leave to future work an investigation as to whether this is a full set of solutions of (7.14).
14 Since the right-hand side is negative, these solutions involve imaginary φ i . This is similar to the instanton solutions in the O(N ) model in 4 − dimensions, which are also imaginary (see [44], Section 2.2, and the following section). basis of real spherical harmonics, then take Dn Y n,i ( x) , i = 1, . . . , D n , 0 , i = D n + 1, . . . , N , where α is a constant. Then φ i φ i = α 2 is indeed a constant on S 6 . The choice ( The on-shell action of these solutions can be obtained by simply plugging (7.15) and (7.16) in the action (2.9) and using −∇ 2 φ i = n(n + 5)φ i . This calculation gives

Instantons close to d = 4
One can similarly find additional classical instantons in d = 4 in the quartic scalar model (2.19). The solutions to the classical equations of motion for the S 4 and R 4 theories are in one-to-one correspondence, and we choose to work on S 4 . Instead of using the quartic action (2.19), it is convenient to write the theory on S 4 with the help of the auxiliary Hubbard-Stratonovich field σ: Performing the path integral over σ (or, at the classical level, solving for σ from its equation of motion and plugging the solution back into (7.20)), one recovers the quartic model in (2.19), so the theories (2.19) and (7.20) are indeed equivalent. The equations of motion following from (7.20) are As in 6-d, we should look for solutions for which σ is a constant on S 4 . 16 The equation The classical action evaluated on the instanton solution presented above can be obtained by simply plugging (7.22) into the action (7.20) and noticing that after using the φ i equation of motion, only the last term in (7.20) survives. Thus, one obtains S n = − 2π 2 (n + 1) 2 (n + 2) 2 3g . (7.24) The analysis above was in the classical theory (7.20)  At leading order in large N , this expression matches (7.13), provided that we identify k = n + 1, as required to match (7.22) with σ = −k(k + 1).

Acknowledgments
We On S 4 × R we write the partition function as where we put tildes on all quantities so we do not confuse them with the analogous quantities in the S d computation.
To place the theory on S d−1 × R we proceed as follows. We use coordinates (t,p) for parameterizing this space, and (τ, ρ,n) for parameterizing the moduli space. Then take: P = sinh t ,p , cosh t , X = cosh ρ sinh τ ,n sinh ρ , cosh ρ cosh τ .

(A.2)
In these coordinates, the instanton profile is The most symmetric configuration is at ρ = τ = 0, where We will do the computations at this point in the instanton moduli space.

A.1 φ determinant
The operator whose eigenvalues we should compute is To evaluate it, we can use the formula that the regularized determinant of the operator .
For the case of interest to us, k = 1, this gives For us, we have modes with a 2 = n(n . Then we find This can be rewritten as Quite nicely, D n (d−1)−D n−2 (d−1) = D n (d−2)+D n−1 (d−2) so this expression is identical to (4.6). Thusf (A.11)

B Computation of one-loop determinants
In order to compute various one-loop determinants it is useful to define a general function where the degeneracies D n (d) are given by D n (d) = (2n+d−1)Γ(n+d−1) . For integer α = k the function f α (d) coincides with the "classical action" of the σ k = −k(k + 1) saddle point in the large N theory. For k = 1 we have the instanton discussed in the main text of the paper, and k = 0 just corresponds to the perturbative vacuum, so that f α=0 is free energy of a free conformal scalar. To compute this function, we can differentiate it by α to obtain Then using the formula ∞ n=0 Γ(n + a) n!
we finally get .

(B.5)
After using the identity Γ(x)Γ(1 − x) = π/ sin(πx), this may be also rewritten as Applying this formula to the case α = 1, corresponding to the σ = −2 instanton, we see that the second integral in the formula above vanishes, and the first integral reproduces the free energy of a free conformal scalar (4.9) on S d−2 , so that in agreement with (4.7).
It is also useful to define another functioñ where the sum goes over n from 2 to +∞ (i.e., in the case of the instanton solution, it excludes the negative and zero modes). The computation off α (d) is similar to the formulas above, we just have to exclud in (B.3) the n = 0 and n = 1 terms. We obtaiñ .
The integral over x may have some divergencies, which should be regularized by taking principal value of the integral. Note that the ratio of determinants log R d , defined in Section 4.3, can be written as (B.10) So, using the integral (B.9) for d = 5 one finds which is the result given in Section 4.3.

B.1 Fluctuation determinants in d = 6 −
We can now apply the above formulas to the calculation of determinants of fluctuations around the classical solution (2.3) in the cubic theory in d = 6− . The analogous calculation for the case of a single scalar field with cubic potential was carried out in [43]. To support the interpretation of the solution (2.3) as the instanton responsible for tunneling from the metastable ground state, it is important to check that even in the presence of the additional N fields φ i , there is still a single negative mode (as well as d + 1 zero modes) as we now show.
Following the discussion in Section 2.1, we know that for the cubic theory, the instanton solution is constant when mapped to S d . Thus, the fluctuation around the classical solutions requires calculating the determinants of the following operators: where σ c = −12/g 2 , as discussed in the Section 2.1, and z = g 1 /g 2 .
Let us first examine the presence of negative eigenvalues. For the σ fluctuations, the calculation is identical to the one in [43], and for σ = 6 − , we find a single negative mode for n = 0. Let us check that there are no additional negative modes coming from the φ fluctuations. The eigenvalues of M φ at d = 6 − are: where we defined ζ = 1 2 √ 1 + 48z. We see that in order for all eigenvalues to be positive, we must require: Or, equivalently, However, the analysis of perturbative fixed points in the cubic O(N ) theory [31,33] shows that the ratio g 1 /g 2 varies from 1/6 at N = ∞ to about 1/8.9 at N = N crit . Therefore, M φ does not contribute additional negative modes.
The determinants of the fluctuation operators may be evaluated explictly using the functions defined in the previous section. Excluding the n = 0 and n = 1 modes which may be treated separately, we can compute the ratio of determinants: Here, the 1/ pole is a UV divergence that comes from the summation over large n (in particular, the n = 0, 1 modes clearly do not affect this UV pole). We can also extract analytically the coefficient of the 1/ pole in the φ determinant. We find, in terms of z = g 1 /g 2 : Since the theory is renormalizable, we expect this divergence to be cancelled by the pertur- where µ is a renormalization scale, and β 2,6d is the beta function for g 2 in d = 6. Using the explicit result for the perturbative beta function found in [31], we get µ (g 2 2 ) bare = 1 g 2 2 + 1 2(4π) 3 3 + N 4 g 3 1 g 3 2 − g 2 1 g 2 2 = 1 g 2 2 + 1 2(4π) 3 3 + N 4z 3 − z 2 .

(B.22)
We can then see that the 1/ pole precisely cancels the one coming from the fluctuation C Thermal mass on S 1 × R d−1 In this section we briefly discuss the calculation of the free energy for the large N critical theory on S 1 × R d−1 , where S 1 is the thermal circle of circumference β = 1/T . We will use a formal dimensional regularization so that all power-like divergences are automatically subtracted away. Starting from the Lagrangian of the critical theory (3.1) and integrating out the φ i fields, we get a path-integral over σ with action S σ = N 2 log det(−∂ 2 + σ) = N F(σ) . (C.1) At large N , we can evaluate the free energy by extremizing with respect to σ, assuming the saddle point occurs for constant σ. Evaluating the φ one-loop determinant one finds where V d−1 is the (infinite) volume of the plane R d−1 . To recover the result for the free theory, one should set σ = 0 in this expression, which gives, using dimensional regularization which is the well-known result. To obtain the free energy for the interacting fixed point, we need to extremize (C.2) with respect to σ. We can compute the σ derivative as Writing coth x 2 = 1 + 2 e x −1 , we have The first integral is convergent for any d, and the second one can be computed by dimensional regularization. After a change of variables in the first integral, we obtain Let us check this result in the case d = 3. Using the integral we get dF dσ = − V 2 4π log 2 sinh β √ σ 2 . (C.8) So we find that the value of σ extremizing F is √ σ * = 2 β log 1 + √ 5 2 (C.9) and, integrating (C.8) in σ, we obtain the free energy of the 3d critical theory which is in agreement with the result of [72].
Let us now consider the critical theory in d = 5. Starting from (C.6), we need to evaluate the integral ∞ β √ σ dy y 2 − β 2 σ e y − 1 = 2Li 3 (e −β √ σ ) + 2β √ σLi 2 (e −β √ σ ) . (C.11) So we get dF dσ = V 4 8π 2 β 2 Li 3 (e −β √ σ ) + β √ σLi 2 (e −β √ σ ) + β 3 σ If we assume that the integration contour can be taken so that it passes through both saddle points, then integrating (C.12) in σ one may get a real free energy. It would be interesting to study the thermal theory further, and clarify the relation to the non-perturbative instability on S d or R × S d−1 that we discussed in this paper.