Conformal bounds in three dimensions from entanglement entropy

The entanglement entropy of an arbitrary spacetime region $A$ in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, $F(A)$. For general theories, the value of $F(A)$ is minimized when $A$ is a round disk, $F_0$, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity $F(A)/F_0$ is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies $C_{ \scriptscriptstyle T} / F_0 \leq 3/ (4\pi^2 \log 2 - 6\zeta[3]) \simeq 0.14887$ for general CFTs. We verify the validity of this bound for free scalars and fermions, general $O(N)$ and Gross-Neveu models, holographic theories, $\mathcal{N}=2$ Wess-Zumino models and general ABJM theories.

The ratio of the trace-anomaly coefficients characterizing any unitary conformal field theory (CFT) in four dimensions is bounded, above and below, by the free scalar and Maxwell field results, respectively [1,2] c a Maxwell ≤ c a ≤ c a free scalar , the numerical values being 18/31 and 3 respectively.Roughly, these "Hofman-Maldacena" (HM) bounds follow from imposing the positivity of the energy flux escaping at null infinity for states resulting from a local insertion of the stress tensor on the vacuum.Analogous constructions in general spacetime dimensions d ≥ 3 give rise to constraints involving correlators of the stress tensor [3,4].For odd-dimensional CFTs there is no trace anomaly and the coefficients a, c are not defined.A somewhat canonical general-dimension version of c is provided by the stress-tensor two-point function coefficient, C T , which is proportional to c in d = 4. On the other hand, a generalization of a which departs from stress-tensor correlators follows from the entanglement entropy (EE) universal coefficient across a round (hyper-)spherical entangling surface, which we denote by F 0 .Again, in d = 4 one finds F 0 ∝ a, and hence the analogy.In odd-dimensional theories, this quantity coincides with the Euclidean free energy on the round sphere, F 0 = − log Z S d [5,6].Also, in d = 3 it defines a renormalization group monotone for general theories [7][8][9].
In this letter we present strong evidence that the quotient C T /F 0 satisfies bounds analogous to (1) for general three-dimensional CFTs, namely, [3] ≃ 0.14887 . . .
(2) These are particular cases of more general conjectural bounds involving the EE of arbitrary regions in d = 3.Given some entangling region A, this is given, for a general CFT, by S 3d (A) = c 0 • perimeter(∂A)/δ − F (A) + O(δ), where c 0 is a non-universal coefficient, δ is a UV cutoff, and F (A) is a dimensionless universal coefficient of non-local nature.Naturally, the round-disk case anticipated above corresponds to F 0 ≡ F | ∂A=S 1 .Recently, it has been proved in [10] that F 0 minimizes F (A) for any given CFT, namely, F (A)/F 0 ≥ 1 with F (A) = F 0 ⇔ A = round disk.Consequently, F 0 provides a canonical normalization for F (A).With these provisos in mind, we are ready to formulate the conjecture which is the central proposal of this letter.
♦ Conjecture: for general CFTs in three dimensions, the universal coefficient in the entanglement entropy, F (A), of an arbitrary region A normalized by the disk result, F 0 , is bounded above by the free scalar result and below by the free Maxwell field one.Namely, we conjecture that holds for general entangling regions and CFTs.
The lower bound is in fact equivalent to the number n of connected components in the boundary of A, and was proved in [10].The Maxwell field saturates the lower bound and so do topological theories.The rest of the letter will be devoted to provide evidence in support of the above conjecture and to extract various consequences.
Hints from four dimensions: In d = 4, the EE universal term is local in nature and appears as the coefficient of a logarithmic divergence.It is given by a linear combination of two theory-independent local integrals over the corresponding entangling surfaces, which appear weighted by the trace-anomaly coefficients a, c.The expression reads [11,12] where W ∂A is the so-called Willmore energy [13] of ∂A and K ∂A is an integral involving a quadratic combination of extrinsic curvatures of ∂A.Observe that we normalized the expression by a following the analogy with the three-dimensional case [14].Now, W ∂A and K ∂A are positive definite and positive semidefinite respectively, so it is straightforward to prove that a conjecture analogous to eq. ( 3) in four dimensions, namely, is trivially equivalent to the HM bounds of eq. ( 1).Since these have been rigorously proven in [2], eq. ( 5) is also true in general.In this case, the local nature of S 4d log (A) limits the content of eq. ( 5) to be exactly equivalent to the one of the HM bounds.On the other hand, our threedimensional conjecture (3) contains much more information, as F (A) does not have a closed geometric expression dependent on just a few coefficients which may be valid for general theories.
On the definition of F (A): Going back to three dimensions, let us start by observing that a direct computation of F (A) from the EE formula using a lattice regularization does not produce unambiguous results in general.This has to do with the fact that it is not possible to resolve the characteristic scales of region A with a precision better than the UV cutoff, e.g., we cannot distinguish R from R(1 + aδ) with a ∼ O(1).This uncertainty pollutes F (A) via the area-law term, F → F −a•c 0 •perimeter(∂A), and the situation cannot be improved by making R larger in the lattice.
In order to define F (A) rigorously we can make use of mutual information [15][16][17][18].Given some region A with characteristic scale R, consider two concentric regions A − and A + with the same shape as A, defined by moving a distance ε/2 inwards and outwards, respectively, along the normal direction to ∂A.Then, the mutual information I(A + , A − ) tends, in the ε/R ≪ 1 limit, to twice the EE of A, providing a well-defined notion of F in the continuum, namely, The robustness of this way of defining F (A) has been previously exploited in several papers [10,18,19] and we will use it henceforth.
Orbifold theories and multicomponent regions: Let us consider the case of orbifold theories O -namely, theories obtained from the quotient of some complete parent theory C by some finite symmetry group G.For them, the mutual information is given by [20] where n is the number of connected boundaries of A and |G| is the number of elements of G.For A ± formed by more than one connected components O are the differences of the relative entropies between the reduced density matrix on the region, and the tensor product of the density matrices reduced on each of its components.By monotonicity these differences are positive semi-definite.
Hence, we can obtain F (A) for a given orbifold theory in terms of the complete theory one using eq.( 6).One finds From this, it is easy to argue that orbifolding tends to decrease the value of F (A)/F 0 , in agreement with our conjecture.Indeed, consider a region with arbitrary topology.In that case, we have The third inequality follows from n ≤ (F (A)/F 0 )| C , proved in [10], the second is a consequence of eq. ( 8), and the first follows from the semi-positivity of S(ρ Hence, the quotient for the parent theory is always greater than the one of the orbifold.Similarly, in all cases the lower bound is provided by the number of connected boundaries of the region.Therefore, as far as our conjecture is concerned, any conclusions which hold for complete theories are also valid for orbifolds of such theories. The same happens for infinite symmetry groups.In that case, log |G| is replaced by a divergent contribution, and the quotient saturates the lower bound appearing in eq. ( 9), namely, (F (A)/F 0 )| O = n.This implies, in particular, that the Maxwell field, which is an orbifold of the free scalar theory under the group R implementing ϕ → ϕ + λ [21] has where n is the number of connected boundaries of the region.More precisely, for the Maxwell field we get ) that diverges with the regularization scale δ [22].The same saturation (10) holds for topological models, for which F (A) = γ n for some constant γ.Hence, the lower bound in our general conjecture (3) is not only consistent but fully equivalent to the general inequality n ≤ F (A)/F 0 .
Let us now try to motivate the upper bound.
Regions with disconnected components and large separations: In case there is a theory which provides an upper bound for F (A)/F 0 for general CFTs and arbitrary regions, this must be given by the free scalar.Indeed, consider an entangling region A consisting of two disconnected subregions where I is the mutual information.Now, assume that A 1 and A 2 are both disk regions.Then, dividing both sides by F 0 and noticing that divergences cancel in both sides of the equality, one is left with notice that in the long-distance regime the free scalar provides the greatest possible value of I(A 1 , A 2 ).Indeed, on general grounds, one has I ∼ ℓ −4∆ where ℓ is the separation between regions and ∆ is the smallest scaling dimension of the model.This is minimized by the free scalar in general dimensions, ∆ free scalar = (d − 2)/2, which saturates the corresponding unitarity bound -see e.g., [23].This means that F (A)/F 0 is absolutely maximized by the free scalar in that case.If one replaces now A 1 and A 2 by arbitrary shapes with characteristic lengths much smaller than ℓ, the inequality eq. ( 3) also holds provided it holds for A 1 and A 2 individually.
Regions with disconnected components and thin deformations on a null cone: Additional evidence follows from the so-called pinching property.Consider the causal cone H associated to some disk region A. Parametrizing the cone by some angular coordinate θ and an affine parameter s ∈ (0, L), where L is the position of A, the region f (δ, ϵ) ≡ {(θ, s)/|θ − θ 0 | < ϵ, s > δ} around some arbitrary direction θ 0 is a sector of a conical frustrum -see Fig. 4 in [24] for a drawing.Then, the region A 1 (δ, ϵ) ≡ H −f (δ, ϵ) corresponds to the causal cone of A with such frustrum removed.A 1 has a boundary which corresponds to the original disk boundary for all θ except for |θ − θ 0 | < ϵ, in which case it is given by the boundary of f .Now, the pinching property establishes that, given some other arbitrary region B, [24][25][26][27] namely, for interacting CFTs -including Generalized Free Fields [28]-the mutual information vanishes identically when we make the tip of f approach the tip of the cone and then we make the conical sector arbitrarily thin.On the other hand, taking the same limits in the case of free CFTs -in the sense of being fields satisfying Quotients c/a in d = 4 and CT /F0 in d = 3 for various CFTs.For visual clarity, each diagram is normalized by the free scalar result.For d = 4, the unitarity bounds are known to be saturated by the free scalar and the Maxwell field, respectively.The holographic result, the free fermion and the EMI model (dashed orange) are also shown.For supersymmetric theories, the band of allowed values is smaller and appears displayed in pale orange.In d = 3 the theories saturating the conjectural bounds are also the free scalar and the Maxwell field, for which CT /F0| Maxwell = 0.Besides the free fermion, the EMI model and holography, we also present the range of values covered by various other theories: the O(N ) models for general N (brown band), the Gross-Neveu models for general N (purple band), the N = 2 Wess-Zumino model with superpotentials X 3 (orange line), X N i ZiZi for general N (pale brown band), SQED (green line) and general ABJM models (pale green band with diagonal lines).Red bands correspond to non-allowed values.
a local linear equation of motion-we are instead left with the mutual information of the original disk region with B. Hence, considering the entanglement entropy for A ≡ A 1 (δ, ϵ) ∪ A 2 where A 2 is a disk region, we have where by A 1 in the second line we mean the disk region which results from fully removing f .This holds regardless of the relative separation between A 1 and A 2 .Hence, it is obvious that in this case F (A)/F 0 is smaller for any interacting CFT than for any free one.If the construction is repeated using regions other than disks, the same conclusion holds again as long as the individual regions satisfy eq. ( 3).Now, eq. ( 12) does not say anything about the hierarchy between the free theories themselves.How-ever, strong numerical evidence suggests that for arbitrary spatial regions A 1 , A 2 [29].Hence, once again we find that the free scalar provides an absolute maximum for F (A)/F 0 in this case.
Small deformations of a disk region: Let us now consider regions with a single connected component.The first obvious case is the one of slightly deformed disks.We can parametrize their boundary by the radial equation where ϵ ≪ 1.Then, at leading order in ϵ, we have [30][31][32] F (A) where C T is the coefficient which controls, for a general CFT, the flat-space stress-tensor two-point function charge, namely, where I µν ≡ δ µν − 2x µ x ν /x 2 .Now, noting that the coefficient which accompanies ϵ 2 is positive semidefinite, applying our conjecture (3) to the deformed-disks case we are left with a conjecture for the quotient of charges C T /F 0 , namely, with eq. ( 2).In that expression, the lower bound becomes trivial, as for the three-dimensional Maxwell field this quotient simply vanishes.As anticipated in the introduction, an inequality of this type is highly reminiscent of the four-dimensional HM bounds for the trace-anomaly coefficients c/a -see Fig. 1.
As it turns out, both C T and F 0 have been computed for a plethora of three-dimensional CFTs and we can test the validity of eq. ( 2) in all those cases.In the appendix we have gathered the results, and in Fig. 1 we have plotted them together.We observe that all considered theories satisfy the conjectural bounds.In particular, one finds a similar hierarchy as in the four-dimensional c/a case, with the free scalar [9,[33][34][35] representing the upper bound, the free fermion [9,[33][34][35] taking a lower value, holographic theories [36,37] an even lower one, and the Maxwell field providing the lowest possible one (zero in the three-dimensional case).Explicitly, we have: where we have also included the result corresponding to the so-called "Extensive Mutual Information" (EMI) model [16,38].
Amongst the interacting theories considered, we have the Gross-Neveu O(N ) UV fixed points models [9,39,40], for which it is possible to find values of N which are both FIG. 2. We plot the EE universal coefficients corresponding, respectively, to a corner region of opening angle θ, and an ellipse of eccentricity e -both normalized by F0-as functions of those parameters for: a free scalar (blue), a free fermion (red), the EMI model (dashed orange) and holographic Einstein gravity (gray).In all cases, the free scalar one lies above the curves of all the rest of theories.The Maxwell field is a trivial lower bound of constant value (0 and 1, respectively) in both cases.
greater and smaller than the free fermion one, the whole range being 0.0854 ≲ C T /F 0 | GN, O(N ) ≲ 0.094 ∀ N .On the other hand, for the Wilson-Fisher fixed points of the scalar O(N ) models [9,39,[41][42][43], one finds that the free scalar result is always an upper bound, for arbitrary values of N .The range is 0.1409 ≲ C T /F 0 | O(N ) ≤ C T /F 0 | free scalar ∀ N .Additional theories considered include various supersymmetric N = 2 Wess-Zumino models as well as general U (N ) k ×U (N ) −k ABJM models [44], for which we find -using results from [45][46][47][48][49][50][51][52][53]-that 0 ≤ C T /F 0 | U (N ) k ×U (N ) −k ABJM ≤ 3/(2π 2 log 4) ≃ 0.10963 for all N and k -see Fig. 3 in the appendix.In all cases, the conjectural bounds are respected.It would be certainly interesting to test the conjecture for additional theories.
Ellipses and corners: Moving from the perturbeddisks regime, values of F (A)/F 0 for more complicated regions exist in some cases, at least for a few theories.In particular, there exist results for free scalars and fermions, the EMI model, as well as for holographic theories in the case of ellipses of arbitrary eccentricity [10] [54].The results are shown in Fig. 2, where it is clear that the free scalar always takes the greatest value.The lower bound provided by the Maxwell theory is always trivially satisfied by all theories since one has (F (e) /F 0 ) Maxwell = 1 ∀ e.
In Fig. 2 we have also presented results for the same set of theories in the case of corner regions of opening angle θ [35,[55][56][57][58][59][60][61].In that case, F (A) builds up a logarithmic divergence weighted by some function a(θ) which, normalized by F 0 , inherits the same hierarchies as in eq. ( 3).Again, the free scalar curve -which also coincides with the one of the large-N limit of the Wilson-Fisher O(N ) model [62]-is above all others.On the other hand, there exists a general lower bound for a(θ) constructed in [63] and given by a(θ)/F 0 ≥ π 2 C T 3F0 log[1/ sin(θ/2)].In the case of the Maxwell field, the right-hand-side is just zero, so again we find consistency with the lower bound in eq. ( 3).Computations of the corner function with θ = π/2 have been performed using numerical methods for the O(N ) models with N = 1, 2, 3 [64][65][66][67][68].In all cases, the result is very close to the free scalar one, but the precision achieved does not seem to allow for a trustworthy quantitative comparison [69].
Both for ellipses and corners, the behavior in the regime in which the region becomes very sharp -i.e., for e, θ → 0, respectively-is controlled by the universal coefficient characterizing the EE of a thin strip.Given such a strip of dimensions L, r with L ≫ r, one finds The coefficient κ is yet another quantity characterizing any given three-dimensional CFT.It is not known to be related with any other coefficient defined beyond EE, so our general conjecture (3) predicts additional independent bounds on the possible values of κ/F 0 .Using the free scalar values of κ computed in [35], one finds The values of κ are also known for free fermions [35], the EMI model [16], as well as for holographic theories dual to Einstein gravity [70].In each of those cases, one finds ] 2 ≃ 0.2285, always in agreement with eq. ( 18).Naturally, using eq.( 18) we can obtain putative bounds for κ for any CFT for which F 0 is known.Evaluating this coefficient for additional theories would be another way of testing our general conjecture.
Discussion: In this letter we have presented evidence in favor of a new conjecture for the EE universal coefficient of general three-dimensional CFTs.As we have seen, the conjecture fits very well with previous results like the HM bounds in d = 4 as well as with the fact that F (A)/F 0 is bounded below by the number of connected boundaries of A for general theories.Naturally, it would be very interesting to find a proof (or a counterexample) to our conjectures.This would entail a better understanding of what makes the free scalar theory special from an entropic point view.
An obvious question is whether our conjecture may also extend to higher dimensions.In d = 5, an analogous putative upper bound corresponding to a free scalar would imply -via the perturbed spheres EE [31], The analogous bound on the strip coefficient would be In both cases, the lower bound provided by the Maxwell field would always be trivially satisfied, since F 0 diverges for that theory [71].It is easy to check that both eq.( 19) and eq.( 20) are satisfied for free fermions as well as for holographic theories.A related question is whether or not the round S 3 is the entangling surface with the smallest F (A) in d = 5.A study of the d = 6 case would also be interesting.This would be trickier than in d = 4 since there are four trace-anomaly coefficients rather than two, and the geometric expression of S 6d log (A) is considerably more involved [72,73].

Free theories
The exact values of C T and F 0 for free scalars and fermions are given by [9,[33][34][35] so the ratios are

Extensive mutual information model
The "Extensive mutual information model" [16] follows from considering a general expression for the mutual information that satisfies all the known general axioms for this quantity in a general QFT [38].In additional to these, it satisfies the condition of being an extensive function of its arguments.This is equivalent to impose that the tripartite information vanishes for any regions A, B and C, this is, The model is equivalent to a free fermion in d = 2 [74] but the identification is lost for higher dimensions [38].This can be easily checked from the inequivalence of the quotients c/a and C T /F 0 displayed in Figure 1.For the latter, the numerical value is given by In spite of differing from an actual theory for d ≥ 3, the EMI model still satisfies all known requirements for a valid mutual information, providing an useful toy model for many purposes -see e.g., [ .

Theories with a holographic Einstein dual
For holographic theories dual to Einstein gravity in the bulk, one finds [36,37] where L ⋆ is the AdS radius and G is the Newton constant.This result is reproduced by supersymmetric gauge theories with a holographic dual in the large N limit.

O(N ) models
There are also available results for the Wilson-Fisher fixed points of the O(N ) model both for large N and for small values of N .We have for the first two orders in the large-N expansion [9,41] and from this Hence, the result falls inside the window allowed by the conjectural bounds.For finite values of N , the result becomes smaller and tends to deviate from the free scalar result.
again within the range.

Gross-Neveu models
There are also results for the UV fixed point of the Gross-Neveu models.In particular, in the large limit we have [9,40] and hence which is also comfortably within the range.In order to probe small values of N , we can use the approximation found in [39] and the results obtained in [40] for C T | GN, O(N ) .One finds, for instance which once more lie within the range.The values tend to grow slightly as N increases, but they stop doing so at some point -the maximum seems to be (conservatively) lower than 0.094.For instance, for N = 100 one finds, from the large N approximation, Observe that as we vary N , in some cases the value of the ratio is greater than the free fermion one, and in others it is smaller.This is different from the situation encountered for the O(N ) models, in which case the ratio is smaller than the free scalar one ∀ N .The free energy and central charge corresponding to the critical Wess-Zumino model, also known as supersymmetric Ising model [78], -which has a cubic superpotential W = X 3 -are given by [39,79] which falls within the range.On the other hand, for the critical point of a Wess-Zumino model with superpotential W = X N i=1 Z i we have, in the large N limit [79] For small values of N , on the other hand, we have [79] As we can see, in all cases the values fall within the range.The case N = 2 is particularly interesting, as it corresponds also to the XY Z model [39] -defined by a superpotential W = XY Z-which is in turn mirror symmetric to N = 2 SQED [80,81].For the latter two models, we have [82-85] where we used the definition [82] ℓ From this, one finds which agrees well with the result from [79] for the XZ(2) model presented above.We also have results for the fixed point corresponding to a superpotential W = The results are actually identical to the ones obtained for N free chiral multiplets and also satisfy the bounds.
As N increases, the upper bound tends to the holographic value (A5) -shown in gray.For clarity we also show the quotient corresponding to the free scalar We can also check the bound in the U (N ) k × U (N ) −k ABJM theory [44].Using results from [45][46][47][48], for N = 1 we find where C T | U (1) k ×U (1) −k ABJM does not depend on k.Observe that at large k, the quotient tends to 0, saturating the lower bound (2).For non-Abelian ABJM theories, the explicit expressions of C T and F 0 become more complicated.Using those computed in [48] for N = 2 we find where, again, as k grows, the quotient approaches 0. In both instances, we observe that the bound is satisfied.We also checked explicitly that this is true for N = 3, 4 with k = 3, whose expressions for C T and F 0 can be found in [48][49][50].On the other hand, in the large N limit, the values of both C T and F 0 are expressed in terms of the Airy function [51][52][53].In this case, the quotient tends to the holographic one (A5), as checked for several values of small and large k.In Figure 3 we plot some values of C T /F 0 U (N ) k ×U (N ) −k ABJM corresponding to the cases mentioned above.

Holographic higher-curvature theories
Consider a holographic higher-curvature gravity in four dimensions defined by a Lagrangian density where L is a length scale, G is the Newton constant, µ n are dimensionless couplings and R (n) are higher-curvature densities of order n.The result for F 0 can be obtained by evaluating the Lagrangian on the AdS background as where L(x) stands for the on-shell Lagrangian of the corresponding theory evaluated on pure AdS with radius L ⋆ , which we expressed in terms of the quotient x ≡ L 2 /L 2 ⋆ .The AdS radius and the action scale L are related on-shell by the equation [90] where, in order to take the derivative with respect to x, L(x) must be expressed in terms of powers of x up to an overall 1/L 2 .For instance, for pure Einstein gravity one has L(x) = 3(1−2x)/(8πL 2 G) and xL ′ (x)/2 = −3x/(8πL 2 G) so eq.(A30) simply implies x = 1, i.e., L = L ⋆ .On the other hand, for higher-curvature theories which have second-order equations on maximally symmetric backgrounds, C T can be computed as [91] Hence, for the ratio of interest we find where we made use of eq.(A30).Naturally, for pure Einstein gravity, this reduces to eq. (A5), since in that case L ′′ (x) = 0. Using our conjectural bounds (2), one finds the following conditions on the higher-curvature couplings of this class of theories More generally, the bounds can be used to put constraints on the possible values of the higher-curvature couplings of other holographic theories for which C T and F 0 are available -e.g., [92,93].Note however that the meaning of such bounds is somewhat unclear since, generically, higher-curvature theories give rise to instabilities when considered beyond the perturbative effective field theory regime.
FIG. 1.Quotients c/a in d = 4 and CT /F0 in d = 3 for various CFTs.For visual clarity, each diagram is normalized by the free scalar result.For d = 4, the unitarity bounds are known to be saturated by the free scalar and the Maxwell field, respectively.The holographic result, the free fermion and the EMI model (dashed orange) are also shown.For supersymmetric theories, the band of allowed values is smaller and appears displayed in pale orange.In d = 3 the theories saturating the conjectural bounds are also the free scalar and the Maxwell field, for which CT /F0| Maxwell = 0.Besides the free fermion, the EMI model and holography, we also present the range of values covered by various other theories: the O(N ) models for general N (brown band), the Gross-Neveu models for general N (purple band), the N = 2 Wess-Zumino model with superpotentials X 3 (orange line), X N i ZiZi for general N (pale brown band), SQED (green line) and general ABJM models (pale green band with diagonal lines).Red bands correspond to non-allowed values.
For instance, for N = 20 one finds C There are also available results for lower values of N .For the Ising model, corresponding to N = 1, one has[39,42,43] C T | Ising ≃ 0.9466 C T | free scalar , F 0 | Ising ≃ 0.957 F 0 | free scalar , so C T F 0 Ising ≃ 0.1468 ,(A9)which lies again within the range.Similarly, for N = 2 and N = 3 one finds[39,43]

FIG. 3 .
FIG. 3. Values of /F0 U (N ) k ×U (N ) −k ABJM for several values of N and k, namely: i) for k = 1 and k = 2 we represent the quotients for N = 1, . . ., 10 as circles and triangles respectively.ii) for k = 3 we include the quotients for N = 1, 2, 3 and 4 with squares.iii) For k = 10, 20 and 30 we plot the values N = 1, 2, 8, 9, 10, represented by pentagons, hexagons and heptagons respectively.All values we explored are contained in the blue region, whose lower bound is given by CT /F0 U (N ) k ×U (N ) −k ABJM ≥ 0 -coinciding with the value for the Maxwell theory shown in green-and its upper bound is