Positive geometry in the diagonal limit of the conformal bootstrap

We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the geometric understanding of the bootstrap equations for unitary theories leads to cyclic polytopes for which the faces can all be written down and, in principle, the intersection between the unitarity polytope and the crossing plane can be systematically explored. We find that in higher dimensions, the natural structure that emerges, due to the inclusion of spin, is the weighted Minkowski sum of cyclic polytopes. While it can be explicitly shown that for physical theories, the weighted Minkowski sum of cyclic polytopes is not a cyclic polytope, it also turns out that in the large conformal dimension limit it is indeed a cyclic polytope. We write down several analytic formulae in this limit and show that remarkably, in many cases, this works out to be very good approximation even for $O(1)$ conformal dimensions. Furthermore, we initiate a comparison between usual numerics obtained using linear programming and what arises from positive geometry considerations.


Introduction
In a CFT, the four point function of identical scalars with dimension ∆ φ can be written as sum over conformal blocks of exchange operator O ∆, of scaling dimension ∆, spin with OPE coefficient and C ∆, > 0 for unitary theories. Crossing symmetry implies This is the usual starting point for modern studies of the conformal bootstrap. One can consider the diagonal limit of this equation which is obtained by setting u = zz, v = (1 − z)(1 −z) and taking the limit z →z. Thereby we obtain a problem in 1-variable. Following the numerical approach based on linear programming, pioneered in [1][2][3] and recently reviewed in [4], one can study this equation in any dimensions. Somewhat surprisingly 1 , for d = 2, retaining only scalar operators one finds a "kink" in the plot of the allowed leading scalar operator dimension when plotted against ∆ φ .
In figure 1, we give a comparison between the numerics obtained in the diagonal limit and what is obtained in the full problem 2 . The kink in the diagonal problem is at ∆ φ ≈ 0.09, ∆ 2 ≈ 1.2 and adding other operators with spin does not appear to change the location drastically (although we have not attempted to make a thorough comparison). The full problem [5] in which there are off-diagonal derivatives, needs the inclusion of higher spin blocks before a kink appears. Furthermore, compared to the full problem where the kink appears at ∆ φ ≈ 0.125, ∆ 2 ≈ 1, the kink in the diagonal limit is displaced. Nonetheless, one can ask what causes the kink in this instance. In the full problem as reviewd in [4], the kink arises due to decoupling of certain operators (which happens for physical theories like the 2d Ising model due to null state conditions).  Figure 1: Plot of bounds on dimension of leading scalar in the OPE of φ × φ for d = 2 using linear programming, keeping only scalars in the spectrum in the diagonal limit. The blue line is for 10 derivatives in the functional while the black is for 20 derivatives. In each case truncating the spectrum does not change the plot much. The green line is obtained in the non-diagonal limit [5], including spins, while the red cross is the location of the 2d Ising model. The red line indicates what arises from considering the intersection of the N = 2 cyclic polytope obtained using scalars in d = 2 and the crossing plane.
Thus the diagonal limit (being somewhat simpler since it is a one variable problem) appears to be a good playground to investigate various features of the conformal bootstrap. Recently the one variable bootstrap in d = 1 has been studied and a deep connection with the alternative approach to the bootstrap advocated in [6][7][8][9][10][11][12] and based on crossing symmetric Witten diagram basis expansion [8,12], has been found [13]. Another development that has happened last year is an attempt to connect the usual conformal bootstrap program to the positive geometry of cyclic polytopes [14]. In this paper, we will endeavour to further this connection.
The work of [1] already made use of the polytope structure underlying the bootstrap equations.
The idea was to rewrite the bootstrap equations in the form ∆, C ∆, F d,∆, (z,z) = 1 , (1.4) where F d,∆, 's are certain linear combinations of the direct and crossed-channel blocks given by where u = zz and v = (1 − z)(1 −z). Then, taking derivatives of the above equation around z,z = 1 2 , one assumes a certain spectrum where the lowest scalar has dimension ∆ S and asks if the crossing condition can be satisfied. Using linear programming, one can rule out potential spectra and obtain bounds on ∆ S amongst other quantities. The starting point in [14] is somewhat different. One first expands the correlator (in [14] it was done in d = 1), in terms of the direct channel blocks. The key novel insight provided in [14] is that if ∆ is not too small, then this expansion has the structure of a cyclic polytope. The reason why this is important is that the face structure of the cyclic polytope has been fully classified. This enables one to study the intersection of the crossing plane with the polytope systematically and obtain various non-trivial insights about a potential CFT satisfying the bootstrap constraints.
In this paper we will extend the work of [14] in several directions. We will consider the diagonal limit of the bootstrap equations in arbitrary dimensions. Rather than an expansion in terms of SL(2,R) blocks, this entails an expansion in terms of the so-called diagonal blocks worked out in [15].
We will show that one can work out a systematic 1/∆ expansion of the blocks 3 which enables an analytic study of the results of [14]. Furthermore, the cyclic polytope picture in [14] enables us to initiate a comparison of how the kink in fig.1 is obtained in the approach of [1] and from the cyclic polytope perspective. Compared to the d = 1 study in [14] where the sum over spectrum only involved the conformal dimension, here we will also have to sum over spin. This creates an interesting difficulty.
While for each spin, there is a cyclic polytope structure, when we sum over spins we get what is called a weighted Minkowski sum of cyclic polytopes [17]. The resulting polytope is not necessarily a cyclic polytope and one can easily check that for known CFTs like the 2d/3d Ising models, it is not. For low number of vertices, one can of course grind out the face structure. But that would miss the point made in [14]. What we find however, is that in the large ∆ limit the weighted Minkowski sum is again a cyclic polytope! This means that at least in this limit, we can begin understanding (even analytically) the bootstrap problem using the positive geometry of cyclic polytopes as in [14]. 3 The efficacy of the 1/∆ expansion has also been recently discussed in [16]. This paper is organized as follows. In section 2, we review known results about the diagonal limit of conformal blocks in arbitrary dimensions and present formulae for the leading order large ∆ limit of the blocks. In section 3, we review salient features of polytopes, cyclic polytopes and conditions for the intersection of the crossing plane with the polytope. In section 4, we investigate the positivity criteria for cyclic polytopes both numerically and analytically. In sections 5 and 6 we consider the N = 1 and N = 2 polytopes and present asymptotic formulae for the bounds in [14] as well as analogous results in higher dimensions. In section 7 we initiate a study of numerical bootstrap using the cyclic polytope picture but closer in spirit to the approach [1] which used linear programming.
We conclude in section 8. Some appendices support various calculations in the main text. We will reserve the notation d for spacetime dimensions and D for the dimensionality of the polytope.

Diagonal limit of conformal blocks
In order to use the technology presented in [14] in higher dimensions, one would like to consider the diagonal limit of higher dimensional conformal blocks. This was worked out in [15].

Exact blocks
Unlike d = 1, where there are no spins, the higher dimensions the spectrum of operators are labelled by two quantum numbers: ∆ and and hence a sum over these two quantum numbers 4 , . (2.1) The 3 F 2 above can again be of course again decomposed in terms of a sum over the SL(2, R) blocks.
We will not however venture in that direction. We note specifically, for the scalar operators, .
We plot the second derivative of F d,∆, as a function of z in fig.2   at least one operator with ∆ =2 ≤ ∆ T for some ∆ T without necessarily having a scalar ∆ ≤ ∆ S .

Asymptotic blocks
In this paper, at various stages we will use the large-∆ limit of the diagonal conformal blocks. The result we will need was worked out by G. N. Watson in 1914 and can be found in [18]: where Z = z 2 4(z−1) . In terms [15] of Z = − 4ρ 2 (1−ρ 2 ) 2 , or z = 4ρ (1+ρ) 2 , with 0 ≤ ρ ≤ 1, this is very simply Under the crossing transformation z → 1 − z, ρ → This variable naturally arises in the large ∆ asymptotics [1]. The blocks are analytic in the z plane except for the cut (1, ∞). ρ maps the cut plane to a unit disc and can also be thought of as lightcone coordinates in higher dimensions [19]. The general form is that the 1/∆ n term in the series inside the bracket comes with p n (ρ 2 , r, d)/(1 − ρ 2 ) n where p n (ρ 2 , r, d) is a polynomial in ρ 2 , r, d of degrees n, n, 2n respectively. Many subleading orders can be systematically worked out following [18]. An interesting thing to notice is that the sub-leading term for r = 0 changes sign when d > 6. We do not know if this is of any significance. Now using eq.(2.3) in eq.(2.1), we get where Z = z 2 4(z−1) . Then doing the r rum and using the Saalschutz formula we finally get to the leading order, In terms the ρ variable, we note We plot below the ratio of exact and asymptotic blocks. From fig.(3) it is clear that eq.(2.6) is a reasonable approximation (upto 1% error for even ∆ ∼ O(1)). In many calculations below 5 , we will make use of these asymptotic blocks and derive analytical formulae which explain several numerical results arising from using the exact blocks.
3 Polytopes: Unitary CFT and crossing equation

Usual approach of CFT
In this section, we will summarize certain key geometric ideas given in [14] as well as [1] which will be used in the paper. To start with, we consider the geometric meaning of the sum rule presented in eq. (1.4) in the diagonal limit.  The vector space is made of even derivatives 6 , while the odd derivatives vanish due to crossing symmetry in (3.1). In terms of the basis, (3.1) can be written as a set of infinite relations for m ≥ 0, The key idea is to identify the linear functional, such that, (1.4) can be checked via numerical methods such as Linear Programming. Refraining from the details which are explained nicely in [1], the essence of the approach is: • Assume a spectrum consistent with unitarity and assuming that scalars have dimension ∆ ≥ ∆ S .
• A solution to the linear programming problem gives a set of λ 2m such that ∆, C ∆, λ 2m F 2m ∆, > 0. In other words for such a spectrum it is not possible to satisfy eq.(3.3) since this is nothing but a linear combination of the equations in eq.(3.3).
• Reduce ∆ S such that no solution to the linear programming problem is found. This corresponds to a potentially allowed spectrum.

Positive geometric approach of CFT
The approach taken by [14] is somewhat different and makes use of the known faces of what is called a cyclic polytope. Instead of starting with eq.1.4, one can equivalently consider the convex cone C or more specifically the convex hull of vectors built from the conformal blocks themselves.
Specifically, one truncates the Taylor series of the four-point function, which in turn is expanded in terms of the direct channel conformal blocks, upto some say 2N + 2 terms around z = 1/2 . Therefore A lies in the positive span of the block vector G d,∆, where C ∆, > 0 for unitary theories and, The dimensionality D of the polytope is D = 2N + 1. To make P 2N +1 explicit, we note that the above equation 3.6 tells us that since we can trivially multiply both sides by a non-zero number (absorbing this into the C ∆, 's on the rhs), we can equivalently consider the expansion of A → t A.
Similarly the cone spanned by G d,∆, is same as G d,∆, → α ∆, G d,∆, with α ∆, > 0 since we can again absorb this into the OPE coefficients. In other words we can expand Now plugging back in to the equation 3.6 gives α ∆, C ∆, ≡ C ∆, = t. Therefore we can write the equation 3.6 as . Also note that ∆, λ ∆, = 1 which defines convex hull of G d,∆, which is a polytope in R 2N +1 . Therefore more generally we can think of the equation (3.6) as projective polytope It is also possible to define the polytope by a collection of hyperplanes W aI , the facets, that cut out the polytope by the linear inequalities. Here W a defines a collection of hyperplanes and outline the structure of the polytope. It is difficult to check whether a given vector lies inside or outside a polytope. This is because the representation of A is non-unique. By contrast the facet definition does allow us to practically check whether or not a given A is inside or not by simply checking the inequalities. Now given a polytopes in P D , the facets for the convex hull of a set of vectors v i 7 are given by The corresponding facets defining the boundary of the polytope are, There is a class of polytope which vertices have an ordering v 1 , . . . v n where n can be larger than D, such that [14].
Such structures can be analytically computed for any P D and form the cyclic polytope. The facets of a cyclic polytope are given by [14] D odd: . (3.14) 7 v i or simply " i " in determinants are short hand notation for the components of G d,∆i, i

Positivity of the blocks
From the point of view of the CFT spectrum, the block vectors can be ordered simply in terms of increasing ∆. This would mean that the spins are not necessarily ordered. Additionally there are two more vectors v 0 = (1, 0, . . . , 0) and v ∞ = (0, . . . , 0, 1) which represent the identity operator and the ∆ = ∞ operator as a part of the spectrum. In order to show that block vectors G d,∆, form a cyclic polytope we proceed as below: First we have an ordered set of vectors Then in terms of the condition for positivity discussed in [14], the condition for a cyclic polytope is In order to proceed from here we define, We can show that the condition for positivity discussed in [14], can be written in a more generic format, given by 8 , where |K i | gives the determinant of a i × i matrix and |K i | = 1 for i < 0. With this, as we have checked rigorously, all the positivity criterion discussed in [14] can be reproduced viz.
and so on.

The crossing equation
So far we have discussed the notion of cyclic polytopes which is relatively simpler to compute analytically and associated with the notion of a collection of block vectors given in equation (3.6). However in the numerical bootstrap, the convex hull of vectors is defined with respect to the crossing equation and not directly with the blocks themselves. In order to connect the concept of cyclic polytope to bootstrap, we will need to put in the ingredient of "crossing" in the projective picture as well.
Crossing in the diagonal limit reads This relates odd A 2n+1 in terms of the even ordered derivatives A 2n . This in turn defines a hyperplane X[∆ φ ] which is a (2N + 2) × (N + 1) matrix in P 2N +1 . For example [14], for N = 2 or Now we have both ingredients for bootstrap in the projective picture.
• A convex hull of vectors built from 2N + 2 components of a block with a natural notion of ordering. This defines a cyclic polytope in P 2N +1 . Call this U[∆].
The consistent solution of bootstrap entails finding U The question now is given a , what are the conditions needed for the intersection with X[∆ φ ]. As argued in [14], a k−plane intersects with a D−dimensional polytope with a D − k face at a point given by, For a point inside the polytope and satisfying the intersection property above, A further simplification occurs when one of the vertex vectors is the identity operator v 0 or the infinity operator v ∞ since this reduces the dimensionality of the problem. Another simplification is when projecting through specific planes obtained by performing a GL(D + 1) transformation, In terms of the intersection problem, X , . . . reduces to a D−k+1 dimensional determinant, reducing the rank by k − 1.

Positivity criterion for U[∆ i ]
Here we discuss the positivity criterion of U[∆ i ] in diagonal limit. As mentioned in (3.18), a sufficient condition for positivity of the block vector is g i > 0. Note however, that g i < 0 does not necessarily mean that the cyclic polytope structure is absent; it simply means that in this case one needs to check (3.15) directly.

Scalar blocks
We plot below numerical evaluation of g i for the scalar block. From figure [4] it is clear that: 2 . The minima of the negative g i 's shift upwards for increasing i 9 . From ∆ d−2 2 , analysis we can track the ∆ for which the minimum goes above the axis, which is indicated approximately in figure 4.

limit anaytic results
We can explain some of the features of g i analytically by using the asymptotic blocks in section 2.2.
For the scalar block we can parametrize the large ∆ block as: Using this and after coaxing mathematica a bit, we find For d = 2 we have not noticed any feature like this. g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 (a) g i vs ∆ for d = 1 g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 (c) g i vs ∆ for d = 3 g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 (d) g i vs ∆ for d = 4 Notice that the correction begins at 1/∆ 2 . This leads to the conclusion that the n-dependence is only through the last term in the above equation and the rest of the pieces are universal. Also observe that B(z) in the block does not enter the final expression. We find This explains why in the numerical plots, we found g i 's approaching 2 √ 2 ≈ 2.828. Furthermore, since for non-zero spins, the leading order block is the same (up to some overall dependent factor which drops out in g i ), for non-zero spins too we will find the same result at leading order, namely g i ≈ 2 √ 2.
See Appendix A for derivation in general case. This essentially means: For large ∆ operators, the structure of the conformal block expansion is that of a cyclic polytope in any dimensions.
We then explicitly have for the = 0 case This means that the universal part is while the non-universal n-dependent part is To be explicit Notice that c 1 ( 1 2 ) changes sign at d = 6. Furthermore notice that

Weighted Minkowski sum
The Minkowski sum of set of vectors X i , i = 1, 2, . . . , n , is defined as the set of vectors which can be written as the sum of a vector in X i [17] X 1 + X 2 + · · · + X n := {x 1 + x 2 + · · · + x n |x i ∈ X i , ∀i} (4.11) Note that Minkowski sum of polytopes is also a polytopes. Now if P i , i = 1, 2, . . . , n are polytopes Δ g i , d=2, ℓ=2 g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 Figure 5: g i vs ∆ for d = 2, = 2 in R d then their weighted Minkowski sum P (λ i ) is defined as [17] P (λ i ) = λ 1 P 1 + λ 2 P 2 + · · · + λ n P n ; λ i ≥ 0, ∀i (4.12) As we mentioned the condition g i > 0 is sufficient but not necessary. When g i < 0 we resort to We have found for each spin upto some small caveat pertaining to scalars close to the unitarity bound (see appendix (E)) that there is a cyclic polytope structure for unitary theories. After adding spin, the resulting polytope is P (c ) = c 0 P 0 + c 2 P 2 + c 4 P 4 + . . . , where P is the cyclic polytope structure following from the spin-block vectors. So we get the weighted Minkowski sum [17] of cyclic polytopes which itself is not necessarily a cyclic polytope (as we explicitly demonstrate in the Appendix B). However if we work with the approximate blocks, as we have shown above, the result is again a cyclic polytope irrespective of spin sum.
This is the motivation for looking at the large ∆ limit in order to derive some intuition about what the cyclic polytope picture can teach us about bootstrap.

N = 1: Bounds on leading operator
A general constraint arising from U{∆} ∩ X[∆ φ ] is then given by a 2n × 2n matrix whose last n entries are the columns from the basis vectors of the unitary polytope. As a simple example, we can demonstrate some generalizations of the bounds for N = 1 in (projecting through identity) in [14, eq. 7.9] for higher dimensions. The crossing plane X is one-dimensional, The polytope is 3-dimensional. The facets can be found by using (3.14) Here the identity operator G d,∆ 0 ,0 = (1, 0, 0, 0) and ∞ is G d,∆∞,0 = (0, 0, 0, 1). No other operator in between ∆ i and ∆ i+1 and we order them as ∆ i < ∆ i+1 . The crossing plane X intersects with the facets if and only if equation (3.23) is satisfied. As we mentioned earlier it is sometimes simple and useful to take the projection through some special point [14].

Projection through identity
We project our three-dimensional cyclic polytope through the identity G d,∆ 0 , 0 = (1, 0, 0, 0) . As mentioned in [14] that the necessary conditions one can get by forcing X, 0, ∆ to be zero . The crossing plane intersects the unitary cyclic polytope at two points which are given by the two solutions This tells us that there must exist at least one operator with dimension ∆ such that This also implies that the gap between the identity and the leading operator has to be smaller than ∆ + . Below we tabulate ∆ + , ∆ − for various ∆ φ : It is possible to further constrain the ∆ + and ∆ − including spin analysis from the technique of N = 2 analysis given in section (6). We will study this in detail in a forthcoming publication [21]. In   Table 4: List of ∆ + , ∆ − for various ∆ φ , d = 4 appendix (D) we have made a preliminary study of the effects of adding spin to the analysis of bound on the dimension of the leading scalar.

5.2
∆ + and ∆ − for large ∆ φ Using eq (2.1) with the approximations of the 3 F 2 given in section 2.2, 10 we can find analytic solutions for ∆ + and ∆ − for large ∆ φ : This formula is even very good approximation for ∆ φ > 1. For comparison we give a plot for ∆ + and ∆ − vs ∆ φ calculated using exact block and eq (5.5). 10 For O( 1 ∆ n φ ) one has to add O( 1 ∆ n ) corrections to eq (2.6) using [18].
We can also expand around ∆ Φ = ∆ φ + a One can easily see from Figure (6) that eq (5.6) with a = 0 i.e 5.5 is very good approximation for ∆ φ > 1, while it deviates a lot from the exact numbers for ∆ φ < 1. Also 5.6 is very good approximation even for ∆ φ < 1 for the choice of a = 1. One can optimize the value of a for better numerical agreement for ∆ φ < 1, which we have not done here. where F d,∆, (z) is given in equation (3.1). We can write ∆,

Interpretation of ∆ + and ∆ − from numerical bootstrap
From figure (8), it is clear that ∂ 2 z F d,∆,0 (z)| z=1/2 changes its sign at indicated values. At least there should be one operator below ∆ + (the larger value) in order to satisfy equation (5.8).
A peculiar feature that is captured by both the analysis we are doing in this section as well as the polytope method, is that there is a third root of the N = 1 determinant (or equivalently ) which allows for a small region d−2 2 < ∆ < ∆ * close to the unitarity bound as an allowed region. However, it seems to us that this is somewhat artificial and we will not comment on this further. The main point is that there could always be operators below ∆ − as well, but there must be at least one operator (which need not be the leading scalar operator in the OPE which happens if  there are operators 11 ∆ < ∆ − ) such that ∆ − < ∆ < ∆ + for there to be any chance for a solution.

N = 2: Constraints on the first two primary operators
In this section, we consider N = 2 i.e., the 5-dimensional polytope. The crossing plane is given in equation (3.21) and the boundaries can be found using 3.14. As before, for simplicity and usefulness 11 A simple example is the case where in a free theory in d = 4 we have the external operator to be φ 2 so that the leading scalar operator is φ 2 itself. From table (4), it is clear that since ∆ φ 2 = 2 we have ∆ φ 2 < ∆ − .    See table 1 and table 4 for values of ∆ + and ∆ − . The indicated values in the plot is exactly matched with the value given in the tables we will view the geometry projected through identity [14]. The constraints for the first two operators ∆ 1 , ∆ 2 is as follows. For ∆ 1 < ∆ − we should have ∆ 2 < ∆ + , otherwise there will be no operators between (∆ − , ∆ + ) as we observed in the N = 1 case. And also ∆ 1 > ∆ + is not allowed if ∆ 1 is the leading operator. As explained in [14], the necessary conditions one would obtain from the observation that for ∆ − < ∆ 1 < ∆ + , ∆ 2 must be below the curve X, 0, ∆ 1 , ∆ 2 = 0 (see figure (9) 3.23 is not satisfied. Explicitly We show in fig. 9 how the curve X, 0, ∆ 1 , ∆ 2 = 0 changes if we put the second operator to be one with spin (there is an assumed ordering and in this instance we are assuming that the second operator is one that carries spin, assuming that the cyclic polytope structure is preserved). In figure (9.a) we have taken ∆ φ = 2, d = 3 , both ∆ 2 and ∆ 1 blocks are taken to be scalar block for this case. In figure (9.b), we have taken the 2d ising model ∆ φ = 1 8 and used the spin = 2 block for the operator ∆ 2 i.e we used G 2,∆ 2 ,2 instead of G 2,∆ 2 ,0 . One can see the plot is only slightly different if one uses scalar block i.e G 2,∆ 2 ,0 for the operator ∆ 2 . Had the ∆ 2 operator been a scalar (say by demand) in this case, we would be forced to conclude that ∆ 2 = 4 is not allowed thereby leading to a contradiction with the known result. Thankfully, from the plot, the spin-2 bound allows for ∆ = 2 which is the stress tensor and should be treated as the second operator.

Kink from positive geometry
In this section we will explain how the red line in fig.1 was obtained. We consider 10 scalar operators ∆ i where ∆ 0 is the identity operator and ∆ 9 is the infinity vector. ∆ 1 = ∆ S and ∆ i , i ≥ 2 are chosen randomly to be above 12 ∆ S . The intersection conditions eq.(3.23) are now checked. For N = 1 we find essentially the same results as obtained in sec. 5.1. However for N = 2 the result is different. We find that there is a kink type feature in the plot as in fig.1. In fig.1 below the red line, there is always at least some region where the intersection conditions are satisfied (but not above the line). So why does a kink arise?
For concreteness, let us examine the signs of the determinants arising from the intersection of the The plots in fig.11 are for det 3 , det 4 . Above ∆ φ ≈ 0.08,there is a small region below ∆ S ≈ 1.178 where both are negative but below ∆ φ ≈ 0.08, the region where they are both negative jumps further left giving rise to the kink in fig.1.
In other dimensions, there are also kinks. For d = 1, the kink appears ∆ ∼ 0, for d = 3, the kink appears around ∆ ∼ 0.26 and for d = 4, it appears near ∆ ∼ 0.5 . The latter two are for non-unitary operators and hence we will not consider them further. When we put in operators with spin, while the qualitative features not affected, the exact location of the kink moves around slightly. However, we have not investigated this in detail as the faces in this case are those of the weighted Minkowski sum which is not that of a cyclic polytope necessarily. If we repeated this analysis with the asymptotic blocks, the kink is no longer as sharp as in fig.1. We leave the search for an analytic (asymptotic) equation for the kink via this approach for future work. This is not impossible: a strategy would be to examine carefully the location of the largest zeroes in det 3 , det 4 by plugging in the asymptotic blocks and studying analytically when they cross each other. We have looked at this problem and our preliminary analysis can be found in appendix C. Another question that we leave for the future is exploring higher N constraints to see what effect it has on the kink.

Discussion
In this paper we have extended the work of [14] in several directions. First, we have considered the diagonal limit in arbitrary dimensions and have shown that in the large ∆ limit, even in the presence of spin, the geometric structure of the expansion in terms of the diagonal blocks of [15] is that of a cyclic polytope. We presented various analytic results and also initiated a comparison between the usual numerical bootstrap of [1] and the positive geometry constraints. We conclude with some possible future directions: • One can consider more systematically the higher N generalizations of the analysis presented in this paper, especially with the kink picture, also including the contributions from higher spins.
This can be done irrespective of the polytope being a cyclic one. It will also be interesting to see if there is some systematic classification of the facets of the weighted Minkowski sum that we found in this paper. More ambitiously, one can also try to develop a systematic asymptotic (in 1/∆) analytic equation for the kink-in our context we have made a preliminary attempt in appendix C.
• There is still work to be done in order to connect this geometric picture to the usual numerical bootstrap. The main direction to proceed from here is the generalization of the geometric approach to the non-diagonal case. The usual numerical bootstrap takes the input of both the dimension and spin for the generic off-diagonal blocks (i.e the block is a function of both z and z). In that sense the bound obtained from numerical bootstrap is more constraining. As already pointed out in [14], for the off-diagonal case, the simple picture of the polytopes does not apply and it generalizes to tensor products of the simple polytope picture we have handled so far. It would be interesting to see what the generalization has in store for us and how much can still be done analytically using the tensor product of (cyclic) polytopes.
• We can also consider the alternate formulation of the bootstrap in [7][8][9][10][11][12] using the Polyakov blocks which are manifestly crossing symmetric and ask about the OPE consistency in terms of the geometric picture. In that case the consistency conditions arise by demanding the absence of certain spurious poles. These conditions in the diagonal limit schematically read ∆, A Derivation of g i ≈ 2 √ 2 in ∆ d−2 2 , limit, general case.
Using the large ∆ limit of the diagonal blocks in (2.6), we can grind out the positivity criterion analytically at least in the ∆ d−2 2 , limit. Step-wise, are the block vectors. Now using Leibniz theorem we can calculate F m, , in a simple closed form expression, U is HypergeometricU or Tricomi confluent hypergeometric function. Some of the K 2N +1 (d, ∆, ) are, For the leading order |K i |, a general formula for large ∆, then looks like, Now using eq.(3.18) one can easily compute g i analytically, to obtain All the g i 's are positive for large ∆ for any d and any . Therefore we conclude that unitary polytope U{∆ i } is a cyclic polytope in the limit ∆  3.85515 × 10 9 7 1.15493 × 10 12 8 −9.71096 × 10 12 9 6.31945 × 10 14
Other possible orderings do not change the conclusion. It may be possible that some complicated combination of ∆ and can lead to cyclic polytope structure in even small ∆ range which we have not found so far.  Therefore, in some approximate sense P (c ) is also a cyclic polytope.
Also we have explicitly calculated the faces of the polytopes e.g. table (6) and we find that P (c ) is not a cyclic polytopes and for each separately they from cyclic polytopes. And we find that P (c ) is cyclic polytopes if we use (2.6).

C Analytic kink analysis
Here will briefly elaborate on our attempts to track down the location of the kink, in the cyclic polytope picture in section 7, analytically. By plugging in the asymptotic blocks to the 1/∆ S order which can be easily found using eq. (2.3), one can easily find that the the largest zero of det 3 (obtained by expanding the determinant to O(∆ 2 S )) which we will denote by ζ 0 is given by On the other hand, the location of the largest zero of det 4 is given by a 5-th order polynomial equation (obtained by expanding the determinant to O(∆ 2 S )) whose roots cannot be found analytically. However, we are interested in crossing of the zeros of det 3 and det 4 so we will just put the largest zero of det 4 as and Taylor expand in δ to obtain a simpler equation. This equation now (in the context of the analysis in section 7) depends on the ∆ 8 . We can expand in the limit where ∆ 8 1 and find an approximate equation whose (positive) root gives us the value of ∆ φ at the kink. This works out to be Considering the fact that we have used only the 1/∆ S approximation of the block and we are trying to track down a kink at ∆ S ∼ O(1), it is not surprising that we do not get a very precise location which is closer to (∆ φ , ∆ S ) = (0.08, 1.2) but nonetheless this would be a good starting point to see if a systematic solution can be worked out by retaining higher order 1/∆ S terms.

D Adding spin and kink
In this appendix we consider the effect of adding spin to the analysis of bounds on the leading scalar dimension ∆ S . In the diagonal limit, it seems that position of spin does not move much, even if we add spins. See figure (11). We will study this effect in detail in an upcoming work [21]. (a) ∆ S vs ∆ φ for d = 2 using linear programming, 10 derivative in eq (3.3).     Knowing g 1 > 0, g 2 > 0 it is then recursively true that for positivity of D dimensional polytope it is sufficient that g i > 0, ∀i, 1 ≤ i ≤ D.

Note
This analysis shows how K i (d, ∆, ) appears. We have checked analytically that our definition of g i 's in terms of K i (d, ∆, ) eq (3.18) is same given in [14]. Even though we don't have first hand proof of it. We guess the formula by observing first three g i 's and verified it till g 8 that it is true.
Also it is important to mention that Mathematica has some numerical issues evaluating the determinants eq (3.15). It gives very small negative numbers (∼ 10 −50 ) or very big negative numbers (∼ 10 50 ) when the spacing δ j is very small even if in the region of ∆ for which g i 's are positive. That this should not happen is clear from the above analysis.