The Geometry of the Modular Bootstrap

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on $R^+{\otimes}\mathbb{Z}$, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as constant computation time with increasing central charge. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above $\Delta^*_{gap}$, where $\frac{c{-}1}{12}<\Delta^*_{gap}<\frac{c}{12}$, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

For theories whose physical observables can be constrained uniquely via systematic imposition of unitarity, locality and symmetries, one often finds that the observable can be identified with a well defined mathematical entitie.The most well studied example is the amplituhedron for the scattering amplitude of maximal super Yang-Mills [1], where the amplitude is identified as the canonical form on a positive geometry.Other examples include the associahedron for bi-adjoint ϕ 3 theory [2], the amplituhedron for ABJM theory [3,4], the cosmological polytope for the wave function of conformally coupled scalars [5].
In some cases general physical principles are not sufficient to uniquely determine specific theories, but instead allow a landscape of possibilities.Remarkably, even this "theory space" is in many cases an interesting geometry.A natural question is then how much of allowed space is saturated by known theories, and if the remaining space either contains new unknown theories, or implies some new principle must shrink the space further.
One such example is the space of Wilson coefficients in the low energy effective field theory of a consistent UV completion.After the pioneering result in [6], it was recently uncovered that all coefficients satisfy an infinite number of inequalities, forming the so called EFThedron [7].Bounds on these coefficients were obtained in various ways, using both numerical and analytical methods [8][9][10][11][12][13].Work on the EFT bootstrap benefited from renewed interest in CFT bootstrap initiated in [14], which was able to constrain the conformal dimensions of low lying operators through the method of optimal functionals. 1  The geometric nature of the CFT bootstrap was also explored in [18,19], finding many similarities with the EFThedron.
In this paper we import recent lessons of the EFT bootstrap and apply them to obtain a geometric interpretation of the modular bootstrap for 2D CFTs [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].This bootstrap applies the constraint that the CFT can be consistently defined on arbitrary Riemannian manifolds, and so its partition function must satisfy modular invariance of the torus.It was introduced by Hellerman to obtain bounds on the lowest primary [37], yielding c 6 for c ≫ 1, where c is the central charge.This bound was subsequently improved through a series of works [38][39][40], with the strongest current result c 9.1 , obtained in [41].These bounds are weaker than expected from holography where the Banados-Teitelboim-Zanelli (BTZ) black holes [42] serve as the lowest state of the vacuum theory, having ∆ = c 12 .While it is now generally believed that the modular bootstrap is not sufficient to improve the bound much further beyond c 9.1 , several open questions remain, such as the exact implications of integer degeneracy, which has not been systematically studied except in rational CFT [43].
One perhaps surprising aspect we will quickly observe is that the modular CFT and EFT bootstraps are in fact almost identical mathematical problems.Both bootstraps can be solved efficiently via SDPB, as was done in [9], but also by phrasing them as moment problems [44].
To see just how closely related the EFT and CFT bootstraps are, we will show the Taylor coefficients of the partition function Z(τ, τ ), expanded around the self dual point τ = i, can be put in a form very similar to the EFT dispersion relations derived in [13] Z CFT p,q = i n i ∆ p i J q i g EFT p,q = i p i m p i J q i . (1.1) On the CFT side, ∆ i = h i + hi ≥ ∆ gap is the scaling dimension of state i, and n i ≥ 0 is the (rescaled) degeneracy of this state.Modular invariance implies that the spin J i = h i − hi is an integer, and also that particular linear combinations of Z p,q vanish.On the EFT side, m i ≥ M and J i are the mass and spin of UV state i, with M being the UV scale, unitarity implies p i ≥ 0, while crossing symmetry requires linear combinations of the g k,q to vanish.The vanishing of these linear combinations are what we will refer to as modular planes, or null planes, and imposing these to high order is one of the main challenges in both bootstrap programs.We will label the highest total derivative order imposed by Λ max ≡ max[p + q].Bounds obtained are always valid for any Λ max , but they become stronger the higher in Λ max one can go, until at some point they converge.Because eq.(1.1) includes spin integrality as a constraint, solving it is referred to as the spinning bootstrap.
A much simpler problem is ignoring spin altogether, and considering only a subspace of this problem which is referred to as the spinless bootstrap.Crucially, eq.(1.2) constitute a single variable moment problem, i.e. what are the sufficient conditions on Z for which the RHS of eq.(1.2) exists.The solution to such single moment problem is given in terms of positivity conditions on Hankel matrices.Similarly, the expressions in eq.(1.1) constitute double moment problems, with solutions in terms of generalized Hankel matrices.The moment problem formulation provides valid and analytic answers to the bootstrap.At low order in Λ max , the conditions can even be solved in closed form, but are typically not optimal.However, such solutions may reveal fine details that numerical methods potentially miss.Besides investigating low order closed form results, in this paper we also take a step further in the moment problem approach, and develop a new method to solve the (generalized) Hankel matrix positivity constraints to high order, by phrasing them as a semi-definite program (SDP).This allows us to compare the moment problem solutions of the bootstrap to traditional SDPB approaches at high derivative order.Since positivity conditions of the (generalized) Hankel matrices are necessary conditions, the space or spectrum that are ruled out attains a certificate of in-feasibility, irrespective of the order of derivative-or spin-truncation.
As an example, we test this setup by computing the gap and comparing to previous results.Our approach allows us to directly compute bounds at arbitrary c, for both spinless and spinning bootstrap.In the current paper we have computed up to c = 1200 and Λ max = 11 for spinning and c = 2000 and Λ max = 43 for spinless.Evaluating at c ≫ 1 does not introduce any slow down in computation time, which demonstrates the utility of our approach for large c bootstrap.Much higher Λ max can be reached by implementing our program on a cluster.
For spinless, the gap we find is considerably weaker compared to [41], which computed up to Λ max = 2000.However, our method allows directly checking much higher c, reducing errors due to extrapolation.For the spinning bootstrap, the state of the art was given in [39], at c = 20 and Λ max = 175.Our results support the findings in [39] that imposing spin integrality does not substantially improve bounds on the gap at large central charge.
Next we study the space of consistent CFT, defined in terms of the space of allowed Z p,q .In general, this space is unbounded.However, if the spectrum has a gap that is above some threshold, we find the space transitions from unbounded to bounded.This motivates us to term the threshold the critical gap ∆ * gap .As an example at Λ = 11, the upper bound for the partition function evaluated at the self-dual point Ẑ0 is shown below in Figure 1 for c = 100.We also notice the appearance of kinks between ∆ * gap and the maximal gap.These kinks become ubiquitous as c ≫ 1 , and their position appears stable with respect to derivative truncation, suggesting correlation with physical theories.
In the last part of the paper we take the first steps towards imposing the integer degeneracy condition.While on general grounds it is believed this condition will not affect the gap bounds at large central charge [39], we will show that it drastically reduces allowed space at c ∼ 1, even when considered at very low derivative order.We will impose integrality by viewing the partition function as a Minkowski sum of individual states, similar to the method proposed by some of the authors to impose the unitarity upper bound in the EFT bootstrap [45].Focusing on the spinless case given by eq.(1.2), the "allowed space" of each state i contributing to the partition function is a curve, parameterized by ∆ i ∈ [∆ gap , ∞).The allowed space for the whole partition function is therefore a Minkowski sum of such curves, each rescaled by a number n i .We will show that the boundaries of this problem are simple to determine in arbitrary dimension.This more constrained geometry is non-convex, lying inside the previously derived convex geometry, generally touching the original boundary only in isolated points.
The intersection with modular planes is however more difficult to perform in general.For one null plane, this intersection can be solved analytically, while for more null planes it must be solved numerically.We solve the intersection with two null planes, and find that the integrality condition already excludes extra regions compared to traditional spinless or spinning bootstraps at much higher derivative order.For c = 1, we find the boundaries significantly approach the points corresponding to the known free boson theories.This is shown in Figure 2 below.It is important to note that non-integer bootstraps can only produce convex regions, while at least this c = 1 case demonstrates that the full picture must necessarily include integrality.
Circle branch, R=n/2 At large c, the relative effect of integrality with just two null planes appears to wane.However, it is generally expected that high c also requires more null planes for convergence, so computing higher order intersections would be required before making a definitive statement on the role of integrality at high c.
To summarize our main results: • The spinless/spinning modular bootstrap can be formulated as a single/double moment problem, which is solved in terms of Hankel matrices • At low derivative order, the Hankel positivity can be solved in closed form, while at high order via SDP.This provides some advantages over SDPB, such as eliminating the spin truncation convergence check, and stable efficiency with increasing c.
• A critical gap exists, above which the allowed space becomes bounded from all directions.Above this gap the space exhibits numerous kinks • The integer degeneracy condition leads to a non-convex geometry that significantly reduces allowed space even at very low derivative order.
Added note: During the preparation of this draft, we became aware of the work [46] where integrality was imposed over states within a small range of conformal dimensions around the gap.One then requires that at least one integer exists between the lower and upper bounds of the spectral density.It was found that this condition slightly improves the gap at small central charge.In our work, integrality is imposed across the entire spectrum.
The paper is organized as follows.In Section 2 we introduce the torus partition function of 2D CFT, which is our object of study.In Section 3 we describe how the single and double moment problems can be solved in terms of Hankel matrices, and in Section 4 we show how the partition function can be related to such moment problems.We propose an SDP that can solve positivity of Hankel matrices, and explore bounds on the gap, twist gap, as well as introduce the notion of a critical gap.In Section 5 we explore the integrality constraint, which we impose by constructing partition function space via iterative Minkowski sums.We compute the intersection with two modular planes, and compare to results in the previous sections and known theories at c = 1.We conclude in Section 6.

Setup
The torus partition function in 2D CFT with central charge c > 1 is given by where τ the torus modulus, and n h, h is the degeneracy of the state with scaling dimensions ∆ = h+ h and spin J = |h− h|, i.e. it is a positive number with n 0, 0 = 1.The functions χ h (q) are the Virasoro characters, which sum up the contribution to the partition function from the descendants of each Virasoro primary.With c > 1, they are given by χ 0 (q) = q − (c−1) 24 1 − q η(q) , χ h (q) = q h− (c−1) where χ 0 (q) is the vacuum character and η(q) is the Dedekind eta function, satisfying η(− 1 τ ) = √ −iτ η(τ ).For c = 1, degenerate characters appear, which require subtraction, spoiling the form of eq.(2.1).We leave this generalization to future work, and in this paper focus only on partition functions of the above form, which generally means c > 1, but also includes a subset of free boson theories at c = 1, as will be discussed in detail later.
Modular invariance on the torus simply implies invariance under the action of PSL(2, Z) on (τ, τ ): 3 The group is generated by S and T transformations In terms of the representation in eq.(2.1),T invariance is ensured by the spin J being integer.One can simply focus on S invariance Z(τ, τ ) = Z(− 1 τ , − 1 τ ).This constraint is usually studied by expanding around the S self-dual point implying [37]: That is, the Taylor expansion allows one to discretize the constraint: instead of a functional identity one now has a vector (which can be taken to be infinite dimensional) identity.
In the bootstrap program, one imposes modular invariance on eq.(2.1) and extracts constraints on n h, h and gaps in h, h.Here, we would like to approach from another viewpoint, by considering the "space" of consistent partition functions, defined as Z(τ, τ ), that admits a representation of the form eq.(2.1), and respects modular invariance.We begin by discretizing eq.(2.1) around the self-dual point: where The space of partition functions is now defined by the allowed range for the set of Taylor coefficients By matching order by order in Taylor coefficients on both side of the equality in eq.(2.7), we have: where d is the degree of truncation in the Taylor expansion.Since n h, h > 0, the above simply states that the partition function [Z] must lie within the convex hull of tensored vectors (⃗ χ h , ⃗ χh).The convex hull of vectors is a polytope, and here we essentially have the geometry of tensored polytopes.To understand the space we will need to find the boundaries that "carve" out the hull.

The convex hull of moments
So far the components for the vectors ⃗ χ h , ⃗ χh on the RHS of eq.(2.9) are polynomials in h.As we will soon see, after a series of simple GL transformations, these can be transformed into points on a d-dimensional moment curve, i.e. each vector is a collection of monomials with increasing degree: The convex hull of such moments is well discussed in mathematical literature, and is termed the moment problem.In its most general form the moment problem identifies the necessary and sufficient conditions on a vector ⃗ y = (y 0 , y 1 , • • • , y n ), such that admits a solution ρ(x) that is non-negative.Here I is some region on the real axes, which falls into three categories: Hamburger I ∈ (−∞, +∞), Stieltjes I ∈ [0, +∞) and Hausdorff I ∈ [0, 1] moment problem.Equivalently, these conditions are then the co-dimension one boundaries for the convex hull of points on a moment curve.For finite n, this is called the truncated moment problem, and sufficient conditions can be given phrased in terms of the symmetric Hankel matrix of y n .One can generalize to the bi-variate moment problem, which will be relevant for the spinning bootstrap.In this case, we ask for conditions on [y] n×n such that admits a non-negative solution for ρ(x, y).For the bi-variate problem, only necessary conditions are known for the most general truncated moment problems, if we require a finite number of constraints.However, we will numerically demonstrate that these conditions lead to constraints that are close to the true boundary.

The single moment problem
Let us first review the geometry associated with the convex hull of moment curves, or the single moment problem: As shown in [47], the image of the hull is carved out by the statement that the symmetric Hankel matrix K[⃗ y], defined as is a positive semi-definite matrix, which implies its leading principle minors are nonnegative.If we further constrain ourselves to the convex hull of half moment curves, where x i ∈ R + , we then have that the Hankel and shifted Hankel matrix are totally positive matrices, i.e. all minors are non-negative.The independent conditions are given as: That these are necessary conditions can be derived as follows: the Hankel matrices can be written as where Sandwiching K n with a random vector one easily sees that ⃗ v T K n ⃗ v ≥ 0, and hence K n is a positive semi-definite matrix.The shifted Hankel can be written as: If x ∈ R + then xρ(x) ≥ 0, and one similarly concludes that K shift n is a positive semi-definite matrix.
Since the boundaries are associated with singular Hankel matrices, this implies that the boundary is parameterized by the convex hull of finite elements.Consider n = 2p+1, where the space is carved out by The space is projective P 2p+1 and the co-dimension one boundary is naturally associated with DetK p = 0 or DetK shift p = 0. We can readily read off the spectrum on these boundaries where ∆ i represents a state with (1, ∆ i , ) and ∆ gap corresponds to the state (1, 0, 0, • • • , 0).The set of states in eq.(3.10) can be easily understood from the fact that ∆ ∞ do not contribute to K p while ∆ gap do not contribute to K shift p .The free parameters that parameterize the hull are the 2p parameters (∆ i , ρ i ) and ρ ∞ or ρ gap , with one degree of freedom removed due to the projective nature.
where ρ(x, x) ≥ 0 for x, x ∈ I exists.For each fixed row or column, we can construct a Hankel and shifted-Hankel matrix which are positive semi-definite from our previous argument.Furthermore, consider the following "generalized Hankel matrix" where ( ⃗ for any real vector V, we conclude that K[Y] is a positive semi-definite matrix.If we further restrict that x, x ∈ R + implies that the shifted generalized Hankel matrices, either by one column or one row, must also be positive semi-definite.For the bi-variate moment problem, the vanishing of the minors for the generalized Hankel no longer leads to simple constraint on the number of elements in the hull.To see this consider Det y (0,0) y (0,1) y (0,1) y (0,2) .( This can vanish with an infinite number of states with distinct x but the same x. The sufficient conditions are less understood for the bi-variant moment problem.On general grounds, we expect that the boundaries are parameterized by finite number of elements in the hull.That is, the degrees of freedom that parameterize the boundary should be the (ρ i , x i , xi )s of the finite elements.For the single moment problem, the Hankel matrix being singular puts a bound on the number of elements, and thus they serve as the boundary or define limiting points to the boundary.For generalized Hankel of the double moment, this is no longer true.At truncated order, we no longer expect that the singular generalized Hankel describes the true boundary.However, if we have a collection of singular Hankel matrices, one can infer a bound on the number of elements.Indeed, this expectation is verified when compared to numerical SDPB results in the next section.

The modular-hedron
We are now ready to map the modular bootstrap problem to the moment problem.It will be more convenient to begin with the reduced partition function Ẑ(τ, τ ), defined as As the pre-factor |τ | 1/2 |η(τ )| 2 is invariant under S-transform τ → −1/τ , the reduced partition function retains its invariance.Similarly, the expansion is now defined on the reduced characters, which are independent of the η functions: where Again we expand both sides of eq.( 4.2) around the self-dual point.First for the reduced characters, where X = (c−1−24h)π, X = (c−1−24 h)π and a n,q are numbers independent of (c, h, h).
Up to a universal pre-factor α c,h ≡ (−1) , we see that the χ(n) h s ( χ(n) h ) are proportional to a polynomial of degree n in h ( h).As an example h , • • • ), via a universal c dependent GL transformation M (c) we can rotate it into a moment form.For example, truncating at the third derivative order : Applying the same double expansion for the partition function and the vacuum character, eq.( 4.2) can be reorganized as: where ñh, h = n h, he . The subscript for [ Z] h, h indicates the double moment is in (h, h).We see that after the double GL rotation, the vacuum-subtracted partition function is given by the convex hull of a tensor product of half-moment curves.We will term this space the "modular-hedron".It is sometimes more convenient to consider states labeled by (∆, J ).The geometry for In this basis, the geometry is associated with the tensor product of a half-and a discrete moment curve, where J ∈ Z.
The spinless modular polytope A simple limit of the geometry is the case where we only consider z(q,0) , i.e.where the spin information is suppressed.Then the double moment reduces back to the single moment.This corresponds to the spinless geometry where we consider the torus moduli being purely imaginary, τ = iβ and We then have the modular-polytope where we identify The character representation implies that the Taylor coefficients of the partition function must lie inside the hull of tensored moment curves as in eq.(4.7).This geometry encodes the positivity of the degeneracy number, as well as unitarity h, h ≥ 0. Modular invariance is then reflected in the additional requirement that ẑ(i,j) = 0 for i+j ∈ odd.
That is, the discretized partition function lives on a "modular subplane" The geometry for the modular bootstrap is an intersection problem, taken between the modular plane [ Ẑ] mod with the modular-hedron [ Ẑ].The boundary of this intersection will be determined by the boundaries of the modular-hedron.Restricting ourselves to the spinless geometry, i.e. the modular-polytope, this immediately implies that for d = 2p+1, the boundary of the intersection would be given by p states, as observed in [41].
In the following, we will systematically analyze the geometry using both analytical and numerical methods.We will also impose further constraints on the spectrum such as, • A gap for either ∆ or min[h, h].
• Integrality for the degeneracy number.
This will allow us to glean information with regards to the twist-gap, gap, and bounds on the value of the partition function at the self-dual point.

The semi-definite program setup
Here we summarize the constraints developed in the previous sections as follows: • Modular-hedron geometry: where [ Z] h, h is defined in eq.(4.7), and K is defined in eq.(3.12).
• Modular invariance: These constraints, together with a linear function of ẑ(i,j) to be extremized, constitute a semi-definite program (SDP).In particular, we begin with the relevant Hankel matrices parameterized by the modular plane, i.e. ẑ(i,j) with i+j = even, with suitable deformations to incorporate a gap for the spectrum.For fixed values of gap, this sets up a feasibility problem for SDP.In addition to finding the optimal value for the discretized partition function, it is also possible to obtain the optimal gap.The idea is to make an assumption about the gap and then find a set of ẑ(i,j) that satisfies all the constraints.If a certificate of infeasibility to the problem is found, the proposed spectrum is then ruled out.See Appendix B for a detailed discussion of SDP and the feasibility problem.
In our notation, the traditional SDPB approach starts with the modular constraint: One then constructs linear combinations of f a,b (h, h) that are positive above a gap.Each such functional rules out any spectrum whose lowest twist (or dimension) lives above the corresponding gap.The combination that gives a minimum gap is then the optimal functional.Translated to our geometry, what the SDBP probes is the intersection geometry of a co-dimension n-modular plane, where n is the number of modular constraints, with the infinite dimensional double moment problem.By comparing the result to our geometric bounds, it provides a measure of how far the generalized Hankel constraints are from the true boundary.

Bounding the gap
As a warm up, using the semi-definite programming setup in the previous subsection, we study the gap and twist gap at fixed derivative order up to Λ = 43.This is far from Λ ∼ 2000 studied in [41], and we do not expect novel bounds.The main purpose is to demonstrate the feasibility of this approach and comparing spin-less vs spinning results as well as the effects of integrality of spins.
To impose a gap on the spectrum, recall that total positivity of the Hankel matrices stems from its representation as a convex hull: (4.17) If the spectrum has a gap, say ∆ gap , then we immediately see that also constitute a definite positive matrix.In terms of the original coordinates, we can for example write: For a fixed truncation order Λ max , i.e. the total degree in h and h, we solve the feasibility problem for various different values of the gap.For the spinless bootstrap, Λ max is just the highest degree in ∆.This enables us to determine an upper bound for ∆ gap , and similarly for the twist-gap.
It is also straightforward to impose integrality on the spins.We will define J ≡ h− h ∈ Z. Suppose the spin J does not take value between l and l + 1, the following matrix will be positive: By imposing the positivity of K[ Z] l,l+1 for each pair of adjacent spins (l, l + 1), one effectively requires that J take only integer values.In practice, to implement these constraints, one considers only a finite number of spins −l max ≤ l ≤ l max .It is however worth noting that the Hankel matrix positivity, in contrast with the polynomial program approach of SDPB, gives valid upper bounds on ∆ gap for any spin-truncation l max and truncation order.Increasing l max only improves the bound.This is because the positivity of the truncated Hankel matrices is always a necessary condition for the double-moment geometry, and we are bounding the gap by ruling out configurations of the spectrum.

Bounding the gap of scaling dimension
Let us being with the simplest setup, the spin-less boostrap at Λ = 3 where the geometry is in P 3 .Let us consider the geometry { Ẑ0 , Ẑ1 , Ẑ2 , Ẑ3 }, where we can analytically solve the boundary.Because of modular invariance, there are the only three independent parameters {∆ gap , Ẑ0 , Ẑ2 }, with Ẑ0 = ẑ(0,0) , Ẑ2 = 2ẑ (1,1) This space is carved out by the union of the following equation sets, where the explicit entries of the 2 × 2 Hankel matrix K 1 = K 1 [⃗ y] are given by, where ⃗ G vac is the contribution from the vacuum character and M (c) is the associate GL transformation that converts the character expansion into the moment form.They are explicitly given as: where c 1 = π(c−25), c 2 = π(c−13) and c 3 = π(c−1) respectively.Now the boundary of the space { Ẑ0 , Ẑ2 } is given by, In Figure 3 we show an example of the allowed region of { Ẑ0 , Ẑ2 } in c = 1 and ∆ gap = 0. First, note that the region is unbounded with the boundaries given by either K 1 or K twist ∆ 1 being rank 1.As we will see, we can actually solve for optimal ∆ gap analytically in c → ∞ limit.
Since we expect that beyond the gap there are no consistent solutions, i.e. the space of partition function is empty.This means that the boundaries in P 3 becomes degenerate, det K 1 = det K twist ∆ 1 = 0.This condition is satisfied if there is only one state.the gap state.Then we can rewrite (4.23) as, Next focus the second and fourth row of this vector equation, The zero on the RHS indicates that the two vectors are collinear, Now ω(∆ gap ) is a degree 3 polynomial in ∆ gap , we can analyse the c → ∞ behavior by first assuming ∆ gap = a 1 c + a 0 + a 1 c + . . ., then solve the largest root ∆ * order by order.At leading order we have, (4.29) Since ∆ gap will be the largest zeros, a 1 = 1 6 .Then we can continue to solve the second order equation, The first two order of ∆ gap when c → ∞ are, This precisely matches with the early result in [37], where the O(c 0 ) term precisely evaluates to the numeric value 0.473695 quoted there in.
In Figure 4 we display the results for optimal gap at higher truncation order (Λ max = 11) with varying central charges.We compare between the geometric modular-polytope/hedron approach and SDPB.For the spinless analysis, we see that the modular polytope matches with the spinless SDPB.This is expected since for the case of a single moment, the Hankel positivities are necessary and sufficient conditions for the truncated moment problem.For the spinning analysis we see that at finite central charge, for example c ∼ 10 in the inset of Figure 4, the result of modular-hedron is slightly weaker than the spinning SDPB.Again this is reflecting that for a finite truncation order, the positivity of the Hankel matrices need not be sufficient condition for the moment geometry.However, such difference appears to be negligible at large central charges c ≫ 1.In fact, as we extend to large central charge, there is negligible difference between the spinless and spinning bootstrap.
In Figure 5 we give the result for various different truncation orders up to Λ max = 43 and extrapolate Λ max to infinity with the fitting function ∆ gap /c = A + B exp (−DΛ max ).However, we found that the extrapolated result is not stable.For instance, at central charge 2000, the extrapolated result improves when Λ max increases from 39 to 43.Shown in Figure 6, the fitting function derived from Λ ≤ 39 data deviates from the Λ max =43 data point.This indicates that the extrapolated result has yet to converge, and it will continue to decrease as Λ max increases.
We continue to fit the extrapolated result and obtain an estimation of ∆ gap /c at the limit c → ∞.We fit the data within the range 500 ≤ c ≤ 2000.The details of the fitting  functions are described in Table 1 which is also used in [41].The estimation differs since different functions were used.However, both fitting functions estimated that the ratio ∆ gap /c is approximately 1/6.60 ∼ 1/6.44 at the infinite c limit.We expect that the result will improve as we increase the Λ max truncation.

Bounding the twist-gap The twist-gap corresponds to min[h, h] ≥ h gap , which means
h ≥ h gap and h ≥ h gap .Translated to our geometry, one modifies the Hankel in eq.(4.13) to the twisted Hankel K[ Ẑhgap ] by identifying Similarly, to impose h ≥ h gap , We demonstrate the results in Figure 7, again for various central charges.Here, we compare the result from the modular-hedron geometry with or without integer spin constraints.As evident in the plot, the difference is negligible when c > 10.At c = 100 the difference is less than 0.1%, so for sufficiently large central charge, spin integrality will not significantly improve the bounds.

Two sided bounds for Z 0 and critical gap
So far, we've demonstrated how to extract the optimal gap or twist gap from our geometric approach.It is also interesting to focus on the region of intersection itself, representing the "space" of partition functions.For this purpose it will be sufficient to consider the modular-polytope ⃗ Z spinless eq.(4.10).Once again we simply have that the Hankel and twisted-Hankel must be positive: First we claim that Z spinless q is bounded from both sides.Recall that modular invariance implies that ẑ(m,n) = 0 and m+n = odd.Due to the GL transformation, Z spinless q for q = odd will be linear combination of Z spinless i with i < q.Now, starting with Z spinless 0 , every time we introduce two more derivatives to the partition function, the Hankel conditions are also enlarged to include two new inequalities linear/quadratic in the new variable Z spinless n , respectively.The highest unshifted Hankel determinant reads which means the term linear in the new variable Z spinless n is non-negative.On the other hand, the highest twisted Hankel determinant reads det

This gives a quadratic inequality whose leading coefficient for (Z
, which is non-positive.Thus the former condition gives a lower bound for Z spinless n , while the later an upper bound. Therefore the boundedness of the space boils down to Z spinless 0 , which is always bounded from below since it is definite non-negative.As we will see there is a critical gap ∆ * gap above which Z spinless 0 develops an upper bound.Once again let us consider the P 3 modular polytope geometry in more detail.The constraints det K twist ∆ 0 ≥ 0 and det K 0 ≥ 0 in (4.22) take the form  From (4.35), we see that when the coefficient of Ẑ0 turns negative the constraint becomes an upper bound on Ẑ0 .This occurs when the gap is above a critical value, In this case, Ẑ0 will be bounded from two sides, where det K 1 is linear and det K twist ∆ 1 is quadratic in Ẑ2 .The lower bound for Ẑ0 in (4.37) ensures that the coefficient for Ẑ2 is positive in the first inequality above, implying a lower bound.From the second inequality, the discriminant for the quadratic polynomial in Ẑ2 is always positive as long as Ẑ0 satisfies (4.37), leading to an upper bound for Ẑ2 .We see that the space of { Ẑ0 , Ẑ2 } is finite once the gap of the spectrum satisfies (4.36).Figure 8 shows an example at c = 1, ∆ gap = 1/10.As we increase ∆ gap , the region shrinks, until it vanishes completely when ∆ gap reaches the maximal allowed value.

Critical gap and HKS bound
The presence of a critical gap, above which the partition function is bounded, can be derived from the Hartman, Keller, Stoica [20,27] (we will call it HKS bound from now on) bound.To see this, we start with (6.6) in [27] 2  where Z[β] is defined in (4.9) and Z L (β) comes from summing over all the contributions from states having scaling dimensions below ∆ H , Here the separation between low and high energy states can be arbitrary as long as the threshold ∆ H is larger than c−1 12 .Now we take the self-dual point limit β → 1. (4.39) becomes,   For c = 12, the upper bound is more sensitive to the integer-spin constraints.As seen in Figure 12, the spinning bootstrap yields stronger result compared with the spinless bootstrap even at lower derivative order.Imposing the spinning constraints, the sharp kinks turn into smooth bumps.However, for large central charge, imposing integer-spin constraints does not significantly change the upper bound of Ẑ0 , and the kinks survive.The phenomenon that higher central charge is insensitive to the integer spin constraint is consistent with our earlier observation on bounding the gap at large central charge.
In summary, we find that the the space of partition functions is bounded once the gap of the spectrum is above the critical gap.The value of critical gap lies between c−1 12 and c 12 , and we observe the presence of stable kinks emerging which are stable under increase of derivative truncation.If we conjecture that the stable kinks point to physical theories, the fact that they only appear below c/12 for sufficiently large central charge is consistent with the expectation that the physical theories can only admit ∆ gap ≤ c/12 in the large c limit expected from holography.

Integer degeneracy number
So far, we have only considered the convex geometry implied by imposing the positivity of the degeneracy number.In this section we explore implications of requiring the degeneracy number to be an integer.We will solve this problem geometrically by using the method introduced in [45,48], which constructs the allowed space iteratively via Minkowski sums of all possible states contributing to the partition function.Returning to eq.( 4.1) we now require n h i , hi ∈ N. We will focus on the spinless problem in ∆ space, which amounts to studying: where Ẑk = p+q=k k q Ẑ(p,q) , and similar for the χ(k) .Explicitly G vac is given in eq.( 4.24), and where X = (c − 1 − 12∆ i )π.Unlike the case where n i is simply assumed to be positive, the exponential factor e (c−1−12∆ i ) π 6 can no longer be scaled away if we wish to impose n i ∈ N. Our first goal is to find the boundaries of the space ( Ẑ0 , Ẑ1 , Ẑ2 , . ..).We focus on the space of consecutive couplings, as only in this space the boundaries have no nontrivial self intersections.Once we have the boundaries, all that remains is computing their intersection with the modular planes, Ẑodd = 0.The intersection with one plane (Λ max = 2) can remarkably be solved in closed form, and we are able to also numerically solve the intersection with two null planes (Λ max = 3).We will find that already these results are in some regions stronger even than the spinning bootstrap at large Λ max .Intersections with even more null planes would impose even stronger constraints, and can still be carried out using our methods, but computation time may be high.

A Minkowski sum of curves
First let us briefly explain how the allowed space of Ẑk can be viewed as a Minkowski sum of curves and introduce some notation.Slightly rewriting eq.(5.2), define assuming wlog ∆ j ≥ ∆ i ≥ ∆ gap for j ≥ i.Any given configuration of n i and ∆ i corresponds to a particular point in partition function space, and so the allowed space for Ẑk , for some given gap ∆ gap , is simply the image of ∆ i ≥ ∆ gap and n i ∈ N3 under eq.(5.4).Our task is therefore to compute this image, and in particular to find its boundaries.
It is in fact straightforward to show that the problem reduces to finding boundaries only for configurations of the type (5.5) The crucial observation is that any A(n i )(∆ i ) can be obtained from some A(0, 1, 1, 1, . ..)(∆ ′ i ) by simply fixing the ∆ ′ i appropriately.For instance , ∞, . ..) .
(5.6) States with large ∆ (in general, with ∆≫c) have vanishing contribution to the partition function.For N → ∞, the converse is immediately also true.Therefore we conclude that the space A(n i )(∆ i ) is equivalent to the space A N →∞ .We will obtain the space for A ∞ by constructing A N iteratively, and taking the limit N → ∞.
Next we discuss how the image of ∆ i may be computed.

Two dimensions
Let us begin with the 2D space ( Ẑ0 , Ẑ1 ), and N = 1, ie. the space of just one state satisfying ∆ ≥ 0, plus the vacuum.We have This space is a curve, paramaterized by ∆ 1 ≥ 0, which we plot in Figure 14 as the blue curve.For N = 2 we now have Since we need the image of both ∆ 1 and ∆ 2 , this is simply the Minkowski sum of the previous curve A 1 with another curve χ.The result is a 2D surface, parameterized by the two ∆ i .It is straightforward to plot the space, and one finds a 2D region with three boundaries, given by A(0; 1), A(1; 1) and A(0; 2).This is shown in Figure 14a.
In general, boundaries of Minkowski sums can be of two types.The first type are associated to bounds of the parameters themselves.Indeed A(0, 1) can be obtained from A(0, 1, 1) by taking ∆ 2 = ∞, and A(1, 1) can be obtained by taking ∆ 1 = 0.The second type of boundaries can be obtained by extremizing in one direction, keeping the other directions fixed.This can done via Lagrange multipliers, which in 2D entails solving4 We find the extremal solution ∆ 1 = ∆ 2 , corresponding to the third boundary A(0, 2).In fact, even in higher dimensions and arbitrary number of curves, solutions obtained via extremizing will always be of the type ∆ i = ∆ j , ensuring remarkably simple boundaries.
It is important that these boundaries have no intersections, except at the parameter bounds, ∆ = 0 or ∆ = ∞.Again we emphasize this is only valid for our particular curves χ and for spaces Ẑk with consecutive k.As a counter-example, one can check that in the space ( Ẑ0 , Ẑ2 ) the boundaries do have nontrivial self-intersections. A(0,1) A(1,1) A(0,3) Now let us move on to A 3 = A(0; 1, 1, 1).Using the fact that the boundary of a Minkowski sum of objects is included in the Minkowski sum of the objects' boundaries, we simply compute the Minkowski sum of the boundaries of A 2 and an extra curve.The resulting space is given in Figure 14b, where the boundaries are A(0; 3), A(0; 1), A(1; 1), and A(2; 1).
(5.10)This is shown in Figure 15, with the lower boundary being pushed successively downward.In the N ≫ 1 limit, the lower boundary (corresponding to A(0, ∞)) becomes a straight vertical line.Meanwhile the upper boundary is simply the same periodic structure, repeated infinitely.We therefore obtain the boundaries of A ∞ in the 2D space { Ẑ0 , Ẑ1 } as upper bdy : A(n 0 , 1) , n 0 = {0, 1, 2, . ..} , lower bdy : A(0, ∞) . (5.11) We also compare the c = 1 and c = 2 boundaries of this space in Figure 16.
Taking N → ∞ the upper boundary becomes A(n 0 , 1, ∞), which is a curved infinite strip in the Ẑ2 direction for each n 0 .The three components of the upper boundary can be parameterized as: with x ≥ 0 an independent parameter.This parameterization makes the relation between the geometry of 2D and 3D clear: projecting out Ẑ2 , we are left with A(n 0 , 1), so the upper boundary from 3D becomes the lower boundary in 2D.

Integrality with a gap
The presence of a gap further reduces the space.There are two possible independent changes to be made.
First, if we only assume that all states satisfy a bound ∆ ≥ ∆ gap , the boundary structure remains unchanged, but the range of parameters is now ∆ ∈ [∆ gap , ∞).This does not automatically assume the state corresponding to ∆ gap is actually present.Rather, the space contains all theories in which the lowest state is anywhere above this gap.
Second, if we know that the state ∆ gap itself is present, the space is modified further.This is only true for integer degeneracy, since if some state is present, it must have degeneracy equal to at least 1, and so must have a finite contribution to the partition function.Therefore, if we assume a gap state is present, all boundaries we found previously must have n 0 ≥ 1.This amounts to translating the whole space by the vector χ(∆ gap ).

Null planes
In this geometrical approach, modular invariance acts a series of null planes intersecting the geometry we described earlier.The intersection with the first null plane at Ẑ1 = 0 was sketched in Figure 18, and in fact can be solved analytically in terms of the Lambert W k (x), or product log function.This function is the solution to the equation and has two real branches for x ≥ −e −1 : the k = 0 branch for x ≥ 0, and the k = −1 branch for x < 0. Intersecting a boundary of the form A(0, n 1 , 1) with this null constraint implies solving giving which at c = 1 we find gives valid solutions for n 1 ≥ 2. This solution corresponds to the green intersection line in Figure 19.For higher c, the first intersection occurs at higher n 1 , with n 1 = 7 at c = 2 for example.
The intersection with higher order null planes can only be solved numerically.For instance, intersecting the 4D geometry upper boundaries with two null planes requires solving the intersection A (1) (n 0 , 1, n 2 , 1) = A (3) (n 0 , 1, n 2 , 1) = 0 . (5.18) We accomplish this in Mathematica by placing ∆ 3 on a grid, and using NSolve to find ∆ 1 and ∆ 2 for all values of n 0 , n 2 , and ∆ 3 .Since high values of ∆ are exponentially suppressed, we can optimize the search by limiting to low values.Experimentally we find ∆ < 10 is sufficient.Once a valid solution for some values (n 0 , n 2 ) is found, the complete space can be quickly determined using the fact that consecutive values for n 0 , n 2 correspond to neighboring boundaries.We show the result for c = 1 with zero, one, and two null plane intersections in Figure 19, comparing with the case n i ∈ R as well.We observe that the integrality condition rules out significant regions of allowed space, and only intersects the original boundary in isolated points.For even higher null constraints, the task becomes computationally intensive, and would require more specialized approaches that can efficiently find roots of polynomialexponential equations.Note that we are not claiming the bounds due to integrality have converged at Λ max = 3.However, they are still valid bounds, as increasing the number of null plane intersections can only reduce the allowed space further.
Integrality at large c As we increase c, keeping Λ max = 3, we find that the relative importance of the integrality condition diminishes.In Figure 20 we compare the boundaries for c = 1 and c = 2.However, it is known from the usual bootstrap approaches that as c increases, we have to also increase Λ max for bounds to converge.Our current analysis is therefore not sufficient to make any claim on the effect of integrality at large c, but leaves open the possibility for significant effects at large also c.
Bounding the gap By increasing ∆ gap , eventually the null planes no longer intersect the geometry, allowing us to find bounds on the gap.With two null planes and c = 1, we find a gap ∆ gap ≤ 0.605, compared with ∆ gap ≤ 0.615 from the usual spinless bootstrap with no restriction on degeneracy, when using the same number of null planes.Using more null planes, the spinless bootstrap gap bound converges to ∆ gap ≤ 0.603, while the spinning bootstrap converges to ∆ gap ≤ 0.5, saturating the free boson theory.Since the bound is already saturated by the spinning bootstrap there is no improvement to be found, however we find it encouraging that the integer condition is stronger than the spinless bootstrap at equivalent derivative order.This leaves open the possibility that at large Λ max the integer condition improves the spinless result, and for large c it may improve also the spinning bootstrap bound, given the reductions in allowed theory space we found previously.

Comparing with non-integer bootstraps and known theories at c = 1
We can now compare the integer bootstrap at Λ max = 3 with the SDP bootstrap at much higher orders, and also to the known c = 1 free boson theory.
For c = 1, the character representation in eq.(2.1) is no longer valid.This is because for c = 1, primaries with h = n 2 and h = (n + 1 2 ) 2 , n ∈ Z have extra null states that requires removal, introducing minus signs in the usual character expansion.Therefore, in general for c = 1, our geometry requires modification to include these special primaries.
However, for compactified free bosons at special radius its partition function can be positively expanded on the usual characters.Indeed its reduced partition function is given by [49] with a, b ∈ Z.To express this partition function in the form of eq.(2.1), we need to subtract the vacuum (1 − q)(1 − q), and for general R the term q q, corresponding to a state (h, h) = (1, 1), appears with a minus sign.However, for R = n 2 , n ≥ 2, n ∈ Z, the term becomes positive.Therefore, it becomes a valid question if these specific theories can be bootstrapped.There are three interesting cases we can consider for the bootstrap, depending on our assumptions of the gap state.∆ ≥ 0, no gap state assumed We compare the case where the only assumption on the scaling dimensions is ∆ ≥ 0, but we do not assume a state with ∆ = 0 is necessarily present.In other words, the allowed region includes all possible theories with any gap.This is shown in Figure 21.First, we explore convergence from usual SDPB without imposing integrality.We find that spinless converges quickly around Λ max = 15, while spinning has still not converged in the bottom region even at Λ max = 25.It appears plausible that this lower boundary will converge precisely on the circle branch theories.
Next we plot the boundary due to imposing the integrality (up to Λ max = 3) on the spinless bootstrap.We observe this boundary is stronger in many regions compared to both the spinless and spinning bootstrap at much higher order.Since the circle branch points can only be saturated by non-convex conditions, it is possible that integrality at high Λ max may converge exactly on these isolated points.∆ ≥ 0.1, gap state not present It is also instructive to explore this space when we require all ∆ ≥ 0.1 as used for Figure 8.Here we are still not assuming the presence of any particular gap state, only that it must be somewhere above 0.1.In the circle branch, the lowest state in scaling dimension corresponds to a = 1, b = 0, implying that the circle branch satisfies ∆ ≥ ∆ gap if R ≤ (4∆ gap ) − 1 2 , leading to three allowed theories at R = {1, 3/2, 2} for this particular gap.∆ gap = 0.1, gap state present In Figure 24 we plot the allowed region of theories which have the gap state at ∆ gap = 0.1.For this particular gap these is no known theory to exist.We see that the integer condition rules out an even larger piece of the region allowed by the spinning bootstrap.
Figure 24: Gap state has ∆ gap = 0.1, and is assumed to be present, significantly pushing the integer boundary.There are no know theories satisfying our constraints for this gap.
Linear combinations of circle branches As observed in [39], it is possible to take linear combinations of circle branch theories which have positive expansion on non-degenerate characters for any R. Let us focus on the space for ∆ ≥ 0.1, where the only allowed partition functions compatible with our assumptions are for R = {1, 3/2, 2}, and consider the combination (5.21) It is easy to verify that for particular ranges of x, y, z the above partition function has positive expansion on the non-degenerate characters, as well as the correct vacuum normalization.Clearly, for generic values of R it will not satisfy integer degeneracy, and so does not constitute a valid CFT, but it will be compatible with the spinning bootstrap.We plot this in Figure 25, and note this simple linear combination covers most of the space except a corner, at least for Λ max = 25.This is similar to what was observed in the EFT bootstrap, where a linear combination of two theories spanned most of allowed space [9].We can also view this result in the other direction: the bootstrap, based only on convex conditions like positivity, will never be able to rule out the region covered by this linear combination.Adding the integrality condition is therefore necessary to reduce this space further, and we can see that this occurs already at Λ max = 3.

Conclusions
In this paper, we elucidate the geometry behind the modular bootstrap, where the constraint of modular invariance is imposed on the derivative expansion of the torus partition function around the self dual point.We show that the character representation of the torus identical curves, which leads to a non-convex geometry.Intersecting this geometry with up to two modular planes (Λ max = 3) we find that integrality rules out regions allowed by the spinning bootstrap at much higher Λ max .We speculate that at high Λ max , integrality could in fact reduce the space to precisely the free boson theories.On bounding the gap, we find integrality is stronger than the spinless bootstrap at equivalent derivative order, but at high Λ or high c it is still not clear what the effect of integrality is.The reductions in theory space support the possibility that integrality should also improve bounds on the gap, but direct computation is required for a definitive conclusion.
There are several questions we have not yet answered about integrality.First would be pushing the derivative order, ie. the number of null plane intersections.Since the boundaries are known to arbitrary dimension, this problem is purely computational, where one has to solve systems of polynomial-exponential equations.More efficient and specialized algorithms could potentially increase Λ max , hopefully to the point where bounds begin to converge and become relevant also at large c.Finally, we have not explored the combination of imposing both integer spin and integer degeneracy.Our approach of finding boundaries via Minkowski sums can still be used in principle, but organising the geometry in such a high dimensional space becomes a very difficult task.Solving both integer spin and integer degeneracy to high order in Λ max would finally answer if the modular bootstrap alone can fix ∆ gap = c 12 , as expected from holography, and more generally how the space of known theories compares to the space of allowed theories.
It should also be feasible to apply our approach to superconformal characters.For N=2 super conformal symmetry, the states are now labeled by (h, h, Q, Q), where Q is the charge under the U(1) R symmetry, and the partition function is given by sums over BPS and non-BPS representations (see see [38,51] for details).For the spinless bootstrap, the space is given by four distinct moment curves, subject to modular constraints that interwine the four curves.It will be interesting to see if the space of consistent partition function also exhibits kink like behaviour with some critical gap.Carving out the theory space Seen in section 4, the problem of bounding the partition function can be viewed as a semi-definite program, where the optimization variable x consists of the discretized partition functions, i.e. the vectorization of (2.8).In this basis the equality constraint (B.3), which corresponds to the modular plane, becomes trivial.We simply remove the vanishing z (i,j) with odd i+j and reduce the dimension of the problem by roughly half.Therefore, we almost never need to work with equality constraints explicitly.A (l)

B Explicit details for SDP
i and C (l) are symmetric matrices that depend on the gap and the central charge, and L counts the number of positivity constraints, which is equal to the number of generalized Hankel matrices.Solving (B.1) then enables us to bound arbitrary linear combinations of the discretized partition functions.
Bounding the gap To find the maximal possible value of, say, the gap of the scaling dimension ∆ gap , we start from a small value of ∆ gap and gradually increase its value until no solution x that satisfies the constraints (B.2) can be found.Therefore, to obtain an upper bound for ∆ gap , one needs a certificate of infeasibility.To achieve this, we can introduce an auxiliary variable t which measures the violation of the positivity, and solve the following optimization problem: Suppose the optimal value t * is positive, the problem is infeasible; otherwise, the problem is feasible.
Certifying unboundedness Suppose we obtain an optimal solution to (B.1), it automatically serves as a certificate of boundedness of c T x.However, the optimization problem (B.1) may also be unbounded, such as when the gap ∆ gap is tuned to some value below the critical gap.The following feasibility problem can then be served as a certificate for Since if such a point y exists, then x + αy for any α > 0 will again be a feasible point of problem (B.1), implying that the objective function is unbounded from below.
Implementation There are many freely available solvers for semi-definite programs, for example SDPA [53] and SDPB [54].Designed especially for polynomial matrix programs, the SDPB solver is widely used in the bootstrap community.Ordinary semi-definite programs are a special case of polynomial programs, which we frequently encounter in this paper.While SDPB is highly efficient and optimized for polynomial programs, it is overkill for solving ordinary SDPs.Instead, we needed a solver that could handle not only large number of spins but also large matrix sizes.To this end, we developed SDPJ, a native Julia SDP solver based on the primal-dual interior point method, which is parallelized and supports arbitrary precision, with a modified parallelization architecture to suit our needs.
The code is publicly available on GitHub [55].The choice of parameters in this paper is shown in Table 2. Table 2: Choice of parameter for SDPJ in this work.See [55] for the definition of the parameters.

3 5 4
Two sided bounds for Z 0 and critical gap 20 Comparing with non-integer bootstraps and known theories at c = 1 34 1 Introduction

Figure 2 :
Figure 2: Comparing regions carved out by SDPB with the region carved out by the (spinless) degeneracy integrality condition at Λ max = 3.The integrality condition rules out new regions in the space, and significantly approaches the circle branch theories.

Figure 6 :
Figure 6: Bounds on ∆ gap /c vs. Λ max at central charge equals 2000.The fitting function is obtained from Λ max ≤ 39 data.The Λ max = 43 data point deviates from the fitting function.This implies that the extrapolated result has not stabilized.

Figure 7 :
Figure 7: Bound on the twist gap vs. central charge c at different derivative orders.

Figure 9 : 12 Figure 10 :
Figure 9: Estimation of the critical gap, where the fitting function ∆ * gap /c = a + bΛ −p max is used.a = 0.076063 is obtained for c = 12, and a = 0.082466 for c = 100.In both cases we have p ≈ 0.6.

Figure 11 :
Figure 11: Upper and lower bounds on Z (0,0) plotted against the gap for c = 12.The space vanishes when reaching the maximal gap.The upper bound grows super-exponentially, becoming infinite at the critical gap.Notice the kink near ∆ gap = c 12 , which divides the allowed region of ∆ gap into two parts.

Figure 14 :
Figure 14: Allowed space of two states A 2 and respectively three states A 3 .Note A 2 is a subspace of A 3 .We have fixed c = 1 for this figure.

Figure 15 :Figure 16 :
Figure 15: The lower boundary converges to a vertical line for N → ∞

Figure 17 :
Figure 17: Boundaries for the sum of three curves.The co-dimension 1 boundaries intersect along co-dimension 2 boundaries, which in turn are just the previous boundaries of the 2D space.

Figure 18 :
Figure 18: Boundaries of A ∞ in three dimensions.Ẑ1 = 0 is the first modular plane, intersecting the lower boundaries along the green line.This intersection can be worked out in closed form.

Figure 19 :
Figure 19: Intersection of the geometry with 0, 1, and 2 null planes for c = 1, comparing integer and non-integer degeneracy boundaries.

Figure 21 :
Figure 21: Regions carved out by SDPB, for c = 1, with no assumption on the scaling dimension except ∆ ≥ 0. The black dots represent the circle branch of the free boson theory, evaluated at particular R = n 2 , n ∈ N. The spinless bootstrap has already converged around Λ max = 15, while the spinning bootstrap region is still shrinking.At large Λ max we conjecture the lower boundary will reach the circle branch theories.

2 Δ≥0 2 Figure 22 :
Figure 22: Comparing regions carved out by SDPB with the region carved out by the (spinless) degeneracy integrality condition at Λ max = 3.The integrality condition significantly approaches the circle branch theories.

Figure 25 :
Figure 25: The space covered by eq.(5.21) covers most of the space allowed by the spinning bootstrap.The integrality condition starts ruling out some of this space.

Table 1 :
[41]d on ∆ gap vs. central charge c for truncation order Λ max = 11.We observe that for c > 100 there is very little difference between the four bootstraps.Bound on ∆ gap vs. central charge c for different truncation orders Λ max .The color dots are the data points of truncation order Λ max = 7, 11,15, 19, 23, 27, 31, 35, 39, 43.The fast bootstrap data are extracted from[41].It is clear the bounds have not yet converged, but we are able to directly verify the c ≳ 2000 regime.The details of the fitting function.