The unitary subsector of generalized minimal models

We revisit the line of non-unitary theories that interpolate between the Virasoro minimal models. Numerical bootstrap applications have brought about interest in the four-point function involving the scalar primary of lowest dimension. Using recent progress in harmonic analysis on the conformal group, we prove the conjecture that global conformal blocks in this correlator appear with positive coefficients. We also compute many such coefficients in the simplest mixed correlator system. Finally, we comment on the status of using global conformal blocks to isolate the truly unitary points on this line.

where c is the central charge. The power of this infinite-dimensional symmetry was perhaps most famously demonstrated in [1] with the discovery of the minimal models. In addition to providing an exact solution, representation theory of the Virasoro algebra enabled [2][3][4][5] to show that these models are the only unitary CFTs in two dimensions with c < 1. However, it has become known more recently that one can see hints of the special role played by minimal models without exploiting Virasoro symmetry at all [6][7][8][9][10][11][12][13]. The method in question is the numerical bootstrap, which uses only the global conformal transformations -two copies of sl (2) in this case. Exclusion plots, based on crossing symmetry and unitarity, are shown in Figure 1 where a straight line containing the minimal models is clearly visible. 1 A kink is present at the Ising point (∆ σ , ∆ ) = 1 8 , 1 but, unlike in the three-dimensional case [14][15][16], this kink does not sharpen into an island when three correlators are used to restrict the number of relevant operators. Following [17], it is worthwhile to see which  features of the 1 8 ≤ ∆ σ ≤ 1 2 solution can be predicted analytically. The goal of this work is to put the one-correlator upper bound on a more rigorous footing and to explain why the three-correlator upper bound is unchanged.
In order to have a unitary 2D CFT with c < 1, it is necessary that all primary operators have conformal weights equal to Kac's formula h r,s (c) for some (r, s). The Kac table of degenerate weights is given by For generic values of m, this leads to an infinite discrete spectrum. All OPEs are finite, but as we raise the values of r and s, these sums become arbitrarily long. A special situation occurs when m is an integer. This precisely describes a central charge for which h r,s (c) = h m−r,m+1−s (c). The Kac table for these doubly degenerate weights can be shown to truncate, allowing us to consider only 0 < r < m and 0 < s < m + 1. This leads to a finite number of primary operators and, as it turns out, a unitary theory. This theory, called a (unitary) minimal model, is often denoted M(m + 1, m). Since Figure 1 only shows a kink for m = 3, it is evident that non-integer values of m are still important for the bootstrap. Analytically continuing the unitary minimal models in this way is not new. In [18,19], the generic M(m + 1, m) was found to be a solvable consistent theory and referred to as a generalized minimal model. 2 It obeys the 2D CFT axioms of associativity and Virasoro symmetry but not unitary [20]. The upper bound from the bootstrap is realized by the generalized minimal model four-point function σσσσ if we identify σ ≡ φ 1,2 and ≡ φ 1,3 . 3 The authors of [17] observed squared OPE coefficients of quasiprimaries in this correlator that were all positive. If this conjecture is correct, we must conclude that in an arbitrary M(m + 1, m) theory, the non-unitarity is mild enough that it cannot be diagnosed from the correlator of four σ operators. What this means is that for a nondegenerate O appearing in the σ × σ OPE, the state O(x 1 )σ(x 2 )σ(x 3 ) |0 on the cylinder will have a positive norm. Similarly, within a degenerate subspace, all such norms will sum to a positive number. We use the terminology that σσσσ comprises a unitary subsector of the theory since other OPEs, e.g. σ × , are needed to construct negative norms. As we will show, the conjecture can indeed be proven with the help of a new formalism in [21,22] for deriving sl(2) block expansions. This proof, along with a systematic look at the other two correlators, forms the main result of this work. 4 The summary of results in Table 1 Correlator 3 < m < 4 4 < m < ∞ σσσσ All coefficients ≥ 0 All coefficients ≥ 0 σσ One checked coefficient < 0 All checked coefficients ≥ 0 Infinitely many coefficients < 0 All checked coefficients ≥ 0 Table 1: The status of three-correlator sl (2) block coefficients in the generalized minimal models. Statements about σσσσ apply rigorously to the full set of coefficients. For the other two correlators, we have manually decomposed them up to order 15. shows that for 3 < m < 4, σσσσ is the only unitary subsector we have found in the sense described above. However, it appears that the generalized minimal models with m > 4 have larger unitary subsectors that include the other two four-point functions. The last line shows a surprising tension with Figure 1. Despite the fact that displays significant unitarity violation for m < 4, the line (1.4) in the three-correlator exclusion plot is uninterrupted. The numerics are telling us that there is a partial solution to crossing, other than M(m + 1, m), which fills in this region. Using the properties of minimal models, we will show that the existence of this solution can be concluded from a simpler numerical setup. It would be nice to eventually find a fully analytic construction.
Before deriving the results in Table 1, it is instructive to consider the lower bound which appears on the left side of Figure 1. This gives us a more straightforward opportunity to use the techniques in [21,22]. The explicit solution for σσσσ along this line was found in [23], which focused on its special role in the non-unitary (severe truncation) bootstrap of [24][25][26]. 5 This solution exhibits Virasoro symmetry with a central charge given by but no Virasoro identity block. To find an analytic explanation for why this four-point function appears in the unitary bootstrap, one must be able to show that this vacuum decoupling is the only sign of non-unitarity that appears at the level of a single correlator.
In other words, one must be able to repeat the logic of [17] and find positive squared OPE coefficients for all of the quasiprimaries that do appear. These will be seen as very large coefficients by the numerics because the algorithm used for Figure 1 fixes λ σσI = 1. This is indeed what we find from the extremal functional method of [29]. This paper is organized as follows. In section 2, we study the lower bound (1.5) as a warm-up. In this case, it is particularly easy to invert the OPE and find that all sl(2) block coefficients are positive. The methods involved prepare us for our main interest, which is the upper bound (1.4). Turning to this upper bound in section 3, we calculate the global block coefficients summarized in Table 1. In the case of σσσσ , we find closed-form expressions.
In showing that they are positive, we prove the conjecture made in the appendix of [17]. For the other two correlators, we expand them to high order recursively and conjecture that the sl(2) block coefficients are positive again when 4 < m < ∞. Even though the Virasoro blocks in appear to have positive sl(2) expansions everywhere, the unitarity violation for 3 < m < 4 arises because of the coefficients multiplying the Virasoro blocks themselves. In section 4, we go back to the bootstrap and discuss what these patterns in the OPE coefficients mean for the results in Figure 1. In particular, we perform a semianalytic treatment of the problematic correlator. The result is that one does not need to perform a three-correlator bootstrap to predict that (1.4) survives -the search for an upper bound may be reduced to a one-correlator problem. Before concluding, in section 5, we discuss other 2D theories which might be possible to study using more correlators or more assumptions on the spectrum.

The lower line: A warm-up
The four-point function along the line (1.5) consists of a single Virasoro block V (h i , h, c; z). It was found in [23] via the Coulomb gas formalism which writes the central charge as 5 The fact that it also appears in the unitary bootstrap has not received much attention. In [27], it was mentioned that the lower bound at ∆ σ = 1 8 was somewhat close to ∆ = 1 6 . Interestingly, the region below the bound coincides with the region where standard OPE maximization techniques cannot constrain the central charge [28]. c = 1 − 24α 2 0 and places a background charge of 2α 0 at infinity. This allows a number of four-point functions to be realized as correlators of vertex operators with additional insertions of screening charges. The simplest of these is a correlator of four scalars that all have charge α 0 2 . Since the neutrality condition for this is satisfied without any screening charges, one finds the manifestly crossing symmetric Our task now is to expand g(z) into sl(2) blocks: This is guaranteed to be an expansion in even integers due to the Bose symmetry of the σ ×σ OPE. To proceed by the brute-force approach, we expand the hypergeometric function and switch the order of two sums.
We may now compare the inner finite sums to the Taylor coefficients of (2.2), given by . Since the lower triangular system for c n yields to back-substitution, Rather than using this recursive procedure, we will now review a method for computing the c n directly. The blocks, defined in (2.3), are eigenfunctions of the conformal Casimir It is well known that D is self-adjoint on [0, 1] with respect to the measure z −2 . The authors of [21] used this fact to develop the Sturm-Liouville theory of this operator and construct the orthogonal eigenfunctions It is convenient to set h = 1 2 + α in which case (2.7) becomes a function Ψ α (z) which is even in α. In order for it to have a finite norm, α cannot be real. We must go to imaginary dimension space and take α ∈ iR. 6 The result is that to any four-point function f (z), we may associate a densityf (α) =f (−α) via the invertible transform .

(2.8)
It is now clear that OPE coefficients may be read off from the residues off (α) whenever its poles are on the real axis. A formula that [21,22] derived using this method is We will use this in the current section and the next one. Specializing (2.9) to the four-point function (2.2), we immediately find There are two ways to assess the positivity of (2.10). The first is to recall the definition of a continuous Hahn polynomial [30]. Clearly, is a valid rewriting of (2.10). 7 Suppressing their parameters, the polynomialsP n (a, b, c, d; x) satisfy the following recurrence relation: .
(2.13) 6 There is another name for this space as evidenced by the title of [21]. 7 In the notation of [31], we would write c n = 2 −n n! Q 4 3 ∆σ+n n,0 For our parameters, we may easily check that A n +B n +a+x = 0. It then follows by induction that all c 2k+1 vanish. Once we know this, (2.13) is effectively a two-term recursion. Seeing a positive constant of proportionality iñ we conclude that the sequence c 2k decays to zero monotonically from above. It is therefore imperative that the bootstrap single out a region that includes (1.5). To derive this without referring to continuous Hahn polynomials, one may instead express c n in terms of gamma functions. This is possible because Watson's theorem [32], applies whenever p = 2q in (2.9). Although (a + b − 2c) < 1 is usually needed for convergence, we may drop this requirement for a hypergeometric function that terminates. This analysis does not explain why (1.5) saturates the lower bound in the one-correlator result of Figure 1. However, it is encouraging that bounds of this form in one dimension have been proven in [33]. The Coulomb gas formalism does not yield an obvious way to solve for the correlators σσ , or even to verify that they exist. Because the three-correlator plot in Figure 1 excludes this line, any theory to which σσσσ could extend would have to be highly non-unitary.
In this section, we have seen two methods for proving that global block coefficients in (2.2) are positive. One uses Watson's theorem and the other uses a recurrence relation for orthogonal polynomials. We will need both of these methods when we prove positivity in the generalized minimal models. Before moving on, there is an interesting way to check our results in a spacetime with Minkowski signature. Even though (2.2) is not strictly a correlation function in a unitary theory, it is still bounded in the Regge limit. Its sl(2) block expansion should therefore be calculable with the conformal Froissart-Gribov formula [34] which in our case reads . (2.16) To define the double-discontinuity, we must treat z,z as independent variables and rotate around thez = 1 branch point. Since this can be done in two ways, we subtract the average from our four-point function to find dDisc [g(z)g(z)] = 2 sin 2 π∆ σ 3 (zz) Performing the factored integrals yields a spectral density given by with poles at ∆ − = 4 3 ∆ σ + 2n. As none of these are integers, the correct prescription for finding OPE coefficients is to simply take the residue [34].
Although it is not obvious, we have checked that (2.19) is equal to c n c n+ by using Watson's theorem twice.

The upper line: One and three correlators
Each state |h r,s dual to a degenerate operator in a generalized minimal model has a null descendant |χ r,s at level rs called a singular vector. Four-point functions may be calculated once the necessary singular vectors are known. We simply convert χ r,s to a differential operator with the Ward identities and use the fact that this operator must annihilate any correlation function involving φ r,s in the first position. The resulting equation, known as a BPZ differential equation [1], has rs linearly independent solutions representing the exchanged multiplets. One may read off their dimensions by looking at the O(z h ) behaviour as z → 0. While expressions for singular vectors are generally non-trivial, with some appearing only recently [35][36][37], the ones we need are relatively simple: It is clear that we may confine ourselves to the border of the Kac table when studying  Fusion rules Weights correlators of σ ≡ φ 1,2 and ≡ φ 1,3 . The set of operators φ 1,s which closes under fusion is called the Verlinde subalgebra. We will continue to parametrize the generalized minimal model by ∆ σ -the horizontal axis of Figure 1. For convenience, Table 2 summarizes the OPEs that are important for the simplest mixed correlator system.

All global block coefficients in the simplest correlator
We will now derive new expressions for the squared OPE coefficients in σσσσ along (1.4). Positivity, as predicted by [17], will then follow from methods analogous to those in the last section. Since this correlator solves a second-order BPZ equation, we may write it as where the functions of z are Virasoro blocks. 8 The specific operator, read off from (3.2), is Acting on (3.3) with (3.4), we arrive at a PDE in terms of (z 1 , z 2 , z 3 , z 4 ). To reduce it to an ODE, we map these points to (0, z, 1, ∞) by a global conformal transformation: Well known solutions, which have the expected asymptotic behaviour, are First, let us look at the identity block. After a quadratic transformation, the hypergeometric function becomes a series in z 2 4z−4 . This makes the formula (2.9) applicable if we set p = 2n and q = n.
Exchanging the two sums, we find (3.8) for the non-vanishing global block coefficients. 9 Since p = 2q, this hypergeometric is in a form that can be treated with Watson's theorem. When we apply this to the c 2k , one pole in the numerator cancels another in the denominator. 10 Above, we can easily see three gamma functions that will cancel when we multiply by We have also written 1 Γ(n) 2 in a limiting form for later convenience. After substituting (3.9), one must use the identities until each term of (3.8) only depends on n through the Pochhammer symbol. This leads to The block can be analyzed in the same way. Doing so will in fact be easier since we will not have to pass to the δ → 0 limit. Starting from we may use (2.9) with p = 1+4∆σ 3 + 2n and q = 1+4∆σ 6 + n. This leads to which again allows us to use Watson's theorem. Employing (3.10) to perform the sum, we arrive at A useful observation about the hypergeometric functions in (3.11) and (3.14) is that they both have a parameter excess of 1. This means that they are Wilson polynomials [30]. 11 Invoking this notation, our results are As with continuous Hahn polynomials, there is a recurrence relation that the Wilson polynomials satisfy.
Appendix A uses this recursion to solve for the asymptotic behaviour of Wilson polynomials and prove that the ones in (3.16) are positive.

Some global block coefficients in the other correlators
Given our success at explaining the one-correlator results, the next logical step is to find the global block decompositions applicable to three correlators. We will start with . Although this is another four-point function of identical scalars, the main difference compared to σσσσ is the lack of a closed-form Virasoro block. Because a generic only has a null descendant at level 3, we will have to work with a third-order BPZ equation which does not have simple solutions analogous to (3.6). This makes the formula (2.5) important for finding low-lying OPE coefficients. The singular vector that must annihilate (1,5) (z) has four terms that can be read off from (3.2). The Virasoro commutation relations reduce it to the three term expression 4 The null state condition that follows from (3.18) and (3.19) is To approximate the Virasoro blocks that solve this, it will be helpful to use the Frobenius method. 12 Inserting the ansatz G(z) = ∞ k=−∞ b k z r+k , we may reindex the sum so that all terms carry the same power of z. This gives a recurrence relation for the coefficients.
We will set b 0 = 1 and b k = 0 for all k < 0. The values of r that make this consistent (called roots of the indicial equation) are h 1,1 , h 1,3 and h 1,5 as expected. We find them by demanding that b 0 drop out of (3.21) when k = −1. For each value of r, it is straightforward to iterate (3.21) and then feed the results into (2.5). Some of the OPE coefficients that follow from this are written in Table 3. Due to the appearance of the upper bound (1.4) in the three-correlator bootstrap, we expect the coefficients to be positive when 1 ≤ ∆ ≤ 2, at least up to some high order. Figure 2 shows that this is indeed the case. We should also be able to find positive squared OPE coefficients in the mixed correlator. The four-point function showing that the first five are all positive on the interval satisfies second-order and third-order BPZ equations. For simplicity, we will consider the second-order equation which has the recurrence relation  (1, 1), (1, 3) and (1, 5) contributions found by taking the × OPE twice.
It is easily seen that r ∈ {h 1,2 , h 1,4 } is the solution of the indicial equation for (3.24). Because the product σ × no longer has Bose symmetry, the procedure by which we extract the conformal block expansion this time is somewhat different. We must include dimension  n c  (1, 2) and (1,4) contributions found by taking the σ × OPE twice. differences in the hypergeometric function and sum over all integers whether even or odd.
The lower triangular system from this leads to the recursion Some low-lying global block coefficients found with (3.26) are listed in Table 4. While all of them are non-negative above the tricritical Ising value ∆ σ = 1 5 , there is actually one that takes on negative values for 1 8 < ∆ σ < 1 5 as shown in Figure 3. We may explain this by noticing that φ 1,4 is also φ 3,1 in the minimal model M (5,4). This field has exactly one quasiprimary descendant at level 3. The presence of a null state is therefore enough to conclude that c σ (1,4)σ 3 = 0. Alternatively, we can take the OPE in the other channel by permuting operator positions in (3.22). This yields the global block coefficients in Table 5 which are related to the λ σσO λ O CFT data. There is no reason for these numbers to be positive, but we expect to be obeyed. 13 When σ × σ and × share a set of operators S with the same quantum numbers, the left-hand side is a product of sums. The right-hand side is a sum of products and therefore smaller by the arithmetic-geometric mean inequality. A departure from (3.27) would be a violation of unitarity since the matrix would not be positive-definite. , which we could imagine to have a small effect, we have found that these coefficients are nonnegative when 1 8 ≤ ∆ σ ≤ 1 2 . However, this is only meaningful if the same property holds for the structure constants that unite holomorphic and anti-holomorphic halves of a four-point function. A single C (r 3 ,s 3 ) (r 1 ,s 1 )(r 2 ,s 2 ) < 0 for instance would give rise to an infintie number of negative contributions in φ r 1 ,s 1 φ r 2 ,s 2 φ r 1 ,s 1 φ r 2 ,s 2 , severely complicating the interpretation of Figure 1.

Virasoro block coefficients
In a given correlation function, the Virasoro block coefficients that appear must be compatible with crossing symmetry and single-valuedness. By briefly reviewing the method of [38][39][40], we will show that this condition is enough to fix them uniquely. For definiteness, consider the correlator in the generalized minimal model with central charge 13 21 . The holomorphic part of this function comes from the kernel of the operator (3.19), which is three-dimensional. Solving (3.21) has given us a basis for this kernel in which each function vanishes at z = 0. We could equally well have chosen any of the regular singular points 0, 1 and ∞, corresponding to the s, t and u channels for Virasoro blocks. When going from z = 0 to z = 1, there is a special matrix F called the crossing matrix (or fusion matrix ) 13 For another quick check of our results, ∆ 2 c holds as it must by the Ward identity.
It is a special case of the crossing kernel which applies to theories with a continuous spectrum. 14 Since it represents a change of basis, the expression for F is unique. In this case, it is given by which was obtained in [43] via the Coulomb gas formalism. Later reviews are [23,44]. In the following, we will use G 1 , G 2 and G 3 to denote G (1,1) , G (1,3) and G (1,5) respectively. The constraints from (3.29) are best phrased in terms of a metric on the space of conformal blocks; G(z,z) = W ab G a (z)G b (z). We have three conditions that W ab must satisfy: The first of these is an obvious consequence of having unit-normalized operators. The second comes from writing in terms of s-channel blocks and setting it equal to G(z,z). The third ensures that the four-point function has trivial monodromy under z → e 2πi z,z → e −2πiz . This rules out a non-diagonal metric since we only find h r,s weights that differ by integers at special values of m. 15 We build solutions out of the left eigenvectors of F , which have eigenvalues of 1, 1 and −1 respectively. Clearly F 2 = 1, which follows from z ↔ 1 − z being an involution, requires all eigenvalues to be ±1. The following form is invariant under two multiplications by F : There are six off-diagonal components of W that need to vanish. If we set c 21 = c 12 , (3.31) becomes manifestly symmetric and we can use three more parameters, c 11 , c 12 and c 22 , to make W diagonal. As the single remaining parameter, c 33 is used to rescale W so that its leading component is 1.
We may summarize this discussion by stating that there is no freedom in the threecorrelator bootstrap equations once the Virasoro blocks involving σ and are specified.
14 Several interchangeable terms have proliferated over the years. When replacing the z → 1 − z map with z → 1 z , the words crossing and fusion become exchange and braiding. Outside the CFT context, one says that a linear ODE has a monodromy matrix or connection matrix. For ODEs that have less structure than a BPZ equation, finding this matrix is often a difficult problem. 15 It is well known that this happens in discrete minimal models. Some of the non-diagonal theories so constructed also satisfy the stronger requirement of modular invariance. (3.32) The last coefficient in (3.32) is clearly the one that was rederived in [17]. Plotting these in Figure 4, we see that C (1,5) (1,3)(1,3) < 0 for the generalized minimal models between M(4, 3) and M (5, 4). This reveals a problem with our strategy for proving that the allowed region in Figure 1 must be large enough to include (1.4). Constructing the generalized minimal model solution to crossing symmetry only accomplishes this in the one-correlator case. We must therefore conclude that there is at least one other way to extend the unitary subsector σσσσ into a consistent three-correlator system. This solution to crossing should have positive squared OPE coefficients wherever it exists, not just in 1 5 ≤ ∆ σ ≤ 1 2 . As the solution might be very different from the theories discussed above, it is worth using the numerical bootstrap to see what else can be learned about it.

Lessons for the bootstrap
We saw in the last section that above central charge 7 10 , the generalized minimal models exhibit the restricted notion of unitarity that allows them to appear in Figure 1. On the other hand, for 1 2 < c < 7 10 , they are highly non-unitary at the level of three correlators; the global coefficient c σ (1,4)σ 3 and the Virasoro coefficient C (1,5) (1,3)(1,3) both become negative in this region. Working around this problem, the machinery of the bootstrap has filled in this region with another solution whose σ × σ OPE agrees with that of a generalized minimal model. In this section, we will give an intuitive argument for why this should be possible. Beyond this, we will discuss two issues related to the replacement solution.
This first is whether it can be found uniquely. A technique called the extremal functional method is often used to extract a unique solution to crossing symmetry and unitarity whenever a dimension gap or OPE coefficient is extremized [29,[45][46][47][48][49][50]. Based on this, we might expect to find a single line of exotic solutions that smoothly joins the M(m + 1, m) line at ∆ σ = 1 5 . We will actually find the opposite -a boundary of Figure 1 that has many possible choices for the local CFT data outside σ × σ. To reconcile this with the standard lore about extremality, one has to remember that the bootstrap equations take on a more intricate form when there are multiple correlators.
The second is the prospect of excluding the above solution with further numerics. One reason for doing this with global conformal blocks is simply the technical challenge posed by Virasoro conformal blocks. There has indeed been recent progress in using the full Virasoro symmetry to carve out c > 1 CFTs [51]. However, tractable four-point functions with extended supersymmetry appear to be limited to those of BPS operators [52,53]. Global blocks were therefore a necessary ingredient of [54], a program which aims to constrain the space of superconformal theories using external operators in long multiplets. There has also been recent interest in conformal theories that have no locality and therefore no Virasoro algebra [55][56][57]. These provide a different motivation for shrinking the regions in Figure 1.

Reduction to one correlator
The well known bootstrap constraints for three correlators with Z 2 symmetry take the form of five crossing equations. As reviewed in Appendix B, the vector of equations has one component for σσσσ , one component for and three components for σσ . Given a generic solution to crossing, it is easy to see that four (three) sum rules will break when an even (odd) operator is removed from the theory. However, there is a pleasing non-generic property that holds for generalized minimal models; × contains more operators than σ × σ. Because of this, only one crossing equation is disturbed when we remove the φ 1,5 conformal family. This is the source of almost all unitarity violation in the system built from φ 1,2 and φ 1,3 . Since negativity of c σ (1,4)σ 3 only affects spinning operators with ∆ > 45 8 , it is possible that the numerics are largely insensitive to it [58]. Assuming that problems with the mixed correlator are negligible, we will focus on as the single condition that needs to be restored. Once a solution to (4.1) is found, one can incorporate it into the three-correlator problem by choosing λ σσO = 0 for new operators. 16 Figure 5: Dimension bounds for irrelevant operators in (4.2). The dotted line shows the dimension of the primary scalar φ 1,5 whose multiplet needs to be replaced for 3 < m < 4.
Checking the solvability of (4.1) for real λ O is the simplest numerical bootstrap problem. Emphasizing the contributions of operators that are already present, we may write Keeping operators with ∆ ≤ 30, we have used the results of the last section to approximate the right-hand side of (4.2). In the sum over operators, there will be a continuum of irrelevant scalars not in φ 1,1 ∪ φ 1,3 which begins at some gap ∆ * > 2. If this is the only set of scalars on the left-hand side of (4.2), we are dealing with the set S = {∆ > ∆ * }. Alternatively, we could allow the dimension ∆ to appear again and enlarge it to The second choice is the one applicable to Figure 1. However, in Figure 5, we consider the first choice as well. This is because it is possible to rederive Figure 1 under the requirement that is non-degenerate [16]. As numerical accuracy is improved, we expect the blue line to precisely meet the dotted line at m = 4. 17 In Figure 5, the red curve tells us that (1.4) is admissible whenever we treat (∆ σ , ∆ ) as allowed dimensions and perform a two-parameter scan. The blue curve tells us that (1.4) will still be admissible when we fix as a single operator at angle θ = arctan λσσ λ and scan over (∆ σ , ∆ , θ). Finally, one may contemplate the effect of imposing θ = π 2 which is one consequence of Kramers-Wannier duality in the Ising model. Although this question cannot be answered with Figure 5, we have found that (1.4) persists yet again. In all three cases, (1.4) is not just an allowed line -it is the maximal allowed line. From this, we must conclude that many solutions to crossing, labelled by values of ∆ * between the red and blue lines, lie along the bound on the left side of Figure 1. This signals the presence of a flat direction, e.g. a bound in (∆ σ , ∆ , θ) space which is independent of θ. A flat direction in the modular bootstrap was previously seen in [59]. The main argument for unique extremal functionals comes from [46], in which the multi-correlator bootstrap equations were augmented with angles for each operator in OPE space. To extract a spectrum in this formulation, one would have to look for zeros of these functionals on the entire (∆, θ) plane. We suspect that the flat direction here corresponds to a zero being achieved on a codimension-one locus. Figure 6: Tensor product theories that are allowed in both sides of Figure 1. Since we must have ∆ ≥ 1, all other tensor products involving the free field vertex operators necessarily lie to the right.
Because the generalized minimal model line (1.4) is allowed by the bootstrap, there are several lines in the interior of Figure 1 that must be allowed as well. These can be constructed through one or more tensor products. If we multiply two generalized minimal models for instance, the only non-trivial operator whose dimension lies to the left of ∆ σ = 1 is σ ⊗ σ. Writing its OPE schematically, includes two relevant operators. In order for these to have the same scaling dimension, the M(m + 1, m) ⊗ M(m + 1, m ) product must have m = m . Expanding the search to include free theories and generalized free theories, it is a simple exercise to check that the lines in Figure 6 all have one relevant Z 2 -odd scaling dimension and one relevant Z 2 -even scaling dimension. Evidently, it is not possible to isolate particular minimal models in a three-correlator bootstrap by specifying the number of relevant operators. One has to consider more stringent assumptions or add more correlators. The former approach was discussed already in [60], where it was found that the one-correlator Ising kink sharpens considerably when scalars are restricted to lie in S = {∆ > 3} ∪ {∆ = ∆ }. This kink becomes an island when a similar restriction is made for a three-correlator system. Following [61], it is likely that one can obtain this island from a single correlator by imposing large gaps in the spin-0 and spin-2 sectors. Negative for m < 4 Table 6: Virasoro block coefficients (other than the ones in Figure 4) appearing in four-point functions made from σ, and σ . Only one is non-negative for all m ≥ 3.

Coefficients Signs C
A more ambitious goal is to produce islands under minimal assumptions by introducing a third external scalar. Taking this external scalar to be odd, plots along the lines of Figure 4 offer some preliminary insight. 18 Regions where all C (r 3 ,s 3 ) (r 1 ,s 1 )(r 2 ,s 2 ) > 0 are likely to survive, but as Table 6 shows, there can be several negative structure constants with more than three correlators. We have seen that the negative constant C (1,5) (1,3)(1,3) in the threecorrelator system was innocuous because it did not appear in any mixed correlators. It is therefore encouraging that the coefficients C  (1,4) , which have first-order zeros, participate in σσσ σ and σ σ respectively. Lest we become too encouraged, it is important to note that ∆ σ is defined as the starting point for a continuum of irrelevant operators. This represents a fundamental difference as compared to the one-correlator and three-correlator analysis. It remains to be seen whether we can still derive strong bounds from a scan over two isolated scaling dimensions and one non-isolated scaling dimension.

Conformal manifolds
Even though the tensor product lines above all have ∆ σ > 1 4 , there could be other CFTs in 1 8 < ∆ σ < 1 5 with sufficiently few relevant operators to survive the constraints of σ, and σ . An interesting possibility, that would complicate the search for islands, is a continuous line of theories ending somewhere close to the Ising model. We may search for an example by using the extremal functional method on solutions to crossing that involve a scalar Φ of dimension 2. 19 This provides another opportunity to predict a mixed correlator result without actually bootstrapping more than one correlator.
The first step is to maximize λ 2 σσΦ in the range 1 8 < ∆ σ < 1 2 with the constraint that all scalar dimensions are above 2∆ σ . Each point saturating the bound yields a spectrum with the marginal deformation Φ. To see that this set of solutions is not a conformal manifold, we may check that the central charge varies with ∆ σ . Specifically, it reaches a minimum value of c ≈ 1.12. It is then straightforward to force our putative theories to have this central charge (or any larger value) by taking in the usual one-correlator crossing equation. To prevent (4.4) from being undone by a second copy of the stress tensor, we have imposed the gap ∆ T > 2.1 on spin-2 operators. Performing a second pass in this way, we have found low-lying dimensions and OPE coefficients with examples plotted in Figure 7. With constant c and ∆ Φ , these CFT data appear to satisfy all constraints of a conformal manifold that are directly accessible to σσσσ . Despite this, there is a method from [29] that can be revived to gain indirect information about the next four-point function, which we will call . The key is that the dimensions in σ × σ are also the dimensions in any OPE between identical scalars. Given a sufficiently long list of scaling dimensions, there is no reason why a search for optimal OPE coefficients has to be done for σ × σ rather than × . We confirm this in Appendix B, by taking an approximate Ising model spectrum and performing the same type of fit that was done in [29]. Our results show that the estimation of λ 2 σσO could have been extended to λ 2 O without the three-correlator crossing equations. The fitting procedure therefore differs from the primal method of [45], which is more accurate but limited to the direct correlators under study. Assuming that the method can be trusted, at least for 2D theories, we now have access to × coefficients including λ Φ . This allows another property of conformal manifolds to be tested. When CFTs are parameterized by an exactly marginal operator with coupling g, there is a complicated set of differential equations that their local data must satisfy [64,65]. The simplest of these is d∆ immediately follows. The two sides of this equation have been plotted in Figure 8. Even though λ Φ is not known with high precision, the noise in the data seems much too small to   explain the violation of (4.5). We must conclude that if a conformal manifold in 1 8 < ∆ σ < 1 2 allowed by Figure 1 exists, it is not privileged enough to be found by this one-correlator exercise. It would be very interesting to find other applications for the dual method in [29]. Unfortunately, Appendix B shows that results become much less reliable above d = 2. A possible reason for this was given in [47] which noticed a surprising preference for doubletwist operators in the 3D numerical bootstrap. To review, double-twist families have the following schematic form for → ∞: The lightcone bootstrap requires them to appear in any CFT with a twist gap [66,67]. When d > 2, every operator except the identity has positive twist and the extremal functional method is able to find spectra dominated by (4.6). The absence of other operators interferes with our ability to fit OPE coefficients. Convsersely, 2D theories have no need for doubletwist operators as the identity, stress tensor and higher-spin currents all have τ = 0. The quality of our fit suggests that this resolves the main source of bias in the extremal spectrum.

Supersymmetric minimal models
So far, we have been concerned with finding the necessary correlators to single out theories that are minimal with respect to the Virasoro algebra. However, the unitary representations of the N = 1 super-Virasoro algebra also admit a discrete series for central charges below that of the free field.
actually describe two algebras since fermions do not have to be periodic in radial quantization. When the indices on G r are integers, (4.7) is the Ramond superalgebra, otherwise it is the Neveu-Schwarz superalgebra. The super-Virasoro minimal models, which contain representations of each, have a Kac formula given by If r − s is even (odd), this is a Neveu-Schwarz (Ramond) degenerate weight [68]. In either case, it is degenerate at level rs/2. It is clear by inspection that G ± 1 2 generate a subalgebra of (4.7) that is independent of c. For integer indices, on the other hand, no global subalgebra exists. A numerical bootstrap approach is therefore most readily accessible for the Neveu-Schwarz sectors of N = (1, 1) theories.
In the global algebra, which is osp(2|1), primary operators may be written as superfields; Φ(z, θ) = φ(z) + θψ(z). The superspace distance, which enters in correlation functions, is Z ij ≡ z i − z j − θ i θ j . Even though cross-ratios in R d all involve at least four points, invariant combinations in superspace may be built using three points as well. The quantity is invariant under osp(2|1) [69]. As a result, the three-point function depends on more than just an OPE coefficient. The general expression for the chiral half is where ζ is an arbitrary Grassman number. Until recently, such extra parameters were eliminated by restricting the superconformal bootstrap to correlators of BPS operators [12,13]. We may indeed impose shortening conditions on (4.10), but due to the small amount of supersymmetry, this would require us to give up a lot. To be annihilated by a supercharge, each external operator would have to be the identity in at least one of the osp(2|1) factors. It is therefore preferable to leave (4.10) in its most general form and use superconformal blocks that include unknown coefficients reflecting the presence of ζ.
The authors of [54] computed some of the necessary blocks and introduced a framework that still allows the bootstrap to proceed. Their idea is to consider an entire multiplet at once, with the external correlators involving all combinations of a primary and its superdescendants. When this is carried out for N = (1, 1), the allowed regions will have to include all points corresponding to the SM(m + 2, m). The strongest statement we can make from this is that the line ∆ = 8 3 ∆ σ (4.11) must be inside the bound for ∆ σ < 1 8 . This comes from choosing the Neveu-Schwarz fields σ ≡ φ 2,2 and ≡ φ 3,3 . Right at ∆ σ = 1 8 , we find ourselves in the c = 1 model where φ 3,3 = φ 1,3 and the entire level-3 2 subspace decouples. To see this, we may check that is the unique quasiprimary state. Computing the norm and setting h → h 3,3 , we find with a first-order zero. This is the behaviour that we saw for m = 3 in the bosonic case, but now it occurs for m = 4. The tricritical Ising model, which has no φ 3,3 operator, lives at the point (∆ σ , ∆ ) = 1 10 , 1 10 .
The above calculation shows that a global block coefficient in the supersymmetric generalized minimal model line becomes negative for m < 4. If we are to see an associated kink, this line must saturate the bound on operator dimensions from the long multiplet bootstrap. This brings us to a crucial difference between M(m + 1, m) and the Neveu-Schwarz sector of SM(m + 2, m). In the former case, we saw the correct saturation with (1.4). The same cannot hold for (4.11) because it is strictly below the line for vertex operators. Writing a CFT vertex operator as e iqφ(z) and an SCFT vertex operator as e iqΦ(z,θ) , the two important properties are ∆ ∝ q 2 and additivity of q. These lead to ∆ = 4∆ σ which is allowed by the one-correlator region of Figure 1. Due to the restriction on the number of relevant operators, the three-correlator region omits this line until ∆ σ = 2 9 . It is therefore clear that treating four copies of the same multiplet with the methods of [54] is not enough. If our goal is to see a minimal model kink, the N = (1, 1) bootstrap will require multiple correlators at the superspace level. Since each of these must separately expand to a mixed correlator system, the resulting problem is likely to be numerically intensive.

Conclusion
Through a combination of analytic techniques and one-correlator numerics, we have explained an important aspect of Figure 1 -we have shown that the constraints on four-point functions of relevant operators are not strong enough to exclude the generalized minimal models. Constraints that are strong enough may be used in the conformal or superconformal bootstrap, but only when the number of four-point functions exceeds the maximum system size that has been tested to date.
Our analysis proceeded correlator-by-correlator. For σσσσ , the results were exact. The expressions (3.11) and (3.14) ensured that the global block coefficients involved were all positive. For σσ , all but one coefficient appeared to be positive upon using the approach of [17] -we simply expanded in sl(2) blocks to high order and conjectured that the pattern continues to hold. For , two pieces of the M(m + 1, m) line had to be treated separately. In 4 < m < ∞, it was enough to compute coefficients again and see that there were no obvious signs of unitarity violation. However, 3 < m < 4 required an exotic solution having only partial overlap with the CFT data of a generalized minimal model. It was possible to see evidence of this in the one-correlator bootstrap because of a special property of φ 1,5 operators, namely their absence from the superselection sector σ × σ. This had interesting implications for the uniqueness results in [46].
A possible future endeavour is to put all correlators on the same footing as σσσσ . In order to do this, one does not necessarily have to solve for c were ultimately shown to be positive using only the recursion for Wilson polynomials. Instead, the main challenge in extending the positivity proof is expressing global block coefficients as solutions of one recurrence relation instead of two. Our current approach, based on the BPZ equation, is awkward in this respect. It uses the Frobenius method to evaluate Taylor coefficients and then feeds these into a second recursion to obtain global block coefficients. It is worth checking if hypergeometric identities can be used to derive a recursion that operates on global block coefficients directly. It would also be interesting to take a closer look at super-BPZ equations. Our discussion surrounding the vanishing norm (4.13) can be made more systematic if we also check how other osp(2|1) blocks appear. BPZ differential equations in superspace have been studied in [68,69] and some of them are second-order. This is exactly what we need to go beyond recurrence relations and apply the decompositions in [21,22]. Additionally, these methods could be applicable to the KZ equations associated with extended chiral algebras. A realistic hope is using them to explain a numerical bound in [70] which interpolates between W 3 -minimal models.
The last possibility we have discussed is an extension of the extremal functional method -reviving the fit in [29] to estimate more OPE coefficients than the ones that are known to high precision. While we only used this to demonstrate a null result, it would be interesting to find a further use for it in two dimensions. Complications in higher dimensions arise due to the privileged role of double-twist operators in the numerical bootstrap; a result that is not fully understood [47]. One should be able to get a sense of how robust it is by studying alternative bootstrap algorithms such as the one in [71].

Acknowledgements
This work was partially supported by the Natural Sciences and Engineering Research Council of Canada. Numerical results in this paper were obtained using the high-performance computing system at the Institute for Advanced Computational Science at Stony Brook University. I am grateful for discussions with Shai Chester, Liam Fitzpatrick, Anton de la

A Linear difference equations
We have regularly encountered linear recursions with three terms such as (2.13) and (3.17). Asymptotic analysis of sequences obeying these relations is a well understood subject, going by the name Birkhoff-Trjitzinsky theory. The following theorem summarizes a number of results from it [72]. 3. Otherwise, consider the roots α 1 and α 2 of the indicial equation α(α − 1)ρ 2 + (a 1 α + a 2 )ρ + b 2 = 0, ordered according to α 2 ≥ α 1 . If α 2 − α 1 / ∈ Z ≥0 , the solutions satisfy y j (n) ∼ ρ n n α j ∞ s=0 c s,j n s . 4. Otherwise, let m = α 2 − α 1 . The asymptotic expansion for the first solution is unchanged from the previous case but for the second solution we must use y 2 (n) ∼ ρ n n α 2 ∞ s=0 ds n s − dm n m + c log(n)y 1 (n). Taking (A.1) to be the recurrence relation for Wilson polynomials, we find the roots ρ = 1, α 1 = −2(a + x) and α 2 = −2(a − x) in the third case of the theorem. Each coefficient in (3.16) can then be written as a linear combination of two functions asymptotic to power laws. As seen in Table 7, all of them decay to zero. When comparing to the results of [73,74], one must remember that c  Table 7: Decay rates of the fundamental solutions that comprise two of our main results. We have not included the prefactors in (3.16). These will make the convergence much faster, namely (1/16) n , which can be predicted from the growth rate of K 2n (1).
Our main claim about c σσ(1,1)σσ 2n and c σσ(1,3)σσ 2n -that they are positive for finite ncannot be proven with asymptotics. Instead, we will use a theorem from [75] which bounds the ratio between neighbouring terms in a sequence.
Theorem 2. Let x(n) be a solution of where a(n), b(n), c(n) and d(n) are degree-k polynomials with positive leading terms. Also define f (n) = a(n + 1)d(n + 1) − 2b k a k b(n + 1)c(n + 1) − a k 2b k b(n + 1)d(n + 1). Finally, let m be an integer large enough to guarantee that b(n), c(n), d(n) and f (n) have positive values for n ≥ m. If x(m) x(m−1) > a k 2b k then x(n) x(n−1) > a k 2b k for n ≥ m. Proof. The theorem above is more effective at identifying increasing sequences than ruling out changes of sign directly. Therefore, we will work with x 1 (n) = n 2 y 1 (n) and x 2 (n) = ny 2 (n) -expressions that have been guided by the asymptotics in Table 7.
These are clearly positive for n > 2. Also, by writing out the polynomial for a(n), we find a leading coefficient of a 6 = 2b 6 = 96. It remains to check f (n) or equivalently (n + 2) −1 (2n + 3) −1 (4n + 1) −1 (6n − 4∆ σ + 11) −1 f (n). This is a sixth degree polynomial in which 64∆ σ (∆ σ + 1)n 6 is followed immediately by negative coefficients. From this we see that the critical value of n, beyond which f (n) > 0, increases without bound as ∆ σ → 0. This reflects the fact that the n 2 we introduced is only able to overpower n 2∆σ−2 for strictly positive ∆ σ . Fortunately, the smallest value of ∆ σ that we consider is 1 8 leading to a critical value of n = 44. Since x 1 (44) > x 1 (43), we establish positivity of the entire sequence x 1 (n) by checking its first 44 terms.
Although we do not have closed-form solutions for them, it is possible that c are positive as well. The main hint of this, which we now prove, is that the Virasoro blocks containing them have positive Taylor coefficients around z = 0. The fact that this is a necessary condition follows trivially from expanding g(z) = ∞ n=0 c 2n K r+2n (z). The analogous statement in higher dimensions was proven in [76].
Proposition 2. Let b k be a sequence starting at b 0 = 1 with the rest of the terms given by (3.21). If ∆ > 1, the sequence monotonically increases.
Proof. Defining K = k + r for brevity, we have Although this has four terms, we will convert it to a simpler recursion having only three. We do this by assuming that the piece with b k−1 and b k−2 is bounded by some function of Using (A.6) in (A.5), we find an expression of the form R + (K)b k+1 > R 0 (K)b k +R − (K)b k−1 . We will perform a rescaling to instead write this as For our assumption to be true, the fractional coefficients in (A.7) must exceed the K → K+1 versions of the ones in (A.6). These two conditions each give one side of an inequality for M . The result is 10∆ 2 + 20∆ + 4 < M < 10∆ 2 + 29∆ + 4 which may be satisfied for any ∆ > 0. Having chosen M appropriately, we have moved the problem into the domain of the theorem above. The monotonicity proof for b k is now identical to the one for n 2 c The conformal bootstrap is any technique for demanding that crossing symmetry and unitarity hold for the four-point function: The conformal blocks g . Invariance under (1, i) ↔ (3, k), which relates two channels of crossing symmetry, leads to the following sum rule [14].
To apply this rule, we choose an odd scalar σ and an even scalar and let our external operators run over all admissible combinations of these. Using λ σ O = (−1) λ σO , this yields In ruling out solutions to (B.3), which is a set of five functional equations, we must approximate each row as a finite-dimensional vector. The standard way to do this is to expand around the point (u, v) = 1 4 , 1 4 . We may either take derivatives with respect to z,z, defined by u = |z| 2 , v = |1 − z| 2 , or the diagonal / off-diagonal variables a = z +z, b = (z −z) 2 [29]. We choose a, b and control the order of our derivatives ∂ m+n g ∆, ∂a m ∂b n with two parameters m max and n max : Since half of the derivatives vanish when our conformal blocks are added or subtracted, the resulting number of components is N = (n max + 1)(m max + n max + 1)/2 . (B.6) The three-correlator plot in this work was obtained with (m max , n max ) = (3, 5). All onecorrelator results, on the other hand, use (m max , n max ) = (5, 10). There are two additional parameters needed to turn (B.3) into a concrete bootstrapping problem. One is a cutoff on the number of spins, which we take to be max = 30. The other is the accuracy parameter for a single conformal block, which we take to be k max = 40. This controls how many poles from the triple series appear in the function As explained in [14,77,78], there are algorithms for explicitly constructing each conformal block derivative as a rational approximation: ∂ m+n ∂a m ∂b n F ij;kl ±,∆, (a = 1, b = 0) = χ (∆)P ij;kl;mn ±, Here, P ij;kl;mn ±, is a polynomial with the same degree as χ for m = n = 0. Its degree goes up by one whenever the derivative order is increased. Our task of inputting (k max , max , m max , n max ) and computing a table suitable for approximating (B.3) is accomplished with the program PyCFTBoot [79].
With the truncations described above, problems of this form are tractable with semidefinite programming [15]. In the dual formulation, one wishes to find a linear functional y which sends each term of (B.3) to a positive-definite matrix, thereby certifying that no solution to crossing symmetry exists. For illustrative purposes, we consider a single correlator problem which allows us to drop the ij; kl and ± labels on P ij;kl;mn ±, . We will also drop mn through our understanding that P is a vector with components P i . If we single out the contribution of the identity operator as n, we arrive at the polynomial matrix program (PMP) where we include an objective b for generality.
After all, the crossing equation becomes a contradiction when y solving the above conditions is applied to both sides. If we reshuffle each vector according toP dotting P with a functional whose action on n is 1 becomes the same as dottingP with a functional whose leading component is 1. This is precisely the choice to work with crossing equations projectively as (B.11) becomes k, after reshuffling both sides. Any spectrum satisfying the bottom row can automatically be made to satisfy the top row through a rescaling. From now on, we will denote the polynomial vector in the bottom row of (B.13) by P to rewrite the PMP.
maximize b T y such that P 0 (x) + P (x) T y ≥ 0 for all ≤ max , x ≥ 0 (B.14) Here, x = ∆ − ∆ min . To solve this type of problem efficiently, we use the program SDPB [15]. Because we will see an alternative choice shortly, we briefly review the process by which SDPB translates (B.14) into a semidefinite program (SDP). Positivity of P 0 (x) + P (x) T y on the half-line is equivalent to the requirement that it be equal to where Y is positive-definite and q is a vector of orthogonal polynomials. We have abused notation slightly since the maximum power of x that (B.15) needs to express may be even or odd. Because of this, the first q might have one more component than the second q . It is sufficient to demand this equality on a set of sample points which we denote x k . It is also possible to combine all Y (k, ) into a single matrix Y . Making the identifications In the numerical bootstrap, (B.17) and the primal problem corresponding to it are typically solved together, in order to see which one becomes feasible first [15].

B.2 The extremal functional method
It was shown in [29] that solutions to crossing symmetry may be built by locating the zeros of y. This functional may be found either by ruling out CFTs just outside the allowed region or by maximizing an OPE coefficient just inside it. Ideally, elements of the spinspectrum are dimensions ∆ k such that where P is one of the polynomial vectors appearing in (B.9). On the other hand, numerical errors usually prevent this polynomial from ever reaching zero for real ∆. There are two approaches, both based on the spectrum.py script [47], which make it easy to account for this. The first is to run the script as written after spending several iterations to bring the primal and dual solutions close together. A highly converged functional is needed since spectrum.py assumes that the would-be zeros are close to the local minima of (B.18) which uses only the polynomial numerator. The second is to modify the script to use a nonpolynomial function minimizer, allowing us to multiply (B.18) by the prefactor χ from (B.9). The advantage is that when y acts on a full convolved block, the local minima are closer to the physical ∆ k . As a result, punishing SDPB parameters are no longer required. For the data in Figure 7, we have used the first approach and specified --precision=660 --dualityGapThreshold=1e-75 as the non-default parameters. The rest of the script obtains high precision OPE coefficients directly from the primal solution. 20 To plot the points in Figure 8, we have used an older method which minimizes the error in a set of crossing equations with known scaling dimensions. 21 Once a set of Z stable operators has been found, we may consider a truncated crossing equation of the form Z k=1 a k F k = n . (B.19) When studying a single correlator involving Z 2 -even operators, we make the identifications The point is that φ may denote σ, or indeed any scalar whose self-OPE is well approximated by the Z operators in our set. We regard (B.19) as a set of N > Z linear equations with N given by (B.6). A naive approach is to remove rows corresponding to high derivatives bringing the number of equations down to Z. This often leads to a k coefficients that are negative, even when we use dimensions that are accurate to four digits. To overcome this, we follow [29] and leave one extra equation so that N = Z + 1. With this overdetermined system, we will not be able to find a k that satisfy (B.19) exactly. Rather, we find the a k that minimize the distance between the left and right sides of (B.19). In this fit, the constraint a k ≥ 0 may be specified by hand. If our norm for this is the 1-norm, it is convenient to introduce a vector of positive entries t such that It is amusing to point out that a reader using SDPB may continue to use it for solving (B.23) because every linear program is also a semidefinite program. To do this, the (2Z + 2) × (2Z + 1) and (2Z + 2) × 1 matrices B and c must be enlarged to (4Z + 3) × (2Z + 1) and (4Z + 3) × 1 so that they encode component-wise positivity of y in addition to (B.21). The next necessary step is replacing b with −b (and prepending an arbitrary number) as SDPB's objective is maximized instead of minimized. Finally, the orthogonal polynomials should all be 1. With this choice, constraint matrices in (B.17) simply pull off a diagonal component of Y and c − By = Tr(A * Y ) reads c − By ≥ 0. In any case, whether (B.23) is solved as an LP or SDP, it is clear that the returned y will give us the OPE coefficients λ 2 φφO k .   [29].
To verify that this gives reliable values for a k in two dimensions, we have compared exact and approximate results in the Ising model where × operators are also in σ × σ. Starting with ∆ φ = ∆ σ , the λ 2 σσO coefficients are very close to the ones found in [29]. Changing the external dimension to ∆ φ = ∆ , the same algorithm produces accurate λ 2 O coefficients as well. These are shown in Figure 9.
Turning to three dimensions, exact Ising CFT data are not available, but the tables in [47] have negligible error for our purposes. Despite all the progress in isolating this model, it appears that several spin-operators are still missing from these spectra: the ones that do not fall into [σσ] n , [ ] n and [σ ] n twist families. As evidence of this, Figure 10 shows that OPE coefficients fit with the dual method differ greatly from the ones returned by spectrum.py. To guess where the first missing operator appears, we can take the naive view that twist families exist all the way down to = 0. Constructing one out of an irrelevant operator Φ, we have ∆ ≈ 2∆ Φ + 2n + > 2d. Four-point functions approximated by bootstrap data have already proven useful for conformal perturbation theory and measuring non-Gaussianity [82,83]. As it is significantly affected by the incompleteness of the spectrum above ∆ * ≈ 2d, a fit to the crossing equations must be a less forgiving problem.   Figure 9 for the 3D Ising model. In the fit, we have used the Z 2 -even operator dimensions from [47]. This is already a longer list than anything that is likely to come from a one-correlator bootstrap. Due to a remaining bias in the spectrum, there is strong disagreement between the primal and dual methods.
Although we have not done so in this work, it should also be possible to fit mixed OPE coefficients in a 2D theory. The last two rows of (B.3) would lead to coupled quadratic equations which rapidly become impractical to solve as our system grows. Therefore, we will focus on the third row. If only the identity has been singled out, this is a homogenous equation and its parameters are ambiguous up to a rescaling. This is why it helps to find OPE coefficients in a prescribed order. As long as λ φφφ 2 has been found from the φ × φ fit described above, permutation symmetry of this coefficient may be used to make our equation inhomogeneous. In this case, it takes the form of (B.19) but now with a k = λ 2 Essentially, φ plays the same role that the identity operator played before.