Bootstrapping 3D fermions

We study the conformal bootstrap for a 4-point function of fermions 〈ψψψψ〉 in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the ψ × ψ OPE, and also on the central charge CT . We observe features in our bounds that coincide with scaling dimensions in the GrossNeveu models at large N . We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.

The conformal bootstrap [1][2][3], originally successful in elucidating 2D conformal field theories (CFTs), has recently become a powerful method to constrain the operator algebra of unitary CFTs also in D > 2 spacetime dimensions. The origin of this development is the observation made in [4] that the combined constraints of crossing symmetry and unitarity on a 4-point function of scalars can be explored numerically. This method achieved an impressive degree of success, for example, by enabling accurate determinations of the dimensions of low-lying operators in particular CFTs, such as the 3D Ising [5][6][7][8] and critical O(N ) vector [9,10] models. While the original ideas of [4] have been developed in myriad ways in subsequent works, the basic objects of study have always been 4-point functions of scalar operators. The goal of the present paper is to extend the bootstrap toolbox to study 4-point functions of fermionic operators in 3D CFTs.
There are many motivations for bootstrapping fermionic correlators. Fermionic operators exist in many interesting CFTs, though they do not appear in the operator product expansion (OPE) of scalar operators. Therefore, in order to access this sector of the operator algebra, one must study correlators of fermionic operators. More generally, it is of great interest to apply the conformal bootstrap to 4-point functions of operators with non-zero spin. For example, studying the 4-point function of a global symmetry current would lead to universal bounds on all CFTs that admit the corresponding symmetry, without making any additional assumptions on their operator content. Similarly, bootstrapping the 4-point function of the stress-tensor would allow for the most general constraints, since the stresstensor exists in any local CFT by definition. 1 Our numerical study of fermion correlators can be seen as a small step towards implementing the bootstrap for operators with spin, as in the examples discussed above.
In implementing the bootstrap for correlators of fermionic operators, or of operators with spin more generally, one faces two difficulties. The first is that the explicit form of the conformal block decomposition for higher-spin correlators is not known in general for D > 2 dimensional CFTs. The exception occurs in D = 3, where the conformal blocks for external operators with integer spin can be determined by acting with certain differential operators on the scalar blocks [17]. As we will see, a similar strategy applies for operators with half-integer spin. The second difficulty has to do with the proliferation of conformal invariants that can appear in higher-spin correlators. In particular, the number of conformal invariants grows with the spin of the insertions (see, for instance, [18,19]). As we will explain, a 4-point function of primary operators of spin-1/2 generally depends on 16 independent conformal invariants. Imposing parity symmetry reduces the number of invariants to 8, and this number can be reduced further to 5 if we assume the fermions are identical. In our numerical analysis, we will focus for simplicity on this latter case and derive the conformal block decomposition of a 4-point function of identical fermionic operators in a 3D CFT.

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With the conformal blocks in hand, we then embark on a systematic study of CFTs with a small number of relevant operators. By imposing gaps in the low-lying spectrum of parity-even and/or parity-odd scalar operators, we use the logic originally introduced in [4] to derive constraints on the dimensions of the first few scalar operators and on that of the fermionic operator whose 4-point function we study. We also find a lower bound on the coefficient C T that appears in the two-point function of the canonically-normalized stress tensor.
From these studies we find two exciting results. Firstly, the general bound on the dimension of the leading parity-odd scalar ∆ σ possesses a severe discontinuity at a fermion dimension of ∆ ψ ∼ 1.27. This coincides with a kink in the general bound on the leading parity-even scalar dimension ∆ . Based on these features we conjecture the existence of a fermionic 3D CFT containing no relevant scalar operators. While we do not know of a Lagrangian that would give rise to such a theory, this conjectured "dead-end" CFT would furnish a concrete example of self-organized criticality [20,21] in 3D.
The second result is that, when we allow a second relevant parity-odd scalar in the spectrum with dimension ∆ σ = 2 + δ for small values of δ, the resulting allowed region for (∆ ψ , ∆ σ ) possesses a sharp kink that appears to precisely coincide with the dimensions in the O(N ) Gross-Neveu models at large N . This is natural because in the large-N limit one has the expansion ∆ σ = 2 + 32/(3π 2 N ) + . . .. By tracking this feature at larger values of the gap we reveal information about the small-N Gross-Neveu models, including the N = 1 theory which is expected to have N = 1 supersymmetry. In addition to this sequence of kinks, we observe the emergence of a second discontinuity in the allowed region at (∆ ψ , ∆ σ ) ≈ (1.078, 0.565), which we conjecture could also coincide with a 3D CFT with fermionic operators and a large scalar gap. We believe that fully isolating these theories will require implementing systems of mixed correlators containing fermions and scalars, but based on the results of this study the prospects for learning more about these theories using the conformal bootstrap looks very promising.
The rest of this paper is organized as follows. In section 2 we start by reviewing the embedding space formalism for operators with spin and, using this formalism, derive the conformal block decomposition of a 4-point function of identical fermions. In section 3 we set up the crossing equations and outline the strategy we will follow in our numerical study. Next, in section 4, we present numerical results for bounds on dimensions of scalar operators in theories with fermions that satisfy various assumptions, and also present our lower bound on C T . We end in section 5 with a discussion of our results.
We label the 3D coordinates as x µ , with µ = 0, 1, 2, and use the Minkowski metric in mostly plus signature η µν = η µν = diag(−1, 1, 1). The coordinates x µ transform non-linearly under special conformal transformations. It is therefore convenient to introduce a different set of coordinates that transform linearly under the action of conformal transformations. Since the 3D conformal group is isomorphic to SO(3, 2), we can relate conformal transformations to Lorentz transformations in a 5D spacetime with metric η AB = η AB = diag(−1, 1, 1, 1, −1), where the indices A, B run from 0 to 4. The exact relation between the generators of conformal transformations and the SO(3, 2) generators J AB can be taken to be Let us denote 5D coordinates that transform linearly under SO(3, 2) by capital letters, X A . The way to embed the 3D coordinates x µ in 5D space is through the projective null cone, which is defined as the space of all points X A that satisfy the condition X · X = 0 and are identified up to a rescaling X A ∼ λX A . It will be convenient to use lightcone coordinates X ± = X 4 ± X 3 , and thus write the 5D coordinates from now on as X = (X µ , X + , X − ). The exact relation between x µ and X A is given by Note that the parameterization (2.2) obeys X · X = 0.

Embedding of scalar fields
To find the embedding of fields in 5D spacetime we follow the approach of [26]. Consider first a real scalar primary field φ(x). Its transformations under the conformal group are where ∆ φ is the dimension of φ. We can relate φ(x) to a scalar field Φ(X) defined on the lightcone in 5D as: where x is related to X through (2.2). Explicit calculation then shows that Φ(X) is a 5D Lorentz scalar, i.e. that it transforms under 5D Lorentz transformations as if and only if φ(x) is a primary scalar field in 3D with dimension ∆ φ .

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Note that the 5D field Φ(X) defined in (2.4) is a homogeneous function of X of degree −∆ φ . This property together with 5D Lorentz invariance restricts the form of correlation functions in embedding space. For instance, the two-point function takes the form Using (2.4), we can read off as expected. In a unitary theory, we must have c φ > 0. We will conventionally take c φ = 1 for scalar operators in this work.

Embedding of spinor fields
The above procedure can be applied to primary spinor fields as well. The main difference is that such a field ψ α (x) transforms in a spinor representation of the double cover of SO(2, 1), which is isomorphic to Sp(2, R). Under the full conformal group, it transforms as Here, upper (lower) indices α, β, . . . , represent fundamental (anti-fundamental) Sp(2, R) indices that are raised and lowered with the symplectic form Ω αβ = Ω αβ -see appendix A for our conventions. In 3D, the smallest spinor representation is a 2-component Majorana spinor, which is what we will focus on. If we take the 3D γ matrices to be real, as we do in appendix A, a Majorana spinor has real components, ψ α (x) * = ψ α (x).
In order to efficiently keep track of the 3D spinor indices, it is convenient to introduce a set of auxiliary commuting variables s α and consider the product (2.9) The quantity ψ(x, s) contains the same information as the spinor fields ψ α (x), because the latter can be recovered through ψ α (x) = ∂ ∂sα ψ(x, s). Going to the embedding space, we use the double cover of SO (3,2), which is isomor- For every 3D spinor field ψ α (x), we would like to define a 5D spinor field Ψ I (X) on the lightcone (2.2), where upper (lower) indices I, J, etc. denote fundamental (anti-fundamental) Sp(4, R) indices that are raised and lowered with the symplectic form Ω IJ = Ω IJ . As in 3D, we can also introduce polarization variables S I that help us efficiently keep track of the Sp(4, R) indices, Ψ(X, S) ≡ S I Ψ I (X) . (2.10)

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If we wish, we are free to treat S I as a spurionic 5D field that transforms in the antifundamental of Sp(4, R), and similarly to treat s α as a position-independent spurionic field in 3D that transforms in the anti-fundamental of Sp(2, R) and is invariant under dilatations (in other words, s α is a primary spinor field of vanishing dimension). If we do so, then Ψ(X, S) and ψ(x, s) become 5D and 3D Lorentz scalars, respectively. Just as in (2.4), one can check that the relation implies that Ψ(X, S) is a Lorentz scalar in 5D if and only if ψ(x, s) is a primary field in 3D with dimension ∆ ψ . Since we assumed that s α transforms as a dimension-zero primary field, then we have that Ψ(X, S) is a 5D scalar if and only if ψ α is an Sp(2, R) spinor primary field of dimension ∆ ψ .
To finish the identification between the 3D and 5D spinor fields, we can take 2 Using the conventions of appendix A for the embedding of Sp(2, R) into Sp(4, R), one can check explicitly that this relation implies that if S I is an Sp(4, R) anti-fundamental spinor in 5D, then s α is an Sp(2, R) anti-fundamental spinor primary field in 3D with vanishing dimension, as desired. Notice that S I satisfies the transversality condition which is invariant under Sp(4, R) transformations and is consistent with the lightcone condition X · X = 0. Due to this transversality condition, the 5D spinor field Ψ I is defined on the lightcone only modulo the shifts Ψ I (X) → Ψ I (X) + X I J Θ J (X), where Θ J (X) is an arbitrary spinor on the lightcone.
Ψ(X, S) satisfies the homogeneity property Ψ(aX, bS) = a −∆ ψ −1/2 bΨ(X, S) , (2.14) where a and b are arbitrary and independent. Homogeneity, the transversality condition (2.13), and 5D Lorentz invariance restrict the form of embedding space correlation 2 Equivalently, we could have writtenΨ(X,S) =SIΨ I (X) = ψ(x, s)/(X + ) ∆ ψ and identified such thatΨ I (X) is a 5D spinor if and only if ψ α (x) is a primary spinor field of dimension ∆ ψ . ThenΨ(X) would satisfy the transversality condition X I JΨ J (X) = 0 and the polarizationSI would only be defined modulo shiftsSI →SI + TJ X J I . One can relate this description to the one presented in the main text by taking SI =SJ X J I andΨ I (X) = X I J Ψ J (X). With this in mind, we can relate our formalism to that of [19] for traceless symmetric tensor fields. They define a vector Z A that satisfies transverseness Z · X = 0 and is defined up to gauge redundancy Z → Z + λX. Such a vector can be obtained as Z A = SΓ A ΩS. Note that the gauge redundancy of S gives Z A → Z A + T XΓ A ΩS = Z A + X A (T ΩS), which is the correct gauge redundancy for Z. Similarly Z · X = SXΩS = 0 since S is transverse.

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functions of Ψ(X, S). For example, the only consistent expression for the two-point function is where X I J is defined in (2.13) and Ω IJ = Ω IJ is the Sp(4, R) invariant tensor (see appendix A). Using (2.11) and (2.12) in (2.15), we obtain, as expected (2.17) For Majorana fermions, we have c ψ ∈ R, as can easily be seen by using the Majorana condition ψ α * = ψ α and the fact that complex conjugation interchanges the order of the Grassmann variables. (Recall that we work in a basis where the gamma matrices are real.) In this paper, we will take c ψ = 1 for all external operators.

Embedding of fields of higher spin
The above discussion generalizes to fields of higher spin in a natural way. In 3D, a spinfield is a totally symmetric tensor in 2 spinor indices: O α 1 α 2 ...α 2 (x). It corresponds to an embedding field O I 1 I 2 ...I 2 (X) that is homogeneous of degree −(∆ + ) in X and also totally symmetric in its 5D spinor indices. In index-free notation, we contract all the indices with an auxiliary spinor s in 3D or with a transverse auxiliary spinor S in 5D: O (X, S) = S I 1 S I 2 · · · S I 2 O I 1 I 2 ...I 2 (X) .
(2.18) O (X, S) is also homogeneous in the variable S, with degree 2 . By the same argument that led to (2.11), we must have With the help of (2.12) and one can then reconstruct the correlation functions of O α 1 α 2 ...α 2 (x) from the corresponding formulas in embedding space.
As an example, the two-point function of O (X, S) in embedding space is restricted to take the form Here c O is real if O is real. If is an integer, we also have c O > 0 in a unitary theory for a real operator O .

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For future reference, we record that when is an integer, we could have represented the spin-operator in terms of a rank-traceless symmetric tensor of SO(2, 1), namely O µ 1 ...µ . This tensor is related to O α 1 ...α 2 via It is straightforward to show that for spin-1 operators, the 2-point function in (2.21) can also be written in the more familiar form The analogous expression for spin-2 operators is (2.24) For a conserved current, take O = J with ∆ J = 2 in (2.23), and for the stress tensor take O = T with ∆ T = 3 in (2.24).

Three-and four-point functions
3-point functions between two scalars and a spin-operator of dimension ∆ are uniquely constrained up to an overall coefficient λ φ 1 φ 2 O to be of the form This form follows from homogeneity of degree −∆ 1 , −∆ 2 , −(∆ + ) in X 1 , X 2 , and X 3 , respectively, homogeneity of degree 2 in S 3 , transversality of S 3 with respect to X 3 , and Sp(4, R) invariance. In addition, it is useful to note that {X 1 , X 2 } = (2X 1 · X 2 )1 4 , which, together with X k · X k = 0, restricts the choice of quantities that can appear in between S 3 (· · · )S 3 to what is written in (2.25). Note that so in the case of identical operators Φ 1 = Φ 2 , the 3-point function necessarily vanishes if is odd. Furthermore, for real scalar operators we have , so the structure constants λ φ 1 φ 2 O appearing in (2.25) are real. When we come to 3-point functions containing fermions, we have the new complication that multiple tensor structures can appear. In general, we have

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where the index a runs over all possible 3-point structures r a , to be given shortly. These structures can be divided into those that are even under parity X k → −X k and those that are odd. A basis for the parity-even structures is given by while a basis for the parity-odd structures is given by (2.30) The dependence on (X i , S i ) in (2.27)-(2.30) can be derived from the same reasoning as that presented after eq. (2.25). The new ingredients here are the transversality of S 1 and S 2 with respect to X 1 and X 2 , respectively, as well as the Fierz identities which can be used to show that 3-point structures proportional to the left-hand sides of these equalities would be redundant. When we restrict to the case of identical fermions Ψ 1 = Ψ 2 = Ψ, anti-symmetry under 1 ↔ 2 places further restrictions on which correlators can be nonvanishing. In particular, if is even then we have the constraint λ 4 ψψO − = 0, while if is odd then The ± here only serves as a reminder of the parity of the operators O that contribute to each structure in (2.27).) In other words, even-spin operators have two structures of even parity and one structure of odd parity, while odd-spin operators have a single parity-odd structure. Further, for any spin the Grassmann nature of fermions requires Ψ( where the t I are a basis of tensor structures that can appear in the 4-point function. There are many choices of bases, but one convenient way to organize the structures is in terms of their properties under various exchanges of the coordinates.
In general, we find that there are 8 independent structures of even parity that may appear. Anticipating applications to the bootstrap, we will choose 4 structures to be symmetric under the exchange 1 ↔ 3, and 4 to be anti-symmetric. A basis for the symmetric structures is: and a basis for the anti-symmetric structures is: (2.35) When we restrict to the case that all fermions are identical, Ψ 1 = Ψ 2 = Ψ 3 = Ψ 4 = Ψ, there are additional constraints on the allowed structures coming from exchange symmetries. Some of these are highly nontrivial and lead to the bootstrap conditions discussed in the next section. However, there are also trivial constraints on the allowed tensor structures coming from exchanges that leave the cross-ratios u and v invariant: . Symmetry under these exchanges then forces In other words, restricting to identical fermions means that there are only 5 allowed tensor structures.

Conformal blocks
Now we would like to understand how the 4-point functions described in the previous section can be decomposed into conformal blocks, which sum up the contributions of all descendants of a given primary operator appearing in the Ψ × Ψ OPE. There are many approaches to computing conformal blocks in D > 2, including direct summation [2,[39][40][41][42][43], solving the Casimir differential equation [5,[44][45][46][47], pole expansions [7,9], and evaluating JHEP03(2016)120 monodromy-projected conformal integrals [27,38,48]. We will here adopt the latter formulation, since it will allow us to express the fermion conformal blocks in terms of derivatives of known scalar conformal blocks, similar to the approach of [17,30]. Let us briefly review the conformal block decomposition of a four-point function of scalars φ 1 φ 2 φ 3 φ 4 . Performing the s-channel OPE, one can write the function g(u, v) appearing in the scalar 4-point function (2.32) as a sum of conformal blocks: 3 where the sum runs only over primary operators belonging to both the φ 1 × φ 2 and φ 3 × φ 4 OPEs. As described in [27], the conformal block of O can be obtained from the integral whereÕ is the shadow operator of dimension 3 − ∆ whose indices are contracted with those of O , | M denotes a monodromy projection, and N O is a normalization factor. Similarly, performing the s-channel OPE in the fermion 4-point function (2.33), one can write where the index I runs over 4-point structures, while a, b run over 3-point structures.
Similarly to the scalar case, the outer sum in (2.39) runs over the conformal primaries O that belong to both the ψ 1 × ψ 2 and ψ 3 × ψ 4 OPEs. The inner sum in (2.39) is new in the fermion case; it is present because, for any O, there are several OPE coefficients that need to be specified, as in (2.27). In analogy with (2.38), the conformal blocks appearing in the fermion 4-point function (2.33) can be expressed as blocks can be computed from the known scalar blocks. This is the approach that we take in this paper. Concretely, the parity-even structures in eq. (2.29) can be generated by applying certain linear differential operators to Φ 1 Φ 2 O . In constructing these operators, we define and note a few useful identities: Note that in order for differential operators on X k , S k to be well-defined, they must preserve the ideal generated by the relations X 2 k = 0, X k S k = 0, S k S k = 0. This is indeed true for the operators above, though the derivative ∂ ∂X A k is not well-defined on its own.
It can be checked that the 3-point structures appearing in the Ψ 1 Ψ 2 O 3-point function function can be written in terms of the structure appearing in the 3-point function of two scalars and a spin-operator. Explicitly, we have (2.47) where we defined the differential operators and Π a,b applies a shift to the operator dimensions as Note that D 1 and D 2 generate the parity-even 3-point structures, while D 3 and D 4 generate the parity-odd ones. In addition, the operators D 1 , D 2 , and D 3 are antisymmetric under the exchange 1 ↔ 2, while D 4 is symmetric. Together with (2.26), these symmetry properties imply that in the case of identical fermions Ψ 1 = Ψ 2 , we obtain three-point JHEP03(2016)120 functions that obey the anti-symmetry requirement in X 1 and X 2 provided that is even when we use D 1 , D 2 , and D 3 and that is odd when we use D 4 , in agreement with the discussion following eq. (2.31). Defining the fermion conformal blocks (2.40) are given in terms of the known scalar blocks (2.38) according to the prescription: (2.50) The explicit formulas for g I;ab in terms of g are rather complicated, and we will not reproduce them here. Writing the fermionic blocks as derivatives of scalar blocks is useful for numerical applications. Computing derivatives of scalar blocks is straightforward, for example we use the pole expansion derived in [7] to compute the expansion in radial coordinates [47] to order ρ 60 . Derivatives of fermionic blocks are then obtained as a linear transformation on derivatives of scalar blocks.

3D fermion bootstrap
Let us return to the 4-point function of identical Majorana fermions in a parity preserving 3D CFT. Using (2.12) in (2.33), we can write this 4-point function as where t I = t I (x i , s i ) are the 5 different tensor structures that can appear. Crossing symmetry under 1 ↔ 3 gives a constraint v ∆ ψ + 1 where the minus sign on the right-hand side comes from the Grassmann nature of fermions.
In general t I | 1↔3 = M J I t J is related by some matrix M , but in the previous section we have chosen a basis of 4-point structures such that t I + | 1↔3 = t I + and t I − | 1↔3 = −t I − . In this basis the crossing relation becomes 0 = or, isolating each tensor structure, Now, the functions g I ± (u, v) have a conformal block decomposition: where O ± has parity ±, and we have chosen a basis of parity-odd 3-point structures such that the a = 3 structure only allows even spins and the a = 4 structure only allows odd spins, as before. For brevity, we have written λ a O instead of λ a ψψO , and will henceforth continue to do so. Thus, we can write the crossing equations as . This is the starting point for the usual bootstrap logic. We can exclude assumptions on the spectrum by applying a linear combination of functionals α I : where we look for functionals that satisfy the constraints − a,b=1,2 for all parity-even operators with even for all parity-odd operators with even for all parity-odd operators with odd . Recall that in our conventions, all λ a O are pure imaginary -hence the extra sign in the first line above compared to the usual conditions for scalars. The OPE coefficients of the unit operator are given by λ a 1 = iδ a 1 . We search for functionals satisfying these constraints by approximating the search as a semidefinite program and implementing it in the solver SDPB [8]. Details of this implementation are given in appendix C.

JHEP03(2016)120 4 Results
We can now use the formalism derived in the previous section to derive constraints on the space of CFTs. In particular, we consider CFTs with a Majorana fermion ψ, and focus on scalar operators appearing in the ψ × ψ OPE. We assume a parity symmetry, so that we can distinguish between parity-odd scalars, which we denote by σ, σ , σ , . . . (in increasing order of their dimensions), and parity-even scalars, which we denote by , , , . . . (also in increasing order of their dimensions).
Using the methods described in section 3, we first derive general bounds on the dimensions of these operators, observing sharp discontinuities that we conjecture to coincide with a 3D CFT containing no relevant scalar operators. We then study the consequences of imposing gaps in the scalar spectrum, making direct contact with the Gross-Neveu models (described below) at large N . Finally we study bounds on the coefficient C T appearing in the two-point function of the canonically-normalized stress tensor.

Examples of fermionic theories
While presenting our numerical results, it is useful to keep in mind a few simple CFTs that have fermionic operators: • Free theory. The theory of a free Majorana fermion ψ has Lagrangian where ψ ≡ ψ T (iγ 0 ) is the conjugate spinor. The fermionic operator ψ has dimension ∆ ψ = 1. There are no parity-even scalar operators appearing in the ψ × ψ OPE. The only parity-odd scalar appearing in ψ × ψ is ψψ, which has dimension 2. All correlation functions in this theory can be computed via Wick contractions using the free fermion propagator • Mean Field Theory. Mean Field Theory is a generalization of the free theory that in general does not have a local Lagrangian description. Its operators consist of normal-ordered products of a fermionic operator ψ and its derivatives, except that in this case all correlation functions are computed from Wick contractions using the generalized free field propagator Mean Field Theory is not properly a local QFT because it doesn't have a stress tensor. However, it satisfies the properties of unitarity and conformal symmetry that we study in this work. In the ψ × ψ OPE there are now parity-even scalar operators with dimensions 2∆ ψ + 1, 2∆ ψ + 3, 2∆ ψ + 5, . . . , and parity-odd scalars JHEP03(2016)120 with dimensions 2∆ ψ , 2∆ ψ + 2, 2∆ ψ + 4, . . . . In the limit ∆ ψ → 1, we recover a free theory, plus additional operators proportional to / ∂ψ whose OPE coefficients in the ψ × ψ OPE vanish. As in the free theory, the 4-point function of ψ in Mean Field Theory satisfies crossing symmetry.
• Gross-Neveu(-Yukawa) model. Another 3D CFT with fermionic operators is the critical point of the Gross-Neveu model [49]. In the Gross-Neveu-Yukawa description, one starts with N Majorana fermions ψ i (with i = 1, . . . , N a flavor index) and a parity-odd scalar field φ, with the Lagrangian where g and λ are coupling constants. When N is even, this theory can be studied perturbatively in d = 4 − dimensions (see, for example [50]). It has a critical point that can be achieved by appropriately tuning the scalar mass m 2 (a fermionic mass term is forbidden by parity symmetry). This critical point is believed to survive down to d = 3, where it can also be studied perturbatively in the 1/N expansion [50][51][52][53]. Previous contact between the conformal OPE and the large-N expansion of this model was made in [54].
In the context of this work, we consider a four-point function of the fermionic operator ψ = ψ 1 . (

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(with the parameter h tuned appropriately), which, when setting m 2 = gh/2 and λ = g 2 /8, is nothing but the N = 1 case of the Gross-Neveu-Yukawa model. This Lagrangian has N = 1 SUSY and can be described in terms of a real superfield Σ = φ + θψ + 1 2 θθF with superpotential Note that a superpotential with quadratic or quartic terms in Σ is forbidden by parity symmetry, since Σ is parity-odd. In the IR this theory is believed to be described by the N = 1 superconformal algebra osp(1|4).
Just like for the Gross-Neveu-Yukawa CFT, the dimensions of operators in the N = 1 super-Ising model are not known precisely (but may be experimentally probeable [55]). 4 They must obey, however, relations imposed by supersymmetry such as are the lowest few scalar operators. The relation ∆ = ∆ σ + 1 was used in [59] to derive ∆ σ ≥ 0.565. This inequality was obtained by intersecting the supersymmetric line ∆ = ∆ σ + 1 with the bootstrap bounds derived from the crossing symmetry of unitary Z 2 -invariant CFTs.

Universal dimension bounds
Let us start by computing general upper bounds on the dimensions of scalars appearing in the ψ × ψ OPE. For the moment, we assume only conformal symmetry, parity symmetry, and unitarity.

The lowest dimension parity odd scalar
In figure 1, we plot a universal upper bound on ∆ σ (the lowest dimension parity-odd scalar) as a function of ∆ ψ in any unitary, parity-invariant 3D CFT. The bound starts at the point (∆ ψ , ∆ σ ) = (1, 2), corresponding to the free theory. It then grows monotonically with ∆ ψ up to ∆ ψ ≈ 1.27, at which point a sharp vertical discontinuity occurs, and the bound jumps from ∆ σ ≈ 2.9 to ∆ σ ≈ 7.7. This striking jump suggests that the value ∆ ψ ≈ 1.27 has special significance. We discuss possible interpretations below. At the least, we can conclude that any CFT with a fermionic operator of dimension ∆ ψ 1.27 must have a relevant parity-odd scalar in the ψ × ψ OPE. Conversely, a CFT with no relevant parity-odd scalars in the ψ × ψ OPE must have ∆ ψ 1.27. In addition, we see that any CFT with a fermion of sufficiently low dimension must have a parity-odd scalar in the ψ × ψ OPE of dimension smaller than ≈ 7.7.

The lowest dimension parity-even scalar
In figure 2, we show an upper bound on ∆ (the lowest dimension parity-even scalar) in any unitary, parity-invariant 3D CFT. The bound monotonically increases starting from

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Upper bound on lowest parity-odd scalar σ ∈ ψ × ψ Figure 1. Upper bounds on the dimension of the lowest dimension parity-odd scalar appearing in the ψ × ψ OPE, assuming only conformal symmetry, parity symmetry, and unitarity. The orange region is allowed, and the white region is disallowed. The black dashed line starting at the free theory point (∆ ψ , ∆ σ ) = (1, 2) gives the relation among dimensions specific to Mean Field Theory, while the dashed line starting at (∆ ψ , ∆ σ ) = (1, 0.5) gives the relation among dimensions expected for N = 1 SCFTs, assuming ψ is a superdescendant of σ. These bounds are determined using the procedure described in section 3 (see also appendix C) by performing a binary search in ∆ σ with 10 −3 precision. The parameter Λ defined in appendix C is given by Λ = 23. the point (∆ ψ , ∆ ) = (1, 3) up to a value of ∆ ≈ 5.1. At this point, we encounter a change in slope which occurs at precisely the same value of ∆ ψ as the vertical jump in figure 1.
Note that the free fermion theory does not contain a parity-even scalar of dimension 3, since the only candidate ψ / ∂ψ vanishes by the equations of motion. However, in Mean Field Theory we have ∆ = 2∆ ψ + 1, and hence there exists a continuous family of unitary solutions to crossing symmetry that approach the point (∆ ψ , ∆ ) = (1, 3). By continuity, our bound cannot move below this point, and indeed it attains this optimal value to high precision.

A "dead end" CFT?
The kink near (∆ ψ , ∆ ) ≈ (1.27, 5.1) in figure 2 is reminiscent of the kink in scalar dimension bounds corresponding to the 3D Ising model [5,6]. Hence we might guess that there exists a 3D CFT with a fermion of dimension ∆ ψ ≈ 1.27 whose lowest dimension parity-even scalar has dimension ∆ ≈ 5.1.
The vertical jump in the parity-odd sector is also reminiscent of a feature previously encountered in scalar dimension bounds. Specifically, figure 1 of [7] shows a sharp vertical jump in the bound on ∆ σ as a function of ∆ σ (assuming that ∆ saturates its upper
Reasoning by analogy, figures 1 and 2 lead us to conjecture that there exists a 3D parity-invariant CFT with ∆ ψ ≈ 1.27 and large anomalous dimensions for both the lowest dimension parity-even and parity-odd scalars, perhaps ∆ ≈ 5.1, and 3 < ∆ σ < 7.7. Note that this theory would be a "dead-end" CFT because it has no relevant scalar operators, giving an example of self-organized criticality [20,21]. In particular, it would be completely attractive under RG flow, and hence would require no tuning to reach criticality (assuming Lorentz-invariance is unbroken). 5 We are not aware of a natural candidate Lagrangian for this theory. 6 However, the possibility that we have discovered a new "dead-end" CFT clearly merits further study. 5 Such a theory would also be interesting from the perspective of the AdS/CFT correspondence -an AdS4 holographic dual of this theory would have no tachyonic scalars and hence all moduli would be fully stabilized. We thank Eva Silverstein for emphasizing this point. 6 Several examples of 4D dead-end CFTs were constructed in [60]. Their common feature is that they are chiral gauge theories, where mass terms are forbidden by gauge-invariance.

Imposing gaps: the 3D Gross-Neveu models
With the most general possible assumptions, we have made contact with the free theory, the limit of Mean Field Theory as ∆ ψ → 1, and a conjectured "dead-end" CFT. Meanwhile, the Gross-Neveu models and N = 1 SUSY Ising model lie well inside the allowed regions in figures 1 and 2. To see them, we must input more information. A natural choice for the Gross-Neveu models would be to organize operators according to their O(N ) representations and use the constraints of O(N ) symmetry in the crossing equations, as in [9,10,61]. We leave this investigation to future work. For now, we adopt a simpler procedure: we impose gaps in the operator spectrum and study how the bounds change as a function of the gaps.
Specifically, we will use a lower bound ∆ σ ≥ ∆ min σ as a proxy for N and try to determine (∆ ψ , ∆ σ ) as a function of ∆ min σ . Assuming ∆ σ saturates its lower bound, the dependence of (∆ ψ , ∆ σ ) on ∆ σ should be consistent with table 1 at large N . Note that because we are considering a single component ψ = ψ 1 of the O(N ) vector, all representations of O(N ) appear in the ψ × ψ OPE. In particular, we have σ = φ and σ = ψ (i ψ j) at large N .
In figures 3 and 4, we plot the allowed regions of (∆ ψ , ∆ σ ) assuming ∆ σ ≥ ∆ min σ for several values of ∆ min σ . All allowed regions are consistent with the free theory at (∆ ψ , ∆ σ ) = (1, 2). However, the gap in ∆ σ has the effect of carving out the allowed region below the free theory, revealing new kinks. The positions of these kinks closely track the large-N prediction for the Gross-Neveu models, and hence we conjecture that this family of kinks (in the limit Λ → ∞) interpolates between the 3D Gross-Neveu models. In figure 5, we plot (∆ σ , ∆ ψ ) for the kinks in figure 3, compared with the large-N prediction at 1-loop, finding excellent agreement.
In figure 4, we also see that a new kink appears near (∆ ψ , ∆ σ ) ≈ (1.078, 0.565) when ∆ min σ 2.3. This new kink is quite robust to changes in ∆ min σ . We discuss its possible significance below.

Increasing ∆ σ
At large N , increasing the gap in ∆ σ corresponds to decreasing N . We might hope that for big enough ∆ σ , we could obtain information about the theory with N = 1, namely the N = 1 supersymmetric Ising model. In particular, we should identify a feature in the bound that coincides with the line predicted by supersymmetry ∆ ψ = ∆ σ + 1 2 . Unfortunately, we observe nothing particularly special happening along this line. The kink corresponding to larger N Gross-Neveu models becomes somewhat smooth and crosses the SUSY line when ∆ σ is slightly smaller than 3, see the upper kink in figure 7. If this crossover point corresponded to the N = 1 theory, we would obtain the estimate ∆ ψ ≈ 1.082. However, it is possible that using ∆ σ as a proxy for N ceases to work at smaller N . More directly, the operator ψ (i ψ j) (which we have identified with σ ) does not actually exist when N = 1, so it would be unsurprising if the N > 1 Gross-Neveu kinks are not smoothly connected to the N = 1 theory using ∆ σ as a proxy for N . The precise fate of the Gross-Neveu kinks at small N should become clear when we incorporate the constraints of global symmetry.
However, our study of increasing ∆ σ has revealed an interesting feature in the bound that appears robust: the kink near (∆ ψ , ∆ σ ) ≈ (1.078, 0.565) mentioned in the previous section. Figure 6 shows the space of allowed dimensions assuming ∆ σ ≥ 3 (equivalently, assuming the theory contains exactly one relevant parity-odd scalar), and this kink appears prominently. (In fact, it remains present until ∆ σ 6.) The zoomed-in figure 7 makes clear that the lower feature is not consistent with supersymmetry, while the upper feature (mentioned above) is barely incompatible with supersymmetry for ∆ σ ≥ 3.
Thus, we are led to conjecture the existence of a non-supersymmetric 3D parityinvariant CFT with (∆ ψ , ∆ σ ) ≈ (1.078, 0.565) and exactly one relevant parity-odd scalar. 7 An exotic possibility is that the N = 1 SUSY Ising model does not actually exist -that the RG flow from the free N = 1 theory induced by a Σ 3 superpotential spontaneously breaks SUSY. The theory in the IR of this hypothetical flow would contain a free fermion (by Goldstone's theorem), together with a (possibly empty) interacting sector (perhaps a 3D Ising model). We thank Juan Maldacena and Igor Klebanov for discussions of this point.
The proximity of this theory to the SUSY line may suggest that it is closely related to the N = 1 SUSY Ising model. For example, suppose the N = 1 SUSY Ising model had a parity-even scalar that was slightly relevant, ∆ N =1 3. If is not the top component of a scalar supermultiplet, then deforming the theory by this operator, flowing to the IR, and tuning masses appropriately would yield a non-supersymmetric fixed-point with dimensions very close to those of the SUSY theory. This possibility could be tested with the bootstrap by identifying the N = 1 SUSY Ising theory, determining the dimension and OPE coefficients of , and performing conformal perturbation theory.

Central charge bounds
Having explored bounds on operator dimensions, we finally turn to the "central charge" C T which appears in OPE coefficients of the stress tensor conformal block. We will place a general lower bound on C T as a function of ∆ ψ .
The two-point correlation function of the stress tensor is fixed by conformal invariance to take the form (2.24) (with ∆ = 3) up to an overall coefficient. Let us write
The orange shaded region is allowed, while the white region is disallowed. The black dashed line shows the SUSY relationship ∆ σ = ∆ ψ − 1 2 . The feature corresponding to the Gross-Neveu models at smaller ∆ σ has just crossed the SUSY line near ∆ ψ ≈ 1.082. We observe nothing remarkable when this happens (and the precise dimensions (∆ ψ , ∆ σ , ∆ σ ) are sensitive to Λ). Notice another feature at (∆ ψ , ∆ σ ) ≈ (1.078, 0.565). This feature appears already for ∆ σ ≥ 2.3 and persists all the way to ∆ σ 6 at the same position in (∆ ψ , ∆ σ ) plane.
where T µν can denotes the canonically normalized stress tensor, which participates in the Ward identity for translations as follows: The free boson and free Majorana fermion have C free T = 3/2. The Ward identity (4.8) determines the OPE coefficients λ a T,can in the 3-point functions in eq. (2.27). As we show in appendix D, we have In our setup, however, our normalization of operators appearing in the ψ × ψ OPE depends only on ∆ and and is otherwise independent of the details of the CFT we study. In particular, we can use the normalization of the blocks explained in footnote 3, which is equivalent to requiring that the 2-point function of the stress tensor (parity-even operator with ∆ = 3 and = 2) is normalized as in (2.21) or (2.24) with c O = 1. Comparing (2.24) with (4.7), we find

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and consequently We can put a lower bound on C T as follows. In the sum rule (3.7), we isolate the contribution of the parity-even spin-2 operator with ∆ = 3: where the summation over parity-even operators now excludes the stress energy tensor and the identity operator, whose contributions we wrote separately. We now search for a functional α such that: Here, ∆ is the lower bound on the dimension of a spin-operator, set by unitarity.
Eqs. (4.12) and (4.11) then imply: where we have used λ a 1 = iδ a 1 . Finding a functional α obeying (4.13) places a lower bound on C T . To make the bound as strong as possible, we search for an α satisfying the relations (4.13) that minimizes −α I ± [F I ± 11,0,0 (u, v)]. This is slightly different from our procedure for setting bounds on dimensions, where it was enough just to find a functional satisfying certain constraints. Nevertheless, the additional task of finding a functional whose action on a given vector is minimal can again be efficiently performed using SDPB.
Our central charge lower bound as a function of ∆ ψ is shown in figure 8. We normalize C T by dividing by its value in the free fermion theory, C free T = 3/2. The bound has similar features to analogous bounds on C T coming from scalar 4-point functions in four dimensional CFTs [62][63][64]. As the fermion dimension approaches its free theory value, ∆ ψ → 1, the bound on C T also approaches its free theory value. For larger values of ∆ ψ the bound becomes stronger, reaching a maximum. In this case, the position of the maximum does not coincide with the features observed in the bounds for ∆ σ and ∆ , and does not seem to play an important role as it did in the studies of 3D Ising model. At even greater values of ∆ ψ the bound goes to zero. After that point, the SDP problem described by (4.13) is infeasible, i.e. it is not possible to find an α satisfying the constraints in (4.13). Thus, we obtain no bound on C T for those values of ∆ ψ , beyond the obvious C T ≥ 0.

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Lower bound on C T Figure 8. Lower bound on the central charge C T of a unitary CFT containing a fermion with dimension ∆ ψ . As ∆ ψ → 1, the bound approaches the free theory value of C T . The bound goes to zero at ∆ ψ = 1.47. For larger values of ∆ ψ the SDP is infeasible and therefore does not produce any bound. This bound was computed with Λ = 23.

Discussion
In this work, we set up the 3D fermion bootstrap and explored its numerical implications. We first developed an embedding space formalism suitable for describing fermionic correlators. We found that conformal blocks for identical spin-1/2 operators are given by the action of certain differential operators on conformal blocks for scalars. Using these operators, together with differential operators that relate integer spin correlators to scalar blocks [17], one can further determine the conformal blocks for any 4-point function in 3D.
On the numerical side, we have foremost shown that the bootstrap can extract rigorous constraints from four-point functions of non-scalar operators. We have obtained general bounds on dimensions of low-lying operators in 3D CFTs with fermions and a small number of relevant scalars. We also obtained general bounds on the central charge C T . Our results not only provide rigorous constraints on the operator spectrum of CFTs with fermionic operators, but also show numerous features reminiscent of those found when applying the bootstrap to four-point functions of scalar operators.
One interesting feature revealed by the fermionic bootstrap is the kink in the parityeven bound in figure 2, coinciding with the apparent decoupling of the leading parity-odd scalar in figure 1. We do not yet know the correct interpretation of this feature, but it is intriguing that it may point to the existence of a 3D fermionic CFT with no relevant scalar operators. If such a theory is responsible for the kink in figure 2, it would contain JHEP03(2016)120 a primary spinor operator ψ of dimension ∆ ψ ≈ 1.27, and the lowest parity-even scalar appearing in the ψ × ψ OPE would have dimension ∆ ≈ 5.1. The possibility that these features reveal a "dead-end" 3D CFT that gives an example of self-organized criticality merits further study.
Other interesting features occur when we impose a gap to the second relevant parityodd scalar. By varying its dimension between 2 and 3, we observe a sequence of kinks in the (∆ ψ , ∆ σ ) plane shown in figures 3 and 4. When the gap is very close to 2, their locations match beautifully onto the dimensions in the Gross-Neveu models at large N , seen clearly in figure 5. At larger values of the gap, we expect that the kink locations make precise predictions in small-N Gross-Neveu models. We also observe the appearance of a new discontinuity in the allowed region at (∆ ψ , ∆ σ ) ≈ (1.078, 0.565), which is robust against making the second parity-odd scalar irrelevant.
In order to better understand if these discontinuities correspond to specific CFTs or SCFTs, one could pursue three immediate steps: 8 • One could hope to extend the relation between fermionic conformal blocks and scalar conformal blocks to fractional dimensions. By numerically studying the fermionic crossing-equations in different dimensions, one could compare the evolution of the discontinuities to the results from a perturbative -expansion (similar to [57,65]). • Finally, in order to better understand whether theories live at these discontinuities it would be fruitful to extend our analysis to mixed four-point functions containing both a fermionic operator ψ and a scalar operator φ. This will allow us to impose gaps in the fermionic spectrum, opening up the possibility to obtain isolated islands in the space of operator dimensions, as was seen for scalar correlators in [7,10]. We anticipate that this analysis will be particularly useful for isolating the N = 1 super-Ising model. E.g., these mixed correlators would allow us to determine the fermionic spectrum in the OPE φ×ψ, enabling us to probe the existence of a conserved supercurrent in the spectrum.
We hope to report on these further investigations in future work.

A Group theory for 3D spinors
The 3D Lorentz group SO(2, 1) has a double cover which is SL(2, R) Sp(2, R) SU(1, 1). For us the Sp(2, R) formulation is convenient. It is clear that the smallest irreducible representation is a fundamental of Sp(2, R), which has two real components. This describes a Majorana fermion in 2+1 dimensions. The Lorentz algebra is In the case of SO(2, 1), we take the signature to be η µν = diag(−1, 1, 1) and we have 3 generators J = 1 2 ab M ab = M 12 , K a = M 0 a , where a, b ∈ {1, 2}. In the fundamental representation these generators can be written as which satisfy the algebra and preserve the metric ηM + M T η = 0. Here J performs a spatial rotation and K a perform boosts. As usual, the rotation generators are Hermitian while the boost generators are anti-Hermitian.
The mapping between SO(3, 2) and Sp(4, R) is again realized via a Clifford algebra: and spinors transform in a representation of SO(3, 2) with generators We can construct a real basis for the (Γ A ) I J matrices explicitly as

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The generators (M AB ) I J in the spinor representation satisfy the Sp(4, R) symplectic constraint ΩM AB + (M AB ) T Ω = 0 with the invariant tensor (A. 11) Then in this basis the rotation and boost matrices are block diagonal and are given by: In other words, the upper two components of a Sp(4, R) spinor transform like an Sp(2, R) fundamental, and the lower two components transform like an Sp(2, R) anti-fundamental:

B Gross-Neveu-Yukawa model at large N
In this appendix, we collect known results on the dimensions of low-lying operators in the Gross-Neveu-Yukawa model at its conformal fixed point. The Lagrangian of the Gross-Neveu-Yukawa model was given in (4.4). At the CFT point, one tunes the mass for the scalar field φ to zero, and one can ignore the quartic scalar interaction as well as the kinetic term for φ. After rescaling φ, the Lagrangian takes the form Recall that the Majorana condition in Lorentzian signature is ψ = ψ T (iγ 0 ). In our conventions, γ 0 = iσ 2 , so ψ = −ψ T σ 2 .
B.1 Dimensions of ψ, φ, and φ 2 The dimensions of ψ, φ, and φ 2 have been computed in [51][52][53] at large N . Let us summarize some of these results. The dimension of the fermion operator ψ i is known up to order 1/N 3 : 3) The dimension of φ 2 is To order 1/N , the dimension of the operator φ k can be deduced from the results presented above. Indeed, at leading order in 1/N , the dimension of φ k equals k. At order 1/N , there are only two Feynman diagrams contributing, one scaling as k and one as k(k − 1)/2. We must therefore have for some constants a and b. Comparing with (B.3)-(B.4), we have For k = 3, for instance, (B.6) gives We are also interested in the dimension of the O(N ) symmetric traceless operator ψ (i ψ j) , which appears not to have been calculated in the literature. At leading order in N , this operator has dimension 2. In the rest of this section, we present the first 1/N correction to this result, with the combined answer being

B.3.1 Setup
To derive (B.8), we find it convenient to work in Euclidean signature. The Euclidean Lagrangian is the same as (B.1), with the only difference that we should use Euclideansignature gamma matrices, which can be taken to be γ 0 = σ 2 , γ 1 = σ 1 , γ 2 = σ 3 . The Majorana condition in Euclidean signature is still ψ = −ψ T σ 2 . At N = ∞, the two-point function of ψ is: In momentum space, this is The effective action for φ obtained after integrating out the fermions is

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Since ψψ = −ψ T σ 2 ψ = i αβ ψ α ψ β , we have In momentum space, To leading order in N we can thus use the propagator for ψ: The propagator for φ is D(p) = 1/Π(p), or To compute the anomalous dimension of ψ (i ψ j) , let us consider the particular case

C Implementation in SDPB
In this appendix we provide a description of the numerical implementation of the fermionic bootstrap using SDPB [8]. In order to implement a semi-definite program we limit the space of functionals α I ± over which we search over in section 3, to those taking the form,  independent constraints to SDPB. This can be done numerically. We can view the set of constraints (C.4) as a matrix with rows labeling the constraints and columns labeling the components of a functional, a I ± mn . We then only need to find the linearly independent rows of the matrix. That can be done for example in Mathematica using the built-in RowReduce function. Notice that this step needs to be done only once for a given Λ.
The full description of implementing the polynomial matrix program required to find a I ± mn can be found in the SDPB manual [8]. We have used a Mathematica script to manipulate the fermionic conformal blocks to obtain the matrix input for SDPB. In order to obtain numerically accurate results we have used the parameters presented in table 2 in our SDPB implementation. For Λ = 19 generating the input file required by SDPB takes about 30 minutes (on a single core), while solving each semi-definite program takes 25 minutes (allowed points) or 100 minutes (disallowed points) on an 8 core machine. For Λ = 23 generating the input file required by SDPB takes about 90 minutes while solving each semidefinite program takes 3 hours (allowed points) or 14 hours (disallowed points) on an 8 core machine.

(D.17)
We can compare (D.5) to (D.17) to obtain Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.