The Shadow Formalism of Galilean CFT$_2$

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT$_2$). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the GCFT$_2$ can be regarded as null defect in higher dimensional CFTs.

In particular, the 2d Galilean conformal algebra is isomorphic to the 2d Carrollian conformal algebra and the BMS 3 algebra, the latter of which plays an important role in 3d flat holography [44][45][46][47][48][49][50][51][52][53]. The 2d Galilean conformal algebra consists of two sets of generators, In [54], we initiated the study on bootstrapping the 2d Galilean conformal field theory (GCFT 2 ) based on the global Galilean conformal algebra, and mainly focused on ξ ̸ = 0 sector 2 . 1 The superconformal field theories on superspace, conformal defects on stratified space and p−adic CFT on Qp can also be regarded as examples. 2 For the ξ = 0 sector, the study will appear soon in [55].
We have studied the decomposition of the Hilbert space into the quasi-primary states, have computed the conformal partial waves and checked the consistency of the program by studying the mean field theory in different ways. It turns out that 2d Galilean conformal bootstrap is viable, even though the theory is non-unitary. Our study has revealed a few novel features in Galilean conformal bootstrap. Firstly, there exist multiplet representations in the Hilbert space, which share similar features as the logarithmic multiplets in the logarithmic CFT [56].
To distinguish them, we call these multiplets as boost multiplets. Secondly the boost multiplets satisfy a modified version of the Casimir equations, and appear in the inversion function as the multiple poles rather than the simple poles. Finally, harmonic analysis for the GCFT 2 is quite subtle, since the global Galilean conformal algebra is non-semisimple.
In this work, we would like to continue our study on 2d Galilean conformal bootstrap. We revisit the harmonic analysis and establish the shadow formalism of GCFT 2 . In our former study [54], the technical treatment followed closely the one in CFT 1 [57][58][59], i.e. spectral decomposition of the Casimir operators. To define the Hilbert space properly, we determined the inner product and boundary conditions, and the conformal partial waves supported on the whole cross-ratio plane (x, y) ∈ R 2 . Moreover we showed that the conformal partial waves could not be reached by taking non-relativistic limit on the ones of 2d conformal group, as the normalizable condition and boundary conditions should be analyzed in a way independent of the non-relativistic limit. On the other hand, when using the inversion function, it is only necessary to work in the region (x, y) ∈ (0, 1) × R. This inspires us to develop the shadow formalism for a better understanding of analytic Galilean bootstrap.
The shadow formalism relies heavily on the representation theory of the conformal group.
We will construct the unitary principal series representation of the 2d Galilean conformal group, and then define the shadow transform in GCFT 2 . With the shadow transform, we compute the operator product expansion (OPE) blocks, the conformal blocks and conformal partial waves in GCFT 2 in the framework of shadow formalism. Furthermore, we study several applications of the shadow formalism, including the decomposition of four-point functions in mean field theory, the construction of bilocal actions with Galilean conformal symmetry.
The remaining parts of the paper are organized as follows. In section 2, we give a brief review of 2d Galilean conformal field theory. In section 3, we discuss the representations of 2d Galilean conformal group and define the shadow transform. In section 4, we derive the OPE blocks and Clebsch-Gordan kernels. In section 5, we discuss the conformal blocks and conformal partial waves in the shadow formalism. In section 6, we discuss several applications of the shadow formalism. In section 7, we end with conclusions and discussions. There are a few appendices. In appendix A, we summarize the conventions and notations in this work.
In appendix B, we provide a review on the kinematics and shadow formalism of CFT 1 . In appendix C, we show how to get a Carrollian CFT 2 on a null conformal defect in Lorentzian CFT 3 . In appendix D, we present the details of some calculations.
2 Review of Galilean/Carrollian CFT 2 In this section, we briefly review the kinematical aspects of the two dimensional Galilean conformal field theory (GCFT 2 ), including the symmetry algebra, local operators and correlation functions. For more complete discussions, see [54,55]. In this work, we are going to consider the global Galilean conformal algebra iso (2,1), and in the following will refer to the quasi-primary operators with respect to the local Galilean conformal algebra gca 2 ≃ bms 3 as the primary operators for short.
The global Galilean conformal algebra iso(2, 1) singularly acts on the plane R 2 with coordi- In two dimensions the Galilean conformal symmetries and the Carrollian conformal symmetries are isomorphic due to the coincidence of 2d Carrollian structures and 2d Newton-Cartan structures 3 [74,75]. From physical point of view, the x coordinate serves as the temporal direction in the Galilean geometry. In contrast, the y coordinate serves as the temporal one in the Carrollian case. Hence we may use the terms Galilean and Carrollian interchangeably.

Local operators: singlet and boost multiplet
Singlet and boost multiplet. In GCFT 2 , the primary operators in a boost multiplet inserted at the origin O a = O a (0, 0) can be characterized by the eigenvalues (∆, ξ) of (L 0 , M 0 ) and the rank r of the boost multiplet, where the superscript runs from a = 1, 2, . . . , r. When r = 1 it reduces to the singlet case and the trivial index will be dropped. The definition of a boost multiplet is as follows: the action of dilatation is diagonalized, the two translations act name charge vector field finite transformation  Simply speaking, the primary operators in a boost multiplet share the same scaling dimension.
The action of boost M 0 gives a rank-r upper Jordan block 4 , and equivalently we have descendants form a (generalized) highest weight representation with weight (∆, ξ). This defines a rank-r boost multiplet and we denote it as V ∆,ξ,r in the following.
The infinitesimal transformations of the primary operators are [L n , O a (x, y)] = (x n+1 ∂ x + (n + 1)∆x n + (n + 1)x n y∂ y )δ a b − n(n 5) and the finite transformations are State-operator correspondence. Assuming the conformal invariance of vacuum state |0⟩, the state-operator correspondence (SOC) for a single operator 6 is given by then one can switch between the states and the operators interchangeably. Notice that the order of taking x-limit and y-limit cannot be changed in some circumstances. In the last equality of (2.7), the slope coordinates (x, k) with k = y/x, (2.8) were adopted to resolve the singularity at (x, y) = (0, 0).
There are two types of complete bases of a boost multiplet.
For a rank-r boost multiplet V ∆,ξ,r , the descendant states are and l = n + m is called the level since L 0 |a, n, m⟩ r = (∆ + l) |a, n, m⟩ r . The actions of the generators of iso(2, 1) on the descendant states are L k |a, n, m⟩ r = n! (n − k)! (n + m + ∆ + k(−1 + m + ∆)) |a, n − k, m⟩ r (2.11) where k = −1, 0, 1 and |a, n, m⟩ r = 0 if a > r. This is equivalent to the commutation relation of primary operators (2.5). The mixing between the descendants of |a⟩ r and |a + 1⟩ r happens Out-state and inner product. The physical conjugation relation is the BPZ-like con- The out-state can be defined as And the inner product of primary states is anti-diagonal and contains ⌊ r 2 ⌋ negative norms, which is also a common feature in Logarithmic CFTs. Such an indefinite inner product on the highest weight representation is called the Shapovalov form [79], and has been used to analyse the null states in relativistic CFTs, see e.g. [80][81][82]. In GCFT 2 , the Gramian matrix of this inner product and the null states for boost multiplets are obtained in [55], and it turns out that the ξ = 0 boost multiplets behave drastically different from the ones with ξ ̸ = 0. In most places of this paper we assume the boost charges of exchanged operators are nonvanishing.
Bosonic vs. fermionic. Similar to CFT 1 , the operators can be commutative or anticommutative, and the infinitesimal transformations (2.5) cannot distinguish them. The finite transformations of fermionic primaries are modified by the multiplier c(f, g) = sgn (1 − µx) if the x-SCT is involved, Accordingly the power factors |x ij | α in correlation functions should be replaced by sgn(x ij )|x ij | α .
In most of the following sections we only consider the bosonic operators, and there is no essential difference when discussing fermionic operators.

Ward identities of two-point functions
In this subsection we list the Ward identities of two-point functions, since they will reappear in several circumstances later. The primary operators are denoted as O a i ∈ V ∆ i ,ξ i ,r with ξmatrices (ξ i ) a b , and the default position of O a i is (x i , y i ) unless otherwise specified. The twopoint functions are denoted as K ab := K ab (x 12 , y 12 ) = ⟨O a 1 (x 1 , y 1 )O b 2 (x 2 , y 2 )⟩, where x 12 = x 1 − x 2 , y 12 = y 1 − y 2 and k 12 = y 12 x 12 . In the following Ward identities, the corresponding generators are M 0 , L 0 , M 1 , L 1 respectively. Singlet: Boost Multiplet: There are three types of solutions of these equations, which will be referred to as continuous, exceptional and discrete types for later convenience. The continuous type of solutions is a linear combination of |x 12 | −2∆ 1 k i 12 e 2ξ 1 k 12 and will be reviewed in the next section 2.3. Similar to relativistic CFTs, the two conformal families are forced by the equations of L 1 , M 1 to be identical, ∆ 12 = 0, ξ 12 = 0. We mainly focus on the continuous type in this work.
The exceptional type of solutions exists only when ξ 1 + ξ 2 = 0 and is a linear combination of δ (i) (x 12 )|y 12 | −∆ 1 −∆ 2 +1+i . The two conformal families are not necessarily identical. They appear in e.g. the bilocal actions in section 6.2. This type of solutions also appears in higher dimensional Carrollian and Galilean CFTs [39], and is relevant to the proposed relations between Carrollian CFT and celestial CFT [83,84].
The discrete type of solutions exists when ξ 1 + ξ 2 = 0, ∆ 1 + ∆ 2 ∈ Z and is a linear combination of δ (i) (x 12 )δ (∆ 1 +∆ 2 −2−i) (y 12 ). The further restrictions on the weights from the Ward identities are different from the ones in the exceptional type. They appear in e.g. the inner product of the principal series representations in section 3.4. More complicatedly, the three types of solutions can mix with each other when the conditions on the weights and the charges in different types are satisfied simultaneously.

Correlation functions
Singlets. The two-point functions of singlets are diagonalized as where k 12 = y 12 x 12 . The three-point functions are of the form (2.19) where c 123 are the three-point coefficients and The four-point function can be written as a product of the stripped four-point function x, y) containing the dynamical information and a kinematical factor K (s) (x i , y i ) compensating the conformal covariance of the four-point function Here we are considering the s-channel O 1 ×O 2 → O 3 ×O 4 , and we find the following kinematical factor is convenient for s-channel OPE. As a result the stripped conformal blocks (D.58) depend four-point function is obtained by the permutation (13), where the kinematical factor is, The s − t crossing equation from (2.21) and (2.23) leads to where the crossing region is (x, y) ∈ (0, 1) × R.
Then the inner product interpretation of the four-point function is and its relation to the s-channel stripped four-point function is Boost multiplet. The two-point functions of different boost multiplets vanish. For the same multiplet V ∆,ξ,r its two-point functions form an left-upper triangular matrix where ⟨OO⟩ r=1 is the two-point structure 7 of a singlet.
Symmetry. The Euclidean conformal algebra g E = so(d + 1, 1) and the Lorentzian one 2), are different real slices of the complex Lie algebra so(d + 2, C), hence one's complex 8 representation is naturally the other's representation.
For a classical symmetry group G, the physical projective representation on the Hilbert space corresponds to the linear representation of the universal covering group G. The two groups are related by modding out the fundamental group, G = G/π 1 (G). The classical Euclidean conformal group is G E = SO(d + 1, 1) with π 1 (G E ) = Z 2 , d ⩾ 2, and if spinors are involved we need to consider the double covering group Spin(d + 1, 1).
The fundamental group of the classical Lorentzian conformal group G L = SO(d, 2) is a little bigger: π 1 (G L ) ≃ π 1 (SO(d)) × π 1 (SO(2)) = Z 2 × Z, d ⩾ 3, and unlike the spin group the universal covering group G L is not a linear Lie group, i.e. it cannot be embedded as a linear subgroup of GL(n, C) for any finite n.
Representation. There are various types of representations appearing in relativistic CFT. The first type describes physical operators. The operators located in Lorentzian region In the following for convenience we will call the "operators" with analytic continued weight as virtual operators, since they are not in the physical Hilbert space, only providing a complete basis in the decomposition of correlation functions.
GCFT 2 . Different from the relativistic conformal algebras, the "Wick rotation" of the Galilean conformal algebra iso(2, 1) is isomorphic to itself. This is similar to the case of CFT 1 , as reviewed in section B.
The first type of representations in GCFT 2 includes the singlet and multiplet representations with real weight (∆, ξ) ∈ R. Despite of being non-unitary generically they describe physical operators, like the conformal families in Euclidean CFTs not satisfying the unitary bound. This is acceptable since non-unitary theories are common in Euclidean CFTs, e.g., all the logarithmic CFTs and most of the 2d minimal models.
The second type is the unitary principal series representation of the Galilean conformal group with complex weight ∆ = 1 + is, ξ = ir. The cyclic vectors in the two types of representations follow the same transformation rule, suggesting the viability of the shadow 9 In odd dimensions there are also discrete series appearing in the reduced unitary dual.
formalism in GCFT 2 . The procedure of analytic continuation of weight (∆, ξ) is shown in Figure 1. In the rest of this section we will discuss the principal series representations and the shadow transforms as the starting point of the shadow formalism.
O iri S Re ξ Figure 1: Virtual operators lie on the principal series ξ = iR ̸ =0 . The external and exchanged operators should be analytic continued simultaneously keeping the ratios R i real. The case that the exchanged operator is degenerate ξ = 0 should be handled separately.

Unitary principal series representations
Since the 2d Galilean conformal group is isomorphic to the 3d Poincare group, the "unitary principal series" representations should be identified as unitary irreducible representations of the Poincare group, which has been classified by using the Wigner-Mackey method [85][86][87], see also e.g. [88,89]. To make the shadow transform rigorous, we firstly construct the unitary principal series, then in the next subsection identify them with the tachyonic unitary representation of the Poincare group.
Definition. We define the unitary principal series representation E ∆,ξ of ISO(2, 1) as follows: the representation space is L 2 (R 2 ) ∋ f (x, y), with the inner product and the group action is the same as the one on the singlet primary operators (2.6) but with complex weight (∆ = 1 + is, ξ = ir), r ∈ R ̸ =0 , s ∈ R, where the global Galilean conformal transformations (f, g) are relabeled as (a, b). The infinitesimal transformations are the same as those of primary operators (2.5). The inner product is invariant under the action due to the selected weight, hence defining a unitary representation.
We emphasize that this unitarity is not the physical unitarity, and the conjugation relation on generators is not the BPZ conjugation L † n = L −n , M † n = M −n . Instead, the "Euclidean" conjugation relation is the default anti-Hermitian one,  5) and the commutativity with the group action is A representation is irreducible if any bounded self-intertwining map is proportional to the identity map, and in the following the concept of irreducibility is in this sense, see e.g. [67,90]. There are other definitions of irreducibility, and there can be further subtleties from finite transformations down to the infinitesimal ones. We omit these technical issues for simplicity.
To check E ∆,ξ is irreducible or not, we determine the self-intertwining map K : E ∆,ξ → E ∆,ξ , by requiring K commute with infinitesimal transformations X ∈ iso(2, 1) dx 2 dy 2 K(x 12 , y 12 )X 2 f (x 2 , y 2 ) = X 1 dx 2 dy 2 K(x 12 , y 12 )f (x 2 , y 2 ) (3.8) in which K(x, y) is the distributional kernel. In the above relations, we have used the translation L −1 , M −1 to restrict the kernel depending on (x 12 , y 12 ). After doing integration by parts, the generators M 0 , L 0 , M 1 , L 1 lead to four equations of K(x, y), and they are related to the two-point Ward identities (2.16) by the replacement due to the conformal covariance property of (3.7). The distributional solutions of the equations with respect to L 0 , M 0 are K(x, y) = δ(x)δ(y) + a 1 δ(x)y −1 + a 2 x −2 + a 3 δ ′ (x). (3.10) In the case ξ ̸ = 0, the equation of L 1 restricts a 1 = 0 and the one of M 1 restricts a 2 = a 3 = 0 such that K(x, y) = δ(x)δ(y). (3.11) Hence we conclude that E 1+is,ir , s ∈ R, r ∈ R ̸ =0 is a unitary irreducible representation of the Galilean conformal group.

Relation to tachyonic representations
In this subsection we identify the unitary principal series representation constructed in [M ab , P c ] = −g ac P b + g bc P a . (3.14) Then extending the above relations to the whole Galilean conformal algebra, we get 15) and the conjugation relation (3.4) is preserved. This identification is exactly the same as (C.11). Then the Casimirs are 10 The Casimirs act on the principal series representation E ∆=1+is,ξ=ir as scalars, hence from the first Casimir we find E ∆=1+is,ξ=ir is tachyonic.

Shadow transforms
In this subsection we try to establish the shadow transform of ISO ( (3.20) which is an intertwining map between the two representations If the representations E ∆,ξ and E ∆, ξ are unitary and irreducible, by Schur lemma S is an isomorphism, otherwise the kernel subspace ker S ∈ E ∆,ξ and the image subspace im S ∈ E ∆, ξ can be subrepresentations, or even worse, the integration kernel of S is ill-defined as a tempered distribution. Applying the shadow transform twice S 2 : E ∆,ξ → V ∆,ξ , and in the case that S is an isomorphism, the intertwining kernel K(S 2 ) should be proportional to the δ-distribution, K(S 2 ) = N (∆, ξ)δ(x 12 , y 12 ). (3.23) The prefactor N (∆, ξ) can be calculated as follows y 0 x 12 x 01 x 02 e 2ir x 0 y 12 +x 1 y 2 −x 2 y 1 x 01 x 02 (3.24) where in the second line the integration of y 0 contributes to δ(x 1 − x 2 ), in the third line the simplification is due to x 1 = x 2 and in the last line we change the variable 1 In CFT the factor N −1 (∆, ξ) is proportional to the Plancherel measure [71]. In GCFT 2 we find that the factor N −1 (∆, ξ) is in match with the Plancherel measure of the tachyonic representations [92,93].

Derivation of shadow transforms
In this subsection we give an intrinsic derivation of the shadow transform of GCFT 2 , then discuss the analytic continuation and the inner product.
and if ξ 12 = ∆ 12 = 0 we come back to the discussion of self-intertwining map in section 3.1.
When ξ 12 ̸ = 0, the solution of the first two equations is y 12 x 12 c 1 |x 12 | ∆ 12 −2 + c 2 sgn(x 12 )|x 12 | ∆ 12 −2 . Inner product. However, we need to check whether the integral transform (3.26) is welldefined or not, and this requires us to select the correct weight (∆, ξ). As discussed in section 2.1, the conformal family V ∆,ξ is generated by the smeared states |f ⟩ = dxdy f (x, y)O(x, y) |0⟩ labeled by the wave-function f . The normalizable state, after implementing the integral trans- should be normalizable as well, where K −1 is the kernel of the inverse integral transform.
To answer this question we need to specify the inner product, and it turn out that there are two choices. The ansatz of the inner product of the wave-functions is Following the same trick above, we get the equations of the inner product kernel K ip . They are related to the two-point Ward identities (2.16) by and have two types of solutions. Combining the equations of M 0 , M 1 we have and similarly the equations of M 0 , L 0 , L 1 imply that Physical inner product. For physical operators ∆ ∈ R, ξ ∈ R ̸ =0 , the equations (3.32) and (3.33) are trivial, and the solution is simply the two-point function ⟨OO⟩. This inner is badly-behaved because of the exponential growth. Recall that in relativistic CFTs [94,95], the physical inner product is distribution Wick-rotated from the Euclidean correlator. Inspired by this we can Wick-rotate either the Carrollian time or the Galilean time to the imaginary axis: y = iγ or x = iτ , then for physical weight ξ ∈ R ̸ =0 , the exponential factor e 2iξ γ 12 x 12 = e −2iξ y 12 τ 12 is tamed to a oscillating phase. The two wick-rotations are distinguished by the power factors: |x 12 | −2∆ and (−τ 2 12 +iτ 12 ϵ) −∆ . One of the Wick-rotated integral transform (3.28) should be the analog of the Lorentzian shadow transform, see e.g. [72,96], and for this one the inner product (3.34) cannot be positive-definite since the corresponding highest weight representation contains negativenorm states. This may cause technical difficulties and we leave it for further study.
Inner product of unitary principal series. The equations (3.32) and (3.33) admit an distributional solution K ip (x 12 , y 12 ) = δ(x 12 )δ(y 12 ), (3.36) and the weight is restricted by the original equations of K ip (x 12 , y 12 ) onto the unitary principal series (∆ = 1 + is, ξ = ir). This inner product gives an analog of the Euclidean shadow tranform: analytic continuing the weight to the unitary principal series and replacing the physical inner product by the positive-definite one This can also be understood as choosing the rewriting of the double shadow transform (3.23), as the inner product kernel. Due to the modification of inner product and the selected weight, the representation is E ∆,ξ instead of V ∆,ξ , and the integral transform (3.28) preserves the norm, hence is well-defined.

OPE Blocks and Shadow Coefficients
Before introducing the conformal block expansion and the inversion formula, in this section we discuss the quantities associated with three-point structures, including the OPE blocks 11 , the Clebsch-Gordan kernels and the shadow coefficients. For the four-point functions, the conformal blocks are the two-point functions of OPE blocks [97,98], and the conformal partial waves are the integrals of two Clebsch-Gordan kernels.

OPE blocks
In this subsection, we determine the OPE blocks from the shadow formalism. The idea of OPE blocks are illustrated in CFT 1 in the appendix B.3. The OPE relation can be written as where the derivatives are understood as acting on O k only. The OPE block D encodes all the contributions of the derivative operators in which the prefactor x −∆ 12, 3 12 exp(ξ 12,3 k 12 ) is to give the correct two-point function, and the OPE coefficients and three-point coefficients are related by c 123 = c k 12 δ 3k and δ 12 = c id 12 . In the shadow formalism, the OPE block should be where the integral region is I = (x 1 , x 2 ) × R and the normalization factor N 123 is to ensure that the primary operator contributes to one. The calculation is a bit lengthy and we leave it into the appendix D.1. In the end, the closed form of the OPE block is n (z) = (a + 1) n n! 2 F 1 (−n, 1 + a + b + n; a + 1; For two identical operators, the OPE block gets simplified to In the appendix of [99], the low-level OPE block coefficients of two identical external operators with respect to the BMS algebra was computed by using the recursion relations. Our results of R = 0, ∆ 12 = 0 should match theirs with c M → ∞, and this is indeed true.
Boost multiplets in OPE. Suppose there is a rank-r boost multiplet O a 3 in the singletsinglet OPE O 1 × O 2 , the leading term from the primaries is then inserting the OPE into three-point functions (2.31) we get the relation between threepoint coefficients c a and the OPE coefficients d a := d 123,a , (4.8)

Clebsch-Gordan kernels and shadow coefficients
In this subsection we discuss the Clebsch-Gordan kernel and the shadow coefficient. From the representation theory perspective, the three-point structure is the Clebsch-Gordan kernel [66,100].
in which the kernel K is the infinite-dimensional version of the Clebsch-Gordan coefficient Intuitively the coordinates (x, y) serve as the magnetic quantum numbers and the weight (∆, ξ) serve as the angular momentum quantum numbers. By comparing the conformal covariance of both sides, the kernel K is proportional to the three-point structure Similarly the three-point structure [71,73], In relativistic CFTs, this is also known as the vertex-graph identity or the star-triangle relation [62,101]. The integral in (4.11) can be evaluated explicitly where The y 3 -dependent part in the integrand is a pure phase, hence gives rise to a δ-distribution of x 3 , and the integral gives in which e A 0 gives exactly the exponential part in ⟨O 1 O 2 O 3 ⟩, and Fx |J 0 | is proportional to the power-law part, hence the shadow coefficient can be determined to be Properties of the shadow coefficient. The shadow coefficient is related to the normal- into the three-point function matching with the result (D.9).
The relation between the shadow coefficient and the factor (3.23) is determined as follows.
Consider the doubly shadow-transformed three-point structure and by applying (4.12) twice we get , thus the relation between the shadow coefficients and the factor N (∆, ξ) is which is expected.

Orthogonality of the Clebsch-Gordan kernels
The orthogonality and completeness relations of the Clebsch-Gordan coefficients are re- version of (4.24) for the Clebsch-Gordan kernel should be the equivalence of E 4 and E 4 , and can be determined by the shadow transform once the first term is known. Following the convention of [71] we swap the operators in (4.26) and define the bubble integral of two three-point structures as where The first term. To separate the first term from the shadow term, we suppose r 3 r 4 > 0.
The integration with respect to y 1 , y 2 is of the form like The two equations A 1 = A 2 = 0 decide an algebraic variety with two irreducible components in the space R 8 ∋ (x i , r i ), and the condition r 3 r 4 > 0 selects the component After using δ(r 34 )δ(x 34 ) to simplify the rest part, we find that the substitutions reduce the exponential factor to A 0 = r 3 y 34 and non-exponential part is independent of X 2 . Hence the integration with respect to X 2 gives rise to δ(y 34 ), and we get justifying the first term in (4.26).
The shadow term. Relaxing the assumption r 3 r 4 > 0, the shadow term comes from the integration localized on the second component of and can be determined by the following procedure 12 . Denoting the bubble integral as while the left-hand side can be calculated using the shadow coefficient By comparison we get In summary the bubble integral (4.27) mimicking the orthogonality relation (4.24) is Incompleteness and projector. The infinite-dimensional version of the completeness relation (4.25) should be where we have relabeled the weight as (∆ 0 = 1 + is, ξ 0 = ir), and by (3.25) the factor r 2 is proportional to the Plancherel measure of the principal series. However the set of Clebsch-Gordan kernels is an incomplete basis due to the following reason. Firstly, the Clebsch-Gordan kernel (4.9) corresponds to decomposing the tensor product of two tachyonic representations into another tachyonic one. But there should be massive and massless representations in this tensor product decomposition, since the sum of two spacelike momenta can be timelike or null.
Secondly, according to [92,93], the Plancherel measure of the 3d Poincare group is where c 1 , c 2 are constants depending on the Haar measure, and the two terms count the contributions from tachyonic and massive representations respectively. Combining these two aspects and the orthogonality (4.38), we have where P t and P m are the projection operators of tachyonic and massive representations respectively, P 2 t,m ∼ P t,m , P t · P m = 0, and (4.42)

Conformal Blocks and Partial Waves
In relativistic CFTs, due to the convergence of OPE, the higher-point functions can be reduced to a sum of conformal blocks by applying the OPE relations repeatedly, and the coefficients are products of three-point coefficients. The conformal blocks are completely fixed by the conformal symmetry, depending on the external operators, the specific OPE channel, and the exchanged operators.
This conformal block expansion can be regarded as an on-shell method, since the summation ranges over the physical Hilbert space. The correlation functions can also be expanded into an integral of the conformal partial waves over unphysical unitary principal series -this is the Euclidean inversion formula. Under suitable conditions, the block expansion is recovered from the inversion formula by a contour deformation argument.
In this section we develop the conformal block expansions for four different external singlet operators in Galilean CFT 2 . We first calculate the conformal blocks of exchanged singlets and boost multiplets by solving the Casimir equations, then using the shadow formalism determine the conformal partial waves and establish the inversion formula. The previous results of singlet conformal blocks of the BMS algebra are in e.g. [51,99,102,103], see also the work on BMS torus blocks [104], and singlet conformal blocks with supersymmetric extensions [105]. The s-channel block expansion of a four-point function is where p (s) n = c 12n c 43n , and the conformal block with respect to the primary O n is defined as Similarly the t-channel block expansion is, To further carry out calculations we introduce the stripped version of conformal blocks depending only on the cross ratios by factoring out the kinematical factors then the block expansion of the stripped four-point functions are

Conformal blocks from Casimir equations
In this subsection we derive the Casimir equations of singlet and boost multiplet, then obtain the conformal blocks by solving the Casimir equations.
Conformal blocks of singlets. In GCFT 2 , the conformal blocks of exchanged singlet conformal families are the eigenfunctions of the Casimir differential operators. This originates from the fact that the Casimir elements (3.17) of the Galilean conformal algebra act on the Notice that this is incorrect for boost multiplet and is insufficient for ξ = 0 multiplet [55], where the Casimir equations must be modified appropriately.
In the appendix D.4 we derive the Casimir equations of the stripped conformal block with in which (x, k) = (x, y x ) are the slope coordinates, and the differential Casimir operators are (5.10) Then we solve the two Casimir equations in the appendix D.5, and there are two independent solutions. By checking the s-channel OPE limit x, k → 0, and redefining the normalization to ensure the exchanged primary operator contributes one: g , we find that the solution corresponding to the physical block is 12) and the normalization factor is The other solution is proportional to the shadow block g Conformal blocks of boost multiplets. The conformal blocks of exchanged boost multiplets are related to the ones of the singlets by a derivative relation. Following the logic of the previous subsection, we encounter the obstruction that the Casimir elements acting on the boost multiplets are not scalars, hence do not commute with the projection operators. As in the case of four identical external operators [54] 14 , for a rank-r boost multiplet V ∆ 0 ,ξ 0 ,r , the following operators act as zero, (C i − λ i ) r |n⟩ = 0, i = 1, 2. Denoting the conformal blocks of the modified version of (D.42) is Then using the conjugation relation C i = (K (s) ) −1 C (1+2) i K (s) , we get the Casimir equations of stripped conformal blocks whose solution is a linear combination of ξ 0 -derivative of the singlet conformal block By comparing the s-channel OPE limit, we relate the block coefficients p (s) 0,a with the threepoint coefficients. The s-channel OPE limit of the conformal block (5.16) is while from the leading OPE (4.7) the conformal block behaves as Hence matching the coefficients and using the relation (4.8) we have

Analytic properties of conformal block expansion
In this subsection we explore the analytic properties of the singlet and boost multiplet conformal blocks, and discuss the implications on the conformal block expansion.
Analytic properties of singlet blocks. As a first check of our calculation, taking ∆ 12 = ∆ 34 = ξ 12 = ξ 34 = 0 the conformal block (5.11) agrees with the result in the previous work [54,99], Secondly near the s-channel OPE limit, the conformal block (5.11) can be expanded into where in the bracket it is a double Taylor series of (x, k) counting the contribution of the descendants, and the next-to-leading coefficients are At first glance, it seems that the conformal block in GCFT 2 shares similar analytic structure as the one in CFT 1 , as reviewed in appendix B.4. The s-channel singularity in two cases is controlled by the power factor x ∆ 0 and the rest part is analytic near x = 0.
However there are two additional branch points in the conformal block (5.11) at x = x ± , which are zeros of the function H(x), If x ± ∈ (−1, 1), the contributions from the descendants in (5.21) grow too fast such that the s-channel convergent radius for each individual block is less than one, and the s − t crossing equation can be invalid.
The relation between (R 12 , R 34 ) and x ± is plotted in figure 2. The gray region IV is ruled out since the branch points enter into the s − t crossing region (0, 1) × R. The region III is divided by the curve max(x + , x − ) = −1, and outside the curve, the convergent radius of an individual conformal block is less than one. Ignoring this issue, the crossing equation still holds in (0, 1) × R. In the region I and II, the conformal blocks behave similarly as those in To illustrate these features, we consider the special case ξ 12 = ξ 0 , and x ± = 2 1+R 34 . When R 34 ⩽ 1, the conformal block can be analytically continued from the one with (R 12 , R 34 ) = (0, 0) to (1, R 34 ) along the curve (0, 0) → (1, 0) → (1, R 34 ) while keeping single-valued, and the result is When R 34 > 1, the two roots x ± enter into x ∈ (0, 1), and the conformal block stops being single-valued. Notice that at R 34 = −1 the conformal block is simply the leading factor e −kξ 0 x ∆ 0 , i.e. the contributions of descendant operators are canceled with each other.

12
. When |R 12 | ⩽ 1, the conformal block can be analytically continued along the diagonal line (0, 0) → (R 12 , R 12 ), . And when |R 12 | > 1 the conformal block loses its singlevaluedness. Recovering the kinematical factor, the unstripped version of (5.26) is n,a ∂ a ∂ξ a n G (s) n + . . . . (5.27) x 1 Assuming that the operator spectrum and conformal block coefficients p (s) 0,a are well-controlled, like in the MFT, the s − t crossing equation holds in the region (0, 1) × R. This indicates that under suitable conditions, the OPE convergence region in GCFT 2 is a stripe as shown in figure   3, exhibiting the non-locality of y-direction.

Conformal partial waves and blocks from shadow formalism
In this subsection we derive the conformal partial waves and conformal blocks from the shadow formalism, and discuss their relations.
The conformal partial waves can be constructed as (5.28) The stripped conformal partial waves ψ ∆ 0 ,ξ 0 (x, y) are defined by factoring out the kinematical factor K (s) , Since the stripped conformal partial wave depends only on the cross ratios, we fix the gauge to the standard conformal frame, and find in which The exponential factor is a pure phase due to the analytic continuation ξ 0 ∈ iR ̸ =0 , hence the integration of y 0 gives Dirac δ-distributions of x 0 where the two roots of x 0 are and H(x) is the same as (D.51). Substituting the δ-distributions into the integrand, the resulting conformal partial wave is a combination of two conformal blocks with analytical continued weights in which δ(x 0 −x 0,+ ) contributes to the physical block g ∆ 0 ,ξ 0 (x, k) and the prefactors are simply the shadow coefficients. Notice that in (5.35) the power function parts should be understood as the absolute values since they come from (5.31). Recovering the kinematical factor we get the unstripped conformal partial wave 36) and the shadow partial wave Ψ 2−∆ 0 ,−ξ 0 is proportional to Ψ ∆ 0 ,ξ 0 , by using the identity (4.23).
Relation to conformal blocks. The relation (5.36) between the conformal block and the conformal partial wave is similar to that in relativistic CFT [19,71]. However in our case, when the external operators are identical, the conformal partial wave constructed from the shadow formalism is not the same as the one from the spectral decomposition of the Casimir operators [54]. This could be due to the fact that different boundary conditions at x = 1 lead to different self-adjoint extensions of the Casimir operators, hence the eigenvalues and eigenfunctions are not the same. The analog of (5.35) in relativistic CFT appears in the alpha space approach [107][108][109], where the resulting stripped partial waves are only supported on z ∈ (0, 1).
The conformal partial wave (5.35) is also not supported on the whole cross-ratio plane R 2 .
When the x 0 -roots (5.34) take complex values, the δ-distributions vanish and the corresponding partial wave also vanishes. This reality condition gives the support of conformal partial waves, , where x ± are the zeros of H(x) (5.23), and when x ± / ∈ R, the support is x ∈ R.
The relation (5.36) between conformal partial waves and blocks can be understood from the integral expression (5.2) of conformal blocks. Assuming x 1 < x 2 < x 3 < x 4 and R 12 , R 34 ∈ (−1, 1), inserting the OPE block (4.3) and the normalization factor (4.20) into (5.2) we get, where the integration region is I ij = (x i , x j ) × R. Under the assumption R 12 ∈ (−1, 1), by (4.18) and (4.19), the integration region I 12 can be extended to R 2 . Hence in the second line we can apply the integral expression of the shadow coefficients (4.12) and cancel the factor Since after gauge fixing the physical block comes from δ(x 0 − x 0,+ ), to match the result (5.38) with the first term in (5.2) we must have the inequality, and this is indeed correct.

Inversion formula
In this subsection we discuss the orthogonality of partial waves, and then establish an can be joined by a small semicircle, and the integral along this semicircle vanishes since the Plancherel measure is proportional to ξ 2 dξ.
Orthogonality. The orthogonality of conformal partial waves can be derived from the bubble integral (4.38) in the following way. Denoting Ψ i as the unstripped conformal partial waves of exchanged virtual operators E ∆ i ,ξ i , they admit a natural inner product, which is invariant under the Galilean conformal transformations, The ξ = 0 four-point functions are not in the Hilbert space of the ξ ̸ = 0 conformal partial waves, and should be expanded in a different basis, see [55].
where in the third line we have used the bubble integral (4.38), and the integration of (x 6 , y 6 ) in the second term is done by the shadow coefficient (4.12). In the last line the prefactor A 3-pt is a three-point pairing This integral is an adjustable numerical factor, since the three points can be fixed by the conformal symmetry ISO(2, 1) and there are no integrals and residual symmetries left. We take A 3-pt = 1 2 by rescaling vol ISO(2, 1), and the inner product (5.40) is x i ̸ =x j ,y i ̸ =y j ) with inner product (5.40), and the corresponding projection operator P acting on this space gives a subspace PV . We denote the projected four-point function by the subscript P.
The purpose of the Euclidean inversion formula is to diagonalize the four-point functions and to recover the OPE data. In practice, the information of the s-channel conformal block expansion is stored in the analytic structure near the s-channel OPE limit. If the projected subspace PV contains four-point functions supported on a neighborhood of the s-channel limit, we can expect to read the conformal block expansion from the inversion function (5.44) by the contour deformation procedure. To be concrete, we have the following inversion formula, In the second line we have changed the variables (∆, ξ) → (2 − ∆, ξ) and used the identity where the poles (∆ n , ξ n ) are in the right half-planes, we get the conformal block expansion Projector. Finally we discuss the projection operator. Inserting the inversion function (5.44) into the inversion formula (5.45) and we get the projection operator In the appendix D.6 we show that in the case of four identical external operators, the projector is proportional to (0, 1) × R can be expanded in PV without loss of information. Therefore, we can expect that the s-channel conformal block expansion is captured by the projected four-point functions. In the next section 6.1 we will show that this is correct by the example of mean field theory.

Applications and Generalizations of Shadow Formalism
In this section, we discuss a few applications and generalizations of shadow formalism in GCFT. The first one is to reconsider the decomposition of the four-point functions in Galilean mean field theory, of which a special case has been studied in [54]. The second application is to construct Lagrangian of Galilean MFT. We manage to find a series of bilocal actions, corresponding to the Galilean MFT, with the help of the kernel of the shadow transform.

Decomposition of four-point functions in mean field theory
The mean field theory (MFT), or the generalized free theory, is defined as that all its correlation functions are the Wick contractions of the two-point functions. Regarding the field theory as a stochastic process, the MFT is equivalent to the Gaussian process, and the two-point function is called the covariance function. The MFT provides a simple example of CFT when the two-point function is conformally covariant.
In the relativistic case, the MFT is the leading contribution of a large-N CFT, and corresponds to the free theory in AdS for a holographic CFT [20]. It also gives the leading contribution at large spin in the context of analytic bootstrap [13,14,17]. Finally it relates to the long-range models. The MFT with a fundamental scalar ϕ, ∆ ϕ ̸ = d−2 2 admits an unusual Lagrangian description: the kinematical term contains the fractional Laplacian and is nonlocal, see e.g. [110]. Deformed by a relevant quartic interaction, it flows to the long-range Ising model for a window of the parameters [111].
In GCFT 2 , the MFT [54] is one of the few concrete models besides the BMS free scalar [106] and free fermion [112]. In this subsection we consider the MFT containing two different bosonic singlets ϕ i ∈ V ∆ i ,ξ i . The two point functions are 3) The four-point function of ϕ 1 , ϕ 2 we are interested in is The t-channel ϕ 1 × ϕ 1 → ϕ 2 × ϕ 2 OPE is trivial, and there is only one exchanged operator: the identity operator id. The s-channel ϕ 1 × ϕ 2 → ϕ 1 × ϕ 2 OPE is expected to exchange double trace operators, schematically :ϕ 1 ∂ n ϕ 2 :.
In the following we review the operator construction method [54]. Then using the shadow formalism and the inversion formula we decompose the four-point function (6.4) and obtain the conformal block expansion. The derivation of the conformal block expansion from the inversion formula necessitates a dispersion-like relation on the ξ-plane.
Method of operator construction. The leading and next-to-leading terms in the conformal block expansion was calculated by operator construction in [54]. The leading exchanged conformal family is the singlet with primary operator O L =:ϕ 1 ϕ 2 :. At the next-to-leading order, there are four composite operators {:∂ x ϕ 1 ϕ 2 :, :∂ y ϕ 1 ϕ 2 :, :ϕ 1 ∂ x ϕ 2 :, :ϕ 1 ∂ y ϕ 2 :}. (6.5) Diagonalizing them by L 0 , M 0 , two operators are descendants of :ϕ 1 ϕ 2 :, and the rest two constitute the primaries O a NL , a = 1, 2 in a rank-2 boost multiplet, Here the two-point functions of O a NL are not normalized to the standard form (2.28) and the overall factor is d = 2ξ 1 ξ 2 (ξ 1 + ξ 2 ). Accordingly the block coefficients (5.19) should be divided by d. The three-point coefficients in Then the contributions of these two conformal families to the conformal block expansion are in which the block coefficients are By the operator counting technique, there should be a rank-(n+1) boost multiplet in each order of the conformal block expansion. In the following we derive the conformal block expansion from the inversion formula and justify the result of operator construction.
Inversion function. Following the calculation in relativistic CFT, we firstly analytically continue the external weights (∆ i , ξ i ) onto the unitary principal series, then the inversion function is where the exchanged virtual operator O ∆,ξ is located at (x 0 , y 0 ). In the third line we have used the result of the shadow coefficients (4.12), and in the last line the remaining three-point pairing is a numerical constant A 3-pt (5.42). Then the inversion function is where, Dispersion relation. Notice that c u.p.s. (∆, ξ) has no poles of (∆, ξ), and instead there are five branch cuts anchored at ξ = ±ξ 1 ±ξ 2 , 0 shown in figure 4. This phenomenon is expected as explained in [54]. In the case of four identical external operators, the physical inversion function Figure 4: The analytic structure of the inversion function of MFT for ξ 2 > ξ 1 > 0. There are five branch points at ξ = ±ξ 1 ± ξ 2 , 0 drawn as the wavy lines. The physical cut corresponding to the double-trace operators is anchored at ξ = ξ 1 + ξ 2 . The dashed lines are different analytic continuations to the imaginary axis, and from this ambiguity we read off the discontinuity along the cut ξ = ξ 1 + ξ 2 .
c(∆, ξ) has two branch cuts, and c u.p.s. (∆, ξ) with respect to virtual external operators is the imaginary part of c(∆, ξ) along the physical cut, due to the ambiguity of analytic continuation of weights. The prototypical example of this ambiguity is The real part is recovered by the Kramers-Kronig relation 16 , A natural question is that whether the inversion function can be continued onto the unitary principal series. This requires the external operators of the input four-point functions can be continued according to the scheme ξ i → e iθ ξ i shown in figure 1. In other words we need a family of four-point functions G 4 (λ) depending on λ analytically. In relativistic CFT this is 16 It is also known as the Hilbert transform, and will be reviewed in appendix A.1.
rare since the unitary theories are expected to be isolated points in the space of CFTs. In GCFT 2 we show in the appendix C.2 that the family G 4 (λ) exists at least for real λ, due to the existence of outer-automorphism of the Galilean conformal algebra. We leave this question for further study and return to the discussion on MFT.
Inversion formula. After inserting the inversion function into the inversion formula (5.45), we enclose the ξ-contour along the physical branch cut, then using the relation (6.14) we get As discussed in the appendix A.1, the power factor (ξ − ξ 1 − ξ 2 ) −1−∆+∆ 1 +∆ 2 as tempered distribution possesses simple poles at ∆ n = ∆ 1 + ∆ 2 + n, n = 0, 1, 2, . . . . The remaining factors are analytic at ξ = ξ 1 + ξ 2 and the Taylor expansion is Then separating the ξ-integration of (6.18) into two parts 17 , we find that the part on the interval (ξ 1 + ξ 2 + 1, ∞) contributes no ∆-poles and hence can be dropped when deforming the ∆-contour, where in the last line we have deformed the ∆-contour and pick up the double-trace poles (6.19), and the minus sign is canceled due to the clock-wise order of the contour.
In summary we have the conformal block expansion of the projected four-point function with the block coefficients (6.23). In the crossing region (0, 1) × R, the projected four-point function (6.24) converges rapidly and can be re-expanded as a double Taylor series of (x, k) in the s-channel limit. The expansion matches with that of the four-point function (6.4) order by order. Hence (6.24) should equal to (6.4) in the crossing region. In this way we derive the conformal block expansion of the four-point function and confirm the validity of the inversion formula.

Bilocal actions of mean field theory
There are few GCFT 2 models with concrete actions. One of them that has been thoroughly discussed in the literature is the BMS free scalar model [106] with the action on the plane S = 1 2 dxdy ϕ∂ 2 y ϕ. (6.25) As a worldsheet theory, this model describes the tensionless limit of the free bosonic string, see e.g. [113][114][115] and the ambitwistor strings [116][117][118]. This action can also be realized as a √ T T deformation of the relativistic free scalar [119]. The higher dimensional Carrollian analog of (6.25) was discussed in [36,39,84].
In this subsection we explore the Lagrangian description of Galilean MFT. Inspired by the MFT in AdS/CFT [120] and the long-range models in statistical physics, see e.g. [110], we find a series of bilocal actions labeled by (∆, ξ) corresponding to the Galilean MFT. Moreover at the special value ξ = 0 we get additional actions labeled by (∆ 1 , ∆ 2 ), one of which gives the BMS free scalar (6.25). Starting from the ansatz of bilocal action S = dx 1 dy 1 dx 2 dy 2 ϕ 1 (x 1 , y 1 )K(x 12 , y 12 )ϕ 2 (x 2 , y 2 ) (6.26) with two scalars transforming as ϕ i ∈ V ∆ i ,ξ i , and imposing the Galilean conformal invariance as in table 1 on the action, we get four equations of K, which are related to the two-point Ward identities (2.16) by the shadow replacement (∆ 1 , ξ 1 , Mathematically speaking, the discussion in section 3.4 is on the intertwining maps, and the action (6.26) is an intertwining bilinear form introduced by Bruhat [121], see also chapter 3 of [122]. Intuitively, the finite dimensional analogs for two representations For unitary representations the two concepts are essentially the same. We can use the positivedefinite inner product (V 1 , U 1 ) = g ab V a 1 U b 1 to raise and lower the indices, then the intertwining map and bilinear form are related by K ac = g ab K b c . For non-unitary representations they are not necessarily equivalent.
Bilocal actions. By solving the differential equations of K, we can read the bilocal actions. We find the following distributional solutions for K in two different cases.
Case 1: ξ 1 + ξ 2 ̸ = 0. The last two equations in (2.16) force the scalars to be identical, ξ 1 = ξ 2 , ∆ 1 = ∆ 2 , and the solutions are exactly the kernel of the shadow transform (3.19), where the first one corresponds to bosonic statistics and the second to fermionic statistics.
At special values ξ 1 = 0, ∆ 1 = 3−n 2 , n ∈ Z ⩾0 , one of the power-type distributions should be regularized to the n-th derivative of the δ-distribution δ (n) (x 12 ), and the action is local with respect to x.

Localization of shadow integrals
In this subsection we discuss the localization of the shadow integrals. In the previous sections we notice that the integrals in the shadow formalism are localized to Dirac δ-distributions.
The reason is similar to that of ambitwistor strings [116][117][118]. We call the involved integrals as "shadow integrals" with the following form A(x e , y e , ∆ n , ξ n ) = i∈I dx i dy i ⟨OO⟩⟨OOO⟩ · · · = i∈I dx i dy i F x (x n , ∆ n )e Fy(xn,yn,ξn) , (6.49) where all the operators are virtual singlets O n ∈ E ∆n,ξn , and F x is the collection of power functions and F y is the collection of exponential factors. Borrowing the terminology of Feynman diagram, we label the integrated positions by i ∈ I and call them "internal", (x i , y i ), i ∈ I, and label the remaining positions by e ∈ E and call them "external", (x e , y e ), e ∈ E. Since the exponential factors are bilinear with respect to (y n , ξ n ), we have F y = iA 0 (x n , y e , r n ) + i i∈I y i A i (x n , r n ), (6.50) and the y i -integration gives a δ-distribution localized on (the real points of) the algebraic variety where f ab 's are linear functions of r n . Then the integral (6.49) simplifies to A(x e , y e , ∆ n , ξ n ) = (2π) |I| i∈I dx i δ(A i ) F x (x n , ∆ n )e iA 0 (xn,ye,rn) . A : • OPE block: the factor F x in the shadow integral depends on y ab , then the remaining integral (6.52) contains derivatives of δ-distribution on V.
A : can be constructed as (6.59) and the shadow variety is The conformal partial wave contains four conformal blocks due to the shadow symmetry, implying that V as equations of x a , x b should have four roots, each of which contributes to a conformal block. After eliminating x b we indeed get a fourth order equation of x a . However since the power factors are of the form 5 i=0 |F i | ∆ i , after inserting the solutions of x a , x b , the result is unlikely to be simplified unless ∆ i ∈ Z. We leave this possibility for further study.
For six-point conformal partial waves, there are two types of OPE, named as the comb channel and the snowflake channel. In both cases the shadow varieties V are defined by three equations. As equations of x i , i ∈ I, we check numerically that V has eight different roots, as expected from the shadow symmetry.

Conclusion and Discussions
In this work, we studied the shadow formalism for two-dimensional Galilean conformal field theory. As 2d Galilean conformal group is isomorphic to 3d Poincare group, we are allowed to use the Wigner-Mackey classification of 3d Poincare group to identify the unitary principal series representations and then to construct the shadow transform for GCFT 2 . Using the shadow transform, we computed the OPE blocks, and discussed the Clebsch-Gordan kernels and the shadow coefficients. Moreover, we studied the conformal blocks and conformal partial waves in the framework of shadow formalism.
Furthermore we investigated several applications of shadow formalism, including the revisit of the decomposition of four-point function in Galilean MFT and the construction of a series of bilocal actions of Galilean MFT. In the revisit of the four-point function in MFT, we proposed a new inversion formula, due to different form of conformal partial waves from the one in [54].
The resulting inversion function should be treated carefully and led to the correct conformal block expansion of four-point function. In constructing the bilocal actions of Galilean MFT, we used the intertwining bilinear forms, which obey the Ward identities. In special cases, these actions reduce to those of the BMS free scalar and the homogeneous tensionless fermionic string.
In our study, we came across a kind of integrals, which we called shadow integrals. The shadow integral can be reduced to the integral over an algebraic variety, due to the localization in the y i integration. This remarkable property makes analytic bootstrap in GCFT 2 feasible.
There are several future directions: More on shadow formalism. In this work we mainly focused on the shadow formalism related to singlet representations, and the exchanged boost multiplets were dealt separately by the method of Casimir equations. Firstly it would be interesting to extend the current approach to boost multiplets in GCFT 2 and similarly to logarithmic multiplets in LogCFT.
This requires a way to bypass the unitarity of the "Euclidean" shadow transform. Secondly, when ξ = 0 operators are involved in the shadow formalism there are technical difficulties to be settled. For example, the ξ = 0 shadow transform is not unique due to the existence of different types of solutions of the Ward identities. Besides the shadow formalism, the ξ = 0 subsector are found related to the celestial CFT recently [84], and one may try to explore the role of the ξ ̸ = 0 subsector played in the celestial holography.
GCFT 2 with/as defect. In [123], the lower-point correlation functions of boundary GCFT 2 were determined. Interestingly there are two types of boundaries with different residual symmetries, and in result the coincidence of 2d Galilean and Carrollian conformal symmetries can be distinguished. One could adopt the shadow formalism to the analytic studies of boundary and crosscap GCFT 2 , and even interfaces between Galilean, Carrollian and relativistic CFTs. On the other side, the GCFT 2 itself can be regarded as the null defect in Lorentzian CFT 3 , which deserves to be explored further.
Deformations of MFT. Constructing interacting GCFT 2 is an important task, and the numerical Galilean conformal bootstrap is obstructed by the lack of unitarity in GCFT 2 . The problem is that the highest weight representations contain negative-norm states except for the ξ = 0 singlets, and if the spectrum contains only ξ = 0 singlets, the theory reduces to a CFT 1 . Hence a prior there is no positivity constraint on the bootstrap equations. The unitary representations in the Wigner classification do not correspond to local operators, but are more interesting and can be related to the celestial/Carrollian correspondence, since the analogs in CFTs -the principal series representations have already appeared in the discussions of dS bootstrap and celestial CFTs. In many aspects, the GCFT 2 in this paper is more similar to the logarithmic CFT. It would be interesting to add relevant interactions to the Galilean MFT and to investigate whether there are nontrivial fixed points, which will further serve as the target of the numerical Galilean conformal bootstrap.

Acknowledgments
We would like to thank Peng-Xiang Hao and Zhe-Fei Yu for the participation in the early stage of this project. We are grateful to P. Hao

A.1 Regularization of tempered distributions
In the main text the calculations are undertaken in the framework of tempered distributions, and we have come across the regularization of power-type distributions. In this subsection we provide a mild introduction to the regularization and normalization of distributions by examples, following [124]. For simplicity the functions and distributions are on R.
Regularization. A tempered distribution ϕ ∈ S ′ (R) acting on the rapidly decreasing test function f ∈ S(R) can be formally written as an integral ϕ(f ) = dx ϕ(x)f (x) with kernel ϕ(x). One can imagine f as a Gaussian wave-packet and ϕ as some sharp observable. When the kernel ϕ(x) is a function with singularities, e.g. 1 x−y , the integral is convergent only for a small class V of test functions f ∈ V ⊂ S(R). Then subtracting off all the divergent terms of the integral means extending the domain of ϕ from V to S(R). This procedure is called regularization of distributions.
For example, the regularization of ϕ(x) = 1 x−y can be chosen as This is the Hilbert transform, also the shadow transform of ∆ = 1 2 in CFT 1 as reviewed in the appendix B.2. It is a unitary operator on L 2 (R) 18 with two eigenspaces H 2 + (R) ⊕ H 2 − (R), and as a result the (fermionic) principal series representation with ∆ = 1 2 of SL(2, R) is reducible. The extension is usually not unique. We are interested in the case that a family of distributions ϕ a (x) depends on parameter a analytically. Then the analyticity of a helps us pick out a unique regularization of ϕ a (x). For example, the power-type distribution is convergent if Re a > −1 and hence is analytic with respect to a. If Re a ⩽ −1 the integral is divergent and acquires regularization at x = 0. Inserting a real-analytic test function n! x n and interchanging the order of integration and summation, we have The first term is well-controlled due to the rapid decay of f (x) and is irrelevant to our discussion. The second term implies that (x a + , f (x)) as a function of a is meromorphic in C, with simple poles at a = −1, −2, . . . and residues Res a=−n (x a + , f (x)) = f (n−1) (0) (n−1)! . Then stripping off the test function we find x a + is meromorphic with respect to a ∈ C, with simple poles at a = −1, −2, . . . and residues, Similarly |x| a , |x| a sgn(x) and x a − = θ(−x)|x| a are all meromorphic with respect to a ∈ C. We summarize their analytic structure in the table 3.  distributions   poles residues at x a + −1, −2, −3, . . .
|x| a sgn(x) −2, −4, −6 . . . Normalization of distributions. We can cancel the simple poles of x a + by a suitable Gamma function, and the normalized distribution  16) in which the distributions x −n are understood as |x| −2n and |x| −2n−1 sgn(x) respectively.

Example.
As an example we provide the calculation of (B.15) as follows, where in the first line the integral is regularized by shifting the weight slightly, in the second line the 1d KLT integral [59] is used, and in the last line the δ-distribution comes from regularization of the distribution |x| α . Notice that the KLT integral is equivalent to the star-triangle relation by a special conformal transformation.

B Kinematics and shadow formalism of CFT 1
The 1d CFT appears in the SKY model, conformal defect and lightcone limit of higher dimensional CFT. The discussion of GCFT 2 is similar to that of CFT 1 in many aspects. In this appendix we review the kinematics of CFT 1 , see e.g. [57,107,[125][126][127][128], including correlation functions, the shadow transform, OPE blocks and the conformal block expansion. where n = ±1, 0, and the finite transformations are

B.1 Local operators and correlation functions
where x ′ = f (x) are the conformal transformations.
name charge vector field finite transformation The two-point functions of primary operators and the three-point functions are where c 123 is the three-point coefficient. The four-point function of O i could be written as a product of the stripped four-point function G (s) (x) and a kinematical factor K (s) (x i ) then read the t-channel O 2 × O 3 → O 1 × O 4 kinematical factor by the permutation (13) The 1d Lorentzian conformal group acting on the Lorentzian cylinder is the universal cover SL(2, R). The covering map of these related groups are summarized as The 1d Euclidean conformal algebra is related to the Lorentzian one by the NS-quantization and Wick rotation, The s − t crossing equation from (B.6) and (B.8) is Given a unitary principal series representation E ∆= 1 2 +is , s ∈ R ̸ =0 , we denote the associated shadow representation as E ∆=1−∆ and an operator transforming in E ∆ as O. The shadow is an intertwining map between two representations If the representations E ∆ and E 1−∆ are both irreducible, by the Schur lemma, S is an isomorphism. To check this we apply the shadow transform twice S 2 : E ∆ → E ∆ , In the case that S is an isomorphism, the kernel of S 2 should be proportional to identity, . This kernel K(S 2 ) can be evaluated via the KLT integrals and the prefactor is When ∆ = 1 2 + is, s ∈ R ̸ =0 , the factor N (∆) is finite and nonzero, hence the shadow transform is an isomorphism indeed.
At the position of the poles and the zeros of N (∆), the operators S and S −1 are not isomorphisms. For example, when S is not injective, the kernel subspace ker S is a subrepresentation, and the maximal quotient E ∆ / ker S[E ∆ ] is the discrete series representation, see e.g. chapter 7 of [122].

B.3 OPE blocks
The OPE relation of two primary operators is and the bilocal operator D 12k O k (x 2 ), capturing the resummation of the derivative operators, is called the OPE block [63], Here our convention follows the one used in [129], and is slightly different from the one in [97], which is The relation between two conventions is Notice that the technically safe way of phrasing the OPE relation (B.16) is to introduce the vacuum OPE [65], where R is the operator ordering with respect to a specific quantization scheme.
There are a few equivalent methods of computing the OPE block: the first is to use the compatibility of the OPE relation and the three-point functions, the second is to apply the recursion relation by imposing the symmetries on the OPE relation, and the third is to use the shadow formalism. We first recall the method of the recursion relation and then discuss the OPE blocks from the shadow formalism.
OPE blocks from recursion relations. Shifting the vacuum OPE to the origin and denoting the descendants as |O k , n⟩ := L n −1 |O k ⟩, we have The L 1 's action gives and by using L 1 |O k , n⟩ = n(2∆ k + n − 1) |O k , n − 1⟩ we have Due to the orthogonality of the descendants we get the recursion relation of a n a n (O k ) = (∆ 1k,2 + n − 1) n(2∆ k + n − 1) a is not manifest. For example, to switch from x 2 to x 1 we may use the Kummer identity To make the translational invariance apparent, we can average over the expansion point x 0 by a weight function f 12k , The left hand side transforms as insisting on the Euclidean region, there are extra unphysical contributions to the integral due to the shadow symmetry ∆ → 1 − ∆, and to single out the correct terms we need to introduce the projector of monodromy by hand [69].
If Wick-rotating to the Lorentzian region, the three-point structure We can check the equivalence of the two approaches by evaluating the integral (B.27) directly, where N 12k is some normalization factor to be determined. In the CFT 1 case, the causal diamond degenerates to the interval x 0 ∈ (x 2 , x 1 ) and we have where ∆ 4 = 1 − ∆ k and the coefficient integrals are

B.4 Conformal blocks
Using the OPE relations repeatedly the higher-point functions can be decomposed into a sum of the conformal blocks multiplied by the OPE coefficients. The conformal blocks capture the contributions from the exchanged conformal families. This procedure has been explicitly done in the CFT 1 and CFT 2 case [129] for arbitrary higher-point. There are a few efficient ways of calculating four-point conformal blocks: solving the Casimir equations [130], the recursion relations with respect to ∆ [80], and the shadow formalism [68,71]. We briefly illustrate the method of Casimir equation in CFT 1 .
Settings of conformal block expansion. The s-channel conformal block expansion of four-point function is where the (unstripped) conformal block with respect to primary O n is defined as and p (s) n = c 12n c 43n . To further carry out calculations we need to introduce the stripped version of conformal blocks depending only on the cross ratios by factoring out the kinematical factors then the block expansion of the stripped four-point function is Conformal blocks from Casimir equation. Inserting a complete basis into the fourpoint functions in the radial quantization x 4 > x 3 > x 2 > x 1 , we get where |O 0 | is the projection operator of the conformal family V ∆ 0 and G n,m = ⟨n|m⟩ , n, m ∈ V ∆ 0 is the Gramian matrix. Then the conformal block can be written as a summation over the matrix elements ⟨n| The Casimir differential operator is the representation of the abstract Casimir element of the conformal algebra when acting on the matrix elements ⟨n| O 1 O 2 |0⟩ , n ∈ V ∆ 0 . For element X in the conformal algebra sl(2, R), the corresponding Ward identity is where the differential operators are Hence by repeatedly using (B.40), and the fact that C acts on V ∆ 0 as a scalar ∆ 0 (∆ 0 − 1), we find where the differential operator C (1+2) is Inserting (B.41) into the expression of conformal blocks, we get the Casimir equation Then plugging the definition of stripped conformal blocks into this equation, we get the Casimir equation which has two independent solutions: the physical block and the shadow block g In this appendix we establish the relation between GCFT 2 and null defect in Lorentzian CFT 3 at the kinematical level, see also the discussion in higher dimensions [39] and a related discussion in [78]. We firstly recall the ideas of conformal defect, see e.g. [98,[131][132][133][134][135][136] and the analytic studies in [108,[137][138][139], then discuss the residual symmetry of null defects in the first subsection, the outer automorphism of Carrollian conformal algebra in the second subsection. Notice that this defect picture rules out the infinite-dimensional BMS symmetry, and the result does not contradict to the symmetry enhancement argument in [43], since the latter relies on the existence of local conserved currents, which a defect theory may not have.  The residual conformal group of null defect G d = SO(1, 1) ⋉ SO(2, 1) is exactly the same as one-point stabilizer subgroup of local operators, and G/G d = R 2,1 c , indicating a relation between null defects and points in R 2,1 c . Apparently we can associate the null-cone (x − X) · (x − X) = 0 centered at X ∈ R 2,1 with the point X itself. To see this map from the null defect to the point preserving the symmetry, we consider the set of hyperboloids and hyperplanes in In this set, the elements satisfying X · X − X + X − < 0, > 0, = 0 correspond to the timelike, the spacelike and the null defects respectively. Remarkably the conformal transformations of SO(3, 2) act linearly on M. Now the null-cone X · X − X + X − = 0 in M can naturally be identified with the embedding space of R 2,1 , i.e. R 2,1 c . In this way we establish a 1-1 symmetry-preserving correspondence between the null defects and the points in the conformal in which the null-planes correspond to the points at the conformal boundary of R 2,1 , and the conformal boundary itself corresponds to the spacelike infinity 21 . And as a byproduct we re-derive the embedding space formalism.
On the other side, the residual symmetries ISO(2, 1) act on the defect exactly as the Carrol-

C.1 Null-plane and null-cone defects
In this subsection we discuss three typical configurations of null defects: the null-plane, the lightcone and the conformal boundary. The settings of Lorentzian CFT 3 are summarized in appendix A.
Null-plane. For the defect located at x 0 − x 1 = 0, we use the lightcone coordinates, y ′ = Obviously the residual symmetry of the defect surface contains x-translation, y-translation and dilatation, and we can identify them with the generators of Carrollian conformal algebra, L −1 = P 2 , M −1 = 1 2 (P 0 + P 1 ), L 0 = D. With the null vector (1, 1, 0), the null rotation along the x − y plane, preserves the defect as well. It acts as x → x, y → y + vx, hence suggesting that M 0 = − 1 2 (M 20 − M 12 ).
To get the rest two generators, noticing that the action of the inversion I : x·x , x 2 → −x 2 x·x preserves the defect and acts as the inversion x → − 1 x , y → y x 2 on the defect. Hence we can identify the SCTs as One can check that the six generators form a subalgebra iso(2, 1) ⊂ so (3,2). We summarize the identification of the generators in the following, The last residual symmetry is Lightcone. Another type of null defect locates at the lightcone x · x = 0. Apparently the Lorentz transformations M ab and the three SCTs, which reduce to preserve the lightcone, and should be identified with L's and M 's in iso(2, 1) respectively.
The correct parameterization manifesting the Carrollian conformal symmetry in table 1 turns out to be where c is an arbitrary non-vanishing constant.
The x-dependence in (C.8) is inspired by the N-S quantization and the embedding space 2 maps a CFT 1 on the real line x ∈ R 1 to a CFT 1 on the unit circle θ = 2 arctan x ∈ S 1 , and from the embedding space viewpoint, this corresponds to choosing non-compact vs. compact slicing of the projective lightcone.
In our case the null direction of the lightcone is physical, and the parameterization such that x transforms as linear fractional transformation is The bulk dilatation D b := D acts on the null-cone as x → x, y → e −t y, and its commutation relations are the same as before.
Conformal boundary. Notice that we can exchange the translations and SCTs in R 2,1 by the inversion I, and the lightcone is mapped to the conformal boundary of R 2,1 c . Then the subalgebra inclusion is and D b := −D. In the picture of embedding space of R 2,1 , (X + , X − , X) = (1, x 2 , x a ), the lightcone defect is X − = 0 and the conformal boundary is X + = 0.

C.2 Bulk dilatation as outer automorphism of Carrollian CFT
In relativistic CFT d , the semi-simplicity of the (complexified) conformal algebra so(d+2, C) implies that the outer automorphism group is finite. In Carrollian CFT 2 , the outer automorphism group of both the global and local Carrollian conformal algebras is the multiplicative group, Out(cca 2 ) = R ̸ =0 . The connected component of identity in the group is generated by the bulk dilatation D b , since there is no intrinsic scale in R 2,1 mimicking the AdS radius.
From the null-cone defect picture, the bulk dilatation D b simply rescale the Carrollian time Defining the flowed generators as Q(t) = U (t)QU (−t), U (t) = e tD b , then we have where λ = e t . Defining the flowed highest weight state as |∆, ξ⟩ t = U t |∆, ξ⟩ by M 0 (t) |∆, ξ⟩ t = ξ |∆, ξ⟩ t , we find the boost charge exp where the y 0 -integral is If (4.2) is the correct ansatz for the OPE block, the integral (D.2) should be a homogeneous polynomial of degree (n + m) with respect to x 12 and y 12 I n,m (x 12 , y 12 ) = n+m k=0 a n,m,k (∆ 12 , ∆ 3 , ξ 12 , ξ 3 )x k 12 y n+m−k 12 .

(D.4)
Let us look the first coefficient integral, which is related to the normalization factor and the shadow coefficient. The related y 0 -integral is For virtual operators, the ξ's are purely imaginary, hence this integral is proportional to a δ-distribution where the special point is The interesting thing is that the constraint X ∈ (x 2 , x 1 ) is equivalent to Thus, the first integral is For other coefficient integrals, the computation is straightforward. The higher y 0 -integrals give rise to the derivatives of δ(x) where we have used the analytic continuation ξ n = ir n ∈ iR. Inserting this into the expression of I n,m (D.2), the integration with respect to x 0 turns into the derivatives of order m I n,m (x 12 , y 12 ) = I −1 coefficients We list the coefficients at level 2 in table 5. In the table, the unlisted coefficient a 0,2,2 is Along the diagonal direction the OPE block coefficients are the same up to a binomial coefficient, and this holds at the higher levels. This inspires us to make the simpler ansatz for the OPE block n,m a n,m (x 2 ∂ x 2 + y 2 ∂ y 2 ) n (x 2 ∂ y 2 ) m , (D. 14) in which the derivatives should be understood as acting on O 3 directly, and the coefficients are related to the previous ones by a n,m = a n,m,n+m iff I n,m (x, 0) = x n+m a n,m .

(D.15)
Equivalently there is a recursion relation for I n,m , x∂ y I n,m (x, y) = (n + 1)I n+1,m−1 (x, y) if m ⩾ 1, n ⩾ 0, (D. 16) which can be checked straightforwardly using (D.11). Actually this is originated from the Ward identity with respect to M 0 .
To calculate the leading coefficient a n,m , from (D.11) we read Here the derivative term can be expressed as a Jacobi polynomial. To be concrete, setting In the above, we have used the definition and the Rodrigues' formula for the Jacobi polynomials P (a,b) n (z) = (a + 1) n n! 2 F 1 (−n, 1 + a + b + n; a + 1; We can re-parametrize x 4 as

D.3 Check of the second term in bubble integral
To check the result (4.38), we calculate the second term in (4.34) directly. Assuming where the prefactor is

D.4 Deriving the Casimir equations
We may start from inserting a complete basis into the four-point functions in the radial quantization x 4 > x 3 > x 2 > x 1 , where |O 0 | denotes the projection operator with respect to the conformal family V ∆ 0 ,ξ 0 ,r |O 0 | = n,m∈V ∆ 0 ,ξ 0 ,r G −1 n,m |n⟩ ⟨m| . (D. 35) and G n,m = ⟨n|m⟩ , n, m ∈ V ∆ 0 ,ξ 0 ,r is the Gramian matrix of the inner product. In this way we can rewrite the conformal blocks (5.2) as a summation over the matrix elements ⟨n| O 1 O 2 |0⟩, which are rescaled version of the three-point functions involving the descendants, In principle we can use this projection operator to calculate the conformal blocks directly.

(D.57)
By checking the s-channel OPE limit x, k → 0, and redefining the normalization to ensure the exchanged primary operator contributes one: g (s) ∆ 0 ,ξ 0 ∼ x ∆ 0 e −kξ 0 , we find that the second solution g − (x, k) can be identified to be the physical block 2−∆ 0 ,−ξ 0 (x, k), thus the two solutions respect the shadow symmetry.

D.6 Calculation of the projector of conformal partial waves
In this subsection we show that for four identical external operators the projection operator of conformal partial waves (5.50) is proportional to The overall coefficient is irrelevant to our discussion and has been omitted. For the ϕ 1 ϕ 2 ϕ 2 ϕ 1type external operators, there seems no closed-form of this projector because the associated shadow variety reduces to higher order equations of {x 0 , x ′ 0 , y 0 , y ′ 0 }, and the remaining integration is hard to calculate.
Inserting the definition of conformal partial wave (5.28) into the projection operator (5.50) we have, In the second line the integral region of s 0 has been changed from (0, ∞) to R using (5.45), and we need to select the correct δ-distribution related to the conformal block in the partial wave. In the last line the substitution r 0 y 0 → y 0 , r 0 y 0 → y 0 is used to separate the first three terms in the exponential part, and the factor r 2 0 from N gets canceled so that the integration of y 0 , y ′ 0 , r 0 gives three δ-distributions. The s 0 -dependence is collected into F 0 , and the integration gives δ(F 0 ). The divergent volume factor is kept since we haven't done the gauge fixing procedure. One can check that after fixing (x ′ i , y ′ i ) to the standard conformal frame and renormalizing O ′ 1 (x ′ 4 , y ′ 4 ) as (2.26), the integral is finite. In total the shadow variety V is defined by four equations,