Semi-classical BMS-blocks from the Oscillator Construction

Flat-space holography requires a thorough understanding of BMS symmetry. We introduce an oscillator construction of the highest-weight representation of the $\mathfrak{bms}_3$ algebra and show that it is consistent with known results concerning the $\mathfrak{bms}_3$ module. We take advantage of this framework to prove that $\mathfrak{bms}_3$-blocks exponentiate in the semi-classical limit, where one of the central charges is large. Within this context, we compute perturbatively heavy, and heavy-light vacuum $\mathfrak{bms}_3$-blocks.


Introduction
The Holographic Principle promises a great amount of insight into a possible formulation of quantum gravity [1][2][3]. Over the past two decades, a number of statements concerning quantum gravity could be made rigorous within the framework of the AdS/CFT correspondence [4][5][6]. Low-dimensional examples of the duality, such as the AdS 3 /CFT 2 correspondence, have proven to be valuable toy models; explicit computations are possible on both sides of the duality and can be matched, see [7] for a recent example. In AdS 3 /CFT 2 the central charge of the two-dimensional boundary CFT is related to the inverse of Newton's constant in the bulk theory [8]. Remarkably, semi-classical quantumgravitational phenomena generically emerge from universal properties of two-dimensional conformal field theories at large central charge; in addition to the Virasoro symmetry of the field theory, this statement requires a few reasonable assumptions, including modular invariance, locality, and a sparse spectrum of operators whose conformal dimension remain bounded for large central charge. At finite temperature such CFTs display universal thermodynamics [9] which is in agreement with computations in thermal AdS or hairless BTZ black holes [10]. Similar universal behaviour holds for one-and two-point correlation functions of the bounded operators operators, up to exponentially suppressed correction terms [11]. So-called heavy-light conformal blocks show the approximate thermal nature of heavy operators, which may be viewed as black-hole microstates of the dual gravity side [12]; these blocks may be computed with Wilson-line networks on the dual gravity side [13,14].
However, any conclusion made in AdS 3 /CFT 2 is naturally followed by the question to which extent the insight can be carried over to more realistic cases, such as higher-dimensional models or dualities involving gravity in asymptotically flat spacetimes -the scope of the present work is in connection to the latter aspect.
Flat-space holography is the moniker given to the expected duality between quantum gravity in asymptotically flat cosmologies and BMS-invariant field theories [15,16]. It remains a challenging task to identify an explicit example of such a correspondence; for recent results regarding flat-space holography see for example .
A key ingredient in flat-space holography is the infinite-dimensional Lie algebra bms 3 , which is the asymptotic symmetry algebra of three-dimensional gravity with vanishing cosmological constant, at null infinity [41,42]. The bms 3 algebra is generated by super-rotations L m and super-translations M m , with m ∈ Z, and contains two central charges c L , c M ∈ R [43]. Classical Einstein gravity demands c L = 0; the case c L = 0 emerges in the consideration of so-called chiral gravity [44,45]. Assuming the existence of a flat-space holographic duality, the large central charge limit c M → ∞ of a BMS-invariant field theory should correspond to a semi-classical theory of gravity in asymptotically flat spacetime. Some results regarding BMS-invariant theories concern the bootstrap program [46,47], bms 3 characters and modular invariance [48], torus bms 3 -blocks [49], as well as the computation of heavy-light bms 3 -blocks using the monodromy method [50].
In the context of flat-space holography, heavy-light bms 3 -blocks contain information about probe fields in non-trivial asymptotically flat spacetimes in a dual theory of Einstein gravity. The cosmological solutions generated by heavy operators are then quotients of Minkowski spacetimes and are labelled by their mass M = −1 + 24ξ/c M and angular momentum J/ √ M = 12∆/c M , where ∆ and ξ denote the scaling dimension and rapidity of the heavy operator, respectively [50].
The bms 3 algebra is related to two copies of the Virasoro algebra by suitableİnönü-Wigner contractions: there exist two such contractions, known as the ultra-relativistic and non-relativistic limits; both contractions lead to isomorphic symmetry algebras but with different representation theories [16]. One typically finds unitary (induced) representations by ultra-relativistic contraction of the conformal symmetry algebra [29].
The oscillator construction may be employed in order to express highest-weight representations of the Virasoro algebra in terms of oscillator variables [51]; we review this construction in section 2. This formulation provides an alternative approach to the calculation of various quantities of two-dimensional CFTs, such as correlation functions and conformal blocks, and has recently been employed in the context of the eigenstate thermalisation hypothesis [52] and the proof of Virasoro-block exponentiation [53].
In the present work, we introduce an oscillator construction for highest-weight representations of bms 3 . 1 In section 3 we express the bms 3 generators in terms of oscillator variables and define a suitable measure on the function space. The origin of our expressions is a nonrelativistic limit of a linear-dilaton like theory. The oscillator formalism allows us to prove the exponentiation of bms 3 blocks in the semi-classical limit, which we do in section 4. Moreover, in section 4 we also demonstrate the applicability of the oscillator construction by calculating a perturbatively heavy vacuum bms 3 -block as well as a heavy-light vacuum bms 3 -block.

Oscillator Construction of the Virasoro Algebra
Before approaching the bms 3 oscillator construction it is advantageous to first introduce the oscillator construction and its meaning for two-dimensional conformal field theories. The conformal symmetry algebra decomposes into a direct sum of two copies of the Virasoro algebra: one holomorphic and one anti-holomorphic copy. In the following review we will focus on the holomorphic sector but the analysis straightforwardly extends to the anti-holomorphic sector.

Highest-Weight Representation of the Virasoro Algebra
The generators of the Virasoro algebra, which we denote L vir m , obey the commutation relations where c denotes the central charge and m, n ∈ Z. We consider highest-weight representations of the above Virasoro algebra; such representations are built from a primary state |h with conformal weight h, satisfying L vir 0 |h = h |h and L vir m |h = 0 for m ≥ 1.

Descendant states are generated by basis vectors of the form
with m 1 ≥ . . . ≥ m k ≥ 1. A primary state and its descendants span a vector space called the Verma module V c,h . Imposing that the central charge c is a real number, as well as the adjoint relation L vir m † = L vir −m , there exists a unique Hermitian product q|p on the Verma module V c,h for which a primary state has unit norm. For c, h > 0 and in the absence of singular states the Hermitian product is positive definite and the corresponding representation is irreducible and unitary.

Oscillator Construction
We now employ the oscillator construction to study the highest-weight representation of the Virasoro algebra. In this context we write the Virasoro generators in cursive lowercase and they take the form 2 where k ≥ 1 and the constants λ, µ ∈ R are related to the central charge and conformal weight by 3 In this language, the states of a Verma module V c,h are mapped to functions which depend on a full set of infinitely many representation space variables u n ∈ C with n ∈ N; in accordance with previous literature we will refer to these variables as oscillator variables. Equivalence between the oscillator construction and the formalism presented in the previous subsection follows from the definition where u| ≡ |ū † is a generalised coherent state of the Verma module V c,h , |p is a generic state of the Verma module, and we omit the index of the oscillator variables to indicate a full 2 These generators are obtained by considering a linear dilaton conformal field theory. See appendix A of [52] for further details. 3 To gain access to Verma modules with h < (c − 1)/24 the constant λ must be analytically continued to imaginary values. Similarly, minimal models with c < 1 require an analytic continuation in µ.
set. The presence of a derivative in ℓ 0 makes it natural that the primary state of V c,h maps to a constant, which we choose to be unity, i.e.
The basis vectors |(m 1 , . . . , m k ) ; h , which were defined in equation (2.1.2), are mapped to Given the form of ℓ −k in (2.2.1c), the basis vector on the right-hand side of the above equation is a polynomial in the oscillator variables. The dual function is defined by f q (u) = u|q ≡ q|ū , where |q is a state in V c,h and the overline operation acts as u n →ū n as well as complex conjugation. Such dual functions are acted on byl −m , which results in the action q|L vir n |ū . Note that the barred quantities belong to the same holomorphic sector of the two-dimensional conformal algebra as the non-barred quantities.
Unitarity requires the adjoint property ℓ † m = ℓ −m , which is defined by the Hermitian product; the appropriate expression in the oscillator construction may be found by inserting a completeness relation into the product q|p defined in the previous subsection. The Hermitian product of the Virasoro oscillator construction thus reads with the measure given by where d 2 u n = du n dū n . Using the Hermitian product given in (2.2.7), monomials of oscillator variables form an orthogonal basis which satisfies (2.2.9) The above orthogonality relation proves to be useful when evaluating certain quantities, for instance the Gram matrix [52].

Correlators and Wave Functions
Building upon the Virasoro symmetry and its representations, a two-dimensional conformal field theory is defined in terms of its operators and correlation functions. A holomorphic primary operator O h (z) is defined via the operator-state correspondence as where z is a holomorphic coordinate on the complex plane and the vacuum state |0 is characterised by h = 0 and L vir where L vir m is a differential operator given by Note that L vir m satisfies the Virasoro algebra without a central extension.

Two-point Correlation Functions
The two-point correlation function is given by 0|O h 1 (z 1 )O h 2 (z 2 )|0 ; inserting a complete set of states as defined in equation (2.2.6) results in the expression The above result defines the level-one wave functions where the subscripts h i denote the conformal dimensions of the external operators O h i while h labels the Verma module V c,h . When treated individually we refer to ψ h 2 ;h (z 2 ; u) as the wave function and χ h 1 ;h (z 1 ;ū) as the dual wave function; the relationship between them is 4 which follows from the property that bra-and ket-states of a conformal field theory are related by a coordinate inversion on the complex plane. As a consequence of L vir m |0 = 0 for m ≥ −1, together with (2.3.2), the level-one wave functions satisfy the differential equations Since the level-one wave functions connect oscillator variables and coordinates of the complex plane, we here introduce the superscripts h i and h to emphasise the generators' respective domains of action. Note that the above differential equations allow for non-trivial solutions only if h 1 = h or h 2 = h.

Three-point Correlation Functions and Conformal Blocks
Inserting a completeness relation into the definition of a three-point correlation function, in two different ways, gives The structure of the above expressions may be used to define the level-two wave functions Following the same reasoning as in the previous paragraph the level-two wave functions are related by and they satisfy the differential equations We may now appropriately express conformal blocks in terms of wave functions. Virasoro blocks are defined by the four-point function 0|O where P h denotes the projector onto the Verma module V c,h . It follows directly from the argumentation and definitions presented in this subsection that a conformal block reads The operator-state correspondence allows for the primary and descendant states of the Verma module V c,h to be interpreted as internal operators which arise due to the operator product expansion of the four-point correlation function. Although the differential equations (2.3.7) allow for solutions for general values of h and c, the level-two equations (2.3.11) have known closed-form solutions for the case h i = 1/16 and c = 1, only [51]. Nevertheless, in the semi-classical limit, i.e. c → ∞, it is possible to find approximate solutions for the level-two wave functions, and by extension the Virasoro conformal blocks [53].

Oscillator Construction of the Highest-Weight Representation of bms 3
The bms 3 algebra is a semi-direct sum of one Virasoro algebra and an infinite-dimensional Abelian algebra of super-translations. Hence, in contrast to the Virasoro case, the bms 3 algebra cannot be decomposed into commuting sectors; thus it needs to be treated en bloc. In this section we first review highest-weight bms 3 modules, after which we present our oscillator construction of such modules. We also discuss the computation of correlation functions and bms 3 -blocks in the language of oscillator variables and wave functions.

Modules of the bms 3 Algebra
The bms 3 algebra is generated by L n and M n , and defined by the Lie bracket where c L and c M are central charges, which we take to be non-negative real numbers, and m, n ∈ Z. In the context of asymptotically flat gravity in three spacetime dimensions L n are the generators of super-rotations and the M n generate super-translations. We define a primary state |∆, ξ to be a state that satisfies the eigenvalue equations where ∆ is the scaling dimension and ξ is the rapidity, as well as L n |∆, ξ = 0 and M n |∆, ξ = 0 , is built by acting with an ordered string, of arbitrary length, of operators L −n and M −n with n > 0 on a state |∆, ξ . Hence, the vector space associated to the highest-weight representation of bms 3 is spanned by the basis vectors |(m 1 , . . . , m s ), (n 1 , . . . , n l ); ∆, ξ = L −m 1 · · · L −ms M −n 1 · · · M −n l |∆, ξ , (3.1.5) 5 In the following we will use the short-hand notation (mi) s i=1 ≡ (m1, . . . , ms).
For the highest-weight representation of bms 3 the Hermitian product is not generically positive semi-definite and hence the corresponding representation will not necessarily be unitary. 6 We return to this point at the end of the next subsection.

Oscillator Construction
The expressions for the bms 3 generators in terms of the complex oscillator variables v (1) n and v (2) n , with n ∈ N, may be found by taking a non-relativistic limit of a two-dimensional linear-dilaton like conformal field theory; we refer the reader to appendix A for details. The resulting generators, which we denote in lowercase, read and, for k > 0. The above generators satisfy the bms 3 commutation relations (3.1.1). In the above expressions we have made the identifications 7 (3.2.3b) 6 There exists an exceptional case for cM = 0 and ξ = 0 where the bms3 representation constructed above reduces to a Virasoro highest-weight representation with central charge cL and conformal dimension h = ∆, provided that we take a quotient with respect to the null states M−n|∆, 0 with n ∈ N [54]. 7 If ξ ≥ cM/24 these coefficients are related by complex conjugation,Â k = A * k andB k = B * k ; in this case the adjoint property l † n = l−n holds. However, for ξ < cM/24 all coefficients are real and hence independent of each other; preserving the adjoint property requires analytic continuation, as discussed in section 4.3.1. Unless otherwise stated we will assume ξ ≥ cM/24.
Analogously to the Verma module, a state |p ∈ B c L ,c M ∆,ξ is mapped to a function via where v| ≡ |v † is a generalised coherent state of the bms 3 module, and v denotes the collective set of all oscillator variables v (1) n and v (2) n . 8 It follows from the form of the generators (3.2.1) and (3.2.2) that the requirements for a primary state are fulfilled by a constant function, hence we choose The properties (3.1.2) and (3.1.3) translate to for k > 0. The basis vectors of the form (3.1.4) are thus given in terms of the polynomials n and complex conjugation. The barred functions are acted on with the generatorsl −n andm −n , which in terms of states is expressed as q|L n |v and q|M n |v , respectively.
Finally, the adjoints l † n = l −n and m † n = m −n specify the unique Hermitian product of the oscillator construction where the measure is given by n . The motivation for the form of the measure [d 2 v] ∆,ξ is deferred to the appendix A.2. We arrived at the expression (3.2.7) in the same way as one does for Virasoro, i.e. by inserting a complete set of states of the form into the Hermitian product q|p . The Hermitian product of oscillator monomials is given by the formula which is a result of a calculation that is sketched in appendix A.2. By calculating the Hermitian product of basis vectors we find the Gram matrix; in the oscillator construction a general element of the form (3.1.5) is given by The basis vectors are polynomials in v (1) n and v (2) n , hence to calculate a specific element of the Gram matrix one may advantageously apply the orthogonality relation (3.2.10) to their constituent monomials. For the lowest-level bms 3 Gram matrix we obtain the entries The above matrix components, as well as the second-level Gram matrix entries (which are not displayed here) match the results of [48]. This serves as a check of our oscillator construction of bms 3 .
The bms 3 module is in general not unitary. This can be seen from the lowest-level Gram matrix since at least one of its eigenvalues is negative if ξ = 0; thus the Hermitian product is negative definite, or indefinite, and hence the highest-weight representation is non-unitary.

Correlators and Wave Functions
We define a bms 3 primary operator O ∆,ξ (t, x) by means of the operator-state correspondence where t and x are coordinates on the plane. The vacuum state |0 is defined as the primary state with ∆ = ξ = 0 and L n |0 = M n |0 = 0 for n ≥ −1. The generators of bms 3 act on and the differential operators take the forms [55] Note that L n and M n satisfy the commutation relations (3.1.1) with c L = c M = 0.

Two-point Correlation Functions
Inserting the completeness relation (3.2.9) into the definition of a two-point correlation function of primary operators gives

3.4)
where we have defined the level-one wave functions We will refer to ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) as the wave function and χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) as the dual wave function. The subscripts ∆ i and ξ i label the scaling dimension and rapidity of the external operators, respectively, while ∆ and ξ label the bms 3 module B c L ,c M ∆,ξ . Using that L n |0 = M n |0 = 0 for n ≥ −1, as well as the definitions of the differential operators (3.3.2), we find two sets of differential equations for the wave function for n ≥ −1. By similar arguments we find that the dual wave function χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) is constrained by the set of differential equations for n ≥ −1. Above we use superscripts for the generators l n , m n as well as L n and M n for the same reason as discussed below equations (2.3.7). The wave functions may be determined by solving both sets of differential equations for n ∈ {−1, 0, 1, 2}, which also ensures the validity of the solutions for all n > 2. 9 The differential equations (3.3.6) and (3.3.7) have non-trivial solutions only if ∆ 2 = ∆, ξ 2 = ξ, and ∆ 1 = ∆, ξ 1 = ξ, respectively. 10 We find the following expressions for the level-one bms 3 wave functions, where the coefficients A 1 , B 1 andÂ 1 ,B 1 are given by equations (3.2.3a) and (3.2.3b) for k = 1, respectively. 11 The relationship between the above level-one wave functions is thus where the overline acts in accordance with footnote 7. The above relation can be determined from the expressions (3.3.8) or motivated by the operator-state correspondence for bra-states Inserting the wave functions (3.3.8) in (3.3.4), power expanding, and using the orthogonality relation (3.2.10), we arrive at the expression for the bms 3 two-point correlation function if ∆ 1 = ∆ 2 = ∆ and ξ 1 = ξ 2 = ξ, and zero otherwise. The above expression is in agreement with previously known results [55].

Three-point Correlation Functions and bms 3 -blocks
In terms of our oscillator construction for bms 3 modules, the three-point correlation function reads where we have defined the level-two wave functions by where the shorthand notation ∆ i,j and ξ i,j indicates the dependence on both ∆ i , ∆ j and ξ i , ξ j . The left-and right-hand sides of (3.3.12) are related by an insertion of a complete set of states (3.2.9). The transformation between bra-and ket-states (3.3.10) means that the level-two wave functions are related by (3.3.14) 11 We detail the solution procedure for ψ ∆,ξ (t2, x2; v) in appendix C. Note that the level-one wave functions are completely fixed by the differential equations (3.3.6) and (3.3.7) for n ∈ {−1, 0, 1}. This is connected to the fact that two-point functions are completely determined by the globally well-defined generators Ln and Mn with n ∈ {−1, 0, 1}. Nevertheless, the set of n = 2 differential equations must still be satisfied in order to guarantee a general solution for all n.
Following the same reasoning as in the previous subsection we find that the wave function must satisfy the set of differential equations and similarly the dual wave function must obey for n ≥ −1. Like for the Virasoro case, we have not found any closed-form solutions for the level-two wave functions. Nevertheless, in section 4 we make use of the semi-classical limit in order to approximate solutions for the above set of equations. Finally, bms 3 -blocks are given by where P ∆,ξ is the projector onto the bms 3 module B c L ,c M ∆,ξ , which when restricted to the module acts as the unit operator, and we have inserted a completeness relation.
Without loss of generality we may consider the point configuration such that the bms 3 -block B ∆tot,ξtot;∆,ξ (t, x) is given by Furthermore, in the point configuration (3.3.18) the relationship between the wave function and its dual simplifies to Plugging in the above relation into (3.3.19) and implementing the limits, we reach the compact formula The set of equations (3.3.15) with n = 0 fixes the wave function to be of the form where F (η, ν) is an unknown function, and we have introduced the combinations of oscillatorand spacetime variables where n ∈ N. The dual wave function follows from the transformation (3.3.20). In appendix B.1 we prove that F (η, ν) is uniquely determined by the remaining differential equations.

Semi-classical bms 3 -blocks
In this section we will apply the machinery of our oscillator construction of the highest-weight representation of bms 3 in order to calculate bms 3 -blocks in the semi-classical limit c M → ∞, are the scaling dimension and rapidity of the internal primary operator of each block, while ∆ i , ξ i with i = 1, .., 4 denote the scaling dimensions and rapidities of the external operators. In order to implement the semi-classical limit we express the affected quantities in terms of an auxiliary parameter µ as follows, hence the semi-classical limit corresponds to µ → ∞ while keeping the tilde-quantities fixed.

Semi-classical Differential Equations
The semi-classical limit allows us to evaluate the bms 3 -block, using the integral expression (3.3.21), by means of the saddle-point approximation. In order to express the integrand of (3.3.21) in a form which is suitable for the saddle-point approximation we must consider the contribution due to the exponential function in the measure (3.2.8); we rescale the oscillator variables v  In accordance with our intention to employ the saddle-point approximation, we introduce an exponential ansatz for the wave function (3.3.22) F (σ, κ) = exp µ 2 S(σ, κ) .
were we have defined We arrived at the equations (4.1.4) after dividing by a leading factor of µ 2 and getting rid of the overall exponential function; the resulting terms proportional to 1 µ are sub-leading in the limit µ → ∞ and have hence been dropped, leaving two first-order differential equations for S(σ, κ).

Proof of Exponentiation
Our ansatz (4.1.3) has consequences for the general form of the bms 3 -block. Using the exponential ansatz (4.

.2.2)
Note that the oscillator variables in the sum are those which have been rescaled, providing the overall factor of µ 2 in the exponential function. In the semi-classical limit, i.e. when µ → ∞, the exponent of the exponential function in (4.2.1) is large and hence the integral is dominated by the stationary points of I(t, x; v,v); we may then use the saddle-point approximation in order to evaluate the integral. The stationary points (w,w) are found by extremising I(t, x; v,v) and they satisfy To approximate the bms 3 -block we plug the stationary points into the integrand of (4.2.1), which gives 12 where the overall factor 16n 2 µ 4 in (4.2.1) cancels with the determinant of the Hessian evaluated at the stationary point. The factor t ∆−∆ 1 −∆ 2 may be expressed as an exponential of a logarithm. We have thus shown that the bms 3 -block takes a unique exponential form in the semi-classical limit.
In the following subsections we present two illustrative examples for how to compute bms 3 -blocks using the oscillator construction. We will focus on bms 3 -blocks of the vacuum module B c L ,c M 0,0 , namely the perturbatively heavy vacuum block and the heavy-light vacuum block. Our computations will take advantage of the saddle-point approximation presented in this subsection.

Analytic Behaviour
As a consequence of setting ξ = ∆ = 0 the quantities defined in equations (3.2.3) will have negative expressions under a square root; we remedy this issue by analytically continuing 12 Generically there exits more than one stationary point. However, to keep the formula simple, we assume that there is only a single stationary point (w,w). 13 We set ∆ = ξ = 0 by choosing∆ =ξ = 0 before taking µ → ∞.
these quantities, resulting in the coefficients where we have chosen the branch √ −1 = +i. The absence of factors of i in the above expressions means that the bms 3 generators l n and m n , and by extension the differential equations (4.1.4), are unaffected by the complex conjugation defined with the overline operation; hence the relation between the level-two wave functions (3.3.14) is invalidated. This situation was alluded to in footnote 7. Under the current circumstances we have two independent wave functions which we introduce in a way that is consistent with equation (3.3.21), i.e.
where the superscripts of the wave functions correlate with the superscripts of the coefficients (4.3.1) in their equations. 14 Subsequently, since the n = 0 equations are independent of the coefficients (4.3.1), we use the partial solution (3.3.22) to immediately deduce  The two sets of differential equations for S + and S − must be considered separately.

Solving the Differential Equations
The analysis leading up to the set of differential equations (4.1.4) is general enough that the expressions may serve as equations for the wave functions (4.3.2) if we make the appropriate substitutions. For ψ + ∆ 1 ,ξ 1 ;0,0 (t, x, 0, 0; v) we get 14 Although we will not use it here, the analytic continuation motivates an analogous notation for the bms3 generators, i.e. ln ≡ l + n , mn ≡ m + n andln ≡ l − n ,mn ≡ m − n .
and for ψ − ∆ 3 ,ξ 3 ;∆,ξ (1, 0, 0, 0;v) we get where the tilde references the dependence on fixed quantities, as in equations (4.1.5). S + and S − may be treated as expansions in the infinitesimal parameters∆ 1 ,ξ 1 and ∆ 3 ,ξ 3 , respectively. We choose S + and S − to be of leading order ǫ. 15 The above set of differential equations has the leading order ǫ; hence the terms quadratic in derivatives of S ± are sub-leading of order ǫ 2 and may be dropped.
For the expansions of S ± which we are assuming, the stationary point coordinates (4.2.3) have the leading order ǫ. This means that products of oscillator variables with S ± give rise to terms which are sub-leading when evaluated at the stationary point -such terms will be dropped, too. This reasoning leaves us with the sets of linear differential equations as well as where the omitted terms are indicated by their order at the stationary point. Plugging the relevant expressions into the differential equations (4.3.7b) and (4.3.8b), and integrating, we get where f (σ) and g(σ) arise as constants of integration. The functions f (σ) and g(σ) are determined by integrating the equations (4.3.7a) and (4.3.8a) after the insertion of the above 15 In principle the leading terms of S + (σ, κ) and S − (σ,κ) may be constant functions of order one. Such constant functions may be dropped since they only contribute to the bms3-block as inconsequential multiplicative factors.
results; we get Using the variable transformations (3.3.23) and the rescalings (4.1.1), the leading order solutions for S ± read Note that our solutions for S + and S − are true up to order ǫ 2 when evaluated at the saddle point.

Implementing the Saddle-point Approximation
Evaluating the bms 3 -block according to the formula (4.2.4) requires us to determine I(t, x; w,w) at the stationary point (w,w). Appropriately substituting the solutions (4.3.11) into the formulae (4.2.3) we find that the coordinates of the stationary point are It is practical to consider the terms of I(t, x; w,w), as defined in equation (4.2.2), separately.

3.15)
which follows from the same identifications as for the expression (4.3.13).
Using the above results in the definition where factors of µ have been absorbed into the non-tilde quantities.

Heavy-light Vacuum bms 3 -block
The heavy-light vacuum bms 3 -block has two heavy and two light external operators; the light operators have infinitesimal scaling dimensions and rapidities of order ǫ, while those of the heavy operators are of order one. 16 Continuing with the pairings used in the previous subsection, we assign ∆ 1 , ξ 1 to the light operators and ∆ 3 , ξ 3 to the heavy operators; hence, in the semi-classical limit∆ 1 ,ξ 1 are of infinitesimal order ǫ while∆ 3 ,ξ 3 are of order one. The vacuum block dictates that ∆ = ξ = 0.

Analysing the Differential Equations
The prescription for determining the exponents of the semi-classical wave-function ansätze (4.3.4) is analogous to the procedure in subsection 4.3.2. In fact, the properties of the equations containing∆ 1 andξ 1 are unchanged and hence the solution for S + is identical to the expression (4.3.11a); thus also the coordinates of the stationary point (4.3.12c) and (4.3.12d) 16 Other authors define light operators to scale as order (cM) 0 in the semi-classical limit cM → ∞. Hence a perhaps more correct denomination would be to call the operators under consideration 'perturbatively heavy'.
are the same and of infinitesimal order. However, since∆ 3 andξ 3 are of finite order the discussion below equations (4.3.6) no longer applies to S − ; we must thus re-consider its differential equations and solution.
We motivate an appropriate ansatz for S − (σ,κ) by analysing the lowest non-trivial infinitesimal order of I(t, x; w,w), as given in (4.3.16). As mentioned above, the stationarypoint coordinatesw (i) m are of infinitesimal order ǫ; the coordinates w (i) m are of order one, hence the contribution of the measure to I(t, x; w,w) is of order ǫ. Similar argumentation leads to the conclusion that S + (t, x; w) is of order ǫ, too. Hence, any terms of higher order than ǫ which arise from S − (w) are sub-leading. Since the variablesκ n andσ n are of order ǫ when evaluated at the stationary point, we conclude that S − (σ,κ) should be at most linear in the variables and thus a suitable ansatz is 17 where the coefficients C n and D n depend onξ 3 ,∆ 3 . Plugging the above ansatz into the differential equations (4.3.5) yields The leading terms of the above equations are of order one. Invoking the perspective of the saddle-point approximation, the variablesσ n ,κ n give rise to sub-leading terms of order ǫ; thus terms containing these variables may be dropped. The remaining set of equations takes the form of recurrence relations, i.e.
where the omitted terms are indicated by their order at the stationary point (w,w), and we used the definitions (4.3.1b) forÃ − k andB − k . Note that for k = 1 the above recurrence relations fix the initial values C 1 = D 1 = 0.
The recurrence relations (4.4.3) may be turned into differential equations by using the method of generating functions. To this end, we make the ansätze As a consequence of the above ansätze we have C(0) = D(0) = 0 and the constraints C 1 = D 1 = 0 become the boundary conditions C ′ (0) = 0 = D ′ (0) = 0. Using standard techniques, the recurrence relations (4.4.3) are transformed into two coupled differential equations for C(τ ) and D(τ ), We present the details of the solution procedure of the above differential equations in appendix D; the solutions, to leading order in infinitesimal quantities, read Although the ansatz (4.4.1) is expressed in terms of the coefficients C m and D m , which can be extracted from the functions C(τ ) and D(τ ), we may also continue using the solutions as they are given in (4.4.6).

Implementing the Saddle-point Approximation
Returning to the rescaled oscillator variables (4.1.1) via (3.3.23), and remembering the solution for S + given in equation (4.3.11a), we have Note that the sum in S − starts at n = 2.
Being cognisant of the saddle-point approximation of the bms 3 -block, given in equation (4.2.4), the next step is to evaluate I(t, x; w,w). Making use of the expressions (4.4.8), the coordinates of the stationary point read Plugging the above values into equation (4.3.16) yields hence  where factors of µ are absorbed by the non-tilde quantities. Choosing ∆ 3 , ξ 3 to be infinitesimal; expanding β 3 to first order in ξ 3 ; keeping terms up to second order in infinitesimal quantities, transforms the above result into the perturbatively heavy vacuum bms 3 -block (4.3.17), modulo constant prefactors. 18 Furthermore, (4.4.13) generalises the perturbatively heavy vacuum bms 3 -block presented in [50].

Summary and Outlook
In this paper we presented an oscillator construction of highest-weight bms 3 modules. We used this construction to obtain general expressions for two-and three-point correlation functions and the bms 3 -block in terms of wave functions, where the wave functions must satisfy an infinite set of partial differential equations. We found an expression for the level-one wave function and its dual -we could not, however, obtain closed-form solutions for the level-two wave functions; nevertheless, we proved that they are uniquely determined in generic cases.
We showed the strengths of the oscillator construction by considering the semi-classical limit defined by c M → ∞ while keeping all fractions of the form ∆/c M , ∆ i /c M ; ξ/c M , ξ i /c M and c L /c M fixed. Our oscillator construction allowed us to prove the exponentiation of bms 3 -blocks in this limit. Finally, we explicitly determined the vacuum bms 3 -block in the perturbatively heavy and the heavy-light regimes; the heavy-light vacuum bms 3 -block is in agreement with previous results stemming from holographic computations involving probe particles propagating in flat-space cosmologies within Einstein gravity [50]. It would be interesting to check our predictions against holographic calculations in the semi-classical limit of chiral gravity theories with c L = 0.
The work presented in this paper allows for a number of possible directions which may be explored in the future.
In our proof of uniqueness of the level-two wave functions we could identify a discrete set of values for c L , c M , ∆ and ξ which must be excluded. Analysing the analogous proof for the Virasoro algebra [53], we notice that the set of excluded parameters in this case include the known values of Virasoro minimal models. Following this reasoning, we speculate that this method may be of use for identifying new bms 3 minimal models.
It would be interesting to study bms 3 -blocks beyond the vacuum module, as well as blocks in which all external operators are heavy. In addition, 1/c M corrections to the semi-classical limit are of interest since they give rise to quantum contributions to the bulk gravitational theory. Another avenue worth exploring is the computation of torus bms 3 -blocks in the semi-classical limit; we suspect that the oscillator construction would be beneficial for this task. The study of one-point functions on the torus was recently initiated in [48,49]. We expect that modularity gives similar stringent constraints on the thermodynamics of the dual gravitational theory as in the case of asymptotically-AdS spacetimes.
Furthermore, it would be valuable to have access to an oscillator construction for induced representations of bms 3 , since these representations are unitary [29,56]. As a first step in this direction one could approach an oscillator construction of induced isl 2 modules.
In view of the recent results of [57][58][59], it would also be interesting to obtain an oscillator construction of bms 4 algebras and investigate its relation to conformal field theories on the celestial sphere.
In conclusion, all of the above tasks are of importance in view of a putative holographic duality involving asymptotically flat spacetimes and BMS-invariant field theories. We hope to return to some of these aspects in future work.

A Realising the Oscillatory Construction of bms 3
In section 3 of the main text we state that the oscillator construction of the highest-weight representation of bms 3 may be reached by taking a non-relativistic limit of a two-dimensional linear-dilaton like theory; in this section we detail the procedure. 19 Generalising the linear dilaton theory, the components of the stress-tensor T ≡ T zz and T ≡ Tzz, where the bar denotes quantities belonging to the anti-holomorphic sector of the two-dimensional conformal algebra, may be expressed as We will take the non-relativistic limit at the level of the generators L vir m andL vir m .

A.1 Non-relativistic Limit
In order to implement the non-relativistic limit we introduce a pair of new oscillators Furthermore, we introduce the new constants The eigenvalues of the oscillators a m andâ m are in general complex: the real parts of the eigenvalues are arbitrary and parametrised by λ 1 and λ 2 , respectively; the imaginary parts are fixed by the above expressions. From the transformations (A.1.9) we read off the eigenvalues where the factors of √ 2 are due to a choice of normalisation, and we defined

A.2 Differential Operators and Measure
We set up the oscillator construction of the bms 3 generators (A.1.5) by assigning differential operators to the modes a m andâ m ; we do so in the following way: The adjoint transformation properties a † m = a −m andâ † m =â −m are implied by the Hermitian product with the measure where the normalisation is such that (1, 1) = 1.
The exponential factor of the measure is made plausible by introducing the linear combinations q m = v m . Then a −m only contains q m and a −m only contains ∂ qm ; whileâ −m andâ m contain j m and its derivative, respectively. The measure of the oscillator construction of the Virasoro algebra, given by (2.2.8), corresponds to an assignment akin to (A.2.1a) [52]; hence, replacing u n with q n , the transformation a † m = a −m requires a measure with the exponential factor Taking the overall sign difference between a −m andâ −m into consideration, similar reasoning motivates the exponential factor that allows forâ † m =â −m to be of the form 2) for a general function f (v,v) is made possible by the analytic continuation associated with the mapping n → in. Such an analytic continuation allows us to exploit the behaviour of the complex delta distribution and its derivatives, defined by for any pair of non-negative integers a, b ∈ N 0 . Monomials of complex variables integrate to derivatives of the complex delta distribution according to for any κ ∈ R and v ∈ C; it is then straightforward to determine the normalisation of the Hermitian product, as well as the orthogonality relation for monomials of the oscillator variables

A.3 Generators
To find the oscillator construction of bms 3 we take care of the normal ordering in the bms 3 generators (A.1.8); plug in the differential representations of the oscillators, as given in equations (A.2.1a) and (A.2.1b); use the definitions (A.1.10), and then simplify. Denoting the generators in the oscillator construction by lowercase, we arrive at as well as In (A.3.1a) and (A.3.2a) we have identified the scaling dimension ∆ and rapidity ξ as Furthermore, the above generators satisfy the bms 3 algebra with central charges c L = 2 + 24 µ 2 1 − µ 2 2 and c M = 12 (µ 1 − µ 2 ) 2 . Finally, we will make use of the abbreviations Using the definitions (A.3.3) together with the values of the central charges, the coefficients defined above may be expressed in terms of familiar quantities as where the sign choices for the k-dependent terms are correlated with each other, and the sign choices for the k-independent terms are correlated with each other.

B Proof of Unique Solution
In this section we prove that the level-two bms 3 wave functions admit unique solutions; we show it for finite central charge c M and comment on the semi-classical limit c M → ∞. The discussion for the semi-classical limit may be considered to be part of the proof of exponentiation presented in subsection 4.2. Parts of the analysis below follow methods presented in [53].
B.1 Proof of Unique Solution for F (η, ν) We focus on ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0; v), similar arguments apply to the dual wave function χ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0;v). To get a valid wave function for the point configuration under consideration, F (η, ν) in the ansatz (3.3.22) must satisfy the differential equations (3.3.15) for n ≥ 1. 21 In order to express the differential equations in terms of F (η, ν) it is beneficial to write the bms 3 generators l 0 , m 0 and l k , m k , with k ≥ 1, using the variables η m and ν m ; setting t = 1 and x = 0 for simplicity, we get Moreover, to implement the action of L (∆ 1 ,ξ 1 ) n and M (∆ 1 ,ξ 1 ) n we make use of the identities which result from applying the chain rule to F (η, ν). Taking all of the above quantities together, the differential equations (3.3.15) with n ≥ 1 give rise to the equations We now assume that F (η, ν) takes the form of a power series expansion in the variables η m and ν m , where each F n is a linear combination of every possible linearly independent monomial of the form η j 1 η j 2 · · · η jp ν k 1 ν k 2 · · · ν kq , with the ordering of variables such that j 1 ≤ j 2 ≤ · · · ≤ j p and k 1 ≤ k 2 ≤ · · · ≤ k q , and the sum of the indices satisfies p i=1 j i + q i=1 k i = n. Hence, the number of terms in F n is given by where p(k) is the number of partitions of the number k. We choose the normalisation F 0 = 1. We may view F n as an element of a vector space V n which is spanned by the monomials η j 1 η j 2 · · · η jp ν k 1 ν k 2 · · · ν kq . The dimension of the vector space V n is p 2 (n), i.e. equal to the number of basis monomials. Due to the action of the derivative in the differential operators l k and m k we note that l k (V n ) ⊆ V n−k as well as m k (V n ) ⊆ V n−k for k ≤ n.

B.1.1 Building a Linear Set of Equations
To specify F n , and in turn F (η, ν), we have to know the coefficients in front of the p 2 (n) basis vectors of V n . Our strategy is to show that there are as many coefficients as equations for these coefficients, and that the solution to this set of linear equations is unique.
In terms of F n the equations (B.1.3) read, for k ≤ n, where we have used that l 0 (V m ) ⊆ V m and m 0 (V m ) ⊆ V m for m ∈ N 0 , and the right-hand side contains F n−k because the two sides must belong to the same vector space V n−k . The form of l 0 gives rise to the eigenvalue equation l 0 v = (∆ + m) v for any element v ∈ V m ; this observation allows us to simplify (B.1.6a) further, arriving at with β k,n ≡ ∆ + n − k + k∆ 1 − ∆ 2 . Unfortunately, we cannot constrain m 0 F n−k further since m 0 does not give rise to a simple action on the vector space V m . The set of equations (B.1.7) displays some recursive traits which we will use to our advantage; remembering that F 0 = 1, we may reduce the index of F n until the right-hand side only contains numbers. To this end, we evaluate a string of operators l kq · · · l k 2 l k 1 acting on F n by applying equation (B.1.6a) recursively; we obtain l kq · · · l k 1 F n = β kq,n−K+kq · · · β k 2 ,n−k 1 β k 1 ,n F n−K , (B.1.8) provided that K ≡ q i=1 k i is less than n. Since m 0 does not have a simple eigenvalue equation, we cannot evaluate a string of operators m jp · · · m j 2 m j 1 acting on F n−K unless J ≡ p i=1 j i is equal to n − K; in this case we find where we used that the generators m j i mutually commute so that each product of m j i acts directly on F κ , for the suitable level κ. We may apply m 0 F 0 = ξ recursively to equation (B.1.9), which yields m jp · · · m j 2 m j 1 F n−K = γ jp · · · γ j 2 γ j 1 , (B.1.10) where γ j = ξ + jξ 1 − ξ 1 , for the case J = n − K. The two equations (B.1.8) and (B.1.10) bring us to the general form of repeated action of generators to be 22 m jp · · · m j 2 m j 1 l kq · · · l k 1 F n = γ jp · · · γ j 2 γ j 1 β kq,n−K+kq · · · β k 2 ,n−k 1 β k 1 ,n (B. 1.11) in the case of p i=1 j i + q i=1 k i = n.

B.1.2 Existence and Uniqueness of the Solution
The left-hand side of equation (B.1.11) is a linear combination of the p 2 (n) unknown coefficients of the basis monomials. There are p 2 (n) possible strings of ordered operators m jp · · · m j 2 m j 1 l kq · · · l k 1 with J + K = n. Hence, by considering all actions of the form m jp · · · m j 2 m j 1 l kq · · · l k 1 F n we form a linear system of p 2 (n) equations with p 2 (n) unknowns.
To prove that this system is solvable it is sufficient to show that the matrix M n , whose entries are given by m j 1 · · · m jp l k 1 · · · l kq (η j ′ 1 · · · η j ′ r ν k ′ 1 · · · ν k ′ s ) and which characterises the system of equations, is invertible except for at most a discrete set of exceptional cases; we will do this by arguing that the determinant of M n is generically non-vanishing.
We will continue in a two-step procedure. In the first step we simplify the task by replacing the operators l k by l • k = A k ∂ η k + B k ∂ ν k and m k by m • k = A k ∂ ν k . It is always possible to find a basis such that the simplified matrix M • n has an upper triangular form, with products of A k on its diagonal; we postpone the proof of this statement to the next subsection. The determinant of M • n is the product of its diagonal elements, i.e. products of A k . Using that A k is given by with k ∈ N; this means that there is only a discrete set of values of ξ for which the determinant of M • n vanishes.
As the second step we need to consider the matrix M n corresponding to the equations using the full operators l k and m k . A product of the operators l k , m k can be re-expressed in terms of sums of l • k , m • k and l k − l • k , m k − m • k ; where the differences between the full and simplified operators are independent of ξ and c M . Hence, the determinant of M n and the determinant of M • n have features in common: both are polynomials of the same degree in y ≡ 2ξ − c M /12, and they have the same coefficient of the largest power of y. Consequently, the determinant of M n may vanish only for a discrete set of values y, generically without relation to (B.1.12). We have thus shown that the system of equations for the coefficients of F n admits a unique solution; this means that a solution for F (η, ν) is unique, too.

B.1.3 Proof of Unique Solution for S(σ, κ)
In the semi-classical limit we make the ansatz F (σ, κ) = exp µ 2 S(σ, κ) . The proof of a unique solution for S(σ, κ) goes along the same lines as above. However, there are two differences which we would like to highlight. First, we treat the equations in terms of S(σ, κ), not in terms of F (σ, κ). Hence, the expansion of S(σ, κ) in terms of its variables is reorganised compared to the expansion of F (σ, κ). Second, due to the semi-classical limit we may drop the second derivative terms in l k and m k . Neither of the changes mentioned in this subsection invalidate the arguments above concerning the uniqueness of the wave function. partition just containing n + 1 times the number one, may be obtained from a partition of the number n by raising one of the numbers by one without violating the (decreasing) ordering of the numbers in the partition. For example, the partition (4, 2) of the number 6 can be obtained from the partition (4, 1) of the number 5. By the induction assumption, the matrix M • n is upper triangular and hence also the block corresponding to the partition n. Thus, keeping the ordering of the operators and monomials (but raising one of the indices to match the partition of n + 1) the block of the matrix M • n+1 corresponding to that partition of n + 1 is also upper triangular. By this construction, the diagonal of M • n+1 contains products of A k only.
We are left with the trivial partition 1 + · · · + 1 of n + 1. In this case we have to find a suitable ordering of the operators (m • 1 ) p (l • 1 ) q with p + q = n + 1 and the monomials (η 1 ) r (ν 1 ) s with r + s = n + 1. Since m • 1 is a derivative with respect to ν 1 we have Hence ordering the operators (m • 1 ) p (l • 1 ) n+1−p with increasing p = 0, . . . , n + 1 and the monomials (η 1 ) n+1−r (ν 1 ) r with increasing r = 0, . . . , n + 1 we obtain an upper triangular matrix with powers of A 1 on its diagonal. This completes the proof by induction. Hence we conclude that the matrix M • n has an upper-triangular form for a specific ordering of the operators and monomials, with products of A k on its diagonal.

C Determining the Level-one Wave Function
In this appendix we solve the differential equations (3.3.6) for the level-one wave function. The result for the dual wave function follows from applying the same procedure detailed below to the differential equations (3.3.7).
We first turn our attention to equation ( where we have abbreviated Ψ ≡ ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) for simplicity. The above differential where h ( · · · ) is again an unknown function arising as a constant of integration. With the above expression the wave function takes the form Imposing the boundary condition C ′ (0) = 0 fixes c 2 = −c L √ 3c M . The solution for C(τ ) is thus (D.2.5) In the same way as for D(τ ), evaluating the heavy-light vacuum bms 3 -block requires us to determine the integral t 0 dτ C(τ ) τ . This integral can be solved using partial integration and the partial fraction expansion which simplifies the integrand. The integral is then given by