A BMS-invariant free scalar model

The BMS (Bondi-van der Burg-Metzner-Sachs) symmetry arises as the asymptotic symmetry of flat spacetime at null infinity. In particular, the BMS algebra for three dimensional flat spacetime (BMS$_3$) is generated by the super-rotation generators which form a Virasoro sub-algebra with central charge $c_L$, together with mutually-commuting super-translation generators. The super-rotation and super-translation generators have non-trivial commutation relations with another central charge $c_M$. In this paper, we study a free scalar theory in two dimensions exhibiting BMS$_3$ symmetry, which can also be understood as the ultra-relativistic limit of a free scalar CFT$_2$. Upon canonical quantization on the highest weight vacuum, the central charges are found to be $c_L=2$ and $c_M=0$. Because of the vanishing central charge $c_M=0$, the theory features novel properties: there exist primary states which form a multiplet, and the Hilbert space can be organized by an enlarged version of BMS modules dubbed the staggered modules. We further calculate correlation functions and the torus partition function, the later of which is also shown explicitly to be modular invariant.


Introduction
As the first example of unified space and time, Minkowski spacetime is in some sense the starting point of both quantum field theory and general relativity: it is usually the spacetime background for Lorentzian invariant quantum field theories, and it also provides the simplest solution to Einstein equation. Yet it is still a mystery what the full-fledged quantum theory of gravity on asymptotically Minkowski spacetime should be.
In quest of quantum gravity, one fruitful effort over the past two decades has been holographic duality, whose most well-known incarnation is the AdS/CFT correspondence [2,3,4].
The AdS/CFT correspondence states that gravity in asymptotically Anti-de Sitter spacetime is equivalent to quantum field theory with conformal invariance, and it has become a thriving research field involving many interdisciplinary studies including string theory, black hole physics, condensed matter physics, quantum chromodynamics, and quantum information theory.
It is a tantalizing idea to extend the success of the AdS/CFT correspondence to asymptotically flat spacetime. To this end, asymptotic symmetries play an important role. Recall that in AdS d+1 /CFT d , one basic item in the holographic dictionary is that the asymptotic symmetry of the bulk theory agrees with the global symmetry of the dual field theory, and both are the conformal symmetry in d dimensions. For instance, under the Brown-Henneaux boundary conditions [5], the asymptotic symmetry for Einstein gravity with negative cosmological constant − 1 2 agrees with that of a CFT 2 with central charges c L = c R = 3 2G , where G is Newton's constant in three dimensions. The symmetry argument is especially powerful in the case of AdS 3 /CFT 2 , using which one can provide a microscopic explanation of the Bekenstein-Hawking entropy of the black holes using Cardy's formula [6,7]. The asymptotic symmetry for four-dimensional Minkowski spacetime in Einstein gravity, first studied by Bondi-van der Burg-Metzner-Sachs (BMS) [8,9], is the so-called BMS symmetry. The original BMS symmetry only contains generators that are smooth on S 2 . In a recent resurgence [11,12,10,13], the extended version of BMS group also admits generators that are singular at the south or north poles. The extended BMS symmetry in four dimensions is related to Weinberg's soft theorem and the memory effect [14], and has recently prompted the study of celestial CFTs, see [15,16,17].
A simpler version of the BMS group appears in three dimensions [18,19,20]. Under certain boundary conditions at null infinity, the asymptotic symmetry for flat spacetime in isomorphic to the Galilean conformal algebra (GCA) in two dimensions [22,21,23]. While the GCA can be obtained from the non-relativistic (NR) limit of the two-dimensional conformal algebra, the BMS 3 algebra is the ultra-relativistic (UR) limit, and thus is also an example of a Carrollian algebra [24,25]. Like their CFT 2 cousins, field theories invariant under BMS or Galilean conformal symmetries are highly constrained. In particular, symmetry and other consistency conditions make it possible to initiate a bootstrap program [29,26,27,30,31,28].
It is reasonable to conjecture that the holographic dual of Einstein gravity in asymptotically flat three-dimensional spacetimes is a quantum field theory invariant under the BMS 3 symmetry (BMSFT). As evidence, the torus partition function for BMSFTs has been argued to be modular invariant and a Cardy-like formula can be used to explain the entropy of cosmological solutions with Cauchy horizons [33,32]. Furthermore, entanglement entropy and its holographic dual has been calculated in [34,35,36,37,38]. Other interesting properties in flat holography include geometric Witten diagrams [39], quantum energy conditions [40], etc.
Despite this progress, we know little about the putative dual field theory other than properties that can be implied by the symmetries. In particular, it is important to know if a field theory with BMS invariance really exists at the full quantum level. Thus, it is necessary to construct and study in detail an explicit model of BMSFT. Besides the motivation from flat holography, a model of BMSFT is also interesting from a purely field theoretic perspective, as it provides a playground for further understanding both non-relativistic and ultra-relativistic quantum systems. A Liouville-like theory with BMS symmetries has been constructed in [41,42], which can be obtained from the ultra-relativistic limit of ordinary Liouville theory, or alternatively from the geometric action for the BMS 3 group [43].
In this paper, we study a free scalar BMSFT model in two dimensions with the action The classical theory is invariant under the BMS symmetry (1.1) with zero central charges.
This model also appears in the tensionless limit of string theory [44], and a √ TT deformation of a free scalar CFT 2 [45]. where I is the identity operator, O is a primary multiplet with ∆ = 1, ξ = 0, and V α is the vertex operator with ∆ = 0, ξ = − α 2 2 . Correlation functions between these operators can be calculated explicitly.
• States are organized in terms of an enlarged version of BMS highest weight module dubbed the staggered BMS module.
• The torus partition function can be calculated and is found to be modular invariant, confirming earlier statements based on symmetry arguments [33,32,36,46].
The appearance of primary multiplets and staggered modules are both unexpected. In [22,29], it has been noticed that the commutation relations (1.1) between M 0 and L n imply that the action of M 0 is not diagonal within a BMS highest weight module, and descendants have to form multiplets. In particular, the current T which generates the superrotations, and the current M which generates the supertranslations form a multiplet with conformal weight ∆ = 2 and boost charge ξ = 0. While the algebra (1.1) implies that multiplets at the level of BMS descendants are inevitable, there is no a priori reason that multiplets for primary operators have to exist as well. Our model thus provides the first example of this novel representation.
The appearance of the staggered module is closely related to the subtlety with c M = 0, at which point it was argued that the BMS highest weight module is truncated to the Virasoro module in [22]. Instead of a truncation, however, here the BMS highest weight module is enlarged. The reason for this is that there is an extra quasi-primary K with conformal weight ∆ = 2 in our model that was not assumed to exist in the general argument [22]. The new quasiprimary K forms a BMS triplet together with the Virasoro stress tensor T and supertranslation stress tensor M . This structure is reminiscent of logarithmic CFTs [47,48,49,52,50,51], where the Virasoro stress tensor acquires a logarithmic partner whose presence makes the action of L 0 non-diagonal. In that case, multiplets also appear and the states are also organized into staggered modules [47,52]. It would be interesting to further understand the staggered module from a general analysis, and also to study the implications to holography. We will leave these questions for further study.
Apart from the intrinsic discussion as a two dimensional BMSFT, the model (1.2) with the central charges (1.3) can also be obtained from a free scalar CFT 2 by two steps. From a free scalar CFT 2 with c = 1,c = 1, we can construct the so-called flipped representation of the conformal algebra with central charges c = 1,c = −1. Then our BMSFT model can be obtained from the UR limit of the CFT 2 in the flipped representation. Note that this flippling + UR limit is different from the direct UR limit discussed in [20,42].
Finally, let us briefly comment on the potential gravity dual to this model. It is not directly applicable to Einstein gravity, as the latter has the central charges c L = 0, c M = 3 G . To find BMS 3 with non-vanishing c L , [53,54] considered topologically massive gravity (TMG) which is three dimensional Einstein gravity plus a gravitational Chern-Simons [55,56]. Let the coupling in the Chern-Simons term µ, the central charges are given by To get vanishing c M , we need to take the limit where only the Chern-Simons term persists, which is the so-called flat space chiral gravity theory as discussed in [1].
This paper is organized as follows. Section 2 is a general analysis of BMSFTs. In section 2.1, we first briefly review the general properties of BMSFTs, in section 2.2 we discuss the representations, and introduce a novel highest weight representation where the action of M 0 is a Jordan cell, and in section 2.3 we calculate the correlators. In section 3, we study the free BMS scalar model in detail. We introduce the classical theory and write down its symmetries in section 3.1, perform a canonical quantization with the highest weight vacuum in section 3.2, and calculate correlation functions in section 3.3. In section 4, we arrive at the key result of this paper, the staggered BMS module. We find the operators outside the ordinary BMS highest weight module, and organize them into the staggered module. We further illustrate the properties of this module by diagrams. In section 5, we review the UR limit from twodimensional CFTs to BMSFTs, and point out a subtlety on the consistent plane UR limit.
Then we take the UR limit of the free relativistic scalar model in the flipped vacuum with the central charges c = 1,c = −1 to get the free BMS scalar in the highest weight representation.
In section 6, we calculate the torus partition function of the free BMS scalar model in the highest-weight representations, and find that it is invariant under modular S-transformations.

General properties of BMSFTs
In this section we discuss generic features of BMSFTs. We first provide a short review of the BMS algebra in section 2.1. Section 2.2 is dedicated to the representation theory where we introduce a novel type of highest-weight representations, where the primary states lie in multiplets. In section 2.3 we calculate the correlation functions for general quasi-primary multiplets, paying special attention to multiplets with ξ = 0 which we will encounter later in the free scalar model.

Quick Review
A BMSFT (BMS-invariant field theory) is a two dimensional quantum field theory invariant under the following BMS transformation, Note that although the theory is not Lorentz invariant, there is still a notion of time and space.
For BMSFT, σ should be regarded as a spatial direction, whereas τ is a timelike direction.
This interpretation will become clearer if we view BMSFT as the ultra-relativistic (UR) limit of a CFT 2 , to be described momentarily.
Now let us consider a BMSFT on a cylinder parameterized by the coordinates (τ, σ) with the identification Then the infinitesimal BMS transformation is generated by the Fourier modes, Under the Lie bracket, the generators (2.4) form the BMS algebra The generators that implement the transformations (2.4) on the fields will be denoted as L m and M n , and they form the centrally extended BMS algebra, where c L and c M are central charges. As a side remark, it is found that the asymptotic symmetry of Einstein gravity in asymptotically-flat three-dimensional spacetime is BMS 3 with , while gravitational theories where both central charges do not vanish can be constructed by adding a Chern-Simons term [1,53,54].
The general form of the BMS transformation (2.1) allows the map x = e iσ , y = iτ e iσ (2.9) By analytic continuation, x can be viewed as a holomorphic coordinate on the plane. The above map (2.9) is usually regarded as the map from the cylinder to the plane [44], using which we can further discuss the state-operator correspondence. For later convenience, we also write down the BMS generators on the plane, Let T be the Noether current of the translational symmetry along x, and M be the Noether current of the translational symmetry along y. The BMS charges on the plane can then be written as where denotes the contour integration around the origin on the complexified x plane.
From the algebra (2.8) we expect the following OPEs between the currents, The transformation laws of the currents under the BMS transformatioñ are given by 1 In Eq.(2.18), { , } denotes the usual Schwarzian derivative, and the last term is the so-called BMS Schwarzian derivative [35,36],

Representations
In this subsection we discuss the representations of the BMS algebra (2.8). In the literature, two types of representations have been discussed, the highest weight representation and the induced representation [57]. The induced representation is unitary and can be obtained from a Ultra relativistic limit of the highest weight representation of CFT 2 s. The highest weight representation of BMS algebra is analogous to that of Virasoro, which enables one to adapt techniques of CFT 2 for BMSFT. Despite of the fact that the highest weight representation is non-unitary, several interesting results have been worked out in the highest weight representation including the general structure of correlation function, characters, torus partition function, entanglement entropy, bootstrap, and so on. We will focus on the highest weight representation in this section, and leave the discussion of induced representation to Appendix B. In the following, we first review the usual highest weight representation discussed previously in the literature [57], which we refer to as the singlet version. Then we will introduce a novel multiplet version of the highest weight representation. In section 3 we will see that the multiplet version representation arises naturally in the free scalar model (1.2).
In this subsection we discuss the highest weight representations of the BMS algebra (2.8), which are not unitrary. We first review the usual highest weight representation discussed previously in the literature [57], which we refer to as the singlet version. Then we will introduce a novel multiplet version of the highest weight representation. In section 3 we will see that the multiplet version representation arises naturally in the free scalar model (1.2).

Highest weight representations: the singlet
The singlet version of the highest weight representation of the BMS algebra [57] is a straightforward generalization of the highest weight representation of the Virasoro algebra.
This amounts to considering the BMS module on the plane which consists of primary operators and their descendants. A primary operators at the origin O = O(0, 0) can be labelled by the The descendant operators can be obtained by acting L −n , M −n with n > 0 successively on the primary operators. The primary operator together with its descendants form a highest weight module.

Highest weight representations: the multiplet
In any unitary theory, if two Hermitian operators commute, then we can go to a basis in which the commuting operators are simultaneously diagonalized. Therefore it is natural to consider highest weight representations in CFT 2 , and organize states into Virasoro modules.
For BMSFT, however, several subtleties arise in the highest weight representation of the form (2.21) and (2.22). As noticed in [22], the Kac determinant for the highest weight representation with ξ = 0 is negative, and hence the representation is not unitary. In a state space equipped with an indefinite inner product, a Hermitian operator is not necessarily diagonalizable. In addition, it has been observed in [57] that if we organize the highest weight module in terms of quasi-primaries and their global descendants, the quasi-primaries will generically form multiplets, on which the action of L 0 and M 0 cannot be simultaneously diagonalized within a BMS module even assuming the primary state is a common eigenstate of L 0 and M 0 . These observations then open up the possibility that the action of L 0 and M 0 is not diagonal even on the primary states. This feature is very similar to logarithmic CFTs [47,48,49,52,50,51].
In this case, the representation matrix can be written in the Jordan canonical form. Similar to the discussion in [49], we can choose a basis so that the action of L 0 is diagonal and the action of M 0 is block diagonal, with each block being a Jordan cell. The primary operators in a Jordan chain form a multiplet, which, together with their descendants, form a reducible but indecomposable module. If there are r operators related to each other in a Jordan chain, the multiplet they form will be referred to as having rank r , the same rank as the Jordan block.
The primary operators with diagonal action under M 0 will be referred to as singlets or rank-1 multiplets.
Thus operators of BMSFT can be organized into highest-weight primary multiplets and their descendants. A highest-weight primary multiplet O with rank r is defined by where O a denotes the a-th component of the multiplet O, and ξ is a Jordan cell with rank r and diagonal component ξ, (2.24) The off-diagonal element can also be chosen to be any arbitrary constant, which amounts to introducing a relative scaling among different components.
Similar to CFT 2 , it is also useful to introduce the notion of quasi-primary multiplets, which satisfy (2.23) but only with n = 1 rather than arbitrary positive integers. Quasiprimary multiplets are highest weight states under the action of the global subgroup of the BMS 3 group, which is isomorphic to the Poincaré group in three dimensions.

Correlation functions
Discussions on the correlation functions for quasi-primary operators of the Galilean conformal algebra can be found in [22,28]. The results in [22] directly apply to BMSFT, as the algebras of GCFT and BMSFT are isomorphic. Interestingly, it has been found that multiplets appear generically in GCFTs and BMSFTs [29], a feature also shared by logarithmic CFTs [47,48,51]. Detailed discussions of multiplets in GCFT/BMSFT can be found in [29] which focuses on quasi-primaries with non-vanishing boost charge, i.e. ξ = 0. As we will show later in Section 3, however, our free scalar model also contains a multiplet at the level of primary operators, and it has ξ = 0. In the following, we will first review the correlation functions for singlets and multiplets in the ξ = 0 sector, and then provide results for the ξ = 0 sector.

Singlets
The singlet version of BMS primary operators at the origin is defined by (2.21) and (2.22).
y 12 x 12 , (2.29) y 12 x 12 e −ξ 312 y 31 x 31 e −ξ 231 y 23 where d is the normalization factor of the two-point function, c 123 is the coefficient of threepoint function which encodes dynamical information of the BMSFTs, and

Multiplets
Similar to the discussion of the singlet, local operators corresponding to the highest weight multiplets (2.23) are defined by where O(x, y) denotes a multiplet with rank r, whose components are denoted by O a (x, y) with a = 0, · · · r − 1. The BCH formula now leads to the following transformation law, where we organize the expansion in ascending order of the total power of (x − x) and (ỹ − y), meanwhile put terms with higher power of (ỹ − y) behind.
As proved in [58,29], from an equality for singlets, we can always get the analogue multiplet version by applying the following replacement rule, where F (ξ, O) denotes any expression which explicitly depends on the operator O and its boost charge ξ, and the replacement should be performed on both sides of the equality. In particular, the finite transformation law for the multiplet can be obtained from (2.28) by applying this replacement rule, and the result is Quasi-primaries Quasi-primary operators transform covariantly under the global part of the BMS 3 symmetry, which is isomorphic to the Poincaré group in three dimensions. The infinitesimal transformations satisfy the rule (2.33), but now with n = −1, 0, 1, and the OPE with the stress tensor takes the form where · · · denotes terms more singular than (x −x) −3 . Terms of order (x −x) −3 do not appear due to the conditions coming from M 1 and L 1 . If ∆ = 2, the scaling of each term in the right hand side must be the same as (x − x) −4 from dimensional analysis. On the other hand, the weight of operators should be bounded from below. This means that the most singular term in the OPE must be of order (x − x) −4 . Further using the relation L −1 M = M −1 T , we conclude that quasi-primary multiplets with ∆ = 2 have to satisfy where the function D ab ( y 12 x 12 ) satisfies the following differential equations In the above equation we have omitted the argument of D ab , and prime denotes derivative with respect to the argument y 12 x 12 . Combing the above two equations, we learn that D ab depends on the label a and b only through the the sum a + b, and D ab vanishes whenever one of the indices is 0, where r i = r j = r. Denote q ≡ a + b + 1 − r, and then the most general solution of (2.41) satisfying the conditions (2.43) and (2.44) is then given by where d (r+q−k) are r undetermined integration constants, which can be further fixed by redefining the operators in the multiplet. To do so, we need to find the most general linear The solution can be written as which contains r arbitrary constants c k , k = 0, · · · , r−1. These (r−1) independent parameters c k , k = 1, · · · , r − 1 can be used to eliminate r − 1 degrees of freedom in (2.45), and leave an overall normalization related to the diagonal element c 0 . This simplifies the two-point functions to the following canonical form, where d r is the overall normalization.
In-states and out-states Using the state-operator correspondence 2 on the plane, we can define out-states as the Hermitian conjugate of the operator inserted at infinity, where we have used the transformation rule (2.36) to move the operator from the origin to the point x = ∞, y = 0. Note that the operators on the right hand side of (2.50) are to be understood as acting to the right. For all singlets including the vacuum, the definition of the out-state (2.50) is the same as in CFT 2 . For multiplets, however, the out-state O a | becomes a mixture of operators in the multiplet with indices no bigger than a.
The inner product between different components of a rank-r multiplet primary can then be calculated using (2.48) as, Unlike the case for singlets, for multiplets the two-point function (2.48) is different from the inner product (2.51). This is a characteristic feature of multiplets. From (2.51), we can see that the r-dimensional matrix O a |O b has two different eigenvalues ±1, within which the eigenvalue −1 has algebraic multiplicity r/2 . This means that if the theory contains highest weight multiplets, there must be primary states with negative norm in the theory. Note that this is to be distinguished from earlier discussions of unitarity for the highest weight singlets [21], where descendent states with negative norms have been found assuming primary states have positive norms. Thus, we have found another indication that BMSFT in the highest weight representation is not unitary. In section 6, we will introduce a dual basis which is linear combination of the out-state (2.50) so that their inner products with the in-states are diagonal. The dual basis is useful to define the trace.
Three-point functions The three-point functions involving multiplets can also be determined by the Ward identities. For three multiplets O i , i = 1, 2, 3, the general form of the three point function is given by, where i, j and k label the multiplets, while a, b and c label the components within a multiplet, Note that the three point function (2.52) factorizes into a term A which depends on the boost charges, a term B which depends on the conformal weights, and a structure constant term C ijk;abc . Both A and B are determined by kinematics, while C ijk;abc encodes interactions.
Note that O ia , O jb , O kc can belong to different multiplets of rank r 1 , r 2 , r 3 respectively. The coefficient c (n 1 n 2 n 3 ) ijk encodes the dynamical information of the theory. When r 1 = r 2 = r 3 = 1, the three-point function reduces to (2.30).

Multiplets with ξ = 0
The case with ξ = 0 turns out to be a bit subtle. On the one hand, the action of M 0 ∼ ∂ y acts trivially on singlets with ξ = 0. As a result, correlation functions for singlets with ξ = 0 do not depend on y and they reduce to correlators in chiral CFTs.
On the other hand, if multiplets exist, the action of M 0 is still non-trivial because of existence of off-diagonal elements, and we expect the correlators to have non-trivial dependence on y. To calculate the correlation functions at ξ = 0, we should take the ξ → 0 limit of the correlation functions of the ξ = 0 case. If there are derivatives with respect to ξ, such as in the three point function (2.52), one should take the derivatives first and then take ξ → 0.
As an example, the two-point correlation functions for rank-2 multiplet with ξ = 0 can be written as, In section 3, we will see that such a multiplet with ξ = 0 appears in our free scalar model.
To recapitulate, in this section we have introduced the multiplet version of the highest weight representation of the BMS algebra. The action of M 0 on the primary states is block diagonal, as described by (2.23). The OPE between a rank r multiplet and the stress tensor (2.34) also acquires a non-diagonal term, which mixes different components within the multiplet. Further more, two-point (2.48) and three-point (2.52) correlation functions for multiplets have also been worked out using the BMS symmetry. It is very interesting to further study four point functions, which allows a formulation of bootstrap program with BMS multiplets, similar to the discussion of BMS bootstrap for singlets [27]. The results for multiplets are more complicated, and related discussions will be reported elsewhere [31].

BMS Free Scalar Model
In this section we study a free scalar model which has BMS invariance. In section 3.1 we introduce the classical theory and discuss the classical BMS invariance. In section 3.2 we perform canonical quantization and in section 3.3 we discuss primary operators and correlation functions.

The classical theory
We start with the action on a cylinder parameterized by(σ, τ ) with σ ∼ σ + 2π, The action also appears as part of the worldsheet action in a tensionless limit of string theory [59,60,44,61]. In this paper, however, we will study the model (3.1) as a quantum field theory itself, without embedding it into a larger theory. The equation of motion reads The solutions to the equation of motion (3.2) satisfying the periodic boundary condition on the cylinder can be written in terms of the mode expansion as The reality condition then implies the adjoint relation The conjugate momentum to the field φ is given by 5) and the Poisson bracket is It is not difficult to check that this action is invariant under the BMS transformations (2.1). For infinitesimal transformations parametrized byε(σ) and ε(σ), the field φ transforms as a scalar, The corresponding Noether currents can be obtained from the standard Noether procedure, where the currents T and M are defined as The conservation laws are given by which can also be expressed in terms of the currents as The conservation laws allow us to define the conserved charges as In additional to the BMS symmetries, we also note that there is an affine U (1) symmetry, realized by τ -independent shifts of the field φ parametrized by Λ(σ), The associated Noether current and conserved charge are Interestingly, we note that the current J is proportional to the canonical momentum Π, and that its Sugawara stress tensor is proportional to the current M , As a consistency check, we find these charges indeed implement the transformation (3.9) and (3.17) via the Poisson bracket (3.6), Furthermore, the currents transform as To find the symmetry algebra, we need to first find the symmetry generators, which are mode expansion of the conserved charges (3.16) and (3.20), and can be obtained by expanding the symmetry parameters ε(σ),ε(σ) and Λ(σ) in terms of the Fourier modes, The resulting symmetry generators are These charges form an algebra under Poisson bracket, where The Noether currents T and M corresponding to translations along x and y are The conserved charges on the plane are The internal U (1) current is and the charges are given by

Canonical Quantization
In this subsection we perform canonical quantization to the scalar model (3.1). This amounts to replacing the Poisson bracket (3.6) with the canonical commutation relation which can be equivalently written in terms of the mode operators The commutation relations (3.42) are valid both on the cylinder and on the plane. Henceforth later discussions will be carried out on the plane, unless otherwise specified.
The quantum version of the classical Noether currents T (x, y), M (x) and J(x) that generate translations along x and y and the internal U (1) symmetry now become operators, where the definition of the normal order : · · · : depends on the choice of the vacuum, which will be specified momentarily. Here we would like to keep the normal ordering implicit. The currents can be expanded in Laurent series as which can be inverted to define infinitely many charges as with the Hermitian conjugates given by The charges are the quantum version of the classical charges (3.38) and (3.40) on the plane.

Vacuum in the highest weight representation and the state space
Note that so far we have not specified the vacuum for the model (3.1). We are interested in the vacuum this is invariant under the global symmetries generated by L 0,±1 and M 0,±1 .
That is, the vacuum has to satisfy To describe the vacuum in canonical quantization, we need to translate these conditions in terms of A n 's and B m 's. As [A n , B −n ] = 1, A n and B −n cannot annihilate the vacuum simultaneously. If we let B k 0 |0 = 0 for a given positive integer k 0 , then A −k 0 |0 = 0. Note that the expressions of L ±1 (3.47) contain a term −k 0 A −k 0 B k 0 ±1 . If we require L ±1 to annihilate the vacuum term by term, B k 0 ±1 has to annihilate the vacuum. We can keep using this argument until we arrive at B 0 . Similar arguments also apply to other cases, and we learn that Obviously, neither I and IV nor II and III can happen simultaneously. Therefore there are altogether two physically different choices 3 for the vacuum that is compatible with both the commutation relations and the symmetry condition (3.49). Let us start with entry I listed above, namely Then we have two feasible choices, case II or case III. Choosing I and II leads to the so-called induced vacuum, which we will describe in detail in appendix B, while choosing I and III leads to the highest weight vacuum, as discussed below.
Here in this section, we will focus on the choice with I and III, which combine as the following conditions, It leads to the following normal ordering prescription via creation and annihilation operators, Note that J n = B n , then (3.51) and (3.53) imply that this vacuum is also the highest weight vacuum of the Virasoro-Kac-Moody algebra (3.33). If the free scalar BMSFT is part of the worldsheet theory of tensionless strings, the vacuum (3.51) is the vacuum of a single string with zero momentum in the φ direction of the target space. The momentum can be turned on by considering an eigenstate of B 0 which satisfies
This can be viewed as a coherent state in the Fock space basis, and one can check that this state is also a highest weight state with Putting everything together, we now describe the state space of the free BMS scalar with the choice of the vacuum (3.51). Let i ≡ (i 1 , i 2 · · · ), i ≡ (j 1 , j 2 · · · ), then the state space is spanned by

The quantum BMS algebra
Now we consider the action of other charges on the vacuum, and calculate the resulting algebra. With some straightforward but tedious calculation, we find that the generators (3.47) indeed form the BMS algebra (2.8) with central charges Additionally, the U(1) charges J n together with L n form a U (1) Virasoro-Kac-Moody algebra (3.59) The above results can also be obtained by assuming symmetric ordering in L cyl 0 , then the Casimir energy can then be obtained from ζ-function regularization.

Primary operators and Correlation functions
In this subsection we calculate the Green's function, list the fundamental primary operators, and calculate their correlation functions.
With the mode expansion (3.34) of the fundamental field φ and the choice of the vacuum in the highest weight representation (3.51), we can define the Green's function of φ as, φ(x 1 , y 1 )φ(x 2 , y 2 ) = 0|X(φ(x 1 , y 1 )φ(x 2 , y 2 ))|0 − 0| : φ(x 1 , y 1 )φ(x 2 , y 2 ) : |0 (3.60) where X(· · · ) denotes radial order on the complexified x-plane, related to the time order on the Lorentz cylinder, as explained in the appendix A. Additionally, : · · · : denotes the normal order (3.52) which is compatible with the highest weight vacuum (3.51). From the commutation relation of the modes A n and B n , we learn that only the cross terms contribute, and that the Green's function is given by This provides the following OPE for the fundamental field The OPEs of other operators can then be obtained from (3.62) via Wick contractions. In particular, we note that the OPEs among the stress tensors read,

Primary operators
Now let us find the primary operators in the free BMS scalar field theory. According to the general discussion in section 2, BMS Primary operators in a generic multiplet have the defining property that the OPEs with the stress tensors T and M have to be of the form (2.34). This can be used to find all the primary operators in the free BMS scalar field theory.
We first consider where the pre-factor i makes the operators Hermitian. Their OPEs with the stress tensor read From their OPEs with the stress tensor, (3.68) we can read that the vertex operators are singlet primary operators with ∆ = 0 and ξ = −α 2 /2.
It is interesting to note that α can be either real or purely imaginary, since there are no constraints on the boost charges other than reality. This is different from the usual case of relativistic theory for free scalars which are 2d CFTs, where unitarity requires positive conformal weights hence purely imaginary α's.

Operator basis
Given the state space as described by ( Their descendants are then That is, the states with a single creation operator A −k correspond to the operator ∂ k x φ(x, y), and the states with single B −k correspond to the operator ∂ k−1 x ∂ y φ(x, y). This relation also works for the composite states, namely Putting all the above together, we learn that the state space are generated by acting operators of the form (3.74) on the zero mode states |α 's, the latter of which correspond to vertex operators. Therefore we conclude that in the BMS free scalar model, there exists a complete basis of local operators, which, when inserted at the origin, give all states in the state space spanned by (3.56). As a special case, the identity operator corresponds to I = V 0 . The only fundamental primary

Correlation functions
Correlation functions can be obtained from the OPEs. Let us first consider the rank−2 multiplet (3.64). The two-point functions are given by, Next, the vertex operator V α (3.66) satisfies the following OPE, which implies the correlation functions, .

(3.81)
Using the state operator correspondence, this implies that zero mode background satisfy the following orthonormal condition More generally, we have which means that the two-point functions between them always vanish. Using Wick's theorem, we find the non-vanishing three-point functions are

Truncation at c M = 0?
In this subsection we revisit the general analysis of BMSFTs with c M = 0 [22], which states that the BMS module has a truncation as a Virasoro module. We will show that this statement is true provided that there are no extra quasi-primary operators with ∆ = 2 other than T and M . In this case, the theory does not allow multiplets either. By considering the inner products of the higher descendant states, one can similarly arrive at the conclusion that M i −n · · · |0 , n > 0 are all null states. Then the vacuum is invariant under the action of all the M n 's for arbitrary integer n. This leads to further constrains on the two point functions, where we have used the fact that M is the top component of the rank two multiplet so that the out state M n | is 0|M n according to the definition (2.50). Using the Ward identity, the above condition leads to the following differential equations, From the above argument, one may draw the conclusion that the highest weight representation of the BMS algebra has a truncation to the one of the Virasoro algebra [22], with no appearance of multiplets. In the free scalar model, however, we have explicitly constructed a multiplet with ξ = 0. To understand the apparent discrepancy, let us recall that in the general analysis above, we have assumed that there are no other states in the vacuum module at level two other than L −2 |0 and M −2 |0 . On the other hand, in the free scalar model, there exists a new quasi-primary which is not orthogonal to M −2 |0 , and hence the aforementioned truncation does not happen. We will illustrate this point in more detail in the following subsection.

Enlarged BMS module in the free scalar model
In the general analysis above, we have assumed that the only weight 2 quasi-primary operators in the vacuum module are the stress tensor T and M . However, in the free BMS scalar model, we find that there is another weight 2 state −A −1 A −1 |0 , which corresponds to the operator The existence of (4.3) violates the assumption in the subsection above, so that the obstruction (4.2) of having highest weight multiplets disappears, and the truncation to the Virasoro module will not happen. In fact, the new operator (4.3) will enlarge the highest weight module in our free scalar model. To see this explicitly, we first calculate the OPEs using the φφ OPE (3.62), and the results are as follows, Comparing ( provides no further constraints on the correlation functions. In this case, the representation of the BMS algebra does not truncate to that of a Virasoro algebra as was argued in [22].
Instead, the representation of the BMS algebra is enlarged to the so-called staggered module which we will describe below.
For completeness, we also provide the OPE between K and other primary operators here, The representation In section 3 we discussed the operator basis where the primary operators were found to be I, O and V α 's, and other operators should be organized into one of the three families. We • Removing the side branches, we are left with the main branch, which is the usual highest weight module.
• The first side branch can be viewed as a highest weight module if we mod out the main branch; the second side branch can be viewed as a highest weight module if we mod out both the main branch and first side branch; similarly, the (n + 1)-th side branch can be viewed as a highest weight module if we mod out the first to the n-th side branches as well as the main branch.
• The seed of a side branch can be mapped to the seed of the main branch by lowering operators, whereas there are no raising operators to map the seed of the main branch to seeds of the side branches.
The full structure of the staggered module is schematically depicted in Fig. 1 Finally, applying lowering operators, which run upwards, will add further links between states. For example, there is a blue arrow pointing to northwest on the lower left part of    To recapitulate, we have learnt that BMS field theories with c M = 0 are special and subtle.
In a general BMS field theory, the stress tensors T, M are in a multiplet. Depending on the details of the theories, there are two possibilities for the representation theories as following.
• If T, M are the only quasi-primary operators with ∆ = 2, then the highest weight representation of the BMS algebra reduces to that of the Virasoro algebra. In particular, BMS multiplets should not appear, and all states should have vanishing boost charge ξ = 0.
• If there are other quasi-primary fields with ∆ = 2, so that the multiplet containing T, M is enlarged, then the highest weight representation of the BMS algebra is also enlarged.
BMS primary multiplets can appear, and the truncation does not happen.
For the BMS free scalar model with c M = 0, we indeed find an extra quasi-primary operator K, which provides a seed to enlarge the ordinary highest weight module. We have also explicitly found a rank-2 primary multiplet O. It would be interesting to study general BMS field theories with c M = 0. In particular, the associativity of the operator algebra can be used to constrain the stress tensor multiplet, which can help us to systematically classify this class of theories. We leave this to further work.
5 Ultra-relativistic limit from CFT 2 So far we have been discussing the free scalar model as an intrinsic BMSFT. Alternatively, it is also useful to make connections to relativistic CFT 2 s. Starting from the highest weight representation of a CFT 2 , we can construct the so-called flipped representation as we will describe momentarily. Then BMSFT in the highest representation can be obtained as the Ultra-relativistic limit of CFT 2 in the flipped representation. In this section, we will first revisit the flipping+UR limit of the CFT 2 on the cylinder discussed in [22,46,27,34], and point out a subtlety in the limit on the plane. Then we will discuss the free BMS scalar model as a flipping+UR limit of a free scalar CFT. Starting from a CFT 2 , one can also take the non-relativistic (NR) limit to obtain a Galilean conformal theory, which we will discuss in appendix C. For the BMS algebra and field theories as the UR limit of CFT 2 s, please also see [20,41].

UR limit on the cylinder
Consider a CFT 2 on the cylinder parameterized by σ, t with the periodicity condition The infinitesimal conformal transformations are generated by Conformal transformations are implemented in CFT 2 by the Virasoro generators L + n , L − n which form two copies of the Virasoro algebra The UR limit on the cylinder is defined as so that the speed of light goes to zero, which is the reason why this limit is called the ultrarelativistic limit. Under this limit the conformal transformations become BMS transformations generated by Correspondingly, the Virasoro algebra (5.3) becomes the BMS algebra (2.8) via a Wigner-Inönü contraction [63],

UR limit on the plane
It is also useful to spell out the UR limit on the plane, which we use extensively in the free scalar model. This limit turns out to be more subtle, and our discussion below is different from previous discussions in the literature [21,22,26]. Before the limit, the map from cylinder parameterized by (σ, t) to plane parameterized by (z,z) is given by, and the CFT 2 generators become, Our goal is to find a limit of the coordinates which will give well-defined BMS generators and thus a BMS algebra under the contraction (5.6). One naive guess is to take a limit similar to the one on the cylinder (5.2) z = y + x,z = − y + x, → 0 (5.10) As discussed in [22], an analogy of (5.10) does work in the non-relativistic case and it leads to a well-defined GCA, which we review in the appendix C. However, under the naive limit (5.10), the generators become and hence do not have a well defined UR limit. To get finite generators, we find that the proper UR limit on the plane should be chosen as Note that in terms of the (z,z) coordinates, the CFT 2 generators on the plane can be rewritten This allows us to define BMS generators similar to (5.5) As a consistency check, one can easily verify that the UR limit on the plane (5.12) can also be obtained from the UR limit on the cylinder (5.4) via the BMS plane-to-cylinder map (2.9).

Representations
Starting from a CFT 2 with highest weight representations, the UR limit leads to the induced vacuum. On the other hand, two dimensional conformal algebra allows other representations as well. In [27,46], two types of vacua before the UR limit have been discussed, the highest weight vacuum and the so-called flipped vacuum, which respectively become the induced vacuum and the highest weight vacuum under the UR limit. We will consider the "flipped vacuum → highest weight vacuum" here, and postpone the discussion on the "highest weight vacuum → induced vacuum" to appendix B.
The flipped representation in CFT 2 can be understood as the highest weight representation in the flipped coordinates (5.12) before the UR limit, or equivalently, as an automorphism of the right-moving Virasoro algebra, Under the UR limit (5.6), the flipped representation becomes the highest weight representation of the BMS algebra, Note that the above looks very similar to the NR limit (C.11).

UR Limit of the relativistic free scalar model
In this subsection we show that our free scalar BMSFT (3.1) can be obtained from the UR limit of a free scalar CFT 2 . Consider the free scalar model on the cylinder, Under the UR limit (5.4) together with the corresponding rescaling of the field, the action (5.21) becomes the BMS scalar action (3.1) on the cylinder (σ, τ ) ∼ (σ + 2π, τ ), which we reproduce here, The equation of motion of the relativistic scalar (5.21) can be solved in terms of the mode 1 n (a n e −in(σ+t) −ā −n e −in(σ−t) ). (5.24) with the canonical commutation relations [a n , a m ] = [ā n ,ā m ] = nδ n+m,0 , [a n ,ā m ] = 0, [φ 0 , π 0 ] = i (5.25) Comparing with the mode expansion of the BMS free scalar on the cylinder (3.3) we obtain the relation between modes before and after the UR limit (a n +ā −n ), n = 0, (5.26) As a consistency check, one can verify that under such relation the commutation relations before the UR limit (5.25) indeed become (3.42) after the limit (5.22). Besides, the central charges in the flipped representation of the relativistic free scalar are c = 1,c = −1, due to (5.15). After the UR limit, the central charges become c L = 2, c M = 0, due to (5.7).
This model also enables us to show explicitly why the UR limit on the plane should be (5.12) instead of the naive limit (5.10). If the latter is taken, the plane mode expansion will contain a zero mode part −i log(x)π 0 , which does not exist in the BMS mode expansion (3.34).
This issue, on the other hand, does not appear if we take the proper UR limit (5.12), where the zero mode term in question now becomes −i y x π 0 which further becomes the y x B 0 term in the BMS mode expansion under (5.26). Therefore, the UR limit (5.12) is compatible not only with the plane to cylinder map (2.9), but also with the UR limit of the mode expansion.

Flipped → highest weight representation
As we discussed in general in subsection 5.1.3, if we take the UR limit of a CFT from the flipped representation, we will obtain a highest weight representation in the BMSFT. Now we want to apply this to our relativistic free scalar model. We will provide evidence that this resulting representation of the BMS free scalar theory is just the representation that we discussed in section 3, by showing that the vacuum obtained from this limit agrees with the one (3.62) obtained from intrinsic BMSFT quantization method.
As mentioned earlier, the flipped representation is derived from the usual highest weight representation by the automorphism (5.15) of the right-moving Virasoro. In the free scalar model, this automorphism can be realized at the level of creation and annihilation operators, a n →ā −n , (5.28) Under this automorphism, the "ground" state |0 in the flipped representation is specified by a n |0 =ā −n |0 = 0, n ≥ 0. We recognize this as the BMS highest weight vacuum defined in (3.51). Since we have the same vacuum, all the calculations based on the vacuum should be the same.
In particular, the Green's function on the plane in the flipped representation reduces to that of BMSFT (3.61). To do so, we first compute the Green's function in CFT 2 by summing over all the modes (5.24) on the flipped vacuum (5.29). Using the commutation relations (5.25), this amounts to summing over the 0 F |a n a −n |0 F and 0 F |ā −nān |0 F terms with n > 0, together with a zero-mode term. Further using the map from cylinder to plane (5.8), the Green's function on the flipped vacuum can be written as 32) or more conveniently, in thez =z −1 coordinate, as Under the plane UR limit (5.12), the above Green's function indeed becomes that of the BMSFT in the highest weight vacuum (3.61).
We have just shown that the highest weight representation of our free BMS scalar model comes from the UR limit of the flipped representation in the 2d free relativistic massless scalar model. In the literature, it has also been stated that the UR limit of the highest weight representation in CFT 2 leads to the induced representation in BMSFTs. We will discuss this other limit in appendix B.

Torus partition function
In this section we explicitly calculate the torus partition function for the free scalar BMSFT, and show that it is modular invariant as expected from the general analysis of [46,33,32,36].

Modular invariance of BMSFT
In this subsection we review the derivation of modular invariance for BMSFTs, which helps us to set up the conventions. Those who are already familiar with this story can skip this subsection. The torus partition function for BMSFTs has been shown to be modular invariant, by taking a limit of CFT 2 in [33,46], or intrinsically as in [32,36]. We review the argument of modular invariance for BMSFT, following the intrinsic argument as in Appendix A in [36]. We consider a torus which is determined by two identifications on a two dimensional plane 5 , (canonical) spatial circle : (τ, σ) ∼ (τ, σ + 2π) (6.1) with a ∈ R, b ∈ R. Since the change of orientation of the complexifed σ can be realized by the symmetry (a, b) → (−a, −b), we will only consider a > 0 without loss of generality. 5 It is useful to embed R 2 into C 2 in the subsequent discussions.
The partition function on the above torus is formally a path integral over all fields satisfying boundary conditions specified by the two identifications. Alternatively, the torus partition function can be written as a trace over the state space which is determined by the spatial circle, weighted by the evolution along the thermal circle, where the translational generators are defined on the cylinder with the spatial circle (6.1), which we refer to as the canonical circle. We have also taken the Casimir effect (3.59) into account.
More generally, a torus can be described by the fundamental region on the plane where the trace is taken over the state space, and the translational operators M β S 0 , L β S 0 are both defined on the spatial circle specified by β S , as the subscript and superscript suggest. Note that the modular group is the isometry group acting on the modular parameters, and hence is independent of the theory. In general, the relation (6.8) is a relation between theories with state spaces defined on different spatial circles. For two dimensional quantum field theories with enough symmetries, such as such as CF T 2 , BM SF T , and W CF T , a symmetry transformation can be found to transform the torus such that the spatial circle is transformed back to the original one specified by β S again. For BMSFT on the canonical spatial circle (6.1), such a transformation is a BMS symmetry of the form (2.1), given by, under which the identifications β S = (−2πib , −2πia ), β T = (0, 2π) after the swapping of the cycles becomes, Finally, using the transformation rules (2.18) for the finite BMS transformation (6.9), one can relate the partition functions before and after the S transformation, and find that it is modular . (6.11) Note that the modular group will keep a > 0.

Torus partition function for the free scalar model
In this subsection we explicitly calculate the torus partition function for the free scalar model. The result depends on the choice of the vacuum. We will perform the calculation in the highest weight vacuum and postpone that of the induced vacuum to appendix B.

Intrinsic calculation in the highest weight vacuum
To calculate the torus partition function (6.3), we need to specify both the state space and the explicit definition of the trace. For the highest weight vacuum, the state space is spanned by (3.56) Using the conjugation relations (3.4), the basis for the out states can be written as where the overall sign (−1) j is determined by 14) It is not difficult to verify that inner products between states with different zero mode charges are orthogonal with each other, while the inner products between different states with the same zero mode charge form a non-diagonal matrix N α , namely i , j ; α | i, j; α = δ α,α N α; i j, i j , N α; i j, i j = δ i , j δ j , i (6.15) The fact that the inner product matrix between the out-states (6.13) and the in-states (6.12) is not diagonal requires a more careful definition of the trace. In order to do so, it is useful to introduce a dual basis as where N −1 α denotes the the matrix inverse of N α whose matrix elements are defined in (6.15). One can easily verify that the dual basis is indeed orthonormal to the basis (6.12), such that ∨ i, j; α| i j ; α = δ i, j; i , j δ α,α . (6.17) Then the trace in the partition function (6.3) can be defined by Note that the action of M 0 on | i, j, α is to take the eigenvalue of the boost charge, combined with changing one of the A −k to B −k , so that i k |i 1 , · · · , i k − 1, · · · , j 1 , · · · , j k + 1, · · · ; α .
In the expression of the zero mode part, we have used the fact that |α has zero conformal weight and boost charge ξ = − α 2 2 . Note that the boost charge can be any real numbers, so that we have to integrate along both the real axis and the imaginary axis. The integral can be calculated by analytic continuation from Re(b) = 0. We get the zero mode contribution, Next, we calculate the contribution from the non-zero modes. Note that either the A −k or the B −k operator raises the weight of L 0 by k, which enables us to split the total eigenvalue into a sum of contributions from the A and B modes separately, so that we havẽ where η(ia) is the Dedekind-η function, Using the transformation property of the eta function we indeed obtain the relation (6.11), and hence confirm with the general argument [46,33,32,36] that the BMSFT torus partition function is modular invariant.

From the UR limit
In this subsection, we calculate the torus partition function from the UR limit of a free scalar CFT 2 in the flipped representation. To do so, we first need to work out the CFT 2 partition function in the flipped vacuum. Recall that the torus partition function in the highest weight vacuum of the free scalar CFT 2 reads, where c =c = 1, q = e 2πiτ ,q = e 2πiτ (6.29) One can use (5.28) to map the highest weight vacuum to the flipped vacuum, or equivalentlȳ τ → −τ . (6.30) Using the above map, we obtain the torus partition function in the flipped vacuum 6 , Under the ultra-relativistic limit (5.4), the modular parameters become τ =ã + ib , (6.32) To get the result in the Lorentzian theory, we should further take the analytic continuatioñ a → ia. Taking the limit → 0 with this taken into account, we find that the CFT 2 in the flipped representation torus partition function (6.31) indeed becomes that of the free scalar BMSFT in the highest weight vacuum (6.25), namely where the overall factor comes from the rescaling of the field (5.22).

A Radial quantization in BMSFT
In this appendix we provide a prescription of radial quantization in BMSFTs. We first review the procedure in relativistic quantum field theory as reviewed in [64], and then extend it to BMSFTs.

A.1 Analytic continuation from Euclidean theory to Lorentzian theory
Let us start with the Euclidean correlation function of a relativistic quantum field theory.
which is automatically time ordered as e −Hδt E is unbounded for δt E < 0. To get correlation functions in the Lorentzian theory, one needs to first analytically continue the time direction to the complex plane, with the Euclidean time as the imaginary part, so that the Lorentzian theory corresponds to the real-time theory with t E = 0, and the Euclidean theory corresponds to the imaginary-time theory with t = 0. In the correlators (A.1), one can turn on a real part t with the imaginary part t E fixed, so that the ordering is still controlled by t E . As a final step, we need to take the limit t E = 0 along a chosen trajectory where F(t) is some function of t. Then the ordering in t E is transferred to t. For example, to get the usual time-ordered correlators in Lorentzian QFT, we can choose so that the ordering in the imaginary time t E is the same as that of the real time.

A.2 Radial quantization in CFT 2
Now let us consider a CFT 2 on the Euclidean cylinder (σ, t E ), with the identification After the analytic continuation t E → t E + it, the complex coordinates become and the cylinder to plane map becomes After the analytical continuation, z andz are no longer complex conjugate to each other, and effectively we have extended the theory from C to C 2 . The Euclidean theory can be obtained by imposing z * =z, while the Lorentz theory corresponding to taking |z| = |z| = 1.
In the Euclidean theories with t = 0, the infinitely past t E → −∞ on the cylinder becomes the origin on the plane. Therefore radial quantization at fixed radius on the Euclidean plane corresponds to canonical quantization at fixed time t E on the cylinder, As a result, radial ordering on the Euclidean plane corresponds to Euclidean time ordering on the cylinder, which after a prescription in the form of (A.3) provides an ordering in the real time t.

A.3 Radial quantization in BMSFT
Now we provide a prescription of radial quantization and time ordering in BMSFT. From a CFT 2 on the cylinder with complexified time, we take the UR limit t → τ , with t E fixed, so that Under the UR limit → 0, the cylinder to plane map can be defined as This is equivalent to the cylinder to plane map (5.12) on the unit circle where the latter is defined.
This way we extend the BMSFT to a theory defined on C × R where y x ∈ iR and x takes arbitrary value on the complex plane C, with e t E as the radius. The pure Euclidean theory corresponds to taking τ = 0 on the complexified cylinder (A.9) and (A.10), or equivalently y = 0 on the complex plane (A.11). The Lorentzian theory corresponds to taking t E = 0 in (A.9) and (A.10), or equivalently on the |x| = 1 circle.
In the Euclidean theory, we can perform radial quantization, which again is equivalent to canonical quantization on the cylinder. Correlation functions are radial ordered. Starting from the Euclidean theory on the plane parameterized by x, we can obtain the Lorentzian theory by the following steps. First, we fix t E , and turn on y by using the translational generator M −1 as in (2.25), so that the operators and correlators will all depend on the holomorphic coordinate x, as well as another complex coordinate y. In most part of this paper, we consider the BMSFT on the manifold parameterized by complex coordinates x and y. Next, we impose the relation t E = λτ , so that the origin on the Euclidean plane is mapped to past infinity on the Lorentzian cylinder τ = −∞. This enables us to establish the operator-state correspondence as in (2.49).
Furthermore, the radial ordering X(O · · · O) on the Euclidean plane also corresponds to time ordering of the Lorentzian theory. The final step to get the Lorentzian theory is to take the limit λ → 0, so that the theory is restricted on the unit circle. In many discussions, we will not take this final step explicitly.

B Comments on the induced representation
In this section we make some comments on the induced representation.
Since the BMS algebra is the semi-direct sum of the Virasoro algebra and an Abelian ideal generated by M n 's, one can consider a representation induced from that of the ideal.
In particular, we are interested in the special case when the operators in an indecomposable representation satisfy the following conditions As was discussed in [65,66,67], the induced representation is unitary. It is also possible to discuss multiplets in the induced representation, where either L 0 or M 0 is assumed to be non-diagonalizable, featuring Jordan blocks. We leave this interesting generalization to future study. In this section, we comment on the induced vacuum for the free BMS scalar model. We find an intrinsically defined vacuum that behaves as a direct product state, whereas another induced vacuum from the UR limit of a relativistic scalar is singular.

B.1 Induced vacuum from intrinsic discussion
In section 3.2.1, we discussed how to find a vacuum that is annihilated by L 0,±1 , M 0,±1 , and meanwhile can be described intrinsically by the A n , B n modes. We note that there are two different choices, one of which is the highest weight vacuum. Now we turn to the other choice. Choosing the condition I and II, we get another vacuum satisfying This means that all the A n s are creation operators, and all the B n s are annihilation operators.
Using these conditions, we learn that the action of L n , M n on this vacuum is given by, This vacuum satisfies (B.2), hence is an induced vacuum. To study the property of this vacuum, we calculate the Green's function with respect to x-ordering on this vacuum as φ(x 1 , y 1 )φ(x 2 , y 2 ) = −2πi y 1 θ(x 1 − x 2 ) + y 2 θ(x 2 − x 1 ) δ(x 2 − x 1 ). (B.5) The correlator above tells us that there is no correlation between two points with different spatial coordinates x on the plane, or σ on the cylinder. In other words, this vacuum behaves as a direct product of states living at each point of the spatial slice. which means that every term in M n annihilates the vacuum. There may exist different vacua satisfying (B.2), but not the condition above. The induced vacuum from the UR limit [27] provides an example of this type. More explicitly, taking the UR limit ( In the free scalar model

B.2 Induced vacuum from the UR limit
Now we apply this process to the free scalar. The highest weight vacuum before the UR limit is specified by a n |0 H CF T = 0,ā n |0 H CF T = 0, n > 0 (B.10) From the relation between modes (5.26), one might conclude that under the UR limit → 0, the BMS induced vacuum becomes the vacuum specified by (B.3). However, the UR limit has to be taken more carefully. The reason is as follows. Consider a basis constructed by applying A n s and B n s successively on the UR limit of the highest weight vacuum. By calculating the Gram matrix, we find that none of the basis states become null, and furthermore we cannot find any simple linear combinations of such basis that become null. For example, 0 H |A −n B n |0 H CF T = 1 2 , ∀n = 0, so that (B.3) is not satisfied, and therefore the UR limit of the highest weight vacuum is different from the induced vacuum from intrinsic discussion.
In fact, it is unclear if it is possible to express the UR limit of the highest weight vacuum intrinsically in terms of modes A n s and B n s.
Nevertheless, let us try to study some properties of the induced vacuum from its parent CFT 2 theory and the limiting procedure. To calculate the Green's function of φ, we exploit the relation (5.26) and perform the calculation in terms of the modes a n ,ā n . This is equivalent to taking the UR limit directly from the Green's function in the relativistic free scalar. As a result, we get the Green's function for the BMS scalar φ on the cylinder φ(σ 1 , τ 1 )φ(σ 2 , τ 2 ) = − 1 2 log(2 − 2 cos σ 12 ) (B.11) where the divergence comes from the rescaling of the field in the limit (5.22). Similarly, the UR limit of Green's function on the plane φ(x 1 , y 1 )φ(x 2 , y 2 ) = 1 ΦΦ = − 1 2 (log(x 1 − x 2 ) + log( 1 Putting the issue of divergence aside, the Green's function on the plane has the property that it depends on both complex coordinates x 1 and x 2 , instead of the difference x 1 − x 2 . This is because the induced vacuum (B.1) is not necessarily translationally invariant on the x plane.
The divergence of the Green's function on both the cylinder and the plane reflects the difficulty in finding this induced vacuum intrinsically in terms of annihilation creation operators of our BMSFT model. The divergence, however, is potentially related to the divergence in the oneloop partition of three dimensional gravity discussed in [69].

Torus partition function
We will consider the torus partition function (6.3) of BMS free scalar in the induced vacuum from the UR limit. Since we cannot deal with it intrinsically using the A n , B n modes, it is convenient to express the state space in an orthonormal basis of the CFT 2 , | i, j; α CF T = a i 1 −1 · · ·ā j 1 −1 · · · |α CF T (B.13) where |α CF T is the zero mode contribution in the CFT 2 highest weight vacuum. The behaviour under L 0 , M 0 can be calculated from (5.6), The summation in either the first or the second parenthesis becomes divergent as long as a is real. The similar divergence appears in the calculations of the character of the BMS induced module [68] and the one-loop partition function of the asymptotic flat Einstein gravity in three dimensional spacetime [69], where a imaginary part of a is introduced as a regulator to do the summation.
We end this section by the following comments, • The BMS algebra is the semi-direct sum of the Virasoro algebra and an Abelian ideal generated by the M n s, so one can consider a representation induced from that of the ideal. We are interested in a special kind of induced representation defined by (B.1), with the vacuum specified by the condition (B.2). Note that there may be more than one vacua satisfying this condition (B.2).
• From the mode expansion of our free BMS scalar model (3.34), one can define an induced vacuum intrinsically (B.3) obeying (B.2), which behaves as a direct product state.
• Another vacuum satisfying (B.2) comes from the UR limit of free scalar model, as the original general discussion. This vacuum cannot be expressed intrinsically in terms of the modes A n , B n . It leads to a different theory whose Green's function and torus partition function are both divergent as → 0.

C NR limit from CFT 2
Here we review the non-relativistic (NR) limit of CFT 2 for completeness. We will see that the NR limit on the plane is quite different from the UR limit on the plane.

NR limit on the cylinder
Starting with the same setup as in the section 5.1.1, the NR limit is σ = s, → 0 (C.1) so that the speed of light goes to infinity, which is the reason why this limit is called the non-relativistic limit. The theory is defined on (s, t) after the NR limit. Compared with the UR limit on the cylinder (5.4), the NR limit rescales the σ direction instead of the t direction.

NR limit on the plane
We use the map (5.8) to go to the plane (z,z) before the NR limit, with the CFT 2 generators given by (5.9). Now we can take the plane NR limit (5.10) safely to get the GCA generators, l n = t n +t n = −x n+1 ∂ x − (n + 1)x n y∂ y , m n = (t n −t n ) = −x n+1 ∂ y , (C.5) which is consistent with the GCA cylinder-to-plane map x = e iτ , y = ie iτ σ. (C.6)

Representation
We can consider the usual highest weight representation in CFT 2 , The free scalar model At the level of representation, the UR limit of the flipped representation shares the same features as those of the NR limit of the highest weight representation. However, our BMSFT model (3.1) is different from the NR limit of a CFT 2 .
Starting from a free scalar model in the relativistic theory (5.21), the NR limit (C.1) along with the field rescaling Φ = √ φ leads to a non-relativistic theory with action S = 1 4π dsdτ (−(∂ s φ) 2 ) (C.17) Note that the above action is similar to but different from that of our BMSFT model (3.1).
First, the kinematic term has the wrong sign, and a further Wick rotation of the field φ → iφ is needed to fix the sign. Second, the derivative in the kinematic term is in different directions from the BMS scalar action (3.1). As we perform canonical quantization along the spatial circle, the canonical quantization of the the two limits will be different. Third, after the NR limit, the spatial cycle becomes noncompact, whereas in the UR limit the circle remains compact.