Free field realization of the BMS Ising model

In this work, we study the inhomogeneous BMS free fermion theory, and show that it gives a free field realization of the BMS Ising model. We find that besides the BMS symmetry there exists an anisotropic scaling symmetry in BMS free fermion theory. As a result, the symmetry of the theory gets enhanced to an infinite dimensional symmetry generated by a new type of BMS-Kac-Moody algebra, different from the one found in the BMS free scalar model. Besides the different coupling of the $u(1)$ Kac-Moody current to the BMS algebra, the Kac-Moody level is nonvanishing now such that the corresponding modules are further enlarged to BMS-Kac-Moody staggered modules. We show that there exists an underlying $W(2,2,1)$ structure in the operator product expansion of the currents, and the BMS-Kac-Moody staggered modules can be viewed as highest-weight modules of this $W$-algebra. Moreover we obtain the BMS Ising model by a fermion-boson duality. This BMS Ising model is not a minimal model with respect to BMS$_3$, since the minimal model construction based on BMS Kac determinant always leads to chiral Virasoro minimal models. Instead, the underlying algebra of the BMS Ising model is the $W(2,2,1)$-algebra, which can be understood as a quantum conformal BMS$_3$ algebra.

CFTs, such as state-operator correspondence, operator product expansion (OPE) and modular invanriance, can be shown explicitly in free field theories. Secondly, the free fields can be used to study more generic CFTs, which has free field realizations (see, for example, [1]). Standard examples including the Wakimoto representations of affine current algebras and the vertex operator representations of affine current algebras at level one. In particular, the Coulomb gas formalism [2], which is based on free bosons with background charges, is very useful in study- ing Virasoro minimal models [3]. Besides, the simplest minimal model, the 2d Ising model, in fact have another free field realization in terms of free Majorana fermions. Recently, this fermion-boson duality was generalized to all the minimal models [4].
In the past decade, there has been increasing interests in studying the field theories with nonrelativistic conformal symmetries. These symmetries often partially break the Lorentz symmetry but keep some kind of scale invariance. They include Schrödinger symmetry, Lifshitz symmetry, Galilean conformal symmetry and Carrollian conformal symmetry etc. Especially in two dimensions, the global scaling symmetries could be enhanced to local ones under suitable conditions, which are generated by infinite dimensional algebra [5,6,7,8]. On the other hand, these two dimensional nonrelativistic conformal symmetries play important roles in establishing the holographic dualities beyond the AdS/CFT correspondence [9,10,11,12,6,13,14,15,16]. Among these dualities, the flat-space holography is of particular interest.
The study of the field theory with BMS symmetry, or equivalently the Galilean conformal symmetry, is interesting itself, besides the implication on flat space holography. Firstly the algebra (1.1) could be obtained by taking either the Carrollian (ultra-relativistic) limit or the Galilean (non-relativistic) limit of the usual Virasoro algebra. This is in contrast with the situations in higher dimensions, where two limits lead to different algebras [27]. Secondly, the field theory with BMS symmetry is typically not unitary and exhibit novel features [28,29,30,31]. Nevertheless, the BMS (GCA) bootstrap seems still viable, at least for generalized free field theories. For more studies on the GCA bootstrap, see [29,32,30,33,34] 1 . Even though we have some understanding of various aspects of BMS field theories mainly from symmetry constraints, it is necessary to study concrete BMS models to obtain better picture.
As a first step, one may start from free field theories. In [35], a free scalar theory with the BMS symmetry was carefully studied. In the present work, we would like to study another free theory, the inhomogeneous BMS free fermion theory.
The inhomogeneous BMS free fermion model arise from the tensionless limit of superstring [36,37]. We discuss both of its NS sector and Ramond sector, the quantization and the spectrum. We show that due to an extra anisotropic scaling symmetry, the BMS symmetry gets enlarged to a symmetry generated by a new type of BMS-Kac-Moody algebra with nonvanishing Kac-Moody level. The enlarged algebra is different from the one appearing in the BMS free scalar model, as now the commutators between the u(1) generators and the superrotations are non-vanishing. To describe all the states in the theory we need to enlarged the module as well. Actually we show that there appears naturally BMS-Kac-Moody staggered modules.
Moreover, by studying the operator product expansion of various currents, we notice the emergence of a W (2, 2, 1)-algebra S, which could be taken as the quantum version of the conformal BMS algebra studied in [38]. Interestingly, these BMS-Kac-Moody staggered modules can be taken as highest-weight modules of S.
Another important motivation for this work is to search for the minimal models with BMS symmetry. It is well known that the usual 2d minimal models could be read off from the Kac determinant of the Virasoro algebra [3]. However, direct computation of the Kac determinant of the BMS algebra shows that the possible minimal models must be chiral. Nevertheless, as we will show, there does exist a nontrivial BMS minimal model, the BMS Ising model. This model cannot be simply obtained by taking the non-relativistic (NR) limit of the usual 2d Ising model, though its partition function is exactly the NR limit of the one of the Ising model. We show that the inhomogeneous BMS free fermion indeed gives a free field realization of this BMS Ising model, which is similar to the usual boson-fermion duality between the 2d Ising model and the free Majorana fermion. This BMS Ising model is not a minimal model with respect to the BMS 3 algebra, instead, its underlying algebra is the above W (2, 2, 1)-algebra S. 1 The bootstrap program in these works is based on the global BMS symmetry.

Outline of the paper
The remaining part of this work is organized as follows. In section 2, we give a brief review on the BMS field theories. It includes the symmetry algebra, its representations and the correlation functions. In section 3, we discuss the inhomogenous BMS free fermion in its NS sector. We show that the symmetry algebra is a new type of BMS-Kac-Moody algebra (with non-vanishing Kac-Moody level), which interestingly includes an anisotropic scaling symmetry.
Besides, in order to cover all states in the theory, the corresponding modules need to be further enlarged to BMS-Kac-Moody staggered modules. These modules in fact can be viewed as highest weight modules of a W (2, 2, 1)-algebra S. We also comment on the differences and similarities with the BMS free scalar model. In section 4, we discuss the Ramond sector and the BMS Ising model. We firstly analyze the zero modes and find the degenerate (two) ground states of the Ramond sector. These states are created by two twist operators σ and µ, which realize the "spin operator" in the BMS Ising model. Then we find the fusion rules and calculate the partition function of the theory. In appendix A, we show a similar anisotropic scaling symmetry in the BMS free scalar model. In appendix B, we discuss the bottom-up construction of enlarged BMS algebras, including BMS-Kac-Moody algebras and the quantum conformal BMS algebra. In appendix C, we show that the minimal model construction based on the BMS algebra must be chiral.
Note: while we were finishing this project, we were aware that the same BMS free fermion model was being studied by another group [39]. Their work has many overlaps with the present paper.

Review of BMS field theory
In this section, we give a brief review on BMS field theories (BMSFT) in 2d. More detailed discussions can be found in [28][34] [29][35].

Basics
A two dimensional BMS field theory (or Galilean conformal field theory) is a non-relativistic conformal field theory, which is invariant under the following BMS transformation 2 : The corresponding generators are At the quantum level, the commutators acquire central charges, and we obtain the quantum BMS algebra or the Galilean conformal algebra (GCA): These L n and M n are called super-rotations and super-translations respectively. The above BMS 3 (or GCA 2 ) algebra can be obtained by taking the ultra-relativistic or non-relativistic contraction on the 2d conformal algebra [28].
The OPE among the stress tensor currents are The currents have the following mode-expansions

Representations
At early days, it is believed that the states in BMSFTs could be organized into the BMS primaries and their descendants [28], just as in CFT 2 . However, the recent work on the BMS free scalar model shows that there are BMS staggered modules [35], which include states which are not BMS primaries themselves but also are not descendants of any other BMS primaries.
In fact, we will find such staggered modules in the BMS inhomogeneous free fermion with some novel features. Nevertheless, BMS highest-weight modules always appear in BMSFTs 3 and will be helpful for us to understand the Hilbert space of these theories. We review singlet BMS highest weight modules in the following.
For a BMS singlet highest-weight representation, a BMS primary state |O is an eigenstate of M 0 . Denote the corresponding BMS primary operator as O(x, y), when located at the origin, it is labeled by the eigenvalues (∆, ξ) of (L 0 , M 0 ) (O ≡ O(0, 0)): where ∆ and ξ are referred to as the conformal weight and the boost charge respectively. It also obeys the highest-weight conditions Using the state-operator correspondence, one can write the eigen-equations and highest-weight conditions for a BMS primary state: (2.6) and (2.7) with O replaced by |O and commutators replaced by actions of generators. Acting L −n , M −n with n > 0 successively on the primaries, we get their descendants. The operators at other positions can be obtained by the translation Using the Baker-Campbell-Hausdorff (BCH) formula, the transformation law for primary operators are which can be integrated to get the transformation law One can also define the BMS quasi-primaries, which transform covariantly under the global BMS symmetry generated by L 0,±1 and M 0,±1 . The BMS quasi-primaries are characterized by (∆, ξ) as well, but ξ appears generally in the form of the Jordan blocks, suggesting that there appears naturally the boost charge multiplet [29]. In the BMS free scalar model, the Hilbert space can in fact be organized by BMS quasi-primaries and their global descendants [35] [30].
However, this will not be the case for the inhomogeneous BMS free fermion.

Correlation functions
By requiring the vacuum is invariant under the global BMS symmetry, the correlation functions of quasi-primary operators are well constrained by the Ward identities. For the singlet, the two-point function and three-point function are y 12 x 12 , 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 BMSFT, and The four-point functions of singlet quasi-primary operators can be determined up to an arbitrary function of the cross ratios, where the indices i = 1, 2, 3, 4 label the external operators O i , G(x, y) is called the stripped four-point function and x and y are the cross ratios, (2.15)

Multiplets
As we have mentioned, BMS quasi-primary multiplet naturally appear in BMSFTs. In fact, one can also find BMS primary multiplet in BMSFTs [35]. Since we want to see how the BMS symmetry constrains the correlators, we focus on the general case of the correlators of BMS quasi-primary multiplet.
Similar to the Logarithmic CFT (LCFT), the boost multiplet appears because M 0 acts non-diagonally on the quasi-primary states. Generically, M 0 acts as follows, where O denotes a set of quasi-primary operators in the theory and ξ is block-diagonalized being the Jordan block of rank r. The quasi-primaries corresponding to a rank-r Jordan block form a multiplet of rank r.
Under the BMS transformations (2.1), a multiplet O a of rank r transform as: where a = 0, 1, ..., r − 1 label the (a + 1)-th operators in the multiplet. The two-point functions of the operators in two multiplets can be written in the following canonical form [29] 4 In the above, the indices i, j label the multiplets. When i = j, r ≡ r i = r j and a, b label the (a + 1)-th and the (b + 1)-th operators in the multiplets O i and O j , respectively. One can then use (2.19) to define the out state of a quasi-primaries at infinity, which will be used to calculate the inner product and the Gram matrix, From the two-point function (2.20), the inner product of the quasi-primary states in a multiplet are The general form of the three-point function is where For the stripped four-point function of identical external operators, when the propagating operators have non-zero boost charge ξ r = 0, its global block expansion is ∆r,ξr , ∆r,ξr is the global BMS(GCA) block for a propagating singlet with ξ = 0 (2.28) It can be obtained from the Casimir equation. When the propagating operators have zero boost charge ξ r = 0, due to the emergent null states, the corresponding block is more complicated, one can find it in [30]. Remarkably, it can be written as a weighted sum of building blocks, which turns out to be the one dimensional conformal block with non-identical external operators [30].

BMS free fermion: NS sector
In this section, we study the BMS free fermion theory in its NS sector. In fact, one can find two different types of BMS free fermions arising from the tensionless (super-)string: homogenous and inhomogenous BMS free fermions.The homogenous BMS free fermion theory defined on the plane has the following action [40][36] where ψ 0 and ψ 1 are two fundamental fermions. One can easily find that this theory has only Virasoro symmetry and is simply a theory of two free chiral fermions. In particular, all operators have no y dependence. Hence, we will focus on the more interesting case: the inhomogenous BMS free fermion. In the following we simply call it the BMS free fermion theory.

Inhomogenous BMS free fermion
The action for the BMS free fermion is known as the fermionic part of the inhomogeneous tensionless super-string action defined on the plane. It takes the form [37][41]: Hereinafter we always use the notation that ∂ 0 = ∂ y , ∂ 1 = ∂ x . It is easy to see that this action is invariant under the BMS transformation: together with the following transformation rule for the fundamental fields: This transformation rule indicates that the two fundamental free fermions ψ 1 and ψ 0 form a rank-2 BMS primary multiplet ( 1 2 ψ 0 , ψ 1 ) ⊤ , with dimensions and boost charges (this can be confirmed by their behaviour under the BMS transformation) The equations of motion from the action (3.2) are from which we have the modes expansion: With the help of equations of motion, the stress tensor has the components: It is obvious that: so there are two independent components of the stress tensor. We define the stress tensor operators T and M as: which obey the on-shell relation: (3.14)

Enlarged symmetry and module
In this subsection, we further study the symmetries of the BMS free fermion. We show that there actually exist an additional local symmetry which enlarges the BMS symmetry to a bigger space-time symmetry. This enlarged space-time symmetry turns out to be a new type of BMS-Kac-Moody symmetry: it is generated by an algebra with the commutators between the u(1) generators and the supertranslations being the supertranslations, thus it is different from the ones obtained from ultra-relativistic contractions. Consequently the module structure is enlarged as well.

Enlarged symmetry
The BMS free fermion theory in fact has an enlarged symmetry. From the action (3.2), one can find it is invariant under another dilation symmetry D ′ , which scales the coordinates as Under this scaling transformation, the fundamental fermions has dimension 1 2 and − 1 2 respectively such that the action is invariant under D ′ . Note that this dilation is different from the BMS dilation D which is isotropic and requires the scaling dimension of both fundamental fermions to be 1 2 . The corresponding Noether current of the transformation (3.16) reads where we have introduced the current From the equation of motion, it is clear that ∂ 2 0 E = 0, which has the solution The current is conserved 20) which indicates that i.e. E (1) = M . Therefore, we have In the following we write J 0 D ′ as J. From the action, one can easily find another scaling symmetry D ′′ , However, this symmetry is not independent and can be decomposed into where D is the usual dilation in the BMS algebra.
The scaling symmetry (3.15)(3.16) can in fact be enhanced to a local symmetry [5,6,7], with the fundamental fields transforming as Here we want to find the classical counterpart (that is, without central terms) of this underlying (quantum) symmetry algebra. Denote the generators of the symmetry transformation (3.26) as j n , then j n = −x n y∂ y . (3.28) together with the generators of the BMS transformations (l n , m n ) :   [42,43]. Interestingly, similar commutators could also appear in higher dimensional enlarged Galilean conformal algebras (GCA) [44]. In [44], a possible extension of the GCA (recall that in 2d, the GCA is just the BMS algebra studied in this work) is obtained by adding a so(d) Kac-Moody algebra, and in this extended GCA algebra (equation (3.2) in [44]) there is a commutator which is of Note that the extended GCA algebra in [44] only exists in d > 2. In contrast, the BMS-Kac-Moody algebra (3.30) is defined in 2d. This is feasible because our vector field realisation of J is different from the one in [44] (Eq. (3.4)). A crucial point about our vector field realisation of j n in (3.28) is that it can not be obtained by a contraction, while in [44] it was shown that at least the finite global part of the algebra (Eq. (3.2) in [44]) can be obtained by a contraction. In summary, our result non-trivially extends the enlarged space-time symmetry in [44] for d > 2 to the d = 2 case. Moreover, we find it is just the symmetry algebra of a real BMS field theory: the BMS free fermion.
With this anisotropic scaling symmetry (and the enhanced local symmetry C f ) in hand, one may want to know whether this kind of symmetry can be found in other BMSFTs. In fact, the BMS free scalar model [35] indeed has such a symmetry, as we show in appendix A.
Moreover, in appendix B we show that BMSFTs with such a symmetry (and no other extra symmetry currents) will always have the central charge c M = 0. 5 It is better not to call such algebras as BMS-Kac-Moody algebras because they are generated by the contracted Kac-Moody current so T and M are not basic generators. Thus it is called "Galilean affine algebras" in [42] and "non-Lorentzian Kac-Moody algebras" in [43]. However, to stress that this algebra is an enlarged BMS algebra, we adopt the name "BMS-Kac-Moody algebra" in this paper. 6 We have classified possible BMS-Kac-Moody algebras with one Kac-Moody current in the appendix B.1. More precisely, we show that by considering the most general commutation relations between the BMS algebra and one u(1) Kac-Moody current, the Jacobi identities restrict the enlarged algebra into three types: Type 1 being the one found in [35], Type 2 being the one found in the present work, and Type 3 being some singular cases. This proves the consistency of our algebra from another point of view. Now we discuss the canonical quantization in the NS sector. According to the canonical anti-commutation relations of the fundamental fermion fields and their conjugate momentums, we obtain the anti-commutation relations of the modes defined in (3.7): Because the inhomogenous fermions we consider are real fermions on the cylinder [41], we have the following Hermition condition for these modes: For the NS sector, we must have n ∈ N + 1 2 to satisfy the anti-periodic condition on the cylinder. The vacuum is defined by: It leads to the following prescription for fermionic normal ordering: where X = A, B. Now the stress tensor operators as well as the current E can be written in terms of the normal-ordered products of the operators In the following, we will always treat the normal-ordered products of operators so we omit the normal ordered symbols in T, M, E, and if necessary we useN to denote the normal ordering of composite operators.
The vacuum (3.33) is in fact the BMS-Kac-Moody highest weight vacuum, satisfying: where L n , M m and J k are the BMS-Kac-Moody modes and their explicit expressions will be given in (3.51). The Hilbert space of this theory consist of the following states: Our next goal is to find the module decomposition of this Hilbert space. Firstly, we want to find the BMS-Kac-Moody primary states, which are defined as: and are labeled by the conformal dimension ∆, boost charge ξ and the u(1) charge ε: It turns out that there are four BMS primary states, among which three are BMS-Kac-Moody primaries. The four BMS primary operators are: • The singlet 1 with dimension ∆ = 0, boost charge ξ = 0 and charge ε = 0, the corresponding state is |0 . It is a BMS-Kac-Moody primary.
• The singlet E ≡ − 1 2N (ψ 0 ψ 1 ), with dimension ∆ = 1, boost charge ξ = 0 and charge ε = 0, the corresponding state is |0 . It is a BMS primary but not a BMS-Kac-Moody primary. This is expected because E itself is just the Kac-Moody symmetry current. Note that from the UR limit point of view, this symmetry current is an emergent one because the relativistic counterpart of E is ψψ, which is a non-chiral operator and is not a symmetry current in the relativistic free fermion theory.
Using the anti-commutation relation of the modes, we can calculate the correlators of the fundamental fields and find that ( 1 2 ψ 0 , ψ 1 ) indeed form a doublet (in the canonical form). We stress that the 'energy operator' E is a singlet primary. We may construct triplet primaries (3.42) However, due to the fermionic nature of ψ 0 and ψ 1 , : ψ 0 ψ 0 :=: ψ 1 ψ 1 := 0, so we are left with a singlet. Another interesting property of E is that its modes expansion has y dependence, where the explicit expressions of J n , M n will be given in (3.51). Moreover, the two-point function of the energy operator depends only on x as a two-point correlator of a singlet should behave. Note that this 'energy operator' is the same (same dimension, and being a singlet) as the one obtained by taking the non-relativistic limit of the energy operator in the Ising model.

Enlarged module
To organize all the states in (3.37) into the representations of the algebra, it turns out that not only the underlying symmetry needs to be enlarged to be a BMS-Kac-Moody algebra, but the modules should also be enlarged. It is easy to see this point from many novel nonprimary states in the spectrum. For example, the state is not the descendant of any primary states. In fact, this is similar with the case of BMS free scalar [35]. As a result, we need an extra operator In fact, it is just the existence of this operator that prevent the decoupling of the M n (be null) in the enlarged modules. The theory would reduce to a chiral one if the M n 's get decoupled.
This kind of module is referred to as the staggered module, which is well-known in the study of logarithmic CFT [45]. Note that in the BMS free scalar model, such a K operator, together The first step is to find all the operators in the stress tensor multiplet and write down the operator product expansions (OPEs) among them. In the case at hand, they turn out to be as well as One can certainly construct another operator with ∆ = 2:N (EE), which is the normal ordered product of two E's. However it is not an independent operator since 7 One can see this relation either by writing the OPE of EE up to the constant term, or by showing the corresponding states are identical via the state-operator correspondence.
We can write down the modes expansion of these four operators, where (3.51) The states corresponding to the operators in the stress tensor multiplet are , which satisfy the following relations under the action of L 0 , M 0 and J 0 : Therefore, the stress tensor operators, together with K and ∂ 1 E, form a BMS triplet |T 3 and a BMS singlet |T 1 , where with conformal dimensions, boost charges and u(1) charges It is worth remarking that J 0 provides a u(1) charge, under which the states |K , |M carry the charge 1 and −1, respectively, while |T and |E are neutral. This perspective will be useful in the following discussion on BMS-Kac-Moody staggered modules.
With respect to the Virasoro subalgebra, |K , |T , |M are Viraroso quasi-primaries (|K and |M are in fact Viraroso primaries), as however, |K is not a BMS quasi-primary Finally, with respect to the BMS-Kac-Moody algebra, we have : so only |M is a BMS-Kac-Moody quasi-primary states.
Now, we can describe the structure of a BMS-Kac-Moody staggered module. First of all, let us recall the general definition of a staggered module. A staggered module S for an algebra A is an A-module for which we have a short exact sequence: where H L and H R are referred to as left modules and right modules respectively, both being highest weight modules 9 . There is another central requirement: there must be an element C in the algebra A which is not diagonalisable on the module S. In LCFT, C will always be L 0 .
A staggered module in fact gives a way to "glue" two highest-weight modules. More generally, one could consider indecomposable modules constructed from more than two highest weight modules.
For the staggered modules in the BMS free fermion, the underlying algebra A will be the BMS-Kac-Moody algebra and the element C will be M 0 . As we mentioned above, J 0 can be viewed as a u (1)  As an illustration, we consider the staggered vacuum module V in the BMS free fermion.
In this vacuum module, we have the submodule S 1 , by its definition it can be generated by 10 : S 1 is actually a staggered module. We can see the staggered structure as follows. In (3.65), the set of vectors with a = 0 form a submodule Ω ≡ S 0 (with null states being modded out), which is the BMS-Kac-Moody highest-weight vacuum module. Then we have the following short exact sequence: where However, in the quotient module K 1 , we have Ω ∼ 0, so |K is indeed a primary and it has the following charges The construction of staggered modules in the BMS free field theories is remarkably different from the one in LCFT. In the BMS cases, staggered modules glue infinite number of highestweight modules, while in the LCFT case, staggered modules always glue two highest-weight modules 11 . This remarkable feature indicates that K is in fact a generator of an enlarged algebra.
For general BMS-Kac-Moody staggered modules, we can similarly construct them as S ∞ .
The only difference is that the initial H L will be a general BMS It would be interesting to compare the symmetry and the stress tensor multiplet with the ones in the BMS free scalar model [35] • Firstly, let us compare the symmetry. The BMS free scalar model also has a BMS-Kac-Moody algebra as its underlying symmetry. The difference is that in that case, While in our case, the normal ordered product of the current E gives T : Remarkably, one can also construct M in terms of the current E: not by the Sugawara construction, but by a derivative M = ∂ 0 E. This kind of construction may not apply 12 This bigger algebra is generated by O0 and O1 in [35]. In fact, T and M can be constructed as normal ordered products of O0 and O1 (a Sugawara like construction). It is shown in [35] that this bigger algebra can be obtained by contracting two u(1) Kac-Moody algebras. Note that O1 is not a symmetry current, thus the symmetry algebra of the BMS free scalar is only a subalgebra of this bigger algebra. to general BMS field theories but it really happens in the BMS free fermion case. 13 At first sight, it seems strange that the stress tensor M is a derivative of another current E, which means that M is a global descendent of E. In the CFT case, such an operator cannot be a quasi-Virasoro primary. Interestingly, one can easily verify that M is indeed a BMS quasi-primary, and we had stressed this fact in (3.59), (3.60) and the paragraph between them. Logically, the relation ∂ 0 E = M precisely reflects the fact that there exist a very special enlarged BMS algebra, which enlarge the BMS algebra by the current E.
This algebra is just the new type of BMS-Kac-Moody algebra we found in this paper.
• In the case of the BMS free scalar, T, M and K form a triplet. In our case, roughly speaking, K, T , M still form a triplet. However, due to the existence of the fourth operator ∂ 1 E, the triplet gets modified slightly and there appears another new singlet |T 1 .
• are:   It is easy to work out the OPE of the fundamental fermions to confirm that they form a doublet of BMS-Kac-Moody primaries. The OPEs are of the following forms As a consistent check of the symmetry C f , we can calculate the OPE of J = J 0 D ′ = −E (0) and ψ 0 or ψ 1 : From the leading terms in (3.77) and (3.78) one can directly read that the charge of ψ 0 and ψ 1 under D ′ are 1 2 and − 1 2 , respectively. More precisely, we can calculate the infinitesimal variation of the fundamental fields from the above OPEs, with f (x) = 1 + ω(x). In fact, substituting (3.77) into (3.79), we find which is just the infinitesimal form of (3.27) for ψ 0 . Note that there are no derivative terms with respect to both x and y because Similarly, substituting (3.78) into (3.79), we find which is just the infinitesimal form of (3.27) for ψ 1 .
The OPEs involving the operator K are more complicated, and they present a novel feature: the OPE can not be organized by BMS(-Kac-Moody) quasi-primaries. Firstly, note that K itself is not a BMS quasi-primary. As K is a Virasoro primary, one may try to use only the Virasoro subalgebra to organize the OPE. However, there exist operators in OPEs which are not Virasoro quasi-primaries themselves and are also not the derivatives of any other quasiprimaries appearing in the OPE. For example, let us look at the OPE of M (x)K(x, y), which is of the form (3.83) On the right-hand side, we find an operator (and its corresponding state) which is neither a Virasoro (quasi-)primary as nor the derivatives of any Virasoro quasi-primaries appearing in the OPE. However, K (1) (x, y) can in fact be expressed as a sum of the derivatives of Virasoro quasi-primaries. It turns out that where T , E andN (EEE) are all Virasoro quasi-primaries. Note that while T and E are BMS quasi-primaries,N (EEE) is not. So we conclude that while the OPE can not be organized by the BMS quasi-primaries, it can be organized by the Virasoro quasi-primaries.
To find the corresponding algebra, we must translate (3.83) into the commutators of the modes. Firstly, we have (3.87) Then using the OPE (3.83), we find m are not Virasoro quasi-primary modes, we do not need these commutators to define the algebra. In fact, we only need the commutators among those modes which are not only Virasoro quasi-primary modes, but also the generators. It turns out that these generators include M n , K n ≡ K (0) n and J n . Note that L n is not a generator because L n = 2N (JJ) n , even though we will also write down the commutators involve L n . The commutators among L n , M n and J n were already worked out in   This W -algebra S is in fact a quantum version of the conformal BMS algebra (CBMS) studied in [38]. In [38], it was found that the classical conformal BMS algebra can be defined for generic central charge c. For generic c, T is also a generator of the W -algebra so the classical conformal BMS algebra is of type W (2, 2, 2, 1). Quantum conformal BMS algebra can be obtained by the quantum Drinfeld-Sokolov reduction based on a non-principal sl 2 embedding (see [38,46] or Appendix B) so the central charge c ≡ c L can be generic. In appendix B, we show that when this algebra has a BMS-Kac-Moody subalgebra, the central charge has to be c = 1 and the Kac-Moody level must be k = 1 4 . In this case, the current T is decoupled, and we reproduce the W-algebra S of type W (2, 2, 1). The detailed discussions on this issue can be found in Appendix B. Under this identification, the currents E and K could be interpreted as "super-dilation" and " super-special conformal transformation" operators in the bulk [38].
If we focus on the modes from the beginning, then (3.31) is just the algebra of a complex fermion. The W-algebra S is the maximal bosonic subalgebra of this complex fermion algebra. so when ∆ = 1, there is only one state |E , whose norm is negative, Similarly, when ∆ = 2, the Gram matrix of the four independent states 15 We currently do not find the possible symmetry transformation corresponding to the operator K. This is why we need to use BMS-Kac-Moody staggered modules to organized the Hilbert space. We believe that K can not be realized as a symmetry current in the BMS free fermion. This is very likely also the case for the BMS free scalar model [35].
is not positive definite. As a result, there are also the states with negative norms, for example, |M − |K and |∂ 1 E . The above states lie in the vacuum module V. One can check that there are the states with negative norms in the module F as well.
Finally, we would like to comment on the Hermitian condition (3.100). We have shown that the generators J n , M n , K n and L n form a non-linear W -algebra S. Usually, the W -algebra appearing in a 2d unitary CFT can be equipped with an Hermitian condition, which • gives rise to a positive definite Gram matrix, • is compatible with the structure of the W -algebra.
In the case at hand, such an Hermitian condition 16 actually exists for S: We will show in this section that there is a similar BMS free fermion realization of the BMS Ising model. To find such a realization we essentially need to discuss the Ramond (R) sector to find the 'spin operator' σ.

Twist operators
In the R-sector, the modes number n ∈ Z so the fermions satisfy Firstly, we need to find the degenerate ground states in the Ramond sector. The R-sector ground states are created by the twist operators. For example, in the R-sector of the free 16 This Hermitian condition can be obtained formally by imposing A † n = B−n, B † n = A−n in (3.51). 17 The standard normalization ǫ|ǫ = 1 requires a factor i.
Majorana fermion, one find two twist field 18 σ and µ from a two-dimensional representation of the Clifford algebra coming from the zero modes. They transform into each other when fusing with the two fundamental fermions ψ andψ. In the BMS case, we also need to study the algebra of the zero modes. Recalling that we combine them as (4.4) and then find that they satisfy the same algebra as the modes of free fermions ψ n andψ m : For the zero modes A 0 , B 0 , they are transformed into C 0 , D 0 , which obey the same Clifford algebra just as ψ 0 andψ 0 , The zero modes as well as the fermionic number operator (−1) F , which is used to distinguish the degenerate ground states, can be realized in terms of the Pauli matrices: (4.7) Now we have two twist fields σ and µ, creating two R-sector ground states These two ground states are transformed into each other by the zero modes, (4.9) In terms of A 0 and B 0 , we have (4.10) 18 We will use the same notation σ and µ for the twist operators in the BMS case.
The above twist fields σ and µ can be identified with the 'spin operators' in the BMS Ising model. Now let us calculate the quantum numbers of these twist operators. Since the symmetry algebra is the BMS-Kac-Moody algebra, we need to know their conformal dimensions, boost charges and the u(1) charges. They can be determined by considering the expectation value of the current T , M and E in the R-sector, which can be calculated in two ways: either using the definition of the ground states in the Ramond sector or using the OPE. Comparing the results from these two methods gives the quantum numbers. We first compute the expectation values of the currents directly. Using the commutation relation of the modes, we find so in both of the Ramond ground states, we have (4.14) The short distance behaviour of the above correlators coincides with the corresponding ones in the NS sector, because short-distance behavior is independent of the global boundary conditions. From their definitions, the currents can be realized as: Taking the z − x = ǫ → 0 limit and using (4.11)-(4.14), we find the following expectation values: Next we consider the OPEs between the currents with the spin operators. Because σ and µ are both BMS-Kac-Moody singlet primaries, their OPEs with the stress tensors and the u(1) current are (similar ones for µ).

(4.19)
From the above OPEs (4.19), we have (4.20) Comparing with (4.18), we read the quantum number of the twist fields We had obtained these quantum numbers from the representation theory of the zero modes of S, below (3.98). If we view the twist fields as S-primaries, the corresponding K 0 charge κ is κ = 0, which can also be similarly obtained as ∆, ξ and ε. Note that the values of ∆ and ξ agree with the ones by taking the non-relativistic limit of the Ising model.

Fusion rules
Next we turn to determine the fusion rules. The criterion is to see whether the related three- This fusion algebra as well as the original one (4.22) only concern the BMS primaries and their BMS descendents, so they do not cover all the states in the theory. To include all states, we need to organize the fusions in terms of BMS-Kac-Moody staggered modules or highest-weight modules of S. Therefore the above fusion rules can be rewritten more suitably as follows where the subscript "S" means that they are S-modules. These are the operator spectrum (S-primary operators) and the fusion rules for the BMS Ising model.
Next we consider the structure constants in BMS Ising model. Recall that Ising model has one non-trivial structure constant: C σσǫ = 1 2 . Interestingly, the BMS Ising model with the underlying algebra be S, has no non-trivial structure constant, as can be seen from the above fusion rules. All information of the 4-point correlation function are encoded in the S-conformal blocks.

The partition function
Now we discuss the partition function. Firstly, let us briefly review the modular invariance of the partition function in general BMSFTs. The partition function is defined as 19 :  by taking either NR or UR limit [47], with the modular parameters being related by NR : τ = σ + ǫρ,τ = −σ + ǫρ, (4.29) The BMS modular invariance had been discussed in the literatures, both intrinsically [19][23] [35] and by taking the limit [20][48] [47]. The modular S-transformation is: and the modular T-transformation is: The above partition function can also be viewed as part of a chiral minimal model based on the W -algebra S. In fact, the full local minimal model of S is just the free boson compactified on a S 1 with unit radius r = 1. This rational CFT has the following partition function: 20 It is clear that the partition function (4.32) is not invariant under the modular T-transformation (4.31). This suggest that the BMS free fermion is not well-defined on the torus. This is in fact also the case for the BMS free scalar [35]. Nevertheless, they are consistent theories on the plane.
where χ (S) 0 and χ (S) 1 8 are as above, and . (4.35) Note that, by the fermion-boson duality, this is also the partition function of the Dirac fermion [49] [1]. The corresponding chiral theory includes three S-primaries 1 S , σ S and ǫ S (the Kac spectrum of S), with dimensions Note that we use the same notation 1 S and σ S as in (4.26) since they are exactly the same S-modules. This chiral minimal model has the following fusion rules (omit the trivial ones): We would like to comment on the NR limit of the 2d Ising model. Recall that we have the following relation for the partition function 21 : Ising model While the partition function of the BMS Ising model is exactly the NR limit of the partition function of the Ising model, the details are somehow puzzling. Firstly, if we naively take the NR limit of the Ising model, we find that the spectrum includes three BMS primaries: with dimensions and boost charges respectively One can identify these states with the one in the free field realization as 22 : However, these states, together with their BMS descendants, do not cover all the states we find in the free field realization. More precisely, the operator K can not be read from the NR limit of the Ising model. In fact, under the NR limit, the central charge become c L = c +c = 1, c M = ǫ(c −c) = 0, and the theory reduces simply to a chiral one [28] [35].
However, the above spectrum (4.40) is not the one in a chiral minimal model. Secondly, while we can get the fusion rules from the NR limit: the corresponding null condition can not simply be seen from the NR limit. Finally, in the free field realization, we have shown that the BMS highest-weight module containing 1 ′ (1 f ) and the one containing ǫ ′ (E f ) are related to each other, this fact can not be seen from the NR limit point of view either. These three puzzles can all be resolved by the enlarged symmetry and the enlarged module we find in the free field realization. Thus, the free field realization help us to see how the NR limit of the Ising model makes sense and what is the underlying structure: the underlying algebra changes under the NR limit as: Vir × Vir → S (4.43) Under the NR limit, such a change of the underlying algebra may or may not appear in other minimal models.
It is worthy to emphasize that the BMS Ising model constructed in this section is not a minimal model with respect to the BMS algebra. In fact, the minimal model construction based on the BMS Kac determinant simply reduce to chiral Virasoro minimal models. We show this fact in appendix C.

Conclusion and outlook
In this work, we studied a new kind of free BMS field theories called the (inhomogeneous) BMS free fermion, and then used it to give a free field construction of the BMS Ising model.
The BMS free fermion exhibit many novel features. Firstly, the underlying symmetry is generated by a new type of BMS-Kac-Moody algebra, which is different from those obtained by contractions. In particular, it includes an anisotropic scaling symmetry so the theory has two different scaling symmetries. It is interesting that such a symmetry also appear in the BMS free scalar model (see appendix A). Note that the Kac-Moody current in our case has a non-vanishing level so BMS highest-weight modules are enlarged by this current. This is different from the Kac-Moody current studied in the [35].
Secondly, the module in the BMS free fermion turns out to be BMS-Kac-Moody staggered module, which are similar with the BMS staggered module studied in the BMS free scalar model [35]. In the staggered vacuum module, an extra operator K appears. Unlike the one in the BMS free scalar, here K is not a BMS quasi-primary. As a result, the Hilbert space can not be organized by the BMS quasi-primaries and their descendents. Relatedly, the OPEs can not be organized in terms of BMS quasi-primaries either. Nevertheless, we showed that the OPEs can actually be organized by the Virasoro quasi-primaries, which helped us to read an underlying W -algebra S of the vacuum module. The BMS-Kac-Moody staggered modules appearing in the BMS free fermion can be viewed as highest weight modules of S.
Finally, we studied the Ramond sector to obtain the twist operators, which can be identified It would be interesting to see whether there are other BMS models which could be minimal with respect to some enlarged BMS algebras. For example, as S is the quantum conformal BMS algebra with c = 1, one can try to study generic quantum conformal BMS algebra and search for other possible minimal models. One can also discuss enlarged super-symmetric BMS algebras and the corresponding minimal models. An explicit example (of the algebra) is constructed in [38], which is the super-symmetric version of the classical conformal BMS 3 algebra. This is a W -algebra of type W (2, 2, 2, 3 2 , 3 2 , 1). It is possible that one can construct the super-symmetric BMS Ising model based on this super conformal BMS 3 algebra. A free field realization may be helpful as well in this case.
It will be interesting to see whether the BMS free fermion and the BMS Ising model can be related to the BMS scalar model. It is not clear how to do the bosonization in the BMS case. In particular, it is not clear how to develop the Coulomb gas formalism in the BMS case, which may be helpful to study the BMS Ising model as well as other possible minimal models.

Acknowledgments
We are grateful to Chi-ming Chang, Peng-xiang Hao, Reiko Liu, Wei Song, Yu-fan Zheng for valuable discussions. We would like to thank the participants in the "Third National Workshop on Quantum Fields and String" (Beijing 2022-08) for stimulating discussions. The work is supported in part by NSFC Grant No. 11735001.
A Anisotropic scaling symmetry in the BMS free scalar In this section, we show that a similar anisotropic scaling symmetry also appear in the BMS free scalar model. The BMS free scalar model has the action: In [35], it was shown that the symmetry of this theory is a BMS-Kac-Moody algebra, including the BMS symmetry as well as a u(1) affine current symmetry coming from the y-independent translation of the fundamental scalar: In fact, the action (A.1) has another scaling symmetry, which is anisotropic with the fundamental field transforming as The corresponding Noether current is where M (x) = − 1 2 ∂ 0 φ∂ 0 φ is the stress tensor of the model [35], and we has introduced the current From the equations of motion ∂ 2 0 φ = 0, it is clear that ∂ 2 0 J = 0, and we have The Noether current is conserved i.e. J (1) (x) = M , and We denote J 0 D ′ asJ. One can see that the above analysis is quite similar with the one in the BMS free fermion. Similarly, we can also find another scaling symmetry (denote the scaling in the BMS algebra as D): which is not an independent one since D ′′ λ = D ′ λ −1 D λ . Similar to the BMS free fermion, the scaling symmetry D ′ can be enhanced to a local leading to an affine u(1) symmetry with the chargesJ n , which are the modes ofJ. The generators of C f (denoted asj n ) is the same as j n for C f in (3.28), so fromj n , l n , m n we can obtain the same BMS-Kac-Moody algebra (3.30) without central extensions. One can compute the OPE of the currents to find the corresponding quantum algebra with central extensions.
While it is interesting, further discussions on this symmetry in the BMS free scalar model will be beyond the scope of present work.

B Bottom up construction of enlarged BMS algebras
In this section, we will study enlarged BMS algebras from a bottom up construction. Firstly, we study general BMS-Kac-Moody algebra with a u(1) Kac-Moody current. In particular, we manage to determine all the possible central extensions of the BMS-Kac-Moody algebra (3.30). We find that the central charge c M must be vanishing while c L and the Kac-Moody level k can be arbitrary. Then, we show that when this BMS-Kac-Moody algebra is further enlarged to the quantum conformal BMS algebra (a W-algebra of type W (2, 2, 2, 1)) [38], the central charges and the Kac-Moody level will be uniquely determined by the Jacobi identities.
The resulting W-algebra is exactly S.

B.1 BMS-Kac-Moody algebra
We will study general BMS-Kac-Moody algebras with a u(1) Kac-Moody current. We start with the BMS 3 algebra: where k is the Kac-Moody level. To determine the last commutator [M n , J m ], we need the general formula for the commutator of the modes of two quasi-primaries [51]: c L = 0 or c L = 1 or k = 0. We refer to these singular cases as Type 3. For example, we have: 24 We can do this because we keep the level k undetermined. This normalization is the one used for S.
• Type 3: when k = 0, J and N (JJ) are null states, so we have the following possibility: where α is an arbitrary number. c L and c M can be arbitrary.
There are other possibilities for [M n , J m ] in these singular cases (Type 3). However, these cases are not relevant with the quantum conformal BMS algebra discussed in the following.
Thus we do not list the rest of them.

B.2 Quantum conformal BMS algebra
Now we give an explicit construction of the quantum conformal BMS 3 algebra. It is a Walgebra of type W (2, 2, 2, 1) [38]. We must insert another current K with ∆ = 2 into the algebra, where c ≡ c L (we will always use this notation for c L hereafter). Then we normalize the M |K 27 Even though these commutators vanish, the commutators [Mn, Jm] could includes K so that the CBMS algebra does not has a BMS-Kac-Moody subalgebra. We do not search these details of the generic CBMS algebra in the present work. 28 The BMS Gram matrix and the Kac determinant had been discussed in literature, e.g. [28][47], however, a closed formula for the BMS Kac determinant is absent in literature. We learned the following closed formula from Peng-xiang Hao. In fact, under this condition, there will be more emergent null states and the theory reduce to a chiral CFT [28] [35]. So after modding out the null states, the vanishing condition for the Kac determinant of the remaining states is no longer be the generic one (C.6). Instead, we find the chiral part of standard Virasoro minimal models. As is well-known, while these chiral minimal models has a closed fusion algebra coming from the null states condition, they can not be a full local theory. A crucial point is the loss of modular invariance of their partition functions.
A natural question is whether there exist non-trivial minimal models when the underlying BMS symmetry is enlarged. Let us first review the enlarged BMS algebras or Galilean conformal algebras in the literatures briefly. One way to enlarge the GCA 2 is considering the Galilean (non-relativistic) contractions of various W -algebras. The resulting algebra is known as the Galilean W -algebras, e.g. [42]. In contrast, another way to enlarge BMS 3 (GCA 2 ) is to consider the ultra-relativistic contraction of various W -algebras [56] [57]. As pointed out in [57], there is an interesting feature regarding these two contractions on non-linear W -algebras: while the difference between these two contractions can only be seen from the representations for linear W -algebras 29 , these two contractions lead to different quantum W -algebras when acting on non-linear W -algebras, even though the resulting classical W -algebras are isomorphic.
One may try to construct the minimal models with respect to these NR-or UR-constracted W -algebras. However, the situation will be similar with the one for the BMS 3 (GCA 2 ): one can not obtain any non-trivial minimal models, other than some chiral minimal models with respect to the original W -algebra, which can not be a full local theory. In particular, the generators M n s will always decouple from the module. Thus, to obtain non-trivial minimal models, one essentially needs to enlarge the module to include a special operator K with dimension ∆ = 2, which prevents the decoupling of M n . This is the case in the BMS Ising model we constructed.