Remarks on BMS3 Invariant Field Theories

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As with the Virasoro case, the BMS3 symmetry is infinite dimensional and this fact is responsible for some surprising algebraic features. First of all, it is isomorphic to the two-dimensioanl (2D) Galilean-conformal algebra (GCA), which is an Inönü-Wigner contraction of the Virasoro algebra, see [25–31]. Therefore, we may regard these two algebras as the “same,” but the drawback of these constructions is that we do not have the holomorphicity properties usually available in meromorphic conformal field theories (CFTs) as defined initially by Belavin, Polyakov and Zamolodchikov [32].

We try to remedy this situation using a third realization of the BMS3 algebra: it has been shown that it is also isomorphic to the W (2, 2) algebra [31, 33–37], and that is the viewpoint we adopt in the present work. Roughly speaking, the W (2, 2) algebra is a W algebra defined by the stress-energy tensor and an additional spin-2 field W (z). Similar ideas have been proposed in [38], but with the strong assumptions of that work, the additional spin-2 field was restricted to be the stress-energy tensor. We apply the isomorphism BMS3 ≃ W (2, 2) to study 2D CFTs with this algebra as a chiral symmetry, and then we try to use the powerful holomorphicity properties of meromorphic CFTs. For example, in a generic 2D CFT, we can solve the theory without the use of an action principle or even the equations of motion in a miraculous way known as the bootstrap approach. This formalism is an outcome of what a quantum field theory is: the result of two pieces of data, correlation functions and symmetries. Correlation functions are generally defined from the dynamical details of a particular physical system, while the symmetries impose nontrivial relations among the correlation functions. Evidently, the main role in the bootstrap formalism is played by the infinite-dimensional algebra, which manifests itself as the Ward identities [32, 39–47].

Before going any further, we should remark that although the flat space holography is, probably, a rich source of BMS3 invariant field theories, the CFTs we consider in this work are not expected to be dual to the asymptotically flat spaces that we have mentioned before. The main motivation in this text is toward the construction of CFTs with extended symmetries defined by another spin-2 field of which the modes satisfy the BMS3 algebra. Under this perspective, we should consider that these (elusive) theories are similar to sl(n) Toda CFTs which are endowed with extended Wℓ algebra ℓ ≥ 3 as chiral symmetries; see for example [48, 49] and references therein.

The paper is organized as follows. In section 2, we review the isomorphism BMS3 ≃ W (2, 2). Despite the fact that these results are not new, we try to write this discussion in a chiral CFT language. Additionally, we slightly change the hypothesis of [31] and we do not assume that the zero-mode Lie algebra is necessarily semisimple [50]. We show that outside the semisimple Lie algebras realm it is possible to generate the centrally extended algebra through a Sugawara construction. In the same section, we briefly review the detailed analysis of null states in BMS3 field theories performed in [26]. We pay special attention to some kind of monster in the theory, which is necessary to the main proposal of this work in section 5. Namely, there are zero-norm states that are not null in the usual sense, since they are not, in principle, annihilated by all positive W (2, 2)-algebra modes. In section 3, we study the Ward identities, and we will see that the constraints naively imposed by these identities to the correlation function, also impose that the central charge cM vanishes. As we will argue along this work, a null central charge cM is the worst scenario that we can find; the resulting theory is equivalent to a CFT without extended chiral symmetry at all, and in this case the action of the generators {Wn|n ∈ Z} is completely trivial. In section 4, we avoid the case cM = 0 in our theory by changing some of our hypotheses: we embrace the existence of the seminull states, and we accept the fate of nonunitarity. We will see how we can organize the descendant fields into multiplets, and we also propose a modification of the Ward identity that takes into account the Jordan structure of the theory. We conclude in section 5 and discuss further research directions.

2 2 BMS(3) as a chiral W(2,2) algebra

In this section, we review the isomorphism between the BMS3 and the W(2,2) algebras [35–37, 51]. We mainly follow the notation of [43], but [32, 39–47] may be useful. Let us assume that we have a CFT living in the Riemann sphere 1 CP . Given two states |vi and |wi with conformal weights hv and hw, respectively, the operator product expansion (OPE) between their corresponding vertex operators is

v −n−hv Vv(z)Vw(ζ)= V (V (|vi; z − ζ)|wi; ζ)= V (Vn |wi; ζ)(z − ζ) . (2.1) n h X≤ w Observe that the sum has an upper bound, and it arises from the fact that the theory cannot have any state with negative conformal dimension in a unitary theory. The OPE between a quasiprimary ﬁeld W (z) of conformal dimension 2 and the stress-energy tensor reads

§1) The state |φ2i has conformal weight 2. Then, L0|φ2i =2|φ2i and L1|φ2i = 0. §2) It is easy to show that

L−1|φ2i −n−3 −n−3 V (L−1|φ2i; z)= Vn z = ∂V (|φ2i; z)= (−n − 2)Wnz ; (2.4) n Z n Z X∈ X∈ therefore, Vn(L−1|φ2i)= −(n + 2)Wn . (2.5)

§3) Finally, the state L2|φ2i has conformal weight h = 0. In fact,

L0(L2|φ2i) = ([L0,L2]+ L2L0)|φ2i . (2.6)

Additionally, we know that Vn(|0i)= δn,0 .

Putting all these facts together, we have c [L , W ] = (m − n)W + 2 m(m2 − 1)δ . (2.8) m n m+n 12 n+m,0 Finally, it is clear from (2.1) that the general form of the OPE W (z)W (ζ) is still undetermined, but we can make a consistent choice W (z)W (ζ)=0 ⇔ [Wm, Wn]=0 . (2.9) Relaxing this condition may be interesting from the physical viewpoint, but we keep this choice along this text.

3 Deﬁnition: The W (2, 2) algebra is deﬁned by the generators {Lm, Wn | m,n ∈ Z} satisfying the following commutation relations c [L ,L ] = (m − n)L + 1 m(m2 − 1)δ m n m+n 12 m+n,0 c [L , W ] = (m − n)W + 2 m(m2 − 1)δ (2.10) m n m+n 12 m+n,0 [Wm, Wn]=0 .

The Verma module for the W (2, 2) algebra has been constructed in [35–37] . Furthermore, the algebras (1.2) and (2.10) are clearly isomorphic; therefore, we make the following identiﬁcations: cM ≡ c2 and W (z) ≡ M(z).

The symmetry algebra above can be equivalently deﬁned in terms of the holomorphic ﬁelds

−n−2 −n−2 T (z)= Lnz and W (z)= Wnz (2.11) n Z n Z X∈ X∈ satisfying the OPEs, 1 c 2T (w) ∂T (w) T (z)T (w) ∼ 1 + + 2 (z − w)4 (z − w)2 (z − w) 1 c2 2W (w) ∂W (w) (2.12) T (z)W (w) ∼ + + 2 (z − w)4 (z − w)2 (z − w) W (z)W (w) ∼ 0 .

These expressions have been written down in [51], in which the authors studied a free-ﬁeld realization of the BMS3 algebras in terms of a βγ-ghost system. In their construction, the βγ system generates a BMS3 algebra with central charges (c1,c2) = (26, 0), and the value of c2 may be changed by a twist.

2.1 Sugawara construction

Regarding the BMS3 algebra as a Galilean-Virasoro algebra, the authors of [31] obtained the operators W (z), with central charge c2 = 0, from the same Lie algebra they built the energy-momentum tensor T (z) through a Sugawara construction. Now, we would like to show that, even when we assume that the underlying Lie algebras that generate the operators T (z) and W (z) are diﬀerent, and we shall see that depending on the properties of the Lie algebra, it is possible to generate a nontrivial central charge c2. Suppose that among the vertex operators there exists a complete set of currents

a a a a g = {j (z) ≡ V (|v i; z)|L0|v i = |v i , a =1, ··· , q} . (2.13)

Using (2.1), we write their OPE as KZab Cab jc(ζ) ja(z)jb(ζ) ∼ + c + (jajb)(ζ) , (2.14) (z − ζ)2 (z − ζ)

ab ab where K, Z and C c are constants. The current algebra reads a b ab c ab [jm, jn]= C cjm+n + mKZ δm+n,0 . (2.15)

a b ab c We do not assume that the zero mode algebra g0,[j0 , j0] = C cj0, is semisimple. Therefore, the structure constants ab ba ab C c = −C c do not need to be completely antisymmetric, and we do not impose that Z is proportional to the Cartan- Killing form ab 1 ae bc κ = C cC e , (2.16) N0

4 where N0 is the Dynkin index. Finally, the nondegeneracy of the Cartan-Killing form [52] is not necessary in this setting. In other words, we allow the current algebra to be as arbitrary as possible. We try to build the operator W (z) from a Sugawara construction deﬁned by the currents g as

W (z)= γXab(jajb)(z) . (2.17)

Evidently, Xab = Xba and we allow the possibility Xab ∝ κab, but we do not take it for granted. The properties of the matrices Xab and Zab are determined from the OPEs

W (z)W (ζ) and T (z)W (ζ) (2.18) and the constraints imposed by the BMS(3) algebra:

§1) It is easy to see that the OPE (2.1) implies that the ﬁelds ja(z) are primaries. Then, their transformation is

ja(ζ) ∂ja(ζ) T (z)ja(ζ) ∼ + . (2.19a) (z − ζ)2 (z − ζ)

Now, we are able to calculate T (z)W (ζ) as

γXab dx T (z)W (ζ)= T (z)ja(x)jb(ζ)+ T (z)ja(x)jb(ζ) 2πi x − ζ Iζ (2.20) γKXabZab 2W (ζ) ∂W (ζ) ∼ + + , (z − ζ)4 (z − ζ)2 (z − ζ)

ab ab and we see that c2 =2γKX Z .

§2) Now, we calculate the following OPE:

γXab dx W (z)jc(ζ)= ja(z)jb(x)jc(ζ)+ ja(z)jb(x)jc(ζ) . (2.21a) 2πi x − z Iz Then,

2γKXabZadCbc 2γ KZcaXae + XabCbc Cad je(ζ) W (z)jc(ζ) ∼ d + d e (z − ζ)3 (z − ζ)2 (2.21b) 2γ(KXabZbc∂ja(ζ)+ XabCbc (jajd)(ζ)) + d , (z − ζ)

that is, λc λca ja(ζ) λca ∂ja(ζ)+ λbac(jajc)(ζ) W (z)jc(ζ) ≡ (1) + (2) + (2) (3) . (2.21c) (z − ζ)3 (z − ζ)2 (z − ζ) From this expression it is easy to see that the last term is a bit problematic, since the OPE W (z)W (ζ) will give recursively cubic terms in the currents. For semisimple Lie algebras, the complete antisymmetry of the structure constant implies that this term vanishes trivially. In the present case, if we want to avoid the obvious antisymmetry ab bc choice, the most economical constraint is b X C d = 0. Therefore,

P b abc λ(1) =0 , λ(3) =0 . (2.22)

§3) Using these results, it is easy to compute the OPE:

γXcd dx W (z)W (ζ)= W (z)jc(x)jd(ζ)+ W (z)jc(x)jd(ζ) . (2.23a) 2πi x − ζ Iζ

5 Therefore, γKZabXbcλca 2γXacλcb γXacλcb W (z)W (ζ) ∼ (2) + (2) (jajb)(ζ)+ (2) ∂(jajb)(ζ) , (2.23b) (z − ζ)4 (z − ζ)2 z − ζ

ab bc −3 where the condition b X C d = 0 implies that the term with (z − ζ) vanishes trivially. The condition W (z)W (ζ) ∼ 0 is satisﬁed if K = 0 regardless of the Lie algebra properties, and it gives vanishing central charge ac Pcd db c2 = 0, or if c,d X Z X = 0, that gives nontrivial constraints to the Lie algebras. In the latter case, depending on the underlying details of the Lie algebra, we can generate a nonzero central charge c .1 P 2 We have seen that the condition W (z)W (ζ) ∼ 0 was a bit arbitrary. We can change this OPE taking the condition X = Z−1. Therefore, γK dim(g) 2W (ζ) ∂W (ζ) T (z)W (ζ) ∼ + + (2.24a) (z − ζ)4 (z − ζ)2 z − ζ Kγ dim(g) 2γK γK W (z)W (ζ) ∼ 2γK + W (ζ)+ ∂W (ζ) =2γKT (z)W (ζ) . (2.24b) (z − ζ)4 (z − ζ)2 z − ζ We conclude that W (z)=2γKT (z). In summary, given a stress-energy momentum tensor T (z), possibly, built from a Sugawara approach, we can construct a second quasiprimary ﬁeld W (z) with vanishing central charge c2 provided a nonsemisimple Lie algebra {ja} is given.

2.2 Null states

The complete analysis of generic singular states in the BMS3 invariant field theories, has been performed in [26], and the 2 (2) authors were able to find null vector for special values of the CFT data (h,h ,c1,c2). In this subsection, we review their construction paying special attention to zero-norm states which appear in the representation theory of BMS3 invariant field theories.

We know that the highest-weight representation of a generic Wℓ algebra is deﬁned by

(ℓ) V (ℓ) := span L W |h,h i n ,m ∈ N,n ≥ n ,m ≥ m . (2.25b) h,h  −ni −mj i j i+1 i j+1 j  ni∈~n mj ∈ ~m  Y Y 

As usual, there is a natural L grading 0  

(N) (ℓ) V := span L W |h,h i∈V (ℓ) n + m = N . (2.25c) h,h(ℓ)  −ni −mj h,h i j  ni∈~n mj ∈ ~m i j  Y Y X X 

Then, the Verma module is written as  (N) Vh,h(ℓ) = Vh,h(ℓ) . (2.25d) N Z M∈ In our case, the W-algebra generator has ℓ = 2, and we write the highest-weight states as |hi ≡ |h,h(2)i with (2) L0|hi = h|hi and W0|hi = h |hi. Additionally, there may be states satisfying

(2) ∗ L0|χi = (h + A)|χi , W0|χi = h |χi ,Ln|χi =0 , Wn|χi =0 ∀ n ∈ N , (2.26)

1I would like to thank the anonymous referee for pointing out a mistake in an earlier version of the paper. 2 (2) Their analysis can be easily translated into our notation by the transformation (∆,ξ,C1, C2)theirs ≡ (h,h ,c1/12,c2/12)ours.

6 called null or singular, that are simultaneously primaries and descendants. Null states generate their own module V0, also they are orthogonal to Vh,h(2) and, in particular, to themselves, that is hχ|χi = 0. The important point is that these vectors decouple from the set physical states, and the ﬁnal space is a submodule generated by the quotient Vh,h(2) /V0 [41, 42].

Using these results, we should observe that in the BMS3 algebra, any state of the form |χi = W−n|hi is a zero-norm state, that is, 2 ||χ|| = hv|WnW−n|hi = hv|W−nWn|vi =0 . (2.27)

On the other hand, these states are not, necessarily, annihilated by all the generators Lm, with m ≥ 1. In other words, such states are not necessarily null. Acting with the Virasoro generators, we ﬁnd

L0|χi = L0(W−n|hi) = (n + h)|χi , (2.28a) and (2) c2 2 m 2h + 12 (m − 1) |hi , if m = n Lm|χi = Lm(W−n|hi)=  0 , if m>n> 0 (2.28b) (2)  (m + n)Wm−n|hi , if n>m> 0 Therefore, the condition  c 2h(2) + 2 (m2 − 1)=0 (2.29) 12 needs to be imposed. We need to be careful with the range n>m> 0 since it is not zero, in principle. We say that zero-norm states that are not annihilated by all positive modes are seminull. Therefore, we would like to see under which conditions the semi-null states are singular.

Moreover, it has been shown that if we assume (2.29) with c2 6= 0, the null vector is unique, up to a multiplicative k1 kn constant, and it is of the form W−n1 ··· W−nn |hi, and that when c2 = 0, we recover the Verma module of an ordinary CFT with a symmetry given by the Virasoro algebra only [26,36]. In this last case, the Verma module is reducible, and we can build usual null states using the generators L−n. But observe that even in that case we still have the seminull states, and in this section we would like to pay attention to them.

N = 1 : For the ﬁrst level in the Verma module, we have two states

|χ1i = W−1|hi and |χ2i = L−1|hi. (2.30)

2 i) The vector|χ1i = W−1|hi is a zero-norm state, that is ||χ1|| = 0. Moreover, it is easy to see that

(2) L0|χ1i = (h + 1)|χ1i , W0|χ1i = h |χ1i and Wn|χ1i =0 n ≥ 1 . (2.31) In addition,

c2 2 Ln|χ1i = (n + 1)Wn−1 + n(n − 1)δn−1,0 |hi 12 (2.32) (2) ∗ =2δn,1h |hi , ∀ n ∈ N .

We conclude that this state is seminull, unless we impose h(2) = 0. At this point, the condition (2.29) does not give (2) c2 = 0. Then, if h = 0, W−1|vi is null. ii) Consider now |χ2i = L−1|hi . (2.33) 2 As usual, we have ||χ2|| =2h. In addition,

(2) L0|χ2i = (h + 1)|χ2i , W0|χ2i = |χ1i + h |χ2i , (2.34) and (2) Ln|χ2i =2hδn,1|hi , Wn|χ2i =2h δn,1|hi . (2.35) The state is null just in the case h = h(2) = 0.

7 These conditions can be derived from the 2 × 2 Gram matrix3 hh|L L |hi hh|W L |hi G(1) = 1 −1 1 −1 (2.36) hh|L W |hi hh|W W |hi 1 −1 1 −1 whose determinant det G(2) = −4h(2) vanishes just when h(2) = 0. Furthermore, this condition says that if h(2) > 0 we have a nonunitary theory.

N = 2 : At level 2, we can construct a generic vector of the form

2 2 |µ1i = a1W−2 + a2L−2 + a3L−1W−1 + a4W−1 + a5L−1 |hi , (2.37) and imposing the null vectors conditions, we determine constants ai, see [26].

i) The textbook example of null vector at level N =2 3 |µ i = L − L2 |hi , (2.38) 2 −2 2(2h + 1) −1 is seminull in our case, unless h(2) = 0. ii) We also have a seminull state of the form

|µ3i = (W−2 + a0W−1W−1) |hi , (2.39)

2 where a0 is a constant to be determined. Obviously ||µ3|| = 0 and Wn|µ3i = 0 . Additionally (2) L0|µ3i = (h + 2)|µ3i and W0|µ3i = h2 |µ3i . (2.40) Therefore, we need to calculate

Ln|µ3i = Ln(W−2 + a0W−1W−1)|hi c = (n + 2)W + 2 n(n2 − 1)δ n−2 12 n−2,0 (2.41) h c +2a (n + 1)W + 2 n(n2 − 1)δ W |hi . 0 n−1 12 n−1,0 −1 From this expression, we easily see that i ∗ Lm|hi =0 , ∀ n ∈ N , (2.42a) (2) is identically satisﬁed if h = c2 =0 or

(2) c2 6 h2 = − and a0 = (2.42b) 8 c2 which is consistent with (2.29).

Again, these conditions can be derived from the 4 × 4 Gram matrix 2 2 hh|L2L−2|hi hh|L2L−1|hi hh|L2L−1W−1|hi hh|L2W−1|hi 2 2 2 2 2 2 (2) hh|L1L−2|hi hh|L1L−1|hi hh|L1L−1W−1|hi hh|L−1W−1|hi G =  2 2  (2.43a) hh|W2L−2|hi hh|W2L−1|hi hh|W2L−1W−1|hi hh|W2W−1|hi  2 2 2 2 2 2  hh|W1 L−2|hi hh|W1 L−1|hi hh|W1 L−1W−1|hi hh|W1 W−1|hi   so that 4h + c/2 6h 6h(2) 0 (2) (2) 2 (2) 6h 4h(1+2h) 4h (1+2h) 8(h ) G =  (2) (2)  (2.43b) 4h + c2/2 6h 0 0  (2) 2   0 8(h ) 0 0  (2) (2) 5 (2) (2) (2)  whose determinant det G = 192(h ) (c2 +8h ) vanishes when h =0 or h = −c2/8. 3The order is twice the order of a theory with the Virasoro symmetry only.

8 N = 3 : For completeness, at level 3 we have

3 |νi = W−3 + a1W−2W−1 + a2(W−1) |hi . (2.44)

2 It is easy to see that ||ν|| = 0, Wn|νi =0 ∀n> 0, and

(2) L0|νi = (h + 3)|νi , W0|νi = h |νi . (2.45)

Furthermore, c L |νi = (n + 3)W + 2 n(n2 − 1)δ n n−3 12 n−3,0 n c + a (n + 1)W + 2 n(n2 − 1)δ W 1 n−1 12 n−1,0 −2 h c i (2.46) + a (n + 2)W + 2 n(n2 − 1)δ W + 1 n−2 12 n−2,0 −1 h c i +3a (n + 1)W + 2 n(n2 − 1)δ W 2 |hi , 2 n−1 12 n−1,0 −1 o and the expression ∗ Ln|νi =0 ∀ n ∈ N (2.47a)

(2) is, again, identically satisﬁed for h = c2 = 0 or if the following conditions are satisﬁed

(2) c2 6 h3 = − , a1 = and a2 =0 . (2.47b) 3 c2 Obviously, these conditions are consistent with (2.29). Using similar techniques we can determine null states at arbitrary levels.

(2) (2) Remark: Compatibility of all states (and single valuedness of h ) imposes that h = c2 =0. In other words, if one (2) sets the semi-null states to zero, the central charge c2 and the weights h associated to the BMS3 symmetry are both zero. It is easy to see that the quotient Vh,h(2) /V0 is, equivalently, deﬁned as a space where the generators Wm act trivially, that is Wn|vi 7→ 0 ∀n ∈ Z . (2.48)

Therefore, Vh,h(2) /V0 ≃Vh,0 ≃Vh, that is, the ﬁnal module is the Verma module solely generated by {L−n, n ≥ 1}.

(2) 2 ∗ In summary, we have seen that for 24h = −c2(n − 1), n ∈ Z , the Verma module is reducible and that if we want to avoid seminull states, we need to take c2 = 0. In that case, the resulting module is isomorphic to the states of a Virasoro algebra [26, 36]. However, we have two strong hypotheses in our construction4: unitarity and the absence of zero-norm states in the CFT spectrum. Therefore, in order to avoid trivial central charges and W-weights we need to take seriously the existence of seminull states and leave aside the positivity of the Hilbert space.

Additionally, it is obvious that given the level-1 states |χ1i = W−1|hi and |χ2i = L−1|hi we have the following Jordan structure (2) (2) W0|χ1i = h |χ1i , W0|χ2i = h |χ2i + |χ1i , (2.49)

(2) where |χ1i is evidently a seminull state for generic values of h . It is easy to see that this structure also appears for higher-level states; therefore one might try to consider each pair of states, e.g., (|χ1i, |χ2i), as “conjugate” vectors. We return to all these points in section 4 and propose some solutions to the problems mentioned above.

4There is another important hypothesis: the factorization of the CFT into holomorphic and antiholomorphic sectors.

9 3 Correlation functions

We would like to study the constraints imposed by the BMS3 extended symmetry to the correlation functions of a unitary CFT. We will see that the Ward identities for two- and three-point functions trivialize the action of the Wn generators. In other words, the results of this section are equivalent to the following statement:

No-go: It is not possible to deﬁne a unitary ﬁeld theory with extended symmetry given by W (2, 2) and nontrivial actions of the generators Wn.

Let us rewrite the OPE between the algebra generators and primary ﬁelds as

(ℓ) (ℓ) V (W (ℓ)|vi; ζ) V (W |vi; ζ) V (W |vi; ζ) W (ℓ)(z)V (ζ)= 0 + −1 + ··· + 1−ℓ , (3.1a) v (z − ζ)ℓ (z − ζ)−1+ℓ (z − ζ) where 1 1 V (W (ℓ)|vi; ζ) ≡ W (ℓ)V (ζ)= dz W (ℓ)(z)V (ζ) , (3.1b) −n −n v 2πi (z − ζ)n+1−ℓ v Iw and W (1)(z) ≡ T (z) and W (2)(z) ≡ W (z). It is easy to see that the Ward identities are

n hk ∂k hT (z)V1(z1) ··· Vm(zm)i = 2 + hV1(z1) ··· Vm(zm)i (3.2a) (z − zk) z − zk k X=1 n (2) hk W−1 hW (z)V1(z1) ··· Vm(zm)i = 2 + hV1(z1) ··· Vm(zm)i . (3.2b) (z − zk) z − zk k ! X=1 Regularity imposes the conditions 1 1 lim T (z)= and lim W (2)(z)= . (3.3) z→∞ z4 z→∞ z4 Using the Ward identities, we can write the correlators for descendants as

hL−nV (w)V1(z1) ··· Vm(zm)i = L−nhV (w)V1(z1) ··· Vm(zm)i (3.4a) hW V (w)V (z ) ··· V (z )i = W hV (w)V (z ) ··· V (z )i , (3.4b) −n 1 1 m m b −n 1 1 m m 5 where c n n−1 (1 − n)hk 1 L−n = (−1) n − n−1 ∂k (3.4c) (w − zk) (w − zk) k X=1 b n (2) n−1 (1 − n)hk 1 k W−n = (−1) n − n−1 W−1 . (3.4d) (w − zk) (w − zk) k ! X=1 c (2) The modes associated to W (z) are not (at the quantum level) diﬀerential operators. Therefore, given a null vector |χh ,c2 i associated to highest-weight state |h,h(2)i, the constraints that this state imposes to the correlation functions are not, contrary to the usual Virasoro null states, diﬀerential equations. For example, using the results of the previous section, the null states at level 2 and 3 give, respectively, the two conditions

6 2 hχ(z)φ1(w1) ···i = W−2 + W−1 hVh1 (z)φ1(w1) ···i =0 c2 (3.5) 6 hµ(z)φ (w ) ···i = W + W W hV (z)φ (w ) ···i =0 , 1 1 −3 c −2 −1 h1 1 1 2 5We use the notation k W−1hV1(z1) ··· Vk(zk) ··· Vn(zn)i ≡ hV1(z1) ··· W−1Vk(zk) ··· Vn(zn)i .

10 which in present state seem to be completely algebraic.

Let us assume the existence of an SL(2, C) ∪{W−1, W0, W+1} invariant vacuum |0i. Using V (w)= V (|0i; w)= 1 and taking w = 0 we have the equations n n n 2 ∂kGm =0 , (hk + zk∂k) Gm =0 , 2zkhk + zk∂k Gm = 0 (3.6a) k=1 k=1 k=1 Xn X n X n k (2) k (2) k 2 W−1Gm =0 , hk + W−1zk Gm =0 , 2zkhk + W−1zk Gm =0 , (3.6b) k k k X=1 X=1 X=1 where Gm = hV1(z1) ··· Vm(zm)i. The T -Ward identities impose the usual conditions for the one-, two- and three-point functions, and the W -Ward identities impose further restrictions that we shall verify now. The one-point functions are zero, so let us start with the two-point functions.

Two-point functions: Equations (3.6b) can be organized as the linear system

0 11 hV1(z1)V2(z2)i ~ (2) (2) R · V =  h1 + h2 z1 z2 h(W−1V1(z1))V2(z2)i =0 , (3.7) (2) (2) 2 2   2(h z + h z ) z z hV1(z1)(W−1V2(z2))i  1 1 2 2 1 2      (2) (2) 2 (2) (2) and nontrivial solutions exist iff det R = − h1 − h2 (z1 − z2) = 0. Therefore, h1 = h , and we conclude that the two-point functions are c12δh1h2 δ (2) (2) h1 h2 hV1(z1)V2(z2)i = . (3.8) 2h1 z12 (2) This condition also appears in [48,49] for the W3 algebra, except that in the present case the W0 eigenvalues hk must be equal. (2) (2) (2) Given that h1 = h2 ≡ h must be satisfied in order to get a nontrivial correlation function, the first condition in the Ward identity (3.7) is hW−1V1(z1)V2(z2)i = −hV1(z1)W−1V2(z2)i , (3.9a) and the remaining equations form the system

(2) 2h hV1(z1)V2(z2)i + z1hW−1V1(z1)V2(z2)i + z2hW−1V1(z1)V2(z2)i =0 (3.9b) (2) 2 2 2h (z1 + z2)hV1(z1)V2(z2)i + z1hW−1V1(z1)V2(z2)i + z2hW−1V1(z1)V2(z2)i =0 . Putting all these facts together, we ﬁnd

(2) h c12δh1h2 δ (2) (2) h1 h2 hV1(z1)W−1V2(z2)i = . (3.9c) 2h1+1 z12

Three-point functions: Let us turn our attention to the three-point functions. From (3.6b), one gets the linear system

3 k W−1hV1(z1)V2(z2)V3(z3)i =0 (3.10a) k X=1 3 (2) k hk + W−1zk hV1(z1)V2(z2)V3(z3)i = 0 (3.10b) k X=1 3 (2) k 2 2hk zk + W−1zk hV1(z1)V2(z2)V3(z3)i =0 . (3.10c) k X=1

11 It is an underdetermined system: three equations and four unknowns, that is, hV1V2V3i, hW−1V1V2V3i, hV1W−1V2V3i and hV1V2W−1V3i.

i) The ﬁrst condition obviously gives

hV1(z1)V2(z2)W−1V3(z3)i = −hW−1V1(z1)V2(z2)V3(z3)i − hV1(z1)W−1V2(z2)V3(z3)i . (3.11)

ii) Inserting the last expression into the second and third equations of the linear system gives

1 2 (z1 − z3)( W−1) + (z2 − z3)( W−1) hV1(z1)V2(z2)V3(z3)i = (2) (2) (2) (3.12a) = −(h1 + h2 + h3 )hV1(z1)V2(z2)V3(z3)i and

2 2 1 2 2 2 (z1 − z3)( W−1) + (z2 − z3)( W−1) hV1(z1)V2(z2)V3(z3)i = (2) (2) (2) (3.12b) = −2(h1 z1 + h2 z2 + h3 z3)hV1(z1)V2(z2)V3(z3)i , respectively.

iii) From these last conditions, we ﬁnd

(2) h(W−1V1(z1))V2(z2)V3(z3)i≡ ∆(zk; hk )h(V1(z1))V2(z2)V3(z3)i (2) (2) (2) (2) (2) (2) (3.13) C123 2(h1 z1 + h2 z2 + h3 z3) + (z2 + z3)(h1 + h2 + h3 ) = , h12+1 h13+1 h23 z12 z13 z23 where we have used the textbook result for the three-point function [41, 42, 45].

Using these three expressions, we can write the equations (3.6b) as the homogeneous system

∆ 11 hV1(z1)V2(z2)V3(z3)i ~ (2) (2) (2) S · U =  h1 + h2 + h3 + z1∆ z2 z3 hV1(z1)(W−1V2(z2))V3(z3)i =0 , (3.14) (2) (2) (2) 2 2 2   2(h z + h z + h z )+ z ∆ z z hV1(z1)V2(z2)(W−1V3(z3))i  1 1 2 2 3 3 1 2 3     and again, taking the determinant of S and setting it to zero, we have

(2) (2) (2) −4(z2 − z3)(h1 z1 + h2 z2 + h3 z3)=0 , (3.15) which must be satisﬁed for any value of z1,z2, and z3; then,

(2) (2) (2) h1 = h2 = h3 =0 . (3.16)

Therefore, the action of the Wn modes in a unitary CFT algebra is trivialized.

4 Nonunitary CFTs

As we have seen in the previous sections, if assume that the BMS3 invariant CFTs are unitary, we ﬁnd very unpleasant results. First of all, we cannot generate the W (z) operators with nontrivial central charges c2 following a modiﬁcation of the Sugawara construction based on semisimple Lie algebras. Additionally, the null states and the correlation function analysis indicate that the action of this symmetry algebra is trivial. One important aspect of our study is the presence of zero-norm vectors, called seminull, which are not orthogonal to the whole Verma module. We know that in a unitary CFT, null states generate their own Verma module, and that the

12 set of “physical” vectors is drastically reduced. In other words, we take the quotient of the whole Verma module by the space of null states, that is, V(h,c)/V0. Unitarity is, evidently, an important property in fundamental theories, but we must be careful with its role, and not overestimate its importance in general CFTs. First of all, nonunitary theories have their own place in the realm of condensed matter systems [53]. Additionally, unitarity of the vector space of physical states is much more important than the unitarity of the theory itself. For example, the gauge fixed plus ghost action is generally nonunitary (evidently, this action is not obtained from first principles); on the other hand, the BRST cohomology must, evidently, be positive definite. It is not clear at the present stage whether the presence of the seminull states necessarily gives a nonunitary CFT, but let us assume that it is the case. Therefore, we would like to discuss the resulting aspects of nonunitarity in our (2) (N) analysis; that is, we would like to see the results the condition h 6= 0. Consider again the states |λ1 i = W−N |hi and (N) |λ2 i = L−N |hi. The Jordan structure (2.49) is generalized to

(N) (N) (N) (N) L0|λ1 i = (h + N)|λ1 i ,L0|λ2 i = (h + N)|λ2 i (4.1) (N) (2) (N) (N) (2) (N) (N) W0|λ1 i = h |λ1 i , W0|λ2 i = h |λ2 i + N|λ1 i .

2 Z Given an inﬁnite set of nilpotent variables {θn | θn =0 , ∀ n ∈ }, we deﬁne the state

(2) |ΦN i :=|hN ; h + θN i (2) (2) =|hN ,h i0 + θN |hN ,h i1 (4.2a) (N) (N) ≡|λ1 i + θN |λ2 i ,

(N) where in the second line we have written the Taylor expansion in θN . Let us say that |λ2 i is the Virasoro state and (N) |λ1 i is its seminull partner. Hence, we rewrite the Jordan structure (4.1) as

(2) L0|ΦN i = (h + N)|ΦN i , W0|ΦN i = (h + NθN )|ΦN i . (4.2b)

Evidently, we still have states of the form L−~nW− ~m|hi, where N = |~n| + |~m|, but using the commutation relations, we ′ ′ (N ) (N ) ′ can write all these cross terms as descendants of |λ1 i and |λ2 i with N < N in the following way: at each level N, (N) let us deﬁne the operators Wn as W(N) n := Wn + NθN Ln , (4.2c) where the role of the index n =1, ··· ,N will be explained soon. Observe that the state |ΦN i is written as

W(N) |ΦN i := −N |hi . (4.2d)

Let us consider the ﬁrst states. It easy to see that |Φ0i = |hi. At level 1, we have n = 1; therefore,

W(1) |Φ1i = −1|Φ0i = |λ1i + θ1|λ2i , (4.3) where |λ1i = W−1|Φ0i and |λ2i = L−1|Φ0i have conformal weights h1 = h + 1. Therefore

L0|λ1i = h1|λ1i L0|λ2i = h1|λ2i (4.4) (2) (2) W0|λ1i = h |λ1i W0|λ2i = h |λ2i + |λ1i .

We see that the operator W0 cannot be diagonalized and we can say that |λ2i is the Jordan partner of |λ1i. Finally, we (2) already know that L1|λ1i =2h |hi. W(2) W(2) At level 2, we have the operators −1 and −2. From (4.2d), one has

W(2) (2) (2) |Φ2i = −2|Φ0i = |λ1 i +2θ2|λ2 i , (4.5)

13 (2) (2) where |λ1 i = W−2|Φ0i and |λ2 i = L−2|Φ0i. Evidently, there are other states at level 2 in the Verma module. The W(2) W(1) remaining ﬁelds can be built by applying the operator −1 in the state |Φ1i = −1|Φ0i, that is,

W(2) 2 2 −1|Φ1i = W−1 + θ1W−1L−1 +2θ2L−1W−1 +2θ2θ1L−1 |Φ0i . (4.6)

W(2) Therefore, all level-2 Verma module is organized in the multiplet {|Φ2i, −1|Φ1i}. Similarly, the level-3 can be organized (3) (3) (2) in {|Φ3i, W−1 |Φ2i, W−1 W−1 |Φ1i}. All in all, we see that the Virasoro states and their seminull partners can be seen as the (N) (N) (N) building blocks of this construction and that we can organize all Verma module in terms of the set {Wn , |λ1 i, |λ2 i}. Evidently, we have some redundancies to be removed, for example L−nW−m|Φ0i and W−mL−n|Φ0i for N = n + m that are equivalent modulo the state W−N |Φ0i. Finally, using the operator-state map we deﬁne

The similarity between this theory and the structures of the Logarithm CFTs (LogCFTs) [54–56] has been pointed out in [26,34,37,57,58], but the relation between them is not straightforward, and as the results of this section show, the Jordan structure in the present case is associated to the seminull descendant ﬁelds. The BMS3 invariant CFTs are much more similar to the theories considered in [59, 60]. In that case, the authors argued that, contrary to the unitarity CFTs, the existence of zero-norm states does not mean the reduction of the Verma module and that its degeneracy, that is, the presence of seminull states should be compensated by the some extension (or alien) states |χ˜i such that

hχ˜|χi=0 6 , hχ˜|χ˜i=0 6 , (4.8) where |χ˜i is a “partner” of the zero-norm state |χi. Then, for each seminull state |χi, we have an extended vector |χ˜i that generates its own module and needs to be appended to the initial space of states. But this is exactly what we have found in our computations and the only noticeable difference between the present case and [59, 60], is that our zero-norm states are associated with descendant fields, while their construction associates to the seminull states, extension operators that are neither descendant nor primaries. Despite this small difference, the seminull partners (extension operators) are natural features of our constructions, and not mere ad hoc objects.

In particular, it is obvious that the stress-energy tensor T (z) and the BMS3 generator W (z) are descendants. Given the states |Ti = L−2|0i and |Wi = L−2|0i, their vertex operators are

T (z) := V (|Ti; z) and W (z) := V (|Wi; z) . (4.9)

Therefore, we have the pair T (z) := W (z)+ θT (z) , (4.10) and we see that the ﬁelds (T (z), W (z)) belong to the same “multiplet”.

We conclude that the symmetry transformations generated by the BMS3 algebra must be considered simultaneously. As we have seen in the previous section, the W-ward identities impose severe constraints to our theory, namely h(2) = 0, and now we understand this fact as a consequence of our naive attempt to consider the symmetries associated to T (z) and W (z) as independent transformations. Consequently, we need to modify the Ward identities.

Let us start modifying the OPE W (z)Vh(ζ), that is

h(2)V (ζ) (W V )(ζ) + (∂V )(ζ) W (z)V (ζ)= h + −1 h h ; (4.11) h (z − ζ)2 (z − ζ)

14 hence, the operator (3.4d) becomes

n (2) n−1 (1 − n)hk 1 k W−n = (−1) n − n−1 ( W−1 + ∂k) . (4.12) (w − zk) (w − zk) k ! X=1 c Again, this is similar to the LogCFT, where we modify the Ward identity by a nilpotent operator [55, 56]. Let us reconsider the expressions (3.4b) and (3.6b). Using (4.12); imposing the invariance of the vacuum under {Ln, Wn|n = −1, 0, +1}, that is W−nGm = 0, and ﬁnally using the equations (3.6a), we have

n c k W−1Gm =0 (4.13a) k=1 Xn (2) k hk − hk + W−1zk Gm =0 (4.13b) k=1 Xn (2) k 2 2zk(hk − hk)+ W−1zk Gm =0 . (4.13c) k X=1 With this new identity, we can repeat the computations of the previous section for the two-point functions to see that they (2) are not modified. For the three-point functions we find the condition hk = hk. The nice aspect now is that we can compute four-point functions using the usual technology we have in any CFT with Virasoro symmetry. The four-point function is written in terms of conformal blocks z z G (x)= Cp Cp F(p|x) , x = 12 34 , (4.14) 4 12 34 z z p 13 24 X and the crossing symmetry can be imposed via the bootstrap approach [41, 42]. Evidently, the conditions imposed by the W-Ward identities are “essentially” algebraic, given that we do not have a clear understanding of the nature of the operators Wn in a general W algebra. We have dealt with the existence of the seminull states by a simple modification of the W-Ward identity. But evidently, we need to get rid of the proper null states, since these fields must be absent in the final Verma module. In practice, we are very familiar with these null states, and we know from many standard examples, see [41, 42, 45], that these singular vectors give further conditions to the correlation functions. For example, using the state (2.37) and supposing that it is indeed null, its associated vertex operator χ(z) can be inserted in a generic correlation function. Then,

2 2 a1W−2 + a2L−2 + a3L−1W−1 + a4W−1 + a5L−1 hV (w)V1(z1) ··· Vm(zm)i =0 . (4.15) (2) 2 Consider now that the Wc weightb is given byb 24ch = −cc2(n − 1)b for some n ∈ Z; then null vector is unique [36]. For a particular example, we assume that n = 2; then, the condition

6 W + W 2 hV (w)V (z ) ··· V (z )i = 0 (4.16) −2 c −1 1 1 m m 2 is deﬁned by the state (2.39). c c

5 Conclusions and further directions

In this paper, we explored the equivalence BMS3 ≃ W (2, 2) to remark some aspects of meromorphic CFTs with extended chiral symmetries given by the BMS3 algebra. Initially, we tried to study the structure of these generic theories using the

15 results and insights of unitary theories, for example, the Sugawara construction. The drawback of this approach is that it the algebraic aspects of the nonsemisimple Lie algebra are, at the present stage, very elusive. To study these theories, we have seen that there are certain zero-norm vectors that are not necessarily annihilated by the modes {Ln, Wm}, with n,m > 0. The removal of these states provided us with a relation between the W weights (2) h and the central charge c2. In the particular case of vanishing central charge, c2 = 0, it is well known that the Verma modules for these theories are equivalent to the Verma modules of ordinary CFTs [36]. We turned our attention to the analysis of Ward identities, which are, in a very precise way, the quantum manifestation of the classical symmetries of the system, and they are the cornerstone of any consistent QFT with classical local or global symmetries. Furthermore, the Ward identities for the W symmetry imply that the weights of all states must be zero; therefore, we cannot have a unitary field theory with a nontrivial W symmetry action. Next, we considered nonunitary CFTs and we assumed that seminull states are important in our setting. It is not known whether the presence of seminull states necessarily gives a nonunitary CFT, and it would be interesting to check this condition carefully. In any case, the presence of these states gives a rich structure to the theory, and we organized the Verma module through a nilpotent operator that mixes the BMS3 generators. Moreover, we have shown that in the nonunitary setting the W-Ward identities do not trivialize the action of the symmetry operators. There are many directions for future research. One can try to perform an mathematical analysis of the conditions we considered in section 4, in particular, one can try to understand the similarities between the BMS3 theories and LogCFTs. Additionally, since we do not have an interpretation for the operators Wn, there is no good reason to believe that these operators respect the holomorphic and antiholomorphic factorization of the theory (for example, these operators could act as a multiplication by an antiholomorphic function); therefore one could try to relax this condition. One can try to change the OPE W (z)W (ζ) and see if one can find a consistent CFT with extended algebra, or add- higher spins operators in this construction and see how the Verma module is modified. Remember that in the usual Wℓ algebra approach, one considers just one additional primary operator W (ℓ)(z) for each integer ℓ ≥ 3. Then, the study we considered in this paper is part of a bigger problem: the study of CFTs with extended symmetries associated with several quasiprimary operators of the same conformal dimension.

It is known that the Wℓ algebras are the symmetry of the Aℓ−1 Toda systems, and in particular, the Virasoro algebra W2 ≡ Vir is the symmetry algebra of the simplest Toda system, the Liouville CFT, see [48, 49, 61] and references therein. In [15], the relation between some “ﬂat space limit” of the Liouville theory and BMS3 symmetry has been analyzed; therefore, it would be interesting to see if we can connect the present construction to some deformation of the Liouville CFT. Even more speculatively, it would be interesting to see if conformal blocks of these elusive theories are connected to the partition functions of four-dimensional theories, `ala AGT correspondence [62, 63]. We hope to address some of these points in a future work.

Acknowledgments

I would like to thank Shahin Sheikh-Jabbari, Eoin O´ Colg´ain, Thiago Fleury, and Carlos Cardona for discussions, correspondence, and critical reading of the manuscript. I would also like to thank the anonymous referees for valuable comments, criticism, and suggestions. This work was supported by the Korea Ministry of Education, Science and Technol- ogy, Gyeongsangbuk-Do and Pohang City Independent Junior Research Groups at the Asia Pacic Center for Theoretical Physics.

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