The BMS3 Lie algebra belongs to a one-parameter family of Lie algebras obtained by centrally extending Abelian extensions of the Witt algebra by a tensor density representation. In this paper we call such Lie algebras ĝλ⁠, with BMS3 corresponding to the universal central extension of λ = −1. We construct the Becchi–Rouet–Stora–Tyutin (BRST) complex for ĝλ in two different ways: one in the language of semi-infinite cohomology and the other using the formalism of vertex operator algebras. We pay particular attention to the case of BMS3 and discuss some natural field-theoretical realizations. We prove two theorems about the BRST cohomology of ĝλ⁠. The first is the construction of a quasi-isomorphic embedding of the chiral sector of any Virasoro string as a ĝλ string. The second is the isomorphism (as Batalin–Vilkovisky algebras) of any ĝλ BRST cohomology and the chiral ring of a topologically twisted N = 2 superconformal field theory.

Topics
Lie algebras, Algebraic structures, Symmetry algebra, Conformal field theory, BRST cohomology, BRST quantization, String theory, Superconformal algebras

Two-dimensional conformal field theories (2D CFTs) have been of immense physical and mathematical interest ever since they were discovered to appear as worldsheet descriptions of string theory and various condensed matter systems. The rigorous algebraic formulation of 2D CFT led to the birth of vertex operator algebras,1 whose significance in mathematics first became prominent due to the construction of the monster vertex algebra by Frenkel et al.2 and its usage in the proof of the monstrous moonshine conjecture by Borcherds.3 

The seminal paper4 by Belavin et al. not only revealed the crucial role played by the Virasoro algebra in such theories, but also generated a huge amount of interest in the study of extended conformal algebras. These are the symmetry algebras of field-theoretic extensions of 2D CFTs, obtained by adding a set of fields of some conformal weights, which contain the Virasoro algebra as a subalgebra. The most celebrated and well-known examples of these are the affine Kac–Moody algebras and the superconformal algebras, which lie at the heart of Wess–Zumino–Witten (WZW) models5 that describe strings propagating on Lie groups and superstrings, respectively. However, the resulting symmetry algebra after the extension need not be a Lie algebra; the most notable example of this is the W3 algebra, introduced by Zamolodchikov.6 The Becchi–Rouet–Stora–Tyutin (BRST) cohomology of the W3 algebra was studied in detail by Bouwknegt et al.7 

In recent years, there has been an increased interest in studying certain Abelian extensions of the Witt/Virasoro algebras and their corresponding representation and field theories. In particular, from the field theory perspective, the extension of 2D CFTs by a spin-2 quasiprimary field M(z) which has regular operator product expansion (OPE) with itself has garnered attention, as the symmetry algebra of one such field theory, known as the three-dimensional Bondi-Metzner-Sachs (BMS3) algebra, was shown to appear as the symmetry algebra of the tensionless closed bosonic string worldsheet.10–12 In mathematics, (a special case of) this was introduced as the W(2, 2) algebra by Zhang and Dong.13 This led to various works on the representation theory of the BMS3 algebra.14–18 More recently, the representation theory of numerous other extensions of the Virasoro algebra have been studied. Some examples are the (twisted) Heisenberg–Virasoro algebra,19 mirror Heisenberg–Virasoro algebra,19,20 N = 1 super-BMS algebra,21 BMS Kac–Moody algebra and Ovsienko–Roger algebra.22 What is noteworthy about all these algebras is that they are all special cases (or their minimally supersymmetric extensions) of the Lie algebra W(a,b)⁠, which is constructed via the semi-direct sum of the Witt algebra W and its tensor density modules I(a, b), for (a,b)∈C2 (see Refs. 23 and 24). Explicitly, W(a,b)=⨁n∈ZCLn⨁CMn with Lie bracket
(1.1)
For our purposes, we restrict ourselves to a,b∈Z⁠, and since W(a+a′,b)≅W(a,b) for all a′∈Z⁠, it suffices to consider the one-parameter family of Lie algebras gλ:=W(0,λ)⁠, where λ∈Z⁠. (This one-parameter family of algebras was shown to appear as the near horizon symmetry algebra of non-extremalblack holes in Ref. 25. In their work, the parameter s comes from a choice of boundary condition, where s = −λ.)

In this paper, we consider 2D CFTs whose symmetry algebra is gλ⁠. As usual, this statement is merely a reformulation of the Lie algebra gλ in terms of fields in one formal variable admitting certain OPEs. Nonetheless, with such a field-theoretic formulation, we may then consider its Becchi–Rouet–Stora–Tyutin (BRST) quantisation. In this paper, we explicitly build the BRST operator for all gλ field theories as the semi-infinite differential of the Lie algebra gλ⁠.

The notion that the BRST cohomology of various 2D CFTs coincides with the semi-infinite cohomology of the underlying symmetry Lie algebra, relative to its centre, is not a new one, particularly since the work of Frenkel et al.,26 in which they explicitly computed the spectrum of the bosonic string as the (relative) semi-infinite cohomology of the Virasoro algebra with values in the Fock module. This interplay between the physical spectrum of states and an algebraic structure corresponding to Lie algebras has been a useful and powerful tool in both mathematics and physics. The purely algebraic approach to BRST cohomology through the construction of the semi-infinite wedge representation of the Lie algebra at hand is very instructive in letting us build free field realizations of that algebra in terms of fermionic bc-systems. This is done by repackaging the findings from semi-infinite representation theory into generating functions of one variable. While this formulation may simply be regarded as a trick which allows one to use OPEs instead of the cumbersome infinite sums in semi-infinite cohomology theory to perform mathematical computations, it also admits a natural field-theoretic interpretation, where those generating functions are precisely the fields that generate the 2D CFT of the bc-systems. On the other hand, these bc-systems would be the Faddeev–Popov ghosts that one would introduce in the BRST quantisation of a 2D CFT whose symmetry algebra is the (central extension of the) Lie algebra with which we started.

This approach of recasting mode algebras as fields will underpin the entire paper, which is structured as follows. Section II will introduce definitions for 2D CFTs and show the field-theoretic formulation of gλ⁠. In Sec. III, we review the notion of semi-infinite cohomology of Z-graded infinite-dimensional Lie algebras in general, and then construct the semi-infinite wedge representation of gλ explicitly to demonstrate how fermionic bc-systems are simply the field-theoretic formulation of semi-infinite cohomology. We then construct the BRST operator of gλ field theories, which indeed coincides with the semi-infinite differential of gλ⁠, and requires that the matter sector of such theories have (Virasoro) central charge 26 + 2(6λ2 − 6λ + 1) to be square-zero. Using these constructions, we present a set of embedding theorems in Sec. IV, which relate the relative semi-infinite cohomology (a.k.a. BRST cohomology) of the Virasoro and gλ algebras and the chiral ring of a twisted N = 2 superconformal field theory (SCFT). These results present isomorphisms of homotopy Batalin–Vilkovisky (BV) algebras, which are “stronger” than the isomorphisms of graded vector spaces which one would expect from the semi-infinite analogue of Shapiro’s lemma (proven by Voronov in Ref. 27). In Sec. V, we study the special case of gλ=−1 in detail, since it is isomorphic to the (centreless) BMS3 algebra. We go beyond semi-infinite representations and argue why a square-zero BRST operator for the BMS3 algebra cannot exist for cM ≠ 0. We present two physical realizations of chiral BMS3 field theories - the ambitwistor string8,9 and the gauged Nappi–Witten string.28 Finally, in Sec. VI, we summarise our results for generic λ∈Z⁠, address the implications of these results for BMS3 field theories (i.e., the case λ = −1), with reference to the caveat of BMS3 symmetry appearing in a non-chiral manner in tensionless strings, and present some ongoing and potential extensions to our work.

In this section, we set up some notation and terminology with regards to 2D CFTs and introduce the class of Lie algebras W(a,b)⁠. We then show how to set up extended CFTs which admit gλ≅W(0,λ) symmetry, which will be the focus of this paper.

We refer the reader to the standard Refs. 2 and 29–34 on this subject for more details.

Definition 2.1.

A meromorphic 2D CFT or vertex operator algebra (VOA) is given by the following data:

  • A complex vector superspace V=⨁n∈ZVn0̄⊕Vn1̄⁠, which has a Z2-grading and a Z-grading that are compatible with each other, spanned by elements known as states. The Z-grading is known as conformal weight.

  • An injective linear mapping sending a state A ∈ V to a field A(z) = End V[[z, z−1]] known as the state-field correspondence. For all A ∈ Vh, A(z) = ∑Anz−n−h.

  • A linear map ∂: Vh → Vh+1 such that (∂A)(z):=ddzA(z)⁠.

  • A set of bilinear brackets [−, −]n: V ⊗ V → V, labelled by n∈Z⁠, defined by the operator product expansion (OPE),
    (2.1)
    where the summation index n ≪ ∞ indicates that there are only a finite number of singular terms (those with n > 0) in the sum, satisfying
    • Identity: There is a distinguished state called the vacuum 1∈V00̄ such that limz→0A(z)1=A for all A ∈ V and ∂1=0⁠. Thus, for all A ∈ V,
      [1A]n=A,n=0,0otherwise.
    • Commutativity: For all A, B ∈ V,
      [AB]n−(−1)|A‖B|+n[BA]n=∑l⩾1(−1)l+1l!∂l[AB]n+l.
    • Associativity: For all A, B, C ∈ V,
      [[AB]pC]q=∑l⩾q(−1)l−qp−1l−q[A[BC]l]p+q−1−(−1)|A‖B|∑l⩾1(−1)p−lp−1l−1[B[AC]l]p+q−l.
    The normal ordered product of two fields A(z) and B(z) is their n = 0 bracket and is denoted [AB]0 ≕ (AB). For convenience, we denote nested normal-ordered products as (ABC) ≔ (A(BC)).

  • A Virasoro element T=∑n∈Z Lnz−n−2∈V20̄ such that

    • [T T]n>4 = 0, [TT]4=12cL1⁠, [T T]3 = 0, [T T]2 = 2T and [T T]1 = ∂T, where c∈C is known as the central charge.

    • For all A ∈ Vh, [T A]2 = hA and [T A]1 = ∂A. If in addition [TA]n⩾3 = 0, then A(z) is a primary field with conformal weight h. If [T A]3 = 0, but for some n > 3, [T A]n ≠ 0, then A(z) is a quasiprimary field.

From this point, meromorphic 2D CFTs as given in this definition will just be referred to as CFTs. CFTs admit rich mathematical structure and thus a myriad of useful properties to probe them. Some essential ones are listed and derived in Subsection  1 of the  Appendix. In particular, (P8) together with (D5) imply that the modes {Ln}n∈Z obey the Virasoro algebra, and that V contains a graded representation of the Virasoro algebra with central charge cL, where the grading element L0 ∈ End V is diagonalisable [due to (D1)]. If the field theory is generated by just T, we are in the usual case of non-logarithmic, meromorphic 2D CFT, whose symmetry algebra is that of the modes of the field T(z) (i.e., the Virasoro algebra).

Of course, using the formalism of Definition 2.1, we can consider field theories which are not generated by T(z) alone, but instead by T(z) and an additional set of fields {Wi(z)}i∈I of weight hi, for some finite set I, with OPEs that obey the axioms in (D4). Such field theories are still CFTs, since the generator of the conformal symmetry, T(z), is still one of the generators. The symmetry algebra of any such field theory is then the mode algebra of the fields T(z) and {Wi(z)}i∈I⁠. It will contain the Virasoro algebra as a subalgebra by construction. Hence, such CFTs are called extended CFTs.

We now define a well-known example of CFTs called bc-systems.35,36 In string theory, these often appear in the ghost sector of the theory.

Definition 2.2.
A weight (1 − λ, λ) bc-system is a 2D CFT formed from two bosonic (resp. fermionic) primary fields b(z) and c(z) of weights 1 − λ and λ with OPEs
(2.2)
where ϵ = ±1 if b(z) and c(z) are fermionic (resp. bosonic). The Virasoro element is
(2.3)
with central charge −ϵ2(6λ2 − 6λ + 1). The fields admit the following mode expansions:
(2.4)
where bn and cn are endomorphisms of the underlying vector space of states of the CFT. Given a bc-system, a vacuum state |q⟩ ∈ V of charge q, where q∈Z+1/2 for the Neveu–Schwarz (NS) sector and q∈Z for the Ramond (R) sector, is given by the conditions
(2.5)
The space of states is built from the modes bn and cn that act non-trivially on the chosen vacuum.

Remark 2.3.

Conventionally, bosonic bc-systems are referred to as βγ-systems. We will also adopt this convention. The corresponding space of states with vacuum choice |q⟩ will be called Vqβγ. For the fermionic bc-systems, the space of states will either be referred to as Vbc or Λ∞⋅, the latter notation denoting the space of semi-infinite forms, introduced in the next section.

We will revisit bc-systems when we construct semi-infinite cohomology and discuss BRST quantisation of extended CFTs.

Definition 2.4.
The one-parameter family of Lie algebras gλ:=W(0,λ) has underlying vector space generated by {Ln,Mn}n∈Z and is defined by the Lie bracket
(2.6)
Unless mentioned otherwise, we take λ∈Z⁠. Since there is a surjective Lie algebra homomorphism gλ→W to the Witt algebra, we may pull back the Gelfand–Fuchs cocycle
(2.7)
to gλ⁠. This allows us to centrally extend gλ to ĝλ⁠, for all λ∈C⁠. Explicitly, the Lie bracket on ĝλ is given by
(2.8)

Remark 2.5.

For λ = −1, 0, 1, there exist other possible central charges coming from other central extensions of these Lie algebras (see Ref. 23, Theorem 2.3).

Definition 2.6.
Let ρ:ĝλ→End V be a Z-graded ĝλ-module in the category O (see Definition 3.11), where ρ(L0) ∈ End V is the grading element. Define generating functions,
T(z)=∑n∈Zρ(Ln)z−n−2andM(z)=∑n∈Zρ(Mn)z−n−(1−λ).
These have the following OPEs:
(2.9)
(2.10)
(2.11)
Using (P8), one can show that the above OPEs are equivalent to the commutator of the modes ρ(Ln) and ρ(Mn) obeying the ĝλ algebra (2.8), with the central element cL acting as some multiple of the identity on the space V. For convenience, this multiple is also called cL, which is then referred to as the central charge of the representation (D5). Thus, a ĝλ field theory is any CFT generated by fields T(z) and M(z) which admit the OPEs (2.9)–(2.11). Equivalently, ĝλ field theories are extended CFTs with symmetry algebra ĝλ⁠.

This section aims to elucidate the relationships between the BRST quantisation of extended CFTs whose symmetry algebras are Lie algebras and the relative semi-infinite cohomology of the underlying symmetry algebra. For brevity, we will drop the “relative” once this notion is introduced. We will write down free field realizations of gλ in terms of both fermionic and bosonic bc-systems. As we will show, the former is simply the field-theoretic formulation of the semi-infinite wedge representation of ĝλ⁠; the reason for needing the “hat” will be clarified in this section. It will also enter our expression for the BRST current, whose zero mode (i.e., BRST operator of the ĝλ field theory) coincides with the semi-infinite differential of ĝλ⁠.

We provide a review of semi-infinite cohomology as explained in Refs. 26 and 37. Some key proofs are provided in Subsection 2 of the  Appendix. After building up the framework in general, we show the explicit computations of the semi-infinite wedge representation of gλ⁠.

1. Building the space of semi-infinite forms

Let g=⨁n∈Z gn be a Z-graded Lie algebra over C⁠, with dimgn<∞∀n∈Z⁠. Let g±:=⨁±n>0 gn⁠. Let {ei}i∈Z be a basis for g such that if ei∈gn (for some i,n∈Z⁠), then either ei+1∈gn or ei+1∈gn+1⁠. Let g′=⨁n∈Zgn′ be the restricted dual of g with gn′=gn*=Hom(gn,C)⁠. Let {ei′}i∈Z be the dual basis for g′⁠, where ei′(ej)=δij⁠.

We may define a Clifford algebra Cl(g⨁g′) with respect to the dual pairing ⟨−,−⟩:g′×g→C⁠, defined by x′,x:=x′(x)⁠, as follows. For any x+x′∈Cl(g⨁g′)⁠, we have the following relation between the product “·” of the algebra and the dual pairing:
(3.1)
Polarizing this relation we obtain, for a more general combination of elements,
(3.2)

Definition 3.1.
The space of semi-infinite forms Λ∞⋅ is the space spanned by monomials
(3.3)
where i1 > i2 > ⋯ and ∃N(ω)∈Z such that ik+1 = ik − 1 ∀ k > N(ω).

Definition 3.2.
For all x∈g⁠, x′∈g′⁠, we define the contraction ι(x)∈End Λ∞⋅ and exterior or wedge product ε(x′)∈End Λ∞⋅ through their actions on monomials as follows:
(3.4)
(3.5)
where the hat denotes omission.

The following is the result of a simple calculation.

Lemma 3.3.
For all x,y∈g and x′,y′∈g′, the following (anti)commutation relations hold:
(3.6)

Proposition 3.4.

Λ∞⋅ admits a Clifford module structure over Cl(g⨁g′)⁠.

2. Constructing the semi-infinite wedge representation

Definition 3.5.
Just like with any other Lie algebra, we can define the adjoint representation of g via the linear map ad:g→End(g)⁠,
(3.7)
Similarly, the coadjoint representation of g is given by the linear map ad′:g→End (g′) such that ∀x,y∈g⁠, y′∈g′⁠,
(3.8)
so adx′y′=⟨y′,[x,−]⟩∈g′ indeed.

The most natural guess for a representation ρ:g→End Λ∞⋅ is the generalization of the coadjoint action to semi-infinite monomials:
(3.9)

Proposition 3.6.
The following commutation relations hold for all x,y∈g, y′∈g′:
(3.10)

Equation (3.9) is well-defined except for x∈g0⁠, in which case, the infinite sum does not truncate to a finite one. For a sensible definition of ρ:g→End(Λ∞⋅)⁠, we would like to fix the bad behavior for x∈g0⁠. We start by defining a vacuum semi-infinite form ω0 such that for all x∈gn⁠, y∈g−n where n∈Z\{0}⁠, ρ([x, y])ω0 is proportional to ω0. The standard way to construct such a vacuum is by choosing i0 such that ei0′∈gm0⟹ei0+1′∈gm0+1 and then letting
(3.11)
Hence, ω0 is the ordered wedge product of the dual basis elements spanning ⨁n≤m0gn′⁠, for some m0∈Z⁠. Then for a given ω0, choose a β∈g0′ such that β([g0,g0])=0⁠, and define ρ(x)ω0 = ⟨β, x⟩ω0 for all x∈g0⁠. By demanding that the anti-commutation relations (3.10) hold, we may extend such an action of ρ to all of g⁠. Explicitly,
(3.12)
where we have defined the normal-ordered product with respect to the vacuum ω0 as
(3.13)
This is not ρ as defined in Ref. 26, but we will prefer this version (also used in Refs. 37 and 38)as it simplifies many explicit calculations since it is easier to work with the adjoint rather than the coadjoint action. Note that for all x∈gn and y∈g−n for n ≠ 0, we have
ρ([x,y])ω0=⟨β,[x,y]⟩ω0=−∂β(x,y)ω0,
where ∂ is the differential in Lie algebra cohomology. Hence, the infinite sums have indeed been tamed and, more specifically, ρ(x, y)ω0 is proportional to ω0 up to a factor determined by some coboundary. This is more than just an observation. It is closely related to the extent to which ρ:g→End(Λ∞⋅) fails to be a Lie algebra representation, characterised by the following proposition.

Proposition 3.7

(Ref. 26, Proposition 1.1). There exists a two-cocycle γ∈H2(g) depending on the choice of vacuum ω0 and β such that

  1. γ(gm,gn)=0∀m+n≠0⁠,

  2. [ρ(x), ρ(y)] = ρ([x, y]) + γ(x, y).

If γ is a coboundary, then there exists a choice of β for a given ω0 such that γ=0∈Λ2(g).

Proposition 3.7 tells us that the failure of (3.12) to be a representation is characterised by a cocycle that is non-trivial in cohomology. This is in line with the fact that any failure that is characterised by a coboundary could be absorbed by an appropriate choice of β. Also note that γ is non-zero only on the zero-graded part of g×g⁠. This is also to be expected; for x∉g0⁠, (3.12) reduces to the generalised coadjoint action, so one should not expect it to fail as a representation outside the zero-graded part.

If γ is a representative of a non-trivial class in H2(g)⁠, we are obstructed from making Λ∞⋅ a g-module. This obstruction can only be overcome by instead working with ĝ⁠, the central extension of g constructed using γ. This allows us to view γ as a coboundary instead, which can then be set identically to zero by an appropriate choice of β, as stated in Proposition 3.7.

3. Gradings

There exist two natural gradings one can define on Λ∞⋅⁠.

Definition 3.8.
∀x∈g⁠, x′∈g′⁠,
(3.14)
Fixing Degω0∈Z⁠, this defines the grading Deg on Λ∞⋅⁠. We will sometimes refer to this grading as the ghost number, the name being motivated by BRST quantisation in physics.

Since Deg ρ = 0, this makes Λ∞m:={ω∈Λ∞⋅|Degω=m} a g-module ∀m∈Z⁠.

Definition 3.9.
∀x∈gn⁠, x′∈gn′⁠,
(3.15)
Fixing degω0∈Z⁠, this defines the grading deg on Λ∞⋅⁠. In the context of CFT, this is referred to as the conformal weight.

Let Λ∞m;n:={ω∈Λ∞m|degω=n} and Λ∞⋅;n:={ω∈Λ∞⋅|degω=n}⁠. For all x∈gk⁠, ρ(x):Λ∞m;n→Λ∞m;n+k⁠. Hence, deg makes Λ∞m and Λ∞⋅ graded g-modules.

Definition 3.10.

The category O0 comprises graded g-modules M=⨁n∈Z Mn such that dimMn<∞ and for all n > n0, dimMn=0⁠, for some n0∈Z⁠.

Regardless of how deg ω0 is fixed, the structure of Λ∞⋅ and the construction of deg is such that dimΛ∞*;n<∞ and is zero for all n > n0 for some n0∈Z⁠. Hence, Λ∞*;n∈O0⁠.

Definition 3.11.

The category O⊃O0 comprises graded g-modules M=⨁n∈Z Mn such that the g+-submodule {U(g+)v|v∈M}⁠, where U(g+) denotes the universal enveloping algebra of g+⁠, is finite dimensional for any v∈M⁠. We often abbreviate this last condition by saying that the g+-action is locally nilpotent.

4. Semi-infinite complex

Consider an arbitrary graded g-module M∈O0 with representation π:g→EndM⁠. Let deg v = n for all v∈Mn⁠. Defining deg(v ⊗ ω) ≔ deg v + deg ω turns M⨂Λ∞⋅ into a Z-graded vector space, with each graded subspace being finite dimensional. Then θ:g→End(M⨂Λ∞⋅) given by θ(x) = π(x) + ρ(x) makes M⨂Λ∞⋅ a module in category O0⁠.

Definition 3.12.
The semi-infinite differential d is given by
(3.16)

Proposition 3.13.

d2 = 0.

This can be proven by using the result by Akman38 that the statement of Proposition 3.13 is equivalent to the representation θ:g→End(M⨂Λ∞⋅) being given by
(3.17)
Furthermore, the proof of the nilpotence of the semi-infinite differential is one of the most illuminating examples for highlighting the computational power of the OPE-oriented field-theoretic formulation of semi-infinite cohomology when working with specific examples of g⁠. In our case, we will be working with g=gλ⁠, and the nilpotence of the semi-infinite differential, which in the relative subcomplex to be defined below will be referred to as the BRST operator, is shown by an OPE computation that is much simpler than that done with the mode expansion (3.16).

Definition 3.14.
{M⨂Λ∞⋅,d} is a (graded) complex
⋯ ⟶d M⨂Λ∞m−1 ⟶d M⨂Λ∞m ⟶d M⨂Λ∞m+1 ⟶d ⋯
and the corresponding cohomology H∞⋅(g;M) is known as the semi-infinite cohomology of g with values in M⁠. Explicitly,
(3.18)

The differential raises Deg by 1 and leaves deg unchanged, so one can consider the complex for each deg too
⋯ ⟶d M⨂Λ∞m−1n ⟶d M⨂Λ∞mn ⟶d M⨂Λ∞m+1n ⟶d ⋯.
Then H∞m(g;M)=⨁n∈ZH∞m;n(g;M)⁠, where
(3.19)
As mentioned previously, what we refer to as “semi-infinite cohomology” is actually relative semi-infinite cohomology, which we define next.

5. The relative subcomplex

Let h⊂g0 be a subalgebra. We define a subspace
(3.20)
Equation (3.17) implies
θ(x)w=0⟺dι(x)+ι(x)dw=ι(x)dw=0∀w∈C∞⋅(g,h;M).
Consequently, for any w∈C∞⋅(g,h;M)⁠, ι(x)dw = 0 and θ(x)dw=dι(x)+ι(x)ddw=0⁠, so dC∞⋅(g,h;M)⊆C∞⋅(g,h;M)⁠.

Definition 3.15.
Let
C∞m(g,h;M):={w∈M⨂Λ∞m|ι(x)w=θ(x)w=0∀x∈h}.
The subcomplex relative to h is the complex {C∞⋅(g,h;M),d}
⋯ ⟶d C∞m−1(g,h;M) ⟶d C∞m(g,h;M) ⟶d C∞m+1(g,h;M) ⟶d ⋯.

The cohomology of this relative subcomplex is denoted H⋅(g,h;M)⁠.

Lemma 3.16.
When h=z, the centre (or a central subalgebra) of g, acts on M by scalars, H∞⋅(g,z;M) is non-trivial only if
π(z)=−ρ(z)=−⟨β,z⟩∀z∈z.

In this paper, when g is the symmetry algebra of an extended CFT, we call H⋅(g,z;M) the BRST cohomology of the g field theory, where π:g→M obeys Lemma 3.16 and M is referred to as the matter sector of the g field theory. Note that Lemma 3.16 is equivalent to that of anomaly cancellation setting the central charge of the matter sector, or the critical dimension, of string theories (e.g., when g=Vir⁠). This will be made clearer in Subsection III B dedicated to working through the construction of the semi-infinite cohomology of gλ⁠.

Let gλ=⨁n∈Z (gλ)n⁠, where (gλ)n=CLn⨁CMn and likewise for the restricted dual gλ′⁠. We choose the ordered basis
(3.21)
We now build the representation ρ:gλ→End(Λ∞⋅) according to (3.12) using the basis (3.21), β = 0, and the normal-ordering prescription dictated by the vacuum semi-infinite form
(3.22)

Lemma 3.17.
Using the relations
adLn Li=(n−i)Ln+i,adLn Mi=−(i+λn)Mn+i,adMn Li=(n+λi)Mn+i,adMn Mi=0,
we have the following:
(3.23)
(3.24)

Corollary 3.18.

For the choice β = 0, ρ(Ln⩾0)ω0 = ρ(Mn⩾0)ω0 = 0.

With (3.23) and (3.24) at hand, we proceed to compute the failure of ρ:gλ→End(Λ∞⋅) to be a representation. That is, we compute the 2-cocycle in Proposition 3.7 and check whether it can be made identically zero by an appropriate choice of β. If this cannot be done, then Λ∞⋅ is at best a representation of a central extension of gλ by that 2-cocycle. This simply requires the computations [ρ(Ln), ρ(L−n)] − ρ([Ln, L−n]) and [ρ(Ln), ρ(M−n)] − ρ([Ln, M−n]) acting on ω0, since this is the only way we get something non-trivial. Doing so, we observe that part of this failure is proportional to the Gelfand–Fuks cocycle 2.7, and therefore is not a cohomologically non-trivial contribution that can be absorbed by a different choice of β. Thus, we have the following theorem (see the  Appendix for a proof).

Theorem 3.19.

The space of semi-infinite forms on gλ, Λ∞⋅, is a representation of ĝλ, where ĝλ is the central extension of gλ by the Virasoro cocycle. The central element acts on Λ∞⋅ as ρ(cL)=−26+2(6λ2−6λ+1)IdΛ∞⋅⁠.

Remark 3.20.

Recall from 2.5 that for λ = −1, 0, 1, there exist other possible central charges coming from other central extensions of these Lie algebras. We show that these must be zero for ρ:ĝλ→Λ∞⋅ to be a well-defined representation. See the  Appendix for more details.

1. The semi-infinite wedge representation as bc-systems

We now construct the field-theoretic formulation of ρ:ĝλ→End(Λ∞⋅)⁠. The main result is summarised in the following proposition.

Proposition 3.21.
The ĝ-module structure of the space of semi-infinite forms Λ∞⋅ of ĝλ is equivalent to a free field realization of a ĝλ field theory in terms of two independent bc-systems of weights (2, −1) and (1 − λ, λ), generated by (b, c) and (B, C), respectively. The resulting field theory has ĝλ symmetry generated by the fields
(3.25)
where Tbc and TBC are given by (2.3).

Proof.
We start with the space of semi-infinite forms and make contact with bc-systems as follows. Let
(3.26)
and define the generating functions
(3.27)
(3.28)
These fields, together with their construction of their respective Virasoro elements as described in Definition 2.2, satisfy the properties of weight (2, −1) and weight (1 − λ, λ) fermionic bc-systems, respectively.
The key principle is to construct generating functions
(3.29)
from the fields b(z), c(z), B(z), and C(z). This can be done by looking at ρ(Ln) and ρ(Mn) in more detail. Using (3.26) in Lemma 3.17,
(3.30)
(3.31)
Hence, we may ask: what normal-ordered products of b(z), c(z), B(z) and C(z) have modes (3.30) and (3.31)? The answer is straightforward for Tgh:
(3.32)
where Tbc and TBC are given by 2.3. Thus, the form of Tgh is exactly what one would expect when considering the total Virasoro element of two independent bc-systems.
The answer to the earlier question is not as obvious for Mgh, but the form of Tgh is quite instructive in helping us guess what terms should be there. ρ(Ln) has one term with b and c modes and another with B and C modes. The corresponding field Tgh, whose n-th mode is ρ(Ln), is a linear combination of weight 2 fields formed from the normal-ordered products of one b and one c, and one B and one C. Now consider ρ(Mn). Since only B and c modes appear in (3.31), we infer, based on the form of Tgh in relation to ρ(Ln), that the most general expression for the corresponding field Mgh is
(3.33)
for some a1,a2∈C⁠. A quick computation reveals that a1 = λ − 1 and a2 = −1. Thus,
(3.34)
Now that we have our fields Tgh and Mgh, we compute their OPEs (using Mathematica30,39) and arrive at Eqs. (2.9)–(2.11) with cL = −26 − 2(6λ2 − 6λ + 1), as expected. This shows that our field-theoretic formulation of the semi-infinite wedge representation of ĝλ is consistent and thereby completes the proof.□

2. The BRST quantisation of ĝλ field theories

We are now ready to explain the BRST quantisation of ĝλ field theories in the language of semi-infinite cohomology. This quantisation procedure requires the construction of a square-zero BRST operator, which is precisely the semi-infinite differential (3.16). The resulting BRST cohomology with respect to this operator is then H∞⋅(ĝλ,cL;M)⁠. According to Lemma 3.16, π:ĝλ→EndM needs to be a category O representation with π(cL) = −ρ(cL) = 26 + 2(6λ2 − 6λ + 1) to ensure that the BRST operator is square-zero. The field-theoretic formulation is complete when M is regarded as the matter sector of the ĝλ field theory generated by
(3.35)
These obey (2.9)–(2.11). Tmat is the Virasoro element of this ĝλ field theory with central charge π(cL) that cancels that of the ghost sector of the theory, which is precisely the semi-infinite wedge representation ρ:ĝλ→End(Λ∞⋅)⁠.

We summarise the construction of the BRST operator in the following theorem.

Theorem 3.22.
Let (Tmat, Mmat) as constructed via (3.35) generate a ĝλ field theory with central charge cmat and (Tgh, Mgh) be the fields (3.32) and (3.34). The zero mode d of the BRST current
(3.36)
otherwise known as the BRST operator, is square-zero if and only if cmat = 26 + 2(6λ2 − 6λ + 1). This is the field-theoretic restatement of Lemma 3.16. Define Ttot ≔ Tmat + Tgh and Mtot ≔ Mmat + Mgh. Then,
(3.37)
This is the field-theoretic restatement of the fact that the semi-infinite differential (3.16) obeys (3.17).

Corollary 3.23.

Equation (3.37) along with the fact that L0tot acts semi-simply on M⨂Λ∞⋅ implies that all non-trivial BRST cohomology resides only in the zero-eigenvalue eigenspace of L0tot⁠.

The BRST cohomology of a topological conformal field theory (TCFT) has more structure than just that of a graded vector space. It is actually a Batalin–Vilkovisky algebra (see Refs. 40–42). Therefore we need to be precise when we talk about isomorphisms of BRST cohomology. In this section we will exhibit some BV-algebra isomorphisms of BRST cohomologies. These are “stronger” than vector space isomorphisms. The main idea is to show that these isomorphisms preserve the extra structure (i.e., that of a BV algebra) manifestly, without reference to the exact details of the structure. In this section, we present a couple of embedding theorems relating the BRST cohomology of the Virasoro, ĝλ and the twisted N = 2 superconformal algebras.

The first embedding theorem can be stated as follows.

Theorem 4.1.
Let M be a Vir-module with central charge 26. The BRST cohomology of a CFT with matter sector M (i.e., generated by TM) is isomorphic, as a BV algebra, to the BRST cohomology of a ĝλ field theory with Vqβγ⨂M as its matter sector, where the ĝλ-module Vqβγ is the space of states of the βγ-system with any vacuum choice |q⟩. More succinctly,
(4.1)
as BV algebras.

To prove this, we need two ingredients. Evidenced by its appearance in the theorem statement, the first is the free field realization of ĝλ in terms of an appropriately weighted βγ-system. This can be obtained by a straightforward generalization of the same construction for the BMS3 algebra which was done in Ref. 43.

Lemma 4.2.
There exists a free field realization of every ĝλ field theory in terms of a weight (1 − λ, λ) βγ-system given by
(4.2)
where Tβγ is constructed according to Definition 2.2 and has central charge cL = 2(6λ2 − 6λ + 1).

Proof.

Computing the OPEs of (Tβγ, β) using the properties given by (D4) shows that the embedding (4.2) indeed satisfies (2.9)–(2.11), with the Virasoro central charge cL = 2(6λ2 − 6λ + 1).□

The second ingredient is the Koszul CFT.

Definition 4.3.

A Koszul CFT is spanned by a β̃γ̃-system and a b̃c̃-system, each of weight (1 − μ, μ). Together with the differential dKO:=(c̃β̃)0 and some choice of vacuum |q⟩, a Koszul CFT describes a differential graded algebra spanned by the modes {β̃n,γ̃n,b̃n,c̃n}n∈Z⁠. The cohomology of the differential graded algebra described by any Koszul CFT with respect to the differential dKO, denoted HKO⋅⁠, is called the chiral ring of a Koszul CFT.

Lemma 4.4.
The chiral ring of a Koszul CFT is 1-dimensional. That is,
(4.3)
where |vac〉q:=|−q〉β̃γ̃⊗|q〉b̃c̃⁠.

Proof.

This follows from the Kugo–Ojima quartet mechanism (see Ref. 44).□

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1.

We construct an explicit inner automorphism of the Lie superalgebra structure on End(Vβγ⨂M) which splits d, the BRST operator of ĝλ [zero mode of (3.36)], into that of the Virasoro CFT, dVir, and a Koszul differential dKO. This is analogous to Ishikawa and Kato’s proof for the embedding of the bosonic string into the N = 1 superstring.45 

Let (TM+Tβγ,β) describe the matter sector of the ĝλ field theory, where the βγ-system is of weight (1 − λ) (see Lemma 4.2). Let (Tgh, Mgh) given by (3.32) and (3.34) describe the ghost sector. Starting with (3.36) and using Tmat=Tβγ+TM and (3.32),
(4.4)
where we have defined
(4.5)
(4.6)
The BRST operator of the Virasoro CFT, dVir, the Koszul differential, dKO, and the BRST operator of the ĝλ field theory, d, are the zero modes of JVir⁠, JKO and J⁠, respectively.
Next, we construct the weight 1 bosonic field
(4.7)
whose zero mode generates the similarity transformation, which performs the splitting of d as intended:
(4.8)
Thus, by the Künneth Formula,
(4.9)
as BV algebras. Finally, Lemma 4.4 implies that the only linearly independent state of the weight (1 − λ, λ) βγ and BC-systems, which is non-trivial in dKO-cohomology is the choice of vacuum. This completes the proof.□

The statement of Theorem 4.1 is a result of embedding the semi-infinite complex of the Virasoro algebra with values in some c = 26 Vir-module, M⁠, into the relative semi-infinite complex of the ĝλ algebra with values in the ĝλ module Vβγ⨂M⁠, where {Mn}n∈Z act trivially on M⁠. The embedding is therefore specific to a choice of ĝλ-module. Indeed, this is what the field-theoretic formulations describes too.

On the other hand, the next embedding theorem is not an embedding of complexes but rather a construction of a twisted N = 2 superconformal algebra using the modes of the complex formed from tensoring the semi-infinite complex of ĝλ with the Koszul complex. It holds for any choice of ĝλ-module with central charge such that the BRST operator is square-zero.

The theorem we present in this subsection arose from testing the conjecture made in Ref. 46 and then refined in Ref. 47. This conjecture states that every topological conformal field theory (TCFT) is homotopy equivalent to a twisted N = 2 SCFT.47 It originated from a search for a “universal string theory.”48 We refer the reader to Refs. 47, 49, and 50 for a definition and/or review of TCFTs and N = 2 SCFTs.

Throughout this subsection, let (b̃,c̃,β̃,γ̃) form a Koszul CFT as described in Definition 4.3. Let (b, c, B, C) form the ghost sector of a ĝλ field theory as described by expressions for Tgh and Mgh in (3.32) and (3.34), and let (TM,MM) generate the matter sector with central charge 26 + 2(6λ2 − 6λ + 1). We may then state the theorem as follows.

Theorem 4.5.

The BRST cohomology of a ĝλ field theory given by (TM,MM) is isomorphic as a BV algebra to the chiral ring of a twisted N = 2 SCFT. In other words, for any ĝλ-module with cL = 26 + 2(6λ2 − 6λ + 1), there exists a twisted N = 2 SCFT whose chiral ring is isomorphic to the semi-infinite cohomology H∞⋅(ĝλ,cL;M) of ĝλ relative the center.

The proof simply involves constructing the TCFTs corresponding to ĝλ field theories and Koszul CFTs (presented as the following two lemmas), taking their tensor product, and constructing a twisted N = 2 SCFT whose chiral ring is the tensor product of that of the gλ̂ and Koszul TCFTs. Doing so shows that the conjecture in Ref. 47 holds true for all ĝλ field theories.

Lemma 4.6.
The TCFT given by the fields
(4.10)
describe a ĝλ field theory with matter sector M. Its BRST cohomology is now taken with respect to the operator Q:=[G+,−]1⁠.

Lemma 4.7.
The TCFT given by the fields
(4.11)
is a twisted N = 2 SCFT description of a Koszul CFT. The cohomology is taken with respect to the differential QK:=[GK+,−]1=dKO⁠.

It is worth reiterating that the above lemmas do not present any new information about the ĝλ field theory and Koszul CFT; they are simply repackagings of the existing data of the field theories. Equipped with these lemmas, we are ready to present the proof.

Proof of Theorem 4.5.
On its own, the ĝλ TCFT in Lemma 4.6 cannot be modified using the available fields to obtain a twisted N = 2 SCFT. However, this becomes possible once we tensor the ĝλ TCFT with a Koszul TCFT. From the field content of this larger TCFT, we may assemble
(4.12)
where
(4.13)
By computing OPEs, we can check that these fields indeed describe a twisted N = 2 SCFT. The chiral ring of any twisted N = 2 SCFT is the cohomology taken with respect to the differential QN=2:=[GN=2+,−]⁠. In this case, due to (P3), the total derivative term in GN=2+ does not contribute to QN=2⁠. Thus QN=2=Q+QK indeed. JN=2+ is also a sum of J and JK⁠, up to extra terms that are trivial in QN=2-cohomology due to Lemma 4.4. Hence, all four fields of the twisted N = 2 SCFT are, up to cohomologically trivial terms, equal to the corresponding fields in the tensor product of the ĝλ and Koszul TCFTs. Thus, by the Künneth formula, the chiral ring of the twisted N = 2 SCFT given by (4.12) is isomorphic to the tensor product of the BRST cohomology of the ĝλ field theory given by (TM,MM) and the chiral ring of the Koszul CFT. Since the latter is spanned by just its vacuum state, this completes the proof.□

The universal central extension of gλ when λ = −1 is isomorphic to the BMS3 algebra. This algebra was first introduced to the Lie algebra and VOA literature by Zhang and Dong13 as the W(2, 2) algebra. Since the BMS3 algebra is the symmetry algebra of the closed tensionless string,10,12 λ = −1 is an interesting case to explore in more detail. However, the tensionless string does not admit a holomorphically factorisable field-theoretic description, so the BMS3 algebra does not appear as the symmetry algebra of the chiral part of some full CFT. This is contrary to how the ĝλ algebra appears in our field theoretic descriptions. Nonetheless, the results we present are intrinsic to the Lie algebra, and not the field theory it describes. The field-theoretic formulation only serves as a computational tool for the construction of the semi-infinite cohomology of the Lie algebra and the results that follow. How one wishes to extrapolate these findings on the BMS3 algebra to BMS3 field theories is a separate matter, on which we shed some light in the last section.

Let us remind ourselves that the BMS3 algebra is the vector space ⨁n∈ZCLn⨁CMn with Lie bracket
(5.1)

Now consider a Z-graded BMS3-module M with central charges denoted (cL, cM). An important consequence of Theorem 3.19 is that there does not exist a BRST operator for the BMS3 algebra when cM ≠ 0. The calculation performed for generic λ to construct ρ:gλ→End Λ∞⋅ proves that this must be the case. A closer look at the calculation (see the  Appendix) shows that this is feature is specifically due to {Mn}n∈Z forming an Abelian ideal of gλ⁠, corroborating the fact that the central extension which one needs to use is always the Virasoro one for any value of λ, including the cases λ = −1, 0, 1 where other central extensions exist. Nonetheless, for λ = −1, we present another argument as to why this must be the case by going beyond the construction of the semi-infinite wedge representation of gλ⁠.

Consider a BMS3 field theory generated by some T and M. Purely from the perspective of gauge theory, we would need to introduce two sets of ghosts - one for T and the other for M, of appropriate weights, to gauge the BMS3 symmetry of the theory. These are the weight (2, −1) bc-system and weight (1 − λ, λ) BC-system, respectively, where the latter is also of weight (2, −1) for λ = −1. These should themselves assemble into some Tgh and Mgh which generate BMS3 symmetry via their OPEs. Proposition 3.21 already tells us how to do this for (cL, cM) = (−52, 0). However, we now step away from semi-infinite wedge representations and consider, more generally, any normal-ordered products of the fields b, c, B and C to obtain a bosonic weight 2 field which is quasiprimary with respect to Tgh = Tbc + TBC as given in (2.3). The table below summarises every possible weight 2 bosonic term that one could form from the fields of the two bc-systems.

No. of b, c, B, CTerm
1 None 
2 (b∂c), (∂bc), (b∂C), (∂bC), (B∂c), (∂Bc), (B∂C), (∂BC) 
3 None 
4 (bcBC) 
No. of b, c, B, CTerm
1 None 
2 (b∂c), (∂bc), (b∂C), (∂bC), (B∂c), (∂Bc), (B∂C), (∂BC) 
3 None 
4 (bcBC) 
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Terms with 5 or more b, c, B, C that are bosonic and weight 2 will necessarily vanish. Taking the most linear combination of these fields
(5.2)
and enforcing the OPEs
(5.3)
we obtain the following result.

Proposition 5.1.
There exists a realization of a BMS3 field theory in terms of two weight (2, −1) bc-systems with central charges (−52, −cM) for any nonzero value of cM given by
(5.4)

Proposition 5.1 shows that cM ≠ 0 is achieved through a term that is quartic in the fields of the bc-systems. It is the emergence of this term in the ghost sector of a BMS3 field theory which causes the BRST operator to no longer be square-zero.

Proposition 5.2.
Let (T, M) generate a BMS3 field theory with cL = 52 and cM ≠ 0. Let (Tgh, Mgh) be given by (5.4). Then the zero mode of the BRST current
(5.5)
is not square-zero. Alternatively, there does not exist any BRST differential such that db = M + Mgh.

We present string theories which can be studied as chiral BMS3 field theories from the perspective of the worldsheet. An example would be the bosonic ambitwistor string in D-dimensional Minkowski spacetime.9 Its worldsheet description is given by D weight (0, 1) βγ-systems labelled by a spacetime index μ ∈ {0, 1, …, D − 1}. The Virasoro element Tamb is what we would expect, while the weight 2-primary Mamb is constructed from the normal-ordered products of the weight 1 primaries γμ:
(5.6)
(5.7)
This is a BMS3 field theory with central charge (cL, cM) = (2D, 0). Thus, the 26-dimensional ambitwistor string with (cL, cM) = (52, 0) admits a sensible BRST complex from which we can compute BRST cohomology. This is consistent with both the fact that its spectrum should emerge as the non-relativistic contraction of two copies of the Virasoro algebra12 and that its critical dimension is 26.8,9,12
Another realization one could consider comes from the Nappi–Witten string.28 Consider the complexification of the Nappi–Witten algebra for convenience, generated by P±, I and J. The Lie bracket on these generators is
(5.8)
and the invariant inner product is
(5.9)
and zero otherwise. Without any extra effort, we may consider a higher dimensional analogue (also considered in Ref. 51) given by the Lie algebra nw2n+2 generated by {Pa±}a∈{1,…,n}⁠, I and J, with Lie bracket
(5.10)
and invariant inner product
(5.11)
and zero otherwise. These translate into the OPEs of the corresponding currents
(5.12)
(5.13)
(5.14)
As usual, the modes of each of the weight 1 fields P1±(z),…,P2n±(z),I(z),J(z) obey the affinization nŵ2n+2 of nw2n+2⁠. Hence, we may build a Virasoro element via the Sugawara construction (as done in Ref. 28 for n = 1)
(5.15)
with central charge 2n + 2. Likewise, one can also construct a weight 2 primary
(5.16)
Msug from weight one currents as well. Thus, (Tsug, Msug) given by (5.15) and (5.16) give a realization of a BMS3 field theory via the Sugawara construction applied to the higher dimensional generalization of the Nappi–Witten algebra. This realization has central charges (cL, cM) = (2n + 2, 0). Hence, setting n = 25 indeed gives a BMS3 field theory of central charge cL = 52. Of course, one could also pick any n∈N and tensor this theory with another CFT of appropriate central charge to give a total matter sector Virasoro element with cL = 52.

As an aside, it is interesting to note that the generalised Nappi–Witten algebras nw2n+2 are bargmannian,52 and sigma models constructed from these are WZW models for strings propagating on bargmannian Lie groups. Gauging the symmetry generated by the null element, I, yields a new class of non-relativistic string models where the string propagates on a Lie group with a bi-invariant galilean structure. The full BRST quantisation of such string theories would then require the gauging of the extension of the Virasoro algebra by this weight 1 primary field I(z). This is precisely the algebra ĝλ=0⁠, with I(z) taking the role of M(z). The construction of such non-relativistic string models is part of ongoing work.

By staring at (5.16), one might easily infer that it is actually possible to obtain realizations of ĝλ for all λ ≤ −1 from nŵ2n+2⁠. Explicitly, this realization is given by
(5.17)
Naturally, one could also do this with a weight (1, 0) or (0, 1) βγ-systems and take normal-ordered products of the weight 1 field to construct M. Hence, in general, we can construct gλ≤−1 field theories out of gλ=0 field theories. These are summarised in the following embedding diagram (Fig. 1).
FIG. 1.
A diagram summarising the different explicit constructions of ĝλ≤0 field theories from weight (1, 0) βγ-systems and field theories with nŵ2n+2 symmetry.
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A diagram summarising the different explicit constructions of ĝλ≤0 field theories from weight (1, 0) βγ-systems and field theories with nŵ2n+2 symmetry.

FIG. 1.
A diagram summarising the different explicit constructions of ĝλ≤0 field theories from weight (1, 0) βγ-systems and field theories with nŵ2n+2 symmetry.
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A diagram summarising the different explicit constructions of ĝλ≤0 field theories from weight (1, 0) βγ-systems and field theories with nŵ2n+2 symmetry.

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Coming back to the cases λ = −1 and λ = 0, there exists a construction of a BMS3 field theory with cM ≠ 0 out of a central extension of ĝλ=0⁠, given in Ref. 17, Theorem 7.1.

We have shown that for any chiral ĝλ field theory,

  1. There exists a free-field realization in terms of a weight (1 − λ, λ) βγ-system with central charge cL = 2(6λ2 − 6λ + 1).

  2. There exists a free-field realization in terms of a weight (2, −1) bc-system and a weight (1 − λ, λ) BC-system with central charge cL = −26 − 2(6λ2 − 6λ + 1). This free-field realization is the field-theoretic formulation of the semi-infinite wedge representation of ĝλ and is the ghost sector of the ĝλ field theory.

  3. There exists a square-zero BRST operator if and only if the central of the matter sector cL = 26 + 2(6λ2 − 6λ + 1).

Taking a closer look at the case λ = −1, where ĝλ may be further centrally extended to the BMS3 algebra, we have shown that for any extended chiral CFT, which admits BMS3 algebra symmetry, the BRST quantisation of such field theories demands that the theory have central charge (cL, cM) = (52, 0). This can be proved in two ways:

  1. Algebraically formulate the BRST operator and Faddeev–Popov ghosts as the semi-infinite differential and the semi-infinite wedge representation of the BMS3 algebra. The resulting ghost system forms a chiral BMS3 field theory with (cL, cM) = (−52, 0). The vanishing of the total central charges, required for the BRST operator to be square-zero fixes the matter sector of the BMS3 field theory to have (cL, cM) = (52, 0).

  2. Take a completely field-theoretic approach and construct the most general chiral BMS3 field theory from the bc-systems that appear as ghosts in the gauging procedure of BMS3 symmetry. Doing so, one finds that obtaining a cM ≠ 0 realizations gives rise to quartic term that prevents the resulting BRST operator from being square-zero. It also forbids Mtot from being BRST-exact, for all possible BRST operators. This once again forces the ghost sector to admit BMS3 symmetry with (cL, cM) = (−52, 0), and we obtain the same conclusion as in the first approach.

Now, can we still consider the notion of BRST quantisation of field theories that admit BMS3 algebra symmetry with cM ≠ 0? As it stands, the answer is yes. The findings here only rule out the situations in which cM ≠ 0 is not possible, namely chiral BMS3 theories. Field theories which admit BMS3 symmetry in a manner that is not holomorphically factorisable may still admit some consistent notion of BRST quantisation with cM ≠ 0 through the formalism of full CFTs.53 In particular, we must consider that the BRST cohomology of such theories is not just the semi-infinite cohomology of the underlying symmetry algebra of the theory.

Another point to consider is the notion of “flipped” vacua in CFT (e.g., Refs. 8, 54, and 55). Such vacua can be the starting points of physically valid constructions of tensionless string spectra, as argued by the authors of Ref. 12. More specifically, we need to pay attention to the fact that the Virasoro automorphism Ln → −L−n, c → −c does not lift to a VOA automorphism. Hence, without any further assumptions, theories constructed as a result cannot be studied in a rigorous manner using existing algebraic 2D CFT techniques. In particular, CFTs with a normal vacuum in one sector and a flipped one in the other cannot be probed in this way. Intuitively, this is because boundedness conditions that naturally occur (e.g., highest weight or smooth modules of the underlying symmetry algebra) no longer exist when both of these vacua are put together in the same system. We need an alternative formalism (i.e., some sort of “flipped” VOA) to rigorously encapsulate the modified normal-ordering with respect to these flipped vacua. Perhaps such a formalism exists, using which one can write down a different “BRST quantisation” procedure which admits the existence of a square-zero BRST operator for cM ≠ 0. This would be particularly relevant to the case of tensionless strings, because the BMS3 symmetry not only emerges in a non-chiral manner, but also in a way that mixes the positive and negative modes of the two copies of the Virasoro algebra that appears in the parent tensile closed string theory (i.e., an “ultra-relativistic” contractio12).

Despite the aforementioned caveats preventing us from directly applying our results to tensionless string theory, there do exist string-theoretic realizations of the chiral BMS3 algebra. We have highlighted two such realizations in this paper:

  1. The ambitwistor string, given by (5.6) and (5.7),

  2. the Nappi–Witten string, given by (5.15) and (5.16).

A logical next step would be to seek other physical realizations of this BMS3 algebra, such as in terms of free bosons and fermions. These would be intrinsic constructions, rather than as limits of parent theories such as those considered in Refs. 54, 56, and 57.

Naturally, one could also consider realizing BMS3 using affine Kac–Moody currents. Sugawara constructions which are compatible with Galilean contractions have been explored in Refs. 58 and 59, but again, we can look for more general ones that need not necessarily be compatible with contraction procedures. Doing so, one finds that although the end product is a field-theoretic description of a Lie algebra (i.e. BMS3), the conditions coming from this construction are not Lie algebraic. In particular, M need not be built from an invariant tensor. Nonetheless, one could impose this as a condition and then try classifying all the Lie algebras from which one could build the BMS3 algebra via the Sugawara construction as a result. Some early progress in this regard, such as the construction from the (generalised) Nappi–Witten algebra, looks promising.

Finally, one could consider the BRST quantisation of super-BMS3 field theories. This could mean either the minimally supersymmetric extension of a BMS3 field theory by a spin-3/2 fermionic field or the algebra obtained from the contraction of two copies of N = 1 super-Virasoro algebras.60 

J.M.F.-O. would like to acknowledge a useful conversation with Tim Adamo about the ambitwistor string. J.M.F.-O. has spoken about this work in the Erwin Schrödinger Institute, during his participation at the Thematic Programme “Carrollian Physics and Holography.” He would like to thank the organizers, particularly Stefan Prohazka, for the invitation and the hospitality. J.M.F.-O. has also spoken about this at the Niels Bohr Institute and he would also like to thank Niels Obers and Emil Have for the invitation and their hospitality. G.S.V. would like to thank Arjun Bagchi and David Ridout for insightful comments on an earlier draft of this paper, Yi-Zhi Huang and Yuto Moriwaki for sharing their expertise on full VOAs, and Ross McDonald, Christopher Raymond, and Ziqi Yan for thought-provoking discussions. G.S.V. is supported by a Science and Technologies Facilities Council studentship (Grant No. 2615874).

The authors have no conflicts to disclose.

José M. Figueroa-O’Farrill: Conceptualization (equal); Formal analysis (equal); Supervision (supporting); Writing – original draft (equal); Writing – review & editing (equal). Girish S. Vishwa: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1. Properties of meromorphic 2D CFTs

Listed below are some key properties:

  • For all A, B, C ∈ V,
    [A[BC]p]q=(−1)|A‖B|[B[AC]q]p+∑l⩾1q−1l−1[[AB]lC]p+q−l.
  • For all A ∈ V, [A, −]1 is a super-derivation over all other [−, −]n. That is,
    [A[BC]n]1=[[AB]1C]n+(−1)|A‖B|[B[AC]1]n.
    A special case of this is the derivation [T, −]1 = ∂.

  • [∂AB]n = −(n − 1)[AB]n−1 and [A∂B]n = (n − 1)[AB]n−1 + ∂[AB]n.

  • (∂A)n = −(n + hA)An, where A(z)=∑n Anz−n−hA⁠.

  • The brackets [−, −]n have conformal weight −n. That is,
    [−,−]n:Vi⨂Vj→Vi+j−n.

  • [AB]n=An−hAB⁠.

  • The modes (AB)n in the expansion of the normal-ordered product of A∈VhA and B∈VhB⁠, (AB)(z)=∑n (AB)nz−n−hA−hB⁠, are given by
    (AB)n=∑l≤−hAAl Bn−l+(−1)|A‖B|∑l>−hABn−l Al.
  • There exists a Lie superalgebra structure on the modes, given by
    [Am,Bn]:=Am Bn−(−1)|A‖B|Bn Am=∑l⩾1m+hA−1l−1[AB]lm+n.

Proof of (P1) and (P2).
First, we relabel l → l + q in the first summation of (D4) and l → l + 1 in the second summation to rewrite the (D4) as
(A1)
Now consider the sum
(A2)
Using (A1), we may write each term in the sum above as follows:
[[AB]pC]q=∑k⩾0(−1)kp−1k[A[BC]q+k]p−k+(−1)|A‖B|+p[B[AC]k+1]p+q−k−1,[[AB]p−1C]q+1=∑k⩾0(−1)kp−2k[A[BC]q+1+k]p−1−k−(−1)|A‖B|+p[B[AC]k+1]p+q−k−1,⋮[[AB]1C]p+q−1=[A[BC]p+q−1]1−(−1)|A‖B|[B[AC]1]p+q−1.
Adding each term above after multiplying with the appropriate factor given in (A2), we notice that all terms, except for the k = 0 and k = p − 1 terms in the expansion of [[AB]pC]q⁠, cancel out. Hence, we are left with
(A3)
Rearranging, we obtain (P1). Setting q = 1 then proves (P2) right away.□

Proof of (P3).
From the definition of ∂,
(∂A)(z)B(w):=ddzA(z)B(w)=ddz∑n≪∞[AB]n(w)(z−w)n=∑n≪∞−n[AB]n(w)(z−w)n+1.
But (∂A)(z)B(w) itself admits an OPE
(∂A)(z)B(w)=∑n≪∞[∂AB]n(w)(z−w)n.
Equating equal powers of z − w gives
[∂AB]n+1=−n[AB]n⟺[∂AB]n=−(n−1)[AB]n−1.
In a similar manner, or by using (P2) for ∂ = [T, −]1, we get
[A∂B]n=(n−1)[AB]n−1+∂[AB]n.
□

Proof of (P4).
Using (P2) for [T, −]1, we have
[T∂A]2=[TA]1+∂[TA]2=(hA+1)∂A.
Hence, (∂A)(z) admits a mode expansion
(∂A)(z)=∑n(∂A)nz−n−(hA+1).
On the other hand,
(∂A)(z):=ddzA(z)=ddz∑nAnz−n−hA=∑n−(n+hA)Anz−n−hA−1.
Equating the two mode expansions gives (∂A)n = −(n + hA)An.□

Proof of (P5).
Let A∈VhA and B∈VhB⁠. By (D5), this means [TA]2 = hAA and [TB]2 = hBB. Thus,
[T[AB]n]2=[A[TB]2]n+[[TA]1B]n+1+[[TA]2B]nby(P1)
=hB [AB]n+[∂AB]n+1+hA [AB]nby(D5)
=(hA+hB−n)[AB]nby(P3).
This proves that [−, −]n indeed has conformal weight −n.□

Proof of (P6).
We have, by (D4),
limw→0A(z)B(w)1=A(z)B=∑nz−n−hAAnB.
At the same time,
limw→0∑n[AB]n(w)(z−w)n1=∑nz−n[AB]n.
Equating equal powers of z in the two expressions above gives [AB]n=An−hAB as desired.□

Proof of (P7).
Using (D4), we can write [(AB)C] as
(A4)
Using properties (P5) and (P6),
[(AB)C]=(AB)q−hA−hBC,[A[BC]l]q−l=Aq−l−hA Bl−hBC,[B[AC]l]q−l=Bq−l−hb Al−hAC.
Substituting these back into (A4) gives
(AB)q−hA−hBC=∑l⩾qAq−l−hA Bl−hB+(−1)|A‖B|∑l⩾1Bq−l−hb Al−hAC,∀C∈M.
We may abstract C since it holds true ∀C∈M⁠. Relabelling the first summation with m = q − l − hA and letting n ≔ q − hA − hB gives
∑l⩾qAq−l−hA Bl−hB=∑m≤−hAAm Bn−m=∑l≤−hAAl Bn−l.
Relabelling the second summation with m = l − hA gives
∑l⩾1Bq−l−hb Al−hA=∑m⩾−hA+1Bn−m Am=∑l>−hABn−l Al.
Putting them back together gives us the desired result.□

Proof of (P8).
For all C ∈ V, we may act [Am, Bn] ∈ End V to get
[Am,Bn]C:=Am(BnC)−(−1)|A‖B|Bn(AmC)=Am [BC]n+hB−(−1)|A‖B|Bn [AC]m+hAby(P6)
=[A[BC]n+hB]m+hA−(−1)|A‖B|[B[AC]m+hA]n+hBby(P6)
=∑l⩾1m+hA−1l−1[[AB]lC]m+n+hA+hB−lby(P1)
=∑l⩾1m+hA−1l−1[AB]lm+nCby(P5)  and  (P6).
Since it holds for any C ∈ V, we obtain the desired result.□

2. Some proofs in semi-infinite cohomology

Proof of Proposition 3.4.
Define κ:Cl(g⨁g′)→End(Λ∞⋅) via κ(x + x′) ≔ ι(x) + ɛ(x′). We need to show that (κ(x+x′))2=(x+x′)⋅(x+x′)=⟨x′,x⟩IdΛ∞⋅⁠,
(κ(x+x′))2=ι(x)2+ε(x′)2+ι(x)ε(x′)+ε(x′)ι(x)=[ι(x),ε(x′)]=⟨x′,x⟩IdΛ∞⋅,
where the last equality follows from Lemma 3.3.□

Proof of Proposition 3.6.
We will perform calculations on monomials and the argument extends to all semi-infinite forms by C-linearity.
[ρ(x),ε(y′)]ei1′∧ei2′∧⋯=adx′y′∧ei1′∧ei2′∧⋯+∑k⩾1y′∧ei1′∧ei2′∧⋯∧adx′eik′∧⋯−y′∧∑k⩾1ei1′∧ei2′∧⋯∧adx′eik′∧⋯=adx′y′∧ei1′∧ei2′∧⋯=ε(adx′y′)ei1′∧ei2′∧⋯,[ρ(x),ι(y)]ei1′∧ei2′∧⋯=ρ(x)∑k⩾1(−1)k−1⟨y,eik′⟩ei1′∧ei2′∧⋯∧eik′̂∧⋯−ι(y)∑k⩾1ei1′∧ei2′∧⋯∧adx′eik′∧⋯=∑k⩾1(−1)k−1−⟨y,adx′eik′⟩ei1′∧ei2′∧⋯∧adx′eik′̂∧⋯=∑k⩾1(−1)k−1⟨adxy,eik′⟩ei1′∧ei2′∧⋯∧eik′̂∧⋯=ι(adxy)ei1′∧ei2′∧⋯.
□

Proof of Proposition 3.7.
Naively, the failure of ρ:g→Λ∞⋅ to be a representation, given by [ρ(x), ρ(y)] − ρ([x, y]) = γ(x, y), is encapsulated by some bilinear form γ:g×g→C which is non-zero only if x∈gn and y∈gn⁠, for all n∈Z⁠. We deduce the fact that γ(x, y) = γ(y, x) from the antisymmetry of the commutator and Lie brackets on the LHS. The fact that is encapsulated by a 2-cocycle in γ follows from the Jacobi identity of both g and the associative algebra on End Λ∞⋅ given by the commutator bracket,
(A5)
Hence, γ obeys the cocycle condition. The fact that this failure is only important up to an equivalence class in cohomology is reinforced in the second statement of Proposition 3.7, proved as follows: If γ is a coboundary, there exists α∈g′ such that γ = ∂α and recall that γ(x, y) = ∂α(x, y) = −α([x, y]). In particular, this means that α∈g0′⁠. We currently have a ρ:g→End(Λ∞⋅) that obeys 3.7. Let us define a new representation ρ̃:g→End(Λ∞⋅) given by ρ̃(x):=ρ(x)−⟨α,x⟩⁠. Then,
[ρ̃(x),ρ̃(y)]:=[ρ(x)−⟨α,x⟩,ρ(y)−⟨α,y⟩]=[ρ(x),ρ(y)]=ρ([x,y])+γ(x,y)(by proposition 3.7)
=ρ([x,y])+∂α(x,y)(by definition of γ)
=ρ([x,y])−⟨α,[x,y]⟩=ρ̃([x,y])(by definition of ρ̃).
Any ω0 defines ρ:g→Λ∞⋅ satisfying Proposition 3.7 using some choice of β. If ∃α∈g0 such that γ = ∂α, then one can make the modification β→β̃:=β−α so that γ = 0.□

Proof of Theorem 3.19.
We first recall our choice of basis (3.21), vacuum (3.22) and the statement of Lemma 3.17. Choosing n > 0, it follows that
[ρ(Ln),ρ(L−n)]ω0=ρ(Ln)ρ(L−n)−ρ(L−n)ρ(Ln)ω0=ρ(Ln)ρ(L−n)ω0.
First, we simplify ρ(L−n)ω0 using (3.13) as follows:
ρ(L−n)ω0=−∑i≤1(n+i)ι(Li−n)ε(Li′)−∑i≤1(i−λn)ι(Mi−n)ε(Mi′)+∑i>1(n+i)ε(Li′)ι(Li−n)+∑i>1(i−λn)ε(Mi′)ι(Mi−n)ω0=∑i=2n+1(n+i)ε(Li′)ι(Li−n)+∑i=2n+1(i−λn)ε(Mi′)ι(Mi−n)ω0.
Thus, as expected, the infinite sums in ρ(L−n) truncate to finite ones when acting on the vacuum ω0. Splitting each normal-ordered term in ρ(Ln) according to the normal-ordering prescription (3.13), we may expand ρ(Ln)ρ(L−n)ω0 into the eight terms as done below:
ρ(Ln)ρ(L−n)ω0=∑j≤1∑i=2n+1(n−j)(n+i)ι(Lj+n)ε(Lj′)ε(Li′)ι(Li−n)ω0+∑j≤1∑i=2n+1(n−j)(i−λn)ι(Lj+n)ε(Lj′)ε(Mi′)ι(Mi−n)ω0−∑j≤1∑i=2n+1(j+λn)(n+i)ι(Mj+n)ε(Mj′)ε(Li′)ι(Li−n)ω0−∑j≤1∑i=2n+1(j+λn)(i−λn)ι(Mj+n)ε(Mj′)ε(Mi′)ι(Mi−n)ω0−∑j>1∑i=2n+1(n−j)(n+i)ε(Lj′)ι(Lj+n)ε(Li′)ι(Li−n)ω0−∑j>1∑i=2n+1(n−j)(i−λn)ε(Lj′)ι(Lj+n)ε(Mi′)ι(Mi−n)ω0+∑j>1∑i=2n+1(j+λn)(n+i)ε(Mj′)ι(Mj+n)ε(Li′)ι(Li−n)ω0+∑j>1∑i=2n+1(j+λn)(i−λn)ε(Mj′)ι(Mj+n)ε(Mi′)ι(Mi−n)ω0.
Lines 2, 3, 6 and 7 vanish because the annihilation operators coming from the expansion of ρ(Ln) can freely (up to a sign) commute past the other operators to act on the vacuum without the addition of any other non-trivial terms. Lines 5 and 8 are also vanishing because the non-trivial contribution we get from commuting the annihilation operators past the others is a δj,−n+i term, which is never non-zero for the values i and j take in those sums. Thus, the only non-zero contributions are from lines 1 and 4. After performing the necessary commutations, we are left with
ρ(Ln)ρ(L−n)ω0=∑j≤1∑i=2n+1−(n−j)(n+i)ι(Lj+n)ε(Li′)δj,i−n ω0+∑j≤1∑i=2n+1(j+λn)(i−λn)ι(Mj+n)ε(Mi′)δj,i−n ω0=∑i=2n+1(i−2n)(n+i)ι(Li)ε(Li′)+i+(λ−1)n(i−λn)ι(Mi)ε(Mi′)ω0=∑i=2n+1(i−2n)(n+i)+i+(λ−1)n(i−λn)ω0.
On the other hand, ρ([Ln, L−n])ω0 = 2nρ(L0)ω0 = 0. Thus, any non-zero contribution to [ρ(Ln), ρ(L−n)]ω0 is either from a coboundary term (i.e., a different choice of β), which, according to the form of (3.12), must be proportional to L0′ or a cohomologically non-trivial cocycle term, which implies that we would need to centrally extend our Lie algebra to make ρ a representation. This non-zero contribution is precisely the finite sum we have obtained above, which we now evaluate:
∑i=2n+1(i−2n)(n+i)+i+(λ−1)n(i−λn)=∑i=2n+12i2−2ni−2+λ(λ−1)n2=23n3+3n2+133n−n3−3n2−2+λ(λ−1)n3=−13−2−λ2+λn3+133n=−73−λ2+λn3+133n=−73+16−16−λ2+λn3+136n+136n=−136(n3−n)+16(−1−6λ2+6λ)n3+136n=−136(n3−n)−1122(6λ2−6λ+1)n3+136n+1122(6λ2−6λ+1)n−1122(6λ2−6λ+1)n=−11226+2(6λ2−6λ+1)(n3−n)+112(26−2(6λ2−6λ+1)n.
Thus, we have manipulated [ρ(Ln), ρ(L−n)]ω0 into the above form containing two terms. The first terms is proportional to the Gelfand–Fuks cocycle given by 2.7, while the second term is proportional to n. The presence of the first term indicates that we need to centrally extend gλ to ĝλ using the Gelfand–Fuks cocycle in order to make ρ a Lie algebra representation on Λ∞⋅⁠. The proportionality factor is precisely the action of the new central charge on Λ∞⋅⁠, i.e.,
(A6)
The contribution proportional to n can be absorbed by modifying our initial naive choice of β = 0 to
(A7)
For brevity, we reiterate the fact that ρ:ĝλ→Λ∞⋅ defines a representation of ĝλ on Λ∞⋅⁠:
[ρ(Ln),ρ(L−n)]ω0=ρ([Ln,L−n])ω0⟺−11226+2(6λ2−6λ+1)(n3−n)+112(26−2(6λ2−6λ+1)n=2nρ(L0)ω0+112n(n2−1)ρ(cL).
Substituting (A6) and (A7) above, we see an agreement of the LHS and RHS, demonstrating the validity of our calculations. For generic λ∈Z⁠, in particular, for λ ≠ −1, 0, 1, gλ has no other cohomologically non-trivial 2-cocycles. Hence, it suffices to check the failure of ρ on just the Ln generators.
On the other hand, when λ = −1, 0, 1, we need to perform additional checks on other pairs of generators. Taking λ = −1, we may repeat the exact calculation above for the centreless BMS3 algebra. As expected, we get ρ(cL) = −52, which is in agreement with the general case (A6). However, gλ=−1 admits a second cohomologically non-trivial 2-cocycle
γM(Ln,Mm)=112n(n2−1)δm+n0.
Hence, we need to check if this 2-cocycle γM is required to ensure that ρ does not fail as a representation on Λ∞⋅⁠. Once again, choosing the same basis (3.21), vacuum (3.22) and β as given in (A7) with λ = −1,
[ρ(Ln),ρ(M−n)]ω0=ρ(Ln)ρ(M−n)−ρ(M−n)ρ(Ln)ω0=ρ(Ln)ρ(M−n)ω0.
Using Lemma 3.17 and (3.13),
ρ(M−n)ω0=−∑i≤1(n+i)ι(Mi−n)ε(Li′)+∑i>1(n+i)ε(Li′)ι(Mi−n)ω0=∑i=2n+1(n+i)ε(Li′)ι(Mi−n)ω0.
For clarity, we explicitly write out the four terms in ρ(Ln)ρ(M−n)ω0:
ρ(Ln)ρ(M−n)ω0=∑j≤1∑i=2n+1(n−j)(n+i)ι(Lj+n)ε(Lj′)ε(Li′)ι(Mi−n)ω0−∑j>1∑i=2n+1(n−j)(n+i)ε(Lj′)ι(Lj+n)ε(Li′)ι(Mi−n)ω0−∑j≤1∑i=2n+1(j−n)(n+i)ι(Mj+n)ε(Mj′)ε(Li′)ι(Mi−n)ω0+∑j>1∑i=2n+1(j−n)(n+i)ε(Mj′)ι(Mj+n)ε(Li′)ι(Mi−n)ω0.
By the same arguments, we are only left with line 3:
ρ(Ln)ρ(M−n)ω0=∑i=2n+1i−2n(n+i)ι(Mi)ε(Li′)ω0.
However, this time, the RHS is zero since the contraction and wedge operations that appear in the RHS are not canonically dual to each other, and thereby anti-commute freely. Thus, we do not need to modify gλ=−1 through the addition of the second non-trivial 2-cocycle γM to make ρ a representation on Λ∞⋅⁠. One can perform identical calculations with λ = 0 and λ = 1 as well, since those are the only other values for λ, which dimH2(gλ)>1⁠. It then follows that ρ:ĝλ→End Λ∞⋅ indeed defines a Lie algebra representation for all λ∈Z⁠.□

Proof of Theorem 3.22.
As mentioned earlier, the computation of the square of the BRST operator is one of the most prominent examples of the computational power of the field-theoretic formulation of semi-infinite cohomology. To compute d2 would require the simplification of the product of two infinite sums, each of which is a nested infinite sum of products of the modes bn, cn, Bn and Cn. Such an immensely tedious calculation is greatly simplified as follows. We first notice that for any Y∈M⨂Λ∞⋅⁠,
d2Y=[J[JY]1]1[by(P6)]
=−[J[JY]1]1+∑l⩾10l−1[[JJ]lY]2−lby(P1)
and hence d2Y=12[[JJ]1Y]1⁠. Computing the OPE of J with itself, we get
(A8)
We demand that ker[[JJ]1,−]1=M⨂Λ∞⋅ by enforcing that this map is zero on all generators of the ĝλ field theory. Doing so enforces cmat = 4(7 − 3λ + 3λ2) = 26 + 2(6λ2 − 6λ + 1). This is exactly in agreement with the statement of Lemma 3.16. Notice that this makes [JJ]1 a total derivative, so that by (P3), d2Y=[[JJ]1Y]1=0 for all Y∈M⨂Λ∞⋅ indeed.□

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