Bosonizations of $\widehat{\mathfrak{sl}}_2$ and Integrable Hierarchies

We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.


Introduction
Vertex operators and vertex (operator) algebras are powerful tools for studying infinite-dimensional Lie algebras, their representations, and generalizations [LW,FK,B,FLM,K2,FB,LL]. For any even integral lattice L, one constructs the lattice vertex algebra V L associated to L. When L is the root lattice of a finite-dimensional simply-laced Lie algebra g, this gives the Frenkel-Kac construction of a level one representation of the corresponding affine Kac-Moody algebraĝ (see [FK,K1,K2]).
In this paper we construct a different vertex operator realization of the affine Kac-Moody algebra sl 2 of an arbitrary level k. We start with the Wakimoto realization of sl 2 , which can be viewed as an embedding of the associated affine vertex algebra in the vertex algebra generated by a pair of charged free bosons a + , a − (also known as a βγ-system) and another free boson b (which generates the Heisenberg algebra); see [W, F]. The Friedan-Martinec-Shenker bosonization of a + , a − then gives us an embedding in a certain lattice vertex algebra V L (see [FMS, Wa, A]). The resulting realization of sl 2 has appeared previously in [FS], and is also related to the ones in [FF, JMX]. Under some assumptions, we prove the uniqueness of this realization by classifying all such embeddings of sl 2 in a lattice vertex algebra. We Date: July 22, 2014. 2010 Mathematics Subject Classification. Primary 17B80; Secondary 17B69, 37K10, 81R10. then consider a twisted representation M of the vertex algebra V L (cf. [L, FLM, FFR, D, BK]), and obtain on M a representation of sl 2 of level k. This representation does not appear to exist elsewhere in the literature and may be of independent interest.
Let V be a highest-weight representation of an affine Kac-Moody algebraĝ, and Ω 2 ∈ End(V ⊗ V ) be the Casimir operator that commutes with the diagonal action ofĝ (see [K1]). Consider the equation where λ ∈ C is a constant such that the equation holds when τ is the highest-weight vector v ∈ V . Then (1.1) holds for any τ in the orbit of v under the Kac-Moody group associated toĝ (see [PK]). Equivalently, (1.1) is satisfied for all τ such that τ ⊗τ is in theĝ-submodule generated by v ⊗ v. In the case when V provides a vertex operator realization ofĝ, such as the Frenkel-Kac construction, after a number of nontrivial changes of variables one can rewrite (1.1) as an infinite sequence of non-linear partial differential equations called the Kac-Wakimoto hierarchy [KW]. The action ofĝ allows one to construct some particularly nice solutions to these equations called solitons. For example, the Korteweg-de Vries and non-linear Schrödinger hierarchies are instances of Kac-Wakimoto hierarchies related to different realizations of sl 2 (see [K1]). In this paper we investigate the hierarchy (1.1) arising from the Friedan-Martinec-Shenker bosonization of the Wakimoto realization of sl 2 , which we call the Wakimoto hierarchy. The Casimir operator Ω 2 is replaced with one of the operators from the coset Virasoro construction, which still commutes with the diagonal action of sl 2 (see [GKO, KR]). We write the equations of the Wakimoto hierarchy explicitly as Hirota bilinear equations, and we find the simplest ones. This is done both in the untwisted case when V is a Fock space contained in V L , and the twisted case when V = M is a twisted representation of V L . The new phenomenon is that these are representations of sl 2 of any level k, so the level becomes a parameter in the equations of the Wakimoto hierarchy.
The paper is organized as follows. In Section 2, we construct explicitly the embedding of sl 2 of level k in a lattice vertex algebra V L , and we prove a certain uniqueness property of this embedding. The untwisted Wakimoto hierarchy is investigated in Section 3. In Section 4, we determine the action of sl 2 on a twisted representation M of V L , and study the corresponding twisted Wakimoto hierarchy. Throughout the paper, we work over the field of complex numbers.

Wakimoto realization and its FMS bosonization
We assume the reader is familiar with the basic definitions and examples of vertex algebras, and refer to [FLM,K2,FB,LL] for more details. Let us review the Wakimoto realization of sl 2 of level k from [W] in the formulation of [F]. Consider a pair of charged free bosons a + , a − (also known as a βγ-system) and a free boson b with the only non-zero OPEs given by: Then we have a representation of sl 2 of level k defined by: where the normal ordering of several terms is from the right.
Theorem 2.1. Up to the rescaling (2.2), the above formulas (2.1) provide an embedding of sl 2 of level k in the lattice vertex algebra V L if and only if Proof. We need to verify the following OPEs: We compute (where "h.o.t." stands for higher order terms in z − w): e(z)f (w) = :α(z)e δ (z): :γ(w)e −δ (w): This implies Similar computations for e(z)e(w) and f (z)f (w) give Similarly, from (h|α) = 0 we get which gives (α|γ) = k + 1 .
Gathering all the lattice equations so far obtained, we notice that we are free to rescale α → λα and γ → 1 λ γ, which allows us to fix (δ|α) = 1. This then immediately fixes all the other inner products and we obtain the desired result.
Remark 2.2. The above embedding of sl 2 in V L is essentially equivalent to the "symmetric" W (2) 2 algebra of [FS], to which they refer as the "three-boson realization" of sl 2 .
We will expand fields φ(z) in the standard way and call the coefficients φ (n) the modes of φ(z). The lattice vertex algebra V L is equipped with the standard action of the Virasoro algebra (see, e.g., [K2]): where {a i } and {b i } are dual bases of h with respect to the bilinear form (·|·). The Virasoro central charge is equal the rank of L.
This formula remains true for c = −1 if we remove the last term and set γ = −α.
Note that the Gram matrix associated to the lattice L has determinant −2k − 4. Therefore, rank L = 3, unless the level is critical (i.e., k = −2), in which case rank L = 2 and we can set γ = −α.

The untwisted Wakimoto hierarchy
In the previous section, we saw that the modes of e(z) = :α(z)e δ (z): , give a representation of the affine Kac-Moody algebra sl 2 of level k, where

Introduce the bosonic Fock space
The Heisenberg fields α(z), γ(z) and δ(z) acts on B as follows (n > 0): By setting q = e δ , we identify B as a subspace of V L . Then B is preserved by the actions of sl 2 and Virasoro. Introduce the Virasoro field and the Casimir field Proposition 3.1. All modes of Ω(z) commute with the diagonal action of sl 2 , i.e., [ Proof. This follows from the observation that the modes of Ω(z) give rise to the coset Virasoro construction (see [GKO, KR]).
Note that, up to adding a scalar multiple of the identity operator, Ω (1) = Res z zΩ(z) is the Casimir element of sl 2 (see [K1], where it is denoted Ω 2 ). The Kac-Wakimoto hierarchy [KW] is given by the equation where τ is in a certain highest-weight module and λ ∈ C is a constant such that the equation holds when τ is the highest-weight vector. Instead of Ω (1) , we will consider the operator Ω (0) := Res z Ω(z). Since the vector 1 ∈ B satisfies Ω (0) (1 ⊗ 1) = 0, we obtain the following.
Corollary 3.2. Every vector τ ∈ B, such that τ ⊗ τ is in the sl 2submodule of B ⊗ B generated by 1 ⊗ 1, satisfies the equation We will call (3.1) the untwisted Wakimoto hierarchy. Our goal now is to compute explicitly the action of Ω (0) on B ⊗ B. The main step is to simplify e(z) ⊗ f (z). We will use the shorthand notation Recall the elementary Schur polynomials defined by the expansion Explicitly, one has S m (t) = 0 for m < 0, S 0 (t) = 1, and Lemma 3.3. With the above notation, one has Expanding the exponentials in e δ ′ −δ ′′ (z), we obtain which completes the proof.
Following the procedure of the Japanese school [KM, DKM, DJKM] (see also [MJD, KR]), we will rewrite the untwisted Wakimoto hierarchy (3.1) in terms of Hirota bilinear equations. Let us recall their definition.
Definition 3.4. Given a differential operator P (∂ x ) and two functions f (x), g(x), we define the Hirota bilinear operator P f · g to be As above, we will consider We will make the change of variables and similarly for y ′ , y ′′ and t ′ , t ′′ . Then In order to rewrite the equations in Hirota bilinear form, we recall the formula (see e.g. [KR]): Using Lemma 3.3 and the above notation, we obtain: do not commute for i = −j. As usual, the normally ordered product :a ′ (i) g ′′ (j) : is defined by putting all partial derivatives to the right.
Similarly, by switching the single-primed and double-primed terms, we find: The other terms in (3.1) are easy to compute. Recalling that we get: Finally, we observe that Res z L(z) = L −1 and apply Lemma 2.3 to find: Then Res z τ ⊗ L(z)τ is obtained by switching the single-primed and double-primed terms, where In this way, we have rewritten all terms from (3.1) as Hirota bilinear operators. We expand τ as Since the functions τ m do not depend on any of the variablesx i ,ȳ i , t i , q ′ and q ′′ , all coefficients in front of monomials in these variables give Hirota bilinear equations for τ m . Observe that, in order to get (q ′ ) m (q ′′ ) n in (3.1), we need to apply Res z e(z) ⊗ f (z) to the summand On the other hand, 1 2 Res z h(z) ⊗ h(z) and L −1 ⊗ 1 + 1 ⊗ L −1 have to be applied to τ m (x ′ , y ′ , t ′ )τ n (x ′′ , y ′′ , t ′′ ) (q ′ ) m (q ′′ ) n .
Note that for any polynomial P , we have Using this, from the coefficient of 1 in (3.1), we find Similarly, the coefficient ofx 2 1 in (3.1) gives the equation
Explicitly, sl 2 is realized in V L as Observe that h has an order 2 isometry σ given by which preserves the sl 2 subalgebra described above. In fact, σ operates on sl 2 as the involution σ = exp( πi 2 ad e+f ), which acts as Composing the embedding sl 2 ֒→ V L with any σ-twisted representation of V L , we will obtain a representation of sl 2 . We refer the reader to [L, FLM, FFR, D, BK] for twisted modules over lattice vertex algebras. Here we will only need a special case. Recall first the σ-twisted Heisenberg algebra h σ , spanned over C by a central element I and elements a (m) (a ∈ h, m ∈ 1 2 Z) such that σa = e −2πim a (see e.g. [KP, L, FLM]). The Lie bracket on h σ is given by Let h ≥ σ (respectively, h < σ ) be the subalgebra of h σ spanned by all elements a (m) with m ≥ 0 (respectively, m < 0). Consider the irreducible highest-weight h σ -module M = S( h < σ ), called the σ-twisted Fock space, on which I acts as the identity operator and h ≥ σ annihilates the highestweight vector 1 ∈ M.
We will denote by a M (j) the linear operator on M induced by the action of a (j) ∈ h σ , and will write the twisted fields as: One of the main properties of twisted fields is the σ-equivariance In our case, this means that when σa = a the modes a M (j) are nonzero only for j ∈ Z. On the other hand, if σa = −a the modes a M (j) are nonzero only for j ∈ 1 2 + Z. Note that the eigenspaces of σ on h are spanned by δ, α − γ and by α + γ.
We have the inner products and all other inner products are zero. Then we can identify and the action of h σ is given by: Recall that we also have twisted fields corresponding to e ±δ (see [L, FLM]): By definition, these also satisfy the σ-equivariance (4.3). Then the embedding (4.1) allows one to extend Y M to the generators of sl 2 .
The above lemma can also be used to express Y M (f, z), since by (4.2), (4.3). Recall that V L has a Virasoro element ω so that Explicitly, by Lemma 2.3, we have: Then we have an action of the Virasoro algebra on the twisted module M, given by Lemma 4.2. We have: Proof. Recall that for a, b ∈ h, we have Then, as in the proof of Lemma 4.1 above, where p ∈ {0, 1 2 } is such that σa = e 2πip a. Now computing Y M (ω, z), the last term in the above equation will be nonzero only when a = δ and b = α − γ, in which case p = 1 2 and (δ|α − γ) = 2. The twisted version of the Casimir field Ω(z) from Sect. 3 is Note that Therefore, when computing the coefficients in front of integral powers of z in Ω M (z), we can replace the first two terms, [ Proof. We observe that the commutator formula for the modes of twisted fields, is just like the commutator formula in the vertex algebra itself, provided that a is an eigenvector of σ (see e.g. [FLM, FFR, D]). However, in the modes a M (m) of twisted fields, the index m is allowed to be nonintegral.
We already know from Theorem 2.1 that where (a|b) = tr(ab). Thus for a ∈ {h, e + f, e − f }, we have: where m ∈ Z for a = e + f and m ∈ 1 2 + Z for a = h, e − f . Since σ is an inner automorphism of sl 2 , by Theorem 8.5 in [K1] the above modes a M (m) give a representation of the affine Kac-Moody algebra sl 2 . The statement about Ω M (z) follows again from the commutator formula and Proposition 3.1.
Note that the vector 1 ∈ M satisfies Similarly to Corollary 3.2, we have the following.
Corollary 4.4. Every vector τ ∈ M, such that τ ⊗ τ is in the sl 2submodule of M ⊗ M generated by 1 ⊗ 1, satisfies the equation (2) (τ ⊗ τ ) = 0 . We will call (4.4) the twisted Wakimoto hierarchy. As in Sect. 3, we will compute explicitly the action of Ω M (2) on M ⊗ M. We use the same notation as before regarding primed and double-primed objects, , and make the change of variables Introduce the "reduced" Schur polynomials where S m (t) are the elementary Schur polynomials defined by (3.2). Then we can compute the first term in the expression Ω M (2) = 2 Res z z 2 Y M (e, z) ⊗ Y M (e, e 2πi z) (4.5) Lemma 4.5. We have: We finish the proof by finding the coefficient of z −3 in Y M (e, z) ⊗ Y M (e, e 2πi z).
Now, as in Sect. 3, we can express the action of Ω M (2) on τ ⊗ τ in terms of Hirota bilinear operators using formula (3.3). The recipe is and, accordingly, ∂x is replaced with ∂ u , while ∂t is replaced with ∂ w .
Then we obtain: (4.7) Finally, we get from Lemma 4.2: (4.8) Similarly, τ ⊗L M 1 τ is given by the same formula with all primes replaced with double primes.