Bosonizations of sl₂ and Integrable Hierarchies

We construct embeddings of sl₂ 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...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2015
Main Authors:	Bakalov, B., Fleisher, D.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2015
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Cite this:	Bosonizations of sl₂ and Integrable Hierarchies / B. Bakalov, D. Fleisher // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ.

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spelling	Bakalov, B. Fleisher, D. 2019-02-11T21:20:30Z 2019-02-11T21:20:30Z 2015 Bosonizations of sl₂ and Integrable Hierarchies / B. Bakalov, D. Fleisher // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B80; 17B69; 37K10; 81R10 DOI:10.3842/SIGMA.2015.005 https://nasplib.isofts.kiev.ua/handle/123456789/146902 We construct embeddings of sl₂ 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 sl₂; that is, the level becomes a parameter in the equations. This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html. We are grateful to Naihuan Jing and Kailash Misra for many useful discussions, and to the referees for valuable suggestions that helped us improve the exposition. The first author was supported in part by NSA and Simons Foundation grants. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bosonizations of sl₂ and Integrable Hierarchies Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Bosonizations of sl₂ and Integrable Hierarchies
spellingShingle	Bosonizations of sl₂ and Integrable Hierarchies Bakalov, B. Fleisher, D.
title_short	Bosonizations of sl₂ and Integrable Hierarchies
title_full	Bosonizations of sl₂ and Integrable Hierarchies
title_fullStr	Bosonizations of sl₂ and Integrable Hierarchies
title_full_unstemmed	Bosonizations of sl₂ and Integrable Hierarchies
title_sort	bosonizations of sl₂ and integrable hierarchies
author	Bakalov, B. Fleisher, D.
author_facet	Bakalov, B. Fleisher, D.
publishDate	2015
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We construct embeddings of sl₂ 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 sl₂; that is, the level becomes a parameter in the equations.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146902
citation_txt	Bosonizations of sl₂ and Integrable Hierarchies / B. Bakalov, D. Fleisher // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 31 назв. — англ.
work_keys_str_mv	AT bakalovb bosonizationsofsl2andintegrablehierarchies AT fleisherd bosonizationsofsl2andintegrablehierarchies
first_indexed	2025-11-26T02:06:33Z
last_indexed	2025-11-26T02:06:33Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 005, 19 pages Bosonizations of ŝl2 and Integrable Hierarchies? Bojko BAKALOV † and Daniel FLEISHER ‡ † Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA E-mail: bojko bakalov@ncsu.edu URL: http://www4.ncsu.edu/~bnbakalo/ ‡ Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel E-mail: daniel.fleisher@weizmann.ac.il Received July 22, 2014, in final form January 09, 2015; Published online January 14, 2015 http://dx.doi.org/10.3842/SIGMA.2015.005 Abstract. We construct embeddings of ŝl2 in lattice vertex algebras by composing the Waki- moto 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 ŝl2; that is, the level becomes a parameter in the equations. Key words: affine Kac–Moody algebra; Casimir element; Friedan–Martinec–Shenker bosonization; lattice vertex algebra; Virasoro algebra; Wakimoto realization 2010 Mathematics Subject Classification: 17B80; 17B69; 37K10; 81R10 1 Introduction Vertex operators and vertex (operator) algebras are powerful tools for studying infinite-dimen- sional Lie algebras, their representations, and generalizations [3, 12, 13, 14, 19, 25, 26]. For any even integral lattice L, one constructs the lattice vertex algebra VL 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 algebra ĝ (see [14, 18, 19]). In this paper we construct a different vertex operator realization of the affine Kac–Moody algebra ŝl2 of an arbitrary level k. We start with the Wakimoto realization of ŝl2, 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 [11, 30]. The Friedan–Martinec–Shenker bosoniza- tion of a+, a− then gives us an embedding in a certain lattice vertex algebra VL (see [1, 15, 31]). The resulting realization of ŝl2 has appeared previously in [8], and is also related to the ones in [9, 17]. Under some assumptions, we prove the uniqueness of this realization by classifying all such embeddings of ŝl2 in a lattice vertex algebra. We then consider a twisted representation M of the vertex algebra VL (cf. [2, 6, 10, 13, 24]), and obtain on M a representation of ŝl2 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 algebra ĝ, and Ω2 ∈ End(V ⊗ V ) be the Casimir operator that commutes with the diagonal action of ĝ (see [18]). Consider the equation Ω2(τ ⊗ τ) = λτ ⊗ τ, τ ∈ V, (1.1) ?This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html mailto:bojko_bakalov@ncsu.edu http://www4.ncsu.edu/~bnbakalo/ mailto:daniel.fleisher@weizmann.ac.il http://dx.doi.org/10.3842/SIGMA.2015.005 http://www.emis.de/journals/SIGMA/LieTheory2014.html 2 B. Bakalov and D. Fleisher 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 to ĝ (see [29]). Equivalently, (1.1) is satisfied for all τ such that τ⊗τ is in the ĝ-submodule generated by v⊗v. In the case when V provides a vertex operator realization of ĝ, such as the Frenkel–Kac construction, after a number of non-trivial changes of variables one can rewrite (1.1) as an infinite sequence of non-linear partial differential equations called the Kac–Wakimoto hierarchy [22]. The action of ĝ 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 ŝl2 (see [18]). In this paper we investigate the hierarchy (1.1) arising from the Friedan–Martinec–Shenker bosonization of the Wakimoto realization of ŝl2, which we call the Wakimoto hierarchy. The Casimir operator Ω2 is replaced with one of the operators from the coset Virasoro construction, which still commutes with the diagonal action of ŝl2 (see [16, 21]). 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 VL, and the twisted case when V = M is a twisted representation of VL. The new phenomenon is that these are representations of ŝl2 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 ŝl2 of level k in a lattice vertex algebra VL, 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 ŝl2 on a twisted representation M of VL, and study the corresponding twisted Wakimoto hierarchy. The conclusion is given in Section 5. Throughout the paper, we work over the field of complex numbers. 2 Wakimoto realization and its FMS bosonization We assume the reader is familiar with the basic definitions and examples of vertex algebras, and refer to [12, 13, 19, 25] for more details. Let us review the Wakimoto realization of ŝl2 of level k from [30] in the formulation of [11]. 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 a+(z)a−(w) ∼ 1 z − w , a−(z)a+(w) ∼ − 1 z − w and b(z)b(w) ∼ 2k (z − w)2 . Then we have a representation of ŝl2 of level k defined by e(z) = a−(z), h(z) = −2:a+(z)a−(z): + b(z), f(z) = −2:a+(z)2a−(z): + k∂za +(z) + :a+(z)b(z):, where the normal ordering of several terms is from the right. Recall that the boson-boson correspondence, or Friedan–Martinec–Shenker (FMS) bosoniza- tion, provides a realization of the charged free bosons a+, a− in terms of fields in a lattice vertex algebra. Namely, consider the lattice Zα1 + Zα2 with (α1\|α2) = 0 and \|α1\|2 = −\|α2\|2 = 1; then a+(z) = eα1+α2(z), a−(z) = −:α1(z)e−α1−α2(z): (see [1, 15, 31]). Notice that \|α1 + α2\|2 = 0. We apply FMS-bosonization to the Wakimoto realization of ŝl2, and we obtain an embedding in a lattice vertex algebra VL of the general form e(z) = :α(z)eδ(z):, h(z) = β(z), f(z) = :γ(z)e−δ(z):, (2.1) Bosonizations of ŝl2 and Integrable Hierarchies 3 where \|δ\|2 = 0 and α(z), β(z), γ(z) are Heisenberg fields. We will classify all such embeddings, where the lattice L contains δ and h = C⊗Z L = spanC{α, β, γ, δ}. Note that the rescaling e 7→ λe, f 7→ 1 λ f, h 7→ h, λ ∈ C, (2.2) is an automorphism of sl2. Theorem 2.1. Up to the rescaling (2.2), the above formulas (2.1) provide an embedding of ŝl2 of level k in the lattice vertex algebra VL if and only if β = kδ + α− γ, \|δ\|2 = 0, (α\|γ) = k + 1, \|α\|2 = \|γ\|2 = (δ\|α) = −(δ\|γ) = 1. Remark 2.2. The matrix of (·\|·) relative to the spanning vectors {δ, α, γ} of h is G =  0 1 −1 1 1 k + 1 −1 k + 1 1  . Since detG = −2k − 4, we have dim h = rankL = 3 if the level k 6= −2 is not critical. When k = −2 is critical, we have γ = −α and dim h = rankL = 2. Proof of Theorem 2.1. We need to verify the following OPEs: e(z)f(w) ∼ h(w) z − w + k (z − w)2 , h(z)h(w) ∼ 2k (z − w)2 , h(z)e(w) ∼ 2e(w) z − w , h(z)f(w) ∼ −2f(w) z − w , e(z)e(w) ∼ 0, f(z)f(w) ∼ 0. We compute (where “h.o.t.” stands for higher order terms in z − w): e(z)f(w) = :α(z)eδ(z): :γ(w)e−δ(w): ∼ ( −(α\|δ)γ(w)− (γ\|δ)α(z) z − w + (α\|γ) + (δ\|α)(δ\|γ) (z − w)2 )( 1 + (z − w)δ(w) + h.o.t. ) ∼ ( (α\|γ) + (δ\|α)(δ\|γ) ) δ(w) z − w − (α\|δ)γ(w) + (γ\|δ)α(w) z − w + (α\|γ) + (δ\|α)(δ\|γ) (z − w)2 . This implies k = (α\|γ) + (δ\|α)(δ\|γ), h = β = kδ − (α\|δ)γ − (γ\|δ)α. Similar computations for e(z)e(w) and f(z)f(w) give (α\|α) = (δ\|α)2, (γ\|γ) = (δ\|γ)2. Now, h(z)e(w) = h(z) :α(w)eδ(w): ∼ (h\|δ):α(w)eδ(w): z − w + (h\|α) (z − w)2 4 B. Bakalov and D. Fleisher tells us that (h\|δ) = 2, (h\|α) = 0, and a nearly identical computation for h(z)f(w) gives (h\|γ) = 0. Then one checks (h\|h) = (h\|kδ − (α\|δ)γ − (γ\|δ)α) = k(h\|δ) = 2k, as required. Expanding (h\|δ) = 2, we find 2 = (h\|δ) = (kδ − (δ\|α)γ − (δ\|γ)α\|δ) = −2(δ\|α)(δ\|γ), i.e., (δ\|α)(δ\|γ) = −1. Similarly, from (h\|α) = 0 we get 0 = (h\|α) = (kδ − (δ\|α)γ − (δ\|γ)α\|α) = k(δ\|α)− (δ\|α)(γ\|α)− (δ\|γ)(α\|α) = (δ\|α) ( k + 1− (α\|γ) ) , which gives (α\|γ) = k + 1. Gathering all the lattice equations so far obtained, (δ\|α)(δ\|γ) = −1, (α\|γ) = k + 1, (α\|α) = (δ\|α)2, (γ\|γ) = (δ\|γ)2, we notice that we are free to rescale α 7→ λα and γ 7→ 1 λγ, which allows us to fix (δ\|α) = 1. This then immediately fixes all the other inner products and we obtain the desired result. � Remark 2.3. The above embedding of ŝl2 in VL is essentially equivalent to the “symmet- ric” W (2) 2 algebra of [8], to which they refer as the “three-boson realization” of ŝl2. We will expand fields φ(z) in the standard way φ(z) = ∑ n∈Z φ(n)z −n−1 and call the coefficients φ(n) the modes of φ(z). The lattice vertex algebra VL is equipped with the standard action of the Virasoro algebra (see, e.g., [19]): Ln = 1 2 rankL∑ i=1 ∑ m∈Z :ai(m)b i (n−m):, n ∈ Z, where {ai} and {bi} are dual bases of h with respect to the bilinear form (·\|·). Lemma 2.4. When the level k = c− 1 6= −2 is not critical, we have Ln = c− 1 4 ∑ m∈Z :δ(m)δ(n−m): + 1 2 ∑ m∈Z :δ(m)(α− γ)(n−m): + 1 4(c+ 1) ∑ m∈Z :(α+ γ)(m)(α+ γ)(n−m):. This formula remains true for c = −1 if we remove the last term and set γ = −α. Bosonizations of ŝl2 and Integrable Hierarchies 5 Proof. The proof is straightforward, using that for c 6= −1 the Gram matrix G of (·\|·) relative to the basis {δ, α, γ} of h, and its inverse are G =  0 1 −1 1 1 c −1 c 1  , G−1 = 1 2(c+ 1)  c2 − 1 c+ 1 −c− 1 c+ 1 1 1 −c− 1 1 1  . For c = −1, the pair of dual bases for h are: {δ, α} and {−δ + α, δ}. � Note that Y (ω, z) = L(z) = ∑ n∈Z Lnz −n−2, (2.3) where the Virasoro vector ω ∈ VL is given by ω = c− 1 4 δ(−1)δ + 1 2 δ(−1)(α− γ) + 1 4(c+ 1) (α+ γ)(−1)(α+ γ). (2.4) Remark 2.5. The above representation of the Virasoro algebra has a central charge equal to rankL (see, e.g., [19]). On the other hand, the Sugawara construction provides a represen- tation of the Virasoro algebra from ŝl2 of level k 6= −2, with central charge equal to 3k/(k + 2) (see, e.g., [19]). 3 The untwisted Wakimoto hierarchy In the previous section, we saw that the modes of e(z) = :α(z)eδ(z):, f(z) = :γ(z)e−δ(z):, h(z) = kδ(z) + α(z)− γ(z) give a representation of the affine Kac–Moody algebra ŝl2 of level k, where \|δ\|2 = 0, \|α\|2 = \|γ\|2 = (δ\|α) = −(δ\|γ) = 1, (α\|γ) = c := k + 1. Introduce the bosonic Fock space B = C [ x, y, t; q, q−1 ] , where x = (x1, x2, x3, . . . ), y = (y1, y2, y3, . . . ), t = (t1, t2, t3, . . . ). The Heisenberg fields α(z), γ(z) and δ(z) act on B as follows (n > 0): α(n) = ∂xn + c∂yn + ∂tn , α(−n) = nxn, α(0) = q∂q, γ(n) = c∂xn + ∂yn − ∂tn , γ(−n) = nyn, γ(0) = −q∂q, δ(n) = ∂xn − ∂yn , δ(−n) = ntn, δ(0) = 0. By setting q = eδ, we identify B as a subspace of VL. Then B is preserved by the actions of ŝl2 and Virasoro. Introduce the Casimir field, whose modes act on B ⊗B inside VL ⊗ VL, Ω(z) = e(z)⊗ f(z) + f(z)⊗ e(z) + 1 2 h(z)⊗ h(z)− k ⊗ L(z)− L(z)⊗ k, where L(z) is the Virasoro field from the previous section (see (2.3)). Note that Ω(z) = Y (Ω, z) for Ω = e⊗ f + f ⊗ e+ 1 2 h⊗ h− k ⊗ ω − ω ⊗ k, (3.1) with ω given by (2.4). 6 B. Bakalov and D. Fleisher Proposition 3.1. All modes of Ω(z) commute with the diagonal action of ŝl2, i.e., [Ω(z), a(w)⊗ 1 + 1⊗ a(w)] = 0, a ∈ sl2. Proof. This follows from the observation that the modes of Ω(z) give rise to the coset Virasoro construction (see [16, 21]). It can also be derived from the commutator formula from the theory of vertex algebras (see, e.g., [19]): [A(m),Ω(n)] = ∞∑ j=0 ( m j ) (A(j)Ω)(m+n−j), applied to the elements A = a⊗ 1 + 1⊗ a with a ∈ sl2. Using that for a, b ∈ sl2 and j ≥ 0, a(j)ω = δj,1a, a(j)b =  [a, b], j = 0, k tr(ab), j = 1, 0, j ≥ 2, one easily checks that A(j)Ω = 0 for all j ≥ 0. Therefore, [A(m),Ω(n)] = 0 for m,n ∈ Z. � Note that, up to adding a scalar multiple of the identity operator, Ω(1) = Resz zΩ(z) is the Casimir element of ŝl2 (see [18], where it is denoted Ω2). The Kac–Wakimoto hierarchy [22] is given by the equation Ω(1)(τ ⊗ τ) = λ(τ ⊗ τ), 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) = Resz Ω(z). Since the highest-weight 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 ŝl2-submodule of B⊗B generated by 1⊗ 1, satisfies the equation Ω(0)(τ ⊗ τ) = 0. (3.2) We will call (3.2) 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 φ′ = φ⊗ 1, φ′′ = 1⊗ φ, so for example, x′n = xn ⊗ 1; and we identify B ⊗B = C [ x′, y′, t′, x′′, y′′, t′′; (q′)±1, (q′′)±1 ] . Recall the elementary Schur polynomials defined by the expansion exp ( ∞∑ n=1 tnz n ) = ∑ m∈Z Sm(t)zm. (3.3) Clearly, Sm(t) = 0 for m < 0, but it will be convenient to use summation over all integers. Explicitly, one has S0(t) = 1 and Sm(t) = ∑ m1+2m2+3m3+···=m tm1 1 m1! tm2 2 m2! tm3 3 m3! · · · , m ≥ 1. Bosonizations of ŝl2 and Integrable Hierarchies 7 Lemma 3.3. With the above notation, one has Resz e(z)⊗ f(z) = q′(q′′)−1 ∑ i,j,l∈Z Sl ( 2t̄ ) α′(i)γ ′′ (j)Sl−i−j−1 ( −∂̃x̄ + ∂̃ȳ ) , where 2t̄n = t′n − t′′n and ∂̃x̄n = 1 n(∂x′n − ∂x′′n). Proof. Note that e±δ(z) = z±δ(0)e±δ exp ( ± ∑ n>0 δ(−n) zn n ) exp ( ± ∑ n>0 δ(n) z−n −n ) = q±1 exp ( ± ∑ n>0 tnz n ) exp ( ± ∑ n>0 (∂̃yn − ∂̃xn)z−n ) , where ∂̃xn = 1 n∂xn . Then e(z)⊗ f(z) = :α′(z)eδ ′ (z)γ′′(z)e−δ ′′ (z): = :α′(z)γ′′(z)eδ ′−δ′′(z):. Expanding the exponentials in eδ ′−δ′′(z), we obtain eδ ′−δ′′(z) = q′(q′′)−1 ∑ l,m∈Z Sl(t ′ − t′′)Sm(−∂̃x′ + ∂̃y′ + ∂̃x′′ − ∂̃y′′)zl−m = q′(q′′)−1 ∑ l,m∈Z Sl(2t̄)Sm(−∂̃x̄ + ∂̃ȳ)z l−m, which completes the proof. � Following the procedure of the Japanese school [4, 5, 23] (see also [21, 28]), we will rewrite the untwisted Wakimoto hierarchy (3.2) 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 Pf · g to be Pf · g = P (∂u) ( f(x+ u)g(x− u) )∣∣ u=0 , where x = (x1, x2, . . .), ∂x = (∂x1 , ∂x2 , . . .), etc. As above, we will consider τ ⊗ τ ∈ C [ x′, y′, t′, x′′, y′′, t′′; (q′)±1, (q′′)±1 ] . We introduce the following new variables in B ⊗B (which are different from the previous ones in B): xn = 1 2 (x′n + x′′n), x̄n = 1 2 (x′n − x′′n), yn = 1 2 (y′n + y′′n), ȳn = 1 2 (y′n − y′′n), tn = 1 2 (t′n + t′′n), t̄n = 1 2 (t′n − t′′n). Then x′n = xn + x̄n, x′′n = xn − x̄n, ∂x′n = 1 2 (∂xn + ∂x̄n), ∂x′′n = 1 2 (∂xn − ∂x̄n), 8 B. Bakalov and D. Fleisher and similarly for y′, y′′ and t′, t′′. Thus τ ⊗ τ = τ(x′, y′, t′; q′)τ(x′′, y′′, t′′; q′′) = τ(x+ x̄, y + ȳ, t+ t̄; q′)τ(x− x̄, y − ȳ, t− t̄; q′′). In order to rewrite the equations in Hirota bilinear form, we recall the formula (see, e.g., [21]): P (∂x̄)τ(x+ x̄)τ(x− x̄) = P (∂u)τ(x+ x̄+ u)τ(x− x̄− u)\|u=0 = Q(∂u)τ(x+ u)τ(x− u)\|u=0 = Qτ · τ, (3.4) where Q(∂u) = P (∂u) exp  ∞∑ j=1 x̄j∂uj  . Using Lemma 3.3 and the above notation, we obtain Resz e(z)τ ⊗ f(z)τ = q′(q′′)−1 ∑ i,j,l∈Z Sl ( 2t̄ ) :a′(i)g ′′ (j):Sl−i−j−1 ( −∂̃u + ∂̃v ) Eτ · τ, (3.5) where E = exp  ∞∑ j=1 x̄j∂uj + ȳj∂vj + t̄j∂wj  , a′(i) =  q′∂q′ , i = 0, −i(x−i + x̄−i), i < 0, 1 2(∂xi + ∂ui) + c 2(∂yi + ∂vi) + 1 2(∂ti + ∂wi), i > 0, and g′′(j) =  −q′′∂q′′ , j = 0, −j(y−j − ȳ−j), j < 0, c 2(∂xj − ∂uj ) + 1 2(∂yj − ∂vj )− 1 2(∂tj − ∂wj ), j > 0. Note that α′(i) = α(i) ⊗ 1 and γ′′(j) = 1 ⊗ γ(j) commute, while a′(i) and g′′(j) 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 Resz f(z)τ ⊗ e(z)τ = (q′)−1q′′ ∑ i,j,l∈Z Sl ( −2t̄ ) :a′′(i)g ′ (j):Sl−i−j−1 ( ∂̃u − ∂̃v ) Eτ · τ, (3.6) where a′′(i) =  q′′∂q′′ , i = 0, −i(x−i − x̄−i), i < 0, 1 2(∂xi − ∂ui) + c 2(∂yi − ∂vi) + 1 2(∂ti − ∂wi), i > 0, and g′(j) =  −q′∂q′ , j = 0, −j(y−j + ȳ−j), j < 0, c 2(∂xj + ∂uj ) + 1 2(∂yj + ∂vj )− 1 2(∂tj + ∂wj ), j > 0. Bosonizations of ŝl2 and Integrable Hierarchies 9 The other terms in (3.2) are easy to compute. Recalling that h(j) = α(j) − γ(j) + kδ(j), k = c− 1, we get Resz h(z)τ ⊗ h(z)τ = ∑ j∈Z h′(−j−1)h ′′ (j)Eτ · τ, where h′(i) =  2q′∂q′ , i = 0, −i(x−i + x̄−i) + i(y−i + ȳ−i)− ki(t−i + t̄−i), i < 0, ∂ti + ∂wi , i > 0, and h′′(j) =  2q′′∂q′′ , j = 0, −j(x−j − x̄−j) + j(y−j − ȳ−j)− kj(t−j − t̄−j), j < 0, ∂tj − ∂wj , j > 0. Finally, we observe that Resz L(z) = L−1 and apply Lemma 2.4 to find Resz L(z)τ ⊗ τ = c− 1 4 ∑ j∈Z d′(j)d ′ (−1−j)Eτ · τ + 1 2 ∑ j∈Z d′(j)(a ′ (−1−j) − g ′ (−1−j))Eτ · τ + 1 4(c+ 1) ∑ j∈Z (a′(j) + g′(j))(a ′ (−1−j) + g′(−1−j))Eτ · τ, where d′(j) =  0, j = 0, −j(t−j + t̄−j), j < 0, 1 2(∂xj + ∂uj )− 1 2(∂yj + ∂vj ), j > 0. Then Resz τ ⊗L(z)τ is obtained by switching all single-primed terms with double-primed terms, where d′′(j) =  0, j = 0, −j(t−j − t̄−j), j < 0, 1 2(∂xj − ∂uj )− 1 2(∂yj − ∂vj ), j > 0. In this way, we have rewritten all terms from (3.2) as Hirota bilinear operators. We expand τ as τ = ∑ m∈Z τm(x, y, t)qm, τm ∈ C[x, y, t]; then τ ⊗ τ = ∑ m,n∈Z τm(x′, y′, t′)τn(x′′, y′′, t′′) (q′)m(q′′)n. (3.7) Since the functions τm do not depend on any of the variables x̄i, ȳi, t̄i, q ′ and q′′, all coefficients in front of monomials in these variables give Hirota bilinear equations for τm. Observe that, in order to get (q′)m(q′′)n in (3.2), we need to apply Resz e(z)⊗ f(z) to the summand τm−1(x′, y′, t′)τn+1(x′′, y′′, t′′)(q′)m−1(q′′)n+1 10 B. Bakalov and D. Fleisher from (3.7). Similarly, Resz f(z)⊗ e(z) is applied to τm+1(x′, y′, t′)τn−1(x′′, y′′, t′′)(q′)m+1(q′′)n−1 On the other hand, 1 2 Resz h(z)⊗ h(z) and L−1 ⊗ 1 + 1⊗ L−1 have to be applied to τm(x′, y′, t′)τn(x′′, y′′, t′′)(q′)m(q′′)n. If we further specialize to m = n = 0, we get equations for the three functions τ−1, τ0 and τ1. We will find the simplest such equations after setting xi = yi = ti = x̄i = ȳi = t̄i−1 = 0, i ≥ 2. Then (3.5) reduces to Resz e(z)τ−1 ⊗ f(z)τ1 = ∑ i,j≥−1 i+j≤−1 :a′(i)g ′′ (j):S−1−i−j ( −∂̃u + ∂̃v ) E1τ−1 · τ1, where now a′(0) = g′′(0) = −1 and E1 = exp ( x̄1∂u1 + ȳ1∂v1 ) . Similarly, from (3.6) we have Resz f(z)τ1 ⊗ e(z)τ−1 = ∑ i,j≥−1 i+j≤−1 :a′′(i)g ′ (j):S−1−i−j ( ∂̃u − ∂̃v ) E1τ1 · τ−1, where a′′(0) = g′(0) = −1. The remaining terms from (3.2) become zero when applied to τ0 ⊗ τ0. Note that for any polynomial P , we have P (∂u, ∂v, ∂w)τ1 · τ−1 = P (−∂u,−∂v,−∂w)τ−1 · τ1. Using this, from the coefficient of 1 in (3.2), we find[ x1y1(∂u1 − ∂v1) + (x1 + y1) ] τ−1 · τ1 = 0. Similarly, the coefficient of x̄2 1 in (3.2) gives the equation[ x1y1(∂u1 − ∂v1)∂2 u1 + (x1 + y1)∂2 u1 + 2y1(∂u1 − ∂v1)∂u1 + 2∂u1 ] τ−1 · τ1 = 0. 4 The twisted Wakimoto hierarchy We now investigate the integrable hierarchy arising from a twisted representation of ŝl2. Recall from Theorem 2.1 the embedding of ŝl2 of level k in the lattice vertex algebra VL, where L is a lattice containing an element δ such that h = C⊗Z L = span{α, γ, δ} and \|δ\|2 = 0, \|α\|2 = \|γ\|2 = (δ\|α) = −(δ\|γ) = 1, (α\|γ) = c := k + 1. Explicitly, ŝl2 is realized in VL as e = α(−1)e δ, f = γ(−1)e −δ, h = kδ + α− γ. (4.1) Bosonizations of ŝl2 and Integrable Hierarchies 11 Observe that h has an order 2 isometry σ given by σ(α) = γ, σ(γ) = α, σ(δ) = −δ, which preserves the ŝl2 subalgebra described above. In fact, σ operates on sl2 as the involution σ = exp(πi 2 ade+f ), which acts as σ(e) = f, σ(f) = e, σ(h) = −h. (4.2) Composing the embedding ŝl2 ↪→ VL with any σ-twisted representation of VL, we will obtain a representation of ŝl2. We refer the reader to [2, 6, 10, 13, 24] for twisted modules over lattice vertex algebras. Here we will only need a special case. Recall first the σ-twisted Heisenberg algebra ĥσ, spanned over C by a central element I and elements a(m) (a ∈ h, m ∈ 1 2Z) such that σa = e−2πima (see, e.g., [13, 20, 24]). The Lie bracket on ĥσ is given by [a(m), b(n)] = mδm,−n(a\|b)I, a, b ∈ h, m, n ∈ 1 2Z. Let ĥ≥σ (respectively, ĥ<σ ) be the subalgebra of ĥσ spanned by all elements a(m) with m ≥ 0 (respectively, m < 0). Consider the irreducible highest-weight ĥσ-module M = S(ĥ<σ ), called the σ-twisted Fock space, on which I acts as the identity operator and ĥ≥σ annihilates the highest- weight vector 1 ∈M . We will denote by aM(j) the linear operator on M induced by the action of a(j) ∈ ĥσ, and will write the twisted fields as YM (a, z) = ∑ j∈ 1 2 Z aM(j)z −j−1, aM(j) ∈ EndM. One of the main properties of twisted fields is the σ-equivariance YM (σa, z) = YM ( a, e2πiz ) := ∑ j∈ 1 2 Z aM(j)e −2πijz−j−1. (4.3) In our case, this means that when σa = a the modes aM(j) are nonzero only for j ∈ Z. On the other hand, if σa = −a the modes aM(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∣∣∣α± γ 2 ∣∣∣2 = c± := 1± c 2 , ( α− γ 2 ∣∣∣δ) = 1, and all other inner products are zero. Then we can identify M = C[x, t], where x = (x1, x2, x3, . . . ), t = (t1, t3, t5, . . . ) and the action of ĥσ is given by δM(j) = { ∂t2j , j ∈ ( 1 2 + Z ) >0 , −jx−2j , j ∈ ( 1 2 + Z ) <0 ,( α− γ 2 )M (j) = { ∂x2j , j ∈ ( 1 2 + Z ) >0 , −j(t−2j + c−x−2j), j ∈ ( 1 2 + Z ) <0 , 12 B. Bakalov and D. Fleisher( α+ γ 2 )M (j) = { ∂x2j , j ∈ Z>0, −jc+x−2j , j ∈ Z≤0. From here we obtain αM(j) =  ∂x2j , j ∈ ( 1 2Z ) >0 , −j(t−2j + c−x−2j), j ∈ ( 1 2 + Z ) <0 , −jc+x−2j , j ∈ Z≤0. Recall that we also have twisted fields corresponding to e±δ (see [13, 24]): YM ( e±δ, z ) = exp ∓ ∑ j∈( 1 2 +Z) <0 δM(j) z−j j  exp ∓ ∑ j∈( 1 2 +Z) >0 δM(j) z−j j  = exp ± ∑ j∈( 1 2 +Z) >0 x2jz j  exp ∓ ∑ j∈( 1 2 +Z) >0 ∂t2j z−j j  . By definition, these also satisfy the σ-equivariance (4.3). Then the embedding (4.1) allows one to extend YM to the generators of ŝl2. Lemma 4.1. We have YM (e, z) = YM ( α(−1)e δ, z ) = :YM (α, z)YM ( eδ, z ) :− 1 2z YM ( eδ, z ) . Proof. Recall that for a ∈ h, we have a(0)e δ = (a\|δ)eδ, a(m)e δ = 0, m > 0. It follows from (3.13) in [2] that :YM (a, z)YM ( eδ, z ) : = YM (a, z)(−1)Y M ( eδ, z ) = ∞∑ m=0 ( p m ) z−mYM ( a(m−1)e δ, z ) = YM ( a(−1)e δ, z ) + pz−1(a\|δ)YM ( eδ, z ) , where a ∈ h and p ∈ {0, 1 2} are such that σa = e2πipa. Now for a = α+ γ, we have p = 0 and :YM (α+ γ, z)YM ( eδ, z ) : = YM ( (α+ γ)(−1)e δ, z ) . Similarly, for a = α− γ, we have p = 1 2 and :YM (α− γ, z)YM ( eδ, z ) : = YM ( (α− γ)(−1)e δ, z ) + z−1YM ( eδ, z ) , since (α− γ\|δ) = 2. Adding these two equations gives the desired result. � The above lemma can also be used to express YM (f, z), since YM (f, z) = YM ( e, e2πiz ) = YM ( e, e−2πiz ) by (4.2), (4.3). Recall that VL has a Virasoro vector ω given by (2.4). Then we have an action of the Virasoro algebra on the twisted module M , given by YM (ω, z) = LM (z) = ∑ n∈Z LMn z −n−2. Bosonizations of ŝl2 and Integrable Hierarchies 13 Lemma 4.2. We have LMn = c− 1 4 ∑ j∈ 1 2 +Z :δM(j)δ M (n−j): + 1 2 ∑ j∈ 1 2 +Z :δM(j)(α− γ)M(n−j): + 1 4(c+ 1) ∑ i∈Z :(α+ γ)M(i)(α+ γ)M(n−i): + 1 8 δn,0. Proof. Recall that for a, b ∈ h, we have a(m)b = δm,1(a\|b), m ≥ 0. Then, as in the proof of Lemma 4.1 above, :YM (a, z)YM (b, z): = YM (a, z)(−1)Y M (b, z) = ∞∑ m=0 ( p m ) z−mYM (a(m−1)b, z) = YM (a(−1)b, z) + p(p− 1) 2 z−2(a\|b), where p ∈ {0, 1 2} is such that σa = e2πipa. Now computing YM (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 the previous section is (cf. (3.1)): ΩM (z) = YM (Ω, z) = YM (e, z)⊗ YM (f, z) + YM (f, z)⊗ YM (e, z) + 1 2 YM (h, z)⊗ YM (h, z)− k ⊗ LM (z)− LM (z)⊗ k. Note that YM (e, z)⊗ YM (f, z) + YM (f, z)⊗ YM (e, z) = YM (e, z)⊗ YM ( e, e2πiz ) + YM ( e, e−2πiz ) ⊗ YM (e, z). Therefore, when computing the coefficients in front of integral powers of z in ΩM (z), we can replace the first two terms YM (e, z)⊗ YM (f, z) + YM (f, z)⊗ YM (e, z), with 2YM (e, z)⊗ YM ( e, e2πiz ) . Theorem 4.3. The modes of the above twisted fields YM (e, z), YM (f, z) and YM (h, z) pro- vide M with the structure of an ŝl2-module of level k. The modes of ΩM (z) commute with the diagonal action of ŝl2 on M ⊗M , i.e.,[ ΩM (z), aM(m) ⊗ 1 + 1⊗ aM(m) ] = 0, a ∈ sl2, m ∈ 1 2Z. Proof. We observe that the commutator formula for the modes of twisted fields, [ aM(m), b M (n) ] = ∞∑ j=0 ( m j ) (a(j)b) M (m+n−j), is just like the commutator formula in the vertex algebra itself, [a(m), b(n)] = ∞∑ j=0 ( m j ) (a(j)b)(m+n−j), 14 B. Bakalov and D. Fleisher provided that a is an eigenvector of σ (see, e.g., [6, 10, 13]). However, in the modes aM(m) of twisted fields, the index m is allowed to be nonintegral. We already know from Theorem 2.1 that [a(m), b(n)] = [a, b](m+n) +mδm,−n(a\|b)k, a, b ∈ sl2, m, n ∈ Z, where (a\|b) = tr(ab). Thus for a ∈ {h, e+ f, e− f}, we have:[ aM(m), b M (n) ] = [a, b]M(m+n) +mδm,−n(a\|b)k, b ∈ sl2, n ∈ 1 2Z, where m ∈ Z for a = e+ f and m ∈ 1 2 + Z for a = h, e− f . Since σ is an inner automorphism of sl2, by [18, Theorem 8.5] the above modes aM(m) give a representation of the affine Kac– Moody algebra ŝl2. The statement about ΩM (z) follows again from the commutator formula and Proposition 3.1. � Note that the vector 1 ∈M satisfies aM(j)1 = ( e±δ )M (j) 1 = 0, a ∈ h, j ∈ 1 2Z≥0. By Lemmas 3.3 and 4.2, this implies aM(n)1 = LMn 1 = 0, a ∈ sl2, n ≥ 1, while eM(0)1 = −1 21 and LM0 1 = 1 81. In particular, ΩM (n)(1⊗ 1) = Resz z nΩM (z)(1⊗ 1) = 0, n ≥ 2. Similarly to Corollary 3.2, we have the following. Corollary 4.4. Every vector τ ∈M , such that τ⊗τ is in the ŝl2-submodule of M⊗M generated by 1⊗ 1, satisfies the equation ΩM (2)(τ ⊗ τ) = 0. (4.4) We will call (4.4) the twisted Wakimoto hierarchy. As in Section 3, we will compute explicitly the action of ΩM (2) on M ⊗M . We use the same notation as before regarding primed and double- primed objects, x′n = xn ⊗ 1, x′′n = 1⊗ xn, etc. Slightly abusing the notation, we make the change of variables xn = 1 2 (x′n + x′′n), x̄n = 1 2 (x′n − x′′n), tn = 1 2 (t′n + t′′n), t̄n = 1 2 (t′n − t′′n). Then x′n = xn + x̄n, x′′n = xn − x̄n, ∂x′n = 1 2 (∂xn + ∂x̄n), ∂x′′n = 1 2 (∂xn − ∂x̄n), and similarly for t′, t′′. Introduce the “reduced” Schur polynomials Rm(t) = Sm(t1, 0, t3, 0, t5, 0, . . . ), where Sm(t) are the elementary Schur polynomials defined by (3.3). Then we can compute the first term in the expression ΩM (2) = 2 Resz z 2YM (e, z)⊗ YM ( e, e2πiz ) + 1 2 Resz z 2YM (h, z)⊗ YM (h, z)− k ⊗ LM1 − LM1 ⊗ k. Bosonizations of ŝl2 and Integrable Hierarchies 15 Lemma 4.5. We have Resz z 2 YM (e, z)⊗ YM ( e, e2πiz ) = ∑ i,j,l∈Z (−1)jRl(2x̄) ( αM(i/2) )′ (αM(j/2)) ′′Rl−i−j+2(−2∂̃t̄) − 1 2 ∑ j,l∈Z Rl(2x̄) (( αM(j/2) )′ + (−1)j ( αM(j/2) )′′) Rl−j+2(−2∂̃t̄) + 1 4 ∑ l∈Z Rl(2x̄)Rl+2(−2∂̃t̄), where 2x̄n = x′n − x′′n and ∂̃t̄n = 1 n(∂t′n − ∂t′′n). Proof. Using Lemma 4.1, we obtain YM (e, z)⊗ YM ( e, e2πiz ) = :YM (α′, z)YM ( α′′, e2πiz ) YM ( eδ ′−δ′′ , z ) : − 1 2z : ( YM (α′, z) + YM ( α′′, e2πiz )) YM ( eδ ′−δ′′ , z ) : + 1 4z2 YM ( eδ ′−δ′′ , z ) . Then we expand YM ( eδ ′−δ′′ , z ) = exp  ∑ j∈( 1 2 +Z) >0 (x′2j − x′′2j)zj  exp  ∑ j∈( 1 2 +Z) >0 (−∂′t2j + ∂′′t2j ) z−j j  = exp  ∑ j∈( 1 2 +Z) >0 2x̄2jz j  exp − ∑ j∈( 1 2 +Z) >0 2∂̃t̄2jz −j  = ∑ l,m∈Z Rl(2x̄)Rm(−2∂̃t̄)z (l−m)/2. We finish the proof by finding the coefficient of z−3 in YM (e, z)⊗ YM (e, e2πiz). � Now, as in Section 3, we can express the action of ΩM (2) on τ ⊗ τ in terms of Hirota bilinear operators using formula (3.4). The recipe is that τ ⊗ τ gets replaced with Eτ · τ , where E = exp  ∞∑ j=1 x̄j∂uj + ∞∑ i=0 t̄2i+1∂w2i+1  , and, accordingly, ∂x̄ is replaced with ∂u, while ∂t̄ is replaced with ∂w. Then (αM(j)) ′ becomes a′(j) =  1 2(∂x2j + ∂u2j ), j ∈ ( 1 2Z ) >0 , −j(t−2j + t̄−2j)− jc−(x−2j + x̄−2j), j ∈ ( 1 2 + Z ) <0 , −jc+(x−2j + x̄−2j), j ∈ Z≤0, while (αM(j)) ′′ becomes a′′(j) =  1 2(∂x2j − ∂u2j ), j ∈ ( 1 2Z ) >0 , −j(t−2j − t̄−2j)− jc−(x−2j − x̄−2j), j ∈ ( 1 2 + Z ) <0 , −jc+(x−2j − x̄−2j), j ∈ Z≤0. Putting these together, we obtain from Lemma 4.5 Resz z 2 YM (e, z)τ ⊗ YM ( e, e2πiz ) τ = ∑ i,j,l∈Z (−1)jRl(2x̄) :a′(i/2)a ′′ (j/2):Rl−i−j+2(−2∂̃w)Eτ · τ 16 B. Bakalov and D. Fleisher − 1 2 ∑ j,l∈Z Rl(2x̄) ( a′(j/2) + (−1)ja′′(j/2) ) Rl−j+2(−2∂̃w)Eτ · τ + 1 4 ∑ l∈Z Rl(2x̄)Rl+2(−2∂̃w)Eτ · τ. Recall that, by (4.3), γM(j) = (−1)2jαM(j), j ∈ 1 2Z. So, to compute the other terms in ΩM (2), we just need to replace (δM(j)) ′ with d′(j) = { 1 2(∂t2j + ∂w2j ), j ∈ ( 1 2 + Z ) >0 , −j(x−2j + x̄−2j), j ∈ ( 1 2 + Z ) <0 , and (δM(j)) ′′ with d′′(j) = { 1 2(∂t2j − ∂w2j ), j ∈ ( 1 2 + Z ) >0 , −j(x−2j − x̄−2j), j ∈ ( 1 2 + Z ) <0 . Then we obtain Resz z 2 YM (h, z)τ ⊗ YM (h, z)τ = ∑ j∈ 1 2 +Z h′(j)h ′′ (1−j)Eτ · τ, where h′(j) = kd′(j) + 2a′(j), h′′(j) = kd′′(j) + 2a′′(j), j ∈ 1 2 + Z. Finally, we get from Lemma 4.2 LM1 τ ⊗ τ = c− 1 4 ∑ j∈ 1 2 +Z d′(j)d ′ (1−j)Eτ · τ + ∑ j∈ 1 2 +Z d′(j)a ′ (1−j)Eτ · τ + 1 c+ 1 ∑ i∈Z a′(i)a ′ (1−i)Eτ · τ. Similarly, τ ⊗ LM1 τ is given by the same formula with all primes replaced with double primes. This completes the rewriting of the twisted Wakimoto hierarchy in terms of Hirota bilinear operators. Then, as in Section 3, all coefficients in front of monomials in the variables x̄i, t̄i give Hirota bilinear equations for τ . We will find the simplest such equation by setting x̄i = xi+2 = t̄2i−1 = t2i+1 = 0, i ≥ 1. Then a′(j) = a′′(j) = 0 for j < −1 and d′(m) = d′′(m) = 0 for m < −1 2 . Thus we obtain ΩM (2)(τ ⊗ τ) = 2 ∑ i,j≥−2 i+j≤2 (−1)j :a′(i/2)a ′′ (j/2):R2−i−j(−2∂̃w)τ · τ − ∑ j=±1,±2 ( a′(j/2) + (−1)ja′′(j/2) ) R2−j(−2∂̃w)τ · τ + 1 2 R2(−2∂̃w)τ · τ + 1 2 ∑ j=± 1 2 , 3 2 h′(j)h ′′ (1−j)τ · τ − k2 4 ∑ j=± 1 2 , 3 2 ( d′(j)d ′ (1−j) + d′′(j)d ′′ (1−j) ) τ · τ Bosonizations of ŝl2 and Integrable Hierarchies 17 − k ∑ j=± 1 2 , 3 2 ( d′(j)a ′ (1−j) + d′′(j)a ′′ (1−j) ) τ · τ − k k + 2 ∑ i=−1,2 ( a′(i)a ′ (1−i) + a′′(i)a ′′ (1−i) ) τ · τ. Note that, after setting all x̄i and t̄i equal to zero, we have a′(j) = a′′(j) and d′(j) = d′′(j) for j < 0. Also P (∂u, ∂w)τ · τ = 0 for any odd polynomial P , i.e., such that P (−∂u,−∂w) = −P (∂u, ∂w). Let us assume, in addition, that τ is independent of x3, x4, t3 and t5. Then the first term in the above sum simplifies to − 8 45 c2 +x 2 2∂ 6 w1 − 1 3 (t1 + c−x1)2∂4 w1 − 2(t1 + c−x1)∂x1∂ 2 w1 − 4c+x2∂x2∂ 2 w1 − 1 2 ( ∂2 x1 − ∂ 2 u1 ) . The other terms of ΩM (2) are easier to compute and add up to 4 3 c+x2∂ 4 w1 + 2∂u1∂w1 + ∂x2 + 1 2 ∂2 w1 + k2 8 ( ∂2 t1 − ∂ 2 w1 ) + k 2 (∂t1∂x1 − ∂u1∂w1) + 1 2 ( ∂2 x1 − ∂ 2 u1 ) − k2 8 ( ∂2 t1 + ∂2 w1 ) − k 2 (∂t1∂x1 + ∂u1∂w1). Putting these together, we obtain that the coefficient in front of 1 in ΩM (2) gives the Hirota bilinear equation[ − 8 45 c2 +x 2 2∂ 6 w1 − 1 3 ( (t1 + c−x1)2 − 4c+x2 ) ∂4 w1 − 2(t1 + c−x1)∂x1∂ 2 w1 + 1 4 (2− k2)∂2 w1 + (2− k)∂u1∂w1 + ∂x2 ] τ · τ = 0. We then employ the change of variables x2 = t, t1 = x, x1 = y, u = log(τ), which allows us to write the above as the evolutionary equation ut = 8 45 c2 +t 2 ( uxxxxxx + 30uxxxxuxx + 60u3 xx ) + 1 3 ( (x+ c−y)2 − 4c+t )( uxxxx + 6u2 xx ) + 1 2 ( k2 − 2 ) uxx + (k − 2)uxy + 2(x+ c−y)(uxxy + 2uxxuy). Note that at the critical level, k = −2, we have c+ = 0, c− = 1, and the above equation becomes ut = 1 3 (x+ y)2 ( uxxxx + 6u2 xx ) + 2(x+ y)(uxxy + 2uxxuy) + uxx − 4uxy. Another reduction is obtained by assuming uy = 0 and letting y = 0. In this case, we get an order six non-autonomous non-linear PDE, resembling those found in [4], ut = 2 45 (k + 2)2t2 ( uxxxxxx + 30uxxxxuxx + 60u3 xx ) + 1 3 ( x2 − 2(k + 2)t )( uxxxx + 6u2 xx ) + 1 2 ( k2 − 2 ) uxx. 5 Conclusion The Frenkel–Kac construction of the homogeneous realization of the basic representation of ŝl2 provides an embedding of the affine vertex algebra of ŝl2 at level 1 in the lattice vertex algebra VL, where L is the root lattice of sl2 (see [14, 19]). The other realizations of the basic representation 18 B. Bakalov and D. Fleisher of ŝl2 can be viewed then as twisted modules over VL (see [20, 24, 26]). These realizations have spectacular applications, such as combinatorial identities obtained from the Weyl–Kac character formula and integrable systems obtained from the Casimir element (see [18]). In particular, for the principal realization of [26] one obtains the Korteweg–de Vries hierarchy, and for the homogeneous realization of [14] one obtains the non-linear Schrödinger hierarchy. In this paper we have constructed other embeddings of the affine vertex algebra of ŝl2 in lattice vertex algebras VL, for an arbitrary level k (now L is not the root lattice). Then the twisted modules over VL provide new vertex operator realizations of ŝl2 at level k. It will be interesting to understand their representation theoretic significance, and to generalize them to other Lie algebras or superalgebras. In particular, we hope to do this for ŝln, since in this case the Wakimoto realization is known explicitly by [7]. Other relevant works include [1, 8, 9, 17, 31]. As an application of these new vertex operator realizations of ŝl2, we have obtained two hier- archies of integrable, non-autonomous, non-linear partial differential equations. A new feature is that the level k becomes a parameter in the equations. It would be interesting to see if the hierarchies associated to ŝl2 (or more generally ŝln) are reductions of some larger hierarchy, sim- ilarly to how the Gelfand–Dickey hierarchies are reductions of the KP hierarchy. A “bosonic” analog of the KP hierarchy has been constructed by K. Liszewski [27], and it might be related to one of our hierarchies when k = −1. Constructing soliton solutions for our equations is as of yet elusive, and is complicated by the fact that all the fields are bosonic. Acknowledgements We are grateful to Naihuan Jing and Kailash Misra for many useful discussions, and to the referees for valuable suggestions that helped us improve the exposition. 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Phys. 195 (1998), 95–111, q-alg/9708008. http://dx.doi.org/10.1007/BF01391662 http://dx.doi.org/10.1007/BF01391662 http://dx.doi.org/10.1016/0550-3213(86)90356-1 http://dx.doi.org/10.1016/0550-3213(86)90356-1 http://dx.doi.org/10.1016/0370-2693(85)91145-1 http://dx.doi.org/10.1090/S0002-9939-09-09942-0 http://dx.doi.org/10.1090/S0002-9939-09-09942-0 http://arxiv.org/abs/0908.0211 http://dx.doi.org/10.1017/CBO9780511626234 http://dx.doi.org/10.3792/pjaa.57.342 http://dx.doi.org/10.1073/pnas.82.24.8295 http://dx.doi.org/10.1007/978-0-8176-8186-9 http://dx.doi.org/10.1007/978-0-8176-8186-9 http://dx.doi.org/10.1007/BF01940329 http://www.lib.ncsu.edu/resolver/1840.16/7058 http://dx.doi.org/10.1073/pnas.80.6.1778 http://dx.doi.org/10.1007/BF01211068 http://dx.doi.org/10.1007/s002200050381 http://arxiv.org/abs/q-alg/9708008 1 Introduction 2 Wakimoto realization and its FMS bosonization 3 The untwisted Wakimoto hierarchy 4 The twisted Wakimoto hierarchy 5 Conclusion References

Bosonizations of sl₂ and Integrable Hierarchies

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