The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints

The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators.


Introduction
Givental, Milanov, Frenkel, and Wu, showed a in a series of publications [6,8,9,25] that the total descendant potential of an A, D or E type singularity satisfies the Kac-Wakimoto hierarchy [17]. Recently Bakalov and Milanov showed in [2] that this potential is also a highest weight vector for the corresponding W -algebra. For type A Fukuma, Kawai and Nakayama [7] showed that these W constraints can be obtained completely from the string equation. This was used by Kac and Schwarz [14] to show that this A n potential is a unique (n + 1)-reduced KP tau function, if one assumes that it corresponds to a point in the big cell of the Sato Grassmannian.
Uniqueness for type D and E singularities, together with the A case as well, was recently shown by Liu, Yang and Zhang in [20]. They use the results of [2] and the twisted vertex algebra construction to obtain this result. Both constructions use the Kac-Wakimoto principal hierarchy construction of [17].
In this paper we obtain the principal realization of the basic module of type D (1) n as a certain reduction of a representation of D ∞ . The reduction of the corresponding DKP-type (or sometimes also called 2-component BKP) hierarchy gives Hirota bilinear equations for the corresponding tau functions. This gives an equivalent but slightly different formulation of Kac-Wakimoto D n principal hierarchy [17]. The total descendent potential of a D n type singularity satisfies these equations. This approach has 3 advantages: (1) there is a Lax type formulation for this hierarchy; (2) there is a Grassmannian formulation for this reduced hierarchy; (3) one can show that the string equation generates part of the W -algebra constraints. This makes it This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html The principal hierarchy of the affine Lie algebra D (1) n+1 can be described in many different ways [11,17]. Here we take the approach of ten Kroode and the author [21] and describe this hierarchy as a reduction of the 2-component BKP hierarchy, i.e., we introduce two neutral or twisted fermionic fields and obtain a representation of the Lie algebra of d ∞ . We define an equation which describes the corresponding D ∞ group orbit of the highest weight vector. Following Jimbo and Miwa [10] we use a certain reduction procedure, which reduces the group to a smaller group, viz., to the group corresponding to D (1) n+1 in its principal realization and thus obtain a larger set of equations for elements in the group orbit.
Remark 2.1. It is important to note the following. The Kac-Wakimoto principal hierarchy of type D (1) n+1 characterizes the group orbit of the highest weight vector of type D (1) n+1 in the principal realization (see [17,Theorem 0.1] or [11]). Jimbo and Miwa show in [10] that elements of this group orbit satisfy this D (1) n+1 reduction of this DKP or 2 component BKP hierarchy. Since the total descendent potential of a D n+1 singularity satisfies the Kac-Wakimoto hierarchy it is an element in this D (1) n+1 group orbit and hence also satisfies this Jimbo-Miwa D (1) n+1 principal reduction or (n, 1)-reduced DKP hierarchy.
The normal ordered elements : φ a i φ b j : form the infinite Lie algebra of type d ∞ , where the central elements acts as 1, see [21] for more details. The best way to describe the affine Lie algebra D (1) n+1 is to introduce, following [1], ω = e πi n the 2n-th root of 1, and write The fields corresponding to the elements in the Clifford algebra are Then the commutation relations can be described as follows in term of the anti-commutator { , } is the 2n-twisted delta function, e.g. [1]: Then, see [21], the modes of the fields together with 1 span the affine Lie algebra of type D (1) n+1 in its principal realization. The spin module V splits in the direct sum of two irreducible components when restricted to D (1) n+1 . The irreducible components V = V 0 and V 1 correspond to the Z 2 gradation given by Here V 0 is the basic representation, V 1 is an other level 1 module. Both modules are isomorphic.

The DKP hierarchy and its principal reduction
The DKP hierarchy is the following equation on T ∈ V 0 : This equation describes an element in the D ∞ -group orbit of |0 . If one restricts the action on |0 to the loop group of type D (1) n+1 , the orbit is smaller and is given by more equations. The principal reduction, of [3,10] induces the following. If T ∈ V 0 is in this loop group orbit of |0 , it satisfies the (n, 1)-reduced DKP hierarchy for all integers p ≥ 0 However, for us it will be more convenient not only to use the action on |0 but also on |1 and write T a for the action of the loop group on |a , where a = 0, 1. One thus obtains for all integers p ≥ 0, here a, b = 0, 1.

A Grassmannian description
We follow the description of [15]. The Clifford algebra Cl(C ∞ ) has a natural Z 2 -gradation Cl(C ∞ ) = Cl 0 (C ∞ ) ⊕ Cl 1 (C ∞ ), where Cl 0 (C ∞ ) 0 consists of products of an even number of elements from C ∞ . Let Spin(C ∞ ) denote the multipicative group of invertible elements in a ∈ Cl 0 (C ∞ ) such that aC ∞ a −1 = C ∞ . There exists a homomorphism T : Spin(C ∞ ) → D ∞ such that T (g)(v) = gvg −1 . Thus T (g) is orthogonal, i.e., (T (g)(v), T (g)(w)) = (v, w), in fact it is an element in SO(C ∞ ). Let a = 0, 1, then it is easy to verify that Ann(|a ) for a = 0, 1 is a maximal isotropic subspace of C ∞ and hence Ann(g|a ) for a = 0, 1 and g ∈ Spin(C ∞ ) is also maximal isotropic. Hence an element in the D ∞ group orbit of the vacuum vector produces two unique maximal isotropic subspaces. We can say even more, the modified DKP hierarchy, i.e. equation (2.2) with p = 0 and {a, b} = {0, 1}, has the following geometric interpretation, see also [15] for more information, Note that this follows immediately from (2.3). Let e 1 and e 2 be the orthonormal basis of C 2 we identify where we assume that the bilinear form does not change, i.e., We think of t = e iθ as the loop parameter. Now if g corresponds to an element in D (1) n+1 , then Ann(g|a ) satisfies t Ann(g|a ) ⊂ Ann(g|a ), a = 0, 1.

A bosonization procedure
In general there are many different bosonizations for the same level one D n+1 module (see [13] and [21]). Kac and Peterson [13] showed that for every conjugacy class of the Weyl group of type D n+1 there is a different realization. The principal realization first obtained in [12] is the realization which is connected to a Coxeter element in the Weyl group (all Coxeter elements form one conjugacy class). As such the bosonization procedure for this principal realization is unique and well known, see, e.g., [21]. Here we do not take the usual one, but the one which is related to the D n+1 singularities as in the paper of Bakalov and Milanov [2]. This means that we introduce a parameter √ and that we choose the realization of the Heisenberg algebra slightly different from the usual one.
The bosonization of this principal hierarchy consists of identifying V with the space F = C[θ, q a k ; a = 1, 2, . . . , n + 1, k = 0, 1, . . .]. Here θ is a Grassmann variable satisfying . .] and σ(V 1 ) = F 1 = θC[q a k ; a = 1, 2 . . . n + 1, k = 0, 1, . . .]. The Heisenberg algebra, α a k is defined by Then Remark 2.2. Note that in the notation of [2], n = N − 1, and Here the v i form an orthonormal basis of the Cartan subalgebra of the Lie algebra of type D n+1 . Elements e v i are elements in the group algebra of the root lattice of type B n+1 , which has as basis the elements v i . This construction is related to an automorphism ρ, which is a lift of a Coxeter element in the Weyl group and which gives the Kac-Peterson twisted realization [13], see also [21] for more details. ρ acts on the v 1 , v 2 , . . . , v n , v n+1 as follows then (see [2] or [1]) The factors 1 √ 2n and 1 √ 2 in (2.5) follow from the fact that B v 1 ,−v 1 = 4n and B v n+1 ,−v n+1 = 4 (see [2, p. 853] for the definition of these constants).
Let σ be the isomorphism which sends V to F , such that σ(|0 ) = 1 and σ(|1 ) = θ, is the (raising) Pochhammer symbol (N.B. (x) 0 = 1). To describe σφ a (z)σ − 1, we introduce two extra operators θ and ∂ ∂θ , then Now let σ(T 0 ) = τ 0 and σ(T 1 ) = τ 1 θ Using (2.9)-(2.12) we can rewrite the equation (2.2) and thus obtain a family of Hirota bilinear equations on τ a , here p ≥ 0: From now on we will often omit σ. Using Remark 2.1, we obtain that the total descendent potential of a D n+1 singularity satisfies (2.13). x and write ∂ for ∂ x . Then both τ a and Γ b (q, λ)τ a for b = 1, 2, defined in (2.8), will depend on x. We keep the dependence in τ a but remove it in the second term by writing Γ b (x, q, λ)τ a = Γ b (q, λ)τ a e xλ . Next we rewrite (2.2): Divide the first row of W and V by τ 1 and the second by τ 0 , one thus obtains where Then using the fundamental Lemma of [16], equation (3.2) leads to: Taking p = 0 one deduces that and for p > 0 that 2) for p = 0 to some q j k and apply the fundamental lemma then one gets the following Sato-Wilson equations:

Now introduce the operators
Then clearly and one has the following Lax equations: Note that in the important Drinfeld-Sokolov paper [5], in the case of the Coxeter element in the Weyl group of type D, also 2 × 2 pseudo-differential operators appear. The principal realization of the basic representation is definitely related to this Drinfeld-Sokolov hierarchy, see, e.g., [4]. However, a direct relation between our 2 × 2 operators and the ones appearing in [5] is unclear.

The Orlov-Schulman and S operator
Introduce the Orlov-Schulman operator Then [L, M ] = I and the wave function W (λ) satisfies Moreover, We introduce the operator which will play a crucial role in the deduction of the W constraints. S is explicitly given by The string equation and W constraints

The principal Virasoro algebra
The principal realization of the basic representation of type D Using (2.6) we can express L k in terms of the "times" q j k , in particular L −1 is equal to Let τ ∈ V 0 , the string equation is the following equation on τ However, following, e.g., [7], we remove the right-hand side of (4.3) by introducing the shift q 1 1 → q 1 1 − 1. This reduces the string equation to However, this would introduce in the vertex operator Γ 1 + (q, λ) of (2.9) some extra part which fortunately cancels in (2.13). Therefor we will assume that the string equation is of the form (4.4) and that the hierarchy is given by (2.13), where the operators (2.9) do not have this extra term. We will show that if τ is in the D (1) n+1 group orbit of the vacuum vector, hence satisfies (2.1), and τ satisfies the string equation (4.4), i.e., that τ is annihilated by L −1 , that this induces the annihilation of other elements in the W D n+1 W -algebra. We will follow the approach of [24] (see also [23]). For this we use the following. If τ = τ 0 = g|0 satisfies the string equation, then also its companion τ 1 = g|1 , satisfies the string equations. This is because σL −1 σ −1 commutes with the operator θ + ∂ ∂θ which intertwines F 0 with F 1 .

A consequence of the string equation
Assume that the string equation (4.4) L −1 τ a = 0 holds for both a = 0, 1. Then clearly also Now, hence (4.6) turns into We rewrite this as where We will now prove the following Proof . To prove this we first observe that (4.8) is equivalent to where S is given by (3.5). We calculate the various parts of this formula: Substituting these formulas into (4.9) one obtains up to a multiplicative scalar which is exactly equation (4.7).
A consequence of (3.4) and Proposition 4.1: Let τ satisfy the string equation, then for all p, q ≥ 0, except p = q = 0, the following equation holds: We rewrite the formula (4.10), using (3.3): Now using again the fundamental Lemma of [16] this gives

W constraints
Now, let X k = Res w w k ∂ y X(y, w)| y=w , then putting p = 0 in formula (4.14) and using (4.13), one deduces Thus Note that here we abuse the notation, we write X pq for σX pq σ −1 . Consider this as equation in two sets of variables x, q and x , q . Let a = b and set x = x and q = q , This gives Now divide Γ c (q, λ) (X pq τ a ) by τ b , then Now using (4.16), we rewrite (4.15) in the matrix version

This gives
Now multiplying with P (∂)∂ −1 from the right and taking the residue, one deduces that S c (x, q) = 0 for = 1, 2, . . . , hence, from which we conclude that X pq τ a τ a = const.
In order to calculate these constants, we determine [X 01 , X pq ] and [X 11 , X 0q ]. The action of both operators on τ give zero. Now write X pq = X 1 pq + X 2 pq , then From now on we assume n = 1 if a = 2, in particular where a(k) = k n − 1.
Then [X a 01 , X b pq ] = δ ab j>n,k>(q−p)n (4.17) Now, if p − q = 1 the right hand side is normally ordered and we obtain It is straightforward to check that If p − q = 1 we have to normal order the right hand side of (4.17). Note that in that case, the second and third term of the right hand side of (4.17) are equal to 0 and the first term is normally ordered, the last one not. This gives Clearly, if p = 0 the right hand side of (4.18) is equal to 0. For that case, one calculates [X a 11 , X 0q ] = 2qX 0q , so finally we obtain the following result. Note that X p,0 = 0 and let c q = c 1 q + c 2 q , then Theorem 4.3. For all p ≥ 0 and q > 0, one has the following W constraints: It is straightforward to check that for |y| > |z| X a (y, z) = (−) n : φ a (y)φ a (e 2πin z) : y=z and X a pq + δ p,q 2q + 2 c a q+1 = Res z 1 2q + 2 q+1 k=0 q + 1 k c a k z p−k ∂ q−k+1 y (Γ a (y, z)) y=z .
We now want to obtain one formula in which we combine all our W constraints. For this, we first write the generating series of the c a k : Next we calculate for |u| > |z| > |w|,

A comparison with the results of Bakalov and Milanov [2]
Unfortunately we do not obtain all the W constraints of Bakalov and Milanov [2] from the string equation. Kac, Wang and Yan gave a description in [18] of the corresponding W algebra. As is mentioned in [2, Example 2.5], this W algebra is generated by the elements (cf. Remark 2.2) and the element π n+1 := v 1(−1) v 2(−1) · · · (−1) v n(−1) v n+1 .
Our constraints come from the elements ν d , the constraints related to the element π n+1 cannot be obtained from the string equation.

The string equation on the Grassmannian
Using the (4.1)-formulation of L −1 in terms of the elements φ a i , one can show that