The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2014), 007, 19 pages The ( n, W Constraints (cid:63)
Johan VAN DE LEURMathematical Institute, University of Utrecht,P.O. Box 80010, 3508 TA Utrecht, The Netherlands
E-mail:
URL:
Received September 23, 2013, in final form January 09, 2014; Published online January 15, 2014http://dx.doi.org/10.3842/SIGMA.2014.007
Abstract.
The total descendent potential of a simple singularity satisfies the Kac–Waki-moto principal hierarchy. Bakalov and Milanov showed recently that it is also a highestweight vector for the corresponding W -algebra. This was used by Liu, Yang and Zhang toprove 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 Grassmannianformulation of this hierarchy. We show in particular that the string equation induces a largepart of the W constraints of Bakalov and Milanov. These constraints are not only given onthe tau function, but also in terms of the Lax and Orlov–Schulman operators. Key words: affine Kac–Moody algebra; loop group orbit; Kac–Wakimoto hierarchy; isotropicGrassmannian; total descendent potential; W constraints Givental, Milanov, Frenkel, and Wu, showed a in a series of publications [6, 8, 9, 25] thatthe total descendant potential of an A , D or E type singularity satisfies the Kac–Wakimotohierarchy [17]. Recently Bakalov and Milanov showed in [2] that this potential is also a highestweight 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 wasused by Kac and Schwarz [14] to show that this A n potential is a unique ( n + 1)-reduced KP taufunction, 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 recentlyshown by Liu, Yang and Zhang in [20]. They use the results of [2] and the twisted vertexalgebra construction to obtain this result. Both constructions use the Kac–Wakimoto principalhierarchy construction of [17].In this paper we obtain the principal realization of the basic module of type D (1) n as a cer-tain reduction of a representation of D ∞ . The reduction of the corresponding DKP-type (orsometimes also called 2-component BKP) hierarchy gives Hirota bilinear equations for the cor-responding 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 singularitysatisfies these equations. This approach has 3 advantages: (1) there is a Lax type formulationfor this hierarchy; (2) there is a Grassmannian formulation for this reduced hierarchy; (3) onecan show that the string equation generates part of the W -algebra constraints. This makes it (cid:63) a r X i v : . [ m a t h - ph ] J a n J. van de Leurpossible to describe – at least part of – the W -algebra constraints in terms of pseudo-differentialoperators and in terms of the corresponding Grassmannian.This approach, viz. obtaining the principal hierarchy of type D as a reduction of the 2-com-ponent BKP hierarchy, which describes the D ∞ -group orbit of the highest weight vector, wasalso considered by Liu, Wu and Zhang in [19]. They even obtain Lax equations. However, theirLax equations are formulated differently than the ones in this paper. They use certain (scalar)pseudo-differential operators of the second type, where we need not only the basic representationof type D , but also the other level one module. As such we obtain a pair of tau functions τ and τ , which are related. The equations on both tau functions provide (2 × τ is the same.Wu [26] used the approach of [19] to study the Virasoro-constraints, he showed that they canbe obtained from the string equation. Using the (2 × W constraints,but not all. n, D (1) n +1 The principal hierarchy of the affine Lie algebra D (1) n +1 can be described in many differentways [11, 17]. Here we take the approach of ten Kroode and the author [21] and describethis hierarchy as a reduction of the 2-component BKP hierarchy, i.e., we introduce two neutralor twisted fermionic fields and obtain a representation of the Lie algebra of d ∞ . We definean 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 groupto a smaller group, viz., to the group corresponding to D (1) n +1 in its principal realization and thusobtain 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 hierarchyof type D (1) n +1 characterizes the group orbit of the highest weight vector of type D (1) n +1 in theprincipal realization (see [17, Theorem 0.1] or [11]). Jimbo and Miwa show in [10] that elementsof this group orbit satisfy this D (1) n +1 reduction of this DKP or 2 component BKP hierarchy. Sincethe total descendent potential of a D n +1 singularity satisfies the Kac–Wakimoto hierarchy it isan element in this D (1) n +1 group orbit and hence also satisfies this Jimbo–Miwa D (1) n +1 principalreduction or ( n, n be a positive integer, consider the following Clifford algebra Cl( C ∞ ) on the vectorspace C ∞ with basis φ i n , φ i , with i ∈ Z and symmetric bilinear form (cid:16) φ i n , φ j n (cid:17) = (cid:16) φ i , φ j (cid:17) = ( − ) i δ i, − j , (cid:16) φ i n , φ j (cid:17) = 0 . The Clifford algebra has the usual commutation relations: φ i n φ j n + φ j n φ i n = ( − ) i δ i, − j = φ i φ j + φ j φ i , φ i n φ j + φ j φ i n = 0 . We define its corresponding Spin module V with vacuum vector | (cid:105) as follows (cf. [21]): φ i n | (cid:105) = φ i | (cid:105) = 0 , i > , (cid:0) φ + iφ (cid:1) | (cid:105) = 0 . he ( n, φ ai φ bj : form the infinite Lie algebra of type d ∞ , where the centralelements 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 πin the 2 n -th root of 1, and write ϕ ( z ) = (cid:88) m ∈ n Z ϕ ( m ) z − m − , then ϕ ( e πik z ) = (cid:88) m ∈ n Z ω − kmn ϕ ( m ) z − m − . The fields corresponding to the elements in the Clifford algebra are φ ( z ) = (cid:88) i ∈ Z φ i n z − n − i n , φ ( z ) = (cid:88) i ∈ Z φ i z − − i . Then the commutation relations can be described as follows in term of the anti-commutator { , } (cid:8) φ ( z ) , φ (cid:0) e πin w (cid:1)(cid:9) = ( − ) n n − (cid:88) j =0 δ j ( z − w ) , (cid:8) φ ( z ) , φ (cid:0) e πi w (cid:1)(cid:9) = − (cid:88) j =0 δ jn ( z − w ) , (cid:8) φ ( z ) , φ ( w ) (cid:9) = 0 , where δ j ( z − w ) is the 2 n -twisted delta function, e.g. [1]: δ j ( z − w ) = z j n w − j n δ ( z − w ) = (cid:88) k ∈ j n + Z z k w − k − . Then, see [21], the modes of the fields: φ a ( e πik z ) φ b (cid:0) e πi(cid:96) z (cid:1) : , ≤ a, b ≤ , ≤ k, (cid:96) ≤ n − , together with 1 span the affine Lie algebra of type D (1) n +1 in its principal realization. The spinmodule V splits in the direct sum of two irreducible components when restricted to D (1) n +1 . Theirreducible components V = V and V correspond to the Z gradation given bydeg | (cid:105) = 0 , deg φ ± ak = 1 . The highest weight vector of V is | (cid:105) , the highest weight vector of V is | (cid:105) = 1 √ (cid:0) φ − iφ (cid:1) | (cid:105) . Here V is the basic representation, V is an other level 1 module. Both modules are isomorphic. The DKP hierarchy is the following equation on T ∈ V :Res z (cid:0) ( − ) n φ ( z ) T ⊗ φ (cid:0) e πin z (cid:1) T − φ ( z ) T ⊗ φ (cid:0) e πi z (cid:1) T (cid:1) = 0 . This equation describes an element in the D ∞ -group orbit of | (cid:105) .If one restricts the action on | (cid:105) to the loop group of type D (1) n +1 , the orbit is smaller and isgiven by more equations. The principal reduction, of [3, 10] induces the following. If T ∈ V isin this loop group orbit of | (cid:105) , it satisfies the ( n, p ≥ z z p (cid:0) ( − ) n φ ( z ) T ⊗ φ (cid:0) e πin z (cid:1) T − φ ( z ) T ⊗ φ (cid:0) e πi z (cid:1) T (cid:1) = 0 . (2.1)However, for us it will be more convenient not only to use the action on | (cid:105) but also on | (cid:105) and write T a for the action of the loop group on | a (cid:105) , where a = 0 ,
1. One thus obtainsRes z z p (cid:0) ( − ) n φ ( z ) T a ⊗ φ (cid:0) e πin z (cid:1) T b − φ ( z ) T a ⊗ φ (cid:0) e πi z (cid:1) T b (cid:1) = δ a + b, δ p T b ⊗ T a (2.2)for all integers p ≥
0, here a, b = 0 ,
1. J. van de Leur
We follow the description of [15]. The Clifford algebra Cl( C ∞ ) has a natural Z -gradationCl( C ∞ ) = Cl ( C ∞ ) ⊕ Cl ( C ∞ ), where Cl ( C ∞ ) consists of products of an even number ofelements from C ∞ . Let Spin( C ∞ ) denote the multipicative group of invertible elements in a ∈ Cl ( C ∞ ) such that a C ∞ a − = C ∞ . There exists a homomorphism T : Spin( C ∞ ) → D ∞ such that T ( g )( v ) = gvg − . Thus T ( g ) is orthogonal, i.e., ( T ( g )( v ) , T ( g )( w )) = ( v, w ), in fact itis an element in SO( C ∞ ). Let a = 0 ,
1, thenAnn( g | a (cid:105) ) = { v ∈ C ∞ | vg | a (cid:105) = 0 } = (cid:8) gvg − ∈ C ∞ | v | a (cid:105) = 0 (cid:9) = T ( g ) (cid:0) Ann( | a (cid:105) ) (cid:1) . Since Ann( | a (cid:105) ) = C φ + ( − ) a iφ √ ⊕ (cid:77) i> C φ i n ⊕ C φ i , (2.3)it is easy to verify that Ann( | a (cid:105) ) for a = 0 , C ∞ and henceAnn( g | a (cid:105) ) for a = 0 , 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 cansay even more, the modified DKP hierarchy, i.e. equation (2.2) with p = 0 and { a, b } = { , } ,has the following geometric interpretation, see also [15] for more information,dim (Ann( g | a (cid:105) − Ann( g | b (cid:105) )) = 1 , ≤ a (cid:54) = b ≤ . Note that this follows immediately from (2.3). Let e and e be the orthonormal basis of C weidentify φ i n = t i n e , φ i = t i e , (2.4)where we assume that the bilinear form does not change, i.e., (cid:0) t i n e , t j n e (cid:1) = ( − ) i δ i, − j , (cid:0) t i e , t j e (cid:1) = ( − ) i δ i, − j , (cid:0) t i n e , t j e (cid:1) = 0 . We think of t = e iθ as the loop parameter. Now if g corresponds to an element in D (1) n +1 , thenAnn( g | a (cid:105) ) satisfies t Ann( g | a (cid:105) ) ⊂ Ann( g | a (cid:105) ) , a = 0 , . In general there are many different bosonizations for the same level one D (1) n +1 module (see [13]and [21]). Kac and Peterson [13] showed that for every conjugacy class of the Weyl group oftype D n +1 there is a different realization. The principal realization first obtained in [12] is therealization which is connected to a Coxeter element in the Weyl group (all Coxeter elementsform one conjugacy class). As such the bosonization procedure for this principal realization isunique and well known, see, e.g., [21]. Here we do not take the usual one, but the one whichis related to the D n +1 singularities as in the paper of Bakalov and Milanov [2]. This meansthat we introduce a parameter √ (cid:126) and that we choose the realization of the Heisenberg algebraslightly different from the usual one.The bosonization of this principal hierarchy consists of identifying V with the space F = C [ θ, q ak ; a = 1 , , . . . , n + 1 , k = 0 , , . . . ]. Here θ is a Grassmann variable satisfying θ = 0. Let σ be the isomorphism that maps V into F , we take σ ( V ) = F = C [ q ak ; a = 1 , , . . . , n + 1 , k =he ( n, , , . . . ] and σ ( V ) = F = θ C [ q ak ; a = 1 , . . . n + 1 , k = 0 , , . . . ]. The Heisenberg algebra, α ak isdefined by α ( z ) = (cid:88) i ∈ n + n Z α i z − i − := ( − n √ n : φ ( z ) φ (cid:0) e πin z (cid:1) : ,α ( z ) = (cid:88) i ∈ + Z α i z − i − := −
12 : φ ( z ) φ (cid:0) e πi z (cid:1) : . Then (cid:2) α ak , α b(cid:96) (cid:3) = kδ ab δ k, − (cid:96) and (cid:2) α k , φ ( z ) (cid:3) = z k √ n φ ( z ) , (cid:2) α k , φ ( z ) (cid:3) = z k φ ( z ) . Remark 2.2.
Note that in the notation of [2], n = N − α ( z ) = Y ( √ nv , z ) = √ nY ( v , z ) , α ( z ) = Y ( v n +1 , z )and φ ( z ) = 1 √ n Y (cid:0) e v , z (cid:1) , φ ( z ) = 1 √ Y (cid:0) e v n +1 , z (cid:1) . (2.5)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 asbasis the elements v i . This construction is related to an automorphism ρ , which is a lift ofa 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 , v , . . . , v n , v n +1 as follows v (cid:55)→ v (cid:55)→ · · · (cid:55)→ v n (cid:55)→ − v , v n +1 (cid:55)→ − v n +1 , then (see [2] or [1]) Y ( v j , z ) = Y (cid:0) ρ j − ( v ) , z (cid:1) = Y (cid:0) v , e j − πi z (cid:1) ,Y (cid:0) e v j , z (cid:1) = Y (cid:0) e v , e j − πi z (cid:1) , < j ≤ n. The factors √ n and √ in (2.5) follow from the fact that B v , − v = 4 n 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 ( | (cid:105) ) = 1 and σ ( | (cid:105) ) = θ , σα − j − n − k σ − = (cid:126) − q jk ((2 j − / (2 n )) k , σα j − n + k σ − = ((2 j − / (2 n )) k +1 (cid:126) ∂∂q jk , (2.6) σα − − k σ − = (cid:126) − q n +1 k (1 / k , σα + k σ − = (1 / k +1 (cid:126) ∂∂q n +1 k , (2.7)for k = 0 , , , . . . and 1 ≤ j ≤ n , where ( x ) k = x ( x + 1) · · · ( x + k −
1) = Γ( x + k )Γ( x ) is the (raising)Pochhammer symbol (N.B. ( x ) = 1). To describe σφ a ( z ) σ −
1, we introduce two extra operators θ and ∂∂θ , then σφ ( z ) σ − = (cid:0) θ + ∂∂θ (cid:1) √ z − Γ (cid:0) q, z n (cid:1) , σφ ( z ) σ − = i (cid:0) θ − ∂∂θ (cid:1) √ z − Γ (cid:0) q, z (cid:1) , J. van de LeurwhereΓ (cid:0) q, z n (cid:1) = Γ (cid:0) q, z n (cid:1) Γ − (cid:0) q, z n (cid:1) , Γ (cid:0) q, z (cid:1) = Γ (cid:0) q, z (cid:1) Γ − (cid:0) q, z (cid:1) (2.8)with Γ (cid:0) q, z n (cid:1) = exp √ n n (cid:88) j =1 ∞ (cid:88) k =0 (cid:126) − q jk ((2 j − / (2 n )) k +1 z j − n + k , (2.9)Γ − (cid:0) q, z n (cid:1) = exp √ n n (cid:88) j =1 ∞ (cid:88) k =0 − ((2 j − / (2 n )) k (cid:126) ∂∂q jk z − j − n − k , (2.10)Γ (cid:0) q, z (cid:1) = exp (cid:32) ∞ (cid:88) k =0 (cid:126) − q n +1 k (1 / k +1 z + k (cid:33) , (2.11)Γ − (cid:0) q, z (cid:1) = exp (cid:32) ∞ (cid:88) k =0 − (1 / k (cid:126) ∂∂q n +1 k z − − k (cid:33) . (2.12)Now let σ ( T ) = τ and σ ( T ) = τ θ Using (2.9)–(2.12) we can rewrite the equation (2.2) andthus obtain a family of Hirota bilinear equations on τ a , here p ≥ λ (cid:16) λ np − Γ ( q, λ ) τ a ⊗ Γ ( q, − λ ) τ b − ( − ) a + b λ p − Γ ( q, λ ) τ a ⊗ Γ ( q, − λ ) τ b (cid:17) = 2 δ a + b, δ p τ b ⊗ τ a . (2.13) 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 sa-tisfies (2.13). We want to reformulate (2.13) in terms of pseudo-differential operators. For this we introducean extra variable x by replacing q and q n +10 by q + (cid:126) n x and q n +10 + (cid:126) 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 thedependence 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):Res λ (cid:16) W ( λ )diag (cid:0) λ np − , λ p − (cid:1) ⊗ W ( − λ ) T (cid:17) = δ p V ⊗ V T , where W ( λ ) = (cid:18) Γ ( q, λ ) τ i Γ ( q, λ ) τ i Γ ( q, λ ) τ Γ ( q, λ ) τ (cid:19) e xλ , V = (cid:18) τ iτ iτ τ (cid:19) . (3.1)Divide the first row of W and V by τ and the second by τ , one thus obtainsRes λ (cid:16) P ( λ )diag (cid:0) λ np − , λ p − (cid:1) E ( λ ) e xλ ⊗ e − xλ E ( − λ ) T P ( − λ ) T J (cid:17) = δ p I, (3.2)where P ( λ ) = 1 √ Γ − ( q, λ ) τ τ i Γ − ( q, λ ) τ τ i Γ − ( q, λ ) τ τ Γ − ( q, λ ) τ τ he ( n, E ( λ ) = (cid:18) Γ ( q, λ ) 00 Γ ( q, λ ) (cid:19) , J = (cid:18) − i − i (cid:19) . Then using the fundamental Lemma of [16], equation (3.2) leads to:( P ( ∂ )diag (cid:0) ∂ np − , ∂ p − (cid:1) P ∗ ( ∂ ) J ) − = δ p ∂ − I. Taking p = 0 one deduces that P − ∂ − = ∂ − P ∗ J (3.3)and for p > (cid:0) P diag (cid:0) ∂ np , ∂ p (cid:1) P − (cid:1) ≤ = 0 . Now differentiate (3.2) for p = 0 to some q jk and apply the fundamental lemma then one getsthe following Sato–Wilson equations: ∂P∂q jk P − = − (cid:0) B jk (cid:1) ≤ , where B jk = √ n (cid:126) − (2 j − / (2 n )) k +1 P E ∂ j − kn P − if j ≤ n, (cid:126) − (1 / k +1 P E ∂ k P − if j = n + 1 . Now introduce the operators L = P ∂P − , C a = P E aa P − . Then clearly[
L, C a ] = 0 , C a C b = δ ab C a , C + C = I, (cid:0) L np C + L p C (cid:1) ≤ = 0 (3.4)and one has the following Lax equations: ∂L∂q jk = (cid:2)(cid:0) B jk (cid:1) > , L (cid:3) , ∂C a ∂q jk = (cid:2)(cid:0) B jk (cid:1) > , C a (cid:3) . Note that in the important Drinfeld–Sokolov paper [5], in the case of the Coxeter element in theWeyl group of type D, also 2 × × S operator Introduce the Orlov–Schulman operator M = P ExE − P − = P RP − , J. van de Leurwhere R = xI + 2 (cid:126) − ∞ (cid:88) k =0 (cid:32) √ nE n (cid:88) j =1 q jk ((2 j − / (2 n )) k ∂ nk +2 j − + E q n +1 k (1 / k ∂ k (cid:33) . Then [
L, M ] = I and the wave function W ( λ ) satisfies LW ( λ ) = λW ( λ ) , C i W ( λ ) = W ( λ ) E ii , M W ( λ ) = ∂W ( λ ) ∂λ . Moreover, M = ∂P∂∂ P − + 2 (cid:126) − ∞ (cid:88) k =0 (cid:32) √ n n (cid:88) j =1 q jk ((2 j − / (2 n )) k L nk +2 j − C + q n +1 k (1 / k L k C (cid:33) . We introduce the operator S = (cid:18) n M L − n C + 12 M L − C (cid:19) ≤ P, which will play a crucial role in the deduction of the W constraints. S is explicitly given by S = 12 n ∂P∂∂ ∂ − n E + 12 ∂P∂∂ ∂ − E + ∞ (cid:88) k =0 √ n (cid:126) n (cid:88) j =1 q jk ((2 j − / (2 n )) k L n ( k − j − C + 1 √ (cid:126) q n +1 k (1 / k L k − C ≤ P = 12 n ∂P∂∂ ∂ − n E + 12 ∂P∂∂ ∂ − E + 1 √ n (cid:126) n (cid:88) j =1 q j P ∂ j − n − E + 1 √ (cid:126) q n +10 P ∂ − E + ∞ (cid:88) k =1 √ n (cid:126) n (cid:88) j =1 q jk ((2 j − / (2 n )) k L n ( k − j − C + 1 √ (cid:126) q n +1 k (1 / k L k − C ≤ P = 12 n ∂P∂∂ ∂ − n E + 12 ∂P∂∂ ∂ − E + 1 √ n (cid:126) n (cid:88) j =1 q j P ∂ j − n − E + 1 √ (cid:126) q n +10 P ∂ − E − n +1 (cid:88) j =1 ∞ (cid:88) k =0 q jk +1 ∂P∂q jk . (3.5) W constraints The principal realization of the basic representation of type D (1) n +1 has a natural Virasoro algebrawith central charge n + 1. It is given by (see, e.g., [21]) L k = (cid:88) j ∈ Z ( − ) j j n : φ − j n φ j n + k : +( − ) j j φ − j φ j + k : + δ k, (cid:18) n + 116 n + n − n (cid:19) = (cid:88) j ∈ Z
12 : α − n − jn α n + jn + k : + 12 : α − − j α + j + k : + δ k, (cid:18) n + 116 n + n − n (cid:19) , (4.1)he ( n, L ( z ) = (cid:88) k ∈ Z L k z − k − = 12 w − ∂∂w w × (cid:0) ( − ) n : φ ( w ) φ (cid:0) e πin z (cid:1) : − : φ ( w ) φ (cid:0) e πi z (cid:1) : (cid:1)(cid:12)(cid:12) w = z + (cid:18) n + 116 n + n − n (cid:19) z − . Using (2.6) we can express L k in terms of the “times” q jk , in particular L − is equal to σL − σ − = 12 (cid:126) (cid:0) q n +10 (cid:1) + 12 (cid:126) n (cid:88) j =1 q j q n +1 − j + n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 q (cid:96)k +1 ∂∂q (cid:96)k . (4.2)Let τ ∈ V , the string equation is the following equation on τL − τ = ∂τ∂q . (4.3)However, following, e.g., [7], we remove the right-hand side of (4.3) by introducing the shift q (cid:55)→ q −
1. This reduces the string equation to L − τ = 0 . (4.4)However, this would introduce in the vertex operator Γ ( q, λ ) of (2.9) some extra part e − (2 n )2 (cid:126) − √ n λ n +12 n +1 , which fortunately cancels in (2.13). Therefor we will assume that the string equation is of theform (4.4) and that the hierarchy is given by (2.13), where the operators (2.9) do not have thisextra term. We will show that if τ is in the D (1) n +1 group orbit of the vacuum vector, hencesatisfies (2.1), and τ satisfies the string equation (4.4), i.e., that τ is annihilated by L − , thatthis induces the annihilation of other elements in the W D n +1 W -algebra. We will follow theapproach of [24] (see also [23]). For this we use the following. If τ = τ = g | (cid:105) satisfies thestring equation, then also its companion τ = g | (cid:105) , satisfies the string equations. This is because σL − σ − commutes with the operator θ + ∂∂θ which intertwines F with F . Assume that the string equation (4.4) L − τ a = 0 holds for both a = 0 ,
1. Then clearly alsoΓ c − ( λ ) ( L − τ a ) τ b − L − τ b ( τ b ) Γ c − ( λ )( τ a ) = 0 . (4.5)Denote by τ cd = Γ c − ( λ )( τ d ), then (4.5) is equivalent to τ b Γ c − ( λ ) ( L − ) τ ca − τ ca L − τ b ( τ b ) = 0 . (4.6)Now, Γ c − ( λ ) ( L − ) = 12 (cid:126) Γ c − ( λ ) (cid:0) q n +10 (cid:1) + n (cid:88) j =1 q j q n +1 − j + n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 Γ c − ( λ ) (cid:0) q (cid:96)k +1 (cid:1) ∂∂q (cid:96)k , (cid:126) τ ca τ b (cid:0) Γ c − ( λ ) − (cid:1) (cid:0) q n +10 (cid:1) + n (cid:88) j =1 q j q n +1 − j + n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 (cid:32) Γ c − ( λ ) (cid:0) q (cid:96)k +1 (cid:1) τ b ∂τ ca ∂q (cid:96)k − τ ca ( τ b ) q (cid:96)k +1 ∂τ b ∂q (cid:96)k (cid:33) = 0 . We rewrite this as n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 q (cid:96)k +1 ∂ τ ca τ b ∂q (cid:96)k + R abc = 0 , (4.7)where R ab = τ a τ b λ − n − √ n (cid:126) τ a τ b n (cid:88) j =1 λ − j q n +1 − j − √ (cid:126) √ nτ b n (cid:88) (cid:96) =1 ∞ (cid:88) k =0 ((2 (cid:96) − / n ) k +1 λ − nk − n − (cid:96) ∂τ a ∂q (cid:96)k and R ab = τ a τ b λ − − √ (cid:126) τ a τ b λ − q n +10 − √ (cid:126) τ b ∞ (cid:88) k =0 (1 / k +1 λ − k − ∂τ a ∂q n +1 k . We will now prove the following
Proposition 4.1.
The string equation (4.4) induces (cid:18)(cid:18) n M L − n − L − n (cid:19) C + (cid:18) M L − − L − (cid:19) C (cid:19) ≤ = 0 . (4.8) Proof .
To prove this we first observe that (4.8) is equivalent to (cid:18) P ∂ − n E + 12 P ∂ − E (cid:19) ≤ − S = 0 , (4.9)where S is given by (3.5). We calculate the various parts of this formula:12 P a λ − n − √ n (cid:126) n (cid:88) j =1 q j P a λ j − n − = i a − τ a − √ τ − a λ − n − i a − τ a − √ n (cid:126) τ − a n (cid:88) j =1 λ j − n − q j , P a λ − − √ (cid:126) q n +10 P a λ − = i ( − i ) a − τ a − √ τ − a λ − − i ( − i ) a − τ a − √ (cid:126) τ − a λ − . Now 12 n ∂P a ( λ ) ∂λ λ − n = i a − (cid:126) √ nτ − a ∂τ a − ∂λ λ − n = i a − (cid:126) √ nτ − a n (cid:88) (cid:96) =1 ∞ (cid:88) k =0 ((2 (cid:96) − / n ) k +1 λ − n ( k +1) − (cid:96) ∂τ a − ∂q (cid:96)k he ( n, ∂P a ( λ ) ∂λ λ − = i ( − i ) a − (cid:126) √ τ − a ∂τ a − ∂λ λ − = i ( − i ) a − (cid:126) √ τ − a ∞ (cid:88) k =0 (1 / k +1 λ − k − ∂τ a − ∂q n +1 k . Substituting these formulas into (4.9) one obtains up to a multiplicative scalar n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 q (cid:96)k +1 ∂τ a − /τ − a ∂q (cid:96)k + τ a − τ − a λ − n − τ a − √ n (cid:126) τ − a n (cid:88) j =1 λ j − n − q j − (cid:126) √ nτ − a n (cid:88) (cid:96) =1 ∞ (cid:88) k =0 ((2 (cid:96) − / n ) k +1 λ − n ( k +1) − (cid:96) ∂τ a − ∂q (cid:96)k = 0and n +1 (cid:88) (cid:96) =1 ∞ (cid:88) k =0 q (cid:96)k +1 ∂τ a − /τ − a ∂q (cid:96)k + τ a − τ − a λ − − τ a − √ (cid:126) τ − a λ − q n +10 − (cid:126) τ − a ∞ (cid:88) k =0 (1 / k +1 λ − k +3 ∂τ a − ∂q n +1 k = 0 , which is exactly equation (4.7). (cid:4) A consequence of (3.4) and Proposition 4.1:
Proposition 4.2.
Let τ satisfy the string equation, then for all p, q ≥ , except p = q = 0 , thefollowing equation holds: (cid:18)(cid:18) n M L − n − L − n (cid:19) q L np C + (cid:18) M L − − L − (cid:19) q L p C (cid:19) ≤ = 0 . (4.10)We rewrite the formula (4.10), using (3.3): (cid:18)(cid:18) n P R∂ − n − ∂ − n (cid:19) q ∂ np − E P ∗ J + (cid:18) R∂ − − ∂ − (cid:19) q ∂ p − E P ∗ J (cid:19) − = 0 . Now using again the fundamental Lemma of [16] this givesRes λ (cid:32) λ np − (cid:18) n λ − n ∂ λ − λ − n (cid:19) q W ( λ ) E (4.11)+ λ p − (cid:18) λ − ∂ λ − λ − (cid:19) q W ( λ ) E (cid:33) ⊗ W ( − λ ) T = 0 . (4.12)Now let λ k = z , then ∂ z = k λ − k ∂ λ and k λ − k ∂ λ − λ − k = z ∂ z z − , then (4.12) is equivalent toRes z (cid:16) z p ∂ qz (cid:16) z − Γ (cid:0) q, z n (cid:1) τ a (cid:17) ⊗ z − Γ (cid:0) q, − z n (cid:1) τ b − ( − ) a + b z p ∂ qz (cid:16) z − Γ (cid:0) q, z (cid:1) τ a (cid:17) ⊗ z − Γ (cid:0) q, − z (cid:1) τ b (cid:17) = 0 . And this formula inducesRes z (cid:16) ( − ) n z p ∂ qz (cid:0) φ ( z ) (cid:1) T a ⊗ φ (cid:0) e πin z (cid:1) T b − z p ∂ qz (cid:0) φ ( z ) (cid:1) T a ⊗ φ (cid:0) e πi z (cid:1) T b (cid:17) = 0 , Res z (cid:16) ( − ) n z p φ (cid:0) e πin z (cid:1) T a ⊗ ∂ qz (cid:0) φ ( z ) (cid:1) T b − z p φ (cid:0) e πi z (cid:1) T a ⊗ ∂ qz (cid:0) φ ( z ) (cid:1) T b (cid:17) = 0 . (4.13)2 J. van de Leur We have (cid:2) : φ ( y ) φ (cid:0) e πin w (cid:1) : , φ ( z ) (cid:3) = φ ( y ) (cid:8) φ (cid:0) e πin w (cid:1) , φ ( z ) (cid:9) − (cid:8) φ ( y ) , φ ( z ) (cid:9) φ (cid:0) e πin w (cid:1) = ( − ) n n − (cid:88) j =0 δ j ( z − w ) φ ( y ) − δ j (cid:0) z − e πin y (cid:1) φ (cid:0) e πin w (cid:1) , and similarly (cid:2) : φ ( y ) φ (cid:0) e πi w (cid:1) : , φ ( z ) (cid:3) = − (cid:88) j =0 δ jn ( z − w ) φ ( y ) − δ jn (cid:0) z − e πi y (cid:1) φ (cid:0) e πi w (cid:1) . We calculate the action of X ( y, w ) ⊗ (cid:0) ( − ) n : φ ( y ) φ (cid:0) e πin w (cid:1) : − : φ ( y ) φ (cid:0) e πi w (cid:1) : (cid:1) ⊗ δ a + b, δ p X ( y, w ) T b ⊗ T a = Res z z p (cid:40) ( − ) n (cid:32) n − (cid:88) j =0 δ j ( z − w ) φ ( y ) − δ j (cid:0) z − e πin y (cid:1) φ (cid:0) e πin w (cid:1)(cid:33) T a ⊗ φ (cid:0) e πin z (cid:1) T b − (cid:32) (cid:88) j =0 δ jn ( z − w ) φ ( y ) − δ jn (cid:0) z − e πi y (cid:1) φ (cid:0) e πi w (cid:1)(cid:33) T a ⊗ φ (cid:0) e πin z (cid:1) T b + ( − ) n φ ( z ) X ( y, w ) T a ⊗ φ ( e πin z ) T b − φ ( z ) X ( y, w ) T a ⊗ φ ( e πi z ) T b (cid:41) . Thus δ a + b, δ p X ( y, w ) T b ⊗ T a − Res z (cid:16) z p (cid:0) ( − ) n φ ( z ) X ( y, w ) T a ⊗ φ (cid:0) e πin z (cid:1) T b − φ ( z ) X ( y, w ) T a ⊗ φ (cid:0) e πi z (cid:1) T b (cid:1) (cid:17) = w p (cid:0) ( − ) n φ ( y ) T a ⊗ φ (cid:0) e πin w (cid:1) T b − φ ( y ) T a ⊗ φ (cid:0) e πi w (cid:1) T b (cid:1) − y p (cid:0) ( − ) n φ (cid:0) e πin w (cid:1) T a ⊗ φ ( y ) T b − φ (cid:0) e πi w (cid:1) T a ⊗ φ ( y ) T b (cid:1) . (4.14) W constraints Now, let X k(cid:96) = Res w w k ∂ (cid:96)y X ( y, w ) | y = w , then putting p = 0 in formula (4.14) and using (4.13),one deducesRes z (cid:16) ( − ) n φ ( z ) X pq T a ⊗ φ (cid:0) e πin z (cid:1) T b − φ ( z ) X pq T a ⊗ φ (cid:0) e πi z (cid:1) T b (cid:17) = δ a + b, X pq T b ⊗ T a . Thus Res λ λ − (cid:16) Γ ( q, λ ) X pq τ a ⊗ Γ ( q, − λ ) τ b − ( − ) a + b Γ ( q, λ ) X pq τ a ⊗ Γ ( q, − λ ) τ b (cid:17) = 2 δ a + b, X pq τ b ⊗ τ a . (4.15)Note that here we abuse the notation, we write X pq for σX pq σ − . Consider this as equation intwo sets of variables x , q and x (cid:48) , q (cid:48) . Let a (cid:54) = b and set x = x (cid:48) and q = q (cid:48) , This gives X pq τ a τ a = X pq τ b τ b . (4.16)he ( n, c ( q, λ ) ( X pq τ a ) by τ b , thenΓ c ( q, λ ) ( X pq τ a ) τ b = Γ c + ( q, λ ) Γ c − ( q, λ ) τ a τ b Γ c − ( q, λ ) (cid:18) X pq τ a τ a (cid:19) . Now using (4.16), we rewrite (4.15) in the matrix versionRes λ (cid:32) λ − (cid:88) c =1 Γ c − ( q, λ ) (cid:18) X pq τ a τ a (cid:19) P ( λ ) E cc E ( λ ) e xλ ⊗ e − xλ E ( − λ ) T P ( − λ ) T J (cid:33) = X pq τ a τ a I. Let Γ c − ( q, λ ) (cid:18) X pq τ a τ a (cid:19) = ∞ (cid:88) k =0 S ck ( x, q ) λ − k , then Res λ (cid:32) (cid:88) c =1 ∞ (cid:88) k =0 S ck P ( λ ) E cc λ − k − E ( λ ) e xλ ⊗ e − xλ E ( − λ ) T P ( − λ ) T J (cid:33) = X pq τ a τ a I. This gives ∞ (cid:88) k =1 S ck P ( ∂ ) E cc ∂ − k P ( ∂ ) − = 0 . Now multiplying with P ( ∂ ) ∂ (cid:96) − from the right and taking the residue, one deduces that S c(cid:96) ( x, q ) = 0 for (cid:96) = 1 , , . . . , hence, (cid:0) Γ c − ( q, λ ) − (cid:1) (cid:18) X pq τ a τ a (cid:19) = 0 , from which we conclude that X pq τ a τ a = const . In order to calculate these constants, we determine [ X , X pq ] and [ X , X q ]. The action ofboth operators on τ give zero. Now write X pq = X pq + X pq , then X apq = (cid:88) k> ( q − p ) n ( − ) k b pq ( k ) φ a − k n φ a k n + p − q , where b pq ( k ) = (cid:18) k n + 12 − q (cid:19) q − (cid:18) − k n + 12 − p (cid:19) q . From now on we assume n = 1 if a = 2, in particular X a = (cid:88) k>n ( − ) k a ( k ) φ a − k n φ a k n − , where a ( k ) = kn − . X a , X bpq ] = δ ab (cid:88) j>n,k> ( q − p ) n ( − ) k a ( j ) b pq ( k ) (cid:16) δ j − n,k φ a − j n φ a j n + p − q − + δ j, − k +2( q − p +1) n φ a − k n φ a k n + p − q − − δ j,k φ a j n − φ a − j n + p − q − δ j,k +2( p − q ) n φ a − k n φ a k n + p − q − (cid:17) . (4.17)Now, if p − q (cid:54) = 1 the right hand side is normally ordered and we obtain[ X a , X bpq ] = δ ab (cid:88) j> ( q − p +1) n ( − ) j (cid:0) a ( j ) b pq ( j − n ) − a ( j + 2( p − q ) n ) b pq ( j ) (cid:1) φ a − j n φ a j n + p − q − . It is straightforward to check that a ( j ) b pq ( j − n ) − a ( j + 2( p − q ) n ) b pq ( j ) = − pb p − ,q ( j ) , thus (cid:2) X a , X bpq (cid:3) = − δ ab pX bp − ,q , if p − q (cid:54) = 1 . (4.18)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 isnormally ordered, the last one not. This gives (cid:2) X a , X bq +1 ,q (cid:3) = − δ ab ( q + 1) X bq,q − c bq +1 , where c bq +1 = (cid:32) a (2 n ) b q +1 ,q (0) + (cid:88) − n For all p ≥ and q > , one has the following W constraints: (cid:18) X pq + δ p,q q + 2 c q +1 (cid:19) τ a = 0 , for both a = 0 , , where c q = n (cid:88) j =1 − n (cid:18) j n − q (cid:19) q + (cid:18) − j n − q (cid:19) q + (cid:88) j =0 (cid:18) j − q (cid:19) q + (cid:18) − j − q (cid:19) q . he ( n, | y | > | z | X a ( y, z ) = ( − ) n : φ a ( y ) φ a ( e πin z ) := 12 ( yz ) − y n + z n y n − z n (cid:16) Γ a + (cid:0) q, y n (cid:1) Γ a + (cid:0) q, − z n (cid:1) Γ a − (cid:0) q, y n (cid:1) Γ a − (cid:0) q, − z n (cid:1) − (cid:17) and X apq = ( − n q + 1 Res z z p ∂ q +1 y ( y − z ) : φ a ( y ) φ a (cid:0) e πin z (cid:1) : (cid:12)(cid:12)(cid:12)(cid:12) y = z . (4.19)Now ∂ ky (cid:32) ( y − z )( yz ) − y n + z n y n − z n (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z = c ak z − k . Let Γ a ( y, z ) = Γ a + (cid:0) q, y n (cid:1) Γ a + (cid:0) q, − z n (cid:1) Γ a − (cid:0) q, y n (cid:1) Γ a − (cid:0) q, − z n (cid:1) , then X apq = Res z q + 2 q +1 (cid:88) k =0 (cid:18) q + 1 k (cid:19) c ak z p − k ∂ q − k +1 y (Γ a ( y, z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z and X apq + δ p,q q + 2 c aq +1 = Res z q + 2 q +1 (cid:88) k =0 (cid:18) q + 1 k (cid:19) c ak z p − k ∂ q − k +1 y (Γ a ( y, z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z . We now want to obtain one formula in which we combine all our W constraints. For this, wefirst write the generating series of the c ak : ∞ (cid:88) k =0 c ak k ! z k = ∞ (cid:88) k =0 n (cid:88) j =1 − n (cid:18) j n k (cid:19) + (cid:18) − j n k (cid:19) z k = n (cid:88) j =1 − n (cid:16) (1 + z ) j n + (1 + z ) − j n (cid:17) . Next we calculate for | u | > | z | > | w | , ∞ (cid:88) p,q =0 X apq q ! u − p − w q +1 = 12 ∞ (cid:88) p,q =0 Res z u − p − w q +1 ( q + 1)! q +1 (cid:88) k =0 (cid:18) q + 1 k (cid:19) c ak z p − k ( ∂ y ) q − k +1 (Γ a ( y, z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z = 12 Res z u − z ∞ (cid:88) q =0 q +1 (cid:88) k =0 c ak k ! (cid:16) wz (cid:17) k ( w∂ y ) q − k +1 ( q − k + 1)! (Γ a ( y, z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z = 12 Res z u − z ∞ (cid:88) k =0 ∞ (cid:88) (cid:96) =0 c ak k ! (cid:16) wz (cid:17) k ( w∂ y ) (cid:96) (cid:96) ! (Γ a ( y, z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z = 12 Res z u − z n (cid:88) j =1 − n (cid:18)(cid:16) wz (cid:17) j n + (cid:16) wz (cid:17) − j n (cid:19) (Γ a ( z + w, z ) − . z u − z n (cid:88) j =1 − n (cid:18)(cid:16) wz (cid:17) j n + (cid:16) wz (cid:17) − j n (cid:19) = n (cid:88) j =1 − n (cid:18)(cid:16) wu (cid:17) j n + (cid:16) wu (cid:17) − j n (cid:19) − c a . Thus we have Theorem 4.4. For | u | > | z | > | w | , one has the following W constraints: Res z u − z n (cid:88) j =1 − n (cid:18)(cid:16) wz (cid:17) j n + (cid:16) wz (cid:17) − j n (cid:19) Γ ( z + w, z )+ (cid:88) j =0 (cid:18)(cid:16) wz (cid:17) j + (cid:16) wz (cid:17) − j (cid:19) Γ ( z + w, z ) τ a = 0 . We can express this in a different manner. Define q a [ z ] = ∂ y Γ a ( y, z ) (cid:12)(cid:12) y = z and q ar [ z ] = ∂ r − z q a [ z ] , then q r [ z ] = 1 √ n n (cid:88) j =1 ∞ (cid:88) k =0 (cid:32) (cid:126) − q jk ((2 j − / (2 n )) k +1 − r z j − n + k − r − ((2 j − / (2 n )) k + r (cid:126) ∂∂q jk z − j − n − k − r (cid:33) ,q r [ z ] = ∞ (cid:88) k =0 (cid:32) (cid:126) − q n +1 k (1 / k +1 − r z + k − r − (1 / k + r (cid:126) ∂∂q n +1 k z − − k − r (cid:33) , here we use the convention that for m > a ) − m = Γ( a )Γ( a − m ) = ( a − m ) m . Thus X apq q ! + δ pq c q +1 ( q + 1)! = 12 Res z q +1 (cid:88) (cid:96) =0 z p − (cid:96) c (cid:96) (cid:96) ! : S q − (cid:96) +1 (cid:18) q ar [ z ] r ! (cid:19) : , where the S (cid:96) ( x ) are the elementary Schur functions defined by ∞ (cid:88) (cid:96) =0 S (cid:96) ( x ) = exp (cid:32) ∞ (cid:88) k =1 x k z k (cid:33) . Thus we have the following consequence of Theorem 4.4: Corollary 4.5. For | u | > | z | > | w | , Res z u − z (cid:32) n (cid:88) j =1 − n (cid:18)(cid:16) wz (cid:17) j n + (cid:16) wz (cid:17) − j n (cid:19) : e ∞ (cid:80) r =1 q r [ z ] wrr ! :+ (cid:88) j =0 (cid:18)(cid:16) wz (cid:17) j + (cid:16) wz (cid:17) − j (cid:19) : e ∞ (cid:80) r =1 q r [ z ] wrr ! : (cid:33) τ a = 0 . A similar result is described in [2, Section 3.5].he ( n, Unfortunately we do not obtain all the W constraints of Bakalov and Milanov [2] from the stringequation. Kac, Wang and Yan gave a description in [18] of the corresponding W algebra. As ismentioned in [2, Example 2.5], this W algebra is generated by the elements (cf. Remark 2.2) ν d := n +1 (cid:88) i =1 e v i ( − d ) e − v i + e − v i ( − d ) e v i = n (cid:88) i =1 e ρ i ( v ) ( − d ) e ρ n + i ( v ) + (cid:88) i =1 e ρ i ( v n +1 ) ( − d ) e ρ i +1 ( v n +1 ) , d > , and the element π n +1 := v − v − · · · ( − v n ( − v n +1 . Our constraints come from the elements ν d , the constraints related to the element π n +1 cannotbe obtained from the string equation.Since the total descendent potential is a highest weight vector of this W algebra, this means(Theorem 1.1 of [2]) that it is annihilated by all coefficients of the fractional powers of z , wherethe power is ≤ − 1, of all Y ( ν d , z ) and Y ( π n +1 , z ). Now Y (cid:0) ν d , z (cid:1) = 1( d + 1)! ∂ d +1 y ( y − z ) × (cid:32) n (cid:88) j =1 Y (cid:0) e ρ j ( v ) , y (cid:1) Y (cid:0) e ρ j + n ( v ) , z (cid:1) + (cid:88) j =1 Y (cid:0) e ρ j ( v n +1 ) , y (cid:1) Y (cid:0) e ρ j +1 ( v n +1 ) , z (cid:1)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z . Using Remark 2.2 we obtain that Y (cid:0) ν d , z (cid:1) = 1( d + 1)! ∂ d +1 y ( y − z ) × (cid:32) ( − ) n n n (cid:88) j =1 φ (cid:0) e jπi y (cid:1) φ (cid:0) e j + n ) πi z (cid:1) − (cid:88) j =1 φ (cid:0) e jπi y (cid:1) φ (cid:0) e j +1) πi z (cid:1)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = z . (5.1)Using the fact that 1 + ω + ω + · · · + ω k − = 0 for ω (cid:54) = 1 a k -th root of 1, one obtains that allnon-integer powers of z do not appear in (5.1). Hence, Y (cid:0) ν d , z (cid:1) = Res w δ ( z − w ) 1( d +1)! ∂ d +1 y ( y − w ) (cid:0) ( − ) n φ ( y ) φ (cid:0) e nπi w (cid:1) − φ ( y ) φ (cid:0) e πi w (cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) y = w . Using (4.19), we see that the total descendent potential gets annihilated by X pq + δ p,d q + 2 c d +1 , p ≥ , d = 1 , , . . . , which are exactly the constraints appearing in Theorem 4.3. Using the (4.1)-formulation of L − in terms of the elements φ ai , one can show that (cid:104) L − , φ k n (cid:105) = (cid:18) − k n (cid:19) φ k n − , (cid:104) L − , φ k (cid:105) = (cid:18) − k (cid:19) φ k − . (cid:104) L − , t k n e (cid:105) = − (cid:18) t ddt t − (cid:19)(cid:0) t k n (cid:1) e , (cid:104) L − , t k e (cid:105) = − (cid:18) t ddt t − (cid:19)(cid:0) t k (cid:1) e . Now applying the dilaton shift q (cid:55)→ q + 1, then σL σ − changes according to the descrip-tion (4.2) to σL σ − + ∂∂q , and by (2.6) one finds that L − changes into L − + 2 n (cid:126) − α n . Since (cid:104) α n , φ a ( z ) (cid:105) = δ a √ n z n φ ( z ) , we obtain Proposition 6.1. Let W be the point of the Grassmannian which corresponds to the τ -functionthat satisfies the string equation, then W satisfies tW ⊂ W and (cid:18) − ddt + 12 t − + 2 √ n (cid:126) − t n E (cid:19) W ⊂ W. Note that the total descendent potential of a D n +1 type singularity is tau function, thatsatisfies this condition. Vakulenko, used a similar approach in [22]. He showed that the taufunction is unique. However, his action on the Grassmannian seems somewhat strange. Acknowledgements I would like to thank Bojko Bakalov for useful discussions and the three referees for valuablesuggestions, which improved the paper. References [1] Bakalov B., Kac V.G., Twisted modules over lattice vertex algebras, in Lie theory and its Applications inPhysics V, World Sci. Publ., River Edge, NJ, 2004, 3–26, math.QA/0402315.[2] Bakalov B., Milanov T., W -constraints for the total descendant potential of a simple singularity, Compos.Math. (2013), 840–888, arXiv:1203.3414.[3] Date E., Jimbo M., Kashiwara M., Miwa T., Solitons, τ functions and Euclidean Lie algebras, in Mathematicsand Physics (Paris, 1979/1982), Progr. Math. , Vol. 37, Birkh¨auser Boston, Boston, MA, 1983, 261–279.[4] Delduc F., Feh´er L., Regular conjugacy classes in the Weyl group and integrable hierarchies, J. Phys. A:Math. Gen. (1995), 5843–5882, hep-th/9410203.[5] Drinfel’d V.G., Sokolov V.V., Lie algebras and equations of Korteweg–de Vries type, Current Problems inMathematics , Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.,Moscow, 1984, 81–180.[6] Frenkel E., Givental A., Milanov T., Soliton equations, vertex operators, and simple singularities, Funct.Anal. Other Math. (2010), 47–63, arXiv:0909.4032.[7] Fukuma M., Kawai H., Nakayama R., Infinite-dimensional Grassmannian structure of two-dimensionalquantum gravity, Comm. Math. Phys. (1992), 371–403.[8] Givental A., A n − singularities and n KdV hierarchies, Mosc. Math. J. (2003), 475–505, math.AG/0209205.[9] Givental A., Milanov T., Simple singularities and integrable hierarchies, in The breadth of symplec-tic and Poisson geometry, Progr. Math. , Vol. 232, Birkh¨auser Boston, Boston, MA, 2005, 173–201,math.AG/0307176. he ( n, [10] Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. (1983),943–1001.[11] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.[12] Kac V.G., Kazhdan D.A., Lepowsky J., Wilson R.L., Realization of the basic representations of the EuclideanLie algebras, Adv. Math. (1981), 83–112.[13] Kac V.G., Peterson D.H., 112 constructions of the basic representation of the loop group of E , in Symposiumon Anomalies, Geometry, Topology (Chicago, Ill., 1985), World Sci. Publishing, Singapore, 1985, 276–298.[14] Kac V.G., Schwarz A., Geometric interpretation of the partition function of 2D gravity, Phys. Lett. B (1991), 329–334.[15] Kac V.G., van de Leur J., The geometry of spinors and the multicomponent BKP and DKP hierarchies,in The Bispectral Problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes , Vol. 14, Amer. Math. Soc.,Providence, RI, 1998, 159–202.[16] Kac V.G., van de Leur J., The n -component KP hierarchy and representation theory, J. Math. Phys. (2003), 3245–3293, hep-th/9308137.[17] Kac V.G., Wakimoto M., Exceptional hierarchies of soliton equations, in Theta Functions – Bowdoin 1987,Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math. , Vol. 49, Amer. Math. Soc., Providence, RI, 1989,191–237.[18] Kac V.G., Wang W., Yan C.H., Quasifinite representations of classical Lie subalgebras of W ∞ , Adv. Math. (1998), 56–140, math.QA/9801136.[19] Liu S.-Q., Wu C.-Z., Zhang Y., On the Drinfeld–Sokolov hierarchies of D type, Int. Math. Res. Not. (2011), 1952–1996, arXiv:0912.5273.[20] Liu S.-Q., Yang D., Zhang Y., Uniqueness theorem of W -constraints for simple singularities, Lett. Math.Phys. (2013), 1329–1345, arXiv:1305.2593.[21] ten Kroode F., van de Leur J., Bosonic and fermionic realizations of the affine algebra (cid:98) so n , Comm. Algebra (1992), 3119–3162.[22] Vakulenko V.I., Solution of Virasoro conditions for the DKP-hierarchy, Theoret. and Math. Phys. (1996),435–440.[23] van de Leur J., The Adler–Shiota–van Moerbeke formula for the BKP hierarchy, J. Math. Phys. (1995),4940–4951, hep-th/9411159.[24] van de Leur J., The n th reduced BKP hierarchy, the string equation and BW ∞ -constraints, Acta Appl.Math. (1996), 185–206, hep-th/9411067.[25] Wu C.-Z., A remark on Kac–Wakimoto hierarchies of D-type, J. Phys. A: Math. Theor. (2010), 035201,8 pages, arXiv:0906.5360.[26] Wu C.-Z., From additional symmetries to linearization of Virasoro symmetries, Phys. D249