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 principa...
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Johan van de Leur 2019-02-11T17:10:32Z 2019-02-11T17:10:32Z 2014 The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints / Johan van de Leur // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B69; 17B80; 53D45; 81R10 DOI:10.3842/SIGMA.2014.007 https://nasplib.isofts.kiev.ua/handle/123456789/146849 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. 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. I would like to thank Bojko Bakalov for useful discussions and the three referees for valuable suggestions, which improved the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints Article published earlier |
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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.
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The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints / Johan van de Leur // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 26 назв. — англ. |
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2025-11-25T07:13:19Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 007, 19 pages
The (n, 1)-Reduced DKP Hierarchy,
the String Equation and W Constraints?
Johan VAN DE LEUR
Mathematical Institute, University of Utrecht,
P.O. Box 80010, 3508 TA Utrecht, The Netherlands
E-mail: J.W.vandeLeur@uu.nl
URL: http://www.staff.science.uu.nl/~leur0102/
Received September 23, 2013, in final form January 09, 2014; Published online January 15, 2014
http://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 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.
Key words: affine Kac–Moody algebra; loop group orbit; Kac–Wakimoto hierarchy; isotropic
Grassmannian; total descendent potential; W constraints
2010 Mathematics Subject Classification: 17B69; 17B80; 53D45; 81R10
1 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 An 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 cer-
tain 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 cor-
responding tau functions. This gives an equivalent but slightly different formulation of Kac–
Wakimoto Dn principal hierarchy [17]. The total descendent potential of a Dn 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
mailto:J.W.vandeLeur@uu.nl
http://www.staff.science.uu.nl/~leur0102/
http://dx.doi.org/10.3842/SIGMA.2014.007
http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html
2 J. van de Leur
possible to describe – at least part of – the W -algebra constraints in terms of pseudo-differential
operators 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, was
also considered by Liu, Wu and Zhang in [19]. They even obtain Lax equations. However, their
Lax 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 representation
of type D, but also the other level one module. As such we obtain a pair of tau functions τ0
and τ1, which are related. The equations on both tau functions provide (2× 2)-matrix pseudo-
differential operators, with which we can formulate a slightly different, but probably equivalent,
Lax equation. However, in both approaches the equations on the tau-function τ0 is the same.
Wu [26] used the approach of [19] to study the Virasoro-constraints, he showed that they can
be obtained from the string equation. Using the (2× 2)-matrix pseudo-differential approach of
this paper, we recover Wu’s result and even more, the string equation not only produces the
Virasoro constraints but even produces a large part of the Bakalov–Milanov [2] W constraints,
but not all.
2 The (n, 1)-reduced DKP hierarchy
2.1 The principal hierarchy for D
(1)
n+1
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 Dn+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.
Let n be a positive integer, consider the following Clifford algebra Cl(C∞) on the vector
space C∞ with basis φ1
i
2n
, φ2
i
2
, with i ∈ Z and symmetric bilinear form(
φ1
i
2n
, φ1
j
2n
)
=
(
φ2
i
2
, φ2
j
2
)
= (−)iδi,−j ,
(
φ1
i
2n
, φ2
j
2
)
= 0.
The Clifford algebra has the usual commutation relations:
φ1
i
2n
φ1
j
2n
+ φ1
j
2n
φ1
i
2n
= (−)iδi,−j = φ2
i
2
φ2
j
2
+ φ2
j
2
φ2
i
2
, φ1
i
2n
φ2
j
2
+ φ2
j
2
φ1
i
2n
= 0.
We define its corresponding Spin module V with vacuum vector |0〉 as follows (cf. [21]):
φ1
i
2n
|0〉 = φ2
i
2
|0〉 = 0, i > 0,
(
φ1
0 + iφ2
0
)
|0〉 = 0.
The (n, 1)-Reduced DKP Hierarchy 3
The normal ordered elements : φai 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
ϕ(z) =
∑
m∈ 1
2n
Z
ϕ(m)z
−m−1, then ϕ(e2πikz) =
∑
m∈ 1
2n
Z
ω−2kmnϕ(m)z
−m−1.
The fields corresponding to the elements in the Clifford algebra are
φ1(z) =
∑
i∈Z
φ1
i
2n
z
−n−i
2n , φ2(z) =
∑
i∈Z
φ2
i
2
z
−1−i
2 .
Then the commutation relations can be described as follows in term of the anti-commutator { , }
{
φ1(z), φ1
(
e2πinw
)}
= (−)n
2n−1∑
j=0
δj(z − w),
{
φ2(z), φ2
(
e2πiw
)}
= −
1∑
j=0
δjn(z − w),
{
φ1(z), φ2(w)
}
= 0,
where δj(z − w) is the 2n-twisted delta function, e.g. [1]:
δj(z − w) = z
j
2nw
−j
2n δ(z − w) =
∑
k∈ j
2n
+Z
zkw−k−1.
Then, see [21], the modes of the fields
: φa(e2πikz)φb
(
e2πi`z
)
:, 1 ≤ a, b ≤ 2, 0 ≤ k, ` ≤ 2n− 1,
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 = V0 and V1 correspond to the Z2 gradation given by
deg |0〉 = 0, deg φ±ak = 1.
The highest weight vector of V0 is |0〉, the highest weight vector of V1 is
|1〉 =
1√
2
(
φ1
0 − iφ2
0
)
|0〉.
Here V0 is the basic representation, V1 is an other level 1 module. Both modules are isomorphic.
2.2 The DKP hierarchy and its principal reduction
The DKP hierarchy is the following equation on T ∈ V0:
Resz
(
(−)nφ1(z)T⊗ φ1
(
e2πinz
)
T− φ2(z)T⊗ φ2
(
e2πiz
)
T
)
= 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 ∈ V0 is
in this loop group orbit of |0〉, it satisfies the (n, 1)-reduced DKP hierarchy for all integers p ≥ 0
Resz z
p
(
(−)nφ1(z)T⊗ φ1
(
e2πinz
)
T− φ2(z)T⊗ φ2
(
e2πiz
)
T
)
= 0. (2.1)
However, for us it will be more convenient not only to use the action on |0〉 but also on |1〉
and write Ta for the action of the loop group on |a〉, where a = 0, 1. One thus obtains
Resz z
p
(
(−)nφ1(z)Ta ⊗ φ1
(
e2πinz
)
Tb − φ2(z)Ta ⊗ φ2
(
e2πiz
)
Tb
)
= δa+b,1δp0Tb ⊗ Ta (2.2)
for all integers p ≥ 0, here a, b = 0, 1.
4 J. van de Leur
2.3 A Grassmannian description
We follow the description of [15]. The Clifford algebra Cl(C∞) has a natural Z2-gradation
Cl(C∞) = Cl0(C∞) ⊕ Cl1(C∞), where Cl0(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 ∈ Cl0(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
Ann(g|a〉) = {v ∈ C∞|vg|a〉 = 0} =
{
gvg−1 ∈ C∞|v|a〉 = 0
}
= T (g)
(
Ann(|a〉)
)
.
Since
Ann(|a〉) = C
φ1
0 + (−)aiφ2
0√
2
⊕
⊕
i>0
Cφ1
i
2n
⊕ Cφ2
i
2
, (2.3)
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,
dim (Ann(g|a〉 −Ann(g|b〉)) = 1, 0 ≤ a 6= b ≤ 1.
Note that this follows immediately from (2.3). Let e1 and e2 be the orthonormal basis of C2 we
identify
φ1
i
2n
= t
i
2n e1, φ2
i
2
= t
i
2 e2, (2.4)
where we assume that the bilinear form does not change, i.e.,(
t
i
2n e1, t
j
2n e1
)
= (−)iδi,−j ,
(
t
i
2 e2, t
j
2 e2
)
= (−)iδi,−j ,
(
t
i
2n e1, t
j
2 e2
)
= 0.
We think of t = eiθ as the loop parameter. Now if g corresponds to an element in D
(1)
n+1, then
Ann(g|a〉) satisfies
tAnn(g|a〉) ⊂ Ann(g|a〉), a = 0, 1.
2.4 A bosonization procedure
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 of
type Dn+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 Dn+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, qak ; a = 1, 2, . . . , n+ 1, k = 0, 1, . . .]. Here θ is a Grassmann variable satisfying θ2 = 0. Let σ
be the isomorphism that maps V into F , we take σ(V0) = F0 = C[qak ; a = 1, 2, . . . , n + 1, k =
The (n, 1)-Reduced DKP Hierarchy 5
0, 1, . . .] and σ(V1) = F1 = θC[qak ; a = 1, 2 . . . n + 1, k = 0, 1, . . .]. The Heisenberg algebra, αak is
defined by
α1(z) =
∑
i∈ 1
2n
+ 1
n
Z
α1
i z
−i−1 :=
(−1)n
2
√
n
: φ1(z)φ1
(
e2πinz
)
:,
α2(z) =
∑
i∈ 1
2
+Z
α2
i z
−i−1 :=
−1
2
: φ2(z)φ2
(
e2πiz
)
: .
Then [
αak, α
b
`
]
= kδabδk,−` and
[
α1
k, φ
1(z)
]
=
zk√
n
φ1(z),
[
α2
k, φ
2(z)
]
= zkφ1(z).
Remark 2.2. Note that in the notation of [2], n = N − 1,
α1(z) = Y (
√
nv1, z) =
√
nY (v1, z), α2(z) = Y (vn+1, z)
and
φ1(z) =
1√
2n
Y
(
ev1 , z
)
, φ2(z) =
1√
2
Y
(
evn+1 , z
)
. (2.5)
Here the vi form an orthonormal basis of the Cartan subalgebra of the Lie algebra of type Dn+1.
Elements evi are elements in the group algebra of the root lattice of type Bn+1, which has as
basis the elements vi. 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 v1, v2, . . . , vn, vn+1 as follows
v1 7→ v2 7→ · · · 7→ vn 7→ −v1, vn+1 7→ −vn+1,
then (see [2] or [1])
Y (vj , z) = Y
(
ρj−1(v1), z
)
= Y
(
v1, e
2(j−1)πiz
)
,
Y
(
evj , z
)
= Y
(
ev1 , e2(j−1)πiz
)
, 1 < j ≤ n.
The factors 1√
2n
and 1√
2
in (2.5) follow from the fact that Bv1,−v1 = 4n and Bvn+1,−vn+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,
σα1
− 2j−1
2n
−kσ
−1 =
~−
1
2 qjk
((2j − 1)/(2n))k
, σα1
2j−1
2n
+k
σ−1 = ((2j − 1)/(2n))k+1 ~
1
2
∂
∂qjk
, (2.6)
σα2
− 1
2
−kσ
−1 =
~−
1
2 qn+1
k
(1/2)k
, σα2
1
2
+k
σ−1 = (1/2)k+1 ~
1
2
∂
∂qn+1
k
, (2.7)
for k = 0, 1, 2, . . . 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)0 = 1). To describe σφa(z)σ−1, we introduce two extra operators
θ and ∂
∂θ , then
σφ1(z)σ−1 =
(
θ + ∂
∂θ
)
√
2
z−
1
2 Γ1
(
q, z
1
2n
)
, σφ2(z)σ−1 = i
(
θ − ∂
∂θ
)
√
2
z−
1
2 Γ2
(
q, z
1
2
)
,
6 J. van de Leur
where
Γ1
(
q, z
1
2n
)
= Γ1
+
(
q, z
1
2n
)
Γ1
−
(
q, z
1
2n
)
, Γ2
(
q, z
1
2
)
= Γ2
+
(
q, z
1
2
)
Γ2
−
(
q, z
1
2
)
(2.8)
with
Γ1
+
(
q, z
1
2n
)
= exp
1√
n
n∑
j=1
∞∑
k=0
~−
1
2 qjk
((2j − 1)/(2n))k+1
z
2j−1
2n
+k
, (2.9)
Γ1
−
(
q, z
1
2n
)
= exp
1√
n
n∑
j=1
∞∑
k=0
− ((2j − 1)/(2n))k ~
1
2
∂
∂qjk
z−
2j−1
2n
−k
, (2.10)
Γ2
+
(
q, z
1
2
)
= exp
( ∞∑
k=0
~−
1
2 qn+1
k
(1/2)k+1
z
1
2
+k
)
, (2.11)
Γ2
−
(
q, z
1
2
)
= exp
( ∞∑
k=0
− (1/2)k ~
1
2
∂
∂qn+1
k
z−
1
2
−k
)
. (2.12)
Now let σ(T0) = τ0 and σ(T1) = τ1θ Using (2.9)–(2.12) we can rewrite the equation (2.2) and
thus obtain a family of Hirota bilinear equations on τa, here p ≥ 0:
Resλ
(
λ2np−1Γ1(q, λ)τa ⊗ Γ1(q,−λ)τb − (−)a+bλ2p−1Γ2(q, λ)τa ⊗ Γ2(q,−λ)τb
)
= 2δa+b,1δp0τb ⊗ τa. (2.13)
From now on we will often omit σ.
Using Remark 2.1, we obtain that the total descendent potential of a Dn+1 singularity sa-
tisfies (2.13).
3 Sato–Wilson and Lax equations
3.1 Pseudo-differential operator approach
We want to reformulate (2.13) in terms of pseudo-differential operators. For this we introduce
an extra variable x by replacing q1
0 and qn+1
0 by q1
0 + ~
1
2
2n x and qn+1
0 + ~
1
2
2 x and write ∂ for ∂x.
Then both τa and Γb(q, λ)τa for b = 1, 2, defined in (2.8), will depend on x. We keep the
dependence in τa but remove it in the second term by writing Γb(x, q, λ)τa = Γb(q, λ)τae
xλ.
Next we rewrite (2.2):
Resλ
(
W (λ)diag
(
λ2np−1, λ2p−1
)
⊗W (−λ)T
)
= δp0V ⊗ V T ,
where
W (λ) =
(
Γ1(q, λ)τ0 iΓ2(q, λ)τ0
iΓ1(q, λ)τ1 Γ2(q, λ)τ1
)
exλ, V =
(
τ1 iτ1
iτ0 τ0
)
. (3.1)
Divide the first row of W and V by τ1 and the second by τ0, one thus obtains
Resλ
(
P (λ)diag
(
λ2np−1, λ2p−1
)
E(λ)exλ ⊗ e−xλE(−λ)TP (−λ)TJ
)
= δp0I, (3.2)
where
P (λ) =
1√
2
Γ1
−(q, λ)τ0
τ1
i
Γ2
−(q, λ)τ0
τ1
i
Γ1
−(q, λ)τ1
τ0
Γ2
−(q, λ)τ1
τ0
The (n, 1)-Reduced DKP Hierarchy 7
and
E(λ) =
(
Γ1
+(q, λ) 0
0 Γ2
+(q, λ)
)
, J =
(
0 −i
−i 0
)
.
Then using the fundamental Lemma of [16], equation (3.2) leads to:
(P (∂)diag
(
∂2np−1, ∂2p−1
)
P ∗(∂)J)− = δp0∂
−1I.
Taking p = 0 one deduces that
P−1∂−1 = ∂−1P ∗J (3.3)
and for p > 0 that(
Pdiag
(
∂2np, ∂2p
)
P−1
)
≤0
= 0.
Now differentiate (3.2) for p = 0 to some qjk and apply the fundamental lemma then one gets
the following Sato–Wilson equations:
∂P
∂qjk
P−1 = −
(
Bj
k
)
≤0
,
where
Bj
k =
1√
n
~−
1
2
(2j − 1)/(2n))k+1
PE11∂
2j−1+2knP−1 if j ≤ n,
~−
1
2
(1/2)k+1
PE22∂
1+2kP−1 if j = n+ 1.
Now introduce the operators
L = P∂P−1, Ca = PEaaP
−1.
Then clearly
[L,Ca] = 0, CaCb = δabCa, C1 + C2 = I,
(
L2npC1 + L2pC2
)
≤0
= 0 (3.4)
and one has the following Lax equations:
∂L
∂qjk
=
[(
Bj
k
)
>0
, L
]
,
∂Ca
∂qjk
=
[(
Bj
k
)
>0
, Ca
]
.
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.
3.2 The Orlov–Schulman and S operator
Introduce the Orlov–Schulman operator
M = PExE−1P−1 = PRP−1,
8 J. van de Leur
where
R = xI + 2~−
1
2
∞∑
k=0
(
√
nE11
n∑
j=1
qjk
((2j − 1)/(2n))k
∂2nk+2j−2 + E22
qn+1
k
(1/2)k
∂2k
)
.
Then [L,M ] = I and the wave function W (λ) satisfies
LW (λ) = λW (λ), CiW (λ) = W (λ)Eii, MW (λ) =
∂W (λ)
∂λ
.
Moreover,
M =
∂P
∂∂
P−1 + 2~−
1
2
∞∑
k=0
(
√
n
n∑
j=1
qjk
((2j − 1)/(2n))k
L2nk+2j−2C1 +
qn+1
k
(1/2)k
L2kC2
)
.
We introduce the operator
S =
(
1
2n
ML1−2nC1 +
1
2
ML−1C2
)
≤0
P,
which will play a crucial role in the deduction of the W constraints. S is explicitly given by
S =
1
2n
∂P
∂∂
∂1−2nE11 +
1
2
∂P
∂∂
∂−1E22
+
∞∑
k=0
1√
n~
n∑
j=1
qjk
((2j − 1)/(2n))k
L2n(k−1)+2j−1C1 +
1√
~
qn+1
k
(1/2)k
L2k−1C2
≤0
P
=
1
2n
∂P
∂∂
∂1−2nE11 +
1
2
∂P
∂∂
∂−1E22 +
1√
n~
n∑
j=1
qj0P∂
2j−2n−1E11 +
1√
~
qn+1
0 P∂−1E22
+
∞∑
k=1
1√
n~
n∑
j=1
qjk
((2j − 1)/(2n))k
L2n(k−1)+2j−1C1 +
1√
~
qn+1
k
(1/2)k
L2k−1C2
≤0
P
=
1
2n
∂P
∂∂
∂1−2nE11 +
1
2
∂P
∂∂
∂−1E22 +
1√
n~
n∑
j=1
qj0P∂
2j−2n−1E11 +
1√
~
qn+1
0 P∂−1E22
−
n+1∑
j=1
∞∑
k=0
qjk+1
∂P
∂qjk
. (3.5)
4 The string equation and W constraints
4.1 The principal Virasoro algebra
The principal realization of the basic representation of type D
(1)
n+1 has a natural Virasoro algebra
with central charge n+ 1. It is given by (see, e.g., [21])
Lk =
∑
j∈Z
(−)j
j
4n
: φ1
−j
2n
φ1
j
2n
+k
: +(−)j
j
4
: φ2
−j
2
φ2
j
2
+k
: +δk,0
(
n+ 1
16n
+
n2 − 1
24n
)
=
∑
j∈Z
1
2
: α1
− 1
2n
− j
n
α1
1
2n
+ j
n
+k
: +
1
2
: α2
− 1
2
−jα
2
1
2
+j+k
: +δk,0
(
n+ 1
16n
+
n2 − 1
24n
)
, (4.1)
The (n, 1)-Reduced DKP Hierarchy 9
or in terms of the field
L(z) =
∑
k∈Z
Lkz
−k−2 =
1
2
w−
1
2
∂
∂w
w
1
2
×
(
(−)n : φ1(w)φ1
(
e2πinz
)
: − : φ2(w)φ2
(
e2πiz
)
:
)∣∣
w=z
+
(
n+ 1
16n
+
n2 − 1
24n
)
z−2.
Using (2.6) we can express Lk in terms of the “times” qjk, in particular L−1 is equal to
σL−1σ
−1 =
1
2~
(
qn+1
0
)2
+
1
2~
n∑
j=1
qj0q
n+1−j
0 +
n+1∑
`=1
∞∑
k=0
q`k+1
∂
∂q`k
. (4.2)
Let τ ∈ V0, the string equation is the following equation on τ
L−1τ =
∂τ
∂q1
0
. (4.3)
However, following, e.g., [7], we remove the right-hand side of (4.3) by introducing the shift
q1
1 7→ q1
1 − 1. This reduces the string equation to
L−1τ = 0. (4.4)
However, this would introduce in the vertex operator Γ1
+(q, λ) of (2.9) some extra part
e
− (2n)2~−
1
2√
n
λ2n+1
2n+1 ,
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 WDn+1 W -algebra. We will follow the
approach of [24] (see also [23]). For this we use the following. If τ = τ0 = g|0〉 satisfies the
string equation, then also its companion τ1 = g|1〉, satisfies the string equations. This is because
σL−1σ
−1 commutes with the operator θ + ∂
∂θ which intertwines F0 with F1.
4.2 A consequence of the string equation
Assume that the string equation (4.4) L−1τa = 0 holds for both a = 0, 1. Then clearly also
Γc−(λ) (L−1τa)
τb
− L−1τb
(τb)2
Γc−(λ)(τa) = 0. (4.5)
Denote by τ cd = Γc−(λ)(τd), then (4.5) is equivalent to
τbΓ
c
−(λ) (L−1) τ ca − τ caL−1τb
(τb)2
= 0. (4.6)
Now,
Γc−(λ) (L−1) =
1
2~
Γc−(λ)
(qn+1
0
)2
+
n∑
j=1
qj0q
n+1−j
0
+
n+1∑
`=1
∞∑
k=0
Γc−(λ)
(
q`k+1
) ∂
∂q`k
,
10 J. van de Leur
hence (4.6) turns into
1
2~
τ ca
τb
(
Γc−(λ)− 1
)(qn+1
0
)2
+
n∑
j=1
qj0q
n+1−j
0
+
n+1∑
`=1
∞∑
k=0
(
Γc−(λ)
(
q`k+1
)
τb
∂τ ca
∂q`k
− τ ca
(τb)2
q`k+1
∂τb
∂q`k
)
= 0.
We rewrite this as
n+1∑
`=1
∞∑
k=0
q`k+1
∂ τ
c
a
τb
∂q`k
+Rabc = 0, (4.7)
where
Rab1 =
τ1
a
2τb
λ−2n − 1√
n~
τ1
a
τb
n∑
j=1
λ1−2jqn+1−j
0
−
√
~√
nτb
n∑
`=1
∞∑
k=0
((2`− 1)/2n)k+1λ
1−2nk−2n−2`∂τ
1
a
∂q`k
and
Rab2 =
τ2
a
2τb
λ−2 − 1√
~
τ2
a
τb
λ−1qn+1
0 −
√
~
τb
∞∑
k=0
(1/2)k+1λ
−2k−3 ∂τ2
a
∂qn+1
k
.
We will now prove the following
Proposition 4.1. The string equation (4.4) induces((
1
2n
ML1−2n − 1
2
L−2n
)
C1 +
(
1
2
ML−1 − 1
2
L−2
)
C2
)
≤0
= 0. (4.8)
Proof. To prove this we first observe that (4.8) is equivalent to(
1
2
P∂−2nE11 +
1
2
P∂−2E22
)
≤0
− S = 0, (4.9)
where S is given by (3.5). We calculate the various parts of this formula:
1
2
Pa1λ
−2n − 1√
n~
n∑
j=1
qj0Pa1λ
2j−2n−1 =
ia−1τ1
a−1
2
√
2τ2−a
λ−2n −
ia−1τ1
a−1√
2n~τ2−a
n∑
j=1
λ2j−2n−1qj0,
1
2
Pa2λ
−2 − 1√
~
qn+1
0 Pa2λ
− =
i(−i)a−1τ1
a−1
2
√
2τ2−a
λ−2 −
i(−i)a−1τ1
a−1√
2~τ2−a
λ−1.
Now
1
2n
∂Pa1(λ)
∂λ
λ1−2n =
ia−1~
1
2
√
2nτ2−a
∂τ1
a−1
∂λ
λ1−2n
=
ia−1~
1
2
√
2nτ2−a
n∑
`=1
∞∑
k=0
((2`− 1)/2n)k+1λ
1−2n(k+1)−2`∂τ
1
a−1
∂q`k
The (n, 1)-Reduced DKP Hierarchy 11
and
1
2
∂Pa2(λ)
∂λ
λ−1 =
i(−i)a−1~
1
2
√
2τ2−a
∂τ1
a−1
∂λ
λ−1 =
i(−i)a−1~
1
2
√
2τ2−a
∞∑
k=0
(1/2)k+1λ
−2k−3 ∂τ
1
a−1
∂qn+1
k
.
Substituting these formulas into (4.9) one obtains up to a multiplicative scalar
n+1∑
`=1
∞∑
k=0
q`k+1
∂τ1
a−1/τ2−a
∂q`k
+
τ1
a−1
2τ2−a
λ−2n −
τ1
a−1√
n~τ2−a
n∑
j=1
λ2j−2n−1qj0
− ~
1
2
√
nτ2−a
n∑
`=1
∞∑
k=0
((2`− 1)/2n)k+1λ
1−2n(k+1)−2`∂τ
1
a−1
∂q`k
= 0
and
n+1∑
`=1
∞∑
k=0
q`k+1
∂τ1
a−1/τ2−a
∂q`k
+
τ1
a−1
2τ2−a
λ−2 −
τ1
a−1√
~τ2−a
λ−1qn+1
0
− ~
1
2
τ2−a
∞∑
k=0
(1/2)k+1λ
−2k+3 ∂τ
1
a−1
∂qn+1
k
= 0,
which is exactly equation (4.7). �
A consequence of (3.4) and Proposition 4.1:
Proposition 4.2. Let τ satisfy the string equation, then for all p, q ≥ 0, except p = q = 0, the
following equation holds:((
1
2n
ML1−2n − 1
2
L−2n
)q
L2npC1 +
(
1
2
ML−1 − 1
2
L−2
)q
L2pC2
)
≤0
= 0. (4.10)
We rewrite the formula (4.10), using (3.3):((
1
2n
PR∂1−2n − 1
2
∂−2n
)q
∂2np−1E11P
∗J +
(
1
2
R∂−1 − 1
2
∂−2
)q
∂2p−1E22P
∗J
)
−
= 0.
Now using again the fundamental Lemma of [16] this gives
Resλ
(
λ2np−1
(
1
2n
λ1−2n∂λ −
1
2
λ−2n
)q
W (λ)E11 (4.11)
+λ2p−1
(
1
2
λ−1∂λ −
1
2
λ−2
)q
W (λ)E22
)
⊗W (−λ)T = 0. (4.12)
Now let λk = z, then ∂z = 1
kλ
1−k∂λ and 1
kλ
1−k∂λ− 1
2λ
−k = z
1
2∂zz
− 1
2 , then (4.12) is equivalent to
Resz
(
zp∂qz
(
z−
1
2 Γ1
(
q, z
1
2n
)
τa
)
⊗ z−
1
2 Γ1
(
q,−z
1
2n
)
τb
− (−)a+bzp∂qz
(
z−
1
2 Γ2
(
q, z
1
2
)
τa
)
⊗ z−
1
2 Γ2
(
q,−z
1
2
)
τb
)
= 0.
And this formula induces
Resz
(
(−)nzp∂qz
(
φ1(z)
)
Ta ⊗ φ1
(
e2πinz
)
Tb − zp∂qz
(
φ2(z)
)
Ta ⊗ φ2
(
e2πiz
)
Tb
)
= 0,
Resz
(
(−)nzpφ1
(
e2πinz
)
Ta ⊗ ∂qz
(
φ1(z)
)
Tb − zpφ2
(
e2πiz
)
Ta ⊗ ∂qz
(
φ2(z)
)
Tb
)
= 0. (4.13)
12 J. van de Leur
4.3 Some useful formulas
We have[
: φ1(y)φ1
(
e2πinw
)
:, φ1(z)
]
= φ1(y)
{
φ1
(
e2πinw
)
, φ1(z)
}
−
{
φ1(y), φ1(z)
}
φ1
(
e2πinw
)
= (−)n
2n−1∑
j=0
δj(z − w)φ1(y)− δj
(
z − e2πiny
)
φ1
(
e2πinw
)
,
and similarly
[
: φ2(y)φ2
(
e2πiw
)
:, φ2(z)
]
= −
1∑
j=0
δjn(z − w)φ2(y)− δjn
(
z − e2πiy
)
φ2
(
e2πiw
)
.
We calculate the action of
X(y, w)⊗ 1 =
(
(−)n : φ1(y)φ1
(
e2πinw
)
: − : φ2(y)φ2
(
e2πiw
)
:
)
⊗ 1
on the bilinear identity (2.2), using the above formulas one obtains
δa+b,1δp0X(y, w)Tb ⊗ Ta
= Resz z
p
{
(−)n
(
2n−1∑
j=0
δj(z − w)φ1(y)− δj
(
z − e2πiny
)
φ1
(
e2πinw
))
Ta ⊗ φ1
(
e2πinz
)
Tb
−
(
1∑
j=0
δjn(z − w)φ2(y)− δjn
(
z − e2πiy
)
φ2
(
e2πiw
))
Ta ⊗ φ1
(
e2πinz
)
Tb
+ (−)nφ1(z)X(y, w)Ta ⊗ φ1(e2πinz)Tb − φ2(z)X(y, w)Ta ⊗ φ2(e2πiz)Tb
}
.
Thus
δa+b,1δp0X(y, w)Tb ⊗ Ta
− Resz
(
zp
(
(−)nφ1(z)X(y, w)Ta ⊗ φ1
(
e2πinz
)
Tb − φ2(z)X(y, w)Ta ⊗ φ2
(
e2πiz
)
Tb
) )
= wp
(
(−)nφ1(y)Ta ⊗ φ1
(
e2πinw
)
Tb − φ2(y)Ta ⊗ φ2
(
e2πiw
)
Tb
)
− yp
(
(−)nφ1
(
e2πinw
)
Ta ⊗ φ1(y)Tb − φ2
(
e2πiw
)
Ta ⊗ φ2(y)Tb
)
. (4.14)
4.4 W constraints
Now, let Xk` = Resw w
k∂`yX(y, w)|y=w, then putting p = 0 in formula (4.14) and using (4.13),
one deduces
Resz
(
(−)nφ1(z)XpqTa ⊗ φ1
(
e2πinz
)
Tb − φ2(z)XpqTa ⊗ φ2
(
e2πiz
)
Tb
)
= δa+b,1XpqTb ⊗ Ta.
Thus
Resλ λ
−1
(
Γ1(q, λ)Xpqτa ⊗ Γ1(q,−λ)τb − (−)a+bΓ2(q, λ)Xpqτa ⊗ Γ2(q,−λ)τb
)
= 2δa+b,1Xpqτb ⊗ τa. (4.15)
Note that here we abuse the notation, we write Xpq for σXpqσ
−1. Consider this as equation in
two sets of variables x, q and x′, q′. Let a 6= b and set x = x′ and q = q′, This gives
Xpqτa
τa
=
Xpqτb
τb
. (4.16)
The (n, 1)-Reduced DKP Hierarchy 13
Now divide Γc(q, λ) (Xpqτa) by τb, then
Γc(q, λ) (Xpqτa)
τb
= Γc+(q, λ)
Γc−(q, λ)τa
τb
Γc−(q, λ)
(
Xpqτa
τa
)
.
Now using (4.16), we rewrite (4.15) in the matrix version
Resλ
(
λ−1
2∑
c=1
Γc−(q, λ)
(
Xpqτa
τa
)
P (λ)EccE(λ)exλ ⊗ e−xλE(−λ)TP (−λ)TJ
)
=
Xpqτa
τa
I.
Let
Γc−(q, λ)
(
Xpqτa
τa
)
=
∞∑
k=0
Sck(x, q)λ
−k,
then
Resλ
(
2∑
c=1
∞∑
k=0
SckP (λ)Eccλ
−k−1E(λ)exλ ⊗ e−xλE(−λ)TP (−λ)TJ
)
=
Xpqτa
τa
I.
This gives
∞∑
k=1
SckP (∂)Ecc∂
−kP (∂)−1 = 0.
Now multiplying with P (∂)∂`−1 from the right and taking the residue, one deduces that
Sc` (x, q) = 0 for ` = 1, 2, . . . ,
hence,
(
Γc−(q, λ)− 1
)(Xpqτa
τa
)
= 0,
from which we conclude that
Xpqτa
τa
= const.
In order to calculate these constants, we determine [X01, Xpq] and [X11, X0q]. The action of
both operators on τ give zero. Now write Xpq = X1
pq +X2
pq, then
Xa
pq =
∑
k>(q−p)n
(−)kbpq(k)φa− k
2n
φak
2n
+p−q,
where
bpq(k) =
(
k
2n
+
1
2
− q
)
q
−
(
− k
2n
+
1
2
− p
)
q
.
From now on we assume n = 1 if a = 2, in particular
Xa
01 =
∑
k>n
(−)ka(k)φa− k
2n
φak
2n
−1
, where a(k) =
k
n
− 1.
14 J. van de Leur
Then
[Xa
01, X
b
pq] = δab
∑
j>n,k>(q−p)n
(−)ka(j)bpq(k)
(
δj−2n,kφ
a
− j
2n
φaj
2n
+p−q−1
+ δj,−k+2(q−p+1)nφ
a
− k
2n
φak
2n
+p−q−1
− δj,kφaj
2n
−1
φa− j
2n
+p−q
− δj,k+2(p−q)nφ
a
− k
2n
φak
2n
+p−q−1
)
. (4.17)
Now, if p− q 6= 1 the right hand side is normally ordered and we obtain
[Xa
01, X
b
pq] = δab
∑
j>(q−p+1)n
(−)j
(
a(j)bpq(j − 2n)− a(j + 2(p− q)n)bpq(j)
)
φa− j
2n
φaj
2n
+p−q−1
.
It is straightforward to check that
a(j)bpq(j − 2n)− a(j + 2(p− q)n)bpq(j) = −2pbp−1,q(j),
thus [
Xa
01, X
b
pq
]
= −2δabpX
b
p−1,q, if p− q 6= 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 is
normally ordered, the last one not. This gives[
Xa
01, X
b
q+1,q
]
= −2δab(q + 1)Xb
q,q − 2cbq+1,
where
cbq+1 =
(
1
2
a(2n)bq+1,q(0) +
∑
−n<k<0
a(k + 2n)bq+1,q(k)
)
=
n∑
j=1−n
(
j
2n
− q
)
q+1
+
(
− j
2n
− q
)
q+1
.
Clearly, if p = 0 the right hand side of (4.18) is equal to 0. For that case, one calculates
[Xa
11, X0q] = 2qX0q,
so finally we obtain the following result. Note that Xp,0 = 0 and let cq = c1
q + c2
q , then
Theorem 4.3. For all p ≥ 0 and q > 0, one has the following W constraints:(
Xpq +
δp,q
2q + 2
cq+1
)
τa = 0, for both a = 0, 1,
where
cq =
n∑
j=1−n
(
j
2n
− q
)
q
+
(
− j
2n
− q
)
q
+
1∑
j=0
(
j
2
− q
)
q
+
(
− j
2
− q
)
q
.
The (n, 1)-Reduced DKP Hierarchy 15
It is straightforward to check that for |y| > |z|
Xa(y, z) = (−)n : φa(y)φa(e2πinz) :
=
1
2
(yz)−
1
2
y
1
2n + z
1
2n
y
1
2n − z
1
2n
(
Γa+
(
q, y
1
2n
)
Γa+
(
q,−z
1
2n
)
Γa−
(
q, y
1
2n
)
Γa−
(
q,−z
1
2n
)
− 1
)
and
Xa
pq =
(−1)n
q + 1
Resz z
p∂q+1
y (y − z) : φa(y)φa
(
e2πinz
)
:
∣∣∣∣
y=z
. (4.19)
Now
∂ky
(
(y − z)(yz)−
1
2
y
1
2n + z
1
2n
y
1
2n − z
1
2n
)∣∣∣∣∣
y=z
= cakz
−k.
Let
Γa(y, z) = Γa+
(
q, y
1
2n
)
Γa+
(
q,−z
1
2n
)
Γa−
(
q, y
1
2n
)
Γa−
(
q,−z
1
2n
)
,
then
Xa
pq = Resz
1
2q + 2
q+1∑
k=0
(
q + 1
k
)
cakz
p−k∂q−k+1
y (Γa(y, z)− 1)
∣∣∣∣∣
y=z
and
Xa
pq +
δp,q
2q + 2
caq+1 = Resz
1
2q + 2
q+1∑
k=0
(
q + 1
k
)
cakz
p−k∂q−k+1
y (Γa(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 cak:
∞∑
k=0
cak
k!
zk =
∞∑
k=0
n∑
j=1−n
(
j
2n
k
)
+
(
− j
2n
k
) zk =
n∑
j=1−n
(
(1 + z)
j
2n + (1 + z)−
j
2n
)
.
Next we calculate for |u| > |z| > |w|,
∞∑
p,q=0
Xa
pq
q!
u−p−1wq+1
=
1
2
∞∑
p,q=0
Resz
u−p−1wq+1
(q + 1)!
q+1∑
k=0
(
q + 1
k
)
cakz
p−k (∂y)
q−k+1 (Γa(y, z)− 1)
∣∣∣∣∣
y=z
=
1
2
Resz
1
u− z
∞∑
q=0
q+1∑
k=0
cak
k!
(w
z
)k (w∂y)
q−k+1
(q − k + 1)!
(Γa(y, z)− 1)
∣∣∣∣∣
y=z
=
1
2
Resz
1
u− z
∞∑
k=0
∞∑
`=0
cak
k!
(w
z
)k (w∂y)
`
`!
(Γa(y, z)− 1)
∣∣∣∣∣
y=z
=
1
2
Resz
1
u− z
n∑
j=1−n
((
1 +
w
z
) j
2n
+
(
1 +
w
z
)− j
2n
)
(Γa(z + w, z)− 1) .
16 J. van de Leur
Note that
Resz
1
u− z
n∑
j=1−n
((
1 +
w
z
) j
2n
+
(
1 +
w
z
)− j
2n
)
=
n∑
j=1−n
((
1 +
w
u
) j
2n
+
(
1 +
w
u
)− j
2n
)
− ca0.
Thus we have
Theorem 4.4. For |u| > |z| > |w|, one has the following W constraints:
Resz
1
u− z
n∑
j=1−n
((
1 +
w
z
) j
2n
+
(
1 +
w
z
)− j
2n
)
Γ1(z + w, z)
+
1∑
j=0
((
1 +
w
z
) j
2
+
(
1 +
w
z
)− j
2
)
Γ2(z + w, z)
τa = 0.
We can express this in a different manner. Define
qa1 [z] = ∂yΓ
a(y, z)
∣∣
y=z
and qar [z] = ∂r−1
z qa1 [z],
then
q1
r [z] =
1√
n
n∑
j=1
∞∑
k=0
(
~−
1
2 qjk
((2j − 1)/(2n))k+1−r
z
2j−1
2n
+k−r
− ((2j − 1)/(2n))k+r ~
1
2
∂
∂qjk
z−
2j−1
2n
−k−r
)
,
q2
r [z] =
∞∑
k=0
(
~−
1
2 qn+1
k
(1/2)k+1−r
z
1
2
+k−r − (1/2)k+r ~
1
2
∂
∂qn+1
k
z−
1
2
−k−r
)
,
here we use the convention that for m > 0
1
(a)−m
=
Γ(a)
Γ(a−m)
= (a−m)m.
Thus
Xa
pq
q!
+
δpq
2
cq+1
(q + 1)!
=
1
2
Resz
q+1∑
`=0
zp−`
c`
`!
: Sq−`+1
(
qar [z]
r!
)
:,
where the S`(x) are the elementary Schur functions defined by
∞∑
`=0
S`(x) = exp
( ∞∑
k=1
xkz
k
)
.
Thus we have the following consequence of Theorem 4.4:
Corollary 4.5. For |u| > |z| > |w|,
Resz
1
u− z
(
n∑
j=1−n
((
1 +
w
z
) j
2n
+
(
1 +
w
z
)− j
2n
)
: e
∞∑
r=1
q1r [z]w
r
r!
:
+
1∑
j=0
((
1 +
w
z
) j
2
+
(
1 +
w
z
)− j
2
)
: e
∞∑
r=1
q2r [z]w
r
r!
:
)
τa = 0.
A similar result is described in [2, Section 3.5].
The (n, 1)-Reduced DKP Hierarchy 17
5 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)
νd :=
n+1∑
i=1
evi (−d)e
−vi + e−vi (−d)e
vi
=
2n∑
i=1
eρ
i(v1)
(−d)e
ρn+i(v1) +
2∑
i=1
eρ
i(vn+1)
(−d)e
ρi+1(vn+1), d > 0,
and the element
πn+1 := v1(−1)v2(−1) · · ·(−1) vn(−1)vn+1.
Our constraints come from the elements νd, the constraints related to the element πn+1 cannot
be 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, where
the power is ≤ −1, of all Y (νd, z) and Y (πn+1, z). Now
Y
(
νd, z
)
=
1
(d+ 1)!
∂d+1
y (y − z)
×
(
2n∑
j=1
Y
(
eρ
j(v1), y
)
Y
(
eρ
j+n(v1), z
)
+
2∑
j=1
Y
(
eρ
j(vn+1), y
)
Y
(
eρ
j+1(vn+1), z
))∣∣∣∣∣
y=z
.
Using Remark 2.2 we obtain that
Y
(
νd, z
)
=
1
(d+ 1)!
∂d+1
y (y − z)
×
(
(−)n
2n
2n∑
j=1
φ1
(
e2jπiy
)
φ1
(
e2(j+n)πiz
)
− 1
2
2∑
j=1
φ2
(
e2jπiy
)
φ2
(
e2(j+1)πiz
))∣∣∣∣∣
y=z
. (5.1)
Using the fact that 1 + ω + ω2 + · · ·+ ωk−1 = 0 for ω 6= 1 a k-th root of 1, one obtains that all
non-integer powers of z do not appear in (5.1). Hence,
Y
(
νd, z
)
= Resw δ(z−w)
1
(d+1)!
∂d+1
y (y−w)
(
(−)nφ1(y)φ1
(
e2nπiw
)
− φ2(y)φ2
(
e2πiw
))∣∣∣∣
y=w
.
Using (4.19), we see that the total descendent potential gets annihilated by
Xpq +
δp,d
2q + 2
cd+1, p ≥ 0, d = 1, 2, . . . ,
which are exactly the constraints appearing in Theorem 4.3.
6 The string equation on the Grassmannian
Using the (4.1)-formulation of L−1 in terms of the elements φai , one can show that[
L−1, φ
1
k
2n
]
=
(
1
2
− k
2n
)
φ1
k
2n
−1
,
[
L−1, φ
2
k
2
]
=
(
1
2
− k
2
)
φ1
k
2
−1
.
18 J. van de Leur
Then using the identification (2.4) we obtain[
L−1, t
k
2n e1
]
= −
(
t
1
2
d
dt
t−
1
2
)(
t
k
2n
)
e1,
[
L−1, t
k
2 e2
]
= −
(
t
1
2
d
dt
t−
1
2
)(
t
k
2
)
e2.
Now applying the dilaton shift q1
1 7→ q1
1 + 1, then σL1σ
−1 changes according to the descrip-
tion (4.2) to
σL1σ
−1 +
∂
∂q1
0
,
and by (2.6) one finds that L−1 changes into L−1 + 2n~−
1
2α1
1
2n
. Since
[
α1
1
2n
, φa(z)
]
=
δa1√
n
z
1
2nφ1(z),
we obtain
Proposition 6.1. Let W be the point of the Grassmannian which corresponds to the τ -function
that satisfies the string equation, then W satisfies
tW ⊂W and
(
− d
dt
+
1
2
t−1 + 2
√
n~−
1
2 t
1
2nE11
)
W ⊂W.
Note that the total descendent potential of a Dn+1 type singularity is tau function, that
satisfies this condition. Vakulenko, used a similar approach in [22]. He showed that the tau
function 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 valuable
suggestions, which improved the paper.
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1 Introduction
2 The (n,1)-reduced DKP hierarchy
2.1 The principal hierarchy for Dn+1(1)
2.2 The DKP hierarchy and its principal reduction
2.3 A Grassmannian description
2.4 A bosonization procedure
3 Sato-Wilson and Lax equations
3.1 Pseudo-differential operator approach
3.2 The Orlov-Schulman and S operator
4 The string equation and W constraints
4.1 The principal Virasoro algebra
4.2 A consequence of the string equation
4.3 Some useful formulas
4.4 W constraints
5 A comparison with the results of Bakalov and Milanov BM
6 The string equation on the Grassmannian
References
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