Table Of ContentMany Boson Realizations of Universal Nonlinear W -Algebras, Modified
∞
KP Hierarchies and Graded Lie Algebras
3 Feng Yu
9
9
1
Institute for Theoretical Physics, State University of New York
n
a
J Stony Brook, New York 11794, U.S.A.
4
and
1
Department of Physics, University of Utah
1
v
Salt Lake City, Utah 84112, U.S.A.
3
5
0
1
0
3 Abstract
9
/
h
t
p- An infinite number of free field realizations of the universal nonlinear Wˆ (N) (Wˆ (N) )
∞ 1+∞
e
h algebras, which are identical to the KP Hamiltonian structures, are obtained in terms
:
v
of p plus q scalars of different signatures with p − q = N. They are generalizations
i
X
of the Miura transformation, and naturally give rise to the modified KP hierarchies
r
a
via corresponding realizations of the latter. Their characteristic Lie-algebraic origin is
shown to be the graded SL(p,q).
1
Extended W-algebras arise as fundamental symmetries or constraints in 2d confor-
mal field theories and low-dimensional string theories. In these discoveries, a crucial
link, elucidating the rather mathematical formalisms in terms of more intuitive physical
constituents, has been the W-algebras’ free field realization and moreover their intrinsic
connection to simple Lie algebras. On the other hand, W-algebras are isomorphic to
the Hamiltonian structures of the integrable KP (KdV) systems, hence their realiza-
tion naturally provides a field-theoretical construction of the relevant hierarchies. Two
important examples are the Miura transformation in the KdV context [1] for W [2]
N
and the two boson Wˆ realization [3] in the KP basis. The former expresses the W
∞ N
generators, defined as the coefficient functions q (z) of the differential KdV operator due
r
to the identification of W with the second KdV Hamiltonian structure, in terms of the
N
chiral currents of N free bosons~j(z) = φ~′(z) = (j ,j ,···,j ):
1 2 N
N−2 N
Q = DN + q Dr = (D +~h ·~j), D ≡ ∂/∂z (1)
r i
X Y
r=0 i=1
where ~h are the fundamental weight vectors of SL(N) satisfying ~h ·~h = δ − 1. It
i i j ij N
leads to the solvable W-minimal models [4]. Meanwhile, it allows a ~j(z)-constitution
of the KdV hierarchies, which have been called the modified KdV [5]. In the latter
ˆ
case, the generators u (z) of W [6], a general homogeneous nonlinear deformation of
r ∞
W [7] and isomorphic to the second Hamiltonian structure [8] of the KP hierarchy [9],
∞
are realized by a pair of free bosonic currents ¯j(z) = φ¯′(z), j(z) = φ′(z) through the
pseudo-differential KP operator:
∞ 1
L = D + u D−r−1 = D +¯j j (2)
X r D −(¯j +j)
r=0
with
∞
D−1f(z) = (−1)l(∂lf)D−l−1. (3)
z
X
l=0
It was based on eq.(2) that a Wˆ current algebra [10,3] and further an infinite set
∞
of commuting Wˆ charges [11,12] were uncovered in the SL(2,R)/U(1) coset model.
∞
Besides, KP hides much richer algebraic structure. By taking the powers of (2) or the
2
extention of (1) to negative powers
∞
LN = DN + v D−r−1 (4)
r
X
r=−N+1
as basic ingredients, nonequivalent KP Hamiltonian structures Wˆ (N) are generated by
∞
v (z) with spin, s = r + N + 1, ranging from 2 to ∞, which may even be formally
r
generalized to a family of nonlinear W-infinity algebras [13]. However, there lack both
field-theoretical realization and Lie-algebraic interpretation of Wˆ (N).
∞
In the following, we present an infinite number of free field realizations for Wˆ (N), in-
∞
cluding Wˆ ≡ Wˆ (1), in terms of p bosons of one signature plus q bosons of the opposite
∞ ∞
signature with p−q = N. They are generalizations of the Miura transformation (1) and
the two boson Wˆ realization (2). Moreover, under these realizations, Wˆ (N), or rather
∞ ∞
Wˆ (N) by incorporating a spin-1 current, are reducible with each other, thus are all uni-
1+∞
versal nonlinear W-algebras. Correspondingly, we obtain the many boson realizations
of KP, defining the modified KP hierarchies. We further show, via our bosonic realiza-
tions, that the characteristic Lie algebra associated with Wˆ (N) is the graded SL(p,q)
∞
[14]. These results have played a key role in justifying the Wˆ (N)-constraints on the
∞
recently constructed D > 2 topological strings [15].
Let us proceed as usual in the KP basis in which Wˆ (N) can be most conveniently
∞
exhibited. To be generic, we include the v DN−1 term in the N-th KP operator (4),
−N
which is equivalent to extending Wˆ (N) to Wˆ (N) . The complete structure of Wˆ (N)
∞ 1+∞ 1+∞
{v (z),v (z′)} = k(N)(z)δ(z −z′) (5)
r s rs
may be extracted from the bilinear Hamiltonian form
∞
K(P) ≡ (LNP) LN −LN(PLN) = k(N)p D−r−1 (6)
+ + rs s
X
r=−N
where P = Dsp is an arbitrary pseudo-differential operator and the subscript +
s
P
denotes the purely differential part, and the infinite dimensional operator matrix k(N)
rs
were given explicitly in closed form in [12,13]. Consider now the free bosonic currents
j (z) (i = 1,...,p) and ˜j (z) (l = p+1,...,p+q) satisfying the Poisson brackets
i l
{j (z),j (z′)} = δ ∂ δ(z −z′), {˜j (z),˜j (z′)} = −δ ∂ δ(z −z′),
i j ij z l k lk z
{j (z),˜j (z′)} = 0. (7)
i l
3
With p−q = N but q (or p > N) remaining arbitrary positive integers, there exist an
infinite number of transformations of the N-th KP operator:
∞ p p+q
LN = DN + v D−r−1 = (D +j ) (D −˜j )−1. (8)
r i l
X Y Y
r=−N i=1 l=p+1
In this way, the Wˆ (N) currents v are functions of j , ˜j and their derivatives, and the
1+∞ r i l
Wˆ (N) brackets (5) are realized according to (7).
1+∞
To prove the theorem, let us evaluate the brackets between two arbitrary functionals
f(LN) and g(LN) with LN in the form of (8),
H H
p δf δg p+q δf δg
{ f, g} = ( ( )′ − ( )′), (9)
I I I Xi=1 δji δji l=Xp+1 δ˜jl δ˜jl
where δf/δj stands for usual variational derivative. To be concise, introduce
δf ∞ δf
= Dr , (10)
δLN δv
r=X−N r
followed by
δf δf δf δf
Res(δLN ) = δv = ( δj + δ˜j ) (11)
I δLN I Xr rδvr I Xi iδji Xl lδ˜jl
where Res means the coefficient of the D−1 term, so that
δf p p+q δf i−1
= Res( (D +j ) (D −˜j )−1 (D +j )),
j l j
δj δLN
i j=Yi+1 l=Yp+1 jY=1
δf p+q δf p p+l
= −Res( (D −˜j )−1 (D +j ) (D −˜j )−1). (12)
δ˜jl k=Yp+l k δLN iY=1 i k=Yp+1 k
Replacing (δg/δj )′ by the commutator [D +j ,δg/δj ], etc., and substituting (12) into
i i i
(9), we have
p p p+q δf j
{ f, g} = Res{ (D +j ) (D −˜j )−1 (D+j )
I I I i l δLN i
X Y Y Y
j=1 i=j+1 l=p+1 i=1
p p+q δg j
·( (D +j ) (D −˜j )−1 (D+j ))
i l δLN i +
Y Y Y
i=j+1 l=p+1 i=1
p p+q δf j−1
− (D +j ) (D−˜j )−1 (D +j )
i l δLN i
Y Y Y
i=j l=p+1 i=1
4
p p+q δg j−1
·( (D +j ) (D −˜j )−1 (D+j )) }
i l δLN i +
Y Y Y
i=j l=p+1 i=1
p+q p+q δf p k−1
− Res{ (D−˜j )−1 (D +j ) (D −˜j )−1
I l δLN i l
X Y Y Y
k=p+1 l=k i=1 l=p+1
p+q δg p k−1
·( (D −˜j )−1 (D +j ) (D −˜j )−1)
Y l δLN Y i Y l +
l=k i=1 l=p+1
p+q δf p k
− (D −˜j )−1 (D+j ) (D −˜j )−1
l δLN i l
Y Y Y
l=k+1 i=1 l=p+1
p+q δg p k
·( (D −˜j )−1 (D +j ) (D −˜j )−1) }. (13)
l δLN i l +
Y Y Y
l=k+1 i=1 l=p+1
It is easy to see that only the first or last term in each part of (13) survives the summa-
tions, and (13) turns out to be
{ f(LN), g(LN)}
I I
δf p p+q δg p p+q
= Res[ (D+j ) (D −˜j )−1( (D +j ) (D −˜j )−1)
I δLN i l δLN i l +
Y Y Y Y
i=1 l=p+1 i=1 l=p+1
p p+q δf p p+q δg
− (D+j ) (D −˜j )−1) ( (D+j ) (D −˜j )−1 ) ]
i l +δLN i l δLN +
Y Y Y Y
i=1 l=p+1 i=1 l=p+1
δf δg δf δg
= Res[ LN( LN) − (LN ) LN]
I δLN δLN + δLN δLN +
δf δg
= Res[ K( )] (14)
I δLN δLN
where the equality Res[P,Q] = 0 for arbitrary P, Q has been frequently used. Eq.(14)
H
is none but (5) obtained in terms of (7), thus the Wˆ (N) realizations (8) hold.
1+∞
Theabovederivationremainsvalidafteranarbitrarymixing oftheordersofthe(D+
j ) and (D−˜j )−1 terms in (8). It gives rise to modified p+q free boson representations
i l
of Wˆ (N) , in response to different screening charge couplings of j′ and ˜j′ in the spin-2
1+∞ i l
stress tensor v .
−N+1
In terms of eq.(8), allWˆ (N) algebrasare mutually reducible. Namely, the p−1 plus q
1+∞
boson representation of Wˆ (N−1) can be obtained from the p plus q boson representation
1+∞
of Wˆ (N) , by simply, e.g., setting j = 0 and multiplying LN by D−1 from the left:
1+∞ 1
p p+q
L(p−1)−q = D−1Lp−q | = (D +j ) (D −˜j )−1; (15)
j1=0 i l
Y Y
i=2 l=p+1
5
andinturnthelattermaybeobtainedsimilarlyfromthepplusq+1bosonrepresentation
of Wˆ (N−1):
1+∞
p p+q
Lp−q = Lp−(q+1)D | = (D+j ) (D −˜j )−1. (16)
˜jp+q+1=0 Y i Y l
i=1 l=p+1
The point is that, because j and ˜j are free currents, neither constraint effects the
1 p+q+1
Poisson brackets (7) of the rest of the currents so that the composite W-structures from
(15)-(16) can be naturally derived and coincides with (5).
The realizations of Wˆ (N) are acquired by just imposing the second-class constraint
∞
v = j + ˜j = 0 (17)
−N i l
X X
i l
on (8) (and on the versions with j and ˜j terms mixed). They appear, without loss of
i l
elegency, as
p p+q
LN = (D +~h ·~j) (D −~h ·~j)−1 (18)
i l
Y Y
i=1 l=p+1
where we denote the bosonic currents as a vector ~j = (j ,···,j ,−˜j ,···,−˜j )
1 p p+1 p+q
~ ~
projected onto the overcomplete set (h ,h ), that have the inner product
i l
1 1 1
~ ~ ~ ~ ~ ~
h ·h = δ − , h ·h = −δ − , h ·h = , (19)
i j ij l k lk i l
p−q p−q p−q
~ ~
obeying h + h = 0. An equivalent statement is that the Poisson brackets (7) are
i i l l
P P
modified to
1
{j (z),j (z′)} = (δ − )∂ δ(z −z′),
i j ij z
p−q
1
{˜j (z),˜j (z′)} = −(δ + )∂ δ(z −z′),
l k lk z
p−q
1
{j (z),˜j (z′)} = ∂ δ(z −z′). (20)
i l z
p−q
It is obvious that eq.(18) or (8) are a set of generalizations of the Miura transforma-
tion (1) for W or
N
N−1 N
Q = DN + q Dr = (D+j ) (21)
r i
X Y
r=0 i=1
6
for W . With ˜j = j = 0, the former reduces to the latter, and the Dirac
N+1 l i≥p−q+1
brackets for the rest of the currents appear exactly the same as the Poisson brackets.
From the Hamiltonian point of view, the W (W ) embeddings to (6) are natural,
N+1 N
and the constraints v = 0, which are at least linear in ˜j or j , are actually
r≥0 l i≥p−q+1
first class. Combining with the results of last two paragraphs, Wˆ (N) (Wˆ (N)) all serve
1+∞ ∞
as universal nonlinear W-algebras.
Themany bosonrealizations(18)arealsogeneralizationsofthetwo bosonrealization
(2) of Wˆ , though the latter looks a little different. In particular, with p = 2,q = 1, the
∞
mixed version of (18)
L = (D +~h ·~j)(D−~h ·~j)−1(D+~h ·~j)
1 3 2
1 1
= (D +j ) (D +j ) = D +j j (22)
1 2 2 1
D +j +j D +j +j
1 2 1 2
is identical to (2) by setting j ≡ −¯j, j ≡ −j. Note the ¯j, j Poisson brackets from (20)
2 1
are precisely those for a pair of complex free bosons.
Eq.(8) of (18), etc., are automatically various many boson realizations of the KP
operator, for example,
1 1 1
L = (D +j ) (D +j ) ··· (D +j ). (23)
1 (D −˜j ) 3 (D −˜j ) (D −˜j ) 2s+1
2 4 2s
Since Wˆ (N) (Wˆ (N)) are just KP Hamiltonian structures, the KP hierarchy, which is
1+∞ ∞
generated by the involutive Hamiltonian charges H (z) = 1 ResLm (m ≥ 1),
m m
H H
∂L
= [(Lm) ,L] (24)
+
∂t
m
itself enjoys these realizations. The number of bosons (of different signatures) can be
arbitrarily large, thus the realizations are faithful at least formally. We call them the
modified KP hierarchies, containing all the modified KdV hierarchies [5] by reduction.
On the other hand, we may directly propose a set of well-defined integrable~j ≡ (j ,˜j )-
i l
hierarchies
∂~j
= {~j, H } (25)
∂t I m
m
with the right side evaluated according to the Poisson brackets of ~j. By composition
in terms of the realizations, (25) leads to (the equivalent Hamiltonian form of) the
7
modefied KP hierarchies. It enables us to have explicit field-theoretical studies of these
hierarchies.
Letusinturnseek aLie-algebraicinterpretationoftheWˆ (N) algebrasalongwiththat
∞
of the KP hierarchy. In fact, this has been implied in eqs.(18-19). We now verify that
~ ~
the set (h ,h ) satisfying (19) are the weight vectors of the fundamental representation
i l
of the graded Lie algebra SL(p,q). Let ǫ (i = 1,...,p) and δ (l = p+1,...,p+q) are
i l
orthonormal bases of p+q dimensional linear space with positive and negative metrics
respectively. The simple roots of SL(p,q), α (s = 1,...,p+q−1), are chosen as (upto
s
Weyl-group symmetry)
α = ǫ −δ , α = δ −ǫ , (m = 1,...,q),
2m−1 m m 2m m m+1
α = ǫ −ǫ , (n = 1,...,p−q −1). (26)
2q+n q+n q+n+1
The fundamental weights λ (s = 1,...,p+q−1), satifying λ ·α = δ , are correspond-
s s t st
ingly
q 1 q p−q−1 i
λ = (1− )α + α + (1− )α ,
2m−1 2i−1 2i 2q+i
p−q p−q
X X X
i=1 i=m i=1
m
λ = α , (27)
2m 2i−1
X
i=1
q n n n p−q−1 i
λ = (1− )α + i(1− )α + n(1− )α .
2q+n 2i−1 2q+i 2q+i
p−q p−q p−q
X X X
i=1 i=1 i=n+1
It follows that the p + q weight vectors h = λ − λ (λ = λ = 0) are identical
s s s−1 0 p+q
to (~h ,~h ) in (19). Therefore characteristically, Wˆ (N) are associated with the graded
i l ∞
SL(p,q) with p−q = N in terms of the KP Lax operator (18).
An immediate implication is that one may, `a la Drinfeld-Sokolov [1], reconstruct
the (modified) KP hierarchy as well as its Hamiltonian structures Wˆ (N) according to
∞
the (level-1) SL(p,q) ≡ A(p − 1,q − 1) Kac-Moody algebras. One may also obtain
different sets of integrable hierarchies of the KP type and Wˆ -infinity algebras from other
series of graded Kac-Moody algebras. Meanwhile, the SL(p,q) origin of Wˆ (N) allows us
∞
to carry out the quantization of the latter systematically through the Drinfeld-Sokolov
reduction or the Casimir construction [16], besides fromthe transformation(18) directly.
Moreover, with the prescription suggested in Ref.[17], one expects to establish solvable
8
nonlinear Wˆ (N)-gravity models via quantum Hamiltonian reduction of gauged SL(p,q)
∞
WZW models. Finally, we note that by associating the SL(p,p−1) fundamental weight
vectors to the Lax operator of a set of scalar superfields, instead of scalars of different
signatures, one gets super W-algebras of the finite type [18]. Thus naturally, heterotic
W-algebras would arise via relevant SL(p,q) Lax operators. We shall address these
problems elsewhere.
Acknowledgements
The author thanks P. van Nieuwenhuizen and Y.-S. Wu for discussions, and is grate-
ful to C. N. Yang for providing a stimulating atmosphere and support. This work is
supported in part by NSF grant PHY-9008452.
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