A q-deformed Version of the Heavenly Equations
arXiv:hep-th/9405073v2 25 Nov 1995
J. F. Pleba? ski and H. Garc? n ?a-Compe?n a Departamento de F? ?sica Centro de Investi
gaci?n y de Estudios Avanzados del I.P.N. o Apdo. Postal 14-740, 07000, M?xico, D.F., M?xico. e e
Abstract
Using a q-deformed Moyal algebra associated with the group of area preserving di?eomorphisms of th two-dimensional torus T 2 , sdi?q (T 2 ), a q-deformed version for the Heavenly equations is given. Finally, the two-dimensional chiral version of Self-dual gravity in this q-deformed context is brie?y discussed.
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1
Introduction
New directions in mathematical physics seem to converge to Self-dual gravity in di?erent ways. A most interesting formulation is due to Pleba? ski [1]. In n this Ref. it is found that the Self-dual gravity is described by the non-linear second order partial di?erential equation for a holomorphic function (the Heavenly equation). At present time there exists great interest to ?nd a deep relation between Self-dual Yang-Mills gauge theory and Self-dual gravity in dimension four [25]. In these papers Self-dual gravity is obtained from a dimensional reduction of the Self-dual Yang-Mills equations on a 4-dimensional ?at space-time M with in?nite dimensional gauge group SU(∞). This group was taken to be the group of area preserving di?eomorphisms of the two-surface Σ, SDi?(Σ). Some symmetries involved here are in?nite dimensional. In particular the large-N limit of the algebra of Zamolodchikov WN , i.e. W∞ , plays a crucial role. The connection between W∞ algebras and Self-dual gravity
was pointed out for the ?rst time by Bakas in the ?rst paper of Ref [6]. In the present paper we use this fact. In a similar spirit, in the paper [7], we show that working with the Selfdual Yang-Mills ?elds on the 4-dimensional ?at manifold M with the Lie algebra sdi?(Σ)-valued connection 1-form, the ?rst and the second Heavenly equations emerge in a natural manner. Here sdi?(Σ) is the Lie algebra of SDi?(Σ). 2
On the other hand, there was a considerable interest recently in the application of quantum groups and noncommutative geometry in gauge theory [8,9,10]. In Refs. [11,12] the Yang-Mills gauge theory on the classical spacetime was de?ned using the quantum group SUq (2) as the “symmetry group”. In Ref. [13] it is shown how General Relativity can be put in the context of noncommutative geometry. Until now a general theory for Einstein’s theory of gravity including quantum groups as well as quantum spaces has not yet been considered. This work forms a part of our research for the case of Self-dual gravity. We think that this generalization may include some new insights in a realistic theory of quantum gravity. In this paper we use a q-deformed version of the Moyal algebra associated with the Lie algebra of the group of area preserving di?eomorphisms of the torus T 2 (Σ = T 2 ), sdi?q (T 2 ) [14, 15] in order to show new insights in fourdimensional Self-dual gravity. Speci?cally, we obtain a q-deformed version of the ?rst and the second heavenly equations of Self-dual gravity [1]. This forms part of our attempts to construct the general theory of H and HH quantum spaces with quantum group as the symmetry group in a close philosophy with [16]. The paper is organized as follow, in section 2 we describe the necessary tools and the quantum algebras used in section 3. The section 3 is devoted to obtain a q-deformed version for the Heavenly equations. In section 4 we brie?y discuss the recent two-dimensional chiral formulation of Self-dual
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gravity given by Husain in Ref. [17] in the context of q-deformed algebras. Finally, in section 5, we give our conclusions.
2
Basic Tools and Quantum Algebras
The large N-limit of the algebra WN generated by the primary conformal ?elds with spins 1, 2, ..., N is known as a W∞ algebra. In Ref. [18] Bakas found that this algebra provides the representation of an in?nite dimensional (sub)algebra of the area preserving di?eomorphisms of the plane. Thus, taking the generators W(k,m) (of spin k, k ≥ 0, m ∈ Z) of the W∞ algebra, they satisfy the relation
[W(k,m) , W(l,n) ] = {(l + 1)(n + 1) ? (k + 1)(m + 1)}W(k+l,m+n).
(1)
As is well known, if one takes the large N-limit of the Lie algebra su(N) it can be identi?ed in a natural manner with sdi?(T 2 ), i.e. su(∞) ? sdi?(T 2 ) [19]. The algebra sdi?(T 2 )come given by the Poisson algebra
[Lf , Lg ] = L{f,g} where {f, g} is the usual Poisson bracket and Lf = Taking a basis for the torus 4
?f ? ?x ?y
(2) ?
?f ? . ?y ?x
f → fm = exp(im · x),
g → fn = exp(in · x),
(3)
where m and n are 2-vectors with integer entries, m = (m1 , m2 ) and x = (x1 , x2 ). In this basis the Poisson algebras becomes the algebra sdi?(T 2 )
[Lm , Ln ] = (m × n)Lm+n where m × n = m1 n2 ? m2 n1 .
(4)
A deformation of the Poisson algebra (2) is the Moyal algebra. For the torus, the Moyal algebra reads 1 Sin(κm × n)Lm+n κ
[Lm , Ln ] =
(5)
κ is here the parameter of the deformation. A q-deformed version of this W∞ algebra is
q q q q q q [W(k,m) , W(l,n) ]q = W(k,m) W(l,n) ? q · W(l,n) W(k,m)
q = {(l + 1)(n + 1) ? (k + 1)(m + 1)}W(k+l,m+n)
(6)
where q is a complex parameter, in particular a root of unity, q = exp(ih), where h is a real number. This algebra corresponds to a q-deformed version of sdi?(Σ), namely sdi?q Σ. However it is not very general because it is limited to anti-commute
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generators1 . The Moyal algebra (5) is also a Lie algebra and therefore can be qdeformed. There exist a q-deformation of the Moyal bracket (‘quantum Moyal’) proposed for the ?rst time by Devchand, Fairlie, Fletcher and Sudbery [15]. On the other hand, there is other possible q-deformation of sdi?(Σ). In Ref. [14], Fairlie found such a q-deformation from the construction of algebras of q-symmetrized polynomials in the q-Heisenberg operators P and Q. That is, operators which satisfy the q-deformed Heisenberg algebra P Q ? qQP = iλ. where λ is a real parameter. Fairlie shown [14] that the q-deformed Heisenberg algebra leads to a q-deformation of the Moyal algebra
q n×m · Wm Wn ? q m×n · Wn Wm = (ω m×n/2 ? ω n×m/2 )Wm+n + a · mδm+n,0 (7) where a is a constant 2-vector which characterizes the central extension (see also Ref. [20]). The classical limit of (7) gives precisely the Moyal algebra (5). In this algebra (7), appears two parameters q and ω. If it is taked ω = exp (iκ) and q = exp (ih) → 1(or h → 0) we recover of course the Moyal algebra (5).
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We thank Prof. C. Zachos for pointing out this consequence.
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In order to apply the above q-deformed Moyal algebra is convenient to take m = (m1 , 0), n = (0, n2 ) in Eq. (7). Thus we have
q x · Wm Wn ? q ?x · Wn Wm = (ω x/2 ? ω ?x/2)Wm+n + a1 m1 δ(m1 ,n2 ),0 where x ≡ m1 n2 .
(8)
We will apply the q-deformed Moyal algebra to the compatibility conditions for the usual Self-dual Yang-Mills equations on M (“classical” space) with local coordinates {t, x, y, z} and after this we compare the corresponding results. Rede?ning the local coordinates on M to be, α = t+iz, α = t?iz, β = ? ? x + iy and β = x ? iy the Self-dual Yang-Mills equations are
Fαβ = 0, Fαβ = 0, Fαα + Fβ β = 0 ? ? ?? where
(9)
Fij = ?i Aj ? ?j Ai + [Ai , Aj ] ? and i, j ∈ {α, α, β, β}. The compatibility condition reads ? [?α + λ?β , Aβ ? λAα ] ? [?β ? λ?α , Aα + λAβ ] ? ? ? ?
(10)
= [Aα + λAβ , Aβ ? λAα ]. ? ? 7
(11)
Now, we will work with the bundle
SDi? q (Σ) ? P → M
π
(12)
with the connection 1-form Ai on M taking values precisely on sdi?q (T 2 ), i.e. ? ? ? Φi,r ∈ sdi? q (T 2 ) ?r ?s
Ai = Φi,s
(13)
? where Φ = Φ(α, α, β, β; q) and r, s are local coordinates on the two-dimensional ? torus T 2 . They are the generators of sdi?q (T 2 ) and satisfy the algebra (7) or (8). The compatibility condition in the q-deformed version is [?α + λ?β , Aβ ? λAα ] ? [?β ? λ?α , Aα + λAβ ] ? ? ? ?
= [Aα + λAβ , Aβ ? λAα ]q . ? ?
(14)
This equation will provide us the way of obtaining the q-deformed version of Heavenly equations.
3
The q-deformed Heavenly Equations
We start by substituting Eq. (13) into (14) and use the q-deformed commutation relations (8). Comparing powers of λ at order zero, two and one 8
respectively, we obtain the set of equations Φα,βs ?Φβ,αs + (q x · Φα,r Φβ,ss + q ?x · Φβ,s Φα,rs ) ?(q x · Φα,s Φβ,rs + q ?x · Φβ,r Φα,ss )
? + F (s, α, α, β, β, q) = 0, ?
(15)
Φα,βs ?Φβ,αs + (q x · Φα,r Φβ,ss + q ?x · Φβ,s Φα,rs ) ?(q x · Φα,s Φβ,rs + q ?x · Φβ,r Φα,ss ) ? ? ? ? ? ? ? ? ? ? ? ?
? ? + F (s, α, α, β, β, q) = 0, ?
(16)
Φα,αs ?Φα,αs + Φβ,βs ?Φβ,βs + [(q ?x · Φα,s Φα,rs + q x · Φα,r Φα,ss ) ?(q ?x · Φα,r Φα,ss ? ? ? ? ? ? ?
+q x ·Φα,s Φα,rs )+(q ?x ·Φβ,r Φβ,ss +q x ·Φβ,s Φβ,rs )?(q ?x ·Φβ,s Φβ,rs +q x ·Φβ,r Φβ,ss )] ? ? ? ? ?
? + G (s, α, α, β, β, q) = 0, ? here Φα,rs =
? 2 Φα , ?r?s
(17)
etc.
A.
The q-deformed First Heavenly Equation
Now, assuming Φα = ?,α , Φβ = ?,β , Φα = Φβ = 0, ? ? 9
? ? F = 0 = G, F = F (s, α, β, q), F = F
(18)
being ? = ?(α, β, r, s; q) some holomorphic function of its arguments. Thus, Eqs. (16) and (17) are satis?ed trivially and Eq. (15) gives
q x ·?,αr ?,βss +q ?x ·?,βs ?,αrs ?(q x ·?,αs ?,βrs +q ?x ·?,βr ?,αss )+F (s, α, β, q) = 0 (19) where ?,αrs =
?3? ?α?r?s
etc. After the change of coordinates rF → r, this
equation leads directly to
q x · ?,αr ?,βss +q ?x · ?,βs ?,αrs ?(q x · ?,αs ?,βrs + q ?x · ?,βr ?,αss = 1. (20) Notice that the q-deformed ?rst Heavenly equation is a third order nonlinear partial di?erential equation (TONLPDE). When the parameter q → 1 (or equivalently h → 0), we recover the ?rst heavenly equation in its usual form [1] ?,αr ?,βs ? ?,αs ?,βr = 1.
B. Taking
The q-deformed Second Heavenly Equation
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Φα = θ,s , Φβ = ?θr , Φα = Φβ = 0, ? ?
? ? F = 0 = G, F = F (s, α, β, q), θ = θ(α, β, r, s; q), F = F. Thus, Eqs. (16) and (17) hold trivially and Eq. (15) yields
(21)
q x · θ,ss θ,rrs + q ?x · θ,rr θ,sss ? (q x + q ?x )θ,rs θ,rss + θ,rαs + θ,sβs + F (s, α, β, q) = 0. (22) Now, making the substitution
Θ = θ + rsf,
f = f (α, β, q), f,α = F
(23)
(with Θ = Θ(α, β, r, s; q)) into Eq. (22) we ?nally obtain
(q x · Θ,ss Θ,rrs + q ?x · Θ,rr Θ,sss) ? (q x + q ?x )Θ,rs Θ,rss + Θ,rαs + Θ,sβs = 0. (24) This equation is again a TONLPDE. Taking the limit q → 1 we recover of course the second Heavenly equation [1] Θ,rr Θ,ss ? Θ2 + Θ,rα + Θ,sβ = 0. ,rs
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4
Comments on the q-deformed Chiral Model of the Self-dual Gravity
In this last section we make some comments about the possible generalization of the Husein work [17]. In this paper it is shown that Self-dual Einstein equations are equivalent to the two-dimensional Chiral Model with gauge group, precisely SDi?(Σ). Since the gauge group here is SDi?(Σ) the connection 1-form are sdi?(Σ)valued. Thus, we will generalize some results in [17] using the q-deformed Moyal algebra (8) associated with sdi?q (T 2 ) instead of the Moyal algebra (5) associated with sdi?(T 2 ). The Ashtekar-Jacobson-Smolin formulation of Self-dual gravity [21] leads to a set of equations on the four manifold M4 = K3 × R with local complex coordinates {x0 , x1 , x2 , x3 }. The equations are (see also [22])
DivVia = 0
(25)
1 ?Via = ?ijk [Vi , Vk ]a ?t 2
(26)
where Via are three spatial vector ?elds on K3 and [, ] is the Lie bracket. The solutions of these equations lead to the self-dual metric
g ab = (detV )?1 [Via Vjb δ ij + V0a V0b ]
(27)
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where i, j = 1, 2, 3 and V0a is a vector ?eld used in the decomposition M4 = K3 × R. The Eq. (26) is equivalent to the equations
[T , X ] = [U, V] = 0,
(28)
[T , U] + [X , V] = 0,
(29)
where T = V0 + iV1 , X = V2 ? iV3 , U = V0 ? iV1 and V = V2 + iV3 . Using the gauge freedom we can put in terms of the new coordinates β = x0 + ix1 , α = x2 ? ix3 , u = x0 ? ix1 and v = x2 + ix3 ? a ) , ?β ? a ) . ?α
Ta=(
Xa = (
(30)
As Husein showed [17], the Eqs. (28) and (29) are equivalent to the two-dimensional Chiral Model on the plane with coordinates (α, β). The relation with the ?rst Heavenly equation arises in this context when we take ? ? + ?,αr , ?r ?s
U = ??,αs
(31)
V = ?,βs
? ? ? ?,βr . ?r ?s
(32)
These are precisely the generators of the Lie algebra sdi?(T 2 ) (i.e. U, V ∈ sdi?(T 2 )) just as Eq. (13). As we saw before these generators satisfy the 13
Poisson algebra given by (2). The Eq. (28) leads directly to the ?rst Heavenly equation in the form ?,αr ?,βs ? ?,αs ?,βr = 1. Thus, the natural question arises: what is the modi?cation of the ?rst Heavenly equation when U, V are sdi?q (Σ)-valued? To see this, notice that the only modi?cation in the ?eld equations is
[U, V]q = q x · UV ? q ?x · VU
(33)
where now, U,V depends on the parameter q and ? = ?(α, β, r, s; q). Substituting (31) and (32) into (33) we obtain after a few steps
q x · ?,αr ?,βss + q ?x · ?,βs ?,αrs ? (q x · ?,αs ?,βrs + q ?x · ?,βr ?,αss ) = O(s, α, β; q) (34) where O is some holomorphic function its argument. Taking O(s, α, β; q) = 1 we obtain the q-deformed ?rst Heavenly equation from the q-deformed two dimensional Chiral Model . It is
q x · ?,αr ?,βss + q ?x · ?,βs ?,αrs ? (q x · ?,αs ?,βrs + q ?x · ?,βr ?,αss ) = 1 (35) and corresponds exactly to the result given in Eq. (20) for the Self-dual Yang-Mills case. This equation is the q-deformed version of the Eq. (29) 14
of the ?rst Ref. [17]. Finally, taking the q → 1 we obtain again the ?rst heavenly equation in its usual form.
5
Conclusions
In this paper we have considered some applications of the q-deformed Moyal algebra (8) associated with the group of area preserving di?eomorphisms of the torus T 2 . Since the Heavenly equations are very closely related with this symmetry [23,24], a q-deformed version of these equations is a generalization for they. In particular, we showed that the modi?cation prevents the terms in Eqs. (20,24,34) to be complete derivatives which are integrated out in Heavenly equations. In such a way the q-deformed version of the Heavenly equations is of the third order. As usual, when we take the limit q → 1 we recover the usual ?rst and second Heavenly equations. A similar situation occurs for the two-dimensional Chiral Model representation of the Self-dual gravity. Though the understanding of the q-deformation of in?nite dimensional Lie algebras is still incomplete, the impact in physics is direct. Any advance in this direction will give new insights for Self-dual gravity and perhaps in its quantized version.
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Acknowledgements
We wish to thank Profs. M. Przanowski and Piotr Kielanowski for very useful comments. We are indebted to Profs. I. Bakas, G.Torres del Castillo and C. Zachos for very useful comments and suggestions. One of us (H.G.C.) want to thank Prof. A. Zepeda for much encourage and help. Finally, to CONACyT and SNI for support.
References
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