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W-symmetry and the rigid particle
NUCLEAR PHYSICS B ELSEVIER
Nuclear Physics B436 (1995) 529-541
W-symmetry and the rigid particle E d u a r d o R a m o s 1, J a u m e R o c a 2 Department of Physics, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK Received 9 August 1994; accepted 11 November 1994
Abstract
We prove that W3 is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The W3 generators T and W then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the "zero-curvature gauge", and how to rewrite them in terms of the KdV operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the SKdV operator. We also partially extend our analysis in the supersymmetric domain to the scale-invariant rigid particle by explicitly constructing a supercovariant version of its action.
1. Introduction
Since their appearance in the seminal work of Zamolodchikov, W-algebras have played a central role in conformal field theory and string theory. Nevertheless, many of their properties are not yet fully understood mainly due to their nonlinear character. In particular, a complete understanding of their underlying geometry is still lacking, though many partial results are already available [1,2]. It is clear that simple mechanical systems displaying W-symmetry could be an invaluable tool in this difficult task. On the one hand, they could provide us with some geometrical a n d / o r physical interpretations for W-transformations (W-morphisms), while on the other hand they could give us some hints as to which structures are those
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E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
associated with W-gravity - the paradigmatic example being provided by the standard relativistic particle and diffeomorphism invariance (W2). The connection between W-morphisms and the extrinsic geometry of curves and surfaces is well known by now [1]. Therefore, it seems natural to look for a W-particle candidate among the geometrical actions depending on the extrinsic curvature. The canonical analysis of those models has been carried out by several authors [3]. In particular, Plyushchay studied in [4] the action given by S =af(iK2]
ds,
(1.1)
where a is a dimensionless coupling constant 3 and the extrinsic curvature K is given by d2x . d2x ~ ds2,
K2=gg~, d s e
(1.2)
where gu~ stands for a minkowskian, euclidean, or any x-independent metric. He showed that this dynamical system possesses two gauge invariances, and that one of them is the expected invariance under diffeomorphisms. The main purpose of this paper is to show that these gauge invariances are nothing but W 3. Before entering into more technical matters, we would like to point out that the scale-invariant rigid particle plays an important role in such diverse areas as polymer physics, random surfaces and string theory. In particular, since Polyakov pointed out in [5] the relevance of scale-invariant extrinsic curvature terms in the string action for a possible stringy description of QCD, the rigid particle has been used as a simpler laboratory where some of these ideas could be tested [6,7]. In view of this, we believe that the connection of this system with W-symmetry is interesting from both the physical and mathematical points of view, and that an understanding of W-geometry can find direct application in models of physical relevance. The plan of the paper is as follows. In Section 2 we review some well-known results about the scale-invariant rigid particle that will be useful for our purposes. In Section 3 we present the main results of the paper. It is shown how the equation of motion associated with the scale-invariant rigid particle can be recast in the Boussinesq form, and how W3-invariance follows from this. We also show how the equation of motion of the standard relativistic particle is obtained in the "zero-curvature" gauge, and we comment on the appearance of the KdV operator in this case. In Section 4 we extend the connection between particle models and integrable systems to the supersymmetric domain. The reason to do so, besides that of completeness, is that there is no unique supersymmetrization of the Lax operators associated with generalized KdV hierarchies [8,9], and therefore we expect that these models can shed some light on the physical significance of these different 3 In our convention the coordinates are dimensionless.
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supersymmetrizations. We then show that the equation of motion of the standard spinning particle can be recast in terms of the first super-Lax operator of the N = 1 series of [10]. We also construct the supersymmetric version of the scale-invariant rigid particle. Unfortunately, the expression of the action is quite involved, even when written in terms of superfields, and we have been unable to explicitly display the relationship between its equations of motion and any superintegrable model (although we have no doubt of its existence). We finish with some comments about possible generalizations of this formalism to Wn with n >/4, and we speculate about the possible relevance of our results for the construction of a fully covariant W3-gravity theory.
2. Hamiltonian analysis of the rigid particle We now briefly review some standard results concerning the gauge structure of the action (1.1) which will be useful in the following. In an arbitrary parametrization x~'(t), the lagrangian is given by
~/ --2 x. . -~-
L=a
(2.1)
where ~ ' = dx~/dt and £~_ = £ " - ~ ' ( ~ .k)/~2. For the study of the system in the hamiltonian framework it is convenient to introduce the velocity q~' =k~' as an auxiliary variable in order to avoid the presence of second-order derivative terms in the lagrangian. The lagrangian in these new variables reads L=
V~
+?(q-x)"
(2.2)
We introduce the canonical momenta (P,, p,, ~-~,) associated with the coordinates (x ", q~', y~'). Their Poisson bracket algebra is given by { e " , g~} = ~ .
(2.3)
The definition of the momenta implies a number of primary hamiltonian constraints. Being the lagrangian linear in 2 we obtain the trivial second-class constraints ,/" = -P~" and ~-" = 0, from which we can eliminate the canonical pair (y, rr) by means of the Dirac bracket. The primary first-class constraints are
q ~ l= p ' q ~ 0 ,
q~2=
IpZl-i-~
=0.
(2.4)
Stabilization of these constraints through the standard Dirac procedure leads to the secondary, C3=P'q =0,
¢4 = P . p = 0 ,
(2.5)
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and tertiary, q~5 = p 2
(2.6)
= O,
first-class hamiltonian constraints, which form a closed Poisson bracket algebra. In the constrained submanifold, time-evolution is generated by the hamiltonian H = ~b3 + vl~b 1 + v2~b2.
(2.7)
The functions v a and v 2 have a definite expression in terms of lagrangian quantities, q.q va-q2,
1 ~ 2.2 v2=--a q q l I,
(2.8)
but they should be regarded as arbitrary functions of time in the canonical formalism. Time-evolution becomes unambiguous once we assign to them definite values, which is nothing but a choice of gauge. The hamiltonian stabilization procedure guarantees the consistency of the expressions chosen for u 1 and v 2 with their lagrangian expression (2.8). The presence of two arbitrary functions reveals the existence of a second gauge symmetry in addition to the familiar reparametrization invariance. We shall show in what follows that their symmetry algebra is precisely W 3.
3. Equations of motion, Boussinesq operator and W3-symmetry The hamiltonian equations of motion can be suggestively written as ~ , = q~',
{
0
1~
=
q~'
__
0 1
p/Z .
__V 1 __ O / 2 U 2 / / q 4
0
V2
V1
(3.1)
q~'
Notice that, neglecting the first equation, which simply states the definition of q~', the equations of motion can be casted in the " D r i n f e l d - S o k o l o v " form associated with a particular SL(3) connection. We will come back to this point later on. It will be convenient in what follows to introduce a different and more geometrical parametrization for v I and v 2. Let us define e and A as e 2 = Iq21,
A2 = IKEI.
Notice that e 2 is basically the extrinsic are given by v 1 = ~ / e From (3.1) we can
(3.2)
nothing but the modulus of the induced metric, and A is curvature. In terms of e and A the gauge degrees of freedom and v 2 = Ae3/a. obtain a single equation for the velocity vector q~',
~1"~ + u l i j u + u2(1 ~ + u3q ~ = 0,
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with d ln(e3A)
l/1 = - 2
dt
'
~2
~/~
u2 = e4A2 + 15~-~ + 7 ~ -
~2
~
+ 2 ~ - - 4 - e - --h'
~3
~2/~
(3.3)
~,~2
~
~/~
~
~"
U3 = e4,~A +e3~A 2 - 15~-3 -7e2-- ~- - 2 ~ - - ~ + 10~-~ + 2 ~ - + e-A - e " Notice that u I is a total derivative. This allows us to remove the term with the second derivative by a simple local rescaling of qg. Indeed, if we define
1 yU,
e2A2/3q",
(3.4)
the equation for the new velocity vector y~ can be written in terms of the Boussinesq Lax operator, ~'~" + Tp ~ + ( W + T / 2 ) y ~" = O,
(3.5)
with T and W given by
1 ~2 T = eZA2 - 3 e 2
~3
~
4 ~2
eA
3
5
)t 2
~
j~
"l'-2--e --[- ~-'
~2/~
4 ~/~2
W = e~A2 + 3~--~ + -e2hA + + e2----~ 3
56 ~3 +
2--~ h--~
+-+----
e2
3 eh
eA
3 h2
e
3 h
The algebra of symmetries of equations of the type I_~ = 0, with k a differential operator of the form a n + . . . . has been studied by Radul in [11]. From his general construction it is easily deduced that for the particular case of L = 0 3 + TO + W + 7~/2, which is our case of interest, the symmetry algebra is W 3. Rather than reproducing here the general arguments leading to this result, we will work out the case at hand explicitly. The most general (local) variation of ~ preserving the structure of L, up to equations of motion, is given by (3.6) where the parametrization has been chosen so that E and p denote the parameters associated with diffeomorphisms and pure W3-morphisms respectively. The corresponding transformations for T and W which leave the equations of motion
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invariant are given by 6(~T)T = 2~'+ 2~T + E~?, 6~T)w = 3~W+ eW, s(pW)T = 3f~W + 2p[~', (W)
1 ~(v)
-
5 ..
(3.7) ~5 p-- T" - p ( ~• T3+"'
It is now a long but straightforward exercise to check that these transformations obey the algebra of W3-transformations, i.e. 6 e lr) ~ 8(r)1 = 6~r), ~2 ]
where
E =
EI~ 2-~1E2,
[ 6~r), so(w)] = 8} w),
where
/ / = Et~ - 2~p,
6 w) 6
where
E = ~plP2 T - ~PlP2 + gPlP2 -- (Pl ~"~P2),
Pl
'
02
J
2
•
(3.8) 1 -
..
1
""
where we have assumed for simplicity that the parameters are field independent. The most general case can be equally worked out from the results of [11]. Notice that the consistency of the procedure in our case is based on the fact that T and W, through their dependence in ,~ and e, are gauge degrees of freedom themselves, therefore before any solution to the equations is found they should be given definite values. In this language the W-morphisms given by (3.7) are to be understood as gauge transformations. 3.1. W-symmetry as an invariance of the action
Being that the relation between y~ and q~ is not easily invertible, the previous analysis does not yield the transformation rules for q~' directly. These transformations are interesting by themselves because the action is naturally written in terms of q,. Here is where standard techniques in constrained dynamical systems come to our rescue. The gauge transformations for x u, q~' and y~ can be computed using the general method of [12]. We shall give here a brief outline of the procedure. Let us introduce an operator K which acts on arbitrary functions g(~', ~ , t) in phase space, giving (on-shell) their time-derivative in velocity space:
I where /~(~, ~ ) stands for the function in velocity space obtained from h(~, ~ ) after substituting the momenta by their lagrangian expression: ~ = a L / O ~ . The operator K plays a relevant role in the construction of gauge transformations. In particular, it can be shown that all lagrangian constraints are obtained by applying K to the hamiltonian constraints.
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Let us consider the submanifold V1 defined by the primary lagrangian constraints X a~-_ Kq~a, 0 where 4,a0 are the primary hamiltonian constraints. It can be shown [12] that a hamiltonian function G satisfying K G -~ O, on V1, i.e. K G + ~ r aX~ = O, Ol
can be used to construct Noether gauge transformations in the following way:
~'=
(0o) -~
+ EraTa,
(3.9)
a
where the ~/a form a basis for the null vectors of the hessian matrix 0 2 L / O ~ 2 and have the expression
In practice, it is enough to consider G as a linear combination of hamiltonian constraints with arbitrary infinitesimal functions as their coefficients. Imposing K G -~ 0 on F~ determines some of these functions in terms of the rest. Those that remain arbitrary are to be considered as the infinitesimal parameters of the gauge transformations. Primary hamiltonian constraints can, in fact, be excluded from the previous combination as they have no contribution in the gauge transformations. In the case at hand the 'generator' G and the functions r a are given by G = / 3 ( t ) ~ b 3 + 47(t)q~ 4 + r/(t)~b 5,
A r I =fl + -/3-a--47, e
e
1 r 2 = --e3A/3 + ~j - -47, OL
e
and from (3.9) we obtain the gauge transformations o/
6t~,~x" = / 3 q " + 47e ~ - 4 ~ - 2n3,", 6¢,nq • = B q " +/3Cl ~" -- 47 Y " + a--q"e
+ a - - ' eA4 q~
+ / j --e3 A qTM±,
(3.10)
8t3,ny ~' = O.
A direct computation now shows that the transformations given by (3.10) are not only a symmetry of the action, but also that, as a soft algebra, they reproduce the algebra of W3-transformations if we make the following identifications: e=/3+~e3a -p =
o~ e3 A //.
+ r/e3A "--
6 A
2 e (3.11)
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Notice that in this parametrization the variation of x ~' has a nonlocal expression. This is not entirely unexpected due to the fact that the natural variable for the system seems to be supplied by q~' rather than by the coordinates themselves 4 Moreover, as expected, the variations induced on y~' through (3.10) coincide with the ones obtained via (3.6), which tightens up the formalism, and shows the nice interplay between the hamiltonian and the W-algebraic methods of tackling the problem. There is also a more indirect, yet interesting, way to show the invariance of the action (1.1) under W3-morphisms with the help of the Miura transformation. In order to do so we should return to the expression of the hamiltonian equations of motion in terms of the SL(3) connection. If we write (3.1) as
_a) it is clear that these equations have local gauge invariance under p" q,
~M
P" ~ q~
and A ~ M A M
- 1 - M -
dt
'
with M ~ SL(3). It will be convenient for our purposes to work in the Miura gauge, i.e. - - ~ 1 -- ~ 2
0
0
1 0
~2 1
0 qh
A =
(3.13)
The matrix M ~ SL(3) which brings A to this form is -e 0
M = -~
o /
-ia/e -a/e2A
.
Some straightforward algebra now yields qh
e
3 A
q~2 = -~ -~ + iAe.
iAe,
(3.14)
The key point, for our present interest, is given by the fact that the lagrangian (2.1) can be written as L ~ Im(~ol) ~ Im(q~2). (3.15) The transformation under W3-morphisms of the Miura fields can be computed
4 This is somehow reminiscent of what happens for the case of the two-dimensional massless boson, where the natural variable is supplied by its associated U(1) current.
E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
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using the Kupershmidt-Wilson theorem [13]. If we denote by E the parameter associated with diffeomorphisms, and by p the one associated with pure W3-morphisms, these variations are given by
~,,~91 = ~ [ ~
+,9,
+ l~+
½(¢,p +9,p)
- - 1 ( } 9 1 - - 2~2 -- 92 -b 2922 -1- 29192)p] , d a,.,92 = ~[E92
1.. - ~1( 9 •2 p + 92p) - ¢ , p + 91p - ~p
+½(~b1+~92'' - 2 9 2 + 9 2 - 2 9 1 9 2 ) p ] ,
(3.16)
which is a total derivative! And from this follows directly the invariance of the action. 3.2. The relativistic particle
It is easy to recover the equations of motion from (3.1) whenever it is consistent to impose that since v 2 is proportional to the extrinsic consistently imposed when it can be taken to incompatible with this gauge choice). In that motion collapse to
for the standard relativistic particle the gauge condition v 2 = 0. Notice curvature, this gauge can only be be zero (initial conditions may be case the hamiltonian equations of
£" - -~" = 0,
(3.17)
X ~ --~ e l / 2 x ~
(3.18)
e which is the equation of motion for the relativistic particle. Notice that (3.17) can be suggestively written as £ ~ = 0. It is natural to ask now, in the light of our previous discussion, whether the KdV Lax operator appears in this case. The answer to this question is easily obtained by realizing that because via the redefinition the equations of motion for the new variable can be written as £~ + Tx ~ = 0,
(3.19)
1~ T= 2 e
(3.20)
with 3~ 2 4 e2
and e being a gauge degree of freedom for the relativistic particle, we can apply all the same arguments as above.
4. T h e s u p e r s y m m e t r i c
arena
We shall now show how some of our previous results can be extended to the supersymmetric domain.
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4.1. The spinning particle and S K d V
Since the spinning particle provides the natural world line supersymmetrization of the relativistic particle it is natural to address the question of which supersymmetric Lax operator, if any, appears in this case. The massive spinning particle is described by the action [14] S = -~ [ d t --e + m 2e - O t~ + /31~ - X--O 2 + m x
(4.1)
where ~0~' is the supersymmetric partner of the particle position xG X is the world line gravitino and /3 is an auxiliary fermionic variable which is necessary to introduce in order to deal with the massive case. This model is invariant under N = 1 superdiffeomorphisms [15]. This can be made explicit by writing its action in terms of superfields. Let us first introduce the superfields X ~' and E, of weights 0 and 1, associated with the coordinate x ~' and the einbein e respectively: X u = x ~ + 0 e l / 2 ~ u,
E =e
+ Oel/2 X.
(4.2)
In contrast, we do not have among the initial fields a natural superpartner of/3. In fact, one can show that supersymmetry transformations close on /3 only on-shell. This problem can be circumvented by introducing a new auxiliary variable b and its associated zero-weight superfield, (4.3)
B =/3 + Ob
In terms of these three superfields the action can be written in the form S = ½f dt d 0 ( E - 1 D X D 2 X - B D B + 2 m B E 1 / 2 ) .
(4.4)
Notice that even though D 2 X ~ is not a tensor superfield, the nontensorial term in its transformation under superdiffeomorphisms is proportional to D X ~', thus rendering the action invariant. The matter equations of motion can be easily computed to be D 3 X " - ½ E - 1 D E D 2 X " - ½ E - ' D 2 E D X ~ = O.
(4.5)
If in analogy with the bosonic case we now perform the change of variables (4.6)
X " ~ E I / 2 X ",
we can eliminate the second-order term in (4.5) and the equations of motion read D 3 X ~ + U X u = 0,
(4.7)
with 1( ' D 3 E U = -~ , E
3 DE D2E ) 2 E 2
"
(4.8)
It is remarkable that the first-order term also disappears so that Eq. (4.7) is precisely the first super-Lax operator associated with the N = 1 series of [10]. It
E. Ramos,J. Roca/ NuclearPhysicsB436 (1995)529-541
539
was shown in [16] how this N = 1 series provides us with hamiltonian structures for the SBKP reduction of the SKP 2 hierarchy. In particular, the operator D 3 + U provides us with a hamiltonian structure for the SKdV hierarchy, thus making explicit the connection between the spinning particle and superintegrability.
4.2. The spinning scale-invariant rigid particle In order to perform the world line supersymmetrization of the scale-invariant rigid particle it will prove convenient to rewrite its action (1.1) with the help of two auxiliary fields e and h as follows: ( 1 " ' 2) m 1 2X 2 S = a fJ dt e h + 2 e3h 2 e a .
(4.9)
The equations of motion for e and h imply that their explicit expressions coincide with the ones given in (3.2), thus justifying their name. Moreover, since their equations of motion are algebraic we can substitute them back in (4.9) and recover the usual form of the action (1.1). It is now simple with the experience of the standard relativistic particle to attempt the supersymmetrization of (4.9). First of all, we should introduce an even superfield A associated with A, i.e. A = h + Or/. But this is not enough and we should find an explicit expression of the perpendicular projector preserving supercovariance. This can be readily done by observing that 1 Dg2 3 ' - 2 g2 ' (4.10) with a = (DzX) 2 + 2DXD3X,
(4.11)
transforms under superdiffeomorphisms as a superconnection. Therefore we can define a supercovariant derivative, Vr, acting on superfields q~h, of weight h, with the help of F by
Vr~ h = DcI9h - hFcI9h . With all of this in mind we can directly check that the action given by ( l V4X V3X 1DXD2X ) a f dt dO 2 E3A 2 g A - B D B + 2 B ( E K ) 1/2
(4.12)
(4.13)
is explicitly supercovariant, while reproducing (4.9) when we set the superpartners to zero. Unfortunately, when the supercovariant derivatives are expanded in terms of the superderivatives of X ~' the expression obtained is so involved that it has prevented us from making explicit the connection with superintegrability. 5. Final c o m m e n t s and some conjectures
We hope to have convinced the reader that W3-symmetry does play an important role in the symmetry structure of the rigid particle, and that standard
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E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-54l
techniques in constrained hamiltonian systems and W-algebras can be intertwined as powerful tools for the better understanding of both disciplines. It is also quite clear from our results that the connection between integrability and particle models does not restrict to the rigid particle case and seems to be present in other physically relevant particle models. But, of course, much is still to be done. For example, we do not yet completely understand the appearance of Lie algebraic structures in the problem, which should be the key to generalize these procedures to other W-algebras. Although it seems plausible that a particle model with the action depending on the (n - 1)th-order extrinsic curvature will enjoy Wn-symmetry, explicit computations become untractable and a more powerful machinery should be developed. Under quantization [4] the rigid particle is associated with massless representations of the Poincar6 group with integer helicity, therefore becoming a potential candidate for a particle description of photons, gravitons and higher spin fields. It is a tantalizing possibility that W3-symmetry can play a role in such physical systems. In fact there are some indications that this is indeed the case. It is clear from the transformation rules under pure W3-morphisms that the world line element of the particle has no gauge-invariant meaning 5. This nicely connects with some general arguments [17] implying that it is impossible to assign, even classicaly, a position to a single photon (or similarly, any massless higher spin fields). We would like to finish with a few sentences on the possible relevance of our results to W3-gravity. It was shown in [18] that a naive coupling to gravity of the action (1.1), i.e. considering arbitrary x-dependent metrics, changes the gauge structure of the system, and only reparametrization invariance survives. This is due to the fact that the constraint algebra no longer closes in curved space-time. We believe this to be a signal indicating that the rigid particle is only consistently coupled to W3-gravity. Let us elaborate on this. In the standard particle case, the relevant bundle is the tangent bundle of the manifold, TM. In this case, it is well known how to equip the bundle with a metric structure, and all the powerful and well-understood machinery of riemmanian geometry is at our disposal. In the language of jet bundles TM is nothing but JI(R, M), and it is our understanding that the relevant bundle for the W3-particle is provided by j2(~, M), which is not itself a vector bundle. It is well known that j2(~, M) is an associated bundle to F2M, the frame bundle of second order, and it is our belief that the required structure should be a "natural" structure in F2M, as it is the metric in FM. It is our hope that the action of the rigid particle will provide us with a "W3-1ine element", thus offering some valuable insight about which generalized structures in F2M should be considered. Work on this is in progress.
5 This is of course a well-known fact, although to the best of our knowledge the gauge transformations which render the position vector a gauge-variant object were not displayed in the literature.
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Acknowledgements We would like to thank J.M. Figueroa-O'Farrill and C.M. Hull for many useful discussions on the subject. J.R. is also grateful to the Spanish ministry of education and the British council for financial support.
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Nuclear Physics B436 (1995) 529-541
W-symmetry and the rigid particle E d u a r d o R a m o s 1, J a u m e R o c a 2 Department of Physics, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK Received 9 August 1994; accepted 11 November 1994
Abstract
We prove that W3 is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The W3 generators T and W then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the "zero-curvature gauge", and how to rewrite them in terms of the KdV operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the SKdV operator. We also partially extend our analysis in the supersymmetric domain to the scale-invariant rigid particle by explicitly constructing a supercovariant version of its action.
1. Introduction
Since their appearance in the seminal work of Zamolodchikov, W-algebras have played a central role in conformal field theory and string theory. Nevertheless, many of their properties are not yet fully understood mainly due to their nonlinear character. In particular, a complete understanding of their underlying geometry is still lacking, though many partial results are already available [1,2]. It is clear that simple mechanical systems displaying W-symmetry could be an invaluable tool in this difficult task. On the one hand, they could provide us with some geometrical a n d / o r physical interpretations for W-transformations (W-morphisms), while on the other hand they could give us some hints as to which structures are those
i E-mail E. [email protected]. 2 E-mail J. [email protected]. 0550-3213/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
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E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
associated with W-gravity - the paradigmatic example being provided by the standard relativistic particle and diffeomorphism invariance (W2). The connection between W-morphisms and the extrinsic geometry of curves and surfaces is well known by now [1]. Therefore, it seems natural to look for a W-particle candidate among the geometrical actions depending on the extrinsic curvature. The canonical analysis of those models has been carried out by several authors [3]. In particular, Plyushchay studied in [4] the action given by S =af(iK2]
ds,
(1.1)
where a is a dimensionless coupling constant 3 and the extrinsic curvature K is given by d2x . d2x ~ ds2,
K2=gg~, d s e
(1.2)
where gu~ stands for a minkowskian, euclidean, or any x-independent metric. He showed that this dynamical system possesses two gauge invariances, and that one of them is the expected invariance under diffeomorphisms. The main purpose of this paper is to show that these gauge invariances are nothing but W 3. Before entering into more technical matters, we would like to point out that the scale-invariant rigid particle plays an important role in such diverse areas as polymer physics, random surfaces and string theory. In particular, since Polyakov pointed out in [5] the relevance of scale-invariant extrinsic curvature terms in the string action for a possible stringy description of QCD, the rigid particle has been used as a simpler laboratory where some of these ideas could be tested [6,7]. In view of this, we believe that the connection of this system with W-symmetry is interesting from both the physical and mathematical points of view, and that an understanding of W-geometry can find direct application in models of physical relevance. The plan of the paper is as follows. In Section 2 we review some well-known results about the scale-invariant rigid particle that will be useful for our purposes. In Section 3 we present the main results of the paper. It is shown how the equation of motion associated with the scale-invariant rigid particle can be recast in the Boussinesq form, and how W3-invariance follows from this. We also show how the equation of motion of the standard relativistic particle is obtained in the "zero-curvature" gauge, and we comment on the appearance of the KdV operator in this case. In Section 4 we extend the connection between particle models and integrable systems to the supersymmetric domain. The reason to do so, besides that of completeness, is that there is no unique supersymmetrization of the Lax operators associated with generalized KdV hierarchies [8,9], and therefore we expect that these models can shed some light on the physical significance of these different 3 In our convention the coordinates are dimensionless.
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supersymmetrizations. We then show that the equation of motion of the standard spinning particle can be recast in terms of the first super-Lax operator of the N = 1 series of [10]. We also construct the supersymmetric version of the scale-invariant rigid particle. Unfortunately, the expression of the action is quite involved, even when written in terms of superfields, and we have been unable to explicitly display the relationship between its equations of motion and any superintegrable model (although we have no doubt of its existence). We finish with some comments about possible generalizations of this formalism to Wn with n >/4, and we speculate about the possible relevance of our results for the construction of a fully covariant W3-gravity theory.
2. Hamiltonian analysis of the rigid particle We now briefly review some standard results concerning the gauge structure of the action (1.1) which will be useful in the following. In an arbitrary parametrization x~'(t), the lagrangian is given by
~/ --2 x. . -~-
L=a
(2.1)
where ~ ' = dx~/dt and £~_ = £ " - ~ ' ( ~ .k)/~2. For the study of the system in the hamiltonian framework it is convenient to introduce the velocity q~' =k~' as an auxiliary variable in order to avoid the presence of second-order derivative terms in the lagrangian. The lagrangian in these new variables reads L=
V~
+?(q-x)"
(2.2)
We introduce the canonical momenta (P,, p,, ~-~,) associated with the coordinates (x ", q~', y~'). Their Poisson bracket algebra is given by { e " , g~} = ~ .
(2.3)
The definition of the momenta implies a number of primary hamiltonian constraints. Being the lagrangian linear in 2 we obtain the trivial second-class constraints ,/" = -P~" and ~-" = 0, from which we can eliminate the canonical pair (y, rr) by means of the Dirac bracket. The primary first-class constraints are
q ~ l= p ' q ~ 0 ,
q~2=
IpZl-i-~
=0.
(2.4)
Stabilization of these constraints through the standard Dirac procedure leads to the secondary, C3=P'q =0,
¢4 = P . p = 0 ,
(2.5)
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and tertiary, q~5 = p 2
(2.6)
= O,
first-class hamiltonian constraints, which form a closed Poisson bracket algebra. In the constrained submanifold, time-evolution is generated by the hamiltonian H = ~b3 + vl~b 1 + v2~b2.
(2.7)
The functions v a and v 2 have a definite expression in terms of lagrangian quantities, q.q va-q2,
1 ~ 2.2 v2=--a q q l I,
(2.8)
but they should be regarded as arbitrary functions of time in the canonical formalism. Time-evolution becomes unambiguous once we assign to them definite values, which is nothing but a choice of gauge. The hamiltonian stabilization procedure guarantees the consistency of the expressions chosen for u 1 and v 2 with their lagrangian expression (2.8). The presence of two arbitrary functions reveals the existence of a second gauge symmetry in addition to the familiar reparametrization invariance. We shall show in what follows that their symmetry algebra is precisely W 3.
3. Equations of motion, Boussinesq operator and W3-symmetry The hamiltonian equations of motion can be suggestively written as ~ , = q~',
{
0
1~
=
q~'
__
0 1
p/Z .
__V 1 __ O / 2 U 2 / / q 4
0
V2
V1
(3.1)
q~'
Notice that, neglecting the first equation, which simply states the definition of q~', the equations of motion can be casted in the " D r i n f e l d - S o k o l o v " form associated with a particular SL(3) connection. We will come back to this point later on. It will be convenient in what follows to introduce a different and more geometrical parametrization for v I and v 2. Let us define e and A as e 2 = Iq21,
A2 = IKEI.
Notice that e 2 is basically the extrinsic are given by v 1 = ~ / e From (3.1) we can
(3.2)
nothing but the modulus of the induced metric, and A is curvature. In terms of e and A the gauge degrees of freedom and v 2 = Ae3/a. obtain a single equation for the velocity vector q~',
~1"~ + u l i j u + u2(1 ~ + u3q ~ = 0,
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E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
with d ln(e3A)
l/1 = - 2
dt
'
~2
~/~
u2 = e4A2 + 15~-~ + 7 ~ -
~2
~
+ 2 ~ - - 4 - e - --h'
~3
~2/~
(3.3)
~,~2
~
~/~
~
~"
U3 = e4,~A +e3~A 2 - 15~-3 -7e2-- ~- - 2 ~ - - ~ + 10~-~ + 2 ~ - + e-A - e " Notice that u I is a total derivative. This allows us to remove the term with the second derivative by a simple local rescaling of qg. Indeed, if we define
1 yU,
e2A2/3q",
(3.4)
the equation for the new velocity vector y~ can be written in terms of the Boussinesq Lax operator, ~'~" + Tp ~ + ( W + T / 2 ) y ~" = O,
(3.5)
with T and W given by
1 ~2 T = eZA2 - 3 e 2
~3
~
4 ~2
eA
3
5
)t 2
~
j~
"l'-2--e --[- ~-'
~2/~
4 ~/~2
W = e~A2 + 3~--~ + -e2hA + + e2----~ 3
56 ~3 +
2--~ h--~
+-+----
e2
3 eh
eA
3 h2
e
3 h
The algebra of symmetries of equations of the type I_~ = 0, with k a differential operator of the form a n + . . . . has been studied by Radul in [11]. From his general construction it is easily deduced that for the particular case of L = 0 3 + TO + W + 7~/2, which is our case of interest, the symmetry algebra is W 3. Rather than reproducing here the general arguments leading to this result, we will work out the case at hand explicitly. The most general (local) variation of ~ preserving the structure of L, up to equations of motion, is given by (3.6) where the parametrization has been chosen so that E and p denote the parameters associated with diffeomorphisms and pure W3-morphisms respectively. The corresponding transformations for T and W which leave the equations of motion
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invariant are given by 6(~T)T = 2~'+ 2~T + E~?, 6~T)w = 3~W+ eW, s(pW)T = 3f~W + 2p[~', (W)
1 ~(v)
-
5 ..
(3.7) ~5 p-- T" - p ( ~• T3+"'
It is now a long but straightforward exercise to check that these transformations obey the algebra of W3-transformations, i.e. 6 e lr) ~ 8(r)1 = 6~r), ~2 ]
where
E =
EI~ 2-~1E2,
[ 6~r), so(w)] = 8} w),
where
/ / = Et~ - 2~p,
6 w) 6
where
E = ~plP2 T - ~PlP2 + gPlP2 -- (Pl ~"~P2),
Pl
'
02
J
2
•
(3.8) 1 -
..
1
""
where we have assumed for simplicity that the parameters are field independent. The most general case can be equally worked out from the results of [11]. Notice that the consistency of the procedure in our case is based on the fact that T and W, through their dependence in ,~ and e, are gauge degrees of freedom themselves, therefore before any solution to the equations is found they should be given definite values. In this language the W-morphisms given by (3.7) are to be understood as gauge transformations. 3.1. W-symmetry as an invariance of the action
Being that the relation between y~ and q~ is not easily invertible, the previous analysis does not yield the transformation rules for q~' directly. These transformations are interesting by themselves because the action is naturally written in terms of q,. Here is where standard techniques in constrained dynamical systems come to our rescue. The gauge transformations for x u, q~' and y~ can be computed using the general method of [12]. We shall give here a brief outline of the procedure. Let us introduce an operator K which acts on arbitrary functions g(~', ~ , t) in phase space, giving (on-shell) their time-derivative in velocity space:
I where /~(~, ~ ) stands for the function in velocity space obtained from h(~, ~ ) after substituting the momenta by their lagrangian expression: ~ = a L / O ~ . The operator K plays a relevant role in the construction of gauge transformations. In particular, it can be shown that all lagrangian constraints are obtained by applying K to the hamiltonian constraints.
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Let us consider the submanifold V1 defined by the primary lagrangian constraints X a~-_ Kq~a, 0 where 4,a0 are the primary hamiltonian constraints. It can be shown [12] that a hamiltonian function G satisfying K G -~ O, on V1, i.e. K G + ~ r aX~ = O, Ol
can be used to construct Noether gauge transformations in the following way:
~'=
(0o) -~
+ EraTa,
(3.9)
a
where the ~/a form a basis for the null vectors of the hessian matrix 0 2 L / O ~ 2 and have the expression
In practice, it is enough to consider G as a linear combination of hamiltonian constraints with arbitrary infinitesimal functions as their coefficients. Imposing K G -~ 0 on F~ determines some of these functions in terms of the rest. Those that remain arbitrary are to be considered as the infinitesimal parameters of the gauge transformations. Primary hamiltonian constraints can, in fact, be excluded from the previous combination as they have no contribution in the gauge transformations. In the case at hand the 'generator' G and the functions r a are given by G = / 3 ( t ) ~ b 3 + 47(t)q~ 4 + r/(t)~b 5,
A r I =fl + -/3-a--47, e
e
1 r 2 = --e3A/3 + ~j - -47, OL
e
and from (3.9) we obtain the gauge transformations o/
6t~,~x" = / 3 q " + 47e ~ - 4 ~ - 2n3,", 6¢,nq • = B q " +/3Cl ~" -- 47 Y " + a--q"e
+ a - - ' eA4 q~
+ / j --e3 A qTM±,
(3.10)
8t3,ny ~' = O.
A direct computation now shows that the transformations given by (3.10) are not only a symmetry of the action, but also that, as a soft algebra, they reproduce the algebra of W3-transformations if we make the following identifications: e=/3+~e3a -p =
o~ e3 A //.
+ r/e3A "--
6 A
2 e (3.11)
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Notice that in this parametrization the variation of x ~' has a nonlocal expression. This is not entirely unexpected due to the fact that the natural variable for the system seems to be supplied by q~' rather than by the coordinates themselves 4 Moreover, as expected, the variations induced on y~' through (3.10) coincide with the ones obtained via (3.6), which tightens up the formalism, and shows the nice interplay between the hamiltonian and the W-algebraic methods of tackling the problem. There is also a more indirect, yet interesting, way to show the invariance of the action (1.1) under W3-morphisms with the help of the Miura transformation. In order to do so we should return to the expression of the hamiltonian equations of motion in terms of the SL(3) connection. If we write (3.1) as
_a) it is clear that these equations have local gauge invariance under p" q,
~M
P" ~ q~
and A ~ M A M
- 1 - M -
dt
'
with M ~ SL(3). It will be convenient for our purposes to work in the Miura gauge, i.e. - - ~ 1 -- ~ 2
0
0
1 0
~2 1
0 qh
A =
(3.13)
The matrix M ~ SL(3) which brings A to this form is -e 0
M = -~
o /
-ia/e -a/e2A
.
Some straightforward algebra now yields qh
e
3 A
q~2 = -~ -~ + iAe.
iAe,
(3.14)
The key point, for our present interest, is given by the fact that the lagrangian (2.1) can be written as L ~ Im(~ol) ~ Im(q~2). (3.15) The transformation under W3-morphisms of the Miura fields can be computed
4 This is somehow reminiscent of what happens for the case of the two-dimensional massless boson, where the natural variable is supplied by its associated U(1) current.
E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
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using the Kupershmidt-Wilson theorem [13]. If we denote by E the parameter associated with diffeomorphisms, and by p the one associated with pure W3-morphisms, these variations are given by
~,,~91 = ~ [ ~
+,9,
+ l~+
½(¢,p +9,p)
- - 1 ( } 9 1 - - 2~2 -- 92 -b 2922 -1- 29192)p] , d a,.,92 = ~[E92
1.. - ~1( 9 •2 p + 92p) - ¢ , p + 91p - ~p
+½(~b1+~92'' - 2 9 2 + 9 2 - 2 9 1 9 2 ) p ] ,
(3.16)
which is a total derivative! And from this follows directly the invariance of the action. 3.2. The relativistic particle
It is easy to recover the equations of motion from (3.1) whenever it is consistent to impose that since v 2 is proportional to the extrinsic consistently imposed when it can be taken to incompatible with this gauge choice). In that motion collapse to
for the standard relativistic particle the gauge condition v 2 = 0. Notice curvature, this gauge can only be be zero (initial conditions may be case the hamiltonian equations of
£" - -~" = 0,
(3.17)
X ~ --~ e l / 2 x ~
(3.18)
e which is the equation of motion for the relativistic particle. Notice that (3.17) can be suggestively written as £ ~ = 0. It is natural to ask now, in the light of our previous discussion, whether the KdV Lax operator appears in this case. The answer to this question is easily obtained by realizing that because via the redefinition the equations of motion for the new variable can be written as £~ + Tx ~ = 0,
(3.19)
1~ T= 2 e
(3.20)
with 3~ 2 4 e2
and e being a gauge degree of freedom for the relativistic particle, we can apply all the same arguments as above.
4. T h e s u p e r s y m m e t r i c
arena
We shall now show how some of our previous results can be extended to the supersymmetric domain.
E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-541
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4.1. The spinning particle and S K d V
Since the spinning particle provides the natural world line supersymmetrization of the relativistic particle it is natural to address the question of which supersymmetric Lax operator, if any, appears in this case. The massive spinning particle is described by the action [14] S = -~ [ d t --e + m 2e - O t~ + /31~ - X--O 2 + m x
(4.1)
where ~0~' is the supersymmetric partner of the particle position xG X is the world line gravitino and /3 is an auxiliary fermionic variable which is necessary to introduce in order to deal with the massive case. This model is invariant under N = 1 superdiffeomorphisms [15]. This can be made explicit by writing its action in terms of superfields. Let us first introduce the superfields X ~' and E, of weights 0 and 1, associated with the coordinate x ~' and the einbein e respectively: X u = x ~ + 0 e l / 2 ~ u,
E =e
+ Oel/2 X.
(4.2)
In contrast, we do not have among the initial fields a natural superpartner of/3. In fact, one can show that supersymmetry transformations close on /3 only on-shell. This problem can be circumvented by introducing a new auxiliary variable b and its associated zero-weight superfield, (4.3)
B =/3 + Ob
In terms of these three superfields the action can be written in the form S = ½f dt d 0 ( E - 1 D X D 2 X - B D B + 2 m B E 1 / 2 ) .
(4.4)
Notice that even though D 2 X ~ is not a tensor superfield, the nontensorial term in its transformation under superdiffeomorphisms is proportional to D X ~', thus rendering the action invariant. The matter equations of motion can be easily computed to be D 3 X " - ½ E - 1 D E D 2 X " - ½ E - ' D 2 E D X ~ = O.
(4.5)
If in analogy with the bosonic case we now perform the change of variables (4.6)
X " ~ E I / 2 X ",
we can eliminate the second-order term in (4.5) and the equations of motion read D 3 X ~ + U X u = 0,
(4.7)
with 1( ' D 3 E U = -~ , E
3 DE D2E ) 2 E 2
"
(4.8)
It is remarkable that the first-order term also disappears so that Eq. (4.7) is precisely the first super-Lax operator associated with the N = 1 series of [10]. It
E. Ramos,J. Roca/ NuclearPhysicsB436 (1995)529-541
539
was shown in [16] how this N = 1 series provides us with hamiltonian structures for the SBKP reduction of the SKP 2 hierarchy. In particular, the operator D 3 + U provides us with a hamiltonian structure for the SKdV hierarchy, thus making explicit the connection between the spinning particle and superintegrability.
4.2. The spinning scale-invariant rigid particle In order to perform the world line supersymmetrization of the scale-invariant rigid particle it will prove convenient to rewrite its action (1.1) with the help of two auxiliary fields e and h as follows: ( 1 " ' 2) m 1 2X 2 S = a fJ dt e h + 2 e3h 2 e a .
(4.9)
The equations of motion for e and h imply that their explicit expressions coincide with the ones given in (3.2), thus justifying their name. Moreover, since their equations of motion are algebraic we can substitute them back in (4.9) and recover the usual form of the action (1.1). It is now simple with the experience of the standard relativistic particle to attempt the supersymmetrization of (4.9). First of all, we should introduce an even superfield A associated with A, i.e. A = h + Or/. But this is not enough and we should find an explicit expression of the perpendicular projector preserving supercovariance. This can be readily done by observing that 1 Dg2 3 ' - 2 g2 ' (4.10) with a = (DzX) 2 + 2DXD3X,
(4.11)
transforms under superdiffeomorphisms as a superconnection. Therefore we can define a supercovariant derivative, Vr, acting on superfields q~h, of weight h, with the help of F by
Vr~ h = DcI9h - hFcI9h . With all of this in mind we can directly check that the action given by ( l V4X V3X 1DXD2X ) a f dt dO 2 E3A 2 g A - B D B + 2 B ( E K ) 1/2
(4.12)
(4.13)
is explicitly supercovariant, while reproducing (4.9) when we set the superpartners to zero. Unfortunately, when the supercovariant derivatives are expanded in terms of the superderivatives of X ~' the expression obtained is so involved that it has prevented us from making explicit the connection with superintegrability. 5. Final c o m m e n t s and some conjectures
We hope to have convinced the reader that W3-symmetry does play an important role in the symmetry structure of the rigid particle, and that standard
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E. Ramos, J. Roca / Nuclear Physics B436 (1995) 529-54l
techniques in constrained hamiltonian systems and W-algebras can be intertwined as powerful tools for the better understanding of both disciplines. It is also quite clear from our results that the connection between integrability and particle models does not restrict to the rigid particle case and seems to be present in other physically relevant particle models. But, of course, much is still to be done. For example, we do not yet completely understand the appearance of Lie algebraic structures in the problem, which should be the key to generalize these procedures to other W-algebras. Although it seems plausible that a particle model with the action depending on the (n - 1)th-order extrinsic curvature will enjoy Wn-symmetry, explicit computations become untractable and a more powerful machinery should be developed. Under quantization [4] the rigid particle is associated with massless representations of the Poincar6 group with integer helicity, therefore becoming a potential candidate for a particle description of photons, gravitons and higher spin fields. It is a tantalizing possibility that W3-symmetry can play a role in such physical systems. In fact there are some indications that this is indeed the case. It is clear from the transformation rules under pure W3-morphisms that the world line element of the particle has no gauge-invariant meaning 5. This nicely connects with some general arguments [17] implying that it is impossible to assign, even classicaly, a position to a single photon (or similarly, any massless higher spin fields). We would like to finish with a few sentences on the possible relevance of our results to W3-gravity. It was shown in [18] that a naive coupling to gravity of the action (1.1), i.e. considering arbitrary x-dependent metrics, changes the gauge structure of the system, and only reparametrization invariance survives. This is due to the fact that the constraint algebra no longer closes in curved space-time. We believe this to be a signal indicating that the rigid particle is only consistently coupled to W3-gravity. Let us elaborate on this. In the standard particle case, the relevant bundle is the tangent bundle of the manifold, TM. In this case, it is well known how to equip the bundle with a metric structure, and all the powerful and well-understood machinery of riemmanian geometry is at our disposal. In the language of jet bundles TM is nothing but JI(R, M), and it is our understanding that the relevant bundle for the W3-particle is provided by j2(~, M), which is not itself a vector bundle. It is well known that j2(~, M) is an associated bundle to F2M, the frame bundle of second order, and it is our belief that the required structure should be a "natural" structure in F2M, as it is the metric in FM. It is our hope that the action of the rigid particle will provide us with a "W3-1ine element", thus offering some valuable insight about which generalized structures in F2M should be considered. Work on this is in progress.
5 This is of course a well-known fact, although to the best of our knowledge the gauge transformations which render the position vector a gauge-variant object were not displayed in the literature.
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Acknowledgements We would like to thank J.M. Figueroa-O'Farrill and C.M. Hull for many useful discussions on the subject. J.R. is also grateful to the Spanish ministry of education and the British council for financial support.
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