CURRENT ALGEBRAS IN THE THEORY OF THE CLASSICAL ~ = 2 + i STRING

WITH INTERNAL DEGREES OF FREEDOM

S. V. Talalov

A Hamiltonian theory is proposed for a classical infinite ~=2+I string with distributed Majorana spinor field whose components are real numbers. It is shown that the Poisson structure of the model is determined by two centrally extended current algebras that are in involution.

The Hamiltonian approach in the theory of nonlinear evolution equations, which was first proposed in [i], is today the most powerful tool for investigating the dynamics of classical systems corresponding to such equations [2]. One of the most interesting and promising directions, here is the application of the methods of affine Lie algebras [3,4]. In this paper we consider the Hamiltonian theory of a classical infinite relativistic ~9=2+i string with spinor field distributed along the string. The spinors that define the field are spinors both in the original space-time M~,= and in the corresponding planes tangent to the world surface of the string. The proposed theory contains as a special case the standard ~)=2+I Nambu-Goto string in the light-cone gauge.

i. Thus, we consider a locally minimal world surface in the space Mi,2,

X~=X.(~0, ~l), ~=0, l, 2, - ~ < ~ ~ ~ '<~,

the system of c o o r d i n a t e s ( t ~ 61 ) on the s u r f a c e being conformal ly f l a t :

(a~X)~=~ (1)

where 0• ~•177176 (a)2=a~a"=ao~--a,2--a~ 2. I t i s we l l known [5,6] t h a t in t h i s case the components of the vector X~ will be harmonic functions of the operator 8+3_, and the theory is invariant with respect to conformal transformations of the parameters $ ~ $i:

~• (2) where f+ are arbitrary regular functions such that /+'(~+)]-'(~-)~0. On the surface {X~} there is defined a spinor field ~($0, 61) with components ~(~~ ~), a,]=l, 2. The two types of indices signify that the numbers iy=(~0 ~) define not only spinors in the original space- time M~2 (a is an index) but also spinors in the plane {X~} tangent to the world surface at the point X~(g ~ $i). For convenience, one of the indices will, as a rule, be omitted: the notationUl% a=l, 2, will signify a pair of two-dimensional spinors, and ~j, j = i, 2, a pair of three-dimensional spinors. By definition, we shall assume that the fields p0($0, ~i) are free massless two-dimensional spinor fields. Thus, the equations of motion of the model are

O+O-X~=O, ~=0, i, 2, (3a)

iTJ~F~ a=l, 2, (3b)

where ?~ ~l~i~, O~O/O~.

It is well known (see, for example, [7]) that such equations are characteristic of superstring theory. The spinors ~j, j = i, 2, are Majorana spinors. In the paper we use the purely imaginary representation for the Dirac F matrices in M~,2,

and therefore the components of the spinors in the theory are real numbers [8]. In what follows, we shall for convenience denote ~=~+= and ~F~F- ~, a=1, 2. In this notation, Eqs. (3b) can obviously be written in the form

O• a=t , 2.

Kuibyshev State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. i, pp. 41-48, April, 1989. Original article submitted September 29, 1987.

0040-5779/89/7901-0369512.50 �9 1989 Plenum Publishing Corporation 369

We define a pair of three-dimensional isotropic vectors J~: +

2 ] J = ~ • 1 7 7 ~•177 ~

By means of the vectors J~ we formulate subsidiary conditions that play an important role in the theory. We consider only solutions of the equations of motion (3) for which

a~x~]• (4a )

In c o n j u n c t i o n w i t h t h e c o n d i t i o n (4a) t h e p r o p o s e d t h e o r y g e n e r a l i z e s t h e t h e o r y o f t h e c l a s s i c a l " b o s o n i c " ~ 2 + i Nambu-Goto s t r i n g in t h e l i g h t - c o n e gauge . I n d e e d , i f we choose t h e s o l u t i o n s o f Eqs. (3b) in t h e form W + ~ W - ~ c 0 n s t , t h e n in t h i s c a s e J~ = - J ~ = nt ~, where nP i s a c o n s t a n t i s o t r o p i c v e c t o r . Then t h e c o n d i t i o n s (4a ) a r e w r i t t e n as s t a n d a r d c o n d i t i o n s o f t h e l i g h t - c o n e gauge: 3+X~n~ ~ 0. For c o n v e n i e n c e in t h e f o l l o w i n g c a l c u l a t i o n s , we f i x t h e c o n f o r m a l a r b i t r a r i n e s s o f (2 ) by t h e r e q u i r e m e n t

O• ~, (4b)

where U = c o n s t ~ 0. We emphas ize h e r e t h a t i t i s t h e c o n d i t i o n (4a) and n o t (4b) t h a t i s f u n d a m e n t a l .

We i n t r o d u c e t h e n o t a t i o n xJ~•177 ~, y • 1 7 7 v=0, l, 2, and a l s o z•177177 e~ In a c c o r d a n c e w i t h ( 1 ) , t h e d e f i n i t i o n o f J~ , and (4b) t h e v e c t o r s x• and y• a r e i s o t r o p i c , xiY• = �89 and (z• 2 = - 1 . Th i s means t h a t each o f t h e v e c t o r t r i p l e t s (x+, y+, z+) and (x_ , y_ , z_) d e f i n e s an o r t h o n o r m a l b a s i s in t h e o r i g i n a l spac e M~,~. Le t B+(~+) be m a t r i c e s o f t h e c o o r d i n a t e s o f each o f them in some c o n s t a n t b a s i s ~ (~=~=g~, a, ~=0, t, 2):

/Xo • x~ x ~ ~• (~• = ~ |yo• y,• yA ].

\Zo• z~• z~• /

We denote by ,Tt(~ ~ ~') the matrix of the transition from the + basis to the - basis:

B_ (~_) =~ (~~ ~') B+ (~+). (5)

Using the properties of the vectors of each of the bases B+ and B_~ we can readily show that the matrix $i ~ can be parametrized by the elements Kij($ ~ $~) of a real 2 • 2 matrix K(~ ~ iz), det K = i:

�9 2 Kll 2

i ~\2K~Kn 2KnKm K~IKs~ KnK~ .+ K~IKI~,

and this, of course, is a transparent representation of the (local) isomorphism of the groups SO(I, 2)0 and SL(2, ~). It follows from (5) that

a_ (~-'a+~) =0. ; 7 )

By virtue of (6) the matrix K(~ ~ $i) satisfies the same equation:

a_ (K-'0+K) =0. ( 8 )

Note that Eq. (8) is invarian with respect to the transformations K(~ ~ ~ .... ~:~ ~)= U_(~_)K(~ ~ ~)U+(~+). For K($ ~ $i) we write down the Gauss decomposition [9]:

, + ) ,

~ i/ exp !0 * (9)

Such an expansion exists always except for the cases when

K~(~ ~ ~)------exp (--~ (~ ~ y ) / 2 ) =0o (~.0)

It is obvious that Eq. (I0) determines in the (~0 ~z) plane the set of points of the singularities of the fields ~(~0 ~), ~.~(~0 ~). For other ~0, $I the fields ~, cz• are defined by (9) uniquely, At these (regulsr) points of the (~, ~l) plane we have by virtue of (8) and (9)

-%a+a_~+ (a_o~+) (a+~_) e-,=o, ( i la )

370

Denoting the expression in the square brackets in (llb) by p• we find

- ' /~a + O-~+ p +p-e'=O,

O• =0,

0• =p•

( l lb)

(12a)

(12b)

(12c)

2. We discuss the obtained system (12) from the point of view of the original object. By the definition of the matrix ~ and the field ~ we have

x,+x - "=--~/~ exp (--~). (13)

Thus, ~(~~ ~) de te rmines the f i r s t q u a d r a t i c form of the world s u r f a c e of the s t r i n g X~:

dsx ~ =e-" [ (d~ ~ ) ~- ( d~ ~) ~1.

Further, writing down for the coefficients of the first and the second (bij) quadratic forms of the wo<id surface X~($ ~ $1) the Peterson-Codazzi and Gauss equations, we find [6]

a• (14a)

R~=2e-~O+O_~ = ( b.+b~J ( b . -b ,~) . ( 14b )

Hence, by virtue of (12)

(bu•177 ~', c~const.

In terms of the c o o r d i n a t e s X~($ ~ $1) Eq. (15) g ives [6]

(O•177 *'.

Here we must no te t he fo l lowing . I f the f u n c t i o n s P• van ish nowhere, we have the s t anda rd [6] g e o m e t r i c a l d e s c r i p t i o n of the s t r i n g X~ in terms of a s i n g l e f u n c t i o n ~(~0, %~), which s a t i s f i e s t he L i o u v i l l e equa t ion ( see (12a) ) . In c o n t r a s t , t he s t r i n g c o n f i g u r a t i o n s fo r which the f u n c t i o n s p+($+) can van ish a re not confo rmal ly e q u i v a l e n t c o n f i g u r a t i o n s wi th pi($• ~ cons t . For each such s t r i n g , as was shown above, t he Gauss and Pe t e r son -Codazz i equa t ions reduce to a p a i r of equa t ions : (12a) and (12b). In what fo l lows , we s h a l l r ega rd t h e f u n c t i o n s p+(~+) as dynamical v a r i a b l e s and al low them to van i sh , in i n t e r v a l s too.

(15)

(16)

.

where

Equations (12) are the Euler-Lagrange equations for the action

~ = - ~ ~ ~ (~~ ~') d~ ~ d~',

(U, ~') =v/a+~) (a_~) +p+p_e~-p+a_a+-p_a+a_.

Note that the act ion (17) was proposed for the f i r s t t ime in [10] fo r the const ruc t ion of a model of an interacting scalar, ~, and spinor, ~=(~+,~_)r, ~•177 fields in two-dimensional space-time ($0, $I). It was also shown there that the system (12) is completely integrable and that the Cauchy problem with regular initial data ~(~)~(0, ~), ~(~)~0~0~(0, ~), p~(~), ~•177 ~) is uniquely solvable. In [Ii] the same was established for the case when the initial data have a finite number of singularities of a definite type (poles of ~(~) and a• and logarithmic singularity of ~(~); p~ are always regular)�9 We describe briefly the scheme for solving the Cauchy problem for the system (12). We introduce the matrix

U(~~ [ ~(~,~1)], p+(~+)exp [ ~(~0,~1)2 ] !

- ~l,a~ (~~ ~1)la~~

(17)

(18)

In [i0], the solution of the Cauchy problem for (12a) was constructed by means of the matrix of solutions S(~) of a linear 2 • 2 system:

S' (~)+U(0, ~)S(D =0.

Having at our disposal the solution ~(~0, ~i) const-ucted from the initial data ~, ~, p• we can

371

recover the functions ~• $i) from the initial data a• uniquely. However, in the general (singular) case such "deooupling" does not occur - for the solution of the Cauchy problem and also for the proof of the complete integrability in the sense of [2] it is necessary to regard the system (12) as a single entity. We introduce the following notation. Let A and B be certain 2 • 2 matrices that depend on ~0 ~l Then

A [B] - -A- 'BA- (OA-'/O~')A.

We also introduce the matrices

A + = 0 I ' ~- ' - e ~r162 0 "

In [ii] it was shown that the matrices

O(•176 ~') -A• IF= [u(~ ~ ~') ]] ( i9 )

are always r e g u l a r and fo r them

0~Q(• ~ ~')=0. (20)

The solution of the Cauchy problem for (12) (in the case -~ < ~i < ~) takes in accordance with [ii] the following form. Let Ti($) be the matrices of solutions of the pair of linear regular 2 x 2 systems

T• (~)+Q(~)(~) T• (~) =0. (2! )

Then the matrix K(~ ~ ~l) determined from the fields ~(~0, ~), a• ~) in accordance with (9 ) is*

K (~0, ~) =T_ (~_) T+ -~ (~+). (22)

In accordance with (21) , T+ a re de termined up to a t r a n s f o r m a t i o n T+ + T• = T•177 where (B+)ij ~ const. If we fix~ for example T+, then the other matrix (T_) is uniquely fixed by-the boundary conditions on the fields ~, a• The remaining three-parameter arbitrari- ness in the choice of T•

T•177 det B=I, (23)

exactly corresponds to the well-known Bianchi transformation in the theory of the Liouville equation.

4. We obtain formulas that recover the original variables X~ and P~ from the initial data of the Cauchy problem of the system (12). For this we introduce in M~, 2 a constant basis (~, ~, N~), where a='/~(Ni-~0), ~=~/~(~+~0), ai=~=0. In accordance with [6] we obtain for the derivatives of the world surface X~

/•

and also

e - ' = P+P- [/++1 l ', 7+'I2 where f+ = f• are arbitrary regular functions (note that the conditions p+ ~ const adopted in [6] do not affect the derivation of Eqs, (24)). In our case exp(-~)=K~?+ so that by virtue of (21) and (22) we have, up to a transformation (23),

t = + p+ tl,- p.

]+= t=+, ]+ '=(t ,C)~-, I- t . - ] " - - (t, ,-) ' '

where t~j ~ (Ti)ij~

Hence we find �9 ~. ~ + \ 3 + ~ + +

t z - = - ( t . - ) ~=+ ( t , : ) ~ + t . - t . - n , . By direct verification we show that the three-parameter transformation (23) induces a

*In this paper the matrices K and Q(+) differ from the ones used in [!i] by the trivial transformation K~Ko1, Q(+)~oiQ(+)o,.

(24b)

(25)

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Lorentz transformation of the vectors x• As real Majorana spinors we choose

~F+= ~ - " t . + t~- "

Such a choice is justified by the fact that Lorentz transformations of the vectors X~ lead, apart from the sign, to transformations of the form (23) of the quantities t~j and, thus, the objects ~+ and ~_ defined in accordance with (26) transform as spinors of the space

MI,2.

Thus, we have recovered the original object -- the classical ~)=2+I string {X~} with distributed spinor field T~ -- from the initial data of the Cauchy problem of the system (12). Uniqueness of the recovery will be achieved if besides the equations of motion (3) we consider the boundary conditions on the original variables X~ and T~, for example,

, (; i,t,+i-+}J--(~ I ' t ' - i -+N ) ~i._>.+=. (27) As was shown in [i0], the system (12) has a large symmetry group that generalizes the conformal group (2) in the theory of the Liouville equation. This means that the condi- tions (4b), in contrast to (4a), are not essential (the final equations are conformally invariant!). However, we shall not consider here the general case based solely on the conditions (4a).

5. The Hamiltonian structure on the set of (singular, in general) solutions of the system (12) such that ~', =e', p• as ~i + • is determined as follows [ii]. We set formally

{~(D, ~(~)}=6(~--~), (~• 9•177 (28) and all the remaining brackets of the initial data are zero. Then

(q<• @qc~>(n)}=2n~8'(~-n)+ [n• l~| <• (n)-q<~ (~)ol~] 8(~-~), (29a)

{q<+>(~), | (29b)

where R• ~ gz / z s (2P --1~), P i s t he m a t r i x of a p e r m u t a t i o n , t he square b r a c k e t s denote the commutator of m a t r i c e s , and | and, | a r e the s t a n d a r d l y d e f i n e d [2] t e n s o r o p e r a t i o n s . Note t h a t (29a) r e p r e s e n t a p a i r of second Hami l ton ian s t r u c t u r e s of the n o n l i n e a r Schr6d inge r e q u a t i o n [12 ,13 ] .

We emphasize that the elements of the matrices Q(• are always regular and therefore the brackets (29), in contrast to (28), are the brackets of regular quantities. This provides the basis for taking as the foundation of the original Hamiltonian structure the brackets (29), and not the brackets (28), which are not defined when ~(~), ~(~), ~• have singularities. The Hamiltonian structure (12) is constructed in detail on the basis of (29) in [ii], in which regular canonical action-angle variables are also constructed.

We define the currents j~($):

q<~ (~) = - ~ S• r ~. (30)

Then by virtue of (29)

{J• JJ (N)} =• ~6' (i--q) ~e~J• (~) 8 (~--~I), (31a)

{j+" (~), jZ(~I) } =0. (31b)

The obtained results make it possible to describe the original object - represented by Xp($0, $i) and sa(g0, ~z) that satisfy the equations of motion (3), the subsidiary condi- tions (4), and the boundary conditions (27) -- in terms of the pair of regular decreasing currents j~($). The dynamics of the currents is determined by the equations 8+j~($ ~ Sz) = 0, and the Poisson structure on the set of functionals of j~ can be defined,-for example, as

where the Poisson brackets in the integrand are defined in accordance with (31) and the variational derivatives decrease sufficiently rapidly at infinity. Thus, the Poisson

373

structure of the theory is determined in terms of the Lie operators j~ (where ~F ~ {a~ ?}) by a pair of algebras of the type (so(i, 2)|

The canonical energy-momentum tensor T~V for the action (17) has the form IlO]

O•176176176177177 ~.

The Lie operators corresponding to 8• form the Virasoro algebra, since, as follows from (31),

{0•177 (O§

6. The proposed approach can be regarded as a generalization of the well=kno~.~ geometrical approach to the ~=2+I Nambu-Goto string based on the use of the Liouville equation D~+exp~=0 [5,6,14], the generalization being that the dynamical variables in the theory are not only the coefficients of the first quadratic form (exp(--~(~~ but also those of the second quadratic form of the string world surface. As a consequence, the class of possible motions of the string is extended. For example, the model allows string configurations such that 8• ~ const for ~e~[a• be], which are impossible in the case of the conditions p• ~ const.

Equation (8) can be regarded as a special case of the equation of a SL(2, R)-valued chiral field with anomaly. Such fields with values in a Compact Lie algebra were first considered in [15]. Here the occurrence of central extensions of the current algebras is known [2,16]. In this paper we have, through Eq. (8), established a connection between the proposed string model and the conformal model of two-dimensional field theory [i0,i!]~ In conclusion we note that in the theory the vector (X~) and spinor (~• degrees of freedom are completely on an equal footing.

I thank A. K. Pogrebkov, M. K. Polivanov, and L. O. Chekhov for fruitful discussions.

LITERATURE CITED

I. V. E. Zakharov and L. D. Faddeev, Funktsional. Analiz i Ego Prilozhen., ~, !8 (1971)o 2. L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in the Theory of Soiitons

[in Russian], Nauka, Moscow (1986). 3. A. G. Reiman and M. A. Semenov-Tyan-Shanskii, Dokl. Akad. Nauk SSSR, 251, !310 (1980). 4. N. Yu. Reshetikhin and L. D. Faddeev, Teor. Mat. Fiz., 56, 323 (1983). 5. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka,

Moscow (1979). 6. B. M. Barbashov and V. V. Nesterenko, The Model of a Relativistic String in Hadron

Physics [in Russian], ~nergoatomizdat, Moscow (1987). 7. J. H. Schwarz, Phys. Rep. C, 89, 223 (1982). 8. D. Scherk, in: Geometrical Ideas in Physics [Russian translations], Mir, Moscow

(1983). 9. A. O. Barut and R. Raczka, Theory of Group Representations and Applications,

Warsaw (1977). i0. A. K. Pogrebkov and S. V. Talalov, Teor. Mat. Fiz., 70, 342 (1987). ii. S. V. Talalov, Teor. Mat. Fiz., 71, 357 (1987). 12. P. P. Kulish and A. G. Reiman, Zap. Nauchn. Sem. LOMI, 77, 124 (1978), 13. F. Magri, J. Math. Phys., 1-9, 1156 (1978). 14. B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, Teor. Mat. Fiz,~ 40, 15

(1979). 15. S. P. Novikov, Dokl. Akad. Nauk SSSR, 260, 31 (1981). 16. E. Witten, Commun. Math. Phys., 92, 455 (1984).

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