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Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions
Physics Letters B 265 ( 1991 ) 86-91 North-Holland
PHYSICS LETTERS B
Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions A.O. R a d u l ul. Zigulevska),a 5-1-4. SU-109 457 Moscow. USSR Received 1I December 1990; revised manuscript received 6 June 1991
Explicit formulas for the central extensions for Lie algebras of differential and pseudodifferential operators on the supercircle S~~Nand the general n-dimensional manifold M" are given. Possible applications of these formulas to gravity theories coupled with higher spin fields in two and higher dimensions are discussed.
1. Conformal symmetry in two dimensions [ 1 ] and its various extensions play an important role in quantum field theory and in string theory [ 2 ]. Most of these symmetries (and hypothetically all of them) have a geometrical interpretation as the symmetries of homogeneous G-spaces G / H (the coset construction) [3]. Here G and H denote a Lie algebra and its subalgebra. Usually G and H are connected with finite-dimensional simple Lie algebras. Well-known W : s y m m e t r i e s [ 4 - 6 ] including currents J , ( z ) with spin n + 1 can be interpreted as the coset B,/N~ [ 7 - 1 1 ] . (Here B, denotes a Borel subgroup o f some simple Lie group G, N, denotes a unipotent one.) This interpretation does not naturally explain the origin and geometrical meaning of higher spin transformations. In a previous work [ 12 ] I presented some arguments to confirm that W,-symmetries are connected with the Lie algebra o f differential operators on the circle D O P ( S I)={eo +el 0+e2 02+...+ek 0k} , O=d/d.x,
x~S l ,
which contains fields with arbitrary high spin. As quantum anomalies usually lead to nontrivial central extensions of the algebra o f symmetries, one could ask about them for D O P ( S j ). It is well known that there exists a unique nontrivial central extcnsion of the Lie algebra D O P ( S t ) [ 1386
16 ]. The corresponding two-cocycle can be chosen in the form c(f,,O",g,,O")-
m!n! ff(~)o(m+l) dx. (m+n+l)!.J ..... sI (1)
Here f ~ 7 ) =_O~ (.£,) denotes the nth derivative, the integration is taken over S ~. if we restrict ( 1 ) to functions or vector fields it turns into the well-known Heisenberg and Gelfand-Fuks cocycles: c(f), go) = ~ fg' d x , c(d']~ O,g, 8 ) = ~ j ' f l g ' / d x . This article is devoted to the following problem: how to generalize the cocycle ( I ) to n-dimensional (super)manifolds? Explicit formulas (17), (32) for the cases S ~:x (supercircle) and M" (n-dimensional manifold ) are presented. We will apply these formulas to 2D gravity [ 17-19 ] and its generalizations (supercase, fields with higher spin, n-dimensional case) see refs. [ 2 0 22 ] elsewhere. 2. In this section another form [see ( 1 0 ) ] of cocycle ( 1 ), discovered in ref. [23], will be described. All other formulas for cocycles can be viewed as its suitable generalizations.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 265, number 1,2
PHYSICS LETTERS B
8 August 1991 A
We start with the so-called symbol product
AoB= ~" O?AO~B/o~!
(2)
o~>_.0
A ~ [ l n ~,A]
Here A (x, ~) and B(x, ~) are some "functions" in two variables x and ~, or, more precisely, elements of a commutative ring .~ endowed with two commuting derivatives 0x and 0~. a runs over non-negative integers. ,4 should be endowed with a topology to make the RHS of (2) convergent for arbitrary A and B. Then (2) defines an associative operation ~/®~--,~/. A typical example of ,~ is the ring of the classical (formal pseudodifferential) symbols ~ C L ( S ' ) = I_o~
ak~C°°(S t ) ,
(3)
with the elements being formal Laurent series in ~- t and smooth functions as coefficients. When the symbol product is restricted to the subspace of polynomials in ~, it corresponds to the product of differential operators via ~.al,.~ k--} ~ak Ok. There is the so called noncommutative residue [24,25] on CL(S t ) defined by r e s ( ~ ak~k)=a_,.
(4)
It possesses the obvious property res 0e4 = 0.
(5)
One can define the functional Tr: CL(S t ) - , k (here k is the basic field ~ or C) by T r A = ~ resA d x .
(6)
Here the integration is taken over S t . Using the obvious relation f0~b=0 and (5) one can easily verify that Tr[A,B]=0
(7)
for all A, B e C L ( S t ) . Here [A, B] =AoB-B.A. Another example of ,4 is
-
{
F.
}
akj(x)~k(ln ~)/ ,
(8)
--ot~
(9)
preserves the smaller ring CL(S t ). Now we are in the position to state the main result ofref. [23]: the cocycle ( ! ) can be written in the form
c(A, B) = T r [ i n ~, A] B.
(10)
Here A = ~a,~ ~,B= ~bk~k are differential operators on S t. One can assume A, B to be classical symbols. Then (10) defines a nontrivial two-cocycle on the Lie algebra CL(S t ). A few words about proofs. The formula (10) defines a bilinear map CL(S t ) CL(S t ) --,k. This map defines a two-cocycle on the Lie algebra CL(S ~) if and only if the cocycle equations are satisfied.
c(A,B)+c(B,A)=O,
(11)
c( [A, B], D) + c ( [B, Ol, A) + c ( [O, A ], B) = 0 . (12) Eq. ( l 1 ) is equivalent to the identity Tr[ln ~, A] = 0
CL ( S t ) =
A
defined map CL (S ~) --,CL (S t ). It is easy to see, that this map
for every A e C L ( S t ) ,
(13)
which is proved by direct calculation. Eq. (12) follows from (13), (7) and the Jacobi identity for the Lie algebra CL(S t ). 3. This section is devoted to the supercase. It is well known [26], see also ref. [27 ], that there exists only a finite number of"super-Virasoro" algebras, namely only a finite number of Lie superalgebras of vector fields on the supercircle S ~~x has a nontrivial central extension. The situation is quite different for the Lie superalgebras DOP(S t iN) and CL(S ~is): all of them have nontrivial central extensions, which become trivial when one restricts them to vector fields. To describe these extensions we need some notations. We will consider the Neveu-Schwarz (periodic) case only, all modifications in the Ramond case are left to the reader. Let (x, rh ..... qU), 2 = 0 , r],= l be the coordinate system on S t IN (we denote by ~ the parity of a). Classical symbols on S t in are defined as
with formal Laurent series in ~- ~and polynomials in In ~with coefficients in C°°(SI ). In particular the commutator with In ~ is a well87
Volume 265, number 1,2
CL(S'~X)={A=~
PHYSICS LETTERS B
8 August 1991
c(r, a)=J
ak.,. ........ (x, rl)d,kA'~l...).{<~",
(f~go-g'lf,) dx~0.
Here 0;., denotes the right 2,-derivative. There exists the (right) superresidue on C L ( S tl~) [28]
This contradiction proves (17) to he nontrivial on D O P ( S ~ix) and also on C L ( S 2ix). To investigate the restriction of (17) on vector fields we need some additional definitions. We remind the notations for Lie superalgebras on S ~by ( 1 ) W ( N ) are all vector fields. ( 2 ) K (N) are vector fields multiplying the contact form a = d x + ~rl, dq, by a function, or a Lie superalgcbra of N-extended supersymmctry. Explicitly the K ( N ) can bc described as vector fields of the form
sres ,.1 =a_ ~.~.....j(x, q) ,
F(x,q)
- ~ < k < ~ M , ~, = 0 , 1 } . Here ).",= 1, the 2, correspond to 0/017,. They are multiplied via AoB=
E
(~'A %' O~.,)(O.,""O,,B)/a. '" '.
(14)
cr ~> O , v , = (),1
( 15 )
and the supertrace functional s T r / l = f sresA Ber(x, q) .
E
(16)
~ 1 ...q, (n Ber (x, q) f "........(.~)r/,
(,=0.1
= f .~.,.....,(x) d x . It obeys the property sTr[A,B]=0
forA, B~CL(S]I'~).
Here
[A, B] denotes the s u p e r c o m m u t a t o r A B - ( - I )"TOBA. Similar to the :V=0 casc one can dcfinc CL (S ~1.4), prove that sTr[ln~,A]=0
for every A e C L ( S ~lx),
(17)
satisfies the supercocycle equations. To verify this cocycle being nontrivial on DOP(S~.4.) one can proceed as follows. Considcr a commutative subalgebra in D O P ( S ~ix), • ( I I N ) = { F = f ~ (x)q~...q.4. O,,,...O,,, + f , ( x ) } .
(18)
If (17) were trivial on D O P ( S ~lx) then one could represent it as).( [.4, B] ) for some linear functional ). on D O P ( S ~ .4.), and hence (17) should be zero on (1)( 1 IN). But a calculation shows that 88
(19)
(3) S ( N ) arc vector fields preserving a volume form Bet (x, q). (4) S°(N) with {/"0.,4-~_q),O,,,~s(m)l ~,, ...0,~,F = 0I. There exists an isomorphism W( 1 ) -~ K ( 2 ). Then the Fcigin-Leites theorem [26 ] states that: ( l ) There are two-cocycles on K ( N ) of the form (k, G)~.fv(F. G) Ber (x, q), where v(k: G) can be read from table I. In table 1 and below A' - 0~A, etc. ( 2 ) There are two-eocycles on W ( 1 ) and W (2), (FO, +~p O,, (/O, + ~u0,)
,-,f(-½1."q/+{~oG"(-1)':+(o'%,)
Ber,
(20)
(F O, + qh O,, + ~oz0,:, G 0, + ~, 0,, + ~'2 0,,., )
that A ~ [In ~, A ] maps C L ( S ~14) into itself and verify that c(A, B ) = s T r [In ~,A] B
I)P+' E ( 0 , F ) 0 , ,
0, = 0 / 0 r / , - r/, 0x.
,As usual, the Berezin integration is defined by
j"
O,+~(-
~ f ((o't I//2( - l )~':- q~21//i ( - l )~') Ber.
(21)
d
These formulas define cocycles on S ( 1 ), S (2), S O( l ). $ ° ( 2 ) as well. All these cocycles are nontrivial. No Table 1 N
v(F, G)
0 1 2 3
t"G"' l"Oj G" P'OlO2G' FOr0203G
Volume 265, number 1,2
PHYSICS LETTERS B
other nontrivial two-cocycles on W (N), K (N), S (N), S ° ( N ) for 0 ~ < N < ~ exist. Then a simple calculation shows that when ( 17 ) is
restricted to the corresponding Lie superalgebra of vector fields it is nonzero for O<~N<~2 only, and.for these N it coincides with the Feigin-Leites coo,cles in table 1 and eqs. (20), (21 ). Hence, the only exceptional case is K ( 3 ) : it has a nontrivial two-cocycle, which is not the restriction of ( 17 ). Nevertheless, hypothetically it can be included in the picture by a more subtle consideration, but we do not intend to discuss this question here. 4. This section is devoted to the multidimensional casc, namely to n-dimensional real manifolds. We assume they are compact and orientable. The cocycle is given by the same formula as in the ID case [see (10) ], one needs only to explain the meaning of all notations. The unique nontrivial ingredient is a multidimensional, noncommutative residue discovered by Wodzicki [29]. Other objects are well known [30,31]. The ring of classical (formal pseudodifferential) symbols on an n-dimensional manifold M" consists of formal sums
8 August 1991
tp(A ) = a _ ~ ( x , ~) ct ^ to n- l .
(24)
The form ~0(A ) has zero degree with respect to the dilatations ~ t~. I f one chooses some metric g on M, then the so called cosphere bundle N 2"- t is well defined: the fibre of N 2~-t over an m~M" is the unit sphere in T*,,,M. The noncommutative residue is the following n-form on M: Res(A) =
~
~p(A).
(25)
fibre N 2n- I
I recall the definition of fibre integration. One should take n vectors vt ..... v,,~T,,,M and substitute them in ~0(A), and obtain the ( n - l ) - f o r m i,,,...i~,~o(A) on T ' M , and then integrate it over S n- ~c T,,M. Then Res(A ) does not depend on the particular choice of the metric g. The trace functional on C L ( M ") is defined as T r ( A ) = .I R e s ( A ) .
(26)
Mn
A nontrivial computation shows that it has the property Tr[A,B]=0
foraI1A, B ~ C L ( M " ) .
(27)
The Tr is essentially unique, namely C L ( M " ) = {A = - ~ ~,~'Z < k ak(x,~)}.
(22)
Here x = (x~ ..... x,,) is a coordinate patch for M, ~=(~1 ..... ~,,) is a nonzero covector, ak(x, ~) are functions on the cotangent bundle without a zero section T*M \ 0, satisfying the homogeneity condition
at(x,t~)=tka~(x,~),
t>O,
TrA=0c:-A=~
Bk, G ~ C L ( M " ) . (28)
Let T(x, ~) be an arbitrary elliptic symbol, ord T = p # 0 or, in other words, that two conditions are satisfied: (1) Tt,(x, t~)=tVTt,(x, ~),
(23)
with respect to positive dilatation of the covectors. The multidimensional symbol product is defined by the formula (2), but now x and ~ are vectors (xt, .... x,,), ( ~ ..... ~,,), o~ runs over nonnegative integer vectors (o~ . . . . . o~n), a,>_-0; 0,,=0,'~.'...~," o~!= oG !...c¢,,! As in the I D case (2) defines an associative operation on every topological commutative ring .~/endowed with 2n commuting derivatives 0x, ..... 0~,; 0~...... 0¢,. Let ~ be the natural one-form on T*M (namely c~= Y~, dx,), og=dc~. One can consider the following ( 2 n - l )-form, depending on A:
[B~,C~],
(2)Tp(x,~)#0
p#O,
if~#0.
(29)
We define an extended ring of symbols via
-
CL r ( M " ) =
{
F..
}
ak,l(x, ~) In/Tp .
(30)
0<<.1<<.1.
The theory o f complex powers allows us to define the symbol Ln 7"via d 7": Ln T = dzz .o Re.z<0 and to see that 89
Volume 265, number 1,2
Ln T=ln
Tp+f
PHYSICS LETTERS B
feCL(M").
The operation A ~ [Ln 7, A] maps CL(M") into itself and the properly Tr[LnT, A ] = 0
obeys ord S = - n symbol
and commutes with Ln T. The
has zero trace. Using (28) one can represent it as [ Bk, Ck ]. Then
[ Ln
T, Tr(A)s+~ Tr (S)
=Y.{[[LnT, Bk],C~]+[Bk,
[Bk'Ck]]
Res[Ln
[Ln T, C~]]}.
T,A]B
(32)
M n
satisfies ( I 1 ) and ( 12 ). The cohomology class of this cocycle essentially does not depend on T. More precisely,for all symbols 7", and 7", satisfying (29), the linear combination C7z
is a coboundary. Indeed, S, and ,-72are defined via (31). Then n
cs,=-~ct~,
i= 1, 2,
(33)
cs.,. -c~-2. = Tr[In S,$21_,,_,+f' +f2, A] for some classical symbolsft and f2. But the function In(St_~/S2._~) (x, ~) has zero degree with respect to the dilatations ~-, t~, and hence belongs to the ring CL(M"). Then, using (27) one can represent the RHS of (33) as
90
I wish to thank Yu.l. Manin for his interest in this work, B. Feigin, A. Givental, A. Morozov, A. Marshakov, A. Vasiliev, and I. Zaharevich for useful comments, and I. Vaysburd for numerous discussions. My special thanks go to O. Kravchenko and B. Hesin for information about their result [23 ] before publication.
References
We represent [Ln T, A] as the sum of commutators of the classical symbols, hence it has zero trace. Now one formally verifies that the cocycle
p2C.r, -- Pt
or as coboundary of the one-cochain
( 31 )
Tr(A) S Tr(S)
C.r(A,B)= f
(34)
A~Tr(In St _, +f, + f 2 ) A .
S(x, ~) = T-"/P(x, ~)
[Ln T , A ] =
Tr(ln $21_,S' _, +f, + f 2 ) [ A , B]
forAeCL(M")
is verified as follows. The symbol
A
8 August 1991
[I]A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] A.M. Polyakov, Phys. Left. B 103 ( 1981 ) 207. [3] G. Moore and N. Seiberg, preprint IASSNS-HEP-89/6. [4] A.B. Zamolodchikov, Teor. Mat. Fiz. 65 (1985) 347. [5] S.L. Lukyanov, Funct. Anal. Appl. 22 (1988) 1. [6] S.L. Lukyanov and V.A. Fateev, Zh. Eksp. Teor. Fiz. 94 ( 1988 ) 23. [7]A.A. Belavin, Proc. Second Yukawa Memorial Symp. (Nishinomiya, Japan, 1987), Ser. Proc. in Physics, Vol. 31 (Springer, Berlin, 1989 ) p. 132. 181V.G. Drinfeld and V.V. Sokolov, J. Soy. Math. 30 (1985) 1975. [9] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 ( 1989 ) 49. [ 10] I. Bakas, Phys. Left. B 219 (1989) 283. [ 11 ] A. Alekseev and S. Shatashvili, Nucl. Phys. B 323 (1989) 719. [ 12] A.O. Radul, Pis'ma Zh. Eksp. Teor. Fiz. 50 (1989) 341. [ 13 ] B.L. Feigin, Usp. Mat. Nauk 2 ( 1988 ) 157. [ 14] V.G. Kac and P. Peterson, preprint MIT 27/87 (1987). [ 15] A,A. Beilinson, Yu.l. Manin and V.V. Schechtman, in: Ktheory, arithmetic and geometry, Lecture Notes in Mathematics, Vol. 1289 (Springer, Berlin, 1987 )p. 52. [16] Yu.l. Manin, J. Geom. Phys. 5 (1988) 161. [ 17] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [18]A. Alekseev and S. Shatashvili, Nucl. Phys. B 323 (1989) 719. [ 19 ] J.-Z. Gervais, Phys. Lett. B 160 ( 1985 ) 277. [201E.S. Fradkin and M.A. Vasiliev, Nucl. Phys. B 291 (1987) 141. [211 M.A. Vasiliev, Nucl. Phys. B 307 (1988) 319.
Volume 265, number 1,2
PHYSICS LETTERS B
[22] P. Di Francesco and D. Kutasov, Princeton preprint PUPT 1206 ( 1990 ). [23] O.S. Kravchenko and B.A. Hesin, Funct. Anal. Priloz. 23 (1989) 78. [241Yu.l. Manin, J. Soy. Mat. 11 (1979) 1. [25] M. Adler, Invent. Math. 50 (1979) 219. [26] D.A. Leites and B.L. Feigin, in: Group-theoretical methods in physics, Vol. 1 (1983) p. 269. [27] A.O. Radul, Seminar on supermanifolds, Vol. 21, ed. D. Leites (University of Stockholm, 1986) p. 40.
8 August 1991
[28] Yu.i. Manin and A.O. Radul, Commun. Math. Phys. 98 (1985) 65. [29] M. Wodzicki, Lecture Notes in Mathematics, Vol. 1289 (Springer, Berlin, 1985) p. 320. [30] R.T. Seeley, Proc. Symp. on Pure mathematics, Vol. 10 (American Mathematical Society, Providence, RI, 1987) p. 288. [31] M.A. Shubin, Pseudodifferential operators and spectral theory (Nauka, Moscow, 1978) [in Russian].
91
PHYSICS LETTERS B
Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions A.O. R a d u l ul. Zigulevska),a 5-1-4. SU-109 457 Moscow. USSR Received 1I December 1990; revised manuscript received 6 June 1991
Explicit formulas for the central extensions for Lie algebras of differential and pseudodifferential operators on the supercircle S~~Nand the general n-dimensional manifold M" are given. Possible applications of these formulas to gravity theories coupled with higher spin fields in two and higher dimensions are discussed.
1. Conformal symmetry in two dimensions [ 1 ] and its various extensions play an important role in quantum field theory and in string theory [ 2 ]. Most of these symmetries (and hypothetically all of them) have a geometrical interpretation as the symmetries of homogeneous G-spaces G / H (the coset construction) [3]. Here G and H denote a Lie algebra and its subalgebra. Usually G and H are connected with finite-dimensional simple Lie algebras. Well-known W : s y m m e t r i e s [ 4 - 6 ] including currents J , ( z ) with spin n + 1 can be interpreted as the coset B,/N~ [ 7 - 1 1 ] . (Here B, denotes a Borel subgroup o f some simple Lie group G, N, denotes a unipotent one.) This interpretation does not naturally explain the origin and geometrical meaning of higher spin transformations. In a previous work [ 12 ] I presented some arguments to confirm that W,-symmetries are connected with the Lie algebra o f differential operators on the circle D O P ( S I)={eo +el 0+e2 02+...+ek 0k} , O=d/d.x,
x~S l ,
which contains fields with arbitrary high spin. As quantum anomalies usually lead to nontrivial central extensions of the algebra o f symmetries, one could ask about them for D O P ( S j ). It is well known that there exists a unique nontrivial central extcnsion of the Lie algebra D O P ( S t ) [ 1386
16 ]. The corresponding two-cocycle can be chosen in the form c(f,,O",g,,O")-
m!n! ff(~)o(m+l) dx. (m+n+l)!.J ..... sI (1)
Here f ~ 7 ) =_O~ (.£,) denotes the nth derivative, the integration is taken over S ~. if we restrict ( 1 ) to functions or vector fields it turns into the well-known Heisenberg and Gelfand-Fuks cocycles: c(f), go) = ~ fg' d x , c(d']~ O,g, 8 ) = ~ j ' f l g ' / d x . This article is devoted to the following problem: how to generalize the cocycle ( I ) to n-dimensional (super)manifolds? Explicit formulas (17), (32) for the cases S ~:x (supercircle) and M" (n-dimensional manifold ) are presented. We will apply these formulas to 2D gravity [ 17-19 ] and its generalizations (supercase, fields with higher spin, n-dimensional case) see refs. [ 2 0 22 ] elsewhere. 2. In this section another form [see ( 1 0 ) ] of cocycle ( 1 ), discovered in ref. [23], will be described. All other formulas for cocycles can be viewed as its suitable generalizations.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 265, number 1,2
PHYSICS LETTERS B
8 August 1991 A
We start with the so-called symbol product
AoB= ~" O?AO~B/o~!
(2)
o~>_.0
A ~ [ l n ~,A]
Here A (x, ~) and B(x, ~) are some "functions" in two variables x and ~, or, more precisely, elements of a commutative ring .~ endowed with two commuting derivatives 0x and 0~. a runs over non-negative integers. ,4 should be endowed with a topology to make the RHS of (2) convergent for arbitrary A and B. Then (2) defines an associative operation ~/®~--,~/. A typical example of ,~ is the ring of the classical (formal pseudodifferential) symbols ~ C L ( S ' ) = I_o~
ak~C°°(S t ) ,
(3)
with the elements being formal Laurent series in ~- t and smooth functions as coefficients. When the symbol product is restricted to the subspace of polynomials in ~, it corresponds to the product of differential operators via ~.al,.~ k--} ~ak Ok. There is the so called noncommutative residue [24,25] on CL(S t ) defined by r e s ( ~ ak~k)=a_,.
(4)
It possesses the obvious property res 0e4 = 0.
(5)
One can define the functional Tr: CL(S t ) - , k (here k is the basic field ~ or C) by T r A = ~ resA d x .
(6)
Here the integration is taken over S t . Using the obvious relation f0~b=0 and (5) one can easily verify that Tr[A,B]=0
(7)
for all A, B e C L ( S t ) . Here [A, B] =AoB-B.A. Another example of ,4 is
-
{
F.
}
akj(x)~k(ln ~)/ ,
(8)
--ot~
(9)
preserves the smaller ring CL(S t ). Now we are in the position to state the main result ofref. [23]: the cocycle ( ! ) can be written in the form
c(A, B) = T r [ i n ~, A] B.
(10)
Here A = ~a,~ ~,B= ~bk~k are differential operators on S t. One can assume A, B to be classical symbols. Then (10) defines a nontrivial two-cocycle on the Lie algebra CL(S t ). A few words about proofs. The formula (10) defines a bilinear map CL(S t ) CL(S t ) --,k. This map defines a two-cocycle on the Lie algebra CL(S ~) if and only if the cocycle equations are satisfied.
c(A,B)+c(B,A)=O,
(11)
c( [A, B], D) + c ( [B, Ol, A) + c ( [O, A ], B) = 0 . (12) Eq. ( l 1 ) is equivalent to the identity Tr[ln ~, A] = 0
CL ( S t ) =
A
defined map CL (S ~) --,CL (S t ). It is easy to see, that this map
for every A e C L ( S t ) ,
(13)
which is proved by direct calculation. Eq. (12) follows from (13), (7) and the Jacobi identity for the Lie algebra CL(S t ). 3. This section is devoted to the supercase. It is well known [26], see also ref. [27 ], that there exists only a finite number of"super-Virasoro" algebras, namely only a finite number of Lie superalgebras of vector fields on the supercircle S ~~x has a nontrivial central extension. The situation is quite different for the Lie superalgebras DOP(S t iN) and CL(S ~is): all of them have nontrivial central extensions, which become trivial when one restricts them to vector fields. To describe these extensions we need some notations. We will consider the Neveu-Schwarz (periodic) case only, all modifications in the Ramond case are left to the reader. Let (x, rh ..... qU), 2 = 0 , r],= l be the coordinate system on S t IN (we denote by ~ the parity of a). Classical symbols on S t in are defined as
with formal Laurent series in ~- ~and polynomials in In ~with coefficients in C°°(SI ). In particular the commutator with In ~ is a well87
Volume 265, number 1,2
CL(S'~X)={A=~
PHYSICS LETTERS B
8 August 1991
c(r, a)=J
ak.,. ........ (x, rl)d,kA'~l...).{<~",
(f~go-g'lf,) dx~0.
Here 0;., denotes the right 2,-derivative. There exists the (right) superresidue on C L ( S tl~) [28]
This contradiction proves (17) to he nontrivial on D O P ( S ~ix) and also on C L ( S 2ix). To investigate the restriction of (17) on vector fields we need some additional definitions. We remind the notations for Lie superalgebras on S ~by ( 1 ) W ( N ) are all vector fields. ( 2 ) K (N) are vector fields multiplying the contact form a = d x + ~rl, dq, by a function, or a Lie superalgcbra of N-extended supersymmctry. Explicitly the K ( N ) can bc described as vector fields of the form
sres ,.1 =a_ ~.~.....j(x, q) ,
F(x,q)
- ~ < k < ~ M , ~, = 0 , 1 } . Here ).",= 1, the 2, correspond to 0/017,. They are multiplied via AoB=
E
(~'A %' O~.,)(O.,""O,,B)/a. '" '.
(14)
cr ~> O , v , = (),1
( 15 )
and the supertrace functional s T r / l = f sresA Ber(x, q) .
E
(16)
~ 1 ...q, (n Ber (x, q) f "........(.~)r/,
(,=0.1
= f .~.,.....,(x) d x . It obeys the property sTr[A,B]=0
forA, B~CL(S]I'~).
Here
[A, B] denotes the s u p e r c o m m u t a t o r A B - ( - I )"TOBA. Similar to the :V=0 casc one can dcfinc CL (S ~1.4), prove that sTr[ln~,A]=0
for every A e C L ( S ~lx),
(17)
satisfies the supercocycle equations. To verify this cocycle being nontrivial on DOP(S~.4.) one can proceed as follows. Considcr a commutative subalgebra in D O P ( S ~ix), • ( I I N ) = { F = f ~ (x)q~...q.4. O,,,...O,,, + f , ( x ) } .
(18)
If (17) were trivial on D O P ( S ~lx) then one could represent it as).( [.4, B] ) for some linear functional ). on D O P ( S ~ .4.), and hence (17) should be zero on (1)( 1 IN). But a calculation shows that 88
(19)
(3) S ( N ) arc vector fields preserving a volume form Bet (x, q). (4) S°(N) with {/"0.,4-~_q),O,,,~s(m)l ~,, ...0,~,F = 0I. There exists an isomorphism W( 1 ) -~ K ( 2 ). Then the Fcigin-Leites theorem [26 ] states that: ( l ) There are two-cocycles on K ( N ) of the form (k, G)~.fv(F. G) Ber (x, q), where v(k: G) can be read from table I. In table 1 and below A' - 0~A, etc. ( 2 ) There are two-eocycles on W ( 1 ) and W (2), (FO, +~p O,, (/O, + ~u0,)
,-,f(-½1."q/+{~oG"(-1)':+(o'%,)
Ber,
(20)
(F O, + qh O,, + ~oz0,:, G 0, + ~, 0,, + ~'2 0,,., )
that A ~ [In ~, A ] maps C L ( S ~14) into itself and verify that c(A, B ) = s T r [In ~,A] B
I)P+' E ( 0 , F ) 0 , ,
0, = 0 / 0 r / , - r/, 0x.
,As usual, the Berezin integration is defined by
j"
O,+~(-
~ f ((o't I//2( - l )~':- q~21//i ( - l )~') Ber.
(21)
d
These formulas define cocycles on S ( 1 ), S (2), S O( l ). $ ° ( 2 ) as well. All these cocycles are nontrivial. No Table 1 N
v(F, G)
0 1 2 3
t"G"' l"Oj G" P'OlO2G' FOr0203G
Volume 265, number 1,2
PHYSICS LETTERS B
other nontrivial two-cocycles on W (N), K (N), S (N), S ° ( N ) for 0 ~ < N < ~ exist. Then a simple calculation shows that when ( 17 ) is
restricted to the corresponding Lie superalgebra of vector fields it is nonzero for O<~N<~2 only, and.for these N it coincides with the Feigin-Leites coo,cles in table 1 and eqs. (20), (21 ). Hence, the only exceptional case is K ( 3 ) : it has a nontrivial two-cocycle, which is not the restriction of ( 17 ). Nevertheless, hypothetically it can be included in the picture by a more subtle consideration, but we do not intend to discuss this question here. 4. This section is devoted to the multidimensional casc, namely to n-dimensional real manifolds. We assume they are compact and orientable. The cocycle is given by the same formula as in the ID case [see (10) ], one needs only to explain the meaning of all notations. The unique nontrivial ingredient is a multidimensional, noncommutative residue discovered by Wodzicki [29]. Other objects are well known [30,31]. The ring of classical (formal pseudodifferential) symbols on an n-dimensional manifold M" consists of formal sums
8 August 1991
tp(A ) = a _ ~ ( x , ~) ct ^ to n- l .
(24)
The form ~0(A ) has zero degree with respect to the dilatations ~ t~. I f one chooses some metric g on M, then the so called cosphere bundle N 2"- t is well defined: the fibre of N 2~-t over an m~M" is the unit sphere in T*,,,M. The noncommutative residue is the following n-form on M: Res(A) =
~
~p(A).
(25)
fibre N 2n- I
I recall the definition of fibre integration. One should take n vectors vt ..... v,,~T,,,M and substitute them in ~0(A), and obtain the ( n - l ) - f o r m i,,,...i~,~o(A) on T ' M , and then integrate it over S n- ~c T,,M. Then Res(A ) does not depend on the particular choice of the metric g. The trace functional on C L ( M ") is defined as T r ( A ) = .I R e s ( A ) .
(26)
Mn
A nontrivial computation shows that it has the property Tr[A,B]=0
foraI1A, B ~ C L ( M " ) .
(27)
The Tr is essentially unique, namely C L ( M " ) = {A = - ~ ~,~'Z < k ak(x,~)}.
(22)
Here x = (x~ ..... x,,) is a coordinate patch for M, ~=(~1 ..... ~,,) is a nonzero covector, ak(x, ~) are functions on the cotangent bundle without a zero section T*M \ 0, satisfying the homogeneity condition
at(x,t~)=tka~(x,~),
t>O,
TrA=0c:-A=~
Bk, G ~ C L ( M " ) . (28)
Let T(x, ~) be an arbitrary elliptic symbol, ord T = p # 0 or, in other words, that two conditions are satisfied: (1) Tt,(x, t~)=tVTt,(x, ~),
(23)
with respect to positive dilatation of the covectors. The multidimensional symbol product is defined by the formula (2), but now x and ~ are vectors (xt, .... x,,), ( ~ ..... ~,,), o~ runs over nonnegative integer vectors (o~ . . . . . o~n), a,>_-0; 0,,=0,'~.'...~," o~!= oG !...c¢,,! As in the I D case (2) defines an associative operation on every topological commutative ring .~/endowed with 2n commuting derivatives 0x, ..... 0~,; 0~...... 0¢,. Let ~ be the natural one-form on T*M (namely c~= Y~, dx,), og=dc~. One can consider the following ( 2 n - l )-form, depending on A:
[B~,C~],
(2)Tp(x,~)#0
p#O,
if~#0.
(29)
We define an extended ring of symbols via
-
CL r ( M " ) =
{
F..
}
ak,l(x, ~) In/Tp .
(30)
0<<.1<<.1.
The theory o f complex powers allows us to define the symbol Ln 7"via d 7": Ln T = dzz .o Re.z<0 and to see that 89
Volume 265, number 1,2
Ln T=ln
Tp+f
PHYSICS LETTERS B
feCL(M").
The operation A ~ [Ln 7, A] maps CL(M") into itself and the properly Tr[LnT, A ] = 0
obeys ord S = - n symbol
and commutes with Ln T. The
has zero trace. Using (28) one can represent it as [ Bk, Ck ]. Then
[ Ln
T, Tr(A)s+~ Tr (S)
=Y.{[[LnT, Bk],C~]+[Bk,
[Bk'Ck]]
Res[Ln
[Ln T, C~]]}.
T,A]B
(32)
M n
satisfies ( I 1 ) and ( 12 ). The cohomology class of this cocycle essentially does not depend on T. More precisely,for all symbols 7", and 7", satisfying (29), the linear combination C7z
is a coboundary. Indeed, S, and ,-72are defined via (31). Then n
cs,=-~ct~,
i= 1, 2,
(33)
cs.,. -c~-2. = Tr[In S,$21_,,_,+f' +f2, A] for some classical symbolsft and f2. But the function In(St_~/S2._~) (x, ~) has zero degree with respect to the dilatations ~-, t~, and hence belongs to the ring CL(M"). Then, using (27) one can represent the RHS of (33) as
90
I wish to thank Yu.l. Manin for his interest in this work, B. Feigin, A. Givental, A. Morozov, A. Marshakov, A. Vasiliev, and I. Zaharevich for useful comments, and I. Vaysburd for numerous discussions. My special thanks go to O. Kravchenko and B. Hesin for information about their result [23 ] before publication.
References
We represent [Ln T, A] as the sum of commutators of the classical symbols, hence it has zero trace. Now one formally verifies that the cocycle
p2C.r, -- Pt
or as coboundary of the one-cochain
( 31 )
Tr(A) S Tr(S)
C.r(A,B)= f
(34)
A~Tr(In St _, +f, + f 2 ) A .
S(x, ~) = T-"/P(x, ~)
[Ln T , A ] =
Tr(ln $21_,S' _, +f, + f 2 ) [ A , B]
forAeCL(M")
is verified as follows. The symbol
A
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[I]A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] A.M. Polyakov, Phys. Left. B 103 ( 1981 ) 207. [3] G. Moore and N. Seiberg, preprint IASSNS-HEP-89/6. [4] A.B. Zamolodchikov, Teor. Mat. Fiz. 65 (1985) 347. [5] S.L. Lukyanov, Funct. Anal. Appl. 22 (1988) 1. [6] S.L. Lukyanov and V.A. Fateev, Zh. Eksp. Teor. Fiz. 94 ( 1988 ) 23. [7]A.A. Belavin, Proc. Second Yukawa Memorial Symp. (Nishinomiya, Japan, 1987), Ser. Proc. in Physics, Vol. 31 (Springer, Berlin, 1989 ) p. 132. 181V.G. Drinfeld and V.V. Sokolov, J. Soy. Math. 30 (1985) 1975. [9] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 ( 1989 ) 49. [ 10] I. Bakas, Phys. Left. B 219 (1989) 283. [ 11 ] A. Alekseev and S. Shatashvili, Nucl. Phys. B 323 (1989) 719. [ 12] A.O. Radul, Pis'ma Zh. Eksp. Teor. Fiz. 50 (1989) 341. [ 13 ] B.L. Feigin, Usp. Mat. Nauk 2 ( 1988 ) 157. [ 14] V.G. Kac and P. Peterson, preprint MIT 27/87 (1987). [ 15] A,A. Beilinson, Yu.l. Manin and V.V. Schechtman, in: Ktheory, arithmetic and geometry, Lecture Notes in Mathematics, Vol. 1289 (Springer, Berlin, 1987 )p. 52. [16] Yu.l. Manin, J. Geom. Phys. 5 (1988) 161. [ 17] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [18]A. Alekseev and S. Shatashvili, Nucl. Phys. B 323 (1989) 719. [ 19 ] J.-Z. Gervais, Phys. Lett. B 160 ( 1985 ) 277. [201E.S. Fradkin and M.A. Vasiliev, Nucl. Phys. B 291 (1987) 141. [211 M.A. Vasiliev, Nucl. Phys. B 307 (1988) 319.
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[22] P. Di Francesco and D. Kutasov, Princeton preprint PUPT 1206 ( 1990 ). [23] O.S. Kravchenko and B.A. Hesin, Funct. Anal. Priloz. 23 (1989) 78. [241Yu.l. Manin, J. Soy. Mat. 11 (1979) 1. [25] M. Adler, Invent. Math. 50 (1979) 219. [26] D.A. Leites and B.L. Feigin, in: Group-theoretical methods in physics, Vol. 1 (1983) p. 269. [27] A.O. Radul, Seminar on supermanifolds, Vol. 21, ed. D. Leites (University of Stockholm, 1986) p. 40.
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[28] Yu.i. Manin and A.O. Radul, Commun. Math. Phys. 98 (1985) 65. [29] M. Wodzicki, Lecture Notes in Mathematics, Vol. 1289 (Springer, Berlin, 1985) p. 320. [30] R.T. Seeley, Proc. Symp. on Pure mathematics, Vol. 10 (American Mathematical Society, Providence, RI, 1987) p. 288. [31] M.A. Shubin, Pseudodifferential operators and spectral theory (Nauka, Moscow, 1978) [in Russian].
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