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DiffAT2, and the curvature of some infinite dimensional manifolds
Volume 215, number 4
PHYSlCS LETTERS B
29 December 1988
DiffAT2, A N D T H E C U R V A T U R E O F S O M E I N F I N I T E D I M E N S I O N A L M A N I F O L D S Jens H O P P E
Institut,Eir Theoretische Physik. P.O. Box 6380, D-7500 Karlsruhe, Fed. Rep. Germany Received 20 September 1988
A simple formula is given for the curvature of certain infinite dimensional homogeneous K/ihler manifolds. In the case of Diff,T2/C, C being the centralizer of some fixed area-preserving reparametrisation of the torus, the Ricci tensor diverges but yields zero when employing (-function regularisation; the same holds for some diffeomorphism groups of higher dimensional tori. A superextension of these, and representations thereof, are given, and it is pointed out that, although, Diff+T2 formally contains also a U ( 1) Kac-Moody algebra, Virasoro subalgebras should not be relevant to the study of toroidal membranes. A note is added, commenting on trigonometric algebras.
Area-preserving d i f f e o m o r p h i s m s were first studied in detail by A r n o l d [ 1 ]. M u c h later they a p p e a r as a residual s y m m e t r y group in the theory o f relativistic surfaces [ 2 ]. Recent work in string theories [ 3 ] has m o t i v a t e d the idea o f studying the geometry o f these d i f f e o m o r p h i s m groups. Based on refs. [4,5,3 ], Garreis [ 6 ] showed quite explicitly how to calculate the curvature tensor o f infinite d i m e n s i o n a l h o m o geneous K~ihler manifolds that arise as coset spaces o f an infinite d i m e n s i o n a l Lie group G with respect to some subgroup H. F o r the group o f all (area-preserving) d i f f e o m o r p h i s m s o f the torus he chose H = T 2 and o b t a i n e d a divergent expression for the Ricci tensor, which d e p e n d e d on the central extension parameters. As I would like to point out in this letter, the situation is quite different when " d i v i d i n g o u t " a m a x i m a l abelian subgroup. The Ricci tensor is still divergent (due to doubly infinite, divergent s u m s ) , but can be regularized, using (-function techniques. The regulated Ricci tensor vanishes, and the same holds for some diffeomorphism groups o f higher d i m e n s i o n a l tori. I will first give a rather simple form u l a for the curvature tensor o f the m a n i f o l d s considered in refs. [4,6] in terms o f structure constants (and central extensions), which is quite practical, and also shows how several (separately diverging) terms do not cancel for infinite d i m e n s i o n a l groups, while they do for finite d i m e n s i o n a l ones. The formula is then a p p l i e d to the above cases, i.e. Diff+~T2 (the 706
group o f non-constant s m o o t h r e p a r a m e t r i s a t i o n s o f T 2 that have unit j a c o b i a n ) and its generalisations (a class G+ o f infinite d i m e n s i o n a l Lie groups that allow concrete calculations). I will m e n t i o n central- and super-extensions o f the complexified Lie algebras G~, and finally make some remarks concerning the question o f Virasoro subalgebras inside Diff~ T 2. Let me begin by giving the formula for the Ricci tensor. It reads:
R
r.s
-R - -
rsn
" -- -c / r ;a
t"( _ c"fTrf~l
rnasn
CII
+ J- rr J .-,,l-n - - Y r ] r ¢ Yc,,h i
c,
_,_r,. c .
"Y
rsJ an
•
(1)
The notation, and the requirements for ( 1 ) to be valid, are as follows [4,6]: f l . denote structure constants o f G_, c. a central extension. The generators T++ o f the complexified Lie algebra G c fall into three categories (symbolically d e n o t e d by a = r > 0, a = r < 0, a = a ) , two o f which are the h e r m i t e a n conjugate o f each other ( T+~ = 7",), denoted by G ±, while the third ( H e ) is closed under conjugation. The c o m m u t a t i o n relations
I T , , T/~] = f DnT~, + c , aff~
(2)
are supposed to be consistent with h e r m i t e a n conjugation, so that f~a=_(fr),,
c,=-c*.
(3)
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
Volume 215, number 4
PHYSICS LETTERS B
One further assumes that the commutator of H c With G + does not contain any elements of H c, and the union of H e with G + to be a subalgebra of G c. Finally, c~ should be zero. [ For a general central extension satisfying c ~ = c,.~= c , = 0, and the remaining part having an inverse ,tc,~.c~'= ~'~,.,, the formula for the curvature tensor reads: R,.w n .
n
= t ,c E ~ ~ T n ~ ~Cl iI,.]rtTt~lE t" ijsm__(froCmi_
c
ul
)fsln
{ ,'n I itTn n +),'.Jr,,,-f ,',(f hTCmrC ) + f a/.sf.....
(4)
All Latin indices (apart from "a", which labels the subgroup H ) run over the "positive" sector, all barred ones over the "negative" sector. The reason for writing R,.~,," in ( 1 ) was to indicate that the sum over l must be performed before summing over n (also note that I and/-are not considered as independent summation variables). The order of summation is crucial, as, otherwise, the first and second terms in ( 1 ) would cancel identically (as they still do, of course, when both expressions are separately finite). For the Virasoro algebra ( T~ ~ Lo), the first, second, and last terms are (separately) divergent, but taken together (i.e. summing over n after combining the three terms) they give the well known finite result [5,3,4,6] ( - ~r263+ ~r)~,:," (r, s ~ ) . On the other hand, for finite dimensional Lie algebras the first and second terms always cancel - and in most cases (e.g. if H is a Cartan subalgebra of G, so that I and n may label positive roots, and consequently f ) ' , = 0) the third and fourth terms are zero as well (in this special case of finite dimensional Lie algebras, (1) should be compared with the result of ref. [7 ] ). Finally note, that ( 1 ) gives the Ricci tensor of M = G / H at the origin of M (the unit element of G). Rather than deriving (1) - which can easily be done, using, e.g., ref. [6 ] - I want to calculate it for a homogenous Kiihler manifold of possible physical relevance, namely DiffjT 2/C, where C is a maximal abelian subgroup. To do so, let us look at the corresponding Lie algebra Diff,~T 2. As for any area-preserving diffeomorphism group of a two-dimensional manifold [ 2 ], Diffj T 2 can be identified with the real functions (modulo a constant), on the toms, with the Lie bracket Of 0g [f(~o), g(~o) ] = O~0, 0{02
Of 0g 0~2 0{01 '
(5)
29 December 1988
which in the basis Ym= :--i exp(im(o) reads (m~77 2
\{0}): [Y,., Y.]=(mxn)Y,.+.,
(6)
m ~ 0 stands for (m~, m2), ( m X n ) for (m~nz-n~m2), Y~ = - Y , . , and ~o= (~0,, ~02)~ [0, 2n] 2. At this point I want to shortly digress and introduce a class GA of algebras generalizing (6) [while the discussion of the Ricci tensor will continue with eq. (12) ]: Let Y,,, m~O m~Z a (d>_-2), be a basis of the complex Lie algebra GA, with commutation relations [Y,., 11.] =A,jm, njY,,+..
(7)
One easily checks that (7) defines a Lie algebra, for any antisymmetric d x d matrix A. This algebra comes, of course, from Fourier decomposing functions on a d-dimensional torus T d, whose Lie bracket is defined as
If, g] =A,j OfOjg ,
(8)
and identifying functions that differ only by a constant, but I prefer to take (7) as the starting point. The Lie algebras GA defined by (7) have the nice property that despite the fact that they are infinite dimensional (and not of the Kac-Moody type) they allow concrete calculations, due to the 6-function in the structure constants. But let us first note, that ifA has an eigenvector ae77da ~ O, with eigenvalue zero, all generators Y~a, 2~Z drop out completely (meaning that they also do not appear on the RHS of (7), for if m+n=2a, mAn=O), and one can therefore consider the algebras spanned by all Y,.'s with Am ~ O. However, when looking for central extensions of these algebras, one finds the left-out part "reappearing":
[Y,,,, Y,,l=AominjY,,,+,,+ ~ c(().mO,,,+,,,,.T,, A(=0
(9) with c~,)'E=0,
[T,,T,,]=O=[T,,Y,.]
m~ ( z ~ ) ' = ~ a \ l ~ ,
~, ~, ~_d = {k~_dlAk=O} . I will not make much use of (9), but found it interisting to have an example where infinitely many central extensions are explicitly known. In the same spirit, 707
Volume 215, number 4
PHYSICS LETTERS B
although likely of more relevance to physics, I would like to mention super-extensions of GA (each component o f r ~ 0 m a y be integer or half-integer):
29 December 1988
extension c2"l (I am taking c~ = 0 ) remain. Performing the sum over l a n d / - o n e gets
Rr~rg,s= +6 .... "6rs'R(rl, r) ,
[ Y,,, Y.] =A#m, njY,,+.,
ao
-I-~
R(r,,r)= ~ {Xr, Xs} = ( 1 --~r+s.O) Y,+s+C6r+s,oYo ,
gl=
E 1 ill =
(rln-rnl) 2
--00
[ Y.,, Xr] =Aam, rjX.,,+~+c'm6~+ ,..oAo, [Yo, Y,,I=O=[Yo, X,] , [Ao, Y , , I = 0 = { A o , X,},
[Ao, Y o ] = 0
(10)
(note the "reappearance" of Yo as a central extension in the anti-commutator) and, for r~_d\ {0}, C= 0 and c = 0, the representations thereof (acting on the space of smooth functionsf(~o)+Og(~o), (oe [0, 2hi d, and 0 an anticommuting c-number):
R(r~, r) is obviously ill defined. One m a y define it, however, as the analytic continuation of
R,=
2
Irln--rnl I - '
×(~r--+r ) +O(r--n)~ f)
(14)
Y,, = - i exp (im~0)A~j m; 0/0~0j,
X,,,=OY,, +exp(im(o)O/O0
( 11 )
[this is just the representation in terms of vector fields f.0; that come from a single function: f=Aj;Of, f=exp(im~o) ]. Part of eqs. ( 7 ) - ( 11 ) has probably been studied by the mathematicians, in much more depth, but some of it [e.g. ( 1 0 ) ] m a y b e new. Let me now return to (6), noting that (DiffAT2)C=G, o,) ' and carry on with the calcula--~-10 tion of the Ricci tensor (at the origin) for DiffAT 2 divided by a maximal commuting subalgebra C (which is taken to be the antihermitean part of the space spanned by all Ym,,o m~ # 0 ) . The hermitean conjugation is defined by
Y*,,=-Y ........... .
(12)
(Note that the complex Lie algebra G¢_o~), with commutation relations as in (6), allows several other hermitean conjugations, like e.g. Y~=+Ym,.-m~; these will give rise to Lie groups which are not DiffAT 2. ) Take G + as spanned by iY,,,, .... (m from now on always > 0 ) , G - by iY, ......... and H c by iY,,,,.o ml ¢ 0. Consequently, r, s, l, and n in formula ( 1 ) are twodimensional vector indices r,s ... (with integer components) whose second c o m p o n e n t is positive, while a stands for (m~, 0). The third, fourth and (somewhat surprisingly) also the fifth terms are found to be zero, so that only the ones involving the central 708
from large enough positive e to e = - 2. This yields zero as can easily be seen by calculating the sum S,(a)=
la+nll-"
~
(a=-r,n/r).
(15)
nl = --~ --hi
~a
For a an integer, one gets 2~(~) (by shifting the summation variable), which is zero at ~ = - 2. In the other case ( a = f f l + q , 0 < q < 1, ffjsT/) one gets S,(a)=
~
(k+q)-'+ Z [ k + ( 1 - q ) ] - '
k>~0
=[1+(-)']
k>~0
lB,(q)
n
(16)
(if 1 - ~ = n is a natural number), where B.(q) denotes the Bernoulli polynomial. Hence S_2(a) = 0 for all a, which implies the vanishing of R, for e = - 2. For the algebras G A ( d > 2 ) one can argue in a very similar way. Assuming Y t. . . . . . . - - Y . . . . - m ,
the Ricci tensor may be regulated to be the analytic continuation of
PHYSICS LETTERSB
Volume 215, number 4
+6 .... "firs ~
~d
n>O
×
(r2
nl
[£,,,,£,,l=(m-n)£m+n
Irl"xn-rx'nll-~ --I
nl 'xv L (n/r)rl
+ [bm(1 + 3 a 2) -3f12bm316,,,+n,o,
'x
:)
-n-+r)+O(r-n) n___r ,
(18)
to e = - 2. Performing the sum over any of the components of nl yields 0, in analogy with ( 15 ). Let met now turn to a different question related to DifftT 2. In ref. [ 8 ] it was claimed that realisations of Virasoro algebras with non-trivial central extension exist as subalgebras of Diff~ T 2, and that this should be relevant to membrane theories. I do not believe this to be true. First of all, one still has to find a representation of Diff~ T 2 with non-trivial central extension (this is not done in ref. [8] ). Secondly, one has to make sure that for a given representation the formal processes used to define the Virasoro subalgebra do not depart from the representation (as is the case for the representation considered in ref. [8]; this question has apparently been overlooked in ref. [ 9 ] as well). On the other hand, if one ignored these two problems and argued purely formally, one would not see any clear distinction between the trivial and nontrivial central extensions for the Virasoro subalgebra, which can be seen as follows: Define
L,E,,= ~ ~1 Yk..... T,,, =L,,,o = ~ ~1 Yk.,,,. k~O k evcn
(19)
k odd
Then, using [1I=, Y.] = (m×n)Y=+.+c'm6,,+..o,
[L,E,,, L E] = (m--n)LE,+,, +bm6,,,+~,o,
1 ") k = _ arc-c2 '
[T,,,,LEI=mTm+,,
b = _ ~2n2c2,
£,,, =L~, + (a+flm)T,,,,
(21)
but for k ¢ 0 (i.e. c2~0) there does not exist any unitary highest weight representation of (20); so one should not expect (21) to be physically relevant (note, that the redefinition (21) induces also a nontrivial central extension in the commutator of £m with Tin, proportional to m 2; in contrast with Kac-Moody algebras coming from a non-abelian group, such a term is consistent with the Jacobi identities in the U ( 1 ) case, and will only be restricted when considering representations, or additional structures). First of all, I would like to thank all members of the theoretical physics group in Karlsruhe, and in particular J. Wess, I. McArthur, R. Dick, M. Frank, R. Garreis, R. Kaiser, C. Ramirez, P. Schaller and G. Schwarz for the pleasant atmosphere, many discussions and help. Formula (1) grew out of refs. [4,6] and last year's collaboration with R. Garreis and J. Wess (as well as more recent discussions with R. Kaiser and C. Ramirez). I would like to thank M. Baake, M. Bordemann, Th. Filk; H. Leptin, H. Nicolai, V. Rittenberg, M. Scheunert, N. Warner and D. Zagier for helpful discussions, and the Leibniz foundation for financial support. Finally, I am grateful to I.J. Kim and the people of Nongam for their kind help during the writing of the manuscript.
(9')
and ignoring all questions concerning the infinite sums, one finds
IT,,,, T,,]=km&,+,,.o,
29 December 1988
(20)
i.e. the commutation relations for the semidirect product of a U ( 1 ) Kac-Moody algebra and a Virasoro algebra, with trivial (non-trivial) central extension in the L,~,, ( T,,, ) part (c2 4=0 ). One can now always induce a non-trivial central extension for the Virasoro part by adding to LE,: a piece proportional to mT,,,
Note added. While this paper was in print, an interesting preprint [ 10 ] appeared, that considers algebras with structure constants that are, more or less, sin and cos of the ones appearing in (7) and (10). These algebras should be used, e.g., to regulate toroidal membranes [replacing Dif£~T 2 by S U ( N ) ] . In any case, one may understand the existence of such trigonometric algebras by noting that they simply come from projective representations of abelian groups: Let G be any (discrete) finite, or infinite dimensional, abelian group i.e. D(g~) D ( g 2 ) = exp[ifl(g,, g2)]D(gl, g2). Using fl(g,, g2) = -fl(g2, g,), one finds that Y g = - (ia/2)D(g) and Xg= bD(g) can be thought of as elements of a superalgebra, with structure constants sin(fl(g, g' ) ), appearing in the two commutators, and cos(fl(g, g' ) ), 709
Volume 215, number 4
PHYSICS LETTERS B
a p p e a r i n g in the a n t i c o m m u t a t o r XgXg,'~Xg,Xg. F o r G = Z × Z, a n d fl(In, n) = k(m~ n2 - m z m l ), the resulting superalgebra is the o n e c o n s i d e r e d by F a i r l i e et al. A n actual r e p r e s e n t a t i o n m a k i n g use o f the cocycle ( m ~ n 2 - m 2 n ~ ), a n d t h u s a r e p r e s e n t a t i o n o f the superalgebra, is g i v e n by D ( tn ) = exp ( m ~0x + m 2 i x ) . F o r 7#, a n d f l = ½m~Aijnj, o n e can t a k e e x p ( m i B ~ ) , i f
[ Bi, Bj] =i'L4 o.
References [ 1 ] V. Arnold, Sur la g6om6trie diff6rentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfait. Ann. Institut Fourier, XVI (1966) 319. [2] J. Coldstone, unpublished; J. Hoppe, Quantum theory of a relativistic surface, workshop on Constraints theory and relativistic dynamics (Florence, 1986), eds. G. Longhi and L. Lusanna (World Scientific, Singapore 1987) p. 267.
710
29 December 1988
[3] M.J. Bowick and S.G. Rajeev, Phys. Rev. Lett. 58 (1987) 535. [4] J. Wess, Nonlinear realisations, Kfihler manifolds and the Virasoro manifold, Lectures 7th Scheveningen Conf. on the Mathematical structure of field theory (August 1987 ); B. Zumino, preprint LBL-23056 (1987). [5] D.S. Freed, The geometry of loop groups, in: Infinite dimensional groups with applications, ed. V. Kac (Springer, Berlin, 1985 ). [6] R. Garreis, Unendlich dimensionale Cosetmannigfaltigkeiten und ihre KriJmmungstensoren, Doktorarbeit Karlsruhe Universit~it (January 1988). [ 7 ] M. Bordemann, M. Forger and H. Rtimer, Commun. Math. Phys. 102 (1988) 605. [8] I. Antoniades, P. Ditsas, E.G. Floratos and J. lliopoulos, New realisations of the Virasoro algebra as membrane symmetries, preprints CERN-TH. 4970/88, CRETE-TH88/2. [9] E.G. Floratos and J. lliopoulos, Phys. Lett. B 201 (1988) 237. [10] D.B. Fairlie, P. Fletcher and C.K. Zahos, trigonometric structure constants of new infinite dimensional Lie algebras, preprint ANL HEP PR 88-56/DTP 88-13 (October 1988).
PHYSlCS LETTERS B
29 December 1988
DiffAT2, A N D T H E C U R V A T U R E O F S O M E I N F I N I T E D I M E N S I O N A L M A N I F O L D S Jens H O P P E
Institut,Eir Theoretische Physik. P.O. Box 6380, D-7500 Karlsruhe, Fed. Rep. Germany Received 20 September 1988
A simple formula is given for the curvature of certain infinite dimensional homogeneous K/ihler manifolds. In the case of Diff,T2/C, C being the centralizer of some fixed area-preserving reparametrisation of the torus, the Ricci tensor diverges but yields zero when employing (-function regularisation; the same holds for some diffeomorphism groups of higher dimensional tori. A superextension of these, and representations thereof, are given, and it is pointed out that, although, Diff+T2 formally contains also a U ( 1) Kac-Moody algebra, Virasoro subalgebras should not be relevant to the study of toroidal membranes. A note is added, commenting on trigonometric algebras.
Area-preserving d i f f e o m o r p h i s m s were first studied in detail by A r n o l d [ 1 ]. M u c h later they a p p e a r as a residual s y m m e t r y group in the theory o f relativistic surfaces [ 2 ]. Recent work in string theories [ 3 ] has m o t i v a t e d the idea o f studying the geometry o f these d i f f e o m o r p h i s m groups. Based on refs. [4,5,3 ], Garreis [ 6 ] showed quite explicitly how to calculate the curvature tensor o f infinite d i m e n s i o n a l h o m o geneous K~ihler manifolds that arise as coset spaces o f an infinite d i m e n s i o n a l Lie group G with respect to some subgroup H. F o r the group o f all (area-preserving) d i f f e o m o r p h i s m s o f the torus he chose H = T 2 and o b t a i n e d a divergent expression for the Ricci tensor, which d e p e n d e d on the central extension parameters. As I would like to point out in this letter, the situation is quite different when " d i v i d i n g o u t " a m a x i m a l abelian subgroup. The Ricci tensor is still divergent (due to doubly infinite, divergent s u m s ) , but can be regularized, using (-function techniques. The regulated Ricci tensor vanishes, and the same holds for some diffeomorphism groups o f higher d i m e n s i o n a l tori. I will first give a rather simple form u l a for the curvature tensor o f the m a n i f o l d s considered in refs. [4,6] in terms o f structure constants (and central extensions), which is quite practical, and also shows how several (separately diverging) terms do not cancel for infinite d i m e n s i o n a l groups, while they do for finite d i m e n s i o n a l ones. The formula is then a p p l i e d to the above cases, i.e. Diff+~T2 (the 706
group o f non-constant s m o o t h r e p a r a m e t r i s a t i o n s o f T 2 that have unit j a c o b i a n ) and its generalisations (a class G+ o f infinite d i m e n s i o n a l Lie groups that allow concrete calculations). I will m e n t i o n central- and super-extensions o f the complexified Lie algebras G~, and finally make some remarks concerning the question o f Virasoro subalgebras inside Diff~ T 2. Let me begin by giving the formula for the Ricci tensor. It reads:
R
r.s
-R - -
rsn
" -- -c / r ;a
t"( _ c"fTrf~l
rnasn
CII
+ J- rr J .-,,l-n - - Y r ] r ¢ Yc,,h i
c,
_,_r,. c .
"Y
rsJ an
•
(1)
The notation, and the requirements for ( 1 ) to be valid, are as follows [4,6]: f l . denote structure constants o f G_, c. a central extension. The generators T++ o f the complexified Lie algebra G c fall into three categories (symbolically d e n o t e d by a = r > 0, a = r < 0, a = a ) , two o f which are the h e r m i t e a n conjugate o f each other ( T+~ = 7",), denoted by G ±, while the third ( H e ) is closed under conjugation. The c o m m u t a t i o n relations
I T , , T/~] = f DnT~, + c , aff~
(2)
are supposed to be consistent with h e r m i t e a n conjugation, so that f~a=_(fr),,
c,=-c*.
(3)
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
Volume 215, number 4
PHYSICS LETTERS B
One further assumes that the commutator of H c With G + does not contain any elements of H c, and the union of H e with G + to be a subalgebra of G c. Finally, c~ should be zero. [ For a general central extension satisfying c ~ = c,.~= c , = 0, and the remaining part having an inverse ,tc,~.c~'= ~'~,.,, the formula for the curvature tensor reads: R,.w n .
n
= t ,c E ~ ~ T n ~ ~Cl iI,.]rtTt~lE t" ijsm__(froCmi_
c
ul
)fsln
{ ,'n I itTn n +),'.Jr,,,-f ,',(f hTCmrC ) + f a/.sf.....
(4)
All Latin indices (apart from "a", which labels the subgroup H ) run over the "positive" sector, all barred ones over the "negative" sector. The reason for writing R,.~,," in ( 1 ) was to indicate that the sum over l must be performed before summing over n (also note that I and/-are not considered as independent summation variables). The order of summation is crucial, as, otherwise, the first and second terms in ( 1 ) would cancel identically (as they still do, of course, when both expressions are separately finite). For the Virasoro algebra ( T~ ~ Lo), the first, second, and last terms are (separately) divergent, but taken together (i.e. summing over n after combining the three terms) they give the well known finite result [5,3,4,6] ( - ~r263+ ~r)~,:," (r, s ~ ) . On the other hand, for finite dimensional Lie algebras the first and second terms always cancel - and in most cases (e.g. if H is a Cartan subalgebra of G, so that I and n may label positive roots, and consequently f ) ' , = 0) the third and fourth terms are zero as well (in this special case of finite dimensional Lie algebras, (1) should be compared with the result of ref. [7 ] ). Finally note, that ( 1 ) gives the Ricci tensor of M = G / H at the origin of M (the unit element of G). Rather than deriving (1) - which can easily be done, using, e.g., ref. [6 ] - I want to calculate it for a homogenous Kiihler manifold of possible physical relevance, namely DiffjT 2/C, where C is a maximal abelian subgroup. To do so, let us look at the corresponding Lie algebra Diff,~T 2. As for any area-preserving diffeomorphism group of a two-dimensional manifold [ 2 ], Diffj T 2 can be identified with the real functions (modulo a constant), on the toms, with the Lie bracket Of 0g [f(~o), g(~o) ] = O~0, 0{02
Of 0g 0~2 0{01 '
(5)
29 December 1988
which in the basis Ym= :--i exp(im(o) reads (m~77 2
\{0}): [Y,., Y.]=(mxn)Y,.+.,
(6)
m ~ 0 stands for (m~, m2), ( m X n ) for (m~nz-n~m2), Y~ = - Y , . , and ~o= (~0,, ~02)~ [0, 2n] 2. At this point I want to shortly digress and introduce a class GA of algebras generalizing (6) [while the discussion of the Ricci tensor will continue with eq. (12) ]: Let Y,,, m~O m~Z a (d>_-2), be a basis of the complex Lie algebra GA, with commutation relations [Y,., 11.] =A,jm, njY,,+..
(7)
One easily checks that (7) defines a Lie algebra, for any antisymmetric d x d matrix A. This algebra comes, of course, from Fourier decomposing functions on a d-dimensional torus T d, whose Lie bracket is defined as
If, g] =A,j OfOjg ,
(8)
and identifying functions that differ only by a constant, but I prefer to take (7) as the starting point. The Lie algebras GA defined by (7) have the nice property that despite the fact that they are infinite dimensional (and not of the Kac-Moody type) they allow concrete calculations, due to the 6-function in the structure constants. But let us first note, that ifA has an eigenvector ae77da ~ O, with eigenvalue zero, all generators Y~a, 2~Z drop out completely (meaning that they also do not appear on the RHS of (7), for if m+n=2a, mAn=O), and one can therefore consider the algebras spanned by all Y,.'s with Am ~ O. However, when looking for central extensions of these algebras, one finds the left-out part "reappearing":
[Y,,,, Y,,l=AominjY,,,+,,+ ~ c(().mO,,,+,,,,.T,, A(=0
(9) with c~,)'E=0,
[T,,T,,]=O=[T,,Y,.]
m~ ( z ~ ) ' = ~ a \ l ~ ,
~, ~, ~_d = {k~_dlAk=O} . I will not make much use of (9), but found it interisting to have an example where infinitely many central extensions are explicitly known. In the same spirit, 707
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although likely of more relevance to physics, I would like to mention super-extensions of GA (each component o f r ~ 0 m a y be integer or half-integer):
29 December 1988
extension c2"l (I am taking c~ = 0 ) remain. Performing the sum over l a n d / - o n e gets
Rr~rg,s= +6 .... "6rs'R(rl, r) ,
[ Y,,, Y.] =A#m, njY,,+.,
ao
-I-~
R(r,,r)= ~ {Xr, Xs} = ( 1 --~r+s.O) Y,+s+C6r+s,oYo ,
gl=
E 1 ill =
(rln-rnl) 2
--00
[ Y.,, Xr] =Aam, rjX.,,+~+c'm6~+ ,..oAo, [Yo, Y,,I=O=[Yo, X,] , [Ao, Y , , I = 0 = { A o , X,},
[Ao, Y o ] = 0
(10)
(note the "reappearance" of Yo as a central extension in the anti-commutator) and, for r~_d\ {0}, C= 0 and c = 0, the representations thereof (acting on the space of smooth functionsf(~o)+Og(~o), (oe [0, 2hi d, and 0 an anticommuting c-number):
R(r~, r) is obviously ill defined. One m a y define it, however, as the analytic continuation of
R,=
2
Irln--rnl I - '
×(~r--+r ) +O(r--n)~ f)
(14)
Y,, = - i exp (im~0)A~j m; 0/0~0j,
X,,,=OY,, +exp(im(o)O/O0
( 11 )
[this is just the representation in terms of vector fields f.0; that come from a single function: f=Aj;Of, f=exp(im~o) ]. Part of eqs. ( 7 ) - ( 11 ) has probably been studied by the mathematicians, in much more depth, but some of it [e.g. ( 1 0 ) ] m a y b e new. Let me now return to (6), noting that (DiffAT2)C=G, o,) ' and carry on with the calcula--~-10 tion of the Ricci tensor (at the origin) for DiffAT 2 divided by a maximal commuting subalgebra C (which is taken to be the antihermitean part of the space spanned by all Ym,,o m~ # 0 ) . The hermitean conjugation is defined by
Y*,,=-Y ........... .
(12)
(Note that the complex Lie algebra G¢_o~), with commutation relations as in (6), allows several other hermitean conjugations, like e.g. Y~=+Ym,.-m~; these will give rise to Lie groups which are not DiffAT 2. ) Take G + as spanned by iY,,,, .... (m from now on always > 0 ) , G - by iY, ......... and H c by iY,,,,.o ml ¢ 0. Consequently, r, s, l, and n in formula ( 1 ) are twodimensional vector indices r,s ... (with integer components) whose second c o m p o n e n t is positive, while a stands for (m~, 0). The third, fourth and (somewhat surprisingly) also the fifth terms are found to be zero, so that only the ones involving the central 708
from large enough positive e to e = - 2. This yields zero as can easily be seen by calculating the sum S,(a)=
la+nll-"
~
(a=-r,n/r).
(15)
nl = --~ --hi
~a
For a an integer, one gets 2~(~) (by shifting the summation variable), which is zero at ~ = - 2. In the other case ( a = f f l + q , 0 < q < 1, ffjsT/) one gets S,(a)=
~
(k+q)-'+ Z [ k + ( 1 - q ) ] - '
k>~0
=[1+(-)']
k>~0
lB,(q)
n
(16)
(if 1 - ~ = n is a natural number), where B.(q) denotes the Bernoulli polynomial. Hence S_2(a) = 0 for all a, which implies the vanishing of R, for e = - 2. For the algebras G A ( d > 2 ) one can argue in a very similar way. Assuming Y t. . . . . . . - - Y . . . . - m ,
the Ricci tensor may be regulated to be the analytic continuation of
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+6 .... "firs ~
~d
n>O
×
(r2
nl
[£,,,,£,,l=(m-n)£m+n
Irl"xn-rx'nll-~ --I
nl 'xv L (n/r)rl
+ [bm(1 + 3 a 2) -3f12bm316,,,+n,o,
'x
:)
-n-+r)+O(r-n) n___r ,
(18)
to e = - 2. Performing the sum over any of the components of nl yields 0, in analogy with ( 15 ). Let met now turn to a different question related to DifftT 2. In ref. [ 8 ] it was claimed that realisations of Virasoro algebras with non-trivial central extension exist as subalgebras of Diff~ T 2, and that this should be relevant to membrane theories. I do not believe this to be true. First of all, one still has to find a representation of Diff~ T 2 with non-trivial central extension (this is not done in ref. [8] ). Secondly, one has to make sure that for a given representation the formal processes used to define the Virasoro subalgebra do not depart from the representation (as is the case for the representation considered in ref. [8]; this question has apparently been overlooked in ref. [ 9 ] as well). On the other hand, if one ignored these two problems and argued purely formally, one would not see any clear distinction between the trivial and nontrivial central extensions for the Virasoro subalgebra, which can be seen as follows: Define
L,E,,= ~ ~1 Yk..... T,,, =L,,,o = ~ ~1 Yk.,,,. k~O k evcn
(19)
k odd
Then, using [1I=, Y.] = (m×n)Y=+.+c'm6,,+..o,
[L,E,,, L E] = (m--n)LE,+,, +bm6,,,+~,o,
1 ") k = _ arc-c2 '
[T,,,,LEI=mTm+,,
b = _ ~2n2c2,
£,,, =L~, + (a+flm)T,,,,
(21)
but for k ¢ 0 (i.e. c2~0) there does not exist any unitary highest weight representation of (20); so one should not expect (21) to be physically relevant (note, that the redefinition (21) induces also a nontrivial central extension in the commutator of £m with Tin, proportional to m 2; in contrast with Kac-Moody algebras coming from a non-abelian group, such a term is consistent with the Jacobi identities in the U ( 1 ) case, and will only be restricted when considering representations, or additional structures). First of all, I would like to thank all members of the theoretical physics group in Karlsruhe, and in particular J. Wess, I. McArthur, R. Dick, M. Frank, R. Garreis, R. Kaiser, C. Ramirez, P. Schaller and G. Schwarz for the pleasant atmosphere, many discussions and help. Formula (1) grew out of refs. [4,6] and last year's collaboration with R. Garreis and J. Wess (as well as more recent discussions with R. Kaiser and C. Ramirez). I would like to thank M. Baake, M. Bordemann, Th. Filk; H. Leptin, H. Nicolai, V. Rittenberg, M. Scheunert, N. Warner and D. Zagier for helpful discussions, and the Leibniz foundation for financial support. Finally, I am grateful to I.J. Kim and the people of Nongam for their kind help during the writing of the manuscript.
(9')
and ignoring all questions concerning the infinite sums, one finds
IT,,,, T,,]=km&,+,,.o,
29 December 1988
(20)
i.e. the commutation relations for the semidirect product of a U ( 1 ) Kac-Moody algebra and a Virasoro algebra, with trivial (non-trivial) central extension in the L,~,, ( T,,, ) part (c2 4=0 ). One can now always induce a non-trivial central extension for the Virasoro part by adding to LE,: a piece proportional to mT,,,
Note added. While this paper was in print, an interesting preprint [ 10 ] appeared, that considers algebras with structure constants that are, more or less, sin and cos of the ones appearing in (7) and (10). These algebras should be used, e.g., to regulate toroidal membranes [replacing Dif£~T 2 by S U ( N ) ] . In any case, one may understand the existence of such trigonometric algebras by noting that they simply come from projective representations of abelian groups: Let G be any (discrete) finite, or infinite dimensional, abelian group i.e. D(g~) D ( g 2 ) = exp[ifl(g,, g2)]D(gl, g2). Using fl(g,, g2) = -fl(g2, g,), one finds that Y g = - (ia/2)D(g) and Xg= bD(g) can be thought of as elements of a superalgebra, with structure constants sin(fl(g, g' ) ), appearing in the two commutators, and cos(fl(g, g' ) ), 709
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a p p e a r i n g in the a n t i c o m m u t a t o r XgXg,'~Xg,Xg. F o r G = Z × Z, a n d fl(In, n) = k(m~ n2 - m z m l ), the resulting superalgebra is the o n e c o n s i d e r e d by F a i r l i e et al. A n actual r e p r e s e n t a t i o n m a k i n g use o f the cocycle ( m ~ n 2 - m 2 n ~ ), a n d t h u s a r e p r e s e n t a t i o n o f the superalgebra, is g i v e n by D ( tn ) = exp ( m ~0x + m 2 i x ) . F o r 7#, a n d f l = ½m~Aijnj, o n e can t a k e e x p ( m i B ~ ) , i f
[ Bi, Bj] =i'L4 o.
References [ 1 ] V. Arnold, Sur la g6om6trie diff6rentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfait. Ann. Institut Fourier, XVI (1966) 319. [2] J. Coldstone, unpublished; J. Hoppe, Quantum theory of a relativistic surface, workshop on Constraints theory and relativistic dynamics (Florence, 1986), eds. G. Longhi and L. Lusanna (World Scientific, Singapore 1987) p. 267.
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29 December 1988
[3] M.J. Bowick and S.G. Rajeev, Phys. Rev. Lett. 58 (1987) 535. [4] J. Wess, Nonlinear realisations, Kfihler manifolds and the Virasoro manifold, Lectures 7th Scheveningen Conf. on the Mathematical structure of field theory (August 1987 ); B. Zumino, preprint LBL-23056 (1987). [5] D.S. Freed, The geometry of loop groups, in: Infinite dimensional groups with applications, ed. V. Kac (Springer, Berlin, 1985 ). [6] R. Garreis, Unendlich dimensionale Cosetmannigfaltigkeiten und ihre KriJmmungstensoren, Doktorarbeit Karlsruhe Universit~it (January 1988). [ 7 ] M. Bordemann, M. Forger and H. Rtimer, Commun. Math. Phys. 102 (1988) 605. [8] I. Antoniades, P. Ditsas, E.G. Floratos and J. lliopoulos, New realisations of the Virasoro algebra as membrane symmetries, preprints CERN-TH. 4970/88, CRETE-TH88/2. [9] E.G. Floratos and J. lliopoulos, Phys. Lett. B 201 (1988) 237. [10] D.B. Fairlie, P. Fletcher and C.K. Zahos, trigonometric structure constants of new infinite dimensional Lie algebras, preprint ANL HEP PR 88-56/DTP 88-13 (October 1988).