BRS cohomology and Casimir operators

Volume 188, number 2

PHYSICS LETTERS B

9 April 1987

BRS C O H O M O L O G Y A N D CASIMIR OPERATORS

Herbert NEUBERGER Department of Physics, University of California San Diego, La Jolla, CA 92093, USA and Department of Physics and Astronomy, Rutgers University,Piscataway, NJ 08554, USA Received 6 October 1986

Casimir operators play a central role in the study of cohomology problems for semisimple Lie algebras. An attempt is made to generalize this to strings. This generalization may be useful for studying small oscillations around nontrivial backgrounds.

Symmetry groups play a major role in physics. Usually one deals with a linear space "U that carries a representation of a Lie algebra 5e. Let T,, a = 1, ..., N denote the generators of L#. Ghosts c '~, ot = 1, ..., N are introduced into the problem by enlarging the physical space U t o ~ = ~ ® f~ where f9 is the Grassmann algebra generated by the c a. £# is characterized by the structure constantsf~r: (1)

[ T a, Trl =f~rTa .

A GL(N) group acts on ~ by changing the basis; under this action f, T and c transform as (2,1 ), ( 1,0 ) and (0,1) tensors, respectively. The GL(N) invariant content of Jacobi's identity is

q2=0,

q=c'~z~,

z # = - f ~ r c r O I O c '~ .

(2)

The z , represent ~ on (~. In ~ the T~ are represented by linear operators I,~. This fact can be expressed in a GL(N) invariant way as follows: Q2=0,

Q=p+½q,

p=c"I,~.

(3)

Q operates linearly on W. Often only those sectors of U that are invariant under the symmetry are considered as physical. A similar restriction in "W can also be given. The restriction is explicitly GL(N) invariant: "Wph = k e r ( Q ) / I m ( Q ) .

(4)

~¢'ph is the cohomology space associated with the complex ( "iV, Q) usually denoted by H(W, Q). The 214

ghost number N~ N

Ng = ½ Z [c'~,O/Ocal , ot=l

(5)

is also a GL(N) scalar and induces a direct sum decomposition of ~/¢:phin subspaces of definite ghost number, also denoted by H s' ('IV, Q). In gauge theory applications the above structure is enlarged by the addition of Nakanishi-Lautrup fields [ 1]. In string theory applications the structure of H(~V, Q) itself suffices [2]. For definiteness we shall consider open strings; in this case L# is one single Virasoro algebra. Generalizations to other string models are possible. The action for a free string is Af= ( ~ , Qq~),

(6)

Under the inner product in (6) Q is selfadjoint and Ng is antiselfadjoint. Since [Ng, Q] =Q, Af[t0] 0=:-N~ = - 1 ~. Ar is left unchanged by ~--, • + Q ~v. The variation of Af gives the equation of motion Qq~ = 0. It is therefore reasonable to view the set of physical, distinguishable, classical motionsofthe free string as isomorphic to H - 1/z(~r, Q). It has been shown that indeed this procedure gives exactly all the states of the bosonic open string as known from older formulations [2,3]. When an interaction is added to (6) the equations of motion become nonlinear. The small fluctuations of around a nontrivial solution satisfy a linear equation Q8 • = 0 with 0 2= 0. Not much is known about the 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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structure of the linear space of small oscillations ,1. It is reasonable to assume that some, if not all, such O's can be written in a form similar to Q with the difference that now the explicit realization of the Virasoro algebra and the space ~ are no longer given. It is also reasonable to presuppose that all such realizations have some general properties in common. It is interesting to try to find out how much can be said about H ( ~ , Q) in this general setting. The purpose of the present note is to take a small step in this direction. Our motivation stems from the analogue problem for a semisimple Lie algebra [ 4 ]. The general feature ~ i s required to have in this case is that ~ b e decomposable in only finite dimensional representations of ~ . Semisimplicity implies that ¢" is fully reducible. The consequence of this for H ( ~ , Q) is far reaching: only the singlet subspace ~s ~ ~ can have nontrivial cohomology; H(~#r, Q) = H ( U s ® fg, Q) ~H(f~, q). The reason for this is simple: Semisimplicity implies the existence of the invertible Cartan metric g~p. One derives the following identity:

9 April 1987

holds. The proof is straightforward: By inspection one sees that any generalized Casimir operator can be written as (10)

C=I,~F ~ ,

with the (0,1 ) G L ( N ) tensor Ftransforming contravariantly under if: [ I~, F p ] = - f ~ F

(11 )

~.

Then the G L ( N ) scalar operator Ac (12)

A¢ =OF'UOc c'

does the job. The argument leading to (8) can be further generalized: Any time we have an identity like (9) with a diagonalizable B one does not loose any physics by restricting the analysis to ker(B). This can be generalized even further: the main property of ker (B) was that it was stable under the action of Q as a result of

(9): [Q, B] = 0 = , Q ker(B) = k e r ( B ) .

{Q, Iag'~P O/OcP} =I~,Ipg '~ = C .

The right-hand side of (7) is a Casimir operator; can be decomposed in eigenspaces of C which are stable under Q; C = 0 only on ~s. For any • which obeys C ~ = 2 ~ , 2 ~ 0 and Q~o = 0 we have • =Q[(1/2)I,~g~'PO/OcP]~.

(8)

This finishes the argument. Identity (7) can be extended to generalized Casimir operators u. {Q,A}=B.

(13)

(7)

(9)

If B is ghost independent and (9) is representation independent B is a generalized Casimir operator [ 5 ]; this is evident. Conversely, if B is a generalized Casimir operator there exists an operator A such that (9) *' This structure would be o f physical relevance, for example, if we have a nontrivial solution which admits a "soliton" interpretation; the space o f small fluctuations becomes then, upon semiclassical quantization, an analogue of the "mesonic" excitations. A different situation would correspond to some sort of condensation o f closed strings in the vacuum indicative of a nontrivial background metric; the space of small fluctuations would then provide a first approximation to the open string particle spectrum in the respective background. :z Note that 1 is not a generalized Casimir operator.

Whenever one finds a subspace "IV' c ~¢r Q,ar, ~v' one obtains a relation between the cohomology spaces of ~V', ~ and ~ / ~ ' , each one with a naturally induced coboundary operator [ 6 ]. The best way to phrase the consequences employs the concept of exact sequences and will not be given here because it is not relevant for the rest of this note. It suffices to say, that, not surprisingly, one obtains much more information about H ( "IV, Q) if there also exists a subspace ~¢r" = "WQ~tV" = #~" with ~ = ~ ¢ : ' ~ " . This happens when B is diagonalizable. However, when B is not diagonalizable we only have ker(B) as a stable subspace and the physical consequences are more difficult to obtain. When we go to the Virasoro case we shall restrict our attention to the case where B is diagonalizable. However, even if the structure of the bosonic space is not known, one can find, in the ghost sector operators B which are of importance but, nevertheless, are not diagonalizable. To see this we first define our notation. In ~ we have a realization of the Virasoro algebra by operators Ln, n e Z: [Ln, Lm] = ( n - m ) L , , + m +~c(n3-n)~°+,,,

.

(14) 215

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As usual we allow for a central extension. This is the first problem one encounters when one tries to generalize from the finite, semisimple case. As we shall see later, the second problem for generalization is that the structure constants of the Virasoro algebra have different propertie from those of a semisimple Lie algebra. We have ghosts c" and their conjugates O/Oc"=-~,, known as antighosts. They should not be confused with the gauge theory antighosts, although, in a 2D functional integral framework, they play the same role in the Faddeev-Popov formalism. For n > 0 c" and ~_n are creation operators. ~ has the structure of a Fock space. {c", e,, } = ~ 7 , .

(15)

In fq one has a representation analogous to the z~ of eq. (2)

ln= ~ f~j :(icJ: _~o ,

(16)

ld

where, from (14), we have

J~j=(n-j)6~+j.

(17)

The coboundary operator Q is given by

Q= ~ [c"L, + ½:c'( l, _rio):] .

(18)

9 April 1987

Then, after the action of Q, we have

Q I s ) = Z IOu)® I~tu) P

Vplu>tland 7u <~Z.

(22)

This immediately gives chained sets of Q-stable subspaces of ~V. Such sets are useful for the calculation of cohomologies [ 8 ]. From now on we shall restrict our attention to the case of diagonalizable B-operators. First let us list some general requirements we would like V to have as a reprezentation of the Virasoro algebra [ 9]. What these requirements really are is not known. "V has to be a space in which Lo can be diagonalized; this allows to have free propagation generated by Lo. For almost any I0 ) e ~ , [0 ) # 0 we would like L_k,L_k:...L_k, 10) to be linearly independent once the positive integers ki have been unambiguously ordered. This may not hold on some finite-dimensional subspace of U.We would also like the space of Verma modules to be dense in "U in the sense that identities involving only Ln which hold in any Verma module and satisfy some regularity constraints should hold on "U as a whole. The simplest example of an A, B pair of operators with a diagonalizable B is obtained by picking A = Co:

n

{Q, eo} =Lo + l o . The 1, in eq. (16) satisfy communication relations identical to the L, in (17) but with c = - 2 6 . To get nilpotency, Q2=0, we need c = 2 6 in (14). As an example of a nondiagonalizable B we mention {Q, c o } = ~ kckc -k .

(19)

k=l

ker({Q, cO}) is the space in which each gk acting on the vacuum is paired with a ck, k>~ 1; this subspace plays an important role in the Banks-Peskin construction [7]. Not all Q-stable subspaces which are important are found this way. For example if we write

Is)~,

Is>=l¢)®l~),

Igt ) = c pl ...cp"eq, ...(_q,, IO) , N

/~

l= ~pi,

7= ~ q i .

i=l

i=1

216

(20) (21)

(23)

Lo is diagonalizable by assumption and lo by construction. Nontrivial cohomology is therefore restricted to ker (Lo + lo). It is obvious that if more pairs Ai, B~ are given with all the Bi diagonalizable nontrivial cohomology is restricted to Oiker(Bi). If the B; also commute any linear combination of them is also diagonalizable and one can find an A*=Y~2~Ai such that B*={Q, A*} satisfies ker(B*) = O~ker(B3. Hence, in this case, for the purpose of cohomology computation, the set of pairs {Ai, Bi} can be replaced by the single pair A*, B*. We are now looking for a pair A, B with a diagonalizable operator B which commutes with Lo+lo. Such an operator would be given by an analogue to (7). Strictly speaking such an analogue cannot be found. In the semisimple case one could construct a tensor F as in (11 ) which was linear in the I,~ because of the existence of the Cartan metric g. F'~=g'~aI~s satisfies (11 ) because g obeys

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PHYSICSLETTERSB

f,~gr,~ =f~agyO, .

(24)

It is quite straightforward to see that with the structure constants of eq. (17) a nonvanishing solution of (24) cannot be found (even without requiring g to be symmetric - for our needs this would have been acceptable). This reflects the lack of semisimplicity of the Virasoro algebra. Even if a solution to (24) could be found, we still would have to worry about operator orderings and the nonzero value of the central charge. Nevertheless, it is known that there exist generalized Casimir operators for the Virasoro algebra which would be well defined in spaces with the general properties we assumed to hold for ~ [10]. By inspection one sees that these Feigin-Fuks (FF) operators can be written, up to an additive constant, as Y~n :L,F": with some well defined, algebraic meaning of: :. It is plausible to assume that these F n can be chosen such that the analogue of (11 ) holds, namely

[Ln, F " ] = ( m - 2 n ) F

m-" .

(26)

When an explicit realization of ~ is given one may find there tensors F which satisfy (25 ) and (26) but in (26) the inner product of ~ d o e s play a role and the F ' s cannot be written as functions of the L's only. In any case (25) and (26) suggest to interpret F -k for k>_-1 as destruction operators. One is then led to guess a candidate for a good operator B ~3:

B=(Lo-1)F°+

~, ( F k L k + L _ k F - k ) .

(27)

k=l

The facts that c ~ 0 and that only specific orderings of operators make sense force us to explicitly check whether B commutes with all the L,. We find [L~, B] = ~ ( c - 2 6 ) ( n 3 - n ) F - " ,

ten down:

A=~(nF",

{Q,A}=B.

(29)

n

Since B commutes with Lo + lo any linear combination of A and Co is also useful. It was shown by Freeman and Olive [ 3 ] that, in the trivial background case, finding a special tensor F and (23) allowed to write down a linear combination (A - ~o in their case) which almost completely characterized the cohomology space. Eq. (25) also can be stated as demanding F t o have conformal dimension - 1. It is relatively easy to see how such an operator could be found in the oscillator formalism (see ref. [11] and references therein for the original papers). In the general case an algebraic construction, probably a la FF [ 10], would be needed. At this point it has to be pointed out that B in (27) will not be the most general FF operator. If we write

F°(L) =fo(Lo)+ ~

Z L-IflJ(Lo)Lj,

n=l I l l = l J l = n

(25)

Mso, without any reference to an inner product defined in U one has

L~ = L * _ ~ F ~ = ( F - n ) t .

9 April 1987

(28)

and indeed, for c = 26, this works. Note that the ghosts played no direct role in the appearance of 26 here. They do however when the analogue of (12) is writ~3Here we made a choicefor what one should consideras a 1 in the Virasoro context (see the previous fo0tnote).

III =I1 + / 2 + " - + I k ,

1 <~II <~I2...<~Ik,

LI=LhLI2...LIk,

LI=L~_I

,

(30)

and compare to the FF expression C=Ko(Lo) + ~,

E

L-tKIJ(Lo)Lj,

(31)

n=l I l l = l J l = n

we see that we need to have Ko(1)=0.

(32)

We now restrict our attention to the subspace ~ ' = ~ ' ® f9 where ~ ' contains only Verma modules. Clearly Q ( ~ ' ) c ~ ' . We assumed that V / ~ ' is finite dimensional; therefore the restriction to ~ ' is not severe. We can employ our two pairs of A, B operators also in ~F'. In ker(Lo+/o) Lo has only integer eigenvalues smaller or equal to 1 because of the explicit form of lo (16). Let us denote these eigenvalues by h. We can decompose ~ ' into a direct sum of idecomposable modules. A transverse state is defined as an eigenstate of Lo, [0 > e Y" which satisfies Ln [0 > = 0 Vn>~ 1. Any element of the defining basis of an idecomposable module is either transverse (highest weight vector) or, can be written as a linear combi217

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nation o f LI acting on the highest weight vector ( o r b o t h ) . Since the action o f LI increases h, only m o d ules generated from highest weight vectors with h ~< 1, h~ Z need to be considered. It is reasonable to assume that B only has to obey the m i n i m a l set o f constraints which result from (32) and the known nestings o f Verma modules [ 9,10 ]. This means that Ko(h) has to vanish also on all values o f h which correspond to the eigenvalue o f Lo on a transverse state generated from another highest weight vector with h = 1. F r o m this discussion a n d ref. [ 12 ] we conclude that, u n d e r our assumptions about B, only two types o f i d e c o m p o s a b l e modules contribute to nontrivial cohomology. They are conveniently described in the diagrams below [ 12,9]. The notation is as follows: A Verma module V , denotes all the states generated from a highest weight vector I 0 , ) with Lol0, ) = ~(25-n2)

10, ) .

(33)

D e n o t e "~(/'m----~]U"m, if "~/'m is included in f , , , . Then the two possibilities are ( a ) n ( r o o d ) 6 = 1, 5:

z

.....

~

/;174---

Ull-.,,P---V 5

~n

~1 .....

~

(34)

v19~v13-.*----v7

(b) n (mod) 6=5; %

~

. . . . .

Vn+2-,~--- . . . .

~

VlT..~---vll

-~--v 5

v 19 -.,t----- v 1 3 ~

v7

Note that for all the values o f n which appear in (34), (35) h is an integer. To really establish the above conclusions an explicit construction o f the F " would be necessary. Also, the properties assumed for B would have to be established. It m a y very well be that even in the general case more analysis, possibly finding m o r e pairs o f A, B operators, would finally yield a result completely analogous to the trivial background case. Such a result would be that only the subspace o f ~ a n n i h i l a t e d by all L,, n>_-1 and on w h o m Lo= 1 corresponds to physical states. This would have been o b t a i n e d already here were it not for the existence o f reducible, but not fully reducible, representations o f the 218

9 April 1987

Virasoro algebra. Again, this c o m p l i c a t i o n directly reflects the fact that the Virasoro algebra is not semisimple. The author is on a FASP leave from Rutgers University. This research was supported in part b y the D O E u n d e r the OJI program, grant nr. D E - F G 0 5 - 8 6 ER40265 a n d also under contract nr. DE-AT03-81ER40029. Discussions with K. Bardakci, Z. Qiu a n d P. Wills are gratefully acknowledged. I wish to t h a n k the particle theory group at U C S D for their hospitality.

References [ 1] C. Becchi, A. Rouet and R. Stora; Ann. Phys. (NY) 98 (1986) 287; T. Kugo and N. Ojima, Suppl. Progr. Theor. Phys. 66 (1979) 1; L. Baulieu, Phys. Rep. 129 (1985) 1; [2] M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443; J. Schwartz, Caltech preprint CALT-68-1304; T. Banks, M.E. Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, SLAC report SLAC-PUB-3853; A. Neveu, H. Nicolai and P. West, Phys. Lett. B 167 (1986) 307; E. Witten, Princeton University preprint; W. Siegel and B. Zwiebach, Nucl. Phys. B 263 (1986) 105. [3] M. Peskin and C. Thorn, SLAC report SLAC-PUB-3801; M.D. Freeman and D.I. Olive, preprint imperial/tp/8586/20. [4] C. Chevalley and S. Eilenberg, Trans. Am. Math. Soc. 63 (1948) 85. [ 5 ] G. Racah, Princeton Lecture Notes, CERN report CERN61-8 (1961); B. Gruber and L. O'Raifeartaigh, J. Math. Phys. 5 (1964) 1796; A.M. Perelomov and V.S. Popov, Soy. J. of Nucl. Phys. 3 (1966) 819; 5 (1967) 489. [6] W. Greub, S. Halperin and R. Vanstone, Connections, curvature and cohomology: Vol III, Cohomology of principal bundles and homogeneous spaces (Academic Press, New York, 1976); P. Griffiths and J. Harris, Principles of algebraic geometry p.438 (Wiley, New York, 1978). [7] T. Banks and M.E. Peskin, SLAC report SLAC-PUB-3740 (1985). [8] I.B. Frenkel, H. Garland and G.J. Zuckerman, Yale University preprint (1986). [9] D. Friedan, Nucl. Phys. B 271 (1986) 540. [ 10] B.L. Feigin and D.B. Fuks, Soy. Math. Dokl. 27 (1983) 465. [ 11 ] J. Scherk, Rev. Mod. Phys. 47 (1975) 123. [ 12 ] B.L. Feigin and D.B. Fuks, Funct. An. Appl. 17 (1983) 241; A. Rocha-Caridi, in: Vertex operators in mathematics and physics (Springer, Berlin 1984).