Topological Strings from WZW Models

Home , Superalgebra

CERN-TH.7539/94

hep-th/9412198

Top ological Strings from WZW Mo dels

K. Landsteiner, W. Lerche and A. Sevrin

CERN, Geneva, Switzerland

Abstract

We show that the BRST structure of the top ological string is enco ded in the

\small" N = 4 sup erconformal algebra, enabling us to obtain, in a non-trivial way,

the string theory from hamiltonian reduction of A(1j1). This leads to the imp ortant

conclusion that not only ordinary string theories, but top ological strings as well, can

b e obtained, or even de ned, by hamiltonian reduction from WZW mo dels. Using

two di erent gradations, we nd either the standard N = 2 minimal mo dels coupled

to top ological gravity,oranemb edding of the b osonic string into the top ological

string. We also comment brie y on the generalization to sup er Lie algebras A(njn).

CERN-TH.7539/94

Decemb er 1994

Permanent address: Theoretische Natuurkunde, Vrije Universiteit Brussel, B-1050 Brussels, Belgium.

It seems that the BRST structure of any string theory is enco ded in a (twisted) N =2

sup ersymmetric extension of its gauge algebra. More precisely, the BRST structure of

the b osonic string is characterized bya twisted N = 2 sup erconformal algebra [1], that of

the sup erstring bya twisted N = 3 sup erconformal algebra [2, 3], that of W strings by

atwisted N =2 W algebra [2, 4], etc. The idea is that one can add the BRST current

and the anti-ghost to the gauge algebra, which then b ecomes sup erconformally enlarged.

The BRST-charge itself is then one of the sup ercharges,

dz (cT + :::) ; (1) Q G =

BRST

while the anti-ghost b(z ) is the conjugate sup ercurrent, G (z ). This automatically ensures

that T (z )= fQ ;b(z)g.

BRST

Though this structure is quite general, it b ecomes esp ecially imp ortant for non-critical

strings. Here one can de ne the string theory in an almost completely algebraic way

through quantum hamiltonian reduction [5] of an appropriate sup er-WZW mo del. In or-

der to fully characterize the reduction, one has to sp ecify a sup er-algebra and an emb ed-

ding of sl (2j1) into it. This already uniquely determines the sp eci c extended sup ercon-

formal algebra. Furthermore, one has to cho ose a particular gradation. This determines

the particular free- eld realization, whichmust b e such that it allows for an interpretation

in terms of string theory, ie., it must b e of the form (1).

This approach of constructing string theories has the great advantage that the calcu-

lations are of algorithmic nature, enabling one to obtain the explicit form of the BRST

op erator in a relatively straightforward manner (straightforward at least compared to the

usual trial and error metho d). Another advantage is that it also systematically pro duces

a consistent set of screening op erators, which are needed to prop erly de ne the free- eld

Hilb ert space.

This program has b een explicitly carried out for the non-critical W strings, based on

a reduction of sl (njn 1) [2] and strings with N sup ersymmetries, based on a reduction

of osp(N +2j2) [3]. The generalization to arbitrary emb eddings of sl (2j1) in Lie sup er-

algebras remains to b e done, but we exp ect it to yield a classi cation of at least a very

large class of non-critical string theories, if not of all of them. It has also recently b een

shown that one can revert the reduction in that it is p ossible to reconstruct the underlying

Lie sup er algebra in terms of the eld content of a string theory [6].

Up to now, however, it was not clear how to extend this program to top ological strings

[8]. The matter sector of a top ological string is made up by a particular realization of the

twisted N = 2 algebra with central charge c . This top ological conformal eld theory is

coupled to top ological gravity, which can most easily b e represented by a sup ersymmetric

ghost system consisting of di eomorphism ghosts, b(z ), c(z ), and their b osonic sup er- 1

partners, (z ), (z ) [7]. Out of these ghost systems, one can construct a twisted N =2

algebra with top ological central charge equal to 9. Since the \true" central charge of

each building blo ckvanishes identically and separately (as implied by the twisting), the

BRST-op erator automatically squares to zero indep endent from whether the topological

central charges add up to zero or not. Thus, there is no need to include an extra Liouville

sector . In that sense there is no \critical" central charge for top ological strings. Nev-

ertheless, it is well-known that top ological strings with c =9 have sp ecial prop erties

[9], and indeed, in our construction given further b elow, this \critical" case of top ological

strings will turn out to b e distinguished.

In this letter we complete our program of obtaining all known string theories from

WZW mo dels, by showing how top ological strings can b e constructed from a suitable

reduction of sl (2j2). The p ossibilityofcho osing di erent gradations will enable us to nd

two di erent free- eld realizations of the \small" N = 4 algebra. Both haveinteresting

physical interpretations.

The rst gradation results in N = 2 minimal mo dels coupled to top ological gravity,

represented by a fermionic and a b osonic ghost-system, each with spins (2; 1).

The second gradation results in an emb edding of the non-critical b osonic string into

the top ological string, in the sense that the matter system is represented by the b osonic

string. That is, we will not only go one step further in the program initiated in [3] of

classifying string theories by purely Lie-algebraic metho ds, but we also obtain a new

mechanism of a string emb edding that is de ned intrinsically in terms of hamiltonian

reduction. This emb edding lo oks di erent from the kind of string emb eddings develop ed

in [10, 11 ], where one emb eds string theories with N sup ersymmetries into strings with

N + 1 sup ersymmetries; no obvious connection to hamiltonian reduction could b e made

so far for these emb eddings.

Let us now describ e the BRST-op erator structure of top ological strings is some more

detail. Neglecting the currents from the Liouville sector, the total BRST charge, Q =

BRST

Q + Q , consists of the following two contributions [8]:

s v

Q = (G + G )

s + m + gh

1 1

T + G + G : (2) Q = c T +

gh m gh v m

2 2

This implies that T (z ) is conjugate to b(z ) with resp ect to Q : T (z ) /fQ ;b(z)g, and

v v

to G (z ) with resp ect to Q : T (z ) /fQ ;G (z)g, and suggests that the BRST structure

s s

of the top ological string should b e related to a doubly twisted, N = 4 sup erconformal

It has b een shown [8], however, that the inclusion of a Liouville sector makes the computation of

physical observables more feasible 2

algebra [12 , 3 ].

There are essentially two N = 4 algebras, a \large" one [13], which is obtained from the

hamiltonian reduction of D (2; 1; ), and a \small" one [14], obtained from the reduction of

sl (2j2) (or A(1j1) in a di erent notation) [15]. In [3], top ological strings were related to a

subalgebra of the large N = 4 algebra. However, no obvious connection with the reduction

of D (2; 1; ) could b e found. As we will now argue, the more appropriate structure to

lo ok at is the small N = 4 algebra, which is generated by the energy-momentum tensor,

four sup ercurrents and an su(2) ane Lie algebra.

Therefore our starting p oint is the Lie sup er algebra sl (2j2). It consists of two sl (2)

i i

ane algebras whose currents will b e denoted by E (z ) and J (z ), with i 2f+;0;g. The

fermionic currents transform according to the (2j2)(2j2) representation of the sl (2)sl (2)

subalgebra. Wethus denote the fermionic currents by j (z ), with a; i; j 2f+;g. The

OPE's are explicitly given by

2 0 0 2 0 0

z ; J (z ) J (z )= z ; E (z ) E (z )=

1 2 1 2

12 12

8 8

1 1

0 1 0 1

E (z ) E (z )=z E (z ) ; J (z ) J (z )= z J (z ) ;

1 2 2 1 2 2

12 12

2 2

+ 2 1 0 + 2 1 0

E (z ) E (z )= z + z E (z ) ; J (z )J (z )= z + z J (z ) ;

1 2 2 1 2 2

12 12 12 12

4 4

1 1

0 i 1 i i 1 i

E (z )j (z )= z j (z ) ; E (z )j (z )= z j (z ) ;

1 2 2 1 2 2

a 12 a a 12 a

4 2

1 1

0 i 1 i i 1 i

J (z )j (z )= z j (z ) ; J (z )j (z )= z j (z ) ;

1 2 2 1 2 2

a 12 a a 12 a

4 2

2 1 0 0

z z E (z ) J (z ) ; j (z )j (z ) =

2 2 1 2

12 12 +

4 2

2 1 0 0

j (z )j (z ) = z z E (z )+J (z ) ;

1 2 2 2

+ 12 12

4 2

1 1

z E (z ); j (z )j (z )= z E (z ) ; j (z )j (z )=

2 1 2 2 1 2

12 + 12 +

2 2

1 1

j (z )j (z )= z J (z ); j (z )j (z )= z J (z ) : (3)

1 2 2 1 2 2

+ 12 + 12

2 2

The relevantemb edding of sl (2j1) into sl (2j2) is given bycho osing the b osonic gen-

i 0

erators of sl (2j1) to b e E and J . Note that we still havetwo p ossibilities for cho osing

the fermionic part of the emb edding. This traces back to the curious prop ertyofA(1j1)

that the fermionic ro ots are simultaneously b oth, p ositive and negative. We will see in

the following that this do es not really constitute an ambiguity.

+ +

3 E J E J

= 2, g = We used the metric conventions g = g = 4, g = g = 2 and g

++ ++ 00 00 + +

+ +

=2, g =2. 2, g

++ ++ 3

As was emphasized in [3], the emb edding just determines the kind of sup erconformal

algebra. However, in order to obtain a sp eci c free- eld realization of this algebra that

allows for a string interpretation, one has to cho ose the gradation appropriately. The

gradation will determine a consistent set of rst class constraints on the negatively graded

part, and these constraints imply the presence of a gauge symmetry. The gauge xing

and the construction of the generators of the conformal algebra pro ceed then in a way

quite similar to what was develop ed in [16 ]. That is, although the algebra is indep endent

of the gradation, its realization, and in particular the Miura transformation, dep ends on

it.

Wenow apply this to sl (2j2). The standard gradation, i.e. the one used in [16], is

given by the eigenvalues of ad .However, this gives rise to a \symmetric" free- eld real-

ization that has no interpretation as a string BRST algebra. In order to nd appropriate

gradations, we note that in the classi cation [17 ] of Lie sup er-algebras one distinguishes

between algebras of typ e 2, whose fermionic part carries an irreducible representation of

the b osonic subalgebra, and algebras of typ e 1, where this do es not hold. Typ e 1 sup er

Lie-algebras admit a canonical gradation of the form g = g g g , where b oth

1=2 1=2

g and g are irreducible representations of g . In our case wehavej 2 g and

1=2 1=2 +1=2

j 2 g , while the remainder of the generators b elong to g .

1=2 0

We nd that we can use two di erent gradations, denoted by I and II, to describ e

sensible, string-typ e free- elds realizations. Gradation I is given by the sum of the canon-

0 0

ical gradation with the gradation de ned by the action of ad , whereas gradation

2(E +J )

0 0

II is de ned by the eigenvalues of ad ignoring the canonical one. In the following

(2E +4J )

table, we explicitly give the corresp onding grades for the various elements of sl (2j2):

++ + + ++ + +

+ 0 + 0

E E E J J J j j j j j j j j

+ + + +

3 1 1 1 1 1 1 3

I 1 0 1 1 0 1

2 2 2 2 2 2 2 2

3 1 1 3 3 1 1 3

II 1 0 1 2 0 2

2 2 2 2 2 2 2 2

Let us now rst give a detailed discussion of the hamiltonian reduction of sl (2j2) using

gradation I . The constraints we imp ose on the negatively graded currents are of the form

= 0 with

= E = J ; ;

E J

+ +

= j ; = j + ;

+ +

= j ; = j : (4)

The auxiliary elds, (; ), and their conjugates, (; ), were intro duced in order to avoid 4

that currents which are in the kernel of ad and would b ecome the leading term of the

conformal currents, are constrained to zero (see [3] for more details). The additional

auxiliary elds (; ) are needed to ensure that all constraints are rst class.

The constrained theory derives from the action

Z Z Z

1 2 1

2 2 2

d z str (A) d z ( @ @) d z @ ; (5) S = S [g]+

inv WZW

where S is the Wess-Zumino-Witten action on SL(2j2) and the gauge elds A app ear

WZW

as Lagrange multipliers that imp ose the constraints. The gauge symmetry of the action

is generated by the strictly p ositively graded subalgebra of sl (2j2):

A = @ +[A;]; J = @ +[J;]

+ ++

=0; =2 +

++ +

= ; = 2

+ ++

; = : (6) =

2 2

The kinetic terms of the auxiliary elds follow from requiring gauge invariance and give

rise to the following OPE's

1 1

(z ): (z )=(z ):(z )=z (z ):(z )= z : (7)

1 2 1 2 1 2

12 12

Wenow quantize this theory by rst xing the gauge symmetry by setting A = 0, and

intro ducing the ghosts C and anti-ghosts B :

+ E + J + + ++ + + ++ +

C = c t + c t + t + t + t + t

E + J + + + ++ + + + ++

J J + + +

B = b t + b t + t + t + t + t : (8)

E J + + +

The gauge xing action reads

S = d z (DA + B @C + AfB; C g) ; (9)

g:f:

where D is a Langrange multiplier that imp oses the gauge condition. From this we can

read o the OPE's for the ghosts

+ + + +

b (z ):c (z )=b (z ):c (z )= (z ): (z ) =

1 2 1 2 1 2

E E J J +

1 ++ + + ++

: (10) (z ) = z (z ): (z )= (z ): (z )= (z ): =

2 1 2 1 2 1

12 + + +

The BRST-charge then takes the standard form:

1 1

Q = str C + CJ ; (11)

2i 2

4 a b deg (b)

The sup ertrace is de ned by str (xy )=x y g (1) , where deg (b) = 0 or 1, if it corresp onds to

a b osonic or fermionic generator. 5

where the ghost currents, J = fB; C g, are given by:

1 1

+ + ++ +

E = ; E = ;

gh + gh +

2 2

1 1 1 1 1

0 + + + ++ + + ++

E = b c + + ;

gh E E + + + +

2 4 4 4 4

1 1

+ ++ + +

; J = ; J =

+ gh + gh

2 2

1 1 1 1 1

0 + + + ++ + + ++

J = b c + + ;

gh J J + + + +

2 4 4 4 4

1 1 1 1

+ ++ + + +

j = b + c ; j = b b ;

+gh J + + E +gh E + J +

2 2 2 2

1 1 1 1

++ + ++ + + + + +

b c ; j = c + c ; j =

E + + J gh J E +gh

2 2 2 2

1 1 1 1

+ ++ + + ++ +

j = b + c ; j = b c : (12)

J E E J

gh gh

2 2 2 2

Wenow pro ceed by closely following the metho ds of ref. [16]. Classically, the generators

of the extended conformal algebra are given by the gauge invariant p olynomials. At

the quantum level this translates to the fact the generators of the conformal algebra are

given by the generators of the cohomology of Q on the algebra A, which consists of all

normal ordered pro ducts of fJ J + J ;C;B;;; ; ;;g and their derivatives. The

computation of the cohomology is not very hard due to the presence of a double gradation

of the complex A:

A = A : (13)

(m;n)

m;n2 Z

m+n2Z

The grade of the various elds is given by(m; n), where m is the grade previously given

in the table and m + n is the ghost numb er. The auxiliary elds have grade (0; 0). The

BRST op erator splits into three parts, each of which has a de nite grade:

Q = Q + Q + Q ; (14)

0 1 2

where

+ +

Q = c c ;

E J

2i 2

1 1

+ ++ +

+ ( )+ ; Q =

+ +

2i 2

1 1 1 1

+ + + + +

+ (E + (J + (j c )+c ) ) Q = E J j

E J + gh gh gh

2i 2 2 2

1 1

++ + + + ++

(j + j ) (j + j )+ j ; (15)

+ +gh + gh +

2 2

whichhave grade (1; 0), (1=2; 1=2) and (0; 1), resp ectively.

^ ^ ^

Using the fact that B and (where denotes the substitution of J by J in the constraints 6

(4)) generate a sub-complex, one can argue along lines similar to those in [16] that the

cohomology of Q on A is isomorphic to the one computed on a reduced complex A. The

reduced complex is generated by those elds of A whichhave grades (m; n) with m 0.

b b

Z of A: Wethus intro duce a ltration A , m 2

M M

b b

A A : (16)

(k;l)

k 2 Z

This leads to a sp ectral sequence (E ;d ), r 1, converging to H (A; Q), where we

r r

have E = H (E ; d ). The sequence collapses already after the second step: E =

r r 1 r 1 1

+ + 0

b b

^ ^ ^

H (A; Q ), E = E = H (E ; Q ), where E ' H (A; Q) is generated by fE ; J ; J +

0 2 1 1 1 2

1 1 1 2 2

++ ++ + +

^ ^ ^ ^

; ; j ; j ; j + j ; + g. The explicit form of the generators

+ + +

2 4

now follows from a tic-tac-to e pro cedure and have the structure:

1 1

;m+ ) (0;0) (m;m) (m

2 2

+::: + (17) = +

where the upp er indices refer to the grades of the individual terms and the leading terms

(m;m)

are the generators of E given ab ove. Using exactly the same arguments as

(0;0)

those in [16], one can show that ! is an algebra isomorphism, which is noth-

ing but the quantum Miura transformation. This provides us almost with the desired

free- eld realization. In order to obtain the correct realization, we need in addition to

(0;0)0 (0;0) 1

p erform a similarity transformation (see also [3]) given by = S S , where

S = exp [ dz ]. Bosonizing the Cartan currents

p p

i i

0 0

^ ^

E = @ (' '); J = @(' + '); '(z ):'(z )=ln(z ) ; (18)

1 2 12

4 4

p p

i i

and rescaling the auxiliary elds, ! ; ! , then nally yields the

2 2

following currents of the BRST algebra:

i 1 1

2 2

T = @'@' + (@ ' + @ ') @ + @ +

2 2 2

3 1

+@ @ @ ;

2 2

G = @ ' + i @ ;

@; G = @' + @ +2@ + i

G = (@'@' @ + i @ ' + @ + @ @ )

@ + @ + @) (@ ' + i

p p

(i @( @ ' i @ ' + )) (1 + )@ ;

G = ;

p p

K = @ ' + i @ + + i @ ' +

+ 7

+i @ ' (1 + )@ ;

K = i (@ ' @')+ 2 ;

K = : (19)

They are canonically normalized such that they generate the small N = 4 algebra:

1 c

3 2 1

^ ^

z + z K (z )+z (T + @K )(z ); G (z )G (z ) = G (z )G (z )=

3 2 3 2 + 1 2 + 1 2

12 12 12

3 2

2 1

G (z )G (z ) = 2z K z @K (z );

1 2 2

12 12

1 1

^ ^

K (z )G (z ) = z G (z ); K (z )G (z )=z G (z );

1 2 2 1 2 2

12 12

1 1

^ ^

K (z )G (z ) = z G (z ); K (z )G (z )=z G (z )

3 1 2 2 3 1 2 2

12 12

n 1 2

z K (z ); (20) K (z )K (z ) = z +

n 2 l 1 m 2 lm lm

12 12

where c = 6(1 + ).

The interpretation of the ab ove formulas in terms of string theory evident: the b osons

','together with the fermions , represent a minimal N = 2 matter free- eld real-

ization, with background charge given by i .Furthermore, the string ghost-antighost

pair corresp onds to , , whereas , represents the b osonic ghost-antighost pair of the

top ological string. After twisting the energy-momentum tensor T ! T + @K , and

acquire spin 2 and and spin 1.

The sp eci c realization of the N = 2 sub-system that weinterpret here as matter

system is not essential for the closure of the algebra. In fact, it is easy to see that we

can replace it byany other realization of the N = 2 algebra. In contrast to the stringy

hamiltonian reductions considered so far, we do not get a Liouville system. This seems

to b e natural, since as mentioned in the intro duction, one would not exp ect the Liouville

eld to play an essential role in top ological gravity. Let us also note that the only physical

observable in equivariant cohomology, namely the generator of gravitational descendents,

@ , app ears here naturally as part of the sl (2) Kac-Mo o dy current K . The central

extension of this algebra vanishes and @ decouples from the algebra precisely if = 1

which constitutes the \critical case" c = 9 of top ological strings.

The screening op erators asso ciated with this free- eld realization are easily obtained

from sl (2j2). The following screeners are related to the three simple fermionic ro ots:

1 i

S = dz exp( ')

1 i

S = dz exp( ')

1 i i

p p

S = dz ( + ) exp( ') : (21)

They are needed to de ne the physical Hilb ert space of the theory.

Wenow turn to discussing the reduction for gradation II. Here, the rst-class con-

straints lo ok

= E + ; = J ;

E J

2 4

+ + + +

= j ; = j ;

+ +

1 1

= j + ; = j : (22)

+ +

2 2

The action of the constrained theory is now given by

Z Z

1 1

2 2

d z str (A) d z ( @ + @ @) ; (23) S = S [g]+

inv WZW

and the OPE's for the auxiliary elds are (z ): (z )= (z ): (z )= (z ):(z )=z .

1 2 1 2 1 2

The action is invariant under the following gauge transformations:

A = @ +[A;]; J = @ +[J;]

+ ++ ++ +

=0; =2 + + + (24)

J + E

++ + +

= + ; = 2

+ E

++ + +

= ; =2 : (25)

E +

To quantize the theory,we can use almost the same ghost system as in (8), except that we

+ + + + + +

have to replace the ( ; )-system by( ; ), which ob eys: (z ); (z )=

1 2

+ + +

+ + + +

z . The ghost currents j ;j ;j ;j stay the same as b efore and therefore can

+ +

directly b e taken over from (12). The other ghost currents get changed, however (notice,

e.g., that this time there are no ghost contributions to J ):

1 1 1 1

+ + ++ + ++ + +

E = ; E = + ;

gh + + gh + +

2 2 2 2

1 1 1 1 1

0 + + + ++ + + ++

E = b c + + ;

gh E E + + + +

2 4 4 4 4

1 1 1 1 1

+ + ++ + ++ + + 0

+ + ; c = b J

J + + J + + gh

2 4 4 4 4

1 1

++ + + +

j = c j = b ;

J J +

+gh +gh

2 2

1 1

++ + + +

j = c ; j = b : (26)

gh J gh J

2 2

The most drastic di erence in comparison to gradation I shows up in the structure of the

BRST op erator. Using the new gradation to de ne the complex (13), the BRST charge 9

splits into the following ve parts:

I I

1 1 1 1

+ ++ ++

c ; Q = Q = ;

1 0

J +

2i 2i 2 2

I I

1 1 1

+ + + +

+ ; Q = c c ; Q =

2 3

+ E E

2i 2 4 2i

1 1 1

+ + + + +

Q = c (E + E )+c J + (j + j )

E gh J + +gh

2i 2 2

1 1

++ + + + ++

(j + j ) (j + j ) (j + j ) ; (27)

+ +gh + gh + gh

2 2

3 1

They individually square to zero, and their bigrades are given by(2;1), ( ; ), (1; 0),

2 2

1 1

( ; ) and (0; 1). Moreover, they ob ey the relations fQ ; Q g + fQ ; Q g = fQ ; Q g +

1 3 0 4 2 3

2 2

fQ ; Q g =0. The anti-ghosts B and the constraints form BRST-doublets and we

1 4

can de ne a reduced complex, completely parallel to the treatment of gradation I .To

compute the cohomology on this reduced complex, we note that we can write the BRST-

0 0

op erator as Q + Q with Q = Q . Then this de nes a double complex in the weak

0 i

i=1

sense of [16]. Of course, in the sp ectral sequence wethus have d = Q . But nowwe can

0 00 00

use the splitting given ab ove, such that Q = Q + Q and Q = Q . Pro ceeding in

1 i

i=2

this manner, it is straightforward to compute the cohomology in a stepwise fashion. Since

we use here a rather non-standard decomp osition of the BRST-op erator, it is useful to

denote the non-trivial action of the relevant parts of it on the elds explicitly:

0 + +

Q : [Q ;J ]= c ; [Q ; ]=c ;

0 0 0

J J

1 1 1 1

++ ++ ++ ++

Q : [Q ;]= ; fQ ; g = ; fQ ; g = ;

1 1 1 1

+ +

2 2 2 2

1 1 1 1

0 ++ ++ 0 ++ ++

^ ^

[Q ; E ]= + ; [Q ; J ]= + ;

1 1

+ +

8 8 8 8

1 1

+ + + +

^ ^

fQ ; j g = c ; fQ ; j g = c ;

1 1

+ J J

4 4

1 1 1

+ + +

Q : [Q ; ]= c ; fQ ; g = c ; fQ ; g = c ;

2 2 2 2

E E E

4 4 4

1 1

0 + + + ++ ++

[Q ; E ]= c c ; fQ ; j g = + ;

2 2

E E + + +

4 8 4 8

++ ++ +

+ ; g = fQ ; j

4 8

+ + ++ +

Q : fQ ; g = ; fQ ; g = ; fQ ; j g = c ;

3 3 3 3

+ + J

1 1 1

++ + + ++ ++

fQ ; j g = c ; [Q ; E ]= ;

3 3

J +

2 2 2

1 1 1 1

0 + + 0 + +

^ ^

[Q ; E ]= + ; [Q ; J ]= : (28)

3 3

+ +

4 4 4 4

From these equations we immediately see that (; c ) decouple from the Q -cohomology.

Then, lo oking at Q ,we note that the auxiliary elds ( ; ) lie in BRST-doublets together

1 10

++ ++

with the comp osite op erators ( ; ). This means that these decouple from the

Q -cohomology. In the next step wehavetotakeinto account that in the complex

++ ++

E = H (A; Q ) there are vanishing relations of the form = 0. Therefore

2 1

+ +

^ ^

we nd that the comp osite elds j and j are non-trivial elements of the Q -

cohomology on E . Pro ceeding further one nds that the sp ectral sequence collapses

+ + 0

^ ^ ^

after the fourth step and that the cohomology of Q is spanned by fE ; J ; J +

1 1

++ ++ + +

^ ^ ^ ^

; j ; j ; j ; j ; g. It is surprising that not all the elements that lie in

+ +

4 4

ker(ad ) corresp ond to leading terms in the cohomology. Itishowever not dicult to

see, by following a generalized tic-tac-to e pro cedure, that one picks up the highest weight

+ +

^ ^

auxiliaries , in the tail of j and j . It is also clear that the truncation to zero

grade elds still provides an algebra isomorphism.

In order to end up with a string-typ e free- eld realization, we demand that one of the

sup ercharges is given by just a single auxiliary eld. We could cho ose , or, equally well,

.To b e sp eci c, let us cho ose , and p erform the following similarity transformation:

i h

1 1 1 1

0 0

^ ^

E + @ + J ) : (29) dz ( S = exp

2i 2

Furthermore, we b osonize the Cartan currents:

p p

0 0

^ ^

p p

E = @' ; J = @' ; ' (z ):' (z )= ln(z ) (30)

1 2 i 1 j 2 ij 12

2 2 2 2

Then, nally,we arrive at a second free- eld realization of the small N = 4 algebra that

can attributed to top ological string theory:

1 i(1 ) 1 1

2 2 2 2

p p

T = (@' ) + @ ' (@' ) + @ ' +

1 1 2 2

2 2

3 1 3 1

+@ @ @ @ @

2 2 2 2

1 i(1 ) 1 1+

2 2 2 2

p p

G = ( (@' ) + @ ' (@' ) + @ ' @ )+

+ 1 1 2 2

2 2

1 2

+ 2@ ( @' ) @ +

G = @ 2@

i(1 ) 1 1+ 1

2 2 2 2

p p

(@' ) + @ ' (@' ) + @ ' + G = [

1 1 2 2 +

2 2

2 @ @ + @ + @ @ ]

@ + @ ) @ [ ( 2@ ' ) + ] (1 + )@ ( +

G =

1 i(1 ) 1 1

2 2 2 2

p p

(@' ) + (@' ) K = @ ' @ ' @ @' +

1 2 + 1 2 2

2 2

2 2 11

2 2

+ 2 @ ' + + @ @ (1 + )@

K = 2@ ' +2

3 2

K = (31)

After twisting, the auxiliary elds, ( ; ;)have spin 2 and ( ; ; )have spin 1. From

the form of the generators we see that we can indeed interpret ( ; ) and (; ) as the

ghosts of the top ological string. The rest constitutes the matter sector of the top ological

string and is isomorphic to a particular realization of the ordinary b osonic string! That is,

' is the matter and ' the Liouville system of the b osonic string, and the di eomorphism

1 2

ghosts are given by( ; ). Indeed, mo dulo the terms arising from the top ological ghost-

system, the currents T , G , G and K are exactly the well-known N = 2-currents of

+ 3

the b osonic string [1, 2 ]. This means that the ab ove formulas should b e interpreted as

an emb edding of the non-critical b osonic string into the top ological string. Note that

the critical top ological string, which corresp onds to = 1, coincides with the critical

b osonic string, as can b e seen by the vanishing of the background charges.

For sake of completeness, we present the following screening op erators,

2' ) dz exp(i S =

1 1

1 i

S = (' i' )] dz exp[

2 1 2

1 i

S = (' i' )] ; (32) dz ( +2@ + @ ) exp[

3 1 2

that are necessary to prop erly de ne the free- eld Hilb ert space.

Generalizing the ab ove considerations to top ological W -strings [12 ], one exp ects that

a reduction of sl (njn) should b e relevant. Let us consider the reduction of sl (3j3) in some

more detail. As wewanttohavean N =2 W algebra as a part of the full string BRST

algebra, we need to analyze the emb edding of sl (3j2) into sl (3j3). The emb edding of

sl (2j1) in sl (3j2) is then precisely the one that gives rise to the N =2 W algebra. In

terms of sl (2) representations, we nd that the b osonic part of sl (3j3) will give rise to a

spin 3, two spin 2, two spin and one spin 1 current. The fermionic part yields two sets

5 3

of currents with spins , 2 and .Thus we nd 6 b osonic and 6 fermionic generators.

2 2

However, from a W extension of the small N = 4 algebra one exp ects 8 b osonic and 8

fermionic generators. Wethus see from this simple counting argument that the reduction

of sl (3j3) will not give rise to a naive formulation of top ological W -gravity [12].

It is nevertheless worth trying to go on with the analysis. It is easy to nd a gradation

analogous to the one we used for sl (2j2), where twoofthesl (2) highest weights, one

b elonging to a fermionic triplet and one b elonging to a b osonic doublet, have negative 12

grades. From our discussion of sl (2j2) it is clear that this will result in b osonic and

fermionic ghost systems. Under the U (1)-currentofsl (3j3) (the one which corresp onds

, to the ghost current after the reduction), these two ghosts systems havecharges 1 and

resp ectively, such that b oth would obtain spin 3 in the twisted sup erconformal algebra.

They should therefore represent the high-spin ghost-systems of top ological W -gravity.We

thus exp ect that one obtains from such a reduction top ological W -gravity to b e realized

in a kind of \half-rotated" matter picture, where the low-spin ghosts are decoupled from

the theory. That such a realization exists is in fact quite plausible since we know that the

sp ectrum of top ological W -gravity can b e represented entirely in the matter sector [12 ].

Combining the results of this pap er with [3], we are drawn to conclude that a very

systematic and almost completely algebraic approach to string theory might b e attainable,

at least as far as controlling and classifying the p ossible gauge structures on the world-

sheet go es. Indeed, these two pap ers demonstrate that all known string theories can b e

obtained in a straightforward way from reducing WZW mo dels. As all simple sup ergroups

have b een classi ed, the natural step to takenow is a complete classi cation of string

theories that can b e obtained this way. This is presently under investigation [20]. We

hop e that such a group theoretical approach to string theory will eventually provide clues

to the structure of a \universal" string eld theory that would havevacua corresp onding

to arbitrary gauge groups on the world-sheet.

Acknowledgements We extensively used Kris Thielemans Mathematica package OPE-

defs.m [21] for the computations.

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