Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

SUPERSTRING AND GRADED GRASSMANNIANS M.A. A W A D A Blackett Laboratory, Imperial College, London SW7 2BZ, UK

and A.H. C H A M S E D D I N E l CERN, CH- 1211 Geneva 23, Switzerland

Received 26 February 1988

We introduce the infinite dimensional graded grassmannian manifolds in terms of free field operators, and study their properties. We show the embedding of the graded DiffS ~/S~manifold in the graded grassmannians and discuss some physical applications.

At present, there are two attractive views o f string theory, both based on h o l o m o r p h i c geometry. The first is the formulation o f q u a n t u m string theory as integrable analytic geometry on the universal m o d u l i space o f R i e m a n n surfaces [ 1 ]. The other is based on the concept o f loop space and formulated as a h o l o m o r p h i c vector bundle over the m a n i f o l d D i f f S~/S ~ [ 2 ]. In both cases, there exists a one-to-one e m b e d d i n g o f the base m a n i f o l d into the infinite d i m e n s i o n a l grassmannians [ 3-11 ]. There are various advantages o f working with the grassm a n n i a n s mainly that most c o m p u t a t i o n s b e c o m e algebraic as well as having the p r o m i s e o f providing a nonperturbative t r e a t m e n t for m o d u l i spaces o f all R i e m a n n surfaces, including the infinite genus ones. Recently, we have f o r m u l a t e d closed string theory as h e r m i t i a n geometry on grassmannians [ 11 ] and it is natural to generalize this a p p r o a c h to the closed superstring and the heterotic string. This is not a trivial exercise since the m a t h e m a t i c a l tools that were available in the bosonic case [ 3,4 ] are not fully developed in the supersymmetric one. I n d e e d as we shall see, the known cases [ 12,13 ] turn out to be not physically relevant. In this letter we shall start to build the foundations o f the generalized approach to the superstring by introducing the graded grasmannians ( d e n o t e d by S G r ) in terms o f Bogoliubov transformations on the first quantized Hilbert space o f a F e r m i - B o s e system. This can also be equivalently written in terms o f the supersymmetric ghost system. We study the properties o f these manifolds a n d construct the h a m i l t o n i a n and the currents, a n d c o m m e n t on the possible s u p e r s y m m e t r i c K P hierarchy. We give the e m b e d d i n g o f the graded m a n i f o l d super D i f f S ~/S ~ into S G r and present a simple way o f computing the curvature o f the s u p e r h o l o m o r p h i c vector bundle based on this manifold. The simplest way to obtain the graded g r a s s m a n n i a n manifolds S G r is to make use o f free field operators. Following U e n o and Y a m a d a [ 13 ], let A be the Clifford algebra over C with generators ~z,~, ~ * where a is a graded index (a = 0, 1 ). The indices m and n are taken to be integers or half-integers d e p e n d i n g on whether they are in the R a m o n d sector ( R ) or in the N e v e u - S c h w a r z ( N S ) sector. We impose the following quantization conditions [~',~,, ~'~}=0,

[~,,~, ¢ , ' } = 0 ,

[~,,~, ~ , ~ }- - ~ ° ' ~ , . + n . o

,

(1)

On leave of absence from the American University of Beirut, Beirut, Lebanon. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

437

Volume 206, number 3

PHYSICS LETTERSB

26 May 1988

However, since our main interest is in the superstring, we must find the correspondence with the superconformal treatment [ 14 ]. We introduce the superghost system

B(z, O)=fl(z)+Ob(z),

C(z, O)=c(z)+OT,(z) ,

(13)

with superconformal weightsfand ½- j . respectively. The mode expansions of the component fields are

b(z)= E bnz-n-J,

C(Z)= E CnZ-n-(1-J), /7(Z)= E/7otZ -a-(j-l~2),

~2(Z)= E~otZ -°t-(l/2-j) ,

(14)

where o ~ Z + ½in the NS sector and o ~ Z in the R sector, a n d j = f + ½. The quantization conditions are {b., Cm}=O.+m,0,

[7.,/Tin ] =<~.+m,0,

(15)

with all other (anti) commutators vanishing. The vacuum is defined by bnl0)=0

n>-j,

c.]0)=0

n>~j, /Tc~10)=0

o~>½-j,

7<~10)=0

oz>~j-½,

(16)

For simplicity, we shall not deal with the equivalent charged vacuum which is present due to the bosons/7 and [141. Let H the Hilbert space of all complex square integrable superfields on S ~, the superspace extension of the circle. In analogy with the bosonic case, we can polarize H by the action of B and C fields defined over this superspace. The graded extension of the general linear group GL ( H ) and the restricted unitary group Ures ( H ) can be defined [4]. The subgroup G ( H ) of U ~ ( H ) is now represented by

g=exp(-

efi dz2dO2 efl dzl dO~B( z~ , Ol ) T( z~ , Ol ; Z2, 02 )C( z2, 02 ) ) ,

(17)

where T(z~, 01; z2, 02) is a smooth function on S ~× SL The two expressions for g in eqs. (10) and ( 17 ) coincide if we identify

Fn<:)7 ~ J=Lb(z)J'

Lc(z)

'

T(z~, Ol ; z2, 02 ) = T l°(zl, Z2 ) + 01T oo (zi, z2 ) + T ~ (z~, z2 ) 02 + 0102 T °l (zi, z2 ) ,

(18)

with the weight assignments 2°=j-½,

2°*=3-j,

2'=j,

2'*=l-j.

The conserved charges can now be expressed in terms of superspace integrals of the form

B(z, (,0)IV,, (z, O, (o)dz d~o, ~ C(z, ~) W. (z, ~, O)dz d¢, where the Wn are super-Plucker coordinates. A little thought reveals that the SGr we have defined is larger than what one would need since we know that the superconformal transformations in the the (z, 0) space are of a special type. What one would need is a restriction to reduce the SGr into a submanifold. We shall come back to this point later. Since one of the main advantages of the grassmannians is their intimate link with the KP hierarchy and the moduli space of Riemann surfaces, we must seek a similar relation in the graded case. To do this, we introduce the variables

Xa~'(z) = F, X<~bZ"+x°+x .... l, rt:>0

(19)

and define the hamiltonian 439

Volume 206, number 3

PHYSICS LETTERS B

H ( X ah) = ~ Xah(z)jab(Z) dz,

26 May 1988 (20)

where jat,(z) = :g/a(z)g/b*(Z) : are the currents. These currents form a non-trivial infinite dimensional graded algebra: jab , J"cd).__ , . j - - ( - - 1 )ac+bd( ( -- l j~t,c3bc'ad ~, J ~ + m - ( -- 1 )ad(~adjCb+m )

+ ( - 1 )(a+ 1)(b+ 1)(n--I-/~a_[_/]b* 1 )3aaObcS,,+m,O ,

(21)

where

j,h(z ) = ~j~h z-,,+~,+a .... 1. n

The hamiltonian defined in eq. (20) annihilates the vacuum

H ( X "h) 10> = 0 , and the action on the fields ~ a ( z ) and q/"*(z) is exp [ H ( X ) ] ~" exp [ - H ( X ) ] = e x p [ x a b ( z ) ] ~//b(Z), exp [ H ( X ) ] ~a* exp [ - H ( X ) ] = exp [ - xab(z) ] ~U~ ( z ) ,

(22)

which is clearly non-diagonal. This makes finding the vertex operators for ~'a and ~b. a difficult task. The supertau function is given by

r(X a~',g) = ( 01exp [ H ( X "b) ]g[0 ) ,

(23)

where the vacuum ( 0 [ must be defined in such a way as to absorb any possible vacuum charge. The appearance of variables X "b suggests that the KP hierarchy equations [ 3 ]

L(x,O)W(z,x)=zW(z,x),

OW/Ox,,=(L")+ W

where L(x, 8) is a pseudo-differential operator, ~=~/~x~, and ( L " ) + is the differential part of L ", must be modified by replacing L by a matrix L ~b and W b y W ~. However, as we said before, we are more interested in obtaining the hierarchy connected with superconformal theories. From the lagrangian of the superghost system, one easily finds that the supercurrent is

j(z, O)=B(z, O)C(z, O).

(24)

With this we associate the hamiltonian

Hs¢ (x,, ~,, ) = ~ dz dO X(z, O)j(z, O) ,

(25 )

where

X(z, 0 ) = ~ X,,z"+O ~ 2,,z "-j/2 . n>0

n>O

(26)

The hamiltonian in eq. (20) reduces to that in eq. (25 ) if we set X °° =X~, I = X , ,

X,,lo = 0 ,

X .ol = 2 ~ .

(27)

The (X~, 2 . ) development of the B and Cfields is now diagonal and given by exp [ H ( X . , 2 . ) lB(z, O) exp[ - H ( X ~ , 2. ) ] = e x p [X(z, 0) ]B(z, O), exp [H(X~, 2. ) ]C(z, O) exp[ -H(x~, 2. ) ] = e x p [ - X ( z , O) ]C(z, O) , 440

(28)

Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

Thus, it remains to be seen what kind o f SKP hierarchy is generated with the identifications in eq. (27). However, we now know that the only deformation variables in the differential equations must be the X, and 2n, and not the full set Xg ~'. Such a kind o f SKP hierarchy has been recently given by Manin and Radul [ 12 ], and one would have been tempted to take it as the hierarchy we are seeking. However, by studying the algebra of the currents in the hamiltonian flow in that example, one discovers that it can be obtained from eq. (20), after setting

X°°=X~ l=Xn

n>~l, X~ ° = X m = 2 .

n>l.

(29)

The generators of the M a n i n - R a d u l algebra are O/Ox, O2~, @ and O2~_ 1 and related to the currentjg b by

O/Ùx=J°°+J~ ~, @ = J l ° - - J l°1 ,

02l-- 1 =J2l-'1°l +;0*S2/_l •

~2l--J21~;00_L ;11TS21 ,

(30)

On the other hand, the currents appearing in eq. (24) are given by different, inequivalent combinations ofj~ b. Thus, unfortunately it is not related to the SKP hierarchy o f Manin and Radul. As a side remark, we mention that some information about the function T(zl, 0~; z2, 02) (or equivalenty the matrices T,~'~ ) that comes from embedding the supermoduli space into the grassmannian SGr (such embedding would certainly exist because o f the generality o f the SGr we defined). To do this we calculate the two-superpoint function for the superghosts

G(zl, 01 ; z2, 02 ) = (OlB(zl, 01 )C(z2, 02 )g] 0 ) ,

(31)

which can be calculated using eqs. ( 16 ) and (17) to obtain

G(zl, Oj ; z2,02 ) = (01 - 0 2 )/Zl2 - T(z2, 02; zl, 01 ) ,

(32)

where z12 = z , - z 2 - 0 , 02. On the other hand, the two-superpoint function for the superghost on the supertorus was recently calculated [ 15 ] G(z,,O,

;

z2,02,r, oL) = - 4 D ( ( I m z . 2 ) 2 1 2 Im z'

in

01(Zl2lr')2) q(z' )

(33)

where r' = r + (01 + 02 ) a. Comparing eqs. (32) and (3 3) yields the function T corresponding to a supertorus. We now address the question of the embedding o f the graded manifold super DiffS1/S 1 into SGr ( H ) . In the bosonic case this is achieved via the smooth composite map [ 3 ] Diff S' --*Ures ( H ) ~ G r . ( H ) =

Ure s ( H )

U(H+)×U(H_)

"

(34)

Explicitly the map is

f h ( z ) = h ( g ( z ) )[dg/dz[ j ,

(35)

where f: S j -~S ~, h: S 1--*C and g is the inverse o f f The generalization is

f h( z, O) = h ( g ( z, O), z( z, 0))10v/0zl;

(36)

where f: S 1-~ S l, h : S 1--. C and v= g+ Ox is the inverse o f f while j is the superconformal weight of h. Here we shall not attempt to fully characterize the embedded manifold in terms of S G r ( H ) . The infinitesimal action o f super DiffS ~on h is then [ 14]

5h = ( 6z 31Oz + 60 31OO+f Oe/Oz )h

(37)

where 6Z=Co+0~,

50=~+~Odeo/dz,

e=Sz+080.

(38)

441

Volume 206, number 3

PHYSICS LETTERSB

26 May 1988

To find the Fourier representation of the structure constants of super DiffS 1, we expand h(z, O) = h ° ( z ) + 2Ohm(z) , h°(z) = Z h . z - " ,

h'(z)= • h~z-",

c~

e ° ( z ) = Z e- zn,

n

e~(z) = Z e- z " .

n

(39)

c~

Then we can write 6hA =eRfCA G)hc ,

(40)

where A, B, C are graded indices of the form (m, a ) and fc,~,j) = [ ( j _ ½) m - n ] 6 , . +. . f i . , c j..

=

jcC~J) . , ~ = [ 0 " - 1 )m--fl]~m+pO:, ~ c ,

)a-~n],~.+.~,

J.a

= [ 2 ] ~ ,+ a O pc •

(41)

Note that for f = -3 2, rc~3/2~ JAS are the structure constants for super DiffS l in the adjoint representation and have the appropriate symmetry. " = "C The generators for super DiffS 1can thus be represented by ~LA )~ = J ~'C(j) AB when it acts on fields with weightf With this at hand, we can calculate the curvature of the line bundle whose base space is superDiffS~/S 1 and whose fibres belong to the Fock space of a superfield with spin~, as the pull-back of the bundle whose base space is the image of super DiffS 1/S 1 in SGr, with the same fibres. Since the latter is a superK~ihler manifold, we can write for the curvature W( LA, LB )

=Str( LALB-- ( - 1) ABLBL

(42)

A) ,

where the supertrace, Str, must be performed. The above assignment is only valid for Grassmann manifolds and is one of the advantages of working with this formalism since such calculations are purely algebraic. Using the representation OfLA, eq. (42) can be rewritten as o ) ( L . , LB ) = ~

( -- 1 ) o [facDG)fnDCG) __ ( __ 1 ) ASfBcD(f)fAoCG) ] .

(43)

C,.~O D>O

Substituting eq. (41 ) into eq. (43), and taking care that the indices a can either be integers (in the R sector) or ½integers (in the NS sector), we obtain after a straightforward calculation o)(LA, LB)R=[--O'~--~)m36m+n.O 0

0 ] --40"-- l)°t2d~+#,o '

~o(L., L o ) ~ = [ [ - ( ] - - I ) m 3 + 'gml&.+..o

0

0 ] [ - 4 ( . / - ¼)a2+ I ]a.+a,o "

(44) (45)

The simplicity of the above computation should be contrasted with the complicated one in ref. [ 16 ] where the same answer is obtained for the e a s e l = 3. If we have instead a scalar superficial X u (z, O) on the Fock space of the vector bundle on super D i f f S t / S l, then the curvature of the line bundle can be calculated using the representation of the Virasoro algebra for the superstring field X u [ 1O, 11 ] o~(LA, L8 ) = [S(LA ), S ( L B ) ] - S ( [LA, L8 ] ) .

(46)

Computing the above quantity will give the central extension of the Virasoro algebra which is [ 14 ] og(LA,L.)=I[~d(m3-m)-2am]~m+~,o 0

0 ] [½d(a2-1)-ZalO~,+p,o '

where the terms proportional to 2a arise by redefining the Lo generator by [ 10 ] 442

(47)

Volume 206, number 3

PHYSICS LETTERS B

26 May 1988

L~ = L 0 + l a . O b v i o u s l y , b y t a k i n g a p r o d u c t b u n d l e o f t h e s u p e r g h o s t s y s t e m w i t h j = 3 a n d t h e X ~ s u p e r f i e l d w i t h d = 10, t h e n t h e c u r v a t u r e o f t h e p r o d u c t b u n d l e v a n i s h e s i f a = - ~ i n t h e R s e c t o r a n d a = ¼ i n t h e N S sector. W e c o n c l u d e b y s a y i n g t h a t m a n y p r o b l e m s stay t o b e i n v e s t i g a t e d , s u c h as t h e g e n e r a l i z i n g o f t h e K r i c h e v e r m a p s f r o m t h e s u p e r m o d u l i s p a c e to t h e g r a d e d g r a s s m a n n i a n s a n d t h e t r u e n a t u r e o f t h e S K P h i e r a r c h y . T h i s a n d r e l a t e d issues are u n d e r i n v e s t i g a t i o n .

O n e o f us ( A . C . ) w o u l d like to t h a n k C. G o m e z for m a n y s t i m u l a t i n g d i s c u s s i o n s .

References [ 1 ] D. Friedan and S, Shenker, Phys. gett. B 175 (1986) 287; Nucl. Phys. B 281 (1987) 509. [2] M. Bowick and B. Rajeev, Phys. Rev. Lett. 58 (1987) 535; Nucl. Phys. B 293 (1987) 348. [3] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, J. Phys. Soc. Japan 50 ( 1981 ) 3806. [4] G. Segal and G. Wilson, Publ. IHES 61 (1985) 1; A. Presseley and G. Segal, Loop groups (Oxford, U.P., Oxford, 1986). [ 5 ] S. Saito, Phys. Rev. Lett. 59 ( 1987 ) 1798; Phys. Rev. D 36 ( 1987 ) 1819. [6] N. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Len. A 2 (1987) 119. [7] L. Alvarez-Gaum6, C. Gomez and C. Reina, Phys. Lett. B 190 (1987) 55; CERN preprint TH.4775/87 ( 1987); L. Alvarez-Gaum6, C. Gomez, G. Moore and C. Vafa, CERN preprint TH.4883/87 ( 1987 ). [8] C. Vafa, Phys. Lett. B 190 (1987) 47; T. Jayaraman and K.S. Narain, Rutherford preprint RAL-87-l 11 ( 1987 ). [9] E. Witten, Commun. Math. Phys. 113 (1988) 529. [ 10] J. Mickelsson, Commun. Math. Phys. 112 (1987) 653. K. Pilch and N.P. Warner, MIT reprint CTP No. 1457 (1987). [ 11 ] M. Awada and A. Chamseddine, ETH preprint 87/3 (1987). [ 12 ] Yu.I. Manin and A.O. Radul, Commun. Math. Phys. 98 ( 1985 ) 65. [ 13] K. Ueno and H. Yamada, RIMS-Kokyuroku 554 (1985) 91; Lett. Math. Phys. 13 (1987) 59. [ 14 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 15 ] H. Kanno, N. Nishimira and A. Tamekiyo, Kyoto University preprint KUNS 902 ( 1987 ). [ 16] D. Harari, D. Hong, P. Ramond and V. Rodgers, Nucl. Phys. B 294 (1987) 556.

443