- Email: [email protected]
The Krichever-Novikov operator formalism of the superstring
Volume 216, number 1,2
PHYSICS LETTERS B
5 January 1989
T H E K R I C H E V E R - N O V I K O V O P E R A T O R F O R M A L I S M OF T H E S U P E R S T R I N G
De-Hai ZHANG Center of Theoretical Physics, CCAST (WorM Laboratory), Beijing, P.R. China Institute of Theoretical Physics, Peking University, Beijing, P.R. China and Institute of Theoretical Physics, Academia Sinica, Beijing, P.R. China Received 23 August 1988
The Krichever-Novikov operator formalism on the high genus Riemann surface is used to construct an infinite number of conservation charges and a high-genus vacuum state. The calculation of the superconformal ghost is developed in this paper.
The superstring theory, being regarded as one of the best candidates unifying the electroweak, strong and gravitation interactions, has been researched extensively [ 1 ]. Because non-perturbative theories are not understood presently, these investigations can only be done at the perturbative level [2 ]. One of the tasks of the perturbative research is to calculate the scattering amplitudes of various perturbative orders [ 3]. There are two effective ways to deal with this problem, the path integral [4] and the operator method [ 5 ]. Alvarez-Gaum6 et al. developed the operator formalism of the string and superstring [ 6 ]. On the other hand, Krichever and Novikov presented the operator algebras on the high-genus Riemann surface [ 7 ]. The purpose of this paper is to discuss the operator method in the KricheverNovikov formalism, and to develop the calculation of the boson ghost system. Suppose Z is a Riemann surface of genus g. A conformal field system b(z) and c(z) on E has the weights 2 and 1 - 2 respectively. A whole family of basesf},2)(z) of b(z) considered by Krichever and Novikov can be expanded in the neighborhood of a point P as follows: f~,:°(z)=
~
z "+ .... "s'~a)A},~,)~,
(1)
111=0
(and supermoduli) parameters of Z, and have the convention A ~J = 1. It is known from the RiemannRoch theorem that when n ~ < - S ( 2 ) , ~ a ) ( z ) are holomorphic offP and are marked as f/~_)). (z). When n >/S( 1 - 2 ), f ~2)(z) are meromorphic on Z - P and are denoted as f / ~ ) , ( z ) . The field b(z) can be expanded as b(z)=
+
Z
n~<-S(2)
E
b~-'fl~-').(z)
b~+)fla+)).(z) •
(2)
n~>S(l--2)
When genus g = 0, the above formula degenerates to b(z)=
bnz n-2.
~
(3)
n = --oo
For in the same reason for the field c(z) one has c(z)=
~
=
E
cnz "-1+2 - ( - ) r tjJ~- _2 )) . c.
(4) (z)
n~< - - S ( 1 - - 4 )
+
~
n>~S(2)
6~+)flJ+-~)(z).
(5)
We choose the convention that fl~2)(z) f }d-~)(z) are dual to each other:
and
~ f ~ ) ( z ) f } l - 2 ) ( z ) dz=O,+ ....
(6)
here S(2) = ½g-2(g- 1). A ..... depend on the moduli Permanent address.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
107
Volume 216, number 1,2
PHYSICS LETTERS B
According to the above expansion expressions one can derive the relation between bn(6,,) and bn (cn), b ...... + ; . - s t ~ ........
5 January 1989
One has F< + I -- >F= 1, F < 0 1 0 > F = 1. The definition of the internal product of the fermion system is
(7a)
m=O
6,,= ~
I~l=O
n = --co
~a) .". . . . . . a+st~ A ........
(7b)
These operators are just the Krichever-Novikov ones. The key point of the operator method considered by Alvarez-Gaum6 et al. is to find an infinite number of conservation charges constructed from the holomorphic sections off P and then to define the highgenus vacuum state by using these conservation charges. In our formalism the conservation charges are defined as Q~,~) =
d
'-~ b(z)fl_~_,,dz=b~ +),
n>~S(2) . (8b)
The high-genus vacuum state is defined as Q~,a~lO>=0, Q} ~-a~105 = 0 ,
[7.,fl,.]=g.+m.
(16)
One introduces the c o m m o n numbers x. satisfying x~=x_n and its derivative O.=O/Ox.. The algebra (16) can be realized in the Gauss representation
7 ~ = ( 1 / x ~ ) ( x . WiOn) , fln=(--i/x~)(X_~--iO_n).
(17)
The single eigenfunctions can be obtained as
n>~S(1-2), (8a)
Q~'-a)= ~c(z)fl~_))_, d z = 6 ~ +),
When b-c is a commuting system and adapted as fl-7, one has
y.¢n = 0 ,
(9a)
n~>S(2).
(9b)
(18)
~ , = e x p ( - - ~ lix 2- n J .
(19)
and
p.¢" =0,
n>~S(l-2),
¢~n=exP(½ix 2) ,
One is able to derive the action of the g-function on these eigenfunctions,
In order to write its explicit expression represented
g ( y . ) e x p ( - ~lx,,) l' 2 = (21ri) -W2exp(½ix~) ,
by the zero-genus operators, one must define some kind o f naive v a c u u m first.
and
When b-c is an anti-commuting system, one has
[b,,,c,,,]+ =g,,+ ....
(10)
(20a)
g(fl,,)exP( ½ix~.) = ( 2~zi) - ~/2exp ( - ½ix 2_. ) .
( 20b )
Therefore one regards c,, as the anti-commuting numbers satisfying c,,* -c_,,,- and takes b,,=O/Oc_.. If 2 is a half-integer for the Neveu-Schwarz sectors, one has the definition of the naive vacuum,
Using the Gauss representation one defines the naive vacuum state for a half-integer 2,
b,,lO>F=c,,iO>v=O,
This vacuum is expressed as
n>~½.
(11)
n>~½.
(12)
X (... "~-X23/2"~X2/2 --X~l/2 --X2-3/2 --... As for an integer 2, one has
b,,I->v=c,,l+>v=O,
E,I->B=~.I+>n=0,
(13)
)]
.
(22)
When 2 is an integer, then n is an integer, and the naive v a c u u m is defined as
n>~O.
(21)
10>s=exP[½i
This vacuum can be expressed as
I0 >v =cl/2"c3/2"c5/2 ....
fin 10>a =Y,, I0>B = 0 ,
n>~0,
(23)
and
It can be expressed as
I + >v=Co'Cj "c2... ,
(14a)
I->v=bol+>F.
(14b)
I + > n = g ( 9 ' o ) [ - >B 9
"Y
=exp[½i(... +x~ +x~-x-_~ -x-_2 -... ) 1 , (24a)
108
= (l/,fi)(y.
I - >~ = 5 ( # o ) I + >~
=exp[½i(...
5 January 1989
PHYSICS LETTERS B
Volume 216, number 1,2
~
21
"31-X~dl-X
- - X ~ ) - -")X _ _
2
I
--,,. )] ,
(24b) and
=
~
(A k .... x . . . . ~+s(2)
n= -oo
+A 5~!,. i0 . . . . ~+s(2) ) ,
.(+I-)B=I,
B(010)B=I ,
(25)
where the internal product of the boson system is defined as the integral of the commuting numbers, i
.(0, 102).=
,,=I~I _o~ -dx,, ~
I0, )~'102)..
where A'- - . m , =3,.o , + A (+) --n,m A -(-+) --½A (2> -n,m ---n,m~ __ 2~Zn
=0,
1O> ~ = as(k)%-(2) +,
(27)
"Cs(2)+2 ....
If one uses the naive vacuum state of the fermion system, I0) v can be rewritten in another form,
,
m>0
+!A(~-,~) --m ,
(26)
--oo
With this the high-genus vacuum state can be expressed simply in the Krichever-Novikov operator formalism. For b-c of the fermion system, the highgenus vacuum state is
(30)
m
m=0.
(31)
Using them one obtains the high-genus vacuum state of the boson system, I~)B--exP[½i(... +Xs(2)+, ~2 +Yc~(~) - X~s2 ( 2 ) - i -
,~2
s(a)-z-... ) ] .
If one uses the naive vacuum state of the boson system, I~) B can be rewritten in another form, [- i
2-- 1/2--S(2)
I0)B = A-- I / 2 - - S ( 2 )
=
I]
n=S(2)
1--I
~
n = 2 + 1/2--S(2) m= 1
b_,+;-s(2) x
(-
n = 2 + 1/2--S(2) in~
............. ~+s(2)) 10)v, XA ('~) c
H
n=S(2)
g, exp -
n=2--S(A) m= I
XA ~) - - n , m c n-- m - - 2 + ~s( k ) ) l + ) v ,
b_n+2_s(~)
2eZ
~-
IO>, = (28)
x(
--non
l-s(z)
l-I
n=S(2)
2eZh, F
i
6(9,) exPL
-
H--n+2--S(3.)
A -'2' -n,m
n=2-S(2) t?l= l
X~ ).....
,
2+S(~))] I0)B,
--
When b-c is a boson system adapted by fl-7, one has /Jn-- m + 2 - - S ( 2 )
]
2eZh,
xy . . . . . 10)v =
5(9.) exp[
n=S(2)
(
~,, exp -
(32)
2+S(2))]
I JI- > B ,
~,~7/.
(33)
m=O
n~>S(1-2), ~(+)-
(29a)
~ 7....... 2+s(~)A~)..,.
m=O
n>~S(2).
(29b)
We can introduce the Krichever-Novikov operators in the Gauss representation,
The expression (33) is too complicated to calculate explicitly due to a reason we will learn from a later discussion. The best way is to start from the more simple expression (32), and to separate out the derivatives in the exponent. In order to achieve this goal, one rewrites the expression (32) as
I¢ )B =exp( G+ xT Fx--xTHO+OTKO ) ,
(34)
109
Volume 216, number 1,2
PHYSICS LETTERS B
G=-
where G is a n u m b e r
G=
'(
G,,,,n=S(2)
G,-,,, n=-S(2)+
)
~ ( G,+, (n+ + ,,=o 2)------~
,
(35)
1
p=_
~ (F.+,
~
2(n+_l)tr(FK,)~ (n+2)! / 2H.F
,,=o \ ( n + 2 ) !
and F, H and K are infinite dimensional matrices,
xTFx =
5 January 1989
4FK.K
+ n!(n+2------~+ n [ ( n + 3 ) J
x-,,,-z+s(R)f ,,,.IX-1-z+s(2) ,
I Q = - ~ ( . H"+-! 4FK, ,=o \ ( n + 2 ) ! + n ! ( n + 2 ) J '
x-,,,_x+s(a)H,,,./O_/_a+.s.(a),
K'=-~ K,,+, ,,=o (~--2-) ! "
'
-oo
hi,l=
~
XTHO=
(41)
/iLl= -oo
~
0TK0= hi,l=
0 ..... a+s(x) K,,,jO_l_a+s(a) •
(36)
Moreover, G., F., 11. and K. can be obtained by recurrence relations,
- - ~
G. = 2 tr(F._~K-FK._~ ) ,
Further,
F.=2(HF,,_I - H . _ I F ) , F,,, •/ = ~
* ,,,./
1" m . I
--
H . = 4 ( F . _ I K - F K . _ ~ ) + [H, H._,] ,
n = --S(~.) + ]
n =.S'(A)
H,,,./= ~
I
Kn=2(Kgn_l - K n _ ~ g ) ,
.. ,,,j n=S(2)
K,,,j=
n=-S(2)+
XX ltl, I
I
~
--
\n=S(2)
and for the zero order one has K~,,9")
n=--S(2)+
(37)
Go=0,
Fo=-F,
Ho=-H',
Ko=0.
(43)
I
From ( 4 1 ) - ( 4 3 ) one learns that R is a function of H ' . In order to realize the separation (40), one must choose
here G ('')-
~
--
iii=
iA (+) A'- - n . t l + m --n.n+m
,
K(H')=0.
-oo
~iA
..../ = K(.)
" (+) - - t L t / + / H A' - - t l . n + [ + i A ' --?1,t1+1~1 a f a ( -+ - n). n T I
--
a(+),11 -I- n(+) n,ll ~ x
_
~i
ii I ~
_
+
I
•
(44)
Therefore H ' is the solution ofeq. (44). Using (40), one can have a further separation
F~,'/.] = A '_,.,, +,,,A '_,,.,, +/, H(") ttl.I
(42)
,
(38)
We use the following formula to separate the derivative in the exponent,
exp(xVffx+xTITO) =exp(xTffx)exp(xTITO),
(45)
where i f = ~ (2/4)" .=o ( n - ~ ! if"
(46)
1
Similarl),, one has also
ele//= e x p ( f (A - eZ/eaBAe- 2ae-'~/) dj.) 0
Xe "l+u .
exp (xVFx + xTH ' 0 ) (39)
=exp ( xT F" x )exp ( xT H' O) ,
At the first step, the expression (34) can be separated as
where
exp ( G + x TFx + X THO + 0TK0 )
F"=
= e % x p (x TFx + X TH' 0 ) exp ( (~ + x VFx + x v/~0 )
× e x p [ x T ( H - H ' )0+0TK0] , where 110
(40)
~ (2H')" ,=o ( n + 1)! F .
(47)
(48)
One can derive
exp ( xV H'O )exp ( x r ~x )exp ( -- xT H' O) = exp (xXe - " Fx) .
(49)
Volume 216, number 1,2
PHYSICS LETTERS B
Taking (47), (52), (54) and (56) together, one gets 10) n = e x p ( G +
G+xrF"x+xVe 2" Fx)
5 January 1989
As for the scattering amplitudes, we will need studying the case o f many punctures, which is left to be researched in another paper.
× exp (xTH'O)exp (XT/t0)
×exp[xr(H--H ' )0 + 0TK0 ] • 1 =exp[G+G+xr(F " +e-" F)x] ,
(50)
which is the result we desire. The high-genus v a c u u m state needed by the superstring consists of two parts, the ghost system with 2 = 2 and the super-ghost system with 2 = 3,
I W) = 1~(,~=3/2) ) a ~ I~(,~=2) ) F .
(51)
The measure o f the superstring is 2g--3/2
F(--IF(01
3g-2
b,,IW).
1-] ~(fl-) 1-I n= I/2
m=
(52)
I
The action of b,,, on the latter state is simple. The following formula can be used to calculate the action of ~ ( 7 ( S ) ) on the general states: (2hi)-'/2 8 ( y ( S ) )exp(½ixrMx) × e x p [ ~ix v ( M +
I would like to thank Huang Chao-Shang, Guo HanYing and Song Xing-Chang for a helpful discussion. This research is supported by the Natural Science Foundation o f China.
References [1] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987). [2] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 3 ] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357. [4] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 211. [ 5 ] L. Alvarez-Gaum6,C. Gomez, G. Moore and C. Vafa, CERN preprint CERN TH-4883. [6] L. Alvarez-Gaum6, C. Gomez, P. Nelson, G. Sierra and C. Vafa, CERN preprint CERN TH-5018. [ 7 ] I.M. Krichever and S.P. Novikov, Funk. Anal. Pril 21 ( 1987) 46.
= x/ ½sr ( 1 - M ) S
( 1 - M ) S ' S T ( -1M ) ST(I_M)S
)x I "
(53)
111
PHYSICS LETTERS B
5 January 1989
T H E K R I C H E V E R - N O V I K O V O P E R A T O R F O R M A L I S M OF T H E S U P E R S T R I N G
De-Hai ZHANG Center of Theoretical Physics, CCAST (WorM Laboratory), Beijing, P.R. China Institute of Theoretical Physics, Peking University, Beijing, P.R. China and Institute of Theoretical Physics, Academia Sinica, Beijing, P.R. China Received 23 August 1988
The Krichever-Novikov operator formalism on the high genus Riemann surface is used to construct an infinite number of conservation charges and a high-genus vacuum state. The calculation of the superconformal ghost is developed in this paper.
The superstring theory, being regarded as one of the best candidates unifying the electroweak, strong and gravitation interactions, has been researched extensively [ 1 ]. Because non-perturbative theories are not understood presently, these investigations can only be done at the perturbative level [2 ]. One of the tasks of the perturbative research is to calculate the scattering amplitudes of various perturbative orders [ 3]. There are two effective ways to deal with this problem, the path integral [4] and the operator method [ 5 ]. Alvarez-Gaum6 et al. developed the operator formalism of the string and superstring [ 6 ]. On the other hand, Krichever and Novikov presented the operator algebras on the high-genus Riemann surface [ 7 ]. The purpose of this paper is to discuss the operator method in the KricheverNovikov formalism, and to develop the calculation of the boson ghost system. Suppose Z is a Riemann surface of genus g. A conformal field system b(z) and c(z) on E has the weights 2 and 1 - 2 respectively. A whole family of basesf},2)(z) of b(z) considered by Krichever and Novikov can be expanded in the neighborhood of a point P as follows: f~,:°(z)=
~
z "+ .... "s'~a)A},~,)~,
(1)
111=0
(and supermoduli) parameters of Z, and have the convention A ~J = 1. It is known from the RiemannRoch theorem that when n ~ < - S ( 2 ) , ~ a ) ( z ) are holomorphic offP and are marked as f/~_)). (z). When n >/S( 1 - 2 ), f ~2)(z) are meromorphic on Z - P and are denoted as f / ~ ) , ( z ) . The field b(z) can be expanded as b(z)=
+
Z
n~<-S(2)
E
b~-'fl~-').(z)
b~+)fla+)).(z) •
(2)
n~>S(l--2)
When genus g = 0, the above formula degenerates to b(z)=
bnz n-2.
~
(3)
n = --oo
For in the same reason for the field c(z) one has c(z)=
~
=
E
cnz "-1+2 - ( - ) r tjJ~- _2 )) . c.
(4) (z)
n~< - - S ( 1 - - 4 )
+
~
n>~S(2)
6~+)flJ+-~)(z).
(5)
We choose the convention that fl~2)(z) f }d-~)(z) are dual to each other:
and
~ f ~ ) ( z ) f } l - 2 ) ( z ) dz=O,+ ....
(6)
here S(2) = ½g-2(g- 1). A ..... depend on the moduli Permanent address.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
107
Volume 216, number 1,2
PHYSICS LETTERS B
According to the above expansion expressions one can derive the relation between bn(6,,) and bn (cn), b ...... + ; . - s t ~ ........
5 January 1989
One has F< + I -- >F= 1, F < 0 1 0 > F = 1. The definition of the internal product of the fermion system is
(7a)
m=O
6,,= ~
I~l=O
n = --co
~a) .". . . . . . a+st~ A ........
(7b)
These operators are just the Krichever-Novikov ones. The key point of the operator method considered by Alvarez-Gaum6 et al. is to find an infinite number of conservation charges constructed from the holomorphic sections off P and then to define the highgenus vacuum state by using these conservation charges. In our formalism the conservation charges are defined as Q~,~) =
d
'-~ b(z)fl_~_,,dz=b~ +),
n>~S(2) . (8b)
The high-genus vacuum state is defined as Q~,a~lO>=0, Q} ~-a~105 = 0 ,
[7.,fl,.]=g.+m.
(16)
One introduces the c o m m o n numbers x. satisfying x~=x_n and its derivative O.=O/Ox.. The algebra (16) can be realized in the Gauss representation
7 ~ = ( 1 / x ~ ) ( x . WiOn) , fln=(--i/x~)(X_~--iO_n).
(17)
The single eigenfunctions can be obtained as
n>~S(1-2), (8a)
Q~'-a)= ~c(z)fl~_))_, d z = 6 ~ +),
When b-c is a commuting system and adapted as fl-7, one has
y.¢n = 0 ,
(9a)
n~>S(2).
(9b)
(18)
~ , = e x p ( - - ~ lix 2- n J .
(19)
and
p.¢" =0,
n>~S(l-2),
¢~n=exP(½ix 2) ,
One is able to derive the action of the g-function on these eigenfunctions,
In order to write its explicit expression represented
g ( y . ) e x p ( - ~lx,,) l' 2 = (21ri) -W2exp(½ix~) ,
by the zero-genus operators, one must define some kind o f naive v a c u u m first.
and
When b-c is an anti-commuting system, one has
[b,,,c,,,]+ =g,,+ ....
(10)
(20a)
g(fl,,)exP( ½ix~.) = ( 2~zi) - ~/2exp ( - ½ix 2_. ) .
( 20b )
Therefore one regards c,, as the anti-commuting numbers satisfying c,,* -c_,,,- and takes b,,=O/Oc_.. If 2 is a half-integer for the Neveu-Schwarz sectors, one has the definition of the naive vacuum,
Using the Gauss representation one defines the naive vacuum state for a half-integer 2,
b,,lO>F=c,,iO>v=O,
This vacuum is expressed as
n>~½.
(11)
n>~½.
(12)
X (... "~-X23/2"~X2/2 --X~l/2 --X2-3/2 --... As for an integer 2, one has
b,,I->v=c,,l+>v=O,
E,I->B=~.I+>n=0,
(13)
)]
.
(22)
When 2 is an integer, then n is an integer, and the naive v a c u u m is defined as
n>~O.
(21)
10>s=exP[½i
This vacuum can be expressed as
I0 >v =cl/2"c3/2"c5/2 ....
fin 10>a =Y,, I0>B = 0 ,
n>~0,
(23)
and
It can be expressed as
I + >v=Co'Cj "c2... ,
(14a)
I->v=bol+>F.
(14b)
I + > n = g ( 9 ' o ) [ - >B 9
"Y
=exp[½i(... +x~ +x~-x-_~ -x-_2 -... ) 1 , (24a)
108
= (l/,fi)(y.
I - >~ = 5 ( # o ) I + >~
=exp[½i(...
5 January 1989
PHYSICS LETTERS B
Volume 216, number 1,2
~
21
"31-X~dl-X
- - X ~ ) - -")X _ _
2
I
--,,. )] ,
(24b) and
=
~
(A k .... x . . . . ~+s(2)
n= -oo
+A 5~!,. i0 . . . . ~+s(2) ) ,
.(+I-)B=I,
B(010)B=I ,
(25)
where the internal product of the boson system is defined as the integral of the commuting numbers, i
.(0, 102).=
,,=I~I _o~ -dx,, ~
I0, )~'102)..
where A'- - . m , =3,.o , + A (+) --n,m A -(-+) --½A (2> -n,m ---n,m~ __ 2~Zn
=0,
1O> ~ = as(k)%-(2) +,
(27)
"Cs(2)+2 ....
If one uses the naive vacuum state of the fermion system, I0) v can be rewritten in another form,
,
m>0
+!A(~-,~) --m ,
(26)
--oo
With this the high-genus vacuum state can be expressed simply in the Krichever-Novikov operator formalism. For b-c of the fermion system, the highgenus vacuum state is
(30)
m
m=0.
(31)
Using them one obtains the high-genus vacuum state of the boson system, I~)B--exP[½i(... +Xs(2)+, ~2 +Yc~(~) - X~s2 ( 2 ) - i -
,~2
s(a)-z-... ) ] .
If one uses the naive vacuum state of the boson system, I~) B can be rewritten in another form, [- i
2-- 1/2--S(2)
I0)B = A-- I / 2 - - S ( 2 )
=
I]
n=S(2)
1--I
~
n = 2 + 1/2--S(2) m= 1
b_,+;-s(2) x
(-
n = 2 + 1/2--S(2) in~
............. ~+s(2)) 10)v, XA ('~) c
H
n=S(2)
g, exp -
n=2--S(A) m= I
XA ~) - - n , m c n-- m - - 2 + ~s( k ) ) l + ) v ,
b_n+2_s(~)
2eZ
~-
IO>, = (28)
x(
--non
l-s(z)
l-I
n=S(2)
2eZh, F
i
6(9,) exPL
-
H--n+2--S(3.)
A -'2' -n,m
n=2-S(2) t?l= l
X~ ).....
,
2+S(~))] I0)B,
--
When b-c is a boson system adapted by fl-7, one has /Jn-- m + 2 - - S ( 2 )
]
2eZh,
xy . . . . . 10)v =
5(9.) exp[
n=S(2)
(
~,, exp -
(32)
2+S(2))]
I JI- > B ,
~,~7/.
(33)
m=O
n~>S(1-2), ~(+)-
(29a)
~ 7....... 2+s(~)A~)..,.
m=O
n>~S(2).
(29b)
We can introduce the Krichever-Novikov operators in the Gauss representation,
The expression (33) is too complicated to calculate explicitly due to a reason we will learn from a later discussion. The best way is to start from the more simple expression (32), and to separate out the derivatives in the exponent. In order to achieve this goal, one rewrites the expression (32) as
I¢ )B =exp( G+ xT Fx--xTHO+OTKO ) ,
(34)
109
Volume 216, number 1,2
PHYSICS LETTERS B
G=-
where G is a n u m b e r
G=
'(
G,,,,n=S(2)
G,-,,, n=-S(2)+
)
~ ( G,+, (n+ + ,,=o 2)------~
,
(35)
1
p=_
~ (F.+,
~
2(n+_l)tr(FK,)~ (n+2)! / 2H.F
,,=o \ ( n + 2 ) !
and F, H and K are infinite dimensional matrices,
xTFx =
5 January 1989
4FK.K
+ n!(n+2------~+ n [ ( n + 3 ) J
x-,,,-z+s(R)f ,,,.IX-1-z+s(2) ,
I Q = - ~ ( . H"+-! 4FK, ,=o \ ( n + 2 ) ! + n ! ( n + 2 ) J '
x-,,,_x+s(a)H,,,./O_/_a+.s.(a),
K'=-~ K,,+, ,,=o (~--2-) ! "
'
-oo
hi,l=
~
XTHO=
(41)
/iLl= -oo
~
0TK0= hi,l=
0 ..... a+s(x) K,,,jO_l_a+s(a) •
(36)
Moreover, G., F., 11. and K. can be obtained by recurrence relations,
- - ~
G. = 2 tr(F._~K-FK._~ ) ,
Further,
F.=2(HF,,_I - H . _ I F ) , F,,, •/ = ~
* ,,,./
1" m . I
--
H . = 4 ( F . _ I K - F K . _ ~ ) + [H, H._,] ,
n = --S(~.) + ]
n =.S'(A)
H,,,./= ~
I
Kn=2(Kgn_l - K n _ ~ g ) ,
.. ,,,j n=S(2)
K,,,j=
n=-S(2)+
XX ltl, I
I
~
--
\n=S(2)
and for the zero order one has K~,,9")
n=--S(2)+
(37)
Go=0,
Fo=-F,
Ho=-H',
Ko=0.
(43)
I
From ( 4 1 ) - ( 4 3 ) one learns that R is a function of H ' . In order to realize the separation (40), one must choose
here G ('')-
~
--
iii=
iA (+) A'- - n . t l + m --n.n+m
,
K(H')=0.
-oo
~iA
..../ = K(.)
" (+) - - t L t / + / H A' - - t l . n + [ + i A ' --?1,t1+1~1 a f a ( -+ - n). n T I
--
a(+),11 -I- n(+) n,ll ~ x
_
~i
ii I ~
_
+
I
•
(44)
Therefore H ' is the solution ofeq. (44). Using (40), one can have a further separation
F~,'/.] = A '_,.,, +,,,A '_,,.,, +/, H(") ttl.I
(42)
,
(38)
We use the following formula to separate the derivative in the exponent,
exp(xVffx+xTITO) =exp(xTffx)exp(xTITO),
(45)
where i f = ~ (2/4)" .=o ( n - ~ ! if"
(46)
1
Similarl),, one has also
ele//= e x p ( f (A - eZ/eaBAe- 2ae-'~/) dj.) 0
Xe "l+u .
exp (xVFx + xTH ' 0 ) (39)
=exp ( xT F" x )exp ( xT H' O) ,
At the first step, the expression (34) can be separated as
where
exp ( G + x TFx + X THO + 0TK0 )
F"=
= e % x p (x TFx + X TH' 0 ) exp ( (~ + x VFx + x v/~0 )
× e x p [ x T ( H - H ' )0+0TK0] , where 110
(40)
~ (2H')" ,=o ( n + 1)! F .
(47)
(48)
One can derive
exp ( xV H'O )exp ( x r ~x )exp ( -- xT H' O) = exp (xXe - " Fx) .
(49)
Volume 216, number 1,2
PHYSICS LETTERS B
Taking (47), (52), (54) and (56) together, one gets 10) n = e x p ( G +
G+xrF"x+xVe 2" Fx)
5 January 1989
As for the scattering amplitudes, we will need studying the case o f many punctures, which is left to be researched in another paper.
× exp (xTH'O)exp (XT/t0)
×exp[xr(H--H ' )0 + 0TK0 ] • 1 =exp[G+G+xr(F " +e-" F)x] ,
(50)
which is the result we desire. The high-genus v a c u u m state needed by the superstring consists of two parts, the ghost system with 2 = 2 and the super-ghost system with 2 = 3,
I W) = 1~(,~=3/2) ) a ~ I~(,~=2) ) F .
(51)
The measure o f the superstring is 2g--3/2
F(--IF(01
3g-2
b,,IW).
1-] ~(fl-) 1-I n= I/2
m=
(52)
I
The action of b,,, on the latter state is simple. The following formula can be used to calculate the action of ~ ( 7 ( S ) ) on the general states: (2hi)-'/2 8 ( y ( S ) )exp(½ixrMx) × e x p [ ~ix v ( M +
I would like to thank Huang Chao-Shang, Guo HanYing and Song Xing-Chang for a helpful discussion. This research is supported by the Natural Science Foundation o f China.
References [1] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987). [2] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 3 ] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357. [4] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 211. [ 5 ] L. Alvarez-Gaum6,C. Gomez, G. Moore and C. Vafa, CERN preprint CERN TH-4883. [6] L. Alvarez-Gaum6, C. Gomez, P. Nelson, G. Sierra and C. Vafa, CERN preprint CERN TH-5018. [ 7 ] I.M. Krichever and S.P. Novikov, Funk. Anal. Pril 21 ( 1987) 46.
= x/ ½sr ( 1 - M ) S
( 1 - M ) S ' S T ( -1M ) ST(I_M)S
)x I "
(53)
111