- Email: [email protected]
Quenched string field theory
Volume 202, number 1
PHYSICS LETTERS B
25 February 1988
Q U E N C H E D STRING FIELD THEORY F. JIMENEZ Universidad Complutense, Ciudad Universitaria, E-28040 Madrid, Spain
and G. SIERRA CERN, CH-1211 Geneva 23, Switzerland Received 26 October 1987
We construct a toy model of W i n e n ' s open string field theory by truncating the Virasoro group to its SU (1,1) subgroup. The string field is given in terms of finite-dimensional matrices, which satisfy a non-trivial BRST cohomology. The string gauge group is found to be SO(3). Finally, we perform the gauge fixing of the action, finding an explicit formula for calculating the partition function.
In Witten's open string field theory [ 1 ], the string field A [ x~'(a), c (0") ] is some sort of"infinite-dimensional matrix", whose entries are labelled by the leftand right-handed pieces of the string. Following this analogy, one can consider the • product as a matrix multiplication and the integration f as the trace operation. In this way we encounter in Witten's theory a resemblance of the objects and operations of the ordinary matrix algebra. String fields seem to behave as matrices but they are much more complicated in their structure. The complexity of the string field is in contrast with the simplicity of the axioms it obeys. These considerations have lead us to construct a model where the content of the string field is drastically reduced, while retaining its algebraic properties. The model can then be called a "quenched string field theory" =~, and it is far from being realistic but it may serve as a toy model of the complete theory. Our starting point is to consider not the entire set of Virasoro operators {L,,, neZ}, but the set {L,, n = 0 , + 1} which generates a SU(1, 1 ) subgroup. These operators can be represented in terms of 2 X 2 matrices as
=~We thank Professor E. Alvarez for this suggestion.
58
10 ~ -- 10"3 p+ =½(i0.1--0"2) ,
p_
=½(i0"1 +0"2),
(1)
which satisfy the "truncated Virasoro algebra" [p,,,pm]=(n-m)p,,+
n, m = 0 , + 1 .
....
(2)
the 0"i are the usual Pauli matrices. Following the standard construction [ 2 ] we introduce the ghost and antighost modes cn and b, (n = 0, +, ), which satisfy the anticommutation relations -
(3)
{b,,. c , . } = 6 , , + , . . o .
the other ones being zero. The ground state for the ghost sea. I0 ) . is defined by the conditions bo 1 0 ) = b + 105 =c+ 1 0 ) = 0 .
(4)
The BRST charge is then given by Q=co(po-1)+c+p_+c
p+
- 2 c _ c+ bo + c o b _ c+ +coc_ b+ ,
(5)
which is normal ordered and satisfies Q2= 0. Similarly, the ghost number operator G reads G=½ [Co, b o ] + c _ b + - b _ c +
.
(6)
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 202, number 1
PHYSICS LETTERS B
25 February 1988
Table 1
IA>l ~ IAo,A+,A_ > , g
IF>2-1Fo, F+,F >, ID>3--- I D > , 10> cob IO> c b 10>
b_lO>
col0>
c_lO> coc I0> coc_b_lO>
(10)
where D , A , and F,, ( n = 0 , + , - ) are complex 2 × 2 matrices. The • product of a zero-form with a p-form (p = 1,2,3) is the straightforward generalization of eq. (9), i.e.
From eqs. (4) it follows that there are eight ghost states which can be classified according to their ghost number in four sectors, corresponding to the values g = -+ 3, -+ ½. They are shown in table 1. These states are the basic ingredients of our construction. Our aim is to show how Witten's axiom for the open bosonic string field theory can be realized in this truncated string. Hence we must find out a • product, a derivation Q and an integral f satisfying the following axioms:
as
(A,B) * C=A, (B, C),
(7a)
Q l e > o - I[po, el, [p+, el, [p_, el>, ,
Q(A,B) = (QA) *B+ ( - )" A , Q B ,
(7b)
Q2 = 0 ,
(7c)
or, in other words, Q acts on zero-forms as a Lie derivative. The rest o f the construction proceeds along the same lines: finding a • product and an action of Q such that the axioms (7a) and (7b) hold. The derivation Q is completely specified by eq. (12 ) and the following definitions:
f QA=O
VA,
f A*B=(-)m¢ f B,A,
(7d) (7e)
where ( - )"~= + 1 ( - 1 ) ifA is even (odd). We also know that the string fields are "differential forms" of degree g + ~, the • product is the "exterior product" and the BRST charge is the "exterior derivative". Table 1 suggests to take as zero-forms the following states: le>o=eb_
[0>,
(8)
[e>o*[Ao,A+,A
>l=[eAo, e A + , e A _ > l ,
[Ao,A+,A >~*[E>o-[Aoe, A+e,A_e>l ,
and similarly f o r p = 2 and 3. A less trivial step is to define the operator Q, which must behave as a derivation of the • product. For a start, we define the action of Q on zero-forms
+{p+,Ao},{Po, A ) + { p _ , A o } ) 2 ,
QIF>2-I[po, Fo]+[p+,F ] + [ p _ , F + ] > 3 QID>3--0.
, (13)
The remaining non-trivial • products are given by
]A>j * iF>2 ~ ]AoFo +A+F_ +A_F+ >3 ,
le, >o* [e2>0 = 1¢1e2 >o •
IF>2*IA>,=-IFoAo+F+A_+F A + > 3 .
This product is associative but non-commutative and it reduces, for zero-forms, to ordinary matrix multiplication. The structure of the one-, two- and three-forms can be easily guessed from table 1 and is chosen to be
(12)
QJA>, =_]{p+,A_ }+ {p_,A+ ), {po,A+ }
where e is a complex 2 × 2 matrix, and the subscript " 0 " denotes the degree of the form. As the zero-forms must be closed under the • multiplication, we define the product (9)
(11)
[A>~,IB>,~IA+B_+A_B+,A+Bo+AoB+,
A_ Bo +AoB >2
,
(14)
The fact that Q is a derivation follows automatically from its definition and does not require any property of the p matrices. The nilpotency of Q imposes the following equations on the p matrices: 59
Volume 202, number i
[{p,,,p,,,}, X] = 0 ,
PHYSICS LETTERS B (15)
n, m = 0, + , - and n ¢ m for any complex 2 X 2 matrix X. Eqs. (15) are equivalent to the conditions
(Ao)t = A o ,
25 February 1988 (A+)t = - A _
.
Moreover, from eqs. (22) and (23), we deduce that the gauge parameter E must satisfy et = _ e ,
{p,,, p,,,} ~ 0 o r ~ ,
(23)
(24)
(16)
which are indeed satisfied for the choice (1) but they do not force the matrices {p,} to generate the SU(1,1 ) subgroup of the Virasoro group. We shall, however, assume this to be the case, in order to maintain some contact with the strings. Finally, we define the integral according to
from which we conclude that the "string gauge group" is SO(3). It is easy to see that the free field equation QA = 0 has three independent solutions which are not pure gauge, i.e., the first cohomology group H a(Q) is isomorphic to R 3. We get for the other cohomology groups:
f Iq/>,,=0,
H°~H3~,
p=0, 1,2,
f ID>3=TrD.
(17)
Then eqs. (7d) and (7e) follow from the identity
Tr AB=Tr BA .
(18)
This ends up our explicit realization of Witten's axioms in an extremely simplified version of the strings in terms of finite-dimensional matrices. We may now proceed to study the dynamics of the quenched string. The action for the string field A = IA >, is given by S=j A,QA+ ~gA,A,A,
(19)
where we have included the gauge coupling constant g. This action is invariant under the gauge transformation
5A= ( 1 / g ) Q e - e , A + A , e ,
(20)
where e - I ~) o. The explicit form of the action (19) is in our model S = Tr( {Po, {A +, A_ }} + {p +, {Ao, A_ }} +{p_~ {Ao,A+}}+g{Ao, {A+,A_ }}) ,
(21)
HI ~H2~
3 .
(25)
Thus duality holds as for the entire string. It is convenient to parametrize the string in the following way:
Ao=-½A~at,,
A+=½(iA~'-A~)~r~,,
(26)
whereA,u ( i = 1,2, 3;/z=0, 1,2, 3) are real and ~rt~= (1, ~r). In these variables, the action ( 21 ) takes the form S = 2[p~-(A ,A ° +A3 A° ) +#2" (A2A° +A3 A°)
+p3.(A,A°+AzA°)] +g [2A]~A~ "A3 + 2 A ° A2"A3 +A°(A?A° +A~.A~ +A°A° +A2..42)] ,
(27)
and the gauge transformation (22) reads 6A ° = 0 ,
8Ai=(1/g)~X(pi+gAi),
i=1,2,3,
(28)
where p] = o a (i, a = 1, 2, 3), corresponding to the choice (1), and represent in some sense the background. We may now define an euclidean string partition function as f dA e x p [ - - S ( A ) ] .
(29)
and the gauge transformation reads
6A,,=(1/g) [p,,+gA,, e] .
(22)
We must further impose a reality on the string field which guarantees the reality of the action. The A,, matrices are taken to satisfy the same hermiticity conditions as the p~ matrices, namely, 60
The action S(A) is gauge invariant so that we must factorize the volume of the gauge group. It is interesting to see how this works in our case. Let us first define for convenience the 3 X 3 matrix
~/-p~, +gA~ ,
(30)
Volume 202, number !
PHYSICS LETTERS B
which is the analogue of the field A o + A (Ao = background field) ofref. [3]. The gauge group acts on the matrix ~ on the left: 0,"--'O;" = R % 0~',
(31)
where R% is a rotation matrix. This means that we can gauge away the orthogonal part of 0, going to the symmetric gauge where 0 is a positive definite symmetric matrix, i.e. 0eGI(3, ~ ) / O ( 3 ) .
(32)
The gauge fixing conditions are therefore
8~j + g ~ij = g z,j ,
(37)
as it is done in ref. [3], but in our case one gets a linear term proportional to (1/g)~ivi. The free classical solutions that follow from eq. (39) (settingg= 0) take the form qJi = 0 ,
{~/11 ~kO
~'a = |gt,2
~/12 00 ) 0 ~//33
-~',,
,
=0~-0 3 ,
Z=_f d~,det('~+lg(tr ~ - q J ) ) e x p [ - S ( ~ , ) ] ,
f3 = - 0 1 + 0 2 ,
where
and the Faddeev-Popov determinant reads
3 dq/= I ] d~ui I ] dg%
d.et ~f.~
i- I
-0~
-0~
01+0 2
=det[(tr ~)~-0] = 8 d e t [ } + ½g(tr A - A ) ] .
(34)
In this gauge, the matrix A~ is symmetric and will be denoted by a new symbol g%. The action (27) takes a simple form in the symmetric gauge if we define the vector
D = N + X~ 2 XGI(3, N ) / O ( 3 ) ,
~/2 - ~ - 2 A z ° J , qJ3 \ A° J
(35)
(41)
where ~+ is the positive real axis. The Faddeev-Popov determinant in (39) takes a simple form if we define the matrix Z,j as Z = ~ + ½g (tr ~ - ~ , ) ,
N=
(40)
i~j
and D is the integration region. D is not simply N 9 since we already know that the matrix ~ +g~u belongs to the coset space (32). Moreover, the convergence of the integral (39) imposes the variable ~u3to be positive, while q/~ and g/z can be either positive or negative. We thus find that
)
-02
-03
(39)
D
(33)
022+0 3 -0½
(38)
which justifies the claim that H ~~ ~73. The partition function of our model is therefore
f, = _ ~ ] + ~ 3 ,
=det
25 February 1988
(42)
which is also positive definite and symmetric. The integral (39) then reads 1
Z= ~~ f d~l dq/2 f dq/3 ~2 ~+
reading S(~u) = 2 ( t r ~u-~u)0 ~u,vj
×
+ g [ ~ v,(djk - v~vk) q/~ ' ~ ' ~ + (tr q/2 _ ~ , 2 ) u v i g t j ] ,
(36)
~
dz det Z exp [ - S( ~, Z) ] ,
G1(3~)/O(3)
(43)
where the action S is in the new variables: where we have introduced the unit vector e = (0, 0, 1). It is worthwhile noticing that the quadratic part in eq. (36) can be absorbed into the cubic part by means of the field redefinition
S = - ( 8 / g ) v,~,, + ~g ~,3(~ ,2 +~,~)
+ (4/g) ( t r x 2 - Z 2+Z t r z - ½(tr Z)2),j vi~'j •
(44) 61
Volume 202, number t
PHYSICS LETTERS B
Eq. (43) can be taken as the starting p o i n t for constructing "string a m p l i t u d e s " . Let us now s u m m a r i z e our results. In this article we have constructed a toy model o f W i t t e n ' s open string field theory by considering the SU(1, I) subgroup o f the Virasoro group. The string field is given in terms o f finite-dimensional matrices which satisfy W i t t e n ' s axioms. We have chosen a representation o f these matrices which leads to a non-trivial cohomology. In this representation, the string gauge group is S O ( 3 ) . One should further investigate the d e p e n d e n c e o f this cohomology on the representation o f these finite-dimensional matrices. We have p e r f o r m e d the gauge fixing o f the action, finding a simple formula for the p a r t i t i o n function. It is not
62
25 February 1988
clear to us whether this toy m o d e l has any physical significance, but it m a y still capture some information about the real string. We would like to thank Professor E. Alvarez, Professor C. G6mez, Professor M. G u n a y d i n , Professor J. R a m i r e z - M i t t e l b r u n n and Professor P.K. Townsend for useful discussions.
References [ 1] E. Witten, Nucl. Phys. B 268 (1986) 253. [2] T. Banks, Proc. Trieste Spring School (1986) (World Scientific, Singapore). [ 3 ] G. Horowflz, J. Lykken, R. Rohm and A. Strominger, Phys. Rev. Lett. 57 (1986) 283.
PHYSICS LETTERS B
25 February 1988
Q U E N C H E D STRING FIELD THEORY F. JIMENEZ Universidad Complutense, Ciudad Universitaria, E-28040 Madrid, Spain
and G. SIERRA CERN, CH-1211 Geneva 23, Switzerland Received 26 October 1987
We construct a toy model of W i n e n ' s open string field theory by truncating the Virasoro group to its SU (1,1) subgroup. The string field is given in terms of finite-dimensional matrices, which satisfy a non-trivial BRST cohomology. The string gauge group is found to be SO(3). Finally, we perform the gauge fixing of the action, finding an explicit formula for calculating the partition function.
In Witten's open string field theory [ 1 ], the string field A [ x~'(a), c (0") ] is some sort of"infinite-dimensional matrix", whose entries are labelled by the leftand right-handed pieces of the string. Following this analogy, one can consider the • product as a matrix multiplication and the integration f as the trace operation. In this way we encounter in Witten's theory a resemblance of the objects and operations of the ordinary matrix algebra. String fields seem to behave as matrices but they are much more complicated in their structure. The complexity of the string field is in contrast with the simplicity of the axioms it obeys. These considerations have lead us to construct a model where the content of the string field is drastically reduced, while retaining its algebraic properties. The model can then be called a "quenched string field theory" =~, and it is far from being realistic but it may serve as a toy model of the complete theory. Our starting point is to consider not the entire set of Virasoro operators {L,,, neZ}, but the set {L,, n = 0 , + 1} which generates a SU(1, 1 ) subgroup. These operators can be represented in terms of 2 X 2 matrices as
=~We thank Professor E. Alvarez for this suggestion.
58
10 ~ -- 10"3 p+ =½(i0.1--0"2) ,
p_
=½(i0"1 +0"2),
(1)
which satisfy the "truncated Virasoro algebra" [p,,,pm]=(n-m)p,,+
n, m = 0 , + 1 .
....
(2)
the 0"i are the usual Pauli matrices. Following the standard construction [ 2 ] we introduce the ghost and antighost modes cn and b, (n = 0, +, ), which satisfy the anticommutation relations -
(3)
{b,,. c , . } = 6 , , + , . . o .
the other ones being zero. The ground state for the ghost sea. I0 ) . is defined by the conditions bo 1 0 ) = b + 105 =c+ 1 0 ) = 0 .
(4)
The BRST charge is then given by Q=co(po-1)+c+p_+c
p+
- 2 c _ c+ bo + c o b _ c+ +coc_ b+ ,
(5)
which is normal ordered and satisfies Q2= 0. Similarly, the ghost number operator G reads G=½ [Co, b o ] + c _ b + - b _ c +
.
(6)
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 202, number 1
PHYSICS LETTERS B
25 February 1988
Table 1
IA>l ~ IAo,A+,A_ > , g
IF>2-1Fo, F+,F >, ID>3--- I D > , 10> cob IO> c b 10>
b_lO>
col0>
c_lO> coc I0> coc_b_lO>
(10)
where D , A , and F,, ( n = 0 , + , - ) are complex 2 × 2 matrices. The • product of a zero-form with a p-form (p = 1,2,3) is the straightforward generalization of eq. (9), i.e.
From eqs. (4) it follows that there are eight ghost states which can be classified according to their ghost number in four sectors, corresponding to the values g = -+ 3, -+ ½. They are shown in table 1. These states are the basic ingredients of our construction. Our aim is to show how Witten's axiom for the open bosonic string field theory can be realized in this truncated string. Hence we must find out a • product, a derivation Q and an integral f satisfying the following axioms:
as
(A,B) * C=A, (B, C),
(7a)
Q l e > o - I[po, el, [p+, el, [p_, el>, ,
Q(A,B) = (QA) *B+ ( - )" A , Q B ,
(7b)
Q2 = 0 ,
(7c)
or, in other words, Q acts on zero-forms as a Lie derivative. The rest o f the construction proceeds along the same lines: finding a • product and an action of Q such that the axioms (7a) and (7b) hold. The derivation Q is completely specified by eq. (12 ) and the following definitions:
f QA=O
VA,
f A*B=(-)m¢ f B,A,
(7d) (7e)
where ( - )"~= + 1 ( - 1 ) ifA is even (odd). We also know that the string fields are "differential forms" of degree g + ~, the • product is the "exterior product" and the BRST charge is the "exterior derivative". Table 1 suggests to take as zero-forms the following states: le>o=eb_
[0>,
(8)
[e>o*[Ao,A+,A
>l=[eAo, e A + , e A _ > l ,
[Ao,A+,A >~*[E>o-[Aoe, A+e,A_e>l ,
and similarly f o r p = 2 and 3. A less trivial step is to define the operator Q, which must behave as a derivation of the • product. For a start, we define the action of Q on zero-forms
+{p+,Ao},{Po, A ) + { p _ , A o } ) 2 ,
QIF>2-I[po, Fo]+[p+,F ] + [ p _ , F + ] > 3 QID>3--0.
, (13)
The remaining non-trivial • products are given by
]A>j * iF>2 ~ ]AoFo +A+F_ +A_F+ >3 ,
le, >o* [e2>0 = 1¢1e2 >o •
IF>2*IA>,=-IFoAo+F+A_+F A + > 3 .
This product is associative but non-commutative and it reduces, for zero-forms, to ordinary matrix multiplication. The structure of the one-, two- and three-forms can be easily guessed from table 1 and is chosen to be
(12)
QJA>, =_]{p+,A_ }+ {p_,A+ ), {po,A+ }
where e is a complex 2 × 2 matrix, and the subscript " 0 " denotes the degree of the form. As the zero-forms must be closed under the • multiplication, we define the product (9)
(11)
[A>~,IB>,~IA+B_+A_B+,A+Bo+AoB+,
A_ Bo +AoB >2
,
(14)
The fact that Q is a derivation follows automatically from its definition and does not require any property of the p matrices. The nilpotency of Q imposes the following equations on the p matrices: 59
Volume 202, number i
[{p,,,p,,,}, X] = 0 ,
PHYSICS LETTERS B (15)
n, m = 0, + , - and n ¢ m for any complex 2 X 2 matrix X. Eqs. (15) are equivalent to the conditions
(Ao)t = A o ,
25 February 1988 (A+)t = - A _
.
Moreover, from eqs. (22) and (23), we deduce that the gauge parameter E must satisfy et = _ e ,
{p,,, p,,,} ~ 0 o r ~ ,
(23)
(24)
(16)
which are indeed satisfied for the choice (1) but they do not force the matrices {p,} to generate the SU(1,1 ) subgroup of the Virasoro group. We shall, however, assume this to be the case, in order to maintain some contact with the strings. Finally, we define the integral according to
from which we conclude that the "string gauge group" is SO(3). It is easy to see that the free field equation QA = 0 has three independent solutions which are not pure gauge, i.e., the first cohomology group H a(Q) is isomorphic to R 3. We get for the other cohomology groups:
f Iq/>,,=0,
H°~H3~,
p=0, 1,2,
f ID>3=TrD.
(17)
Then eqs. (7d) and (7e) follow from the identity
Tr AB=Tr BA .
(18)
This ends up our explicit realization of Witten's axioms in an extremely simplified version of the strings in terms of finite-dimensional matrices. We may now proceed to study the dynamics of the quenched string. The action for the string field A = IA >, is given by S=j A,QA+ ~gA,A,A,
(19)
where we have included the gauge coupling constant g. This action is invariant under the gauge transformation
5A= ( 1 / g ) Q e - e , A + A , e ,
(20)
where e - I ~) o. The explicit form of the action (19) is in our model S = Tr( {Po, {A +, A_ }} + {p +, {Ao, A_ }} +{p_~ {Ao,A+}}+g{Ao, {A+,A_ }}) ,
(21)
HI ~H2~
3 .
(25)
Thus duality holds as for the entire string. It is convenient to parametrize the string in the following way:
Ao=-½A~at,,
A+=½(iA~'-A~)~r~,,
(26)
whereA,u ( i = 1,2, 3;/z=0, 1,2, 3) are real and ~rt~= (1, ~r). In these variables, the action ( 21 ) takes the form S = 2[p~-(A ,A ° +A3 A° ) +#2" (A2A° +A3 A°)
+p3.(A,A°+AzA°)] +g [2A]~A~ "A3 + 2 A ° A2"A3 +A°(A?A° +A~.A~ +A°A° +A2..42)] ,
(27)
and the gauge transformation (22) reads 6A ° = 0 ,
8Ai=(1/g)~X(pi+gAi),
i=1,2,3,
(28)
where p] = o a (i, a = 1, 2, 3), corresponding to the choice (1), and represent in some sense the background. We may now define an euclidean string partition function as f dA e x p [ - - S ( A ) ] .
(29)
and the gauge transformation reads
6A,,=(1/g) [p,,+gA,, e] .
(22)
We must further impose a reality on the string field which guarantees the reality of the action. The A,, matrices are taken to satisfy the same hermiticity conditions as the p~ matrices, namely, 60
The action S(A) is gauge invariant so that we must factorize the volume of the gauge group. It is interesting to see how this works in our case. Let us first define for convenience the 3 X 3 matrix
~/-p~, +gA~ ,
(30)
Volume 202, number !
PHYSICS LETTERS B
which is the analogue of the field A o + A (Ao = background field) ofref. [3]. The gauge group acts on the matrix ~ on the left: 0,"--'O;" = R % 0~',
(31)
where R% is a rotation matrix. This means that we can gauge away the orthogonal part of 0, going to the symmetric gauge where 0 is a positive definite symmetric matrix, i.e. 0eGI(3, ~ ) / O ( 3 ) .
(32)
The gauge fixing conditions are therefore
8~j + g ~ij = g z,j ,
(37)
as it is done in ref. [3], but in our case one gets a linear term proportional to (1/g)~ivi. The free classical solutions that follow from eq. (39) (settingg= 0) take the form qJi = 0 ,
{~/11 ~kO
~'a = |gt,2
~/12 00 ) 0 ~//33
-~',,
,
=0~-0 3 ,
Z=_f d~,det('~+lg(tr ~ - q J ) ) e x p [ - S ( ~ , ) ] ,
f3 = - 0 1 + 0 2 ,
where
and the Faddeev-Popov determinant reads
3 dq/= I ] d~ui I ] dg%
d.et ~f.~
i- I
-0~
-0~
01+0 2
=det[(tr ~)~-0] = 8 d e t [ } + ½g(tr A - A ) ] .
(34)
In this gauge, the matrix A~ is symmetric and will be denoted by a new symbol g%. The action (27) takes a simple form in the symmetric gauge if we define the vector
D = N + X~ 2 XGI(3, N ) / O ( 3 ) ,
~/2 - ~ - 2 A z ° J , qJ3 \ A° J
(35)
(41)
where ~+ is the positive real axis. The Faddeev-Popov determinant in (39) takes a simple form if we define the matrix Z,j as Z = ~ + ½g (tr ~ - ~ , ) ,
N=
(40)
i~j
and D is the integration region. D is not simply N 9 since we already know that the matrix ~ +g~u belongs to the coset space (32). Moreover, the convergence of the integral (39) imposes the variable ~u3to be positive, while q/~ and g/z can be either positive or negative. We thus find that
)
-02
-03
(39)
D
(33)
022+0 3 -0½
(38)
which justifies the claim that H ~~ ~73. The partition function of our model is therefore
f, = _ ~ ] + ~ 3 ,
=det
25 February 1988
(42)
which is also positive definite and symmetric. The integral (39) then reads 1
Z= ~~ f d~l dq/2 f dq/3 ~2 ~+
reading S(~u) = 2 ( t r ~u-~u)0 ~u,vj
×
+ g [ ~ v,(djk - v~vk) q/~ ' ~ ' ~ + (tr q/2 _ ~ , 2 ) u v i g t j ] ,
(36)
~
dz det Z exp [ - S( ~, Z) ] ,
G1(3~)/O(3)
(43)
where the action S is in the new variables: where we have introduced the unit vector e = (0, 0, 1). It is worthwhile noticing that the quadratic part in eq. (36) can be absorbed into the cubic part by means of the field redefinition
S = - ( 8 / g ) v,~,, + ~g ~,3(~ ,2 +~,~)
+ (4/g) ( t r x 2 - Z 2+Z t r z - ½(tr Z)2),j vi~'j •
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Eq. (43) can be taken as the starting p o i n t for constructing "string a m p l i t u d e s " . Let us now s u m m a r i z e our results. In this article we have constructed a toy model o f W i t t e n ' s open string field theory by considering the SU(1, I) subgroup o f the Virasoro group. The string field is given in terms o f finite-dimensional matrices which satisfy W i t t e n ' s axioms. We have chosen a representation o f these matrices which leads to a non-trivial cohomology. In this representation, the string gauge group is S O ( 3 ) . One should further investigate the d e p e n d e n c e o f this cohomology on the representation o f these finite-dimensional matrices. We have p e r f o r m e d the gauge fixing o f the action, finding a simple formula for the p a r t i t i o n function. It is not
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clear to us whether this toy m o d e l has any physical significance, but it m a y still capture some information about the real string. We would like to thank Professor E. Alvarez, Professor C. G6mez, Professor M. G u n a y d i n , Professor J. R a m i r e z - M i t t e l b r u n n and Professor P.K. Townsend for useful discussions.
References [ 1] E. Witten, Nucl. Phys. B 268 (1986) 253. [2] T. Banks, Proc. Trieste Spring School (1986) (World Scientific, Singapore). [ 3 ] G. Horowflz, J. Lykken, R. Rohm and A. Strominger, Phys. Rev. Lett. 57 (1986) 283.