The path integral representation of the fermionic string propagator

Nuclear Physics B312 (1989) 178-196 North-Holland, Amsterdam

THE PATH INTEGRAL REPRESENTATION OF THE FERMIONIC STRING P R O P A G A T O R A.V. MARSHAKOV Department of Theoretical Physics, P.N. Lebedev Physical Institute, Academy of Sciences of the USSR, Leninsky pr.53, Moscow 117924, USSR

Received 20 June 1988

The representation for the propagator of the fermoinic Neveu-Schwarz-Ramond string, as a path integral over two-dimensional surfaces with fixed boundary curves, is proposed. It is shown that in the case of pointlike boundary conditions the result of the computation has a natural interpretation and it can be compared with the result of operator calculation. The ultraviolet and infrared singularities of the propagtor are considered.

1. Introduction Superstring theory is now the most real candidate for the role of the fundamental theory of all interactions. The most developed approach in the string theory is the calculation of scattering amplitudes of states on the mass-shell by means of a sum over R i e m a n n surfaces. In fact, computing the scattering amplitudes we consider two-dimensional surfaces with punctures, where the vertex operators, satisfying the o n mass-shell conformal invariance condition, are put in. However, one can also consider the objects, defined as path integrals over R i e m a n n surfaces with boundaries, to which the string states (in general off-shell states) are attached. One of the simplest objects of this type is a string propagator, which has been originally c o m p u t e d in ref. [1] by means of a sum over surfaces [2]. In spite of the difficulties connected with the definition of a non-invariant object, the expression for a p r o p a g a t o r can be useful for the study of the singularities in string theories and their causality properties and also for the attempts of constructing an effective field theory of strings. Calculation of the bosonic string propagator by means of the Polyakov path integral was performed in refs. [4-9]. It should be pointed out that in refs. [6, 7] the one-loop statistical sum was obtained with the help of the sewing procedure for G r e e n functions. This is connected with the problem of divergences in string theory. T h e most consistent consideration of the bosonic string propagator from the point of view of the string field theory is presented in ref. [9]. However, all those papers 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A. V. Marshakov / Ferrnionic string

[79

are devoted to the bosonic string. The propagator of the heterotic string in the Green-Schwarz formulation was computed in ref. [10]. A calculation of the heterotic string propagator starting from the action with two-dimensional supersymmetry was carried out in ref. [11]. The purpose of this paper is to compute the propagator of a fermionic string in the Neveu-Schwarz-Ramond (NSR) formulation, where the action of a string is the action of two-dimensional supergravity [12]. The path integral over supergravity fields is reduced to a finite-dimensional integral (over even and odd modular parameters) like in the case of a fermionic point particle (see refs. [13, 14]). The path integral is taken about some classical string trajectory, the quantum fluctuations give some contribution to the integration measure by means of corresponding determinants. The principal problem is the choice of the boundary conditions for quantum fields. The paper is organized in the following way. In sect. 2 the integration measure for the open Ramond string is calculated. In sect. 3 the result for pointlike boundary conditions is compared with that of operator computation. Sect. 4 is devoted to the open Neveu-Schwarz string. The generalization to the case of closed strings is given in sect. 5. Sect. 6 contains a discussion of the obtained results and the conclusion.

2. The open Ramond string T o calculate the scattering amplitudes in fermionic string theory one should take the sum over all different boundary conditions, i.e. spin structures [15]. This procedure corresponds to the consideration of equal number of bosonic and fermionic states given by GSO projection [16] and realizes the space-time supersymmetry of N S R strings. However, in the case of the propagator we fix some boundary conditions for fermions on the boundaries of the surface. Therefore we can speak only of computation of propagators in different sectors of the fermionic string. Let us start from the calculation of the propagator of the open Ramond string, which in the limit a'---> 0 corresponds to the fermionic particle ( N = 1 one-dimensional supergravity). The action in the theory of fermionic string is the action of two-dimensional supergravity [12]: 1

S= 2~r~'f drd°v/~[-½g"" O"xMO"xM--½i~MTuo"~M -

lu, v = O , 1 ,

M -

M=I ..... D,

] .

(1)

180

A. V. Marshakov / Fermionic string

invariant under local reparametrizations and local supersymmetry: 8x = i ~ , 8e~ = - 2 t o y a

.-

(1

X,,,

(2)

and also under local Weyl, Lorentz and super-Weyl transformations [18]. The R a m o n d sector of the open fermionic string is singled out by the following b o u n d a r y conditions: <(0,

=

~ ( 1 , r ) = eq,2(1, r )

f

~b=

~b2

e = _+1,

(3)

(when o = 0 and o = 1 the sign in the equalities (3) is the same) and similar expressions are used for the gravitino and the parameter of supersymmetry transformation (2). In the case of Ramond boundary conditions the action (1) in the limit turns to the action of a spin-~ particle. To demonstrate this let us choose in (1) the following gauge: ds 2 = e 2 ( r , o ) d r 2 + do 2 , "/~X. = O.

(4a) (4b)

Then one can define the one-dimensional fermion as: C'('O

=

=

,

,

(s)

taking into account the fact that, in the limit when the width of the world-sheet tends to zero, ~b1 and ~P2 become independent of the parameter o. It is necessary to stress that the definition (5) does not contradict the boundary conditions (3); the combination of (5) and (3) does not lead to an inconsisency in the case of Ramond string (but the same is not true in the case of Neveu-Schwarz string). One can write an expression similar to (5) for the gravitino: in combination with (46) it leaves the single G r a s s m a n n function dependent only on the parameter r. Using (4) and (5) and averaging in (1) over o, one obtains (up to the renormalization on finite constants) the action of one-dimensional supergravity [17]:

f01dr½[e

15;M~ M -

i~g~M - iC-1xxMIpM] .

(6)

The action (6) corresponds to a spin-½ particle. It is the open Ramond string that gives the action (6) in the limit.

lgl

A. V. Marshakov / Ferrnionic string

The propagator of the fermionic string can be defined as the following functional integral:

fDefDxfD¢fx'(°f~Dx e x p ( -

S },

(7)

where S is the action (1) (in euclidean formulation), and the integration over xM(o, r) should be done with fixed boundaries, i.e. the surface in x-space is rigidly fixed on two boundary curves (contours). In the case of the open string the x M variables satisfy the free boundary conditions at the two other boundaries. After one has singled out the integral over the group of local symmetry of the action (1), the computation of the covariant path integral over supergravity fields is reduced to the effective finite-dimensional integral over supermoduli space [20, 21]. The quantization can be carried out so that the integration measure is obviously invariant under all the symmetries except Weyl and super-Weyl symmetries, which will be, in general, broken in quantum theory. It is shown in ref. [2] that this way of quantization leads to the dynamical contribution of Weyl and super-Weyl modes in all dimensions except the critical one D = 10. Below we shall stay in the critical dimension, when the dynamical contribution is absent. It should be pointed out that formula (7) defines, in general, the non-invariant quantity (for example, if the integrations over the reparametrization of the boundaries are not introduced [1] the propagator is directly dependent on the parametrization of the boundaries). These questions will not be discussed here, they are considered in detail for the bosonic case in ref. [9], it is sufficient to note that the dependence on the boundary values of Weyl and super-Weyl modes can take place. To compute the integration measure in (7) one can use a Wess-Zumino gauge [19]. In this gauge the action has the following form:

s = f d 2 o ( a x a x + + a+ + + a , + 0 = ~1 ( a 2 q_ i a l ) ,

Ox + 0=1(02

ax +

}¢JTx2),

-- i31) ,

d2o = d r d o ~g

(8)

(the sum over M = 1 . . . . . D is omitted). When passing from (1) to (8) the condition y~Xu = 0 is used (the result of the super-Weyl symmetry 3X. = "y~X) and the fields are redefined on some number constants, for example X = ~/2x. The transformation laws for the supergravity set in the chosen gauge have the following form (3h~ = eb3e~). /*

a

,

-

-

82=

-

+ 8h~

z

,

a2+

aa/= ag- Ixg + aa:, 8X=-gaX+gX~ 0g+ag+g

X,

(9)

182

A.V.

Marshakov / Fermionicstring

where ~, 4, e, g are the infinitesimal symmetry parameters of the action (8) (the and ~ transformations contain reparametrization, Weyl and Lorentz transformations). The variations in (9) mean the motion in the directions orthogonal to the gauge transformations. The exact analysis of those variations that cannot be globally represented as the gauge transformations can be carried out for surfaces without boundaries. In the case of surfaces with boundaries one should consider the supermoduli space derivative from the corresponding one for a double surface. It will be assumed below that the strip which is the world sheet for the open string propagator has as the cylinder the single modular parameter - its length. The variations of the gravitino along the supermoduli direction are characterized by the single Grassmann parameter. The reason is that two components of the gravitino left in the Wess-Zumino gauge are connected by the boundary conditions on the ends of the open string. Then the variations of the gravitino can be written as;

gX = P 3~,

3X = P 3~'.

(10)

Using (9) and (10), one can see that the condition of orthogonality with gauge transformations takes place if ~ = const and p = const. Then from the boundary conditions it follows that p = ~.

(10')

Introducing the integration over ghost and antighost variables one has for the integration measures:

ffdrfd~fDb Dc D~ D~D/3DvDgD~ DX D+ DYe(~, b) × 3 ( ( p , 1 3 ) ) e x p ( - f d 2 o [ 0 X0 X + ~ 0~p + q, O~b + b 0c + b 0~ + / 3 0 y +/~ 07

+o~(~0x+ ibm+ ~90~+ + ax+ 1by+ ½/3ac)]},

(11)

where

(n,b)= f(nb+~3), (0,/3> = f(~B+ 0~). Let us integrate in (11) over ~. Then the expression (11) becomes the "average" of the supercurrent - the sum of supercurrent contributions of ghost and matter fields.

A.V. Marshakov / Fermionicstring

183

One can make the substitution by writing all the fields as the sum of classical and quantum components, where the classical component satisfies the free equations of motion and boundary conditions. When averaging, the contribution of the quantum part vanishes. The zero modes of the antighosts correspond to the supermoduli, so the integration measure over antighosts can be written as: Db Db = db 0 D'b D'/~,

(12a)

Dfl Dfl = dfl o D'fl D'fl.

(12b)

The integration over zero modes is trivial because of the presence of &functions in (11) and as a result one obtains: f o ~ d T ( d e t 0_, det O_l)(det 0-1/2 det 0_1/2) -1

x fd2o [ff0x+ + Ox+ %7 +

+

+ ½Ba¢]c,

× (det - Vlo)-5(det 01/2 det 01/2)5 exp{ --SN},

(13)

where

So,=fd2o[aXc

?Xd]+S *.

S* contains the contribution of both the fermionoic and the ghost boundary terms. The dependence of the propagator upon the boundary values of the fermionic and ghost fields will not be discussed in this paper, that is why the contribution of S* will not be considered (as in the ref. [28]). Let us compute the dependence of the integration measure in (13) on the modular parameter T. At first one can notice that for the determinants in (13) the following relation can be written:

(

det 0_ 1 det O_ 1 ~

det 2 1/2 det 0_ 1/2 -

det/~ _ 1 det N_ i det N 2

)1j2( )1j2 =

det z~_ 1 T

'

det z~_i/2 ) 1 / 2 = ( p 2 r d e t A - 1 / 2 ) 1/2 . det N_ 1/2 det N3/2

(14)

The sign of proportionality means the difference of the right-hand side and left-hand side of (14) which is the exponent of the super-Liouville action. But taking into account the fact that the whole contribution of the super-Liouville action to the

A, V. Marshakov / Ferrnionic string

184

integration measure vanishes in the critical dimension, one can change the sign of proportionality in (14) to the sign of equality. The determinants of the operators /~-1 and / ~ 1/2 can be calculated directly from the eigenvalue equations:

0

[]=

0o 1

-[]],52)

=X(~)(52)'

+

,

0
(15a)

o1<1,

0
a 2
(15b)

To find the eigenvahes X in (15) one should choose some boundary conditions. The usual conditions of fixed boundary are used for 51 and 5 2 parameters: o

= 0,1,

~"= 0, T,

51 = 0,

0152= 0,

52 =

0251 = 0.

0,

(16)

The choice of boundary conditions for the Grassmann parameters is the principal question. To find the eigenfunctions of (15b) let us first remember that for e and we have:

E(0, ~-) = -eg(0, ~-),

e(1, ~ ) = - e l ( l , ~-),

e = _+1.

(17)

Therefore one can introduce the mixed components: c =f~-(~-g),

(18)

which are also the eigenfunctions of the Laplace operator but they satisfy the natural conditions when o = 0,1: e = 0,

0 8 ~ e + = 0,

o = 0,1,

(19)

or vice versa. From (19) it follows that all eigenfunctions E are dependent on o, and e+ can be either dependent on or independent of a "zero modes" e(+°). Let us choose the following boundary conditions: ~(+°)(r = 0) = 0,

c(+°)(¢= T ) = 0,

, (T + T,o) = - { _ ( ~ - , o ) ,

(20a)

(20b)

A.V. Marshakov / Fermionicstring

185

i.e. the "zero modes" satisfy Dirichlet boundary conditions but the other eigenfunctions are considered on the sewed strip (on the cylinder) and satisfy the antiperiodic boundary conditions. These conditions correspond to "asymptotic" supersymmetry, they give the correct result in the limit of the fermionic particle. They also give the correct distribution function for the states in the string spectrum. For the determinants in (14) one has:

detz~-l=

det z~_ 1/2 =

n=l ~ ~1

[

nI~I--l~2n2mI-=I--1T-7 '

(21a)

2,2m+1,22122m2

(21b)

'n'2n2+

1-~I l~I

~]

,n-2n2 +

n = l rn=0

-~-

1-~-I ~--7

rn=l

The last product in the right-hand side of (21b) corresponds to the non-trivial contribution of the "zero modes". The determinants (21) are calculated by regularizing the infinite products and introducing corresponding counterterms into the integration measure. Then, for (14) one obtains:

det0 l d e t O 1 -

t

~r2n2+ n,m=l

- e ,r/12 i~i (1 - e 2~,T) _ ~/(r),

(22a)

n=l

= m=0\ qT"2n2+

det 0-1/2 det 0 1 / 2 - oT~I= 1 ]-I

T2

cx)

- pTe ,,T/12 1-I (1 + e ~"r) 2 =--pTI2R(T ) .

(22b)

n=l

To calculate the determinants of the operators, acting on the fermionic matter fields, the antiperiodic boundary conditions for the quantum fields should be used as in the case of the fermionic particle. Then: det 01/2 det ~1/2- (det/~1/2) 1/2

i~i ( -- ~ n=l m=0

~r2(2m+1)2) 2 ~2n2 +

T 2

-

~2R(T ).

(23)

A.V. Marshakov /

186

Fermionic string

Finally, the determinant of the Laplace operator, acting on the two-dimensional scalar fields, has been previously calculated many times: det - D 0 - T ~ ( T ) ,

(24)

where ~/(T) is defined in (22a). Substituting (22), (23) and (24) into (13) one gets:

TS n=t 1 - e -~nr ] -~_d2°[t~cl OXc, + +d OXd]exp{-Scl}

(25)

In expression (25) and everywhere below the contribution of "classical" ghost fields has been omitted.

3. The pointlike boundary conditions Let us consider formula (25) in the case of pointlike boundary conditions. For the fixed boundary values of the xM(o, r) variables, when r = 0, T, the following expansion can be written: xi,f(o) = x~,fq- ~ Xin'fCOS'll'no. n=l

(26)

The pointlike boundary conditions mean that xi,'f = 0;

n = 1 , 2 ....

(27)

Then the classical action in (25), which in general has the form:

s~, = S ( x c , ) + s*(+~l) - ( x ~ - x ~ ° ) ~ 2T

×{[(xolf 2

qTn

+ ~ n=l i

sinhTrnT

2

2xinx~ }

+

s*(<,), (28)

simplifies up to: (x~-x~o) 2 -

S(xd)l xi.'=o, .>o -

2T

tax) ~

2T

(29)

A. ld Marshakov / Fermionicstring

187

In the expressions (28) and (29) x m are normalized as in (1) but not as in (8). The classical contributions of the matter fields to the supercurrent in (25) is:

l fdao[q, dOx~,+~L~Ox~l =1 f,~,mx r

"+

.=1 sinh ~rnT

× [ ~ bi .f( x . - x ~ . e = " r ) + ~f_n(XfneTrnT --Xin)]fI

(30) where E abi'f Tn e iz'no nET

+i'f(O)

tfi'f( O ) = E ~b,~ i,f-irrno e nEZ

(30')

After choosing the pointlike boundary conditions the expression (30) turns to:

lg'oM(X~--Xio)M

+OM(Ax)M T

(31)

In accordance with the quantization rules one should make the following change after computation of the path integral:

where ,/M are ten-dimensional Dirac matrices and ~'11= 1-IM7 M. For the pointlike boundary conditions from (25) one can obtain:

-Yo T s n~l

1

--

e (Ax)2/2T

e -~"r

0 M foodT

l+e-~"rl"

=-c°nstv'0(XoJ) J0 T M"=I1=

1 e ,x,2/2T.

(33)

Using in (33) the Fourier transformation, restoring the factor (2Tra') -1 and taking

A. V. Marshakov / Fermionic string

188

into account, that:

14 1 - - e ~"r ] = ~ 1 + e -rmT

fi n=l

where Co = 1, tor:

C 1 = 8, C 2 =

f dl°P (2~r) 1°

CNe "n'NT,

(34)

N=O

40, C 3 = 160, C4 = 552 etc., one will get for the propaga-

eiPaxl_~rl 2rra '~7

1

1

7 + 8p-2-q _- - _T_

+ 40

p2 + ( 2 / a ' )

0/'

+ 160

1 p2+ (3/a')

+ 552 p 2

1 + (4/a')

+ ....

/

(35)

The first term in the expansion (35) corresponds to the massless Dirac particle, the other propagating modes satisfy some nonlocal equations of motion. Let us compare expression (35) with the result of the operator calculations. The propagator of the open fermionic string in the operator formalism has the following forms: Fo = f ~ d T exp( -'o

TLo),

(36)

where

1 OLn~d n n4~O

1 Lo=p2+-~N,

i ~bi ~b.). N= ~" (a i ,a~+n i

(37)

n=l

For comparison with (35) the matrix element of (36) in pointlike boundary conditions should be computed. Therefore the pointlike boundary conditions in the operator language have to be defined [22]. The conditions (27) will have the form; 2.Jllx, ~>> = 0, XJ -

^

n > 0

olJ -

aJ

-",

j = 1,...,8,

(38)

n

where index j runs all the D - 2 physical degrees of freedom. The states, satisfying

A. V. Marshakov / Fermionic string

189

(38), are:

-a_na

]lx , a ) ) = Ix) ® exp

1 r/

, 10) ® q ' l u ~ ) ,

(39)

where ai,]0) = 0, n > 0; [a~, a ~ ] _ = nS,+m,08q , Ix ) is the eigenstate of the tendimensional coordinate operator; '~" is the operator constructed from the operators ~ba_n, n > 0 (30'). When only part of the q u a n t u m numbers is fixed in the initial and final states it means that the trace over the others should be taken. In the present case the trace is taken over two-dimensional fermions. This fact corresponds to the choice of the antiperiodic b o u n d a r y conditions in the previous section. Calculating the matrix element of the operator (36) between the initial and final states such as given b y f o r m u l a (39), one obtains:

((xf,allFofomdTexp(_TLo)llxi,

fl))=(t~),~flN~==0 p2 dN + m2u ,

(40)

where a'm2u=N, ( N = 0 , 1 , . . . ) is the spectrum of the open R a m o n d string. The weight n u m b e r d N characterizes the contribution of each mass level to (40) and is the s u m of n o r m s of the physical states at each mass level. The norms of the t w o - d i m e n s i o n a l bosonic states are to be c o m p u t e d directly, using the expansion of (39) over the mass levels:

ai_,,ai n 10) = (1 + ~a l a _ l + aa 2a_2 +

exp ½

. . . .

....

n=l

(41) T h e contribution of the two-dimensional fermionic states to the d N factor is simply their n u m b e r at each mass level. It is a consequence of the fact that the trace over fermions is computed. Hence, for the lowest mass level one obtains: d o = 1, d I = 8,

(~bill~),

d 2= 8+28+4=40,

i=

1 . . . . . 8); 1 i la i (~b~21g2),4,~lfJ~lS2),ya

llO));

d 3 = 8 + 64 + 56 + 32 = 160,

(,~/31~a), +'_2@Lllfa), @'_l@'_x+k_ll~), d 4 =

q,i_ll~2) ®

8 + 64 + 28 + 224 + 70 + 4 + 10 + 32 + 112 = 552,

J ' Io)); 5a_~a_~

etc,

190

A. V. Marshakov / Fermionic string

where = 0,

:

n > o.

Comparing (42) and (34), one can easily see that d u = (40) coincides with the path integral result (35).

C u.

Then the operator results

4. The open Neveu-Schwarz string The results of the previous sections can be easily generalized to the case of the open Neveu-Schwarz string. Unlike (3) the Neveu-Schwarz sector is singled out by the conditions: 1(0,

=

~k1(1, ~') = -e~b2(1, ~'),

(43)

where e = _+ 1, but the signs for a = 0 and o = 1 are opposite. The fact that the components of the two-dimensional fermion are connected by boundary conditions of opposite sign (43) leads to the conclusion that there is no natural reduction of the Neveu-Schwarz string to the one-dimensional case (6). Therefore it seems that the action (6) cannot describe bosonic excitations (this conclusion differs from that of [171). The main difference of the integration measure between the sectors of NeveuSchwarz and R a m o n d is the fact that in the gauge ~ = ~ , X = P~, where p = const and ~ = const, now from the boundary conditions of the (43) type, instead of Off) one has: p = ~,

p = -~,

i.e. p = ~ = 0.

(44)

Hence, in the Neveu-Schwarz sector there is no integration over the supermodular G r a s s m a n n parameter and that is why the averaging of the supercurrent will not appear. Carrying out the calculations similar to that of sect. 2 for the propagator of the open Neveu-Schwarz string, one obtains: det fo °~dT ~

×(det

z~_1 ) 1/2 (det '~- x/2) - t/2(det - t20) -5

~1/2 det 01/2)5 exp( -Scl } .

(45)

Calculating the determinants in (45), one has:

fo°TdT[2Ns(T)]4exp(-Scl}

(46)

A. V. Marshakou

/

Fermionic string

191

where

and

g(T)

the function

double

infinite

products mdT

J 0

is defined

in (22a).

in (46), for pointlike nr,2 ncZO(l

Fe

Regularizing boundary

+ ee*T(n+1/2))2

FI~==,(l - eC21inT)

Representing the infinite (27XX’)-1, one obtains:

product

renormalizing

conditions

1 4

e-(Ax)z/ZT

1

p2 + 1/2Cx’

+ 64

p2 + l/U’

the

the result will be:

(47)

in (47) as the sum and restoring

1 +28

and

the factor

1 + 138

p2 -t 3/2a’

i.e. the sum of the propagating scalar field Green functions with some weight factors. The first term corresponds to the tachyon (m2 = - 1/2(~‘). The weight numbers in (48) can be easily computed as in sect. 3 by operator methods. Let us calculate the factors written in the expression (48): d -l/2

=

1,

d,=8,

(~i1,~lfQ, j=l,...J+

d,=8+56=64,

d ,,,=4+64+70=138,

(~~3,2W~

P-,2~j1,2~k1,2lJt))~

A. V. Marshakou / Fermionic string

192

where tga) • + s/ l ~ 2 ) = 0 ,

the indices of d r Schwarz string a ' m 2r = (48) the propagating propagating fields are and

s-

2, , 2 3, - - -

are integer and half-integer as the spectrum of the Neveu1 71 , 1 , . . . . It should be stressed that in expression r, r = - 7,0, fields are space-time bosons and in expression (35) the space-time fermions.

5. Closed strings The results of the previous sections can be generalized to the case of closed strings. The world-sheet corresponding to the closed string propagator is the cylinder. The structure of the integration measure over the metrics on the cylinder differs from the integration measure over the metrics on the strip only by the conformal Killing vector contribution. The contribution of two-dimensional bosons and bosonic ghosts to the integration measure on the cylinder was computed in the original paper [1]. Left and fight fermion sectors are independent on the cylinder. The boundary conditions can be imposed independently in each s e c t o r - the periodicity or antiperiodicity when the field is transported along one of the simplest non-contractible loops. Therefore the contribution of the two-dimensional fermions to the integration measure on the cylinder is the product of the left and right sectors contributions, unlike [23]. There are altogether three different closed string sectors: NS x NS, NS x R or R x NS and R X R. In Ramond (left and right) sectors one can impose on the gravitino the gauge ~ = ~ or X = P~ and correspondingly ~ = 0 or X = 0 in Neveu-Schwarz sectors. In the case of closed strings unlike (12b) the antighost fl a n d / ~ zero modes are present or absent independently. As a result for the closed string propagators one obtains: (i) N S

X

N S sector

m d T [ det - [ ] ( - - ) ]4 exp{ _ Sd }

f0 det-D(--)=

r' L fl

j fl

-r2(2m+1) 2 ]4 T2 + ~2(2n + 1) 2] .

(50a)

n=O m=O

(ii) N S X R sector

~ dT [det-~ (+)det- ~(--)]2 { 1

fo r'

)

f<,ax , e-So,, ,n.2(2m + 1) 2

det - D (+-)

=

47r 2rt 2 + n=l

T2

T2 m=O

2. (50b)

193

A. V. Marshakov / Fermionic string

(iii) R X R sector o¢ d T

d e t - E](+-)

fo

P(r)

]4

1

_

1 f+~lOX~l+~-T-7

{-~f~lOXc,

f(+c, fc,)}exp{_Scl} (5Oc)

]2

In formulae (50):

p(r)=

.=1 lml m=, L r2 +

4

2m2

(51)

"

Only some particular cases of the expressions (50) will be considered below. Fixing the pointlike boundary conditions and calculating the infinite products we get: (i) N S x N S sector

~

1 -Jr- e-~r(2n+l)T) 16

e

(Ax)2/2T~

(52)

fo ~se2'~rl-["I]So~=(1(l-e 4'~'v) 8

(ii) N S X R sector

f aO

T

I-I

n= 1

1- e

2~rnT

e -(a~)~/2r.

(53)

Expanding the infinite products in (52) and (53) into infinite sums one obtains the representation for the closed string propagators as the sum of propagating string modes. It can be pointed out that in expression (52) the propagating fields are space-time bosons and in expression (53) they are space-time fermions. The expressions (52) and (53) contain infrared singularities - tachyons. However, it is known [24] that the physical spectrum of the NS × R sector is tachyon-free due to the constraint L 0 - L 0 = 0. When computing the bosonic closed string propagator [1] for pointlike boundary conditions the constraint L o - £ 0 = 0 is ensured automatically. But when the trace over two-dimensional fermions is taken in the closed fermionic string this constraint is spoiled owing to the two-dimensional fermionic contribution. This is the reason for the presence of the tachyon in (53). The R × R sector should be considered a little more carefully. Restoring in (50c) the normalization of the x v variables as in (1) one obtains:

T5 n=l

1--~]

T

~

+ T(~b°~°) exp

2~

'

A. V. Marshakov / Fermionicstring

194

where the second term in the braces is a consequence of the presence of the X 2 term in (1) and (8). This second term is necessary to get the result: (iii) R × R sector

-

-

~Mo(xfo)"M~No(xfo)N Jo T5

n=l

1 -- e

2~rnT

e -~x)2/zr.

(55)

The results of this section can be compared with the operator results given, for example, in ref [26]. It can be seen that they differ by the absence of 8 ( L o - L0) in (52), (53) and (55) which ensure the contribution only of the physical closed string spectrum. This 6-function is restored in the one-loop statistical sum, where its appearance is the result of the integration over the real part of the complex modular parameter. But for the propagator the world-sheet has the single real modular parameter. However, the expressions for closed string propagators are useful when studying the ultraviolet singularities connected with the causality properties of strings. In the conclusion of this section let us notice that in the limit a' ~ 0 the R × R sector of the closed fermonic string corresponds to the N = 2 one-dimensional supergravity. It is well-known that only the closed strings contain the graviton in the spectrum and can consistently interact with the external gravitational fields. For the open strings it is impossible to define consistently such interaction. It can explain the fact that the quantum hamiltonian coincides with the classical one for N = 2 one-dimensional supergravity in a curved space. For the N = 1 case there appear some corrections, connected with operator ordering [27]. However, in the first order of perturbation theory the path integral with the classical action (without corrections) for the Dirac particle ( N = 1 theory) gives a result for Green function, coinciding with the expansion of the solution of the corresponding Dirac equation for the Green function in curved space (see ref. [29]).

6. Discussion

The obtained results allow us to analyze the properties of the propagators in the fermionic string theory. The most important point is the structure of the singularities of the expressions for the propagators. The infrared singularities are present to the tachyons in string spectra. Apart from the infrared tachyon singularities these expressions possess ultraviolet singularities. One can observe them in the following way. The calculation of the double infinite products, leading to the results (33), (45), (52), (53) and (55), can be carried out by changing the order of taking the double product [25]. Then it can be easily seen that the main ultraviolet singularity has the

A, V. Marshakov / Fermionic string

195

following form

lim fd~r exp

c---,0 Jc

T

2T" 2~ra'

= lim f d T e x p 2~r ,--.o , [ z~

(56)

The integral in (56) diverges exponentially, when ( A x ) 2 < 47r2a ' .

(57)

l = 27rvra~

(58)

It is important that

is the common fundamental length for all (open and closed, bosonic and fermionic) strings in the critical dimensions. The singularities of (56) are connected with the mass spectrum of string theories, increasing as a linear exponent of mass. It follows that string theory can be classified to the non-localizable theories when we consider its causal properties. The Green function of a non-localizable theory can have a singularity in a spacelike domain but the singular point should be not farther from the light cone than l [28, 30]. When the ends of the propagator are sewed together the ultraviolet singularities should vanish. The reason is the appearance of modular invariance (in the case of closed strings) and a sum over spin structures. From this point of view it would be interesting to understand how the restriction of the interaction domain to the modular figure only appears. Let us, finally, point out that the Green functions of strings can be useful in studying the motion of strings in external fields. The dependence of the propagator upon the fermionic and ghost boundary values can be interesting from the point of view of string field theory and the theory of Riemann surfaces with boundaries. I am grateful to V.Ya. Fainberg for constant attention to the work, fruitful discussions and reading the manuscript. I am also indebted to A. Morozov, M. Soloviev and especially to A. Tseytlin for useful discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

A. Cohen, G. Moore, P. Nelson and J. Polchinski, Nucl. Phys. B267 (1986) 143 A.M. Polyakov, Phys. Lett. 103B (1981) 207, 211 A.A. Tseytlin, Phys. Lett. 168B (1986) 63 T. Lee, Washington Univ, preprints 40048-06 P7, 40048-07, P7 T. Ohrndorf, preprint HD-THEP-87-12 C. Ordonez, M.A. Rubin and R. Zucchini, Rockefeller Univ. preprints C.N. Ragiadakos, Class. Quant. Gravity 4 (1987) 1679 N.I. Karchev, JINR preprint E2-87-739 A.N. Redlich, IAS preprint IASSNS-HEP-87/49

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A. V. Marshakov / Fermionic string

[10] S. Carlip, Nucl. Phys. B284 (1987) 365 [11] C.N. Ragiadakos, Crete Univ. preprint [12] S. Deser and B. Zumino, Phys. Lett. 65B (1976) 369; L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. 65B (1976) 471 [13] V1.S. Dotsenko, Nucl. Phys. B285 FS19 (1987) 45 [14] V.Ya. Fainberg and A.V. Marshakov, Nucl. Phys. B306 (1988) 659 [15] N. Seiberg and E. Witten, Nucl. Phys. B276 (1986) 272 [16] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253 [17] L. Brink, P. Di Vecchia, P. Howe, S. Deser and B. Zumino, Phys. Lett. 64B (1976) 435; L. Brink, P. Di Vecchia and P. Howe, Nucl. Phys. Bl18 (1977) 76 [18] P. Howe, J. Phys. A12 (1979) 393 [19] J.J. Atick, G. Moore and A. Sen, Preprint IASSNS-HEP-87/61, SLAC-PUB-4463 [20] G. Moore, P. Nelson and J. Polchinski, Phys. Lett. 169B (1986) 47 [21] E. D'Hoker and D.H. Phong, Nuel. Phys. B278 (1986) 225 [22] M.B. Green, Nucl. Phys. B103 (1976) 333 [23] J. Polchinski and Y. Cai, Nucl. Phys. B296 (1988) 91 [24] J. Govaerts, in Proc. II Mexican School on Particles and Fields, ed. by J.L. Lucio and A. Zepeda (World Scientific, 1987) [25] E. Alvarez and M. Osorio, preprint CERN-TH.4571/86 [26] A. Le Clair, preprint PUPT-1080 [27] V. de Alfaro, S. Fubini, G. Furlan and M. Roncadelli. Nucl. Phys. B296 (1988) 402; Phys. Lett. B200 (1988) 323 [28] V.Ya. Fainberg and A.V. Marshakov, Phys. Lett. B211 (1988) 81 [29] A.V. Marshakov and V.Ya. Fainberg, JETP Lett., to he published, Pis'ma ZhETF 47 (1988) N°10 [30] V.Ya. Fainberg and A.V. Marshakov, in Proc. of P.N. Lebedev Phys. Inst., to be published