Open strings from closed strings: Period matrix, measure and ghost determinant

Volume 202, number 2 PHYSICS LETTERS B 3 March 1988

O P E N S T R I N G S F R O M C L O S E D STRINGS: P E R I O D MATRIX, M E A S U R E AND G H O S T D E T E R M I N A N T

Jo~o P. R O D R I G U E S Centre for Nonlinear Studies and Physics Department, University of the Witwatersrand, Wits 2050, South Africa

Received 6 November 1987

The construction of the double of a surface with boundaries together with an anticonformal involution is used to relate partition functions of closed strings to partition functions of open strings, at an arbitrary order of the perturbation theory. The specific form that the period matrix must have is derived. It is found that the number of components is halved. The partition function factorizes into the product of Neumann and Dirichlet components. In particular, the measure and the ghost determinant factor into two equal components.

1. Introduction

It is the purpose of this letter to investigate to what extent the partition function o f an open string at any order of the string perturbation expansion can be obtained from knowledge of the factors (measure, ghost determinant and determinant o f the laplacian) mak- ing up a corresponding closed string partition function.

It will be found that the measure and the ghost determinant are (apart from well-defined constant factors) the square root of the measure and ghost determinant of an appriopriate closed string partition function with the period matrix restricted to take a specific form. This specific form is derived in this letter, for an arbitrary order of the perturbative expansion. The relationship between the determinants of the laplacians is discussed.

The path integral approach will be used [ 1 ]. Each order of perturbation theory then corresponds to a surface with boundaries. I will restrict myself to the bosonic open string and to orientable topologies. The basic tool will be the construction o f the double of the surface together with an anticonformal involution. Nonorientable topologies (which I believe can be treated in a similar way with an appropriate involu- t ion) will be left to another communicat ion. Al- though only the partition function is dealt with, amplitudes can be obtained in the manner described in this article.

2. The one-loop case

The partition function for the open or closed bosonic string corresponding to a topology without conformal Killing vectors and in the critical dimension is [2-5,8]

Z = f dt a det(~ua' q}b) (2n) -"v2 de t ' (PTPl) ~/2 detl/2(q/~, ~,b)

- - 1 3

X ) d2o.

In the above, m is the dimension of the moduli space o f the surface, ~a ( a = 1, 2 .... , m) is a fixed basis of ker P~ and the ~ (a= 1, 2 ..... m) are linearly independent metrics in the slice with nonvanishing pro- jections on ker PT. The string tension has been set to 1.

In order to identify the issues, it is instructive to consider the one-loop case. The only orientable topology with boundaries and with Z = 0 is the cylinder. The partition function for the cylinder is given by [2,3]

i dr [(1/,,f2) 2rlr /(2iz) l 2]

× (2n/z) 2r I ~/(2iz) 121 -'3FcK. (2)

The above result has been obtained with the funda-

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mental cell 0~<trl < 1; 0~cr2< 1 of unit area. I have isolated FcK, the contribution of the conformal Killing vectors (not present for higher topologies) and the contribution due to the mapping class group. For this topology this identification is unambiguous [ 4,2 ]. ~/(/2) is the Dedekind eta function

~/( /2)=exp(~ni/2) f i [1 -exp(2n in /2) ] . (3)

It is useful to change variables "c~2r in eq. (2). One obtains oo j'd o 2x//2~ [(1/x/~) zl~?(i~)12]

X [(4n/z) z I ~(iz) 12] -13F~K. (4)

I checked that the above expression corresponds to the choice of a fundamental cell 0 ~< a i < I; 0 ~< a2 < I/2 of area 1/2.

The double of the cylinder is the torus obtained by sewing together two cylinders by their two boundaries. The partition function of the torus is [4]

f d2 $'~ 4] [ ½/2~ I~(~) I F

X [(27t//22) /22 i r/(/2 ) 14]-13GcK. (5)

F is a fundamental domain for the action of SL(2, Z) and/2=/21 + i/22. The question is now the following: could we have deduced eq. (4) knowing each of the factors entering eq. (5) ?

Let me start with the det(P]PL) factor in eq. (4). This factor can be obtained from the corresponding one in eq. (5) by let t ing/2=iz and then taking the square root. The fact that /2 is imaginary can be understood with reference to fig. 1. The cilinder is represented by a fundamental cell of half unit area, and a second is adjoined to the first along the boundaries, which are identified with the a-cycles of the

/ -.-°

Fig. 1. Moduli parameters for the torus and cylinder.

double. In principle, the torus has a complex modu- liparameter/2 pictorially depicted in fig. 1. However, by requiring that any point such as point 2 be obtained by reflection of point 1 on the boundary, one easily establishes that /2 cannot have a real component (/2 changes direction under reflection).

For open strings, it is known [5,2,3,6,7] that det' (P] P1 ) must be computed with mixed boundary conditions: Dirichlet for one, and Neumann for the other. If we consider a basis exp[i(2nn/L)]x for the eigenfunctions of the one-dimensional flat scalar laplacian with period L, one can obtain Neumann and Dirichlet basis by doubling the period and changing to the even and odd real functions cos((nn/L)x) and sin((rtn/L)x), respectively. This occurs along the b- cycle of the torus, and since the a-cycle contribution separates, it follows that

det~(/2=iz) =detr~(z)deto(Z), (6)

where the subscripts C, N, and D refer to closed, Neumann and Dirichlet respectively. Eq. (6) is after all just a statement of change of basis. The square root is now understood since det'(P~P1) is equal to det' 2 for the torus.

Turning now to the determinant of the scalar laplacian, eqs. (4) and (5) seem to imply that

det~ (r) = [ det~ (/2= it) ] 1/2. (7)

This is true, but not trivial. The actual result is that [2]

[deth(z)]2= ~ [(2r02n 2] det~(/2=iz), (8) n~0

and it is only because the z independent product in the above equation is exactly I when zeta function regularization is used [ 2 ] that eq. (7) holds true. (I have checked that eq. (7) also applies for the case of the sphere and the disc. In this case, there is no moduli parameter, of course.) The generalization of eq. (8) to the general case is nontrivial.

The remaining factors are now the area of the surface and the measure. The area of the cylinder is half the area of the double. The measure will be discussed in its full generality in the next section.

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3. The general case: period matrix, ghost determinant and measure [ 8 ]

Consider a open orientable Riemann surface M with g handles and b boundaries Fo, F , ..... Fb_ ~. The double [9] is a compact Riemann surface D of genus h = 2g+ b - 1 without boundary, and with an anticonformal involution 0 with fixed point set 0M. One can [9] and will choose a covering of D by open sets U., with c~ in an index set I w Iow I ' , with the property that: if c~d, U,~ ~ M has local coordinates z~: U=-~ cg such that z~, =zoo0 are the local coordinates of a unique set U., =~(U~), a ' e l ' and vice versa. Ifc~e Io, U . = ~(U~) has real coordinates on 0M and positive imaginary part on U . c~ M. The canonical line bundle of M can now be described as the canonical line bundle of the double with cocycles g,~v = d z J d a e H ~ (D, 0") restricted to satisfy g~p(p)>0 if pe0M and g~/3(P) =g. , /r (~(P)). In this way, any differential w and #*w are sections of the same bundle on D with the same transition functions.

I will fix a symmetric canonical homology basis on D

al , bl , ..., ag, bg, ag+ l , bg+ l , ...,

ag+~,_ l , bg+b_ . , al , , hi , , . . . , ag., bg,, (9)

such that a g + , = F , , n = 1, ..., b - 1 . al, bt . . . . , ag, bg (a~,, b~,, ..., ag,, bg,) are cycles in M (0(M)) (an ex- ample is shown in fig. 2).

They satisfy

O(ai) = a c , ~(bi) = - b c , 1 <~i<~g,

O(a,)=a,, ~ ( b , ) = - b , , g + l < ~ i < g + b - 1 , (10)

in HI(D, 7/). The corresponding basis of holomorphic differentials is normalized in the standard way

fwj=(~i j , fwj=g2i j . (1 la, b) ai bi

"N M

b4

g=l

Fig. 2. Canonical homology basis on the double of a surface M with g= 1 and b=4.

plies that the period matrix takes the specific form

~.= T D * , (13)

where A is a g × g symmetric matrix, B is a g × ( b - 1 ) matrix, C is a g × g antihermitean matrix and D is a ( b - I ) × ( b - 1 ) imaginary symmetric matrix.

This matrix has ½h(h+ 1 ) real components, i.e. exactly half the dimension of the Siegel upper half-plane associated with the double of the surface. The global diffeomorphisms consist now of the subgroup of Sp(4g+ 2 b - 2, Z) = Sp(2h, 7/) that preserves the form of the matrix (13). More precisely, one requires that the global diffeomorphisms commute with the matrix representing the action of~ on the basis of Hi (d, 7/) (eq. (10)). The fundamental domain must then be chosen accordingly.

At this stage, it is useful to write eq. (1)

f (21r) /2( det,(P~pl)l/2 det (~/a , ~,) Z = dta detl/2(~t,, g/t,)

From eq. (1 la) one establishes that the pullbacks satisfy

~)*Wi=Wi, , l <~i<~g,

& W , = W i , g + l ~ i < ~ g + b - 1 , (12)

and this result together with the symmetry of ~ ira-

--13

X ~f d2 a

for the partition function. I am restricting myself to topologies without conformal Killing vectors, for which it is sufficient that the double has no conformal Killing vectors.

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With the construction described above, one verifies that the involution commutes with all operators whose determinants appear in eq. (1'). I therefore further restrict the system by requiring that all tensors (in coordinate-free form) are even under this operation. It is remarkable that this single require- ment is sufficient to describe the boundary conditions of all relevant fields, as it follows from physical considerations [ 5,7]. Namely, the scalars describing the embedding must satisfy Neumann boundary conditions, as required for a description of open strings. The normal component of any vector field must satisfy Dirichlet's boundary conditions. This is what the normal component of the generators of diffeomorphisms connected to the identity must satisfy (the boundary must remain invariant). More importantly one must have

tan~Sgab = 0 (14)

(and also n~V bSg~b= 0) at the boundary. It is known that with these conditions the generalized laplacians have a proper eigenmode expansion [5]. Another consequence is that the conformal factor of the metric is required to satisfy Neumann boundary conditions. This is a consistent choice in the critical dimension [ 7 ].

There is a further consequence that is particularly appropriate for our purposes. First, it follows from the double construction that

z (D) = 2z(M). (15)

In the interior of M, the gauge fixed metric is that of the double. (For definiteness, one can think of it as a constant curvature metric normalized to/~ = - 1, although this is not essential in the following.) The re- quirement that all tensors be even under the involution means that the geodesic curvature must vanish at the boundary, since

k = g~bn~ tcV c t b (16)

(n" has unit length) changes sign across the hound- ary. Therefore this argument shows that for open strings one can choose a gauge-fixed metric with van- ishing geodesic curvature. This is important, since then the Gauss-Bonnet theorem in consistent with eq. (15), which is obtained simply by counting boundaries and handles. In particular when .R = - l, the Gauss-Bonnet theorem and eq. (15) are a simple

consequence of the fact that the area of the double is twice that of M.

We are now in position to generalize the first result of section 1. That is

det'(P~P~ ) = [det~(P~121) (12=z)] ,/2, (17)

where r has the form described in eq. (13). This should be clear, since both components of the generators of diffeomorphisms connected to the identity satisfy the same equation, and one must be chosen to be odd (Dirichlet) and the other even (Neumann). For the closed string, both components have both (even and odd) components.

The area of the double is twice that of M, and for the measure we consider a basis of quadratic differentials ~a, a = 1, ..., 2m on D. m is the dimension of the moduli space of M, and in the absence of conformal Killing vectors, eq. (15) and the Riemann-Roch theorem insure that the dimension of the moduli space of the double D is indeed 2m. One can then further require that this basis be chosen to consist of an even and an odd m-dimensional subbasis, each spanning mutually orthogonal subspaces. One chooses a slice ~a, a = 1, ..., 2m in a similar manner. It has somewhat been overlooked in the literature that condition (14) applies to Teichmiiller deformations as well, restricting the measure on D to the even subspace. One then obtains the following measure on M:

f dt~ Adt 2 ^ . . . ^ d t m 2 m/2

f det(~a, ~b) )1/2

~detl,2 ( ~ ~-~-~e-~2 ( ~ , ~b) ,

i = 1 , 2 .... ,m, a , b = l , 2 , . . . , 2 m , (18)

with the t i restricted to the even subspace. It should be noticed that the expression in brackets is independent of the choice of slice and basis of quadratic differentials on the double of M. Indeed one verifies that in the simple case of the torus and cylinder the meas- ures are indeed related in the above manner (eqs. (4) and (5)).

The following clarification is in order. I have ar- gued that even though the quadratic differentials are a basis for the kernel of P~ (and therefore zero modes ofP~ PT), one can still choose them to be even or odd under the involution. The point is that even and odd eigenmodes can still be related by the conjugation

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operator. The conjugation operator [I0] is defined on forms as

*: w=widxi~*w= -w2dx 1 + w l d x 2. (19)

For symmetric traceless tensors:

*h~ = - ih_.z. (20)

Using isothermal coordinates on the covering described in the beginning of this section, it is then not hard to see that if h~ is even, *hz~ is odd and vice- versa. Moreover the standard inner product is invariant, that is (hz= hz~) = (*h~, *h~). This has been used in eel. (18 ). Finally, one must also verify that if h~ is an even (odd) holomorphic quadratic differential, then *h.z is another linearly independent holomorphic quadratic differential. This follows straight- forwardly from eel. (20) and the discussion following it.

It finally be noted that eq. (15), the Riemann -Roch theorem and the fact that the period matrix (13) has ½h(h+ 1) real components imply that the dimension of the matrix (13) equals that of the moduli space of M exactly for those topologies whose doubles have h = 2, 3 handles. These are precisely the topologies without boundaries for which the dimension of the Siegel upper half-plane coincides with that of moduli space.

der conformal coordinate transformations. There- fore, the difference between detA and deto corresponds to "one-dimensional determinants" associated with those modes that are constant in direc- tions transverse to the boundaries. Indeed, it is immediately seen that such is the case for the cylinder (eel. (8)):

f i [(2n)2n 2] =1. (22) n # 0

Let G~(P, P') be the solution of the diffusion equation associated with the laplacian operator on D, with the initial condition lim,~oG,(P, P') =~(P, P'). Then

G~(P, P')=Gt(P, P')+Gt(P, O(P')), P, P 'eM,

GD(p,P')=G,(P,P')-G,(P,f~(P')), e ,e ' ~M. (23)

Defining the determinants as In det = - ~' (0) with oo

' f f ~(s)=f--- ~ dtt s-I d2x,fgGt(x,x), (24) 0 M

it follows that the difference between In det~ and In detD is given by - ~ - D ( 0 ) with

1 i ~(S)N-D= F(S) dt t '- ' 0

4. Scalar determinant

For the determinant of the laplacian it is still true that

det~(12= z) =det~(z) detD(z). (21)

However, in two dimensions, although corresponding to an eigenfunction obeying Dirichlet boundary conditions (Dirichlet eigenfunction) there always exists an eigenfunction obeying Neumann boundary conditions (Neumann eigenfunction), the converse is not true. On the covering of D by local coordinates described above, the Neumann eigenfunctions that are constant in a direction normal to the boundary although varying along the boundary have no Dirichlet counterpart. Apart from the over- all constant function, they form the kernel of the normal derivative. The normal derivative, with its usual meaning at the boundary, has a global meaning un-

(25)

One expects terms proportional to the length of the boundary to result from the above expression [ 5 ]. Indeed, if all lengths in the cylinder had been multi- plied by L, then the term (22) would become

fi [(2n/L)2n 2] =exp[ -4 ( (0 ) In L], (26) nv~0

giving a finite correction to the string coupling constant. Therefore the key issue is whether eel. (25) (or rather, its derivative at s= 0) contains terms that are dependent on the Teichmiiller parameters of the gauge fixed metric ~.

A calculation based on Selberg trace formula tech- niques is not particularly illuminating. What one would like to establish is the dependence of eel. (25) on the period matrix. This issue is under investiga-

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tion. I believe that whatever methods are devised to tackle this problem, they will have to be used in con- junc t ion with the global properties that have been es- tablished in this article, particulary the restrictions applying to the period matrix as well as symmetry properties under the involut ion.

Acknowledgement

The receipt of a Vice-Chancellor's Research Award is kindly acknowledged.

References

5. Conclusion: Neumann-Dirichlet factorization

By means of the double construction it has been shown that the part i t ion function of closed strings, factorizes, to all orders of per turbat ion theory, into a

product

Zc =ZNZD, (27)

where the N e u m a n n (Dirichlet) part i t ion funct ion contains the fields that are even (odd) under an involut ion operator. ZN is the open string part i t ion function. The period matrix must take a special form

and I showed further that the measure and the ghost de te rminant factor into two equal components .

[ 1 ] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [ 2 ] J.P. Rodrigues, Phys. Lett. B 178 (1986) 350; J. Math. Phys.

28 (1987) 2669. [3] C.P. Burgess and T.R. Morris, Nucl. Phys. B 291 (1987)

285. [4] J. Polchinski, Commun. Math. Phys. 104 (1986) 37. [5] O. Alvarez, Nucl. Phys. B 216 (1983) 125. [6] A. Cohen, G. Moore and P. Nelson, Nucl. Phys. B 267

(1986) 143. [ 7 ] W. de Beer, Conformal counterterms and boundary condi-

tions for open strings, University of the Witwatersrand pre- print CNLS-87-18.

[8] G. Moore and P. Nelson, NucL Phys. B 266 (1986) 58; E. D'Hoker and D.H. Phong, Nucl. Phys. B 269 (1986) 205.

[ 9 ] J. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973).

[ 10 ] H.M. Karkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics ( Springer, Berlin, 1980).

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Open strings from closed strings: Period matrix, measure and ghost determinant