OPEN STRINGS AND THE RELATIVE MODULAR ... - Science Direct

Volume 231, number 4

PHYSICS LETTERS B

16 November 1989

O P E N STRINGS AND T H E RELATIVE MODULAR G R O U P Massimo BIANCHI and Augusto SAGNOTTI D~partimento di Fisica, Universit~ di Roma 11, "Tor Vergata", and INFN. Sezione di Roma, "Tor Vergata ", Via Orazio Raimondo, 1-00173 Rome, Italy Received 11 August 1989

At genus greater than one, the surfaces contributing to open-string partition functions admit the action of a non-trivial subgroup of Sp (2g, l ) . The "relative modular group" is not solely associated with modular-invariant closed-string subdiagrams. In particular, at genus two, combining the degeneration of dividing channels with the action of this group leads to spin-statistics restrictions for open strings, in complete analogy with the closed-stringcase. In addition, it shows that only totally left-right symmetric closedstring models may descend to open-string ones, and essentially induces the GSO-like projections in the open and unoriented channels.

It is by now widely appreciated that the proper way to formulate closed-string models based on free fermions or twisted bosons is by demanding that their partition functions be modular invariant [ 1 ]. Modular invariance is a manifest property of the closed bosonic string. Fermionic models, however, result in a number of contributions from various spin structures. These contributions involve cuts along specified homology cycles, and are mapped into each other by the symplectic modular group Sp(2g, Z). Modular invariance is achieved only at the price of imposing relations among the various spin-structure assignments. Open-string partition functions receive contributions from surfaces with holes and/or cross-caps, and a proper understanding of their consistency may not forego a discussion of modular transformations in this context. We shall see that the symplectic modular group reduces to a subgroup, which plays in this context a role similar to that played by the full Sp (2g, Z ) group for closed strings. This group depends on the individual surfaces, and will be referred to as the "relative modular group". It is a classical result [2 ]~t that surfaces with holes and/or cross-caps always admit closed orientable double covers. The embedding into the double coy,z These ideas were introduced in string theory in ref. [ 3 ].

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ers is described via mirror-like involutions. These reverse the intersection form of the cycles, hence they are termed anti-conformal. In characterizing the relative modular group, it is particularly useful to recall a result that was presented some time ago in ref. [4]. There it was shown that an arbitrary Riemann surface with holes and/or cross-caps always admits a homology basis with invariant a cycles. The anti-symplectic nature of the involution then implies that it acts via a matrix of the form

with A symmetric, and directly tied to the handles and cross-caps of the surface. Following ref. [4 ], such a set of homology cycles will be termed an "identity basis". The form of the involution matrix in eq. (1) is familiar from genus one, where A is one for the M6bius strip, and is zero both for the annulus and for the Klein bottle. Quite conveniently, the result extends to arbitrary genus, and allows a unified treatment of the various surfaces of interest. These differ only in the specific form of the matrix A, and it is easy to convince oneself that all the transformations in Sp(2g, Z) that preserve I are of the form M=(C

(A r0) _ , ) ,

(2) 389

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16 November 1989

with det(,4)= + 1, and 2C=A,4 -

(A T) - td

(3)

a matrix of even integers. Transformations of this kind are trivial at genus one, but not at higher genus, where they basically mix ncighboring tori. There is a simple special case. Surfaces with only holes have J = 0 , and their relative modular group is the G L ( g , 7/) group of arbitrary ,4 matrices with integer inverses. This is rather interesting, since such surfaces do not contain closed-string loops. Rather, the modular transformations mix equivalent open channels. The bosonic string is clearly invariant under these transformations that, on the other hand, pose constraints on models based on free fcrmions or twisted bosons. Our interest in these matters is aimed at collecting a set of constraints on thc coefficients of genus-one partition functions, in a way reminiscent of thc work of ref. [ 5 ]. As for closed strings, we shall content ourselves with quadratic constraints. In the open-string case, these originate from surfaces with double covers of genus two and three. More precisely, we shall consider surfaces thal, in certain limits, degenerate into products of pairs of genus-one surfaces, with an intermediate dilaton exchange. We shall appeal to physical intuition to argue that quadratic constraints, together with factorization, comprise the whole of the problem. There are five surfaces with genus-two covers. Their canonical homology bases are described in ref. [4], whcre the surfaces arc identified via triples giving their numbcrs of handles, holes and cross-caps. For the "toms with a hole" (110), for the "'toms with a cross-cap" ( 101 ) and for the "'pair of pants" (030), the A matrices in the idcntity basis are

,,

1) 1 "

a/°~2)=(01

(5)

Similarly, for the "M6bius strip with a hole" (021), one is to distinguish among the three positions of the cross-cap relative to the holes, that lead to

(6) Strictly speaking, the signs of the entries in these matrices are not all fixed. Therefore, the signs of the entries in the real part of the "period matrix" r are also not fixed [4]. This freedom is important, and is related to the reality conditions on open-string partition functions. At genus two, thc largest available group of A matrices is GL(2,/7). It is clearly rcalized for thc "pair of pants', for which A=0. Fig. I shows this surface, together with its trivial pinching limit, where it factorizes into two annuli, separated by an intermediate open-string exchange. The matrices with determinant equal to one form the group SL(2, ~.), familiar from the gcnus-one case, and the whole of GL(2, 2) may be built by adjoining to the usual generators of SL(2, 2~),

,--(i one more with determinant equal to minus one. Alternatively, one may replace S with the Pauli matrix a~. All these transformations correspond to the manifest invariance under arbitrary interchanges of the

;) (4)

For thc "Klein bottle with a hole" (012), one is to distinguish among three corresponding to the three positions of the boundary relative to the cross-caps. They lead to the thrcc A matrices below: \

~(°12)=

I

'

J1812)=

I

\

" Fig. 1. A "pair of pants'" (030).

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three holes of the surface. In particular, matrices with A equal to T correspond precisely to transformations mixing the two neighboring tori of the double cover of the surface. The solutions to eq. (3) are best discussed by reducing it modulo two, which leads to d( a

hd)-(d

at)L1--0

(modulo2).

(8,

It should be appreciated that eq. (8) results in no constraints not only for the "pair of pants", but for all the three surfaces ofeq. (4). One should notice, however, that the ( ' matrices corresponding to these transformations are not zero both for the "torus with a hole" and for the "torus with a cross-cap". This result is best understood in terms of the standard canonical homology basis [6], termed "ol basis" in ref. [4]. In this context, the " 0 " 1 basis" essentially consists of cycles lying on the two sub-tori of the covers. Hence, the relative modular group emerges as the subgroup of Sp (4, 2~) such that corresponding "genus-one" transformations are acted upon the two images of the original surfaces. Actually, the restrictions on genus-two period matrices [4] have a surprise in store. The two images of the torus may be regarded as carrying, respectively, the left and the right-movers of the closed string. As a result, the group GL(Z, 2v) contains, together with the two standard genusone transformations, under which (T-*r+ 1, *-,'~+ 1 ) and ( r - - , - l / , , f--, - l / f ) , additional ones that interchange left and right-movers, such as (z,--*-f). For the "Klein bottle with a hole" (012), and working with A~z) ofeq. (5), eq. (8) implies that a+d and b+c must both be even. The resulting subgroup, denoted Go in ref. [ 7 ], is not normal in SL (2, 2~) and has index three. These properties are natural in this context. Indeed, under conjugation, Go turns into the subgroups corresponding to the other A matrices of eq. (5). As above, extending to GL(2, 2~) adds one extra image in the lower-half plane. The generators may be taken to be

G'°'2'=(O

~,0 )

G~o.2~= o't

and

(TT)_2

.

(9)

The first one corresponds to the manifest symmetry

16 November 1989

under the interchange of the two cross-caps. The presence of the second generator is perhaps more surprising. It corresponds to a mixing between a crosscap and a hole, and involves some even integers. Thus, cycles corresponding to cross-caps turn into pairs of cycles corresponding to holes, rather than into single ones. This is a general feature of the mixing between unlike structures: holes may mix with handles or cross-caps, but the very nature of the identity basis introduces even integers in the reverse operations. For the "M6bius strip with a hole" (021 ), one may confine attention to the involution with A tz) of eq. (6), since the other cases may be reached via suitable Sp(4, Z) transformations (not within the relative modular group, of course). The result is, again, the (not normal) subgroup Go of ref. [ 7 ]. The two generators are still given in eq. (9), apart from the offdiagonal blocks, fixed by eq. (3). Their interpretation, however, is different. The former corresponds to the manifest symmetry under interchange of the two holes. The latter corresponds to a mixing between a cross-cap and a hole. Clearly, this analysis rests on the familiarity with SL ( 2, 7/) that one has acquired in dealing with closed strings at genus one. In principle, proceeding to higher genera would require solving the constraints implied by eq. (3) with arbitrary large matrices. The genustwo case, however, is quite educational in this respect. It is telling one to look for manifest symmetries under interchanges of cross-caps (or holes) among themselves, or modular transformations of (sets of ) tori, and to supplement these with slightly less obvious symmetries involving suitable mixings of holes and cross-caps. Handles need not be dealt with in the presence of cross-caps: a handle and a cross-cap may be replaced with three cross-caps. Thus, the most general situation at arbitrary genus involves either a chain of holes and cross-caps, to which eq. (9) applies almost directly, or sets of handles in the presence of sets of holes. For instance, let us consider the "torus with two holes" (120) of fig. 2. This surface has a genus-three cover, and serves as a good prototype for the general case of mixings between holes and handles. Moreover, its trivial pinching limit relates the spin-structure assignments on annulus and torus. In the identity basis, one may take 391

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'x\.

..

,

x\

// \

I

"

/--X

'

\\

//

\ "~'"

\\

I

//

Fig. 2. A "torus with two boles" ( 120 ).

A(12o~ =

0

,

(10)

1

and in general one would find a set of such diagonal blocks, one for each handle, together with more zero entries along the diagonal, one for each hole. Two sets of transformations are certainly within the relative modular group. These are the modular transformations of the tori, as well as arbitrary mutual interchanges of the holes, all manifestly compatible with the constraint ofeq. (3). The issue of interest here is whether these obvious symmetry operations are to be supplemented with extra ones mixing handles and holes. Indeed, two such transformations belong to the relative modular group. In the notation of eq. (2), they are given by

A,

'!)

i

,

G=O

,

C2 =

(ll)

0 and by

,4z=

1 0

0 0

.

(12)

16 November 1989

Thus, out of eight naive generators, only three are left, two of which are those of the "torus with a hole". There is more, however, since the presence of the annulus forbids an overall interchange of a and b cycles. This calls for the addition of one more generator, and one is left with two generators mixing torus and annulus, for instance those ofeqs. ( 11 ) and (12). A similar discussion may be carried out for the "torus with two cross-caps" (102) that, in the trivial pinching limit, becomes the product o f a torus and a Klein bottle. Since this surface is described by the same involution as the "torus with two holes", their relative modular groups coincide. The third surface of this kind is the "torus with a hole and a cross-cap" ( 111 ), that factorizes into the product o f a torus and a M6bius strip. It would require a separate discussion, since its involution matrix is different. The result is that, in this case, the transformation ofeq. (12) is still allowed, whereas the matrices of eq. ( I I ) are to be replaced with

A~=

~

0

,

c~=

0

0

There are a number of other surfaces with genusthree covers that, in some limit, factorize into products of contributions to open-string partition functions. For instance, the "double annulus" (040) of fig. 3 exhibits an intermediate closed-string channel, trivial in the surface, but not in the cover. This is to be contrasted with the behavior o f t h e " p a i r of pants" (030) of fig. 1, where the intermediate open-string channel is trivial both in the surface and in the cover. Whereas the latter surface would be a natural setting to discuss open-string factorization, its intermediate

/

/

\

\

in the closed-string case, a redundant set of generators for Sp(2g, Z) is provided by the "genus-one" generators of each sub-torus, together with Dehn twists mixing neighboring tori, such as those about the cycles 7 and 1(7) in fig. 2. In this case, the involution demands that corresponding transformations be acted upon the two images of the surface, and that no transformation be left for the middle annulus. 392

(13)

.

Fig. 3. A "double annulus" (040).

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massless exchange is a vector. This forces the residue to vanish in the pinching limit. On the other hand, the former surface does allow the factorization on genus-one partition functions, via a dilaton exchange. Preserving the spin structures in this intermediate channel restricts the full relative modular group GL (3, 2~) to a GL (2, ~r) subgroup, that coincides with the relative modular group of the "pair of pants". These results may now be applied to the case of main interest to us. This concerns the determination of a set of quadratic constraints on spin-structure assignments implied by the action of the relative modular group on products of genus-one vacuum amplitudes. A reasoning of this kind led Antoniadis, Bachas and Kounnas [5] to include, in the definition of closed-string models, a similar set of conditions. These are sensitive to the phase contributed by the gravitino and the longitudinal fermions under modular transformations in the degeneration limit. We shall now detour from our main line of argument, to show that this "gravitino phase" is precisely as assumed in ref. [511. The genus-two vacuum amplitude for ten-dimensional superstrings was given in ref. [8 ], where it was related to the correlation function of two picturechanging operators. Models with free fermions and non-linear supersymmetry [9] require some minor modifications of the part of the picture-changing operator with ghost number one, that in the general Ddimensional case becomes 10-D

"~

Y(I)=e~(~, OX*'+ 1__~1 xlt~Io')*)

(14)

•

The contributions to the partition function from surfaces with genus-two covers that we would like to consider are of the form

Z(~=2) = f [~ d lm Lj(Rzdet lm r)-n/2 f I~ ~lo(w) i