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Exponential suppression of string interactions on orbifolds
Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
EXPONENTIAL SUPPRESSION OF STRING INTERACTIONS ON ORBIFOLDS Mark W. G O O D M A N ~.2
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Received 1 July 1988
Nonperturbative world-sheet instanton effects can be seen in the classical bosonic contribution to scattering amplitudes on orbifolds. These instanton contributions have the form exp ( -cR 2/a' ) and are exponentiallysuppressed when the orbifold radius is large. Amplitudes whose leading contribution comes from instantons are identified with processes in which the corresponding string states are pinned to the orbifold fixed points, and unpinned amplitudes are shown to lack this suppression at large radius. Blowingup the fixed points of an orbifold eliminates any pinningand with it the corresponding exponential suppression.
1. Introduction
fold space group G generated by the lattice A and elements of the orbifold point group Go g2. The ter-
Compactification of superstring [ 1 ] a n d heterotic string [2 ] theories on flat orbifolds [3] has the advantage of explicit calculability [ 4 ] that is normally absent in C a l a b i - Y a u compactification [5] ~, and a realistic chiral spectrum that does not appear in toroidal compactifications [7]. For compactification from ten to four d i m e n s i o n s the internal space is an orbifold that is the quotient K = T 6 / G o of a six-dimensional torus T 6 - - R 6 / A by a finite symmetry group Go of the lattice A. The elements ge Go will have fixed points g x = x on T 6 that become conical singularities of K, but these singularities do not lead to any inconsistency in the resulting string model. For a compactification with (2,2) superconformal symmetry these singularities can be "blown up" to produce a n o n s i n g u l a r C a l a b i - Y a u manifold [8] and there is a systematic p e r t u r b a t i o n expansion in the expectation value of the blow-up mode, a massless scalar in the twisted string sector [ 9 ]. For our purposes it will be simpler to describe a flat orbifold as the quotient K - - - R 6 / G of a six-dimensional euclidean space u n d e r the action of the orbi-
minology is precisely analogous to that used in crystallography. An element of G is written as a pair (0, (), where 0 is a rotation and ( is a translation, and acts on points in R 6 as x ~ O x + ( . The lattice consists of pure translations ( 1 , 2 ) a n d forms a normal subgroup ( 8 , ~ ) ( 1 , 2 ) ( 0 - ' , - 0 - 1 ~ ) = ( 1 , 8 2 ) . The point group is the quotient G o = G / A . I will assume for simplicity that the point group is abelian, so that the rotation pieces of the group transformations commute: 0 ¢ = ¢0. First we recall some basic facts about the states of a closed string on an orbifold K = K o / G [3,4]. We can describe these states in terms of the covering space K o = R 6 with two simple modifications: we must include strings on Ko that are closed only up to an elem e n t of G and project onto states that are i n v a r i a n t under G. Furthermore, a string X( a + n) = g X ( a ) that is closed up to g is identified with a string
Supported by National Science Foundation grant NSF/PHY82-17853 supplemented by funds from NASA. 2 Address after 1 September 1988: Department of Physics and Astronomy, Rutgers University, Piscataway, NY 08855, USA. ~ For exactly calculable Calabi-Yau compactifications, see ref. [6].
h X ( a + n) = h g X ( a ) = ( h g h - ' ) h X ( a )
( 1.1 )
that is closed up to the conjugate element h g h - ~, so the closed string states separate into sectors that correspond to the conjugacy classes of G. In addition to the identity sector we now have winding sectors where the string is closed up to a lattice vector X ( a + n) = ~2 See Dixon et al. in ref. [4]. The terminology here is adopted from crystallography. 491
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X(a) +2, and a twist sector where the string is closed
B'
D~A"'
up to a group element g = (0, () with a nontrivial rotation 0~ 1, i.e., X(a+z~)=OX(a)+~. The elements conjugate to a given twist element g = (0, () have the form
g'= (~, ~)(0, ~)(¢-', - ~ - ' ~ ) =(0, ( 1 - O ) q - ( I - O ) ( + ( ) .
(1.2)
The rotation 0 is fixed, but the shift vector ~ is determined only up to the sublattice A' generated by vectors of the form ( 1 - ~ ) ¢ or (1-0)~/, so for fixed 0 the sectors correspond to cosets A / A ' . The center-ofmass coordinate XCM of a twisted string must be a fixed point on the torus XCM = OXcM'4-~and these fixed points on the torus are in one-to-one correspondence with the cosets A / A ' . The tetrahedron provides a simple example of an orbifold that is particularly useful for purposes of visualization and illustration. It is constructed from a complex one-dimensional torus defined by X ~ X + 1 ~ X + c~with the Z2 twist X ~ - X, as illustrated in fig. 1. The choice o~=exp(i~/3) produces a regular tetrahedron.
2. Nonperturbative effects
It is useful to separate the orbifold coordinate X(z) into its classical and quantum pieces
X(z) = Xc,(z) +Xqu (z),
(2.1)
where Xcj satisfies the classical equation of motion 0OXcl = 0.
(2.2)
Because the sigma model action for a flat orbifold is quadatic the action separates into a sum
S[ X] =Sc, + S[ Xqu],
(2.3)
where Sd = S [ Xc, ] is the classical action. This allows a simplification of the boundary conditions of the quantum field Xq. by absorbing the shift vectors (into
a
B
b Fig. 1. (a) The fundamental regions for the torus and the tetrahedron are the rhombus A A ' A " A " and the triangle AA'A". (b) Identifyingthe edges of the triangle according to the orbifold twist gives the tetrahedron.
Xqu (exp (2~ri) (z-z,), exp( - 2~ri) ( ~ - za) )
=OaXq,(Z-Za, 2--Za).
(2.4b)
The functional integral takes into account the fact that twisted string states correspond to conjugacy classes of the orbifold space group by summing over elements of the corresponding lattice coset, subject to the constraint that the string boundary conditions match whenever strings split or join. Since the Xd are the only variables that depend on the shift vectors ~a the functional integral factorizes
. . . .
(2.5) Xc, (exp (2~zi) (z-za), e x p ( - 2 h i ) (g-ga))
=OaX~(z-z., z-ga) +~, so that 492
(2.4a)
The classical action is quadratic in the Ca, which are proportional to the orbifold radius R, so the exponent is proportional to R z. The sum over classical configurations is exact in the sigma model coupling
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c~' and reproduces the instanton sum [ 10 ]. For large radius the instanton contribution will be exponentially suppressed like exp ( - cR 2). The formulation of Hamidi and Vafa [4 ] gives a natural expression for the classical contribution to an amplitude involving twisted vertex operators. The idea is to describe the interactions not on the world sheet Z but on its branched cover Y.. Consider for example the interaction at tree level of four twisted states on a Z2 orbifold and focus first on a single quantum fluctuation field Xqu, which has simpler boundary conditions than X~l. Around each twisted vertex operator Xqu has a v/~ branch cut; when transported around the vertex operator it is multiplied by - 1. We may choose the branch cuts to run between pairs of vertex operators, and Z is the two-sheeted Riemann surface of the function y = [ ( z - z , ) ( z z2 ) ( z - z3 ) ( z - z4 ) ] 1/2, on which Xqu is single-valued. The classical field X~.~may be multiple-valued on but it no longer has singular branch points. In fact, 'Z is topologically equivalent to a torus, with the basis cycles as shown in fig. 2, and the boundary conditions on X~q around these cycles are determined from the shift vectors at the twisted vertex operators they encircle. The boundary conditions for X a r o u n d these cycles are
(~dX=u,
~dX=v,
a
l!
(2.6)
where U = ~ l - ~ 2 and v = ~ - ~ 3 and the overall conservation of twist requires ~1 -- ~2-- ~3 + ~4 = 0. T h e classical action in this case is 1 Iru+vt 2 4~o~' r2
S d - -
- 47ro~' u2
~2
+2u'v
_
+v 2
_
.
(2.7)
Using SL (2, C ) invariance to fix the positions of three of the vertex operators to be 0, 1, and oo, the modular parameter r = r~ + it2 of 5~ is related to the remaining free parameter on 2 by [4 ] z2 = 0 4 ( r ) / 0 4 ( r ) .
(2.8)
Note that the classical action is not modular invariant. Modular transformations will in general permute the positions of the vertex operators on the torus ~, and the classical sum is only invariant under the
22 December 1988
@
~b
l,
a
/ Z4
b Fig. 2. (a) The complex plane with cuts at zj, z2, z3, z4, and the two noncontractible basis cycles a and b. (b) The two-sheeted surface represented as a torus, with the branch points and basis cycles indicated.
subgroup of modular transformations that leave the vertex operators fixed. This construction generalizes easily to any process involving twisted string interactions on any abelian orbifold, but is easiest to describe for Z,~ orbifolds. The twisted vertex operators introduce z k/x branch cuts on the world sheet Z, leading naturally to an Nsheeted Riemann surface ,~ that covers Z. Since the group is abelian its action can be completely diagonalized and we may treat each variable X ( z ) independently. When transported around a closed curve in Z these variables may pick up phases X ( z ) - - . e x p (ik/ N ) X ( z ) and the closed curves on ~. are those for which k = 0 for each X. In addition to the local restrictions on X near the branch points, we must also 493
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impose the global identifications (2.4), which relate the values of X on different sheets of,~. For Xd these identifications follow from the local conditions at the branch points and the equation of motion (2.2). It will be important in what follows to know the genus ~ o f ~ in terms of the genus g of E [4]. This involves a simple additive formula for the Euler charcteristic. Let G~ be the subgroup of the orbifold point group that leaves invariant the fixed point of twisted vertex operator V~, and let Ua be the order of G~. First remove the n fixed points form E, lowering its Euler characteristic by n. Removing the fixed points from 5~therefore gives a space with Euler characteristic N ( Z - n ), where Z is the Euler characteristic of Z and Nis the order of the orbifold point group G. The twisted vertex V~ corresponds to N / v~ points on ,~, so the Euler characteristic of S~is
7.=N(z--n)+ ~ N = N [ x -
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~ O//~f = i ~ ( ~ c~ ~ f -
~ O/~ fl)
";
bi
ai
bi
(2.13b)
ai
for closed one-forms oz and f we can calculate the classical action
Sd= 1 Pilmg2uVj"
(2.14)
gO/'
To express this in terms of shifts associated with basic cycles at and bi it is more useful to use a real cohomology basis O/i and ft defined to satisfy
~aj=~fj=dgj,
~fj=~o/j=O.
ai
ai
bi
We can write at and fg as linear combinations of ~ot and ~0t: O/t = ½[du + i Re g2tk(Im g2)kj ~]o)j +c.c.,
~(1-1)1.
flit = - ½i(Im g2),7 ' o~j+ c.c. (2.9) In terms of the genus Z= 2 - 2g this gives
g=Ug-U+l+ Z
(2.10)
.
(2.11)
The homology basis of a genus-g Riemann surface consists o f g pairs of basis cycles at, bt, i= 1, ..., g with intersection numbers ( ai, aj)= ( bt, bj)=0, (at, bj)= fitj. The cohomology basis consists of g holomorphic one-forms cot (the abelian differentials) and their complex conjugates satisfying ~(J)Jm(~tj,
~(J))=~'~,)"
ai
bi
(2.12)
where £2 is the period matrix. The most general form for Xd that satisfies the equation of motion OOXd= 0 is ~Xc,= ~ V~ot,
~Xd= Et 12to3"
(2.13a)
where V~ is a complex vector on the orbifold. Using the formula 494
(2.16)
The expression for gcl in a real basis is dXd = 0Xcl + ~Xcl =
We will concentrate on tree amplitudes with g = 0, in which case 2 o
(2.15)
bi
blto/, ~- V i f t,
(
2.17 )
where ut and v, are simply the shifts in Xd around the cycles a~ and bt, respectively: ~ dXc, = ut,
~dXd=vt.
ai
bi
(2.18)
We can express the complex vectors V~in terms of ut and v, as
Vt=l[ut+i(Imf2),7'(ReI2jkuk--vj)].
(2.19)
Rewriting the classical action in terms of these variables and suppressing indices gives 1
S d = 4no/' {u[Im g2+Re f2(Im £2)-~Re g2]u
--2uReg?(Img2)-lv+v(Img2)-Iv}.
(2.20)
Using the fact that the forms o)s are linearly independent it is easy to show that the matrix Im (2 is positive definite, which insures that the classical action in the form (2.14) is positive whenever any of the vectors Vi is nonzero. For most terms in the sum over classical confil arations these will in general be nonzero and will give instanton contributions that are exponentially suppressed for large R. The only processes involving twisted states for which all the Vt
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vanish identically are those for which all the twisted states come from the same fixed point. These are the only ones for which the action can be exactly zero, and appear to be the only ones that are not suppressed for large R ~3 It is easy to see why some amplitudes should be suppressed. Consider for example the interaction of four twisted states on the tetrahedron, one for each of the fixed points. The leading contribution to this type of process comes from a world sheet that wraps exactly once around the tetrahedron. The selection rules for the string boundary conditions effectively pin the string to the fixed points and force the world sheet to wrap around some noncontractible surface on the orbifold. Since the bosonic string action is just the area, this leads to an exponential factor exp( --cR2/o! ' ) in the amplitude. An explicit calculation for this case, using u = 1 and v=c~ in eq. (2.7), gives Sc~-
R2 47rO~ ' "E2
( I rl 2+rl + 1 ).
(2.21)
This does not go to zero for any value of the modular parameter describing the torus, and in fact has a minimum at z=exp(2~zi/3) where X d = R z is holomorphic and Sol = R 2x/~/4~za' is just the area of the tetrahedron. Integrating over z will give an exponential suppression e x p ( - c R 2) for large R, the single holomorphic instanton contribution [ 10 ]. This description in terms of pinning makes it difficult to understand why all interactions involving more than one fixed point should be exponentially suppressed. In particular, processes for which the total twist of all the string states vanishes separately for each fixed point of the orbifold group are not pinned to the fixed points and the minimum area of a surface spanning them is zero. How can the vanishing of the minimum area for these unpinned amplitudes be reconciled with the positivity of the action (2.15 )? In fact, for unpinned amplitudes the action remains positive and has the form cR2/o~ ', but the coefficient c degenerates to zero at boundaries of the parameter space. This happens when vertex opera~3 These amplitudes may be suppressed for other reasons, for example if they appear with factors of OX~bin which case the contributions with zero classical action vanish, having c3X~q=0 [11].
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tors corresponding to the same fixed point coalesce z~--,z~,. We can see this explicitly for the interaction of two twisted states from each of two fixed points of the tetrahedron. We may choose the unpinned amplitude with ~1=~2=0, ~3-----~4=R. Then u = 0 and v=R in eq. (2.7) and the classical action becomes R2
&,-
4~to~'r2
(2.22)
This goes to zero as ~'2-~Go, corresponding to z:-+0. In that region of parameter space the exponential suppression disappears and we can no longer conclude that the amplitude is exponentially suppressed. A similar effect occurs for unpinned amplitudes on any abelian orbifold; the classical action goes to zero when the vertex operators for each fixed point coalesce.
3. Factorization of unpinned amplitudes As described above, an amplitude is pinned to a given fixed point )(1 if the vertex operators for states twisted about )(1 have nonzero total twist; a loop yl surrounding all these vertex operators has twisted boundary conditions. An amplitude is unpinned if it is not pinned to any fixed point. The region of moduli space where all the vertex operators z~ for the fixed point X~ coalesce corresponds to the region where this loop Ytpinches off, i.e. to the factorization of the amplitude onto the Yl channel. For pinned amplitudes this is factorization onto a twisted state, while for unpinned amplitudes it is factorization onto an untwisted state. In the factorization limit the amplitude decomposes into a product of amplitudes for lower order processes. For each fixed point there is an amplitude containing only vertex operators for states twisted about that point. In addition there is amplitude containing all the untwisted states and the factorized states for each of the loops 7i. The factors involving only a single fixed point are not exponentially suppressed, so any exponential suppression must come from the final factor. For unpinned amplitudes this factor does not contain any twisted states so it cannot be exponentially suppressed; the exponential suppression of these amplitudes should disappear in the 495
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factorization limit. This is verified by explicit calculation below ~4. Blowing up an orbifold fixed point corresponds to giving an expectation value to a particular massless scalar mode twisted about that fixed point. We can calculate amplitudes perturbatively in this expectation value by inserting the corresponding vertex operator into scattering amplitudes. With enough insertions of this vertex operator we can make the total twist about the fixed point vanish, and when we blow up all the fixed points we can find contributions to any amplitude that are unpinned. By calculating to high enough order in the blow-up modes we can find a contribution to any scattering amplitude that is not exponentially suppressed at large radius. It is useful to understand what these results for scattering amplitudes imply for the low-energy effective action of the massless modes. The scattering amplitudes are simply the tree diagrams o f the effective field theory. To recover the effective N-point vertex functions from these amplitudes we must subtract from N-point amplitude the contribution of products of lower-point vertex functions connected by massless propagators. These are precisely the contributions to amplitudes that result from factorization onto massless intermediate states, so these contributions do not modify the effective action. Since the massive states are not included in the low-energy effective field theory, amplitudes factorized on massive states do modify the effective action. Thus, the new contributions of unpinned amplitudes to the effective action come from factorization onto massive untwisted states. These include states with m o m e n t u m in the compact directions of order 1/R, whose propagators behave like 1 / M 2 ~ R 2. Far from being exponentially suppressed at large R, these couplings blow up in that limit. There is a simple explanation for this: at very large R the effective field theory involving only the exactly massless fields breaks down as the mass o f these m o m e n t u m states approaches zero. It is instructive to see how the general arguments based on factorization are born out in the behavior of the classical contribution to the bosonic functional ~4This analysis confirms the conclusions of Cvetic [ 1l ]. Amplitudes containing extra factors of aX are exponentially suppressed away from the factorization limit, while in that limit the factors for each fixed point have OX= 0 and vanish. 496
22 December 1988
integral. The outline of this analysis is as follows: First we construct a homology basis on £ for which the translations ui in X about the cycles ai vanish. This is a basis in which each a cycle encircles only vertex operators from a single fixed point. The classical action then takes the form Scl = vi(Im g2),71 v!.
(3.1)
In the factorization limit z~t, = z ~ - z ~ ~ e x p ( - 2 ) z a b for vertex operators from a single fixed point X 1 the classical action scales as Sd~Sc~/2 and goes to zero as 2 ~ o~. This limit corresponds to the shrinking of some of the a cycles, or alternatively to the stretching of the corresponding b cycles. At large 2 the period matrix behaves like Im £2~2 Im 12o + ~ Im £2,
(3.2)
where the divergent piece 2 Im £2o is nondegenerate on the space o f nonvanishing vectors vi, the vectors corresponding to the stretched b cycles. (Im g2o)scales like 2 ~on that subspace and the action (2.20) goes to zero as 2--+oo. We construct the indicated basis of a cycles using a simple counting argument: We construct a set of linearly independent cycles of the given form and show that there are ~ of them. Again we concentrate on orbifolds with the cyclic point group ZN, but the general abelian case is a trivial extension to a product of such groups. First consider the case where the twisted vertex operators come in M oppositely twisted pairs. Then ~=l-N+-~a_~
1-
(3.3)
is the genus of the branched cover I~ for tree amplitudes. We need to construct ~ linearly independent nonintersecting a cycles. The required cycles are those that circle such a pair of oppositely twisted vertex operators on one of the N sheets of~., like the a cycle in fig. 2a. The number of independent cycles of this form is the total number of such cycles, NM, minus the number of linear dependence relations between them. There are two types of dependence relations. For each sheet there is the relation that the loop surrounding all the vertex operators is contractible, so the sum o f all the cycles on that sheet is a trivial cycle. This gives N relations. For each pair o f vertex opera-
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tors there are an additional N / v ~ relations coming from the fact that a loop that surrounds the pair v~ times is contractible. Among these N + ½E~N/vo relations there is one trivial redundancy: the sum of the relations of the first type is the same as the sum of the relations of the second type. Thus there are 1
1
N -N+
2~U-2~v~
1 =~
(3.4)
independent cycles of the required form. The generalization to arbitrary unpinned amplitudes is somewhat more complicated. To start, we focus on the twisted vertex operators from a single fixed point X~. Suppose that these have twists k~, k2..... k .... i.e. that X is twisted by exp(2nikJN) about the ath vertex operator. These must satisfy Ek~=kaV~-O (mod N). Letting Nz be the order of the group generated by these twists we can construct an Nr-sheeted surface of genus 1 ~(p_~)
N,-
gi = ~
-NI + I
(3.5)
a=l
from these vertex operators alone, with ~, nonintersecting a cycles. Note that the corresponding b cycles also involve only the same subset of vertex operators, so that their shift vectors v~will vanish as well. ]~ will contain N / N t copies of each of these, or ~,~N/NI linearly independent non-intersecting cycles. In addition, we may take the N cycles 7}, i = 1, ..., N, one for each sheet of the cover, that encircle all m vertices on that sheet. These cycles, the only ones that appear in the special case when the vertices come in oppositely twisted pairs, will be the most important in the subsequent analysis. Doing the same thing for each fixed point, we end up with E z~,tN/N, + N nonintersecting cycles. Not all of these are linearly independent, however. On each sheet the sum Z,7} is contractible, and for each fixed point there are N IN, dependence relations among the cycles YS. As before, there is one trivial redundancy between these sets of relations, and we find
(
N)
(~,-1) ~-+N-N+
1=~
(3.6)
independent nonintersecting cycles, as required.
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Next, we demonstrate that the period matrix scales as (3.2) as the cycles y} shrink to zero, i.e. as the vertex operators for each fixed point X~ coalesce zai --, zb. 1 For this, we need to know how the corresponding abelian differentials behave. These differentials are normalized by (2.12 )
~ %=5o. at
It is only the a cycles 75 whose corresponding b cycles have nonzero shift vectors and it is on this subspace that the period matrix must scale linearly with 2. The abelian differentials corresponding to the cycles y~ behave like o ) ~ ( 1 / 2 n i ) ( d z ) / ( z - z I) on the ith sheet (where the cycle 75 lies) for z - z ~ >>zal_zb,l since the integral (2.12) then picks up the properly normalized residue. On the sheets where the other chosen cycles 7~ lie co must have zero residue in order to be orthogonal to those cycles. On the remaining sheets, whose cycles 7~ are linear combinations of the chosen basis cycles, the residues must be such that the contour integral of ~o around any contractible cycle vanishes. For z ~/n branch points, for example, the residue on the nth sheet must be - 1/2hi in order that the total integral Eif~,~oJ around the contractible cycle ~,7} vanish. It is easiest to describe the calculation of the period matrix in the case where the vertices come in oppositely twisted pairs, where all the b cycles correspond to a cycles Y}. Consider one pair of vertices and the set of a cycles for this pair of the form 7~. In the neighborhood of these vertices the corresponding b cycles have the form depicted in fig. 3. They approach the branch cut along the sheet where they intersect their corresponding a cycles and leave on a sheet where they intersect none of the a cycles. The integral of the associated abelian differentials a) along such contours then have the general form
¢
f
die"
(3.7)
bi
J, z 1
z 2
Fig. 3. The behavior of a typical b cycle in the neighborhoodof an oppositelytwisted pair of branch points. 497
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These integrals are logarithmically divergent as r = Izl--*0, but this divergence is cutoff at the scale IZ~a--Z:,[, SO that the corresponding elements of the period matrix behave like
f2i/=_iCi/logl z ,I- z : , l1.
(3.8)
The description in the previous paragraph shows that the matrix C~: has the nondegenerate form
C,,=c
2:
... "..
1
,..
.
Acknowledgement
It is a pleasure to thank Paul Griffin, Rob Myers, and Xiao-Gang Wen for helpful discussions. I am particularly grateful to Lance Dixon for pointing out a serious conceptual error in a previous version of this work.
(3.9)
4. Conclusions The chief virtue of orbifolds for string compactification is the existence of a systematic procedure for calculating scattering amplitudes. These include an exact treatment of the sigma-model, including instantons and other n o n p e r t u r b a t i v e effects as well as a systematic expansion in the blow-up parameters on symmetric (2,2) orbifolds. In practice, the full amplitudes may be difficult to evaluate, but a n u m b e r of general properties can be extracted from selection rules and from the qualitative behavior of the classical contribution. Many contributions to the low-energy effective field theory are exponentially suppressed on large orbifolds, a property which may be useful in analyzing the resulting phenomenology. The criterion which determines which couplings are suppressed has a simple topological interpretation in terms of strings propagating on the corresponding orbifold. Amplitudes for which the string is p i n n e d to any of the fixed points are exponentially suppressed, and other amplitudes ar not. W h e n the orbifold fixed points are blown up the insertion of the blow-up modes can unpin an amplitude, eliminating an exponential suppression that would otherwise be present.
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References [ 1] J. Scberk and J. Schwarz, Nucl. Phys. B 81 (1974) I 18; M. Green and J. Schwarz, Phys. Lett. B 109 (1982) 444; M. Green and J. Schwarz, Phys. geu. B 149 (1984) 117. [2] D.J. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B 256 (1985) 253. [ 3 ] L.J. Dixon, J, Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678: B 274 (1986) 285. [4] S. Hamidi and C. Vafa, Nucl. Phys. B 297 (1987) 465; L.J. Dixon, D. Friedan, E. Martinec and S. Sbenker, Nucl. Phys. B 282 (1987) 13. [5] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [6 ] D. Gepner, Phys. Len. B 199 ( 1987 ) 380;Princeton preprint PUPT-1093. [7] K.S. Narain, Phys. Lett. B 169 (1986) 41; K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369. [8] A. Strominger, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986); D.G. Markushevich, M.A. Olshlanetsky and A.M. Perelomov, Commun. Math. Phys. 111 (1987) 247. [9] L.J. Dixon, in: Proc. Workshop in High energy physics and cosmology (Trieste, Italy, 1987), to be published; M. Cvetic, SLAC report SLAC-PUB-4325,in: Proc. Eighth Workshop on Grand unification (Syracuse, New York, April 1987), to be published; SLAC report SLAC-PUB-4324,in: Proc. Intern. Workshop on Superstrings, composite structures, and cosmology (University of Maryland, March 1987)). [ 10] X.-G. Wen and E. Witten, Phys. Lett. B 166 (1986) 397; M. Dine, N. Seiberg, X.-G. Wen and E. Witten, Nucl. Phys. B 278 (1986) 769; B 289 (1987) 317. [ 1l ] M. Cvetic, Phys. Rev. Lett. 59 (1987) 1795.
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22 December 1988
EXPONENTIAL SUPPRESSION OF STRING INTERACTIONS ON ORBIFOLDS Mark W. G O O D M A N ~.2
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Received 1 July 1988
Nonperturbative world-sheet instanton effects can be seen in the classical bosonic contribution to scattering amplitudes on orbifolds. These instanton contributions have the form exp ( -cR 2/a' ) and are exponentiallysuppressed when the orbifold radius is large. Amplitudes whose leading contribution comes from instantons are identified with processes in which the corresponding string states are pinned to the orbifold fixed points, and unpinned amplitudes are shown to lack this suppression at large radius. Blowingup the fixed points of an orbifold eliminates any pinningand with it the corresponding exponential suppression.
1. Introduction
fold space group G generated by the lattice A and elements of the orbifold point group Go g2. The ter-
Compactification of superstring [ 1 ] a n d heterotic string [2 ] theories on flat orbifolds [3] has the advantage of explicit calculability [ 4 ] that is normally absent in C a l a b i - Y a u compactification [5] ~, and a realistic chiral spectrum that does not appear in toroidal compactifications [7]. For compactification from ten to four d i m e n s i o n s the internal space is an orbifold that is the quotient K = T 6 / G o of a six-dimensional torus T 6 - - R 6 / A by a finite symmetry group Go of the lattice A. The elements ge Go will have fixed points g x = x on T 6 that become conical singularities of K, but these singularities do not lead to any inconsistency in the resulting string model. For a compactification with (2,2) superconformal symmetry these singularities can be "blown up" to produce a n o n s i n g u l a r C a l a b i - Y a u manifold [8] and there is a systematic p e r t u r b a t i o n expansion in the expectation value of the blow-up mode, a massless scalar in the twisted string sector [ 9 ]. For our purposes it will be simpler to describe a flat orbifold as the quotient K - - - R 6 / G of a six-dimensional euclidean space u n d e r the action of the orbi-
minology is precisely analogous to that used in crystallography. An element of G is written as a pair (0, (), where 0 is a rotation and ( is a translation, and acts on points in R 6 as x ~ O x + ( . The lattice consists of pure translations ( 1 , 2 ) a n d forms a normal subgroup ( 8 , ~ ) ( 1 , 2 ) ( 0 - ' , - 0 - 1 ~ ) = ( 1 , 8 2 ) . The point group is the quotient G o = G / A . I will assume for simplicity that the point group is abelian, so that the rotation pieces of the group transformations commute: 0 ¢ = ¢0. First we recall some basic facts about the states of a closed string on an orbifold K = K o / G [3,4]. We can describe these states in terms of the covering space K o = R 6 with two simple modifications: we must include strings on Ko that are closed only up to an elem e n t of G and project onto states that are i n v a r i a n t under G. Furthermore, a string X( a + n) = g X ( a ) that is closed up to g is identified with a string
Supported by National Science Foundation grant NSF/PHY82-17853 supplemented by funds from NASA. 2 Address after 1 September 1988: Department of Physics and Astronomy, Rutgers University, Piscataway, NY 08855, USA. ~ For exactly calculable Calabi-Yau compactifications, see ref. [6].
h X ( a + n) = h g X ( a ) = ( h g h - ' ) h X ( a )
( 1.1 )
that is closed up to the conjugate element h g h - ~, so the closed string states separate into sectors that correspond to the conjugacy classes of G. In addition to the identity sector we now have winding sectors where the string is closed up to a lattice vector X ( a + n) = ~2 See Dixon et al. in ref. [4]. The terminology here is adopted from crystallography. 491
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X(a) +2, and a twist sector where the string is closed
B'
D~A"'
up to a group element g = (0, () with a nontrivial rotation 0~ 1, i.e., X(a+z~)=OX(a)+~. The elements conjugate to a given twist element g = (0, () have the form
g'= (~, ~)(0, ~)(¢-', - ~ - ' ~ ) =(0, ( 1 - O ) q - ( I - O ) ( + ( ) .
(1.2)
The rotation 0 is fixed, but the shift vector ~ is determined only up to the sublattice A' generated by vectors of the form ( 1 - ~ ) ¢ or (1-0)~/, so for fixed 0 the sectors correspond to cosets A / A ' . The center-ofmass coordinate XCM of a twisted string must be a fixed point on the torus XCM = OXcM'4-~and these fixed points on the torus are in one-to-one correspondence with the cosets A / A ' . The tetrahedron provides a simple example of an orbifold that is particularly useful for purposes of visualization and illustration. It is constructed from a complex one-dimensional torus defined by X ~ X + 1 ~ X + c~with the Z2 twist X ~ - X, as illustrated in fig. 1. The choice o~=exp(i~/3) produces a regular tetrahedron.
2. Nonperturbative effects
It is useful to separate the orbifold coordinate X(z) into its classical and quantum pieces
X(z) = Xc,(z) +Xqu (z),
(2.1)
where Xcj satisfies the classical equation of motion 0OXcl = 0.
(2.2)
Because the sigma model action for a flat orbifold is quadatic the action separates into a sum
S[ X] =Sc, + S[ Xqu],
(2.3)
where Sd = S [ Xc, ] is the classical action. This allows a simplification of the boundary conditions of the quantum field Xq. by absorbing the shift vectors (into
a
B
b Fig. 1. (a) The fundamental regions for the torus and the tetrahedron are the rhombus A A ' A " A " and the triangle AA'A". (b) Identifyingthe edges of the triangle according to the orbifold twist gives the tetrahedron.
Xqu (exp (2~ri) (z-z,), exp( - 2~ri) ( ~ - za) )
=OaXq,(Z-Za, 2--Za).
(2.4b)
The functional integral takes into account the fact that twisted string states correspond to conjugacy classes of the orbifold space group by summing over elements of the corresponding lattice coset, subject to the constraint that the string boundary conditions match whenever strings split or join. Since the Xd are the only variables that depend on the shift vectors ~a the functional integral factorizes
. . . .
(2.5) Xc, (exp (2~zi) (z-za), e x p ( - 2 h i ) (g-ga))
=OaX~(z-z., z-ga) +~, so that 492
(2.4a)
The classical action is quadratic in the Ca, which are proportional to the orbifold radius R, so the exponent is proportional to R z. The sum over classical configurations is exact in the sigma model coupling
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c~' and reproduces the instanton sum [ 10 ]. For large radius the instanton contribution will be exponentially suppressed like exp ( - cR 2). The formulation of Hamidi and Vafa [4 ] gives a natural expression for the classical contribution to an amplitude involving twisted vertex operators. The idea is to describe the interactions not on the world sheet Z but on its branched cover Y.. Consider for example the interaction at tree level of four twisted states on a Z2 orbifold and focus first on a single quantum fluctuation field Xqu, which has simpler boundary conditions than X~l. Around each twisted vertex operator Xqu has a v/~ branch cut; when transported around the vertex operator it is multiplied by - 1. We may choose the branch cuts to run between pairs of vertex operators, and Z is the two-sheeted Riemann surface of the function y = [ ( z - z , ) ( z z2 ) ( z - z3 ) ( z - z4 ) ] 1/2, on which Xqu is single-valued. The classical field X~.~may be multiple-valued on but it no longer has singular branch points. In fact, 'Z is topologically equivalent to a torus, with the basis cycles as shown in fig. 2, and the boundary conditions on X~q around these cycles are determined from the shift vectors at the twisted vertex operators they encircle. The boundary conditions for X a r o u n d these cycles are
(~dX=u,
~dX=v,
a
l!
(2.6)
where U = ~ l - ~ 2 and v = ~ - ~ 3 and the overall conservation of twist requires ~1 -- ~2-- ~3 + ~4 = 0. T h e classical action in this case is 1 Iru+vt 2 4~o~' r2
S d - -
- 47ro~' u2
~2
+2u'v
_
+v 2
_
.
(2.7)
Using SL (2, C ) invariance to fix the positions of three of the vertex operators to be 0, 1, and oo, the modular parameter r = r~ + it2 of 5~ is related to the remaining free parameter on 2 by [4 ] z2 = 0 4 ( r ) / 0 4 ( r ) .
(2.8)
Note that the classical action is not modular invariant. Modular transformations will in general permute the positions of the vertex operators on the torus ~, and the classical sum is only invariant under the
22 December 1988
@
~b
l,
a
/ Z4
b Fig. 2. (a) The complex plane with cuts at zj, z2, z3, z4, and the two noncontractible basis cycles a and b. (b) The two-sheeted surface represented as a torus, with the branch points and basis cycles indicated.
subgroup of modular transformations that leave the vertex operators fixed. This construction generalizes easily to any process involving twisted string interactions on any abelian orbifold, but is easiest to describe for Z,~ orbifolds. The twisted vertex operators introduce z k/x branch cuts on the world sheet Z, leading naturally to an Nsheeted Riemann surface ,~ that covers Z. Since the group is abelian its action can be completely diagonalized and we may treat each variable X ( z ) independently. When transported around a closed curve in Z these variables may pick up phases X ( z ) - - . e x p (ik/ N ) X ( z ) and the closed curves on ~. are those for which k = 0 for each X. In addition to the local restrictions on X near the branch points, we must also 493
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impose the global identifications (2.4), which relate the values of X on different sheets of,~. For Xd these identifications follow from the local conditions at the branch points and the equation of motion (2.2). It will be important in what follows to know the genus ~ o f ~ in terms of the genus g of E [4]. This involves a simple additive formula for the Euler charcteristic. Let G~ be the subgroup of the orbifold point group that leaves invariant the fixed point of twisted vertex operator V~, and let Ua be the order of G~. First remove the n fixed points form E, lowering its Euler characteristic by n. Removing the fixed points from 5~therefore gives a space with Euler characteristic N ( Z - n ), where Z is the Euler characteristic of Z and Nis the order of the orbifold point group G. The twisted vertex V~ corresponds to N / v~ points on ,~, so the Euler characteristic of S~is
7.=N(z--n)+ ~ N = N [ x -
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~ O//~f = i ~ ( ~ c~ ~ f -
~ O/~ fl)
";
bi
ai
bi
(2.13b)
ai
for closed one-forms oz and f we can calculate the classical action
Sd= 1 Pilmg2uVj"
(2.14)
gO/'
To express this in terms of shifts associated with basic cycles at and bi it is more useful to use a real cohomology basis O/i and ft defined to satisfy
~aj=~fj=dgj,
~fj=~o/j=O.
ai
ai
bi
We can write at and fg as linear combinations of ~ot and ~0t: O/t = ½[du + i Re g2tk(Im g2)kj ~]o)j +c.c.,
~(1-1)1.
flit = - ½i(Im g2),7 ' o~j+ c.c. (2.9) In terms of the genus Z= 2 - 2g this gives
g=Ug-U+l+ Z
(2.10)
.
(2.11)
The homology basis of a genus-g Riemann surface consists o f g pairs of basis cycles at, bt, i= 1, ..., g with intersection numbers ( ai, aj)= ( bt, bj)=0, (at, bj)= fitj. The cohomology basis consists of g holomorphic one-forms cot (the abelian differentials) and their complex conjugates satisfying ~(J)Jm(~tj,
~(J))=~'~,)"
ai
bi
(2.12)
where £2 is the period matrix. The most general form for Xd that satisfies the equation of motion OOXd= 0 is ~Xc,= ~ V~ot,
~Xd= Et 12to3"
(2.13a)
where V~ is a complex vector on the orbifold. Using the formula 494
(2.16)
The expression for gcl in a real basis is dXd = 0Xcl + ~Xcl =
We will concentrate on tree amplitudes with g = 0, in which case 2 o
(2.15)
bi
blto/, ~- V i f t,
(
2.17 )
where ut and v, are simply the shifts in Xd around the cycles a~ and bt, respectively: ~ dXc, = ut,
~dXd=vt.
ai
bi
(2.18)
We can express the complex vectors V~in terms of ut and v, as
Vt=l[ut+i(Imf2),7'(ReI2jkuk--vj)].
(2.19)
Rewriting the classical action in terms of these variables and suppressing indices gives 1
S d = 4no/' {u[Im g2+Re f2(Im £2)-~Re g2]u
--2uReg?(Img2)-lv+v(Img2)-Iv}.
(2.20)
Using the fact that the forms o)s are linearly independent it is easy to show that the matrix Im (2 is positive definite, which insures that the classical action in the form (2.14) is positive whenever any of the vectors Vi is nonzero. For most terms in the sum over classical confil arations these will in general be nonzero and will give instanton contributions that are exponentially suppressed for large R. The only processes involving twisted states for which all the Vt
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vanish identically are those for which all the twisted states come from the same fixed point. These are the only ones for which the action can be exactly zero, and appear to be the only ones that are not suppressed for large R ~3 It is easy to see why some amplitudes should be suppressed. Consider for example the interaction of four twisted states on the tetrahedron, one for each of the fixed points. The leading contribution to this type of process comes from a world sheet that wraps exactly once around the tetrahedron. The selection rules for the string boundary conditions effectively pin the string to the fixed points and force the world sheet to wrap around some noncontractible surface on the orbifold. Since the bosonic string action is just the area, this leads to an exponential factor exp( --cR2/o! ' ) in the amplitude. An explicit calculation for this case, using u = 1 and v=c~ in eq. (2.7), gives Sc~-
R2 47rO~ ' "E2
( I rl 2+rl + 1 ).
(2.21)
This does not go to zero for any value of the modular parameter describing the torus, and in fact has a minimum at z=exp(2~zi/3) where X d = R z is holomorphic and Sol = R 2x/~/4~za' is just the area of the tetrahedron. Integrating over z will give an exponential suppression e x p ( - c R 2) for large R, the single holomorphic instanton contribution [ 10 ]. This description in terms of pinning makes it difficult to understand why all interactions involving more than one fixed point should be exponentially suppressed. In particular, processes for which the total twist of all the string states vanishes separately for each fixed point of the orbifold group are not pinned to the fixed points and the minimum area of a surface spanning them is zero. How can the vanishing of the minimum area for these unpinned amplitudes be reconciled with the positivity of the action (2.15 )? In fact, for unpinned amplitudes the action remains positive and has the form cR2/o~ ', but the coefficient c degenerates to zero at boundaries of the parameter space. This happens when vertex opera~3 These amplitudes may be suppressed for other reasons, for example if they appear with factors of OX~bin which case the contributions with zero classical action vanish, having c3X~q=0 [11].
22 December 1988
tors corresponding to the same fixed point coalesce z~--,z~,. We can see this explicitly for the interaction of two twisted states from each of two fixed points of the tetrahedron. We may choose the unpinned amplitude with ~1=~2=0, ~3-----~4=R. Then u = 0 and v=R in eq. (2.7) and the classical action becomes R2
&,-
4~to~'r2
(2.22)
This goes to zero as ~'2-~Go, corresponding to z:-+0. In that region of parameter space the exponential suppression disappears and we can no longer conclude that the amplitude is exponentially suppressed. A similar effect occurs for unpinned amplitudes on any abelian orbifold; the classical action goes to zero when the vertex operators for each fixed point coalesce.
3. Factorization of unpinned amplitudes As described above, an amplitude is pinned to a given fixed point )(1 if the vertex operators for states twisted about )(1 have nonzero total twist; a loop yl surrounding all these vertex operators has twisted boundary conditions. An amplitude is unpinned if it is not pinned to any fixed point. The region of moduli space where all the vertex operators z~ for the fixed point X~ coalesce corresponds to the region where this loop Ytpinches off, i.e. to the factorization of the amplitude onto the Yl channel. For pinned amplitudes this is factorization onto a twisted state, while for unpinned amplitudes it is factorization onto an untwisted state. In the factorization limit the amplitude decomposes into a product of amplitudes for lower order processes. For each fixed point there is an amplitude containing only vertex operators for states twisted about that point. In addition there is amplitude containing all the untwisted states and the factorized states for each of the loops 7i. The factors involving only a single fixed point are not exponentially suppressed, so any exponential suppression must come from the final factor. For unpinned amplitudes this factor does not contain any twisted states so it cannot be exponentially suppressed; the exponential suppression of these amplitudes should disappear in the 495
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factorization limit. This is verified by explicit calculation below ~4. Blowing up an orbifold fixed point corresponds to giving an expectation value to a particular massless scalar mode twisted about that fixed point. We can calculate amplitudes perturbatively in this expectation value by inserting the corresponding vertex operator into scattering amplitudes. With enough insertions of this vertex operator we can make the total twist about the fixed point vanish, and when we blow up all the fixed points we can find contributions to any amplitude that are unpinned. By calculating to high enough order in the blow-up modes we can find a contribution to any scattering amplitude that is not exponentially suppressed at large radius. It is useful to understand what these results for scattering amplitudes imply for the low-energy effective action of the massless modes. The scattering amplitudes are simply the tree diagrams o f the effective field theory. To recover the effective N-point vertex functions from these amplitudes we must subtract from N-point amplitude the contribution of products of lower-point vertex functions connected by massless propagators. These are precisely the contributions to amplitudes that result from factorization onto massless intermediate states, so these contributions do not modify the effective action. Since the massive states are not included in the low-energy effective field theory, amplitudes factorized on massive states do modify the effective action. Thus, the new contributions of unpinned amplitudes to the effective action come from factorization onto massive untwisted states. These include states with m o m e n t u m in the compact directions of order 1/R, whose propagators behave like 1 / M 2 ~ R 2. Far from being exponentially suppressed at large R, these couplings blow up in that limit. There is a simple explanation for this: at very large R the effective field theory involving only the exactly massless fields breaks down as the mass o f these m o m e n t u m states approaches zero. It is instructive to see how the general arguments based on factorization are born out in the behavior of the classical contribution to the bosonic functional ~4This analysis confirms the conclusions of Cvetic [ 1l ]. Amplitudes containing extra factors of aX are exponentially suppressed away from the factorization limit, while in that limit the factors for each fixed point have OX= 0 and vanish. 496
22 December 1988
integral. The outline of this analysis is as follows: First we construct a homology basis on £ for which the translations ui in X about the cycles ai vanish. This is a basis in which each a cycle encircles only vertex operators from a single fixed point. The classical action then takes the form Scl = vi(Im g2),71 v!.
(3.1)
In the factorization limit z~t, = z ~ - z ~ ~ e x p ( - 2 ) z a b for vertex operators from a single fixed point X 1 the classical action scales as Sd~Sc~/2 and goes to zero as 2 ~ o~. This limit corresponds to the shrinking of some of the a cycles, or alternatively to the stretching of the corresponding b cycles. At large 2 the period matrix behaves like Im £2~2 Im 12o + ~ Im £2,
(3.2)
where the divergent piece 2 Im £2o is nondegenerate on the space o f nonvanishing vectors vi, the vectors corresponding to the stretched b cycles. (Im g2o)scales like 2 ~on that subspace and the action (2.20) goes to zero as 2--+oo. We construct the indicated basis of a cycles using a simple counting argument: We construct a set of linearly independent cycles of the given form and show that there are ~ of them. Again we concentrate on orbifolds with the cyclic point group ZN, but the general abelian case is a trivial extension to a product of such groups. First consider the case where the twisted vertex operators come in M oppositely twisted pairs. Then ~=l-N+-~a_~
1-
(3.3)
is the genus of the branched cover I~ for tree amplitudes. We need to construct ~ linearly independent nonintersecting a cycles. The required cycles are those that circle such a pair of oppositely twisted vertex operators on one of the N sheets of~., like the a cycle in fig. 2a. The number of independent cycles of this form is the total number of such cycles, NM, minus the number of linear dependence relations between them. There are two types of dependence relations. For each sheet there is the relation that the loop surrounding all the vertex operators is contractible, so the sum o f all the cycles on that sheet is a trivial cycle. This gives N relations. For each pair o f vertex opera-
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tors there are an additional N / v ~ relations coming from the fact that a loop that surrounds the pair v~ times is contractible. Among these N + ½E~N/vo relations there is one trivial redundancy: the sum of the relations of the first type is the same as the sum of the relations of the second type. Thus there are 1
1
N -N+
2~U-2~v~
1 =~
(3.4)
independent cycles of the required form. The generalization to arbitrary unpinned amplitudes is somewhat more complicated. To start, we focus on the twisted vertex operators from a single fixed point X~. Suppose that these have twists k~, k2..... k .... i.e. that X is twisted by exp(2nikJN) about the ath vertex operator. These must satisfy Ek~=kaV~-O (mod N). Letting Nz be the order of the group generated by these twists we can construct an Nr-sheeted surface of genus 1 ~(p_~)
N,-
gi = ~
-NI + I
(3.5)
a=l
from these vertex operators alone, with ~, nonintersecting a cycles. Note that the corresponding b cycles also involve only the same subset of vertex operators, so that their shift vectors v~will vanish as well. ]~ will contain N / N t copies of each of these, or ~,~N/NI linearly independent non-intersecting cycles. In addition, we may take the N cycles 7}, i = 1, ..., N, one for each sheet of the cover, that encircle all m vertices on that sheet. These cycles, the only ones that appear in the special case when the vertices come in oppositely twisted pairs, will be the most important in the subsequent analysis. Doing the same thing for each fixed point, we end up with E z~,tN/N, + N nonintersecting cycles. Not all of these are linearly independent, however. On each sheet the sum Z,7} is contractible, and for each fixed point there are N IN, dependence relations among the cycles YS. As before, there is one trivial redundancy between these sets of relations, and we find
(
N)
(~,-1) ~-+N-N+
1=~
(3.6)
independent nonintersecting cycles, as required.
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Next, we demonstrate that the period matrix scales as (3.2) as the cycles y} shrink to zero, i.e. as the vertex operators for each fixed point X~ coalesce zai --, zb. 1 For this, we need to know how the corresponding abelian differentials behave. These differentials are normalized by (2.12 )
~ %=5o. at
It is only the a cycles 75 whose corresponding b cycles have nonzero shift vectors and it is on this subspace that the period matrix must scale linearly with 2. The abelian differentials corresponding to the cycles y~ behave like o ) ~ ( 1 / 2 n i ) ( d z ) / ( z - z I) on the ith sheet (where the cycle 75 lies) for z - z ~ >>zal_zb,l since the integral (2.12) then picks up the properly normalized residue. On the sheets where the other chosen cycles 7~ lie co must have zero residue in order to be orthogonal to those cycles. On the remaining sheets, whose cycles 7~ are linear combinations of the chosen basis cycles, the residues must be such that the contour integral of ~o around any contractible cycle vanishes. For z ~/n branch points, for example, the residue on the nth sheet must be - 1/2hi in order that the total integral Eif~,~oJ around the contractible cycle ~,7} vanish. It is easiest to describe the calculation of the period matrix in the case where the vertices come in oppositely twisted pairs, where all the b cycles correspond to a cycles Y}. Consider one pair of vertices and the set of a cycles for this pair of the form 7~. In the neighborhood of these vertices the corresponding b cycles have the form depicted in fig. 3. They approach the branch cut along the sheet where they intersect their corresponding a cycles and leave on a sheet where they intersect none of the a cycles. The integral of the associated abelian differentials a) along such contours then have the general form
¢
f
die"
(3.7)
bi
J, z 1
z 2
Fig. 3. The behavior of a typical b cycle in the neighborhoodof an oppositelytwisted pair of branch points. 497
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These integrals are logarithmically divergent as r = Izl--*0, but this divergence is cutoff at the scale IZ~a--Z:,[, SO that the corresponding elements of the period matrix behave like
f2i/=_iCi/logl z ,I- z : , l1.
(3.8)
The description in the previous paragraph shows that the matrix C~: has the nondegenerate form
C,,=c
2:
... "..
1
,..
.
Acknowledgement
It is a pleasure to thank Paul Griffin, Rob Myers, and Xiao-Gang Wen for helpful discussions. I am particularly grateful to Lance Dixon for pointing out a serious conceptual error in a previous version of this work.
(3.9)
4. Conclusions The chief virtue of orbifolds for string compactification is the existence of a systematic procedure for calculating scattering amplitudes. These include an exact treatment of the sigma-model, including instantons and other n o n p e r t u r b a t i v e effects as well as a systematic expansion in the blow-up parameters on symmetric (2,2) orbifolds. In practice, the full amplitudes may be difficult to evaluate, but a n u m b e r of general properties can be extracted from selection rules and from the qualitative behavior of the classical contribution. Many contributions to the low-energy effective field theory are exponentially suppressed on large orbifolds, a property which may be useful in analyzing the resulting phenomenology. The criterion which determines which couplings are suppressed has a simple topological interpretation in terms of strings propagating on the corresponding orbifold. Amplitudes for which the string is p i n n e d to any of the fixed points are exponentially suppressed, and other amplitudes ar not. W h e n the orbifold fixed points are blown up the insertion of the blow-up modes can unpin an amplitude, eliminating an exponential suppression that would otherwise be present.
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References [ 1] J. Scberk and J. Schwarz, Nucl. Phys. B 81 (1974) I 18; M. Green and J. Schwarz, Phys. Lett. B 109 (1982) 444; M. Green and J. Schwarz, Phys. geu. B 149 (1984) 117. [2] D.J. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B 256 (1985) 253. [ 3 ] L.J. Dixon, J, Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678: B 274 (1986) 285. [4] S. Hamidi and C. Vafa, Nucl. Phys. B 297 (1987) 465; L.J. Dixon, D. Friedan, E. Martinec and S. Sbenker, Nucl. Phys. B 282 (1987) 13. [5] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [6 ] D. Gepner, Phys. Len. B 199 ( 1987 ) 380;Princeton preprint PUPT-1093. [7] K.S. Narain, Phys. Lett. B 169 (1986) 41; K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369. [8] A. Strominger, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986); D.G. Markushevich, M.A. Olshlanetsky and A.M. Perelomov, Commun. Math. Phys. 111 (1987) 247. [9] L.J. Dixon, in: Proc. Workshop in High energy physics and cosmology (Trieste, Italy, 1987), to be published; M. Cvetic, SLAC report SLAC-PUB-4325,in: Proc. Eighth Workshop on Grand unification (Syracuse, New York, April 1987), to be published; SLAC report SLAC-PUB-4324,in: Proc. Intern. Workshop on Superstrings, composite structures, and cosmology (University of Maryland, March 1987)). [ 10] X.-G. Wen and E. Witten, Phys. Lett. B 166 (1986) 397; M. Dine, N. Seiberg, X.-G. Wen and E. Witten, Nucl. Phys. B 278 (1986) 769; B 289 (1987) 317. [ 1l ] M. Cvetic, Phys. Rev. Lett. 59 (1987) 1795.