summability of superstring theory

30 July 1998

Ž .Physics Letters B 432 1998 317–325

Summability of superstring theory

Simon DavisDepartment of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK

and School of Mathematics and Statistics, UniÕersity of Sydney, NSW 2006, Australia

Received 20 March 1998Editor: P.V. Landshoff

Abstract

Several arguments are given for the summability of the superstring perturbation series. Whereas the Schottky groupcoordinatization of moduli space may be used to provide refined estimates of large-order bosonic string amplitudes, thesuper-Schottky group variables define a measure for the supermoduli space integral which leads to upper bounds onsuperstring scattering amplitudes. q 1998 Published by Elsevier Science B.V. All rights reserved.

PACS: 03.80; 04.60; 11.17Keywords: Superstring; Moduli; Genus; Partitions; Amplitudes; Exponential

The genus-dependence of superstring scattering amplitudes has been estimated recently using the super-Schottky coordinatization of supermoduli space. The N-point g-loop amplitudes in Type IIB superstring theory

Ž Ž ..Ny1 Ž X X X .3 gy3have been found to grow exponentially with the genus, 4p gy1 P f B , B , B , B , B , B ,K K H H B B

where B , BX , B , BX , B and BX are the bounds of integrals over Schottky group parameters which representK K H H B Bw xdegenerating and non-degenerating moduli respectively 1 . As this genus-dependence differs significantly from

the large-order growth of field theory amplitudes, several arguments in support of the conclusion shall be putforward.

The advantage of using the Schottky parametrization of moduli space in the study of the growth of integralsrepresenting the scattering amplitudes is that the dependence of these integrals on the genus is directly linked tothe limits for each of the Schottky group variables. By introducing a genus-independent cut-off on the length of

w xclosed geodesics, in the string worldsheet metric, to regulate infrared divergences 2 , and then translating thiscondition to restrict the integration region in the fundamental domain of the modular group, it can be shown thatthe sources of infrared and large-order divergences are identical, as they both arise, in particular, from the

1< < < < w xgenus-dependence of the K limit, K ; 3,4 . For the superstring, this cut-off is no longer necessary, andn n g

the entire fundamental region is required for the supermoduli space integrals. The introduction of supersymme-try eliminates the infrared divergences because of the absence of the tachyon in the superstring spectrum, andtherefore, the large-order divergences are eliminated simultaneously. This is a consequence of the tachyon beingthe source of the divergences, rather than the instanton, which could remain in the theory even after theintroduction of supersymmetry.

0370-2693r98r$ – see frontmatter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00658-3

( )S. DaÕisrPhysics Letters B 432 1998 317–325318

The validity of estimates based on the super-Schottky group measure depends on the range of the integration.w xThe use of the super-Schottky group measure 5

2g1 dK dZ dZ 1yKn 1n 2 n n y5det Im TTŽ .Ł 3 1dVV Z yZns1 � 0A BC 1n 2 n

B2 2nK 1y y1 KŽ .n n

=

10B 1 2N py pa` `21y y1 K 1yKŽ . a aX X 1Ž .Ł Ł Ł Ł B 1p N pyaž /ž / 21yKa aps1 ps2 1y y1 Ka Ž . a

with infinitesimal super-projective invariant volume element

dZ dZ dZ 1A B CdVV s PA BC 1 dQA BC

2Z yZ Z yZ Z yZŽ . Ž . Ž .A B C A B C

u Z yZ qu Z yZ qu Z yZ qu u uŽ . Ž . Ž .A B C B C A C A B A B CQ s 2Ž .A BC 1

2Z yZ Z yZ Z yZŽ . Ž . Ž .A B C A B C

and with super-period matrix

1 Z yV Z Z yV Z1m a 1n 2 m a 2 nŽm ,n.TT s ln K d q ln 3Ž .Ým n n m n2p i Z yV Z Z yV Z1m a 2 n 2 m a 1na

defines a splitting of a subset of supermoduli space, since the separation of even and odd Grassmann coordinatesis maintained over the entire region, a necessary condition for integration over the odd moduli. Since the globalobstruction to the splitting of supermoduli space can be circumvented by removing a divisor D of codimensiong

w xgreater than or equal to one from the stable compactification of supermoduli space s MM 6 , the integral over allg

of supermoduli space in superstring scattering amplitudes differ in the large-genus limit, therefore, from theestimates based on these subdomains by a contribution from the divisor, given by tadpoles of massless physicalstates at lower genera. Assuming an exponential dependence up to genus gy1, and multiplying this bound bythe number of degeneration limits corresponding to the splitting of the super-Riemann surface into twocomponents, an estimate of the contribution of the divisor can be made. Eliminating the vanishing of two of the

Ž < .B-cycles by OSp 2 1 invariance of the super-Schottky uniformization, so that the contribution of the divisorwill be bounded by

gw x

2Ny1 3 gy6X X X2 gy2 P 4p gy2 f B , B , B , B , B , B q 2 d A A 4Ž . Ž . Ž . Ž .Ž . ÝK K H H B B i i gyi

is1

where A and A represent amplitudes for the i and gy i components and d is determined by the stringi gyi i

propagator, the length of the connecting tube and the moduli described by coordinates around the punctures ofŽ .the pinched surface. The two terms in Eq. 4 represent the contributions of two different sets of divisors D and0

D .iThe boundary of supermoduli space can also be approached by considering the degeneration of cycles

non-homologous to zero. Restricting to the complex-valued part of the super-Riemann surface, of genus g, the

( )S. DaÕisrPhysics Letters B 432 1998 317–325 319

3g-3 degeneration limits are associated with the pinching of A-cycles, B-cycles, which immediately can beidentified with the divisor D , and C-cycles. The C-cycles can be interchanged with the dividing cycles in a0

decomposition of moduli space based on the length and twist parameters of 3g-3 non-intersecting closedgeodesics on the surface. Consequently, the counting of different partitions of the surface, which shall berequired for an estimate of the total amplitude, is equivalent when the remaining degeneration limits are viewedas the vanishing of C-cycles or dividing cycles.

< < < <Finiteness of the superstring amplitudes in each of the degeneration limits K ™0, ns1, . . . , g, H ™0,n m< <ms2, . . . , gy1, B ™0, ms2, . . . , g, implies that there are bounds associated with the correspondingm

multiplier integrals B , B , B for each value of n and m, and it will be demonstrated that additional integralsK H B

associated with non-degenerating moduli are bounded by BX , BX and BX .K H B

A generic Riemann surface will have l non-degenerating A-cycles, gy l degenerating A-cycles, lK K HŽ .non-degenerating B-cycles, two B-cycles fixed by an SL 2,C transformation, gy2y l degenerating B-cycles,H

l non-degenerating C-cycles and gy1y l non-degenerating C-cycles. Given that the integrals over theB B

non-degenerating moduli are less than BX , BX and BX , upper bound including all degeneration limits is thenK H B

given by

3gy3 g gy2 gy1X X Xgy l l gy2yl l gy1yl lK K H H B BB B B B B BÝ Ý K K H H B Bž / ž / ž /l l lK H Bls0 � 4Part. l , l , lK H B

l ql ql slK H B

3gy3g ! gy2 ! gy1 ! 3gy3 !Ž . Ž . Ž .s Ý Ý

3gy3 ! l ! gy l !l ! gy2y l !l ! gy1y l !Ž . Ž . Ž . Ž .K K H H B Bls0 � 4Part. l , l , lK H B

l ql ql slK H B

=B gy lK BX lK B gy lH B gy lB BX lB 5Ž .K K H B

Since the last sum is bounded by

3gy3 3 gy3X X X X X Xl l l l l l1 2 3 4 5 6B B B B B B s B qB qB qB qB qB 6Ž . Ž .Ý K K H H B B K K H H B Bž /l l l l l l1 2 3 4 5 6l , . . . , l1 6

l q . . . ql s3 gy31 6

the entire expression is less than

g ! gy2 ! gy1 !Ž . Ž . 3 gy3X X XB qB qB qB qB qB 7Ž . Ž .K K H H B B3gy3 !Ž .B q B X qB q B X qB q B X

K K H H B B ˜If the ratio is denoted by K , thenmin B , B , BŽ .K H K

3 gy3X X XB qB qB qB qB qBŽ .K K H H B B 3 gy3˜-K 8Ž .g gy2 gy1B B BK H B

Kg N g gy2 gy1Ž Ž ..and the N-point scattering amplitude is bounded by c c 4p gy1 B B B if c s2 and c s .1 2 K H B 1 2 3

Contributions of degenerate Riemann surfaces to the amplitude can be compared with integrals over theregion of parameter space defined by the condition of genus-independent bounds for the multipliers anddistances between the fixed points. For the latter category of surfaces, the uniformizing super-Schottky groups


< < X < < Xhave multipliers and fixed points which satisfy the inequalities e F K Fe , d F j yj Fd . Integra-0 n 0 0 1n 2 n 0

tion over this range produces the following expression:

gy4 y4 gy22 gy3p 1 1 1 120 0 0 0 0< <Õ j ,j ,j ,u ,u ln y ln yŽ .11 21 1 g 11 1 g X X2 gy2 2 2ž / ž /e e4 d d0 0 0 0

=

gy22Xd qdŽ .gy2 0 0X 2 2 w xd yd 1y contribution from the B multipliers 9Ž .Ž .1 1 mX 24d1

< < Xsince there is a projective mapping of the isometric circles to configurations with d - j -d and the q1 1m 1

integrations can be absorbed in

gy1 g

dq q dq q 10Ž .Ł ŁH H1 i 1 i 2 j 2 jis2 js1

Ž . Ž < .In Eq. 9 , the first factor arises from the Jacobian factor resulting from the residual OSp 2 1 symmetry of thesuper-Schottky parameterization used to select the locations of three of the fixed points j 0 , j 0 and j 0 and11 21 1 g

two of the Grassmann variables u 0 and u 0 , and the final factor on the second line represents the reduction of11 1 g

the integral as a result of requiring non-overlapping of the isometric disks in the region of the complex planeŽ .occupied by the disks. From Eq. 2 , it follows that

1 1 1 u 1 uB Cs Z yZ yu u y qB C B CdVV dj dj dj du du 2 Z yZ 2 Z yZA BC A B C B C A B C A

1 1 dj ydjŽ .A Bq u y qcyclic permutations of A , B ,C 11Ž . Ž .C 2 Z yZ dj dj dj du du duA B A B C A B C

The leading term in the expansion with respect to the Grassmann variables, after excluding terms involvingŽ . Ž .integration over a set of parameters other than 3gy3 fixed points and multipliers and 2 gy2 odd moduli, is

0 0 0 0 0 0j yj j yj j yjŽ . Ž . Ž .11 21 1 g 11 21 1 gq q = c.c. 12Ž .

dj dj dj du du dj dj dj du du dj dj dj du du11 21 1 g 11 21 11 21 1 g 11 1 g 11 21 1 g 21 1 g

< Ž 0 0 0 0 0 . < 2 Ž .The functional dependence of Õ j ,j ,j ,u ,u can be deduced from 12 after relabelling the11 21 1 g 11 1 g

Grassmann variables to remove the repetition arising from cyclic permutations. The B part of the measure ism

Bg g g < < < <dB dB B d B dum m m m mP s 13Ž .Ł Ł Ł3 3 3< <Bms2 ms2 ms2 m2 2B Bm m

1 1X Xm g g< < < < < < < < w xSince j sŁ B , it follows that if d - j -d , the range for B can be chosen to be d ,d .2 m js2 j 2 2 n 2 j 2 2

Because

1lnd g2

lim 1q s lim d 14Ž .2ž /gg™` g™`


the magnitude of the integral depends on the choice of d . If d is fixed to be a finite number greater than zero,2 2lnd 2< <then the lower limit of B for large g, is 1q , and the integral isj g

X X2 2g lnd y lnd lnd y lnd2 2 2 2 y3q qO g 15Ž .Ž .Ł 2g gms2

which, of course, significantly decreases the overall genus-dependence of the integral. However, it is clear that< < < <the integration region can be enlarged so that the lower limit for B is e . The lower limit for j then would˜j 2 m

my 1 < <be e , which tends to zero as m™g. The B integrals are therefore˜ mX

dX 2lnd2 1q lng g g1q g< <d B y1 1 1 1mg s s y - 16Ž .XŁ Ł ŁH 2 gy1lnd< <B e< < e2˜ ˜Bems2 ms2 ms2˜ mm e 1qg

Ž . gy12pThe contribution from the B multipliers is bounded by and therefore it does not affect the overallmgy1e

exponential dependence of the integral over this region in the parameter space.< < < <The limits B ™0 represent the degeneration of C-cycles since j ™0, jsm, . . . , g. The Riemannm 2 j

surface splits into two components of genus my1 and gymq1 and each component appears to be a pointw xfrom the perspective of the other component 7 .

Since modular transformations map A-cycles, B-cycles and C-cycles into each other, genus-dependent lowerd2� < <4 < <limits for the ranges of the variables j would take the form . The lower bound for B would then be2 q2 m jg

1 1d2 g g < <. Since g ™1 as g™`, it follows that the lower limit of the B integral tends to 1. In this case, the2 qž / mg

value of e can be chosen to be any constant less than 1.˜< <In the degeneration limit B ™0, the superstring amplitude is finite, so that the integral of the entirem

1gd2supersymmetric measure over the range 0, will be bounded. Since the B integrals over the2 qž / mg

neighbourhood of the boundary and the interior of supermoduli space are bounded by exponential functions ofthe genus, their sum will possess the same property.

Ž . Ž .Given a holomorphic slice on the subset of supermoduli space, s MM yNN D , with NN D being ag g g

neighbourhood of the divisor, an analytic transformation to the super-Schottky group variables maps thisintegration region to a subdomain of the fundamental region in the super-Schottky parameter space, excluding aneighbourhood of the boundary. If the separation between the neighbourhood of the compactification divisorand the interior region of moduli space were to give rise to genus-dependence in the ranges of thesuper-Schottky variables, computations of the multiplier integrals with genus-dependent limits imply that theystill should have finite upper bounds. This can be seen, for example, by considering the integral

y4< <d K 1n 1s ln 17Ž .H 45 ž /< <K1 n< <K lnn ž /ž /< <Kn

Xe X0with mixed integration limits such as ,e , which gives the result1y 2 q 0g

y41y4X1 1y 1y2 q ln gy lne q ln 18Ž . Ž .04 4 Xž /e0

X Ž .Moreover, as long as e does not asymptotically approach 1 at a sufficiently fast rate as g™`, Eq. 180

implies that the upper bound for the integral will be an exponential function of the genus. This can bedetermined by explicit calculations of the location of the fundamental region of the modular group in Schottky


Ž .parameter space. At genus 1, the fundamental region of SL 2;Z in the upper half-plane is well-known, and the'3 '2p it y2p Im t y 3 p< < < <maximum value of K s e se occurs at the minimum value of Im t , , and equals e . At2

Ž .2 Ž .2 w xhigher genus, the determinant condition can be used to show that Re t q Im t G 1, ns1, . . . , g 8 .nn nn1Since one of the conditions defining the fundamental region of the symplectic modular group is y F2

1 1 32Ž . Ž . w x Ž .Re t F , the squares of the diagonal elements of Re t are restricted to the interval 0, and Im t G .m n nn2 4 4

Positive definiteness of imaginary part of the period matrix implies that the positive root should be chosen,'3Ž .Im t G .nn2

< <The following argument also may be useful in setting upper limits for the range of K . One of then

A B< Ž . < Ž .conditions defining the fundamental region of the modular group is det CtqD G 1 for gSp 2 g;Z .ž /C DSince

y1 y1< < < < < <det CtqD s det Im t det Cy iC Re t Im t y iD Im t 19Ž . Ž . Ž . Ž . Ž . Ž .Ž .< Ž . <the determinant will be greater than one for all C, D when det Im t ) b )1 for some number b, so that

< Ž Ž .Ž .y1 Ž .y1 . <det C y iC Re t Im t y iD Im t will be bounded below when det C / 0 and equal toy1 Ž .tr Im ty1< < < Ž . < < Ž . <det D det Im t G det Im t when Cs0. Moreover, the minimum value of isg

g1 1ln q l 20Ž .Ý nnX

e g0 ns1

where l is the greatest lower bound fornn

j yV j j yV j1n a 2 n 2 n a 1nŽn ,n. ln 21Ž .Ýj yV j j yV j1n a 1n 2 n a 2 na

If eX satisfies the inequality,0

1g1 1g

ln G b y l 22Ž .Ý nnXe g0 ns1

the Schottky group multipliers and fixed points will lie in the fundamental region of the symplectic modularŽ .group. It has been discussed previously how exponentials of the sums in 21 can be estimated for different

w xconfigurations of isometric circles 4 . Moreover, positive-definite symmetric real matrices can be expressed asT Ž .QQ PPQQ , where PP is a diagonal matrix with entries p )0 and QQ is a triangular matrix q , q s0, k) l andk k l k l

q s1, and the fundamental domain of the symplectic modular group is contained in the region defined by thek kw x Ž .inequalities 0 F p F tPp and yt F q F t 8 . These constraints can be applied to Im t , with the valuek kq1 k l

Ž .of t being related to the bound on the sum 21 .Both fermionic and bosonic contributions to one-loop superstring amplitudes can both be positive, since there

Ž .are no odd moduli parameters at this genus, det Im t G 0, and the measure is positive-definite. The four-pointone-loop amplitude contains an integral over the entire Riemann surface, so that s-channel, t-channel andu-channel diagrams are included in the field theory limit of the string diagrams. The square of the absolute value

< Ž . <16 w Ž .xy8of a holomorphic function, f w , arising for fermions, is cancelled by a factor f w for right-movingi y8 iw Ž .xmodes a and the complex conjugate factor f w for left-moving modes a in Type II superstring theories˜n n

w x9 , eliminating a potential divergence in the integral over the modular parameter.The positive fermionic and bosonic contributions to the one-loop amplitudes imply that the superstring

amplitudes could receive contributions growing at nearly identical rates with respect to the genus, thusmaintaining the approximately factorial rate of growth of the bosonic string partition function. However, thisproperty is changed by the odd modular parameters at higher genus. This implies that the determinant of theimaginary part of the super-period matrix is no longer necessarily positive-definite. Even though the remaining


part of the measure consists of the absolute square of a holomorphic function of the supermoduli parameters, themeasure is not necessarily positive-definite and the additional contribution to the amplitudes at higher genusmay not occur. Indeed, supersymmetric theories generally exhibit better large-order behaviour, and this isconfirmed by the growth of the superstring amplitudes.

A recent general study of divergences in perturbation theory reveals that they are linked to the violation ofthe hypothesis of Lebesgue’s Dominated Convergence Theorem concerning the formal manipulation of

w xinterchanging the sum and the integral to obtain the series in the functional integral formalism 10 . Thisproblem is resolved by introducing a functional representation of the characteristic function which cuts off theregion in field space representing the source of the divergences. Specifically, it is demonstrated that there exists

f 4 Ž .a convergent series expansion of the partition function for l theory such that the partial sums Z l tend toN4!

Ž .the exact finite value for l)0. It is also known that Z l is non-analytic at ls0, based on an argumentŽ .similar to that applied to quantum electrodynamics. Even though the partial sums Z l satisfy the hypothesisN

Ž .of the Dominated Convergence Theorem, the non-analyticity of Z l at ls0 arises in the limit N™`.Techniques of this kind have already been used in the regulation of infrared divergences in closed bosonic stringtheory and the introduction of finite ultraviolet and infrared cut-offs in QED with fermions yielding a

w xconvergent perturbation expansion 11 . Since a version of quantum electrodynamics and the non-abelian gaugetheories of the standard model should arise in the low-energy limit of superstring theory, it is useful to have aframework in which non-analyticity in the coupling constant plane still can be derived from convergent orsummable perturbation expansions rather than series with terms increasing in magnitude at a factorial rate withrespect to the order. This provides confirmation of the type of growth that has been obtained in the study ofsuperstring scattering amplitudes.

It has been established that field theory amplitudes arise as limits of string theory amplitudes by identifyingdifferent types of field theory diagrams with corners of moduli space. Amplitudes with two, three, four and five

Ž .gluons at one loop in SU N Yang-Mills theory, for example, have been evaluated using certain open bosonicX w xstring amplitudes in the limit of vanishing Regge slope a ™0 12,13 . In general, the corner of moduli space is

defined by letting the complex string moduli space approach the pinching limit so that they can be mapped toSchwinger proper times. The neighbourhood of the singular point at the boundary of moduli space can be

3 w xidentified when the string worldsheet begins to resemble a specific F diagram 13 . To define the pinchinglimit properly, it is sufficient to cut open all of the loops of an N-point g-loop diagram to form a 2 gqN-pointF 3 tree diagram. The tree consists of a main branch and side branches, each with its own Schwinger proper

� 4time flow. The vertices of the diagram can be selected to be the points z , . . . , z ;j ,j , . . . ,j ,j and the1 N 11 21 1n 2 n

pinching limits are represented by the approach of the vertices towards the branch endpoints z , is1, . . . , N ,B bi

< < w x 3so that zyz <1 14 . Many different labellings correspond to the same F diagram, including thoseBi

� 4 3obtained by interchanging j with j or permuting the g triplets k ,j ,j . Since the relevant F1n 2 n n 1m 2 m

diagrams are identified with the corners of moduli space which are neighbourhoods of components of thecompactification divisor DDsDD jDD , it is sufficient to count the degeneration limits, given by the partitions of0 i

l with each of the addends less than g, gy2 and gy1 and the sum less than or equal to 3gy3. The upperbound on the number of relevant diagrams is then

23gy3 gy1g gy2 gy13 gy3-3 23Ž .Ý Ý ž /ž / ž / ž /l l l gy2K H Bls0 � 4Part. l , l , lK H B

The remainder of the string integral includes surfaces well away from the degeneration limit and thecontribution has been estimated to be an exponential function of the genus. There is no direct identification ofthis region with field theory diagrams, and it is preferable to bound the supermoduli space integral over theregion using the estimates for ranges in the super-Schottky parameter space lying in the interior of thefundamental domain of the modular group.


For a closed bosonic string, the lower bound for the regularized partition function increases at anw xapproximately factorial rate with respect to g 2–4,15 . These results are similar to the calculations of the Euler

w xcharacteristic of the once-punctured moduli space 16–19 , duplicated in the counting of Feynman diagrams inw xrandom surface theory and matrix models 20,21 . However, it is apparent that the magnitudes of the moduli

space and supermoduli space integrals depend essentially on the measure. The change fromg d2K g d2Kn nHŁ in bosonic string theory to H Ł in superstring theory, reflecting thens1 ns1

1 14 213 5< < w Ž .x < < w Ž .xK ln K ln< < < <n nK Kn n

removal of the tachyon singularity in the Neveu-Schwarz sector and the absence of the tachyon in the Ramondsector, is sufficient to render the total amplitude an exponential function of the genus in the latter case, whereas

X2 gg e e0 0< <13 git causes the former integral to increase as , when the range of K is , . Furthermore, the uppernln g g gŽ .Ž .bound 23 on the number of different degeneration limits, implies that the partitioning of the surface into

components only leads to the upper bound being multiplied by an exponential factor, because the counting ofrelevant diagrams in a cell decomposition of supermoduli space is restricted to a neighbourhood of thecompactification divisor.

Superstring elastic scattering in the large-s fixed-t limit has been studied, and the amplitude has beenw xcalculated using covariant loop sewing techniques 22–24 , revealing that the leading term in the expansion in

powers of sy1 factors at g-loop order into a product of gq1 tree amplitudes, multiplied by the expectationvalue of a factorized operator. The eikonal approximation, involving a resummation of these contributions,restores unitarity in the theory and gives an exponential result. Restriction to one copy of the fundamental region

1implies that a factor of should be included in the integration, removing the factorial dependence obtainedŽ .g q1 !

j z y j 1 y j1 n 1 2 n 2 n w xby integrating over the multipliers and the variables r syln , s s ln 4,23,24 . TheseŽ .n nj z y j 1 y j2 n 1 1 n 1 n

results confirm the generic estimates.The estimates of the superstring amplitudes are also consistent with the growth of the special scattering

Ž .2amplitudes obtained from topological field theory. In particular, the g! dependence of amplitudes withw x2 gy2 graviphotons and 2 gravitons 25,26 may be verified by combining the exponential bounds on the

w xsupermoduli space integrals with the factorial bounds on the vertex operator integrals 1 .The insertion of Dirichlet boundaries in string worldsheets has been used to reproduce power-law behaviour

w xassociated with point-like structure in QCD 27–29 . Summing over orientable Riemann surfaces with Dirichletw xboundary conditions by associating the moduli with the positions and strengths of electric charges 30 , one

1y conn.kstrobtains non-perturbative amplitudes of order e A which will only be well-defined if the amplitudes for

the closed surfaces are also summable. Summability of the superstring perturbation series would be confirmedby the extension of the positive-energy theorem to superstring vacua corresponding to supersymmetricbackground geometries such as R10. Thus, the exponential dependence of the closed-surface amplitudes and thedescription of non-perturbative effects are compatible, since the latter can be regarded as additional contribu-tions to the superstring path integral, separate from the sum over closed surfaces. The insertion of the Dirichletboundary is associated with a different class of surfaces, which may be confirmed by considering the effect onthe shift to the vacuum. Although there may be a small amplitude for the non-perturbative instability of theinitial string vacuum state, this still could be consistent with the positive-energy theorem, which might becircumvented through the use of zero-norm boundary states.

Acknowledgements

I would like to thank Prof. M. Green for useful discussions on string perturbation theory. This research hasbeen initiated in the Department of Applied Mathematics and Theoretical Physics at the University ofCambridge in May, 1996, and the hospitality of Dr G.W. Gibbons and Prof. S.W. Hawking is gratefullyacknowledged. The encouragement of Dr H.C. Luckock, which allowed this work could be completed at theUniversity of Sydney, is also much appreciated.


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