Type I superstrings in dimensions less than ten (II): Finiteness

Nuclear Physics B319 (1989) 104-132 North-Holland, Amsterdam

T Y P E I S U P E R S T R I N G S IN D I M E N S I O N S L E S S T H A N T E N (II): Finiteness Z. BERN*

The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark D.C. DUNBAR

Department of Physics, Royal Holloway and Bedford New College, University of London, Englefield Green, Surrey, UK

Received 1 August 1988

One-loop finiteness is studied in new open superstring constructions in D < 10. We show that in D = 2, 6 the dilaton tadpole singularity cancels for a number of models with an appropriate choice of Chan-Paton gauge group. In other dimensions, cancellation of the dilaton singularity is not found within this construction.

1. Introduction

A n i m p o r t a n t step in confronting superstring theory [1, 2] with the world we observe is the construction of string models in dimensions less than the critical dimension. The exploration of these classical backgrounds should provide much insight into the properties of string theory. Recently, type II superstring and heterotic string models, which are theories of closed oriented strings only, have been constructed in dimensions less than ten [3-5]. These constructions are based on two dimensional (super) conformal field theories where, in addition to the string coordinates, X ", and their superpartners, X", there are the internal degrees of freedom [6], chosen so that the total conformal anomaly vanishes. These constructions, which are based on requiring modular invariance, have been carried out in formulations where the internal world sheet degrees of freedom are bosonic [3] or fermionic [4, 5]. Most recent interest in string theory has focussed on closed string theories, partly because closed oriented string models are technically simpler, and partly because the D = 10, Es × E 8 heterotic model was phenomenologically superior to the SO(32) *Address after September 1, 1988: Theoretical Division T-8, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 0550-3213/89/$03.50@ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Z. Bern, D.C Dunbar / Type I superstrings (II)

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type I model. However, new possibilities, which should be investigated, are offered by the direct construction of open string models in lower dimensions. Also, in string field theory [7], open strings appear more fundamental than closed strings alone. In ref. [8], D < 26 oriented open bosonic strings were constructed in the fermionic formulation for internal degrees of freedom. Furthermore, in ref. [9] (hereafter Part I) type I superstring models, which are theories of open and closed unoriented strings, were constructed in D < 10 within the fermionic formulation. Type I superstrings have also been considered in ref. [10] on Z 2 orbifolds. For models of open and closed strings the consistency requirements are not as straightforward as for models of purely closed strings. In particular, as is well known, closed string poles appear in open string amplitudes [11,12]; hence, to be consistent with unitarity, the coupling of open models to closed string models is not arbitrary. This was used in refs. [8,13] for oriented bosonic strings and in Part I for D < 10 unoriented superstrings to determine the open string model which may couple to a given closed string model. This is not sufficient to ensure consistency since, as in D = 10, open string loop amplitudes are not finite for arbitrary C h a n - P a t o n gauge groups. In this paper, we investigate the one-loop finiteness of amplitudes with open string external states in the type I superstring models constructed in Part I. (The case with closed string vertex operators is expected to follow similarly.) As for the well-known D = 10 superstring, we may expect the UV divergences present in the annulus and M6bius diagrams to cancel between the two diagrams for particular choices of the C h a n - P a t o n [14] gauge group. By Jacobi transforming amplitudes, these UV infinities may be interpreted as closed string massless states propagating into the vacuum [11,12]. We examine the relative normalisation of the M~Sbius and annulus contributions, after the Jacobi transformation which makes it more convenient to investigate possible cancellations between the two singularities. We show that, amongst the models in Part I, there exist one-loop finite models in D = 2, 6, 10 with C h a n - P a t o n gauge groups USp(2 D/z) for D = 6 and SO(2 °/2) for D = 2, 10. (The D = 10 model is, of course, the well known SO(32) superstring [1].) The D = 6 models are non-chiral which is consistent with the fact that a chiral USp model would be anomalous in our construction. For other dimensions (including D = 4), cancellation of the singularities between the amplitudes is not found. The construction of type I open strings on Z 2 orbifolds has recently been discussed in ref. [10]; however, the connection between the orbifold and fermionic formulations is not clear. In the construction of ref. [10], no consistent open superstring models were found in D = 6. They proposed a GSO type projection which could act non-trivially on the C h a n - P a t o n factors (as originally suggested in ref. [15]), but we only consider the standard inclusion [14] of C h a n - P a t o n factors. This is sufficient to obtain finite models within our construction. In sect. 2, we briefly review and summarise the construction of D < 10 open superstrings. Based on this construction, in sect. 3, we compute the amplitude for M

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external vectors in order to extract the relative normalisation of the planar to M~Sbius amplitude after the Jacobi transformation. Using the Jacobi transformed amplitudes the problems of finiteness and consistency are addressed in sect. 4. In sect. 5 we give concrete examples of our construction and their one-loop finiteness. The reader may wish to examine these before studying the earlier sections in detail. Finally, we comment on the non-chiral nature of our D = 6 examples from the low energy point of view in sect. 6. Details of an example term in the amplitude and the Jacobi transformations of the various functions are given in the appendices.

2. Review of construction of type I models in D < 10

The partition function of unoriented open string theories, in terms of operators, is described by inserting a projector into the partition function of the open oriented string, i.e.

Zopen--tr{ ° nlT

= tr, °Pen } +

l t r { 60/?°P~n~ }

1

(2.1)

= 1Zannulu s -{- ~ZM&bius ,

where /~open is the open string hamiltonian and ~ is the "twist operator" [1, 2]. These unoriented open string models couple to closed string unoriented models whose spectrum is similarly projected

Z d°s~a - tr

2

= 7Zt°rus -}- ½ZKlein'

(2.2)

where 7~ is the "turnover operator" [1] which interchanges the left and right movers. The various terms may be viewed from a world sheet viewpoint as the contributions from the annulus, MiSbius strip, torus, and Klein bottle surfaces, as indicated by the labels. The starting point for consistency of open and closed systems [8-10] is to require Zt .... to be modular invariant. Modular invariant models of type II closed superstrings have been developed in D < 10 using a variety of formalisms [3-5]. In Part I, type II closed string models, within the fermionic formulation of ref. [4], were truncated to type I closed models and then coupled to unoriented open superstring models. We briefly review this construction here, starting with the type II models. In the formalism of ref. [4], a closed string model is specified by the boundary conditions of the 2N F = 2(14 - D) complex world sheet fermions on a torus world sheet, which are collected in a vector W = ( W k) where k = 1,2 . . . . . 2 N v. The first (second) N F elements specify boundary conditions of the left-moving (right-moving) fermions. The set of boundary condition vectors is generated by a "basis" { W~}. By

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summing over the boundary conditions, the partition function on a torus, described by a complex modular parameter -r, is obtained f

d,/-dT

1}paW'TaWzt.__L__,r,

{mi--1) {mj

JR(Im~.)zZB(I", ~)

rr),

(2.3)

E E ~w Bw" (~,=o) (/~j=o}

where m s is the least integer such that all components of miW i (not summed) are integers, ZB(~', ?) is the bosonic partition function and ~Fw t-'~w`~-, ~) is the partition function for fermions with world sheet space boundary conditions aW (where

a W - a W (mod 1)) and time boundary conditions flW given by r ]

- -

^ left

^ right

_ _

^

z~W('r ~) = Tr[e2~i'Howe2~i~H w e 2"i(w'-Bw)No~ Bw" '

~H,

NF

1

2N v

= I-[ F~Wk (e2~/') k=l

1 |

~k

II

F~Wi(e2~i').

k=NF+I

(2.4)

flW

The functions F ~ k are defined in appendix B and the hamiltonians /~teft /~right ~ff.k ~' 2~ ' fermion number operator vector N ~ and mode expansions are given explicitly in refs. [4, 9]. We follow the summation conventions of ref. [4] where, without explicit summation, aifl i is not summed but aft = 32~agfl,. The dot product is lorentzian for closed string vectors [4] but not for open string vectors. We use the notation of Part I, which is slightly different from that of ref. [4] in that we do not assume the vector W0 (the vector consisting entirely of 1/2's) to be a basis vector although, of course, W0 must still appear in the set of boundary conditions and is given by W0 = Zi ziWi for integers z s. This difference is convenient when truncating the type II closed models to type I closed models. To obtain a consistent modular invariant type II model, it is necessary that ksj satisfying

k s j + k j i = W,. Wj,

w,=0,

I,,+ Ek,jzj + ss- l

(modl), ( mod

1),

J

mjkij = 0,

(mod 1),

2 E k i j z j = 0,

(mod 1),

(2.5)

J exist. The spin factor of a vector in a type II model is defined by s s - WsI + WiNF+I. When such ks / exist, the coefficients of the partition functions in eq. (2.3) are given

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by C aW = 13W

m i

e2~riaSe2~ri(flq'(a)+z~(a))

(2.6)

where epi(a ) is defined by dpi(a ) = E k i j a j + s i +

J

Ekijzj+

W 0 • W i - Wi.aW

(modl).

(2.7)

J

This choice of coefficients yields a modular invariant partition function with a generalised GSO projection. Only states which satisfy Wi.N~:q~i(a )

(modl),

Vi

(2.8)

are not projected. For superconformal invariance, the existence of a supercurrent is necessary. One way to ensure this is to split up the left moving internal part of W as triplets [4] [(a)4-P,(a,,bl,q)

..... (ae, bp,cp)],

(2.9)

where P = (10 - D ) / 2 and a=ai+bi+ci=O

or

1/2.

(2.10)

The right-movers are treated similarly. Generalisations of this condition have been developed [16]. Only the subset of type II models which have symmetry between the left- and right-movers are truncatable to closed type I models. In Part I, this symmetry was ensured by, first, restricting to sets of basis vectors W, of the form (2.11) where W~ = ( W.s,l- Wsl) is a left-right symmetric (LRS) vector and Wa = (W~; Wr ) and Wa = ( w r ; W)) are non-LRS but are "mirror images" of each other, and, second, by imposing the additional constraints on kq, 2kss, = 0

(rood 1),

ks~ + ksa = 0

(modl),

kab + ka7, = 0

(modl),

k ,d, + k ab = 0

(mod 1).

(2.12)

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109

This ensures the total spectrum of the type II model is such that the turnover operator, 7~, can act sensibly; if Is) ® Is') is contained in the total Fock space then so is Is') ® Is) (which for non-LRS sectors means identifying the W a sector with the W a sector). The next step in the construction, as discussed in Part I, is to couple the above type I closed string models to open strings. For open strings, the left- and right-movers are not independent world sheet degrees of freedom but are reflected into each other at the world sheet boundaries so the open string boundary condition vectors are NF component vectors. The o boundary condition vectors appearing in the open string are f l V + U o where the V, are defined by ~k = W,? - wiN F+~, i.e. the difference between the left and right components of closed string Wi, and U0 = ( ( 1 / 2 ) U V ) . In general, different IV/, can give rise to the same V~. The set of time boundary conditions are a subset of q/-- {U such that (U; U) is a closed string LRS sector}, as described in refs. [8, 9]. Sometimes only a subset of can contribute since the closed string projections may remove all LRS states from a LRS sector. The set ~ can be generated by a basis Us where (U,;/2,) = F~j c,jWj. The time boundary conditions actually present in the open string belong to the subset {U0 + E~'~rT~} of ~ , where T~ is a basis for [8,91

suchthat c kikc k --0 ,modl, Vr}

213,

,/ks

The Tr are related to the W, by (TAT¢) = 32i griWi for integers g,,. From eq. (2.11), will be generated by 14/,1 and (W2 + Wr ) and by using eq. (2.12) in eq. (2.13) T will always contain, at least, the subset of q/generated by (WJ + w,r). The resultant fermionic annulus partition function of the open string models, which may be coupled to the model (2.3), is [8, 9] [

- - o.p. .e. n ~"

,u, = N2 ~

\

-i

{,,,T- i}

eZ=,ao,[I-[m:}

{Zo}

,r

,

•

exp(2ertEYrPr(t8

iv=o}

\

r

o

)] F / ~ °rv ++ u uo (~o) ] v 0

[ ~°v+u,--/ 1 = N2 ~-'~ e2~i/~°"Tr[ l°

roT--1 e 2~rivr(°r(B°) Tr'N~°v+v0))], (2.14) l ) t _.-77Y m r yr= 0

{rio}

where Pr(t ~°) = £ g r i k i j t ~ ? i,j

+ Sr -- Tr" (t ~ V q- Uo -

Uo),

(2.15)

arises from tracing the Chan-Paton factors around the two boundaries for groups SO(N) or USp(N), gr is the first entry of T r, and m T r is the least integer N 2

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110

such that the components of m~T~ are all integers. The sum 52Bo is over a restricted set of the fl~ which generates all values of flV only once. For the truncatable models (2.11) the set { ~ fl°V~} = (~,i fli(W/i l - W/r) } = ( ~ a ( f l a - f l s ) ( W 2 War) } SO the set { flV } is generated by (W1 - W~). Hence, for convenience, we take (flo }, which m a y be viewed as the (fl~ } for the open string, equal to the coefficients of the IV, (and not the W~) i.e. (flo} = (flo). The (~.,fl,(W 1 - War) } will generate each distinct value of flV once only. For unoriented open strings, we also have the M6bius partition function obtained by insertion of the twist operator ~ into the trace in eq. (2.14.)

open

(' ~

E

e2~rivr(Or(fl°)-- Tr"£ZBov+VO))~BOV+ Uo

mr .itr=O

{B° }

(2.16) where ~ = - 1 for SO C h a n - P a t o n groups and ~/= + 1 for USp groups, and in the fl°V + Uo sector

~BoV+Uo_^

ei~r~ei~r(/~bo~e

= ei~r~ei~'(Lo

p2/2 EB+ HBov+" Uo EF ~ ) p2/2).

(2.17)

T h e /~bose and /4Bo~7~o are respectively the open string bosonic and fermionic hamiltonians and the terms E B= - ( D - 2)/24 and E v = (U. U - 2Uo. U + N F / 6 ) / 2 are, respectively, the bosonic and fermionic contributions to the vacuum energy Evac(U ) = E v + E B for sector U. The factor ei~vrva0 is necessary to ensure that (1 + I 2 ~ ) / 2 is a physically sensible projection when acting on the states not removed by the GSO projection in the f l ° V + U0 sector. For example, in the U0 sector the states created by raising with an odd number of fermionic oscillators (which include the massless vector) have e i~(L°-p2/2) = +_i, so without the phase factor ~:o = -+ 1/2, (1 + ~Uo)/2 would not act as a physically sensible projection on the U0 sector. We choose ~ o = - 1 / 2 so that the massless gauge bosons lie in the adjoint representation of SO(N), for ~/= - 1 , and U S p ( N ) , for ~ = + 1. Furthermore, in any sector, the GSO projector must keep only states with integer spacing in (mass) 2 so that ~ 2 ~ T ~ o = 1 on all physical states. For example, in the U0 sector, states with an even number of fermionic oscillators (which include the tachyon) must be removed by the GSO projection, which occurs only when Uo is among the boundary condition vectors determining the GSO projection. The M/Sbius amplitude, when transformed into a closed string exchange, contains contributions not corresponding to the closed string physical states unless additional

Z Bern, D.C. Dunbar / Type lsuperstrings (1I)

111

conditions are satisfied, which reduce to T~. T~ = 0

(mod 1)

(2.18)

for periodic and anti-periodic boundary conditions [9]. This condition is a non-trivial constraint, not normally guaranteed by the modular invariance conditions for the closed string. For consistent closed models and where the closed string basis vectors are of the form (Vu; 0), (0; Vu), this follows automatically. One effect of restricting a model by the M/Sbius GSO consistency condition (2.18) is that the vacuum energy of any sector is a multiple of 1 / 2 since Ramond fermions can occur only in groups of 4. Thus, the only potentially tachyonic state is in the U0 sector.

3. Amplitude computation Both the planar and M~bius open string amplitudes contain UV infinities but, as for the well-known SO(32) superstring in D = 10 [17], there is the possibility these infinities may cancel between the two amplitudes. By Jacobi transforming the open string amplitudes these infinities become IR infinities which can be understood as due to the propagation of closed string massless singlets into the vacuum [11, 12]. This makes the analysis of possible cancellations easier. In this section, we calculate and then Jacobi transform an M-vector open string amplitude to examine the relative normalisations after the transformation. The same normalisations are expected for general amplitudes from the properties of the physical vertex operators under Jacobi transformations. Jacobi transformed open string loop amplitudes may be regarded as closed string exchanges between appropriate " b o u n d a r y " and "crosscap" states [B) and [C) [8-10,13,18,19]. The total amplitude may be understood in terms of a total boundary state which is the sum of IB ) and IC). Dilaton infinity cancellation can then be regarded as the cancellation of the massless part of the combined state ( [ B ) + IC ) ) [10,18]. By calculating the relative normalisation of Jacobi transformed amplitudes, we are effectively calculating the relative normalisations of the IB) and IC) states. The open string models determined by eqs. (2.14) and (2.16) always contain vector states in the U0 sector so we consider the scattering of M such states. We attach C h a n - P a t o n factors for the groups S O ( N ) or U S p ( N ) to the boundaries in the usual manner [14]. In the F1 formalism* [2, 20], the contribution to the planar amplitude, where all open string vertex operators lie on the same boundary, from a * When we have a fermion sector with both Ramond space and time boundary conditions then the formalism must be modified with one F2 formalism vertex operator and propagator being used [2, 20]. For simplicity we will calculate the normalisation within the fermion sectors Z~ which have no such term, e.g. Z~0. The Ramond space and time case can be expected to follow similarly, although technically different [2, 21].

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Z. Bern, D.C. Dunbar / Type Isuperstrings (II)

sector with space boundary conditions/3°V + U0, is given by J°lvf+Uplanar °=gMN2rt[l"

" "" tM]'IO "-'~- fR 1 r=l

dDp

V~(ki,f~)co#~°~W~o+G°s°+#e°s@(B°V+ Uo) ,

×e2~iB°'Tr

(3.1)

where N is obtained by tracing over the Chan-Paton factor on the boundary with no vertex operator insertions, t~ is the Chan-Paton factor associated with the ith external state and g is the open string coupling constant. R 1 is the appropriate integration region for the fr on the annulus [21. The vertex operator for gauge bosons is given by

Vi(ki,~i) =~i(Ei" o~ig(~i) -]- ~i'X(~i)ki.X(~i))e iki'X(~i)

(3.2)

with ei a polarisation vector. /3(/3°V+ Uo) is a generalised GSO projector given by

H{1+1

P(I ~ ° V + Go)=

}

~ E e2~ri'/'o'(fl°)e-2~ri~"Tr'~'~°v+u° mr ~,r=l

=

m~

~[~ e2~ivo(/3°)e {v}

2wi'T'~Io°V+Uo.

(3.3)

The MiSbius amplitude similarly is 1

dw

G 1

tM]f0 T f R 2 llr=l

Afl°V+U°

~fdDP~r a

[i=1 (3.4) where^ R 2 is the appropriate region of integration for the ~r on the M/Sbius strip [2] and 9 ~ is the twist operator (2.17). Using standard techniques [2], we may evaluate the trace in the planar amplitude (3.1). After expanding the projector (3.3), this is

Al~°v+~°:gMNtr[qt2 planar

x

m[

ld r(

'M]~0 --~--£ 1 r=l C

~

[o)l/24f(~d)l-D+2

e2~iB°,y[ e2~ivo(13°)FB°V+ Uo( ,~#°v+ Uoz ,,

(3.5)

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113

where v" = In fJ2~ri, and fg¢ov+ uo are the Green functions for the contractions in y T + Uo

eq. (3.1) within a sector with space boundary conditions f l ° V + Uo and time boundary conditions 3'T + U0. For the vector amplitude, the effect of the ghosts is simply to cancel two powers of both the bosonic and fermionic partition functions. The key point for our purposes is the properties of f¢ under the Jacobi transformation. In order to be explicit, in appendix A, we calculate f¢ for a specific term in the contraction (3.1), and demonstrate its transformations. All other terms transform in the same manner since each term contains Green functions with the same total Jacobi weight. Furthermore, the Green functions for arbitrary physical amplitudes can be expected to transform in the same way. A lt°V+ Uo is evaluated similarly, except we must include the twist operator (2.17) MObius

which has the effect of replacing r ' by r ' + 1 / 2 for the oscillators but not for the bosonic zero mode contributions. Evaluating, we obtain ...

AS~£v+~ ° = gMTltr[Gt2

×

m

:)1

[~ dw tM]~o 7 L ~

MFI1 d~r ( -- 2"rr l D/2 r=l T\ ln0a ] [°Y24f(--°a)]

D+2

e2~riB°s £ e2~rivo(B °) ei~tiaov+uo {v)

× e-i,,e~e . . . . oFf°V+ UO(e/~w)f ~ ° v + V0(v/ir, ) yT+ Uo yT+ Uo '

(3.6)

where the MObius Green functions are collected in f¢~' (e.g., see appendix A). These amplitudes are Jacobi transformed into closed string exchanges via the change of variables [11, 25] r' = - 1/'r,

vr' -- - vr/r .

(3.7)

Using the results of the appendices, where the transformations of the individual factors in the amplitudes are given, for a single set of boundary conditions of eq. (3.6), f l ° V + Uo, y T + Uo, the resultant planar amplitude is ~tlor pl . . . .

_

- - g Mqr

M_lNtr [

. el dq f l

tlt2"''tM]Jo

TJ

0

M

1

-1

r=l

×e2,iBOse2~ivo(BO)e2=i(vr+Uo-Vo)(Bov+u o u,,)~vr+u0 (a2)~vr+u0 /~ov+uo-- - ~

(v¢1$)

(3.8) where 5ev~(q 2) - [ q l / 1 2 f ( q 2 ) ] - ° + Z F v U ( q 2 )

(3.9)

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114

is the fermionic contribution and bosonic contribution to the partition function taken together. In eq. (3.8) we have used the transformation properties of explicitly computed, using a specific term, in appendix A. Following Part I, the transformed planar amplitude (3.8) can be rearranged to M

1

A'rT+ U°palnar= gM~rM-1N tr[ttt2"''tM]j ofl--q- I r=lH ~)(Pr+l- Pr)dPr ×

E e-2'~/B°°(v)~eVr+u° (q2)~vr+tJo (nr m~ )1 e2"ive(Bo} /~°v+ Uo -¢~°v+ Uo(.rll"),

(3.10)

where

oo( ) = E

ri

(3.11)

o,g,,v, + s o - vo. ( v r + Uo- Uo)

and we have summed over { 13° }. The transformed Green function, ~, is defined in appendix A. Note that the factors of in q cancel from the transformed amplitude. Similarly, Jacobi transforming the M~Sbius amplitude for a single set of boundary conditions, yields

°, ~ = g M~rM- 12D/2 - 12 - M~Itr[tlt2--- t M ] /•B M~bius

l dq 2 M 1 -1 X fo q - fo r91 [~(~r+l-Pr)dPr(I~r mTr) X e 2~rifl°se 2~riv°(B°)e 2"~i(B°v+Uo uo ). (vT+ Uo- Uo)

Xe2~ri(-B°V+2vT+Uo-Uo).(~o Uo)e i~r(D 2 ) / 2 4

-vr+ Uo

(c

vq)e'~TTV °

uo

-

.

(3.12)

The complicated phases in the above are due to the transformation of F ~°v+ uo. The 7T+

Uo

Green functions, ff~', are transformed into ff with the factor of 2 M in the above (see appendix A). The factor 2 D/2-1 arises from the transformation of [~01/24f(-~)] ~2-D) given in appendix B. By redefining the dummy variables q---, q4 and p/2--* ~,, /IMiSbius takes on a form compatible with Apl. . . . . (This is the Green-Schwarz prescription [17, 22] which we follow here as it preserves unitarity.)

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Z. Bern, D.C. Dunbar / Type I superstrings (II) Specifically, eq. (3.12) becomes M

1

1

~fl°,V = gM~ M 12D/2Tltr[1112. " " tM]foldq "'MSbius T fo I r=lH O(Pr+l-Pr)dPr (I~Ir mY ) X e2,riB°Se2~riYo(fl°)e2,Ti(fl°V+ Uo Uo)'(vT+ Oo Uo) Xe2~ri(-fl°V+2vT+Uo-Uo).(vT+Uo Uo)e-i~(D-2)/24 X e , ~ o v + uo e

ivr(E v

+4E F

]

×~-P°v+2rT+U°(e-i~q2)f~ P°v+zrr+U°(vrlr--1/2). yT+

Uo

-yT+

(3.13)

Uo

Simplifying the phases for periodic and antiperiodic boundary conditions, this may be rewritten as A'P°'VMObiu~=

f~dq

~M 1

gM~r" 12D/z~ltr[td2 """ tM]~o q- fo ~l--l:t O(Vr+, -- v~)dvr ×(I~Ir mT)-le2~i~°Sexp[2~i~y~(p~(fi°)+ Tr" T~)]

X~e~°V+Uo(e iF q 2~zB°v+uo~ )~ LvAr- 1 / 2 ) , yT + Uo

where we used

(3.14)

yT + Uo

2E~T + Vo= -Y'rYrTr" Tr + 2EVo (modl)

[9], as well as

2(fl°V+ Uo - Uo)-(~IT+ Uo - U o ) = 2 f l ° V - T T =2

o ~fioWo.

(modl) (yT; 7T = 0

(3.1s)

a

for periodic or antiperiodic boundary conditions. In the last step in eq. (3.15) we used the closed string consistency conditions (2.5). We now specialise to the case of "square models" [9]; i.e. models where { B°V } = { y T }. For this case a term-by-term cancellation of infinities may occur, as we shall see in the following section. For non-square models, in general, no such cancellation is possible. For comparison with the planar contribution (3.10), it is convenient to interchange the flo and ~/a labels in the MSbius amplitude and then sum over {rio)

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again. This is straightforward for square models with only periodic and antiperiodic boundary conditions since T~ = V, (mod 1). Doing so we rewrite (3.14) as

,~yr+UO=gM~rM_12O/Z,ltr[tlt2...tMlfo i M6bius

M-I

-•-fo

1 V[ O ( u ~ + i - G ) d G r= 1

x(l~IrmT)-le2"ivsy',

exp[-27riY'~/~°(0.(y) + T.. T.)J (B° } L a

x~vr+uo (e-i~q2)~vT+Uo (VAt-- 1/2), `R°T+ Uo

(3.16)

,R°T+ Uo

where we have used [9]

Pa(Y) =

(3.17)

which requires the use of the truncation conditions (2.12). In the next section we will take the annulus contribution (3.10) and MiSbius contribution (3.16) and investigate the possibility of cancellation of the massless closed string singularities between the two contributions.

4. Finiteness The amplitudes (3.10) and (3.16) contain potential singularities as q--0 0, which we identify by Taylor expanding the integrands of these amplitudes. Terms of the form will give rise to infinities when a >/0. The two potential singularities are from tachyon (a > 0) and dilaton (a = 0) tadpoles. In fact, from the Ta • T, = 0 (mod 1) condition (see discussion below eq. (2.18)), the only sector with a potential tachyon singularity is the U0 sector. For the U0 sector, taking the planar (3.10) and M6bius (3.16) amplitudes, specialised to square models ({ V. } = { T~}) with only Neveu-Schwarz and Ramond boundary conditions and extracting this leading singularity, we find

foadq/q 1+~

foidq/q 2

A~Uo planar hachyon

= NgMTrM_I tr[tlt2 ...

tM]f01 dqq foiMr-=~ VI O(G+I-G)dG(I~-Ir mT)

X E e 2~ri`R°O(O)f~UO uo(q2) q _ l ~ ( ~ r , ~ {]~o}

(

`ROT+

")

q--I}'

qO

l

(4.1)

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117

while ~Uo A M/Sbius[tachyon

=2D/2ng%M-ltr[tlt2...tMlfo 1

×

(rIr

mXr

y"

l r~=lO(Pr+l-Vr)dvr

e-2~riB°O(O)e-2~riZ.B°T..T, ei~rNV/4e i~r

{B° }

X f(~zB°~Waote ~UO [ -i~r q2)(e ,~/2q) ~Uo B°r+

Uo(Vr,"r-- 1/2)(e_,~/2q)o(e -i~/2 q ) - l } . (4.2)

The notation (£~', ~ )[q, means to extract the coefficient of qn from (~e, ~,). Since the leading term of the Green functions, ~u°r+~ [qO, is independent of/3 °, as may v0 be explicitly checked from the fermionic Green function (B.7), the summation over /3° will act as a GSO projector on both the planar and MObius terms. The induced projector acting on the planar contribution (4.1) always matches the spectrum of the closed string by construction, so that, whenever the closed string tachyon is ZVo = O. However, for projected from the closed string W0 sector, Apianar[tachyon the MObius amplitude (4.2), unless the MObius GSO consistency condition (2.18) on the T~ vectors is satisfied, the induced GSO projection on the states heading into the vacuum need not correspond to the closed string GSO projection of the LRS states. Hence, the tachyon singularity need not vanish, resulting in infinite amplitudes. Cancellation of AMObius[tachyon 7v0 can only happen if the induced closed string GSO projection in the MObius amplitude (4.2) removes the singularity. For the case where the annulus singularity vanishes, this is always guaranteed by the consistency condition (2.18). The other potential singularities to consider are from the massless closed string singlets. From the Taylor expansion of eq. (3.10) the planar contribution is zIYT + Uo] planar

massless

=NgMrrg-ltr[tltz'''tM]fo X e 2~rivs

fo r~=lO(Vr+l-vr)dvr

Y'~e --2rrifl°0(y) [ ~ V3'T+ ~ U oUo ( q )2

{/~o}

~

~ oT+ Uo

J~YT+UoI

q°~CJ~

~vr'

mr)

"r) q° qO

+~rr+_Uo (q2) q ~vr+v0 (vr, r) qlq°ll BOT+ Uo- ~ 1 BOT+ Uo

(4.3)

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118

while the MSbius contribution from eq. (3.16) is MSbius massless

ldq

1M 1

=2D/2~gMqrM-ltr[tlt2"''tM]fo T fo ×e2~,VSe,~,,r.u~ e

-1

r~=l O(Vr+l-tPr)dvr(rIrmT)

i°r(E~v+_~+E")ei~rNV/4 E

e-2-i/~°o(V)e

2~ri~fl°r~.ra

{3° } f~,~ uo (e-i~r _2~1 X~ ~aT_+Uo~ q )[(e

~7,vT+ Uo t

i~/2q)°~flOT+Uol,Pr'l"--l/2)(e ~/2q)O(e i~/2q) 0

+~ovr+ UOaor+Uo(e-'~ q2) [(e-'"/~q) 1

x~vT+B°T+U°u0( vr'

"C-- 1 / 2 )

(e

,,/2 q)x(e-i~/2 q ) °l) .

(4.4)

In these, non-trivial ~lq-,~[o, terms only appear for the Uo sector (~,, = 0). Examining the ~elq0fClq0 term in AMabius, the GSO projector is modified by the presence of the T~. T~ term which will prevent term-by-term cancellation between Aplanar and AM~bius. Without term-by-term cancellation, finiteness cannot be guaranteed for arbitrary amplitudes. Thus, we see the need for the MSbius consistency condition (2.18), T~. T~= 0 (modl), from the finiteness requirement. With this condition imposed, the integrand of . . . . less can be written as (const.) ×

E {/~°}

(~vV+Uo + - AMSbius ~)I " planar e2~iVSe2"i#°°(v)(N- 2D/2~lei=N~/4ei'~r+~°ei~E:=
X \(..£frT~o ~ o (q2) qof~fl°~+UUo(/)rl' qO r) q- =~f;T;+U°uo(q2) q _ 1 6 J ~ ( P r l ' r ) q 1) (4.5) and the infinities will cancel provided (with 7",. T, = 0 imposed)

(N- 2°/2TleiwUV/4ei~e

i~rEvac(YT~°)) = 0 .

(4.6)

For the U0 sector we find that this reduces to

N - 2O/2TI e iTrNv/4=

0.

(4.7)

If N v = 14 - D = 0 (mod4), or D = 2, 6, 10, then this infinity may be cancelled by choosing the C h a n - P a t o n gauge group appropriately. For D = 2, 10 then e 2 ~ i N v / 8 = - 1 and cancellations will occur for SO(2 D/2) groups (~ = - 1 ) . For D = 6 then e 2 ~ r i N v / 8 = q-1 and hence, cancellations are only possible for USp(2 D/2) groups

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(~/= + 1). In other dimensions, Uo • Uo 4:0 (mod 1), so the induced GSO projector in the M~Sbius amplitude does not match that of the planar amplitude and we cannot obtain term-by-term cancellation. Examining other sectors which also have fldq/q infinities, these infinities also cancel provided e i ~ e - i ~ e v , c ~ v r + v0) = + 1 (4.8) and we must choose e ~ ' ~ to satisfy this equation whenever the closed sector contains massless singlets. For sectors where Evac = 0 the choice of $ = 0 ensures that the massless states lie in the adjoint of the C h a n - P a t o n gauge group. In this section we have restricted the analysis to periodic or anti-periodic boundary conditions only since this simplifies the analysis of the transformed M~Sbius amplitude. Models which use complex boundary conditions [4,16] may be possible. For these models, although more complicated, we still have a U0 sector and the dilaton finiteness condition (4.7) will still apply, when term-by-term cancellation is possible. In the next section we shall illustrate these ideas and results with some concrete examples. 5. Examples of open and closed superstring models The first simple model we consider has closed sectors generated by closed string basis vectors, W 1 = (W0; 0 ) ,

W 1 = (0; W0).

(5.1)

The finiteness of the corresponding open string model is also discussed in ref. [25]. For this model, the closed string consistency conditions (2.5) are only satisfied in dimensions D = 2, 6, 10. In these dimensions, the k,s's which satisfy the consistency conditions (2.5) and the truncation conditions (2.12) are [ [kij] =

kll ½+ ~(14-D)

½+ ½ ( 1 4 - D ) ] kll ,

kllE

(0,1}.

(5.2)

The closed model has four sectors 0, W1, Wi, and W1 + Wi = W0. There are two LRS sectors, W0 and 0, whose states may appear in transformed open string planar and MObius amplitudes. The GSO projection conditions for these two sectors are W1

__ 1

.Nwo= !2'

W 1 • NO= kll + ~(14

Wi'Nwo- 5 -

D),

Wi • No = kll ~- ~(14

-

O).

(5.3)

As expected, the graviton and dilaton appear in the spectrum. (The antisymmetric tensor B~ of the W0 sector is projected from the spectrum when we truncate to an unoriented model.) Since in D = 2,6, Evac(0) = 1, 1 / 2 > 0, there are no massless fermions and this model is not locally supersymmetric in space-time.

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The two choices of kll lead to physically equivalent models. For example, fixing kll = 1 / 2 + (14 - D)/8, and using our construction, the model which can couple to the closed string has open string fermionic annulus partition function (2.14) given by 7open = 1~N 2(Ft~o t/o(¢o) - F~° (~o) _FvOo(w) + ~0 ~ot~0~] (5.4) ~annulus k 1] The corresponding expression for the M/Sbius strip is given by zopen MiSbius

½~N [e-i=/2'~i~'14-°)/24( F u°(°i~ w,.,~t - F~o(e i~ ~0)) [ ~, ~ U0 k ~

=

--e

i~,14

D)/12[~,FOtoi=,.,~Uo, ~ ~ , - F°(e/~ 0:))]

(5.5)

where, as discussed below eq. (2.17) the phases, e i ~ ( 1 4 - 0 ) / 2 4 and e -i7rO4 0)/12, arise because the twist operator does not affect the vacuum energy part of the hamiltonian and we have taken @ 0 = - 1 / 2 and ~0= 0. The U0 sector of this model contains massless vector bosons and scalars in the adjoint representation of the C h a n - P a t o n gauge group. The 0 sector contains only massive fermions in D = 2, 6 as for the closed string. Transforming the open string annulus into a closed string vacuum exchange yields (see appendix B) zopen annulus

=

½N2(Fffo(q2)_ eoVo(q2)_ FuO0(q2) + eO(q2))

= ~N 2 tria~s [ q t?~'+/?~0~'(1 - ( - 1) ~?Wo)] ½N 2 trL,S [ qglef'+flg~'(1 --

-

(-- 1) ~0)]

(5.6)

where the trLRs runs over closed string states which are left-right symmetric in their oscillator content. The projector in the trace leaves LRS states which match those not projected by the closed string projection conditions (5.3), with the given choice of k11, as guaranteed by our construction [8, 9]. The corresponding transformed Mtibius strip is open MObius

=

l~Ne-i,/aeS,,i(14

0)/24(FWo0(e-i=

o [e -i~ - 2 ±~qNem(14- D)/12[~F Uo'

I~Mo-i~r/2o5~i(14-D)/24*r 2'1 .

.

.

.

.

.

q2) _ e iv(14

D)/2FoVO(e i~ q2))

q2)_e-i,(t4-n)/2FoO(e i~q2)) [[o i~r/2,/~1:l~,',%t?~,'tl--e-iCr(14-D)/2(--1)Nwo]]]]

LRS[\ ~ I"

.

"/]

k~

/~/left + ~ r i g h t

_½~Nei,(14 m/12trLRs[(e ,~/2q) .

-o

/

(1_e-i~,14-D)/2(_1)~0)]. (5.7)

For D = 2, 6,10, where the closed string model (5.1) is consistent, the GSO projections on LRS closed string states present in the M6bius amplitude match the closed

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121

string projection conditions (5.3). This matching is an explicit example of the T,. T , = 0 ( m o d l ) induced GSO consistency condition (2.18) for the M/Sbius contribution. We demonstrate the cancellation of the infinities in the Uo sector of this model as for the vector amplitude of sect. 3; other amplitudes follow in a similar manner. Taking the transformed annulus amplitude (3.10), for this model the Uo contribution is (neglecting, for convenience, Chan-Paton factors and normalisations which are common to both the planar and M6bius amplitudes) ~Uo Apl ....

M 1 Uo 2 ~ r~=l [~)(Pr+l--Pr)dPr{~Uo(q )~U°(b'r'T)--~'U°(q2)~cg°(Pr,"t')}

=Nfo 1

(5.8) while for the Mrbius (3.16) we obtain

7Vo = AMrbius

1

1M 1

[-I ~)(Pr+l--Pr)dPr ei~rNv/4e-i~r 2D/2"Ofo Tdq f"0 r=l Uo (e -i~r q 2 ) ~Uo° ~ ( ~'r, e -'~ q2 ) × { o~°Uo

--e i~u~/2~OV°(e i~qe)~o~O(~r,e-i~q2)}.

(5.9)

We now Taylor expand these in q and extract the divergent pieces. For the planar amplitude (5.8) we obtain Apl~U°anar[divergent= Nf0 1 ~

f0 1 hr=l I O(Pr+l--Pr)dlPr

X {(~UU°[q0 - ~g0 ]q0)q 1+ (,~uU(:[ql _ fffg0 [q~)q0

+ 2gv(~ffOlqo +f¢oUoIqO)q0}.

(5.10)

Expanding the MSbius amplitude (5.9) we find AMgbiusldivergen t = 2 D/2TI ei~rNF/4 e i,rf0 1 ZVo

7£ 1 "-1 ~--~10(Ur+l -- Vr) dgr

×{(f~uUoqo--e icrNF/2~g 0 qo)(e

i~/2q) -t

q-(~gOlq,--e-i~rNV/2~gOlq~)q° +2NF(f~U°lqO+e-i~NF/2ffloU°lqo)qO} .

(5.11)

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Z. Bern, D.C. Dunbar / Type I superstrings (ll)

Since ~o°[qO = ~o~°[qO, as can be explicitly checked, the tachyonic singularity always cancels within the planar amplitude; however, only in D = 2, 6,10 (where the closed string is consistent) does the tachyonic singularity cancel within the MSbius amplitude. Furthermore, the dilaton singularity is non-vanishing in both amplitudes and will only cancel between the two terms if

N+2D/271=O

N--2D/2~7=O ( D = 6 )

(D=2,10),

(5.12)

so that the C h a n - P a t o n gauge group necessary for dilaton finiteness is SO(2 D/2)

for

D = 2,10,

USp(2 D/z)

for

D = 6,

(5.13)

in agreement with (4.7). Here the D = 10 case is the well-known finite SO(32) superstring [1,17]. In this example, for D < 10, the lowest mass state of the closed string (0; 0) sector 70 is massive so no dilaton type singularities occur in A70plana~ or AMabiu. Since the above D = 6 string model contains no massless fermions it is uninteresting from a phenomenological point of view. One D = 6 model with massless fermions is described by the closed string basis vectors W1 = (Uo;0),

W1 = (0; Uo),

W2 = ( G ; 0 ) ,

W~ = (0; G ) ,

(5.14)

where G = (0 2,(7, 1 ½,0)2) • After imposing the closed string consistency (2.5) and truncation (2.12) conditions, the kij matrix is

[kij] =

k12

kll k12.

kl~ (k12l-k12.) -~- I

k12 k22

kl ~

kl 2

A2~

(k12 + kl~) + 1

,

kijE

(0,1).

(5.15)

The space-time supergravity of the closed model can be determined by counting the number of gravitinos in the (G; U0) or (U0; G) sectors. After truncation to a type I model these two sectors are identified with each other so we only need consider, say, the sector (Uo; G ) = W1 + W~. From eqs. (2.7) and (2.8), the closed projection conditions in the Wx + W~ sector are Wl" N(uo: a) = ~1 = kl~, WI" N(uo; G) = ~1 = k12 + kll + ,21, W2" N(~/o;c) = q~2= k17 + ke~, w : • U uo; c> = q': =

+ I.

(5.16)

Z. Bern, D.C. Dunbar / Type I superstrings (II)

123

(Note that this sector is non-LRS so there are no additional projection conditions arising from the Klein bottle contributions.) Since the gravitino states are ( b ~ 2 or d~)2)]0 ) ® [0), (i = 1,2), they are kept iff ~1 = ½, ~2 = 0, and q~ = 0, so we require k ~ = k2~ = 1/2,

(for N = 2 SUGRA)

(5.17)

to ensure supersymmetry. If this condition is not satisfied, the spectrum is not supersymmetric. For the supersymmetric case, the remaining Wi projection condition reduces the 2 4 component Ramond vacuum ]0) to 23 components which form two 2 2 component Weyl spinors of opposite chirality. In either case this model is not chiral; it is not possible to obtain an N = 1 theory for any choice of the k~j in this model. As we shall see in sect. 6, the reason for this is tied to anomaly cancellation. The LRS states of this closed string can be found in the amplitudes of consistently coupled open strings. An example of a LRS sector whose states may be identified in the open string amplitudes is the (G; G) = W2 + W~ sector. In this sector, from the general projection conditions (2.7) and (2.8), the kept states satisfy W1 "Nw2+ w2 = Wi " Nw2+ w2 = kxl + kl2 + kl~ ,

(5.18 t

W2" Nw2+ w~ = W i . Nw2 + ~ = k2~ + ~.

Similarly, it is not difficult to work out the projection conditions in other sectors for comparison to the Jacobi transformed open string amplitudes. For the above closed string, the corresponding open string partition function for the annulus, from eq. (2.14), is N 2 zopen annulus -- -

- ro

4

°(

-( rgo( ) + 81F°( -(F~(w)

l -

+

1 + 8182F°(

) +

2F°+

+ 8182FoG(¢0) + 8183Fg(~ ) + 8283Fg+ Uo(tO))

+ F~+ v ° ( ~ ) - 82Fg+ U o ( ~ ) - ~283FG+ u0(~) + 83Fo°++~°( ~ ) ]

(5.191 where 61 = e 2~ik11,

82 =

e 2"i~k12+k~),

83 = e 2~i(k22+k2~) = - e 2~i~k12+k~+k~ . (5.20)

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124

The corresponding Mbbius contribution is given by Z~v~e~ius~-" ¼71N [e-i~r/2ei~r/3( FUo°(ei~rw) - FU°(ei~rw ) - FU°(ei'¢o ) + F~_ uo(ei'w )) -e-

2~ri/3 (FOuo(ei~w)

+ 61FO(ei~w)

+ ~lt~2Fd(ei~r~ ) + 82 FGU~uo(eiWw))

-e iw/6( F~(ei,ko) + 8182F~(ei~,w ) + 8183Faa(ei~lo) +8283FGa+uo(ei~r(.o)) +

e i~r/6(r~+ Uo(ei%~) _ 32Fg+ Uo(ei~co) _

~2~3FGG+Vo(ei~o)

+63FoaduUo°(ei"¢o))],

(5.21)

where we have substituted }uo = - 1 / 2 as necessary, and taken 4a = }c+ Uo= }o = O, which is a choice consistent with eq. (4.8). Since the projectors of the various sectors in both the annulus and Mbbius contributions are (Vo) = 1(1 - e2~iv0~?u0)(1 + e2~'a~?,o),

/3(0) = ¼(1 + 81e2~'ivoiVo)(l+82e2~riGNo), =

+

/ 3 ( G + /do) = 1(1 -

8132 e2"iu°'zOa)(1 + 3233e2'~icu~;), +

(5.22)

the projections are explicitly seen to be physically sensible. From these projectors we can determine the open string spectrum. Massless open string states may be found in the U0, G, and G + U0 sectors. The U0 sector contains the massless vector in the adjoint of the C h a n - P a t o n gauge group, which is never projected, as well as scalar states, while the G + Uo sector contains 23 massless scalar states only if 8283 = - 1. For the open string to be supersymmetric the massless fermions in the G sector must not be projected. From the G sector projector in eq. (5.22) these are not projected if k2 ~ =12, (for N = 2 SUSY) (5.23) or equivalently if ~2~3 = 1. When k2~ = 1/2, the 24 massless states of the Ramond vacuum 10) are truncated by the U0 projection to form two, 22 component Weyl spinors of opposite chirality; hence, we have N = 2 non-chiral super-Yang-Mills. The massless fermions lie in the adjoint of the gauge group, as is necessary for supersymmetry. (This is a consequence of choosing ~G = 0 rather than ~a = 1.) For the choice k2~ = 1/2, the pairing of bosonic and fermionic states indeed continues to all mass levels, and the full partition function vanishes, as may be verified by using the fourth order theta function identity [23], [0~/2(01~-)] 4 - [0~/2(01~-)] 4 [00/2(01,/.)] 4 = 0. (The vanishing of the partition function is dependent on the values

125

Z. Bern, D.C. Dunbar / Type I superstrings (II)

we took for ~o and ~ + uo-) As for the closed string, there is no choice of k/j's which yields N = I chiral supersymmetry. Since the closed string supersymmetry condition (5.17) implies the open string supersymmetry condition (5.23), the space-time supergravity necessarily implies that the super-Yang-Mills is of the same type. It is amusing to see this consistency emerge from our formalism. Under the Jacobi transformation (3.7) the annulus contribution to the partition function becomes N2

~annulus7°pen =

uo 2) ~4-[Fffo°(q2)- FoU°(q2)- F~°(q2) + F~+uo(q

-( F~,(q2) + 81FoO(q2 ) + 6182F0(qZ)

+ 82F0+ u0(q2))

- ( F~o(q z) + 8162FoG(qz) + 8163FaC(qz) + 8263Fca+vo(q2)) + F~+ v0 (q2) _ ~2Fa+ v0 (q2) _ ~2%F~ + vo(q2) + % rg++~0 (q2)].

(5.24) As before, these transformed partition functions may be reinterpreted in terms of traces over the LRS states of the closed string. For example,

(F~o(q2) + 6182Foa(q 2) + 6163Fg(q 2) + 6263Fg+ u0(q2)) •

[

/~left

_ + £rright

×(1 +

"

^

+

(5.25)

From the relation of the 6i to the kq, (5.20), we see that in the W 2 + W 2 sector, the projector in eq. (5.25) keeps the same set of LRS states as the closed string projection conditions (5.18). The relationship between the projections found in the open string and the closed string can be checked to hold for all sectors, as guaranteed from our construction [8, 91. Under the Jacobi transformation (3.7) and the q ~ q4 substitution, the M6bius contribution becomes Z~io~ = ¼nN [e i"/2e-i'/3( Fff'o'(e i, q2) _ FoC,O(e i,, q2) _ F2'o(e-i,, q2) + Fc)) uo(e ,,, q2)) _e2~i/3(FOo( e i~q2) +81FoO(e i, q2) +8182FoO(e i~rq2) +82FU2tjb(e iTrq2))

_eT,,,/6( F~o(e i~ q2) +StSzFoG(e ,~ q2) + 81B3F~;(e i,~q2) + 6283F~C,+tJo(e i~ q2)) ~. Uo+ ~"(e ,,~q2)_82Fg+~o(e , ~ q 2 ) _ 8 ~2 3 EGC+c'°to i'~-z~+8 EC+t4te i, q21)] k~ ~/ J' 3 G+ Uo \

+ e7~rs/6/FG

(5.26)

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Z Bern, D.C. Dunbar / Type I superstrings (II)

These M~Sbius contributions have the same induced GSO projections as the annulus contributions (5.24) as guaranteed by the T ~ - Ta = 0 consistency condition (2.18). The relative phases in eq. (5.26) are, of course, valid only for D = 6. By Taylor expanding in q the U0 part of the combined M6bius and annulus amplitudes, (as in eqs. (4.1-4.4)), we find that the tachyon singularity vanishes from each amplitude due to the induced GSO projection within each amplitude. After expansion, we again find that the condition for the vanishing of the dilaton singularity in the U0 sector is N - 26/27 = 0

(5.27)

as before. Thus, for gauge group USp(23) this model has no divergence in the U0 sector, in agreement with eq. (4.5). There is also the possibility of infinities due to massless states in both the G and G + U0 sector. The G + U0 infinity is removed by the GSO projection in the supersymmetric case, k2~ = 1/2, and cancels between planar and M6bius amplitudes otherwise (for the choice ~a+ u0 -- 0). For the G sector an infinity also arises, for the supersymmetric case only, which cancels, between planar and M6bius amplitudes, when the finiteness condition (5.27) is satisfied. This cancellation only happens with the choice of ~a = 0, as was made in eq. (5.21). The choice ~a = 1 would allow (1 + ~ a ) / 2 to act sensibly but then the finiteness condition in the G sector would require a gauge group of SO(23), which is incompatible with the finiteness requirement for the U0 sector. The choice ~a = 0 also ensures that the fermions lie in the adjoint representation of the gauge group, as necessary for supersymmetry. It is interesting to note that the UV finiteness in the open string sector is not dependent on the existence of space-time supersymmetry. Finally, we comment that it is not difficult to construct more models simply by using other basis vectors.

6. Comments on chirality and anomaly cancellation The reason why the D = 6 models in the previous section proved non-chiral may be understood simply in terms of anomaly cancellation requirements in the low energy field theory. The dilaton finiteness condition (4.7) for D = 6, generically requires a USp group for the types of models we have been considering here. On the other hand, only a C h a n - P a t o n gauge group of SO(2 D/z) is anomaly free for models where all fermions lie in the adjoint represention of the gauge group. In D = 6 the Green-Schwarz anomaly cancellation mechanism [24] requires the absence of trfund(F 4) terms in the anomaly generating function, where this trace is in the fundamental representation. For chiral fermions in the adjoint representation, as in the second model of sect. 5, the anomaly generating function contains a

Z. Bern, D.C. Dunbar / 7~Te I superstrings (lI)

127

Tradjoint(F 4) term, where this trace is in the adjoint representation. These two traces are related by

2 2 Yradjoint(F4) = ( N +B23)trfund(F 4) + 3(trfund(F )) .

(6.1)

Hence, for a chiral USp model, the trfund(F 4) term does not vanish and the Green-Schwarz mechanism will not be able to cancel the anomaly. Since finite models are expected to be anomaly free the second model of sect. 5 could not be both chiral and finite because, irrespective of the choice of kq, the only possible massless fermions are in the adjoint representation of USp(23). In our construction, the compatability of the gauge group with the anomaly cancellation requirement follows from non-trivial conspiracies among the basis vectors and kq's, preventing the breaking of the N = 2 space-time supersymmetry to N = 1 chiral supersymmetry. It would be interesting to derive a fundamental understanding of these conspiracies. Anomaly freedom in the low energy limit, although necessary for a chiral string theory, does not imply the existence of a corresponding string theory; finiteness is the guide to string consistency as also indicated in ref. [10].

7. Conclusions

In this paper we have set up a framework within which to study the one-loop finiteness of the new D < 10 type I superstring models of Part I [9]. These models are constructed within the fermionic formulation for internal world sheet degrees of freedom. The finiteness conditions (4.6) should be taken as an additional consistency rule to be placed on the models of I. We have explicitly constructed several examples of open strings which can be made one-loop UV finite by an appropriate choice of C h a n - P a t o n [14] group factors. In general, our construction will yield finite models in only D = 2, 6,10; in other dimensions cancellation of the singularity is not found. The one-loop open string UV finiteness does not require space-time supersymmetry; finiteness depends on cancellations between oriented and unoriented diagrams, while space-time supersymmetry depends on the choice of generalized GSO projections. As a consistency check, in our examples we have found that the coupling of open strings to closed strings implies the appropriate supersymmetric spectrum of the open string for the type of closed string supergravity. Furthermore, for our D = 6 examples, the finiteness condition for USp(2 3) was automatically found to be consistent with the anomaly cancellation requirements because of an inability to choose the GSO projectors to form a chiral model. It would be interesting to extend our analysis to more general supercurrents, GSO type projections, and boundary condition basis vectors.

128

Z. Bern, D.C. Dunbar / Type lsuperstrings (II)

From these examples, we see that anomaly cancellation in the low energy field theory, although important, is not necessarily the guide to constructing consistent D < 10 open string models. This is also indicated by the results of ref. [10]. It is interesting that in open string theories, at least in the fermionic formulation, there appear difficulties in constructing consistent models directly in D = 4, while there are no difficulties in compactifying open strings within the context of the low energy field theory. The basic reason for this is that, through the Jacobi transformation, consistency in string theories depends crucially on the arrangement of the infinite tower of states, while the low energy field theory depends only on the massless states. This should be taken as a warning about trusting naive low energy field theory compactifications of string theories. The construction of D < 10 type I superstrings has recently been considered for Z 2 orbifold constructions in ref. [10]. The relationship between the fermionic and orbifold constructions is not clear and in the orbifold construction of ref. [10] there are, apparently, no finite superstring models in D = 6 [10]. As this work was being completed, we received preprints [25] which also discuss construction and one-loop finiteness of D < 10 superstring models; however, the finiteness of only a single non-unique model containing no massless fermions was considered, with results for the cancellation of the dilaton infinity in agreement with ours. We would like to thank J.L. Petersen and A. Love for discussions. D.D. would like to thank C. Herzogenrath and The Niels Bohr Institute for hospitality and Z.B. thanks Royal Holloway College, University of London.

Appendix A. Example of term in amplitude The vertex operator (3.2) produces a variety of terms in the total amplitude owing to the different types of field contractions. In this appendix we will compute and then transform a sample term in the amplitude (3.1) to illustrate the Jacobi transformation properties. The other terms in the amplitude transform identically. Specifically, we take the ~. Xk. Xe ikx part of the vertex operator (3.2), as an example, and substitute this into eq. (3.1) and obtain

- rld~° M 1 d~r f dDpTr[ M ] T B°r+ U° = gMN tr[ t,t2.. . t , ] JO -~- £ 1 r~=l T j I-~ e'k`x(~')J~°~ planar = i= 1 × e 2~;¢°s Tr[ I-[ e;. X(f;)k;-X(f;)~0~°T+ uo/303°T + U00) -

(A.1)

I_i=1

This trace may be evaluated using Wick's theorem (see e.g. [12, 20]) giving contribu-

Z. Bern, D.C. Dunbar / Type I superstrings (lI)

129

tions to the Green functions in the amplitude (3.5) ~I3°T+V°(v'[r')=

[ 1 perms: {a} ~

1

^

^t

M

M

]

xG(J~T+UOl)(p" IT/)Hk~bbNE~c ( (yT+U~7~o) \ a21--Pa21 1 b= l c=l ] × [ "I-I <: [ ~ (v/Yl l T')] k"k/]

(A.2)

The permutations are { a } = { a 1, a 2 . . . . . a 2 M ~ 1 , 2 , . . . , 2 M such that

( l = 1 ..... M ) }

a2,*a2,_l+l

a21 l < a 2 1 ,

(A.3)

so as to eliminate double counting and self contractions. P is the order of the permutation, vii = v" - vj', and the definitions of ~b and G are given in appendix B. For a I odd (even), /2=2H,,+a)/2 (/~,,/2) and P= v(a,+l)/2 (v,,/2). Although the index structure of this formula is complicated, it is nothing but an explicit realisation of all the possible Wick contractions. All the other terms in the contraction of M gluon vertex operators take on a similar form [20]. The corresponding contribution to the M6bius amplitude is identical except that ~k--+ ~P~' and ~-~ ~¢~', which are defined in appendix B. Under the Jacobi transformation (3.7) and from appendix B, the Green functions transform as :,,l~V+Uol ,, ,,

{--lnqlMafirr+vo

~

1 - -i~r

~'l~v+v°(v/Ir' r r + Uo

Go~,v, )

=

IT ~ =

2-M(--lnq)M:z - irr

~

(A.4)

~ v + Go

)

flV+2rT+Uo/

.....

tV#Zlr/,+-

- vr + Uo

1/2)

(A.5)

where the transformed Green functions are defined by 1 gT+ Uo

-)" v

g

=

(

perms: {a}

~( G (~°V~O0) l (^

Fig<,

/=1

-- ~

(Tr+ uo)~ v.~,

×

- 2 ~ r i O;(O[r)

with the permutations as described in eq. (A.3).

"

-

- ]

IT ) ~YI1 k~bb(~l~C

(A.6)

Z. Bern, D.C. Dunbar / Type I superstrings (II)

130

Since the Green functions for any amplitude are expected to carry identical Jacobi weight, the relative factors of 2 will be the same for any M-point amplitude.

Appendix B. Definitions and transformation properties The various functions used in the text can be written in terms of the Jacobi O-function

0,~(~1~) --

q(,+o-1/Z)2e2~i(,+o-1/2)(~ ~ 1/2)

~ n~

-- oo

= e2~ri((1/2-v)(1/2+u)+v(v--1/2))qV 2

v+l/4

× l - I ( 1 - q 2 " ) ( 1 - q 2(n+v-1) e2=i(* " ) ) ( 1 - q 2(n v)e-2"i(~ ")) (B.1) n=l

where q2 = exp(2~riT), and 0°(rl T) is commonly named 01(u[~") [23]. Another useful function, related to the Dedekind ~/-function, is

f(q2)=fl(1--q2")=qn=l

~/12~(~.)= q 1/12(0;(01"/')) " 2 T r 1/3

(B.2)

The functions appearing in the text describing partition functions are

F~'(x)~ [x(v2-v+l/6)/zfi

le-2~riu)(l-- xn-Ve27riu)]

= e-2~ri(1/2+u)(1/2 v)0ff(0lln x/2~ri) (In x/ZTri )

(B.3)

for a single world sheet fermion and N F

(B.4) k=l

for a set of N F world sheet fermions.

Z. Bern, D.C. Dunbar / Type I superstrings (II)

131

The relevant open string Green functions are ,

,2

, 01(Ptl ~t )

'=" / , 0{(0lz, ) '

(B.5)

2~riei~,2/,Ol(~'l'r" + 1/2)

(B.6)

~b(U'lT' ) = - - 2 7 t i e

O{(Ol'r'+ 1/2) '

¢~(~'lr') = -

1 0~(p'IT') G'(oI~') G 2 ( # l r ' ) = - 27ri 01(~'1~')02(01~' ) ' G ~e.,(I,. I.~ ).= G . ( ~ ' I T ' + 1/2),

(u, v . 0),

(B.7)

(~, ~ , o).

(B.8)

After the Jacobi transformation the relevant functions for the annulus amplitude are 691/24f(69) =

ql/12f(q2),

--

FW(69 ) = exp(2~ri(U- Uo). ( V - Uo))FV~(q2),

~b(l,']~")

G:(~,' I~")

2'n"2 01(t) I,r ) lnq 0[(0l~- ) ' In q ivr

(B.9)

(B.10) (B.11)

G~_-~(plz).

(B.12)

For the MBbius amplitude the relevant transformed functions are 691/24f(_69)

= ~-

_

FW(e'%0 ) = exp[2~ri((U- /do). ( V - Uo) + ( ( - U -

×exp[-Zrri((V.

V-2V.

2 V ) - Uo). ( V - Uo))]

Uo+ N F/ 6 ) ) ] F _ - Uv+ 2V e-i~V~,

2~r 2 0 l ( v / 2 l r / 4 - 1/2) +~(p'l~') = -2-lnq O{(OIT/4- 1/2) G~'~(#I~ -') -

(B.13)

ql/48f(_v[~)

In q G __u + 2 v (u/2l'r/4 - 1/2). 2vri -v

(B.14)

(B.lS) (B.16)

132

z. Bern, D.C. Dunbar / Type I superstrings (II)

F o r c o m p a r i s o n to p l a n a r c o n t r i b u t i o n s the a r g u m e n t s of the M O b i u s f u n c t i o n s are r e s c a l e d b y ~-/4 ~ T a n d v / 2 --* u. F i n a l l y t h e i n t e g r a t i o n m e a s u r e t r a n s f o r m s as d~0_ ~_~1 d~', O) r=l ~r

1 2~2(--lnr0)M+l

__dq M - 1 q ,=1I~ dr'r"

(B.17)

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