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Four-dimensional type I superstrings withN < 4 supergravity
Volume 242, number 2
PHYSICS LETTERS B
7 June 1990
FOUR-DIMENSIONAL TYPE I SUPERSTRINGS WITH N< 4 SUPERGRAVITY Z. BERN Theoretical Division, MS-P285, Los Alarnos National Laboratory, Los Alamos, N M 87545, USA
and D.C. D U N B A R DAMTP, Liverpool University, Liverpool L69 3BX, UK
Received 5 January 1990; revised manuscript received 27 February 1990
We present rules for constructing four-dimensional type I open superstrings and examine a number of models which are oneloop finite and where the space-time supergravity is N= 4, 2, or 1. The N< 4 models are obtained by breaking the supersymmetry with suitable GSO projectors.
1. Introduction
Superstring theories have attracted great interest in recent years as potential theories of q u a n t u m gravity [ 1 ]. Originally, superstring theories were constructed in ten spatial dimensions, but more recently a variety of schemes have appeared for constructing four-dimensional theories o f closed superstrings [ 2 - 5 ] . The construction o f lower-dimensional theories of open superstrings is more complicated [ 6-12 ] largely due to the lack of a simple symmetry principle, analogous to modular invariance which guarantees much o f the consistency of closed strings. The starting point in our construction o f consistent type I models is type II left-right symmetric (LRS) oriented four-dimensional closed string models, which can be truncated to unoriented models [ 6 ]. The coupling of the open string to the closed string is determined by the well-known property that closed string poles are found in open string one-loop amplitudes [ 13 ]. The basic rules for coupling open strings to closed strings have been presented in refs. [ 6 - 1 0 ]. The construction o f consistent models relies on the use o f a twist operator which acts sensibly on the open string states. For dimensions D - - 2, 6, l 0, the standard twist operator [ 1 ] is sufficient to enable open string constructions [6 ] in the fermionic formulation; however, in D = 4 this twist operator does not lead to a sensible theory. The difference between D = 4 and D = 2, 6, 10 can be traced back to the closed string. For the standard twist operator to be sensible the open string GSO projectors are required to ensure that all states in a given sector are integer spaced in mass. Through the open-closed string coupling, this in turn implies a set of closed string boundary conditions which are consistent only in D = 2, 6, 10. Additionally, with the standard twist operator, the closed string states found in the M r b i u s amplitude will not correspond to the spectrum of the closed string [ 6 ]. In a previous paper [ 14 ], by the use o f a modified twist operator we constructed the first example of a space-time supersymmetric one-loop finite four-dimensional open superstring. In this paper, we present the general formalism and also some explicit examples o f one-loop finite four-dimensional type I superstrings whose massless spectrums are N = 4, 2, l, supergravity coupled to super-YangMills. One-loop finiteness requires a specific choice o f C h a n - P a t o n gauge group [ 15,16 ] and in D ( e v e n ) 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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dimensions the type of models we construct are one-loop finite for Chan-Paton gauge group Sp(2 0/2) or SO ( 2 n/2 ) (depending upon D).
2. General considerations
To maintain superconformal symmetry, superstring theories in dimensions less than ten require internal degrees of freedom which can be represented as world-sheet bosons [2 ], fermions [17,3,4] or even directly as superconformal field theories [ 5 ]. We work with models where the internal degrees of freedom are represented by d = 1 4 - D complex Bardakci-Halpern fermions, but our work can be straightforwardly extended to the other constructions. In ref. [ 6 ] a general formalism for type I superstrings in the fermionic formulation for the internal degrees of freedom was presented. We briefly recall the relevant results in the notation of ref. [ 3 ]. A general closed string model consists of generic bosonic degrees of freedom as well as fermionic degrees of freedom. The boundary conditions of the world-sheet fermions on the torus are specified by vectors W= ( w kl I~k) = ( W I,..., W dl l~l, •.., if, d), where the first and second d components respectively specify the boundary conditions of the left- and right-moving world-sheet fermions [ 3,4 ]. For w k = ½the kth fermion has Neveu-Schwarz boundary conditions while for Wk= 0 the boundary conditions are Ramond. We will restrict ourselves to the case where only NeveuSchwarz or Ramond boundary conditions are allowed, although general rational values of W k are possible. The vectors representing the set of allowed boundary conditions in the model are generated by basis vectors Wi, (which always includes Wo = ((½) d l (½)d) ) SO that this set is given by ~iai Wi = ot W where the a~ are integers. The overbar notation indicates that the values of Whave been shifted by integers so that w k = O, ½. The vectors W~are, of course, not arbitrary but satisfy conditions that ensure the modular invariance of the model [3,4], 2Wi'Wj=0 ,
Wi. Wi=O (mod 1 ) ,
(2.1)
as well as a world-sheet supersymmetry triplet condition. (The dot product between closed string basis vectors is lorentzian with a minus sign for the components describing right movers.) The spectrum of the model is truncated by a generalised GSO projection determined by the Wi and by a set of"structure constants" k~j (which are all 0 or ½and again subject to constraints). The states which are kept under the GSO projection form the spectrum of the model and satisfy W ~ . N ~ = ~ ( o t ) - = ~ k~je~j+ko,+si-W~.~W J
(mod 1 ) ,
(2.2)
where N~-ff is the vector with components given by the fermion number operators N,~w~w~.The statistical factor si is, for a type II string, W] + / ~ (mod 1 ). For ~ - - - Yqc~s~= 0 the states of the sector o~W are space-time bosons while for ~ = ½they are fermions. As shown in ref. [ 6 ], the first step in the construction of open superstrings is to focus on the LRS type II closed strings which are truncatable to unoriented models. Such strings can be generated by a set of basis vectors of the form
Ws=(U~IU~),
Wa=(W~II~),
Wa=(ff',*lWa~),
(2.3)
provided the additional truncation constraints on the structure constants
kas"~-kas=kabdrkt~s=ka~-~kab=O (mod 1)
(2.4)
are satisfied, ensuring that the closed string Fock space is symmetric between left- and right-movers. This allows for the truncation to unoriented closed superstrings which can then be coupled to unoriented open superstrings to form a type I superstring model. The open string models are specified by boundary condition vectors with half the number of entries as closed 176
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string vectors because the left- and right-moving components of the fermions are no longer independent, but are reflected into each other at the boundaries of the world sheet. The set of boundary conditions in the open string determining the GSO projectors corresponds to the subset of the LRS sectors of the closed string ( UI U) whose LRS states are not projected. The U's come in two varieties: from (UslUs) and from W a + W a = ( w k + ff'ka I Wka + ff'~ ) =- ( Ua I Ua). The set of boundary conditions determining the open string sectors are Va = ( Wka -- ffgka) = (Vka) which corresponds to the left-right asymmetric terms in the closed string GSO projectors. In general, these two sets of open string boundary conditions need not be the same. However, only for the case where the two sets of open string boundary conditions are identical, so that the model is "square", will there be a one-to-one correspondence between the sectors of the Jacobi transformed planar and Mrbius amplitudes. This allows for cancellation of the string one-loop potential infinities in general amplitudes. As shown in ref. [ 6 ], for det kss, ~ 0 the Us do not contribute to the open string GSO projections so the open string is square and the open string states kept under the GSO projection are those satisfying U a "N # ~ - ~ o = p a ( fl ) ~ ~.~ ( k ab -'l- k ab ) flb -I- .~a - U a " ( fl U -I- Uo - U o ) , b
(2.5)
where the statistical factor ga is the first entry in Ua. The annulus contribution to the open string partition function is then given by
,-.#U+Uo ._x
zann = N2
- - Y~ exp(2~ifls) exp[2ai~p(fl) ] r~--OV~t~ ) , M {ma}
(2.6)
where FV(r)=
k=l fi
( exp{2~iz[(Vk)2-Vk+~]/2}
× f i {1 - e x p [ 2 z f i z ( n + V k - 1 ) ] exp( --2niUk)}{1 - e x p [ 2 r d z ( n - V k) ] exp(2rdUk)}] / n=l
(2.7)
is the partition function for space boundary conditions Vwith exp [2zfi( U - Uo)"Nu] inserted into the trace over oscillators and z is the imaginary Teichmiiller parameter for the annulus. We have included a factor o f N 2 from the Chan-Paton trace over both boundaries and the normalisation factor M = 2 × (number of sectors), so that every unprojected state contributes with a coefficient of unity. (The extra 2 arises because we are considering unoriented theories where the annulus and M6bius amplitudes combine to give the spectrum.) This annulus vacuum amplitude, as well as more general amplitudes, has the property that when transformed into a closed string exchange [ 13,6,8 ] the terms appearing correspond to physical states of the closed string as guaranteed by our construction. To obtain a M6bius amplitude which contains the same set of closed string poles as the annulus amplitude it is necessary to impose the further conditions on the Ua [ 6 ]
Ua'Ua=O,
2Ua'Ub=O
(rood 1 ) .
(2.8)
When we start with a set of closed string basis vectors satisfying the modular conditions then the second of these conditions is always satisfied although the first is non-trivial. Conversely, if we have a set of Ua satisfying eq. (2.8) then we can always construct a suitable closed string model by taking the closed string basis vectors to be Wa = ( Ua + Uo I Uo), Wa = ( Uo I Ua + Uo ) plus I47o= ( Uo I Uo) if not already present in the set of closed string boundary condition vectors. It is not difficult to check that with conditions (2.8) and the world-sheet supersymmetry triplet condition [ 3 ] such a closed model is consistent. It is important to note that Uo need not be amongst the Ua and indeed in D = 4 Uo" Uo#0 and therefore does not satisfy the M6bius condition (2.8). (Equivalently, W= (Uol0) does not satisfy the consistency conditions 177
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for the closed string. ) This means that the purely R a m o n d boundary conditions are not present in four-dimensional open strings (and, in fact, may only appear in D = 10, 6, 2). The lack o f a Uo GSO projector in four-dimensional open string causes technical difficulties for the standard twist operator = (phase) × exp ( i n H ) ,
(2.9)
where H is the open string hamiltonian. Since only the Uo GSO projector [ 1 - e x p ( 2 n i U o . h r) ] can completely remove the half integer mass levels from the Neveu-Schwarz sector of the superstring (which is always present), without this projector ½( 1 + ~ ) will not act as a sensible projector since it is not possible to choose a phase so that ~ 2 = + 1. Thus, it appears that we have hit an obstacle to the construction of four-dimensional open strings. The solution of this problem is actually quite simple. The idea is to modify the twist operator to include GSO projectors so as to split the Fock space into integer and half-integer mass states before applying the twist. The physical restrictions on the twist operator are that ~ 2 = 1, that it actually flips the string over so that for the vertex operators g2 V ( a = 0)£2*= V ( a = n), and most importantly that it results in a consistent coupling to the closed string. It should also reverse the order of vertices in a tree amplitude,/2 VI... VNI 0 ) ~ VN...VI I0 ) [ 18 ]. In the U sector, the modified twist operator satisfying these requirements takes the form I')v = exp( - niECe) ½r/u{ [ 1 + exp(2niU.Nu) ] exp (ni/(tv) - i [ 1 - e x p ( 2 n i U ' N v ) ] e x p ( n i / t u ) }.
(2.10)
The phase exp( - n i E ~ c) is included to cancel the contribution t o / ~ from the zero-point energy E ~ ¢ = - ~ 4 d + ~dR where dR is the number of complex R a m o n d fermions in the Usector. At first sight, the factor o f i associated with the second term of the twist operator (2.10) may seem peculiar; however, its purpose is to cancel a factor of i which is hidden in exp(ni/~u) when acting on states with odd fermion numbers. It is not difficult to check that this choice of ~ v satisfies the necessary consistency condition above (see for example ref. [ 18 ] ). The ~/v control the C h a n - P a t o n gauge group representations at the various mass levels. For the Neveu-Schwarz Uo sector the twist operator is equivalent to the one given 16 years ago by Clavelli and Shapiro [ 18 ]. Using this form of the twist operator the contribution to the MSbius amplitude is z m o b - N E exp(2nifls) M {a,#} [ 117,,OU+Uo .a_t2flU+Uo
exp[2niap(fl)]exp(-niE~)½rlpv+vo t~_y,oU+Uo f;,,oU+Uo
× t ~-- ~u+ vo - - - ~v--O-W~) - i ~, ~v+vo - ,
,v--o-7~) ] ,
(2.11)
where the argument of the F ' s is (;r+ ½) and the factor N arises from the Chan-Paton trace over a single boundary. Both the annulus and MSbius amplitudes contain infinities which are most easily identified after being transformed into closed string exchanges via the Jacobi transformation z' = - 1/z. The transformation rules for the partition functions are
FV(z) = e x p [2nit U - Uo)" ( V - Uo) ] F V ( z ' ) ,
(2.12)
and
FW(z+½)=exp[4xi(U-Uo).(V-Uo)] exp[2~ri(V.V-~d)] exp[ni(U.U-{d)] -~.u - v - u t, ~~- , + ½ ) •
(2.13)
Thus, under the Jacobi transformation the annulus amplitude becomes v+vo. ,,) , - - ~ exp(2nifls) exp[2z~iotp(/~)] ~~av--ow~tz M {~,a}
zBB= N 2
(2.14)
where the superscript BB is to remind us that this amplitude can be represented as a closed string exchange between two boundary states [ 19,20 ]. By relabelling fl with ot and vice versa this takes on the same form as Z ~ " given in eq. (2.6) except the argument is now z' [6]. (The invariance of the form of the model under Jacobi 178
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transformation, o f course, only holds for the square models under consideration. ) Using eq. (2.13) the MSbius amplitude (2.11 ) after the Jacobi transformation is (following ref. [ 6 ] ) ZBc=
N vac ~ exp (2zrifls) exp [ 2niap (fl) ] exp ( - h i- e ~--O;~o ). ½pvr/TbW~o lvi {a#}
× [-iexp(-niflU.flU) tr,v+u o~'aU-r~-v-+~v)~au+v° +U° . .-cxp~'-inflU.flU) ... t~aU+U°~-~-v~O~.-~)-raU+U° ~lj ,
(2.15)
where the BC superscript is to remind us that this amplitude can be represented as a closed string exchange between a boundary and crosscap state [ 19,20,6] and the argument o f the F ' s is ( ~i z , + ~1 ). To obtain this result we have used various simplifications arising from eq. (2.8) which ensure that the sum over {aa} gives the same closed string states as in the transformed annulus after the standard rescaling ¼z ' ~ r' [ 16 ]. The potential infinities in the v a c u u m amplitudes can be obtained from eqs. (2.14) and (2.15 ) by including the contribution from the bosonic partition function, which yields a relative 2 °/2 between the Jacobi transformed M6bius and annulus contributions (see refs. [ 6,1,21 ] ), and an integration over the Teichmiiller parameter z' = In q/i~. The contribution from a particular state is proportional to f~dq qm,2_ 1, where ms is the mass o f the state (in units o f ~ ) , so that potential infinities occur for states which are tachyonic or are massless in the q--, 0 limit. A proper discussion o f infinity cancellation requires a study of general amplitudes. However, if infinities in the v a c u u m amplitudes cancel term-by-term between the M6bius strip and annulus, infinity cancellation of general amplitudes follows [ 6 ], so here we focus only on the vacuum amplitudes. (Although the vacuum amplitude will vanish because of supersymmetry, the requirement of term-by-term cancellation means that it is sensible to work with these amplitudes. ) Models which are space-time supersymmetric are non-tachyonic (such as the models o f the following section) and so that only potential infinities for these models correspond to massless states. In the Neveu-Schwarz Uo sector o f the cylinders, extracting the leading f~ dq/q from eqs. (2.14) and (2.15 ), including the relative 2 °/2 and summing, yields l
(N2+2°/2rluoN) f d-~qq
(2.16)
0
as the potential infinity. To cancel this infinity we must choose ~/vo= - 1 and N=2 °/2. Infinities may also be found in other sectors, but again these cancel by choosing the ~/'s suitably. The source o f the potential infinities in these sectors corresponds to the LRS states o f the corresponding closed string sectors, through the boundary and crosscap states. However, in some o f the closed string sectors the Klein bottle removes the massless LRS states so there are no physical states for these open string singularities to correspond to. This situation is completely analogous to case o f the standard D = 10 type I superstring, where the closed string (010) sector which couples to the R a m o n d sector o f the cylinder has its left-right symmetric states projected by the addition o f the Klein bottle. (This sector contains the anti-symmetric tensor, which is left-right antisymmetric, necessary to fill out the graviton supergravity multiplet. ) The presence of these unphysical states in the open string indicates the inconsistency of the non-finite strings, even if a renormalization scheme could be constructed to remove the infinities. Indeed these types of unphysical states are the source o f the anomaly in the standard D = 10 superstring with gauge group other than SO (32) [ 20 ]. Since the massless vectors must lie in the adjoint representation of the gauge group, we can determine the gauge group by counting the number o f gauge bosons kept after truncation to unoriented strings under the action of the twist operator. The gauge bosons which lie in the Neveu-Schwarz U0 sector are o f the form (d ~_1/2 or b ~_i/2 ) Ia, b ) where a and b are C h a n - P a t o n vector labels 1.... , N attached to the two open string endpoints. For t/vo = + 1 and t/vo = - 1 the a, b indices are respectively truncated to the symmetric and anti-symmetric components, after summing the annulus and Mrbius contributions, so that the resultant gauge groups are respectively S p ( N ) and S O ( N ) [ 1 ]. Taking the values o f N a n d r/uo required for infinity cancellation we find the low energy 179
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gauge group Sp (4). The total gauge group of the models is the product of the Chan-Paton gauge group cross the gauge group arising from the closed string internal degrees of freedom. For the models we consider here the gauge bosons from the closed string are those necessary to complete the supergravity multiplet. However, when more general superconformal field theories are used for the internal degrees of freedom (with c = 9) we expect a richer structure, although the closed string gauge group will always be much smaller than the gauge groups obtained from heterotic string theories (which typically have rank 22). A step towards constructing more general models has been taken in ref. [ 22 ].
3. Examples of models We now present some one-loop finite four-dimensional models whose massless spectrums are that of N = 4, 2, 1 supergravity coupled to super-Yang-Mills, in order to demonstrate the variety of models possible in our construction as well as to illustrate some of their features. The simplest possible open string model [ 14 ] consists of only the Neveu-Schwarz boundary condition Uo, while the supersymmetric models are obtained by adding additional basis vectors to both the open and closed models which respect the open-closed string coupling. The string theories we consider as examples have closed string sectors determined by the basis vectors Wo=((½)'°l(½)'°),
Wl=(O(0½½)3l(½)l°)
W: = (0(0½ ½) (½0½)21 (½)lO) ,
,
W3 = ( O ( 2"2J ! n ! ~ 2,t~a 2! !2~, 1 ( 1 ) ,0) ,
(3.1)
and Wi, W~ and W~ which respectively are identical to Wb W2 and W3 except that left- and right-movers are interchanged. After following the procedure explained in the previous section and choosing koo= ~ (which projects the LRS states in the 0 sector) so that the models are square, the corresponding open superstrings are determined by the open string basis vectors Uo+ flU where the Ui are U,=(½(½00)3),
U2=(½(I00)(0½0)2),
U3=(½(0½0)2(½00)).
(3.2)
The similar structure of the set of open and closed boundary condition basis vectors is, of course, a reflection of the open-closed coupling. It not difficult to check that these closed string boundary conditions satisfy the required consistency conditions of ref. [ 3 ], while the open string boundary conditions satisfy the M6bius consistency conditions (2.8). The Wi and U~have been chosen so as to introduce space-time fermions into the model as well as to truncate the spectrum to N < 4 supergravity. We denote the models of open and closed strings under consideration as closed
open
M l: 141o, Wl, Wi
UI
M2: Wo, W,, Wi, W2, W~
U~, U2
M3:Wo, W1, W~, W2, W2, W3, W~ U~, U2,/-13.
(3.3)
The spectrum of each of these models, of course, depends on the choice of GSO projectors, which in turn depends on the choice of k a. The first model M 1, whose massless spectrum corresponds to N = 4 supergravity coupled to super-Yang-Mills (for appropriate ka), has already been presented in ref. [ 14 ], so here we only briefly review its features. In the closed string, four gravitinos will lie in both the W, and HI, sectors, which after truncation to an unoriented closed string leaves a total of four gravitinos indicating that this model has N = 4 supergravity. The open string massless spectrum consists of one vector and two scalars in the Uo sector and four fermions in the Uo + U~ sector, 180
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all of which lie in the adjoint representation of the Chan-Paton gauge group indicating that the massless spectrum of the open string is N = 4 super-Yang-Mills. The closed string sector of the next model M2 has four sectors which potentially contain gravitinos, WI, W i, W2 and W~. After the truncation to unoriented models the Wi and Wr sectors are identified so we need only consider W 1 and WE. The W2 projection upon W1 and vice versa cuts the number ofgravitinos in each sector to two. However, as we shall see, there are additional projections which may eliminate all gravitinos from a sector leading to a spectrum containing four gravitinos, two gravitinos, or no gravitinos. This results in three choices for the space-time supergravity N = 4, 2, 0. The interesting new case is when N = 2 which we now describe in some detail. After imposing the closed string consistency [ 3 ] and truncation [ 6 ] conditions (2.4), the "structure constant" matrix which determines the GSO projection (2.2) is given by
I
½
k,,
k,,
k22 k22"]
kll
kit
kli
kl2
k~T
k~
k12 k ~ 2 | ,
k,~
k12+½
[ko]=|k~
kk2~
k22
kl~| ko~{0,½},
(3.4)
k22d
where we have arranged the rows and columns according to the ordering 0, 1, 1, 2,2. We have also set koo= ½, in order to satisfy the det ks,, # 0 squareness condition discussed in the previous section. The spectrum is determined from the closed projection conditions (2.2) which for the W~ sector are
Wo'Nw, =~o = k l l + ½ ,
w~ .Nw, =0, =½, W~ .Nw, =aT =k,T +k,, +½, W2 "Nw, =02 =kl2 +k22 dr ½ , W,) .Nwl =¢2 =k12+k22 + ½ •
(3.5)
(The Wt sector is non-LRS so there are no additional projection conditions arising from the Klein bottle contributions.) Since the gravitino states are 15)® (b]}2 or d]~2)10), where 10) is the degenerate Ramond vacuum, they are unprojected iff ~o + 0T = ½, and ~0o+ qh = ½, so we require k,t = ½ ,
kll +k,2+k22=½
(3.6)
to ensure that gravitinos are found in the W~ sector. The structure of the WE sector is quite similar, with the condition for eliminating the gravitinos from this sector k22 - - 0 .
(3.7)
With these choices of structure constants, after truncation to an unoriented string, the supergravity will be N = 2. For the above closed string, the corresponding contribution of the annulus to the open string partition function, from eq. (2.6), is zann
I ~ T 2 1 [ K'Uo
=v,
[~"Uo+UI
- - I,'t U0
t~2~t
.A..~:'Uo
K'Uo+U2 ,g K ' U o + U 2 U0+U2 Uo+UI --v3a Uo+U 2 --(~2(~3guo+ul+u2)
Jf_[l~eO+Ul+V2.{_(~l..~ I'~t U0
15"Uo
,f~ ~ I ~ U o + U I ,f~ ~ ' U o + U I - l - , . ~ .~ L T ' U 0 + U I --oil U0+UI --°'2zU0+U2 ~L'It"2"~ Uo+UI+U2)
[I~'Uo+U2..J_,.g
- - X.t UO
UO
t ~ , vo--Fvo+vl --~uo-t-u21Jt uo+ui-Fu2]
~,uo+el+v2
O'2A U o + U I
,J~ ..~ l ~ V o + V l + V 2 ..~ ,~ i ~ , u o + e l + v 2 " l --LI2LISXUo+U2 --WlW3~tUo+Ul+U21]
,
(3.8)
where the argument for the F ' s is r and 8 ~ = e x p [ 2 z r i ( k , , + k ~ ) ] ,82=exp[2ui(k~2+k,2)] and 83= exp [ 2~ri (k22 + k2~) ]. For simplicity, we have suppressed the overbars on the boundary condition vectors. 181
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Using the modified twist operator (2.10), the corresponding MObius contribution to the open string partition function is zm°b=~Nexp(~Tri)
[ ~t l v o ( F +
vo - x F•_ uo) +other sectors],
(3.9)
where F u+O_ ,.z vo
,.: uo
--IUo--IUo+UI
FU- ° =F~
w uo
--XUo-FU2
~ ,.: vo
TIUo-I-UIJwU2
,:: vo ,.: vo .a_ r: vo ° -vo+v, -vo+~2 - - vo+v,+v~
_,_ ,.: uo TJt
O
w uo
--Jt
,.: uo .a_ ,.z uo
UI - - ~ I
U21~t
UIq-U2 ~
,.: ~o .a. ,~ vo ~ ,.z vo ,.;, Vo --o - - tq - - v~ - v,+v2
,
(3.10)
with ( r + ½) as the argument of the F ' s . The form of the other sectors is similar to the explicitly given NeveuSchwarz sector. For simplicity, we only will explicitly show the finiteness conditions for the Neveu-Schwarz sector of the cylinders. The other sectors are quite similar. Furthermore, we will only demonstrate the term-by-term infinity cancellation in the vacuum amplitude. For more general amplitudes this is technically more involved but will follow the discussion in ref. [ 6 ]. Under the Jacobi transformation the annulus amplitude goes into itself but with the argument of the F ' s replaced by r' as may be explicitly checked from the transformation rule (2.12). Similarly, the MObius amplitude becomes ZSC=~Nexp(~ni)[_½qvoF~+O
_ ]lquo_l'.,wv0 s e +other ctors]_
,
(3.11)
where the argument of the F ' s is (r' + ½) (after the standard rescaling [ 1 ] ~z' ~ r ' ). The cancellation of the infinities between the annulus and MObius contributions can be observed by direct Taylor expansion of the two partition functions using the definition of the F's. By direct computation it is not difficult to verify the Neveu-Schwarz sector finiteness condition (2.16 ) for this particular model, after including the relative 2 D/2 from the world-sheet bosonic degrees of freedom. The spectrum of this model is determined by the GSO projection conditions in the various sectors Us' N~b~o =P~(fl), i= 1, 2, and by using the supergravity conditions (3.6) and (3.7). The two projection conditions in the Uo sector reduce the number of adjoint massless scalars from four to two, while leaving the vector untouched. The two projection conditions in the Uo + UI sector reduce the number of massless fermions from four to two. These two sectors combine to form an irreducible representation of N = 2 supersymmetry, the Uo + U2 sector under the two projections also contains two fermions, which together with the four scalars in the U o + U~ + U 2 sector again form an irreducible representation of N = 2 supersymmetry. One of the curious features of this model is that the partition function always vanishes independent of the choice o f k o, as may be checked using the Riemann theta function identity [23 ]. This occurs even when the total model (i.e. open plus closed) is not space-time supersymmetric. The last model, M3, is quite similar to the previous M2 model. Again a variety of supergravity spectra are possible. There are four gravitino containing sectors after truncation to an unoriented model, WI, W z , W 3 and W~ + W2 + W3. Each sector has more projections acting on it than in model M2. The additional sectors W3 and W3 serve to reduce the number of potential gravitinos in the W~ sector from two to one. Hence there is a possible maximum N = 4 supergravity, with one gravitino coming from each of the four sectors. Analogous to model M2, it is possible to select the k o so that four, two, one, or none of the sectors contribute gravitinos. (The projections in the four sectors are not entirely independent so it is not possible to arrange for only three gravitinos to survive). The case of interest is when only one gravitino survives projection yielding N = 1 supergravity. Following a similar analysis as for model M2, it is not difficult to check for this case that open string massless states fall into representations of N = 1 supersymmetry. However, this model does not contain Weyl fermions, as expected from the presence of adjoint representation scalars, analogous to the situation with heterotic and type II strings [24,25 ]. Possibly, models with Weyl fermions can be constructed by using higher level Kac-Moody algebras [251.
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4. Conclusions A l t h o u g h we have succeeded i n c o n s t r u c t i n g a n u m b e r o f e x a m p l e s o f o n e - l o o p finite f o u r - d i m e n s i o n a l o p e n superstrings, m u c h w o r k r e m a i n s to be d o n e before o p e n strings are as well d e v e l o p e d as the heterotic string. Besides the straightforward e x t e n s i o n o f o u r work to the other s t a n d a r d f o r m a l i s m s for the i n t e r n a l degrees o f f r e e d o m o f string theories such as lattices a n d orbifolds [ 2,26 ], there are a n u m b e r o f a v e n u e s which s h o u l d be investigated. F o r example, there has b e e n a suggestion to have projectors act n o n - t r i v i a l l y o n the C h a n - P a t o n factors which p o t e n t i a l l y w o u l d lead to a m u c h richer structure for the gauge group [ 12,8 ]. T h e r e is also the q u e s t i o n o f t r y i n g to c o n s t r u c t chiral models, p e r h a p s t h r o u g h the use o f higher level K a c - M o o d y algebras [ 25 ]. A n o t h e r i m p o r t a n t topic is the i n v e s t i g a t i o n o f higher-loop a m p l i t u d e s [ 11 ]. Finally, w i t h i n a n y p a r t i c u l a r f o r m a l i s m , it m a y be possible to classify the set o f all o n e - l o o p finite o p e n string m o d e l s since the n u m b e r o f i n t e r n a l degrees o f f r e e d o m is significantly smaller t h a n for the f o u r - d i m e n s i o n a l heterotic string. In c o n c l u s i o n , we have s h o w n that it is possible to c o n s t r u c t a variety o f f o u r - d i m e n s i o n a l o n e - l o o p finite o p e n superstring theories. In p a r t i c u l a r we have c o n s t r u c t e d m o d e l s which are s p a c e - t i m e s u p e r s y m m e t r i c a n d where the s u p e r g r a v i t y o f the massless states is N = 4 , 2 or 1.
Acknowledgement We w o u l d like to t h a n k L. Clavelli, D . R . T . Jones, D. Lewellen a n d F. Q u e v e d o for v a l u a b l e c o n v e r s a t i o n s . T h i s work was s u p p o r t e d in part by S E R C a n d in part by the U S D e p a r t m e n t o f Energy.
References [ 1] J.H. Schwarz, Phys. Rep. 89 (1982) 223. [2] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 ( 1985 ) 678; Nucl. Phys. B 274 (1986) 285; K.S. Narain, Phys. Lett. B 169 (1986) 41; K.S. Narain, M.H. Sarmadi and C. Vafa, Nucl. Phys. B 288 ( 1987 ) 551; W. Lerche, D. Liist and A.N. Schellekens, Nucl. Phys. B 287 (1987) 477. [3] H. Kawai, D.C. Lewellen and S.-H.H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B 288 (1987) 1. [4] I. Antoniadis, C.P. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87. [5] D. Gepner, Nucl. Phys. B 296 (1988) 757. [6] Z. Bern and D.C. Dunbar, Phys. Lett. B 203 (1988) 109; D.C. Dunbar, Nucl. Phys. B 319 (1989) 72; Z. Bern and D.C. Dunbar, Nucl. Phys. B 319 ( 1989 ) 104. [ 7 ] L. Clavelli, P.H. Cox, B. Harms and A. Stern, University of Alabama preprint UAHEP882; L. Clavelli, P.H. Cox and B. Harms, Phys. Rev. Lett. 61 (1988) 787; L. Clavelli, P.H. Cox, P. Elmfors and B. Harms, University of Alabama preprint UAHEP891. [8] N. Ishibashi and T. Onogi, Nucl. Phys. B 318 (1989) 239; N. Ishibashi, Mod. Phys. Lett. A 4 ( 1989 ) 251. [9] J. Cardy, Nucl. Phys. B 275 (1986) 200. [ 10] G. Pradisi and A. Sagnotti, Phys. Lett. B 216 (1989) 59. [ 11 ] M. Bianchi and A. Sagnotti, Phys. Len. B 211 (1988) 407; preprint ROM2F-89/15. [ 12] J.A. Harvey and J.A. Minahan, Phys. Lett. B 188 (1987) 44. [ 13] D.J. Gross, A. Neveu, J. Scherk and J.H. Schwarz, Phys. Rev. D 2 (1970) 697; E. Cremmer and J. Scherk, Nucl. Phys. B 50 (1972) 222; M. Ademollo et al., Nucl. Phys. B 94 ( 1975 ) 221. [ 14] Z. Bern and D.C. Dunbar, Phys. Rev. Lett. 64 (1990) 827. [ 15 ] J.E. Paton and Chan Hang-Mo, Nucl. Phys. B 10 ( 1969 ) 519. [ 16 ] M.B. Green and J.H. Schwarz, Phys. Rev. Lett. B 151 ( 1985 ) 21. [ 17 ] K. Bardakci and M.B. Halpern, Phys. Rev. D 3 ( 1971 ) 2493. 183
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[ 18 ] L. Clavelli and J.A. Shapiro, Nucl. Phys. B 57 (1973) 490. [ 19] C.G.Callan, C. Lovelace, C.R. Nappi and S.A. Yost, Nucl. Phys. B 293 (1987) 83. [20] J. Polchinski and Y. Cai, Nucl. Phys. B 296 (1988) 91. [21 ] J. Govaerts, Phys. Lett. B 220 (1989) 77; Intern. J. Mod. Phys. A 4 (1989) 4353; N. Marcus and A. Sagnotti, Phys. Lett. B 119 ( 1982 ) 97. [22] Z. Bern and D. Dunbar, preprint LA-UR-89-3893, LTH-249-89, Intern J. Mod. Phys. A, to appear. [ 23 ] J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973 ). [24] L. Dixon, V. Kaplunovsky and C. Vafa, Nucl. Phys. B 294 (1987) 43. [25] D.C. Lewellen, SLAC preprint SLAC-PUB-5023. [26] D. Bailin, D.C. Dunbar and A. Love, Nucl. Phys. B 330 (1990) 124.
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FOUR-DIMENSIONAL TYPE I SUPERSTRINGS WITH N< 4 SUPERGRAVITY Z. BERN Theoretical Division, MS-P285, Los Alarnos National Laboratory, Los Alamos, N M 87545, USA
and D.C. D U N B A R DAMTP, Liverpool University, Liverpool L69 3BX, UK
Received 5 January 1990; revised manuscript received 27 February 1990
We present rules for constructing four-dimensional type I open superstrings and examine a number of models which are oneloop finite and where the space-time supergravity is N= 4, 2, or 1. The N< 4 models are obtained by breaking the supersymmetry with suitable GSO projectors.
1. Introduction
Superstring theories have attracted great interest in recent years as potential theories of q u a n t u m gravity [ 1 ]. Originally, superstring theories were constructed in ten spatial dimensions, but more recently a variety of schemes have appeared for constructing four-dimensional theories o f closed superstrings [ 2 - 5 ] . The construction o f lower-dimensional theories of open superstrings is more complicated [ 6-12 ] largely due to the lack of a simple symmetry principle, analogous to modular invariance which guarantees much o f the consistency of closed strings. The starting point in our construction o f consistent type I models is type II left-right symmetric (LRS) oriented four-dimensional closed string models, which can be truncated to unoriented models [ 6 ]. The coupling of the open string to the closed string is determined by the well-known property that closed string poles are found in open string one-loop amplitudes [ 13 ]. The basic rules for coupling open strings to closed strings have been presented in refs. [ 6 - 1 0 ]. The construction o f consistent models relies on the use o f a twist operator which acts sensibly on the open string states. For dimensions D - - 2, 6, l 0, the standard twist operator [ 1 ] is sufficient to enable open string constructions [6 ] in the fermionic formulation; however, in D = 4 this twist operator does not lead to a sensible theory. The difference between D = 4 and D = 2, 6, 10 can be traced back to the closed string. For the standard twist operator to be sensible the open string GSO projectors are required to ensure that all states in a given sector are integer spaced in mass. Through the open-closed string coupling, this in turn implies a set of closed string boundary conditions which are consistent only in D = 2, 6, 10. Additionally, with the standard twist operator, the closed string states found in the M r b i u s amplitude will not correspond to the spectrum of the closed string [ 6 ]. In a previous paper [ 14 ], by the use o f a modified twist operator we constructed the first example of a space-time supersymmetric one-loop finite four-dimensional open superstring. In this paper, we present the general formalism and also some explicit examples o f one-loop finite four-dimensional type I superstrings whose massless spectrums are N = 4, 2, l, supergravity coupled to super-YangMills. One-loop finiteness requires a specific choice o f C h a n - P a t o n gauge group [ 15,16 ] and in D ( e v e n ) 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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dimensions the type of models we construct are one-loop finite for Chan-Paton gauge group Sp(2 0/2) or SO ( 2 n/2 ) (depending upon D).
2. General considerations
To maintain superconformal symmetry, superstring theories in dimensions less than ten require internal degrees of freedom which can be represented as world-sheet bosons [2 ], fermions [17,3,4] or even directly as superconformal field theories [ 5 ]. We work with models where the internal degrees of freedom are represented by d = 1 4 - D complex Bardakci-Halpern fermions, but our work can be straightforwardly extended to the other constructions. In ref. [ 6 ] a general formalism for type I superstrings in the fermionic formulation for the internal degrees of freedom was presented. We briefly recall the relevant results in the notation of ref. [ 3 ]. A general closed string model consists of generic bosonic degrees of freedom as well as fermionic degrees of freedom. The boundary conditions of the world-sheet fermions on the torus are specified by vectors W= ( w kl I~k) = ( W I,..., W dl l~l, •.., if, d), where the first and second d components respectively specify the boundary conditions of the left- and right-moving world-sheet fermions [ 3,4 ]. For w k = ½the kth fermion has Neveu-Schwarz boundary conditions while for Wk= 0 the boundary conditions are Ramond. We will restrict ourselves to the case where only NeveuSchwarz or Ramond boundary conditions are allowed, although general rational values of W k are possible. The vectors representing the set of allowed boundary conditions in the model are generated by basis vectors Wi, (which always includes Wo = ((½) d l (½)d) ) SO that this set is given by ~iai Wi = ot W where the a~ are integers. The overbar notation indicates that the values of Whave been shifted by integers so that w k = O, ½. The vectors W~are, of course, not arbitrary but satisfy conditions that ensure the modular invariance of the model [3,4], 2Wi'Wj=0 ,
Wi. Wi=O (mod 1 ) ,
(2.1)
as well as a world-sheet supersymmetry triplet condition. (The dot product between closed string basis vectors is lorentzian with a minus sign for the components describing right movers.) The spectrum of the model is truncated by a generalised GSO projection determined by the Wi and by a set of"structure constants" k~j (which are all 0 or ½and again subject to constraints). The states which are kept under the GSO projection form the spectrum of the model and satisfy W ~ . N ~ = ~ ( o t ) - = ~ k~je~j+ko,+si-W~.~W J
(mod 1 ) ,
(2.2)
where N~-ff is the vector with components given by the fermion number operators N,~w~w~.The statistical factor si is, for a type II string, W] + / ~ (mod 1 ). For ~ - - - Yqc~s~= 0 the states of the sector o~W are space-time bosons while for ~ = ½they are fermions. As shown in ref. [ 6 ], the first step in the construction of open superstrings is to focus on the LRS type II closed strings which are truncatable to unoriented models. Such strings can be generated by a set of basis vectors of the form
Ws=(U~IU~),
Wa=(W~II~),
Wa=(ff',*lWa~),
(2.3)
provided the additional truncation constraints on the structure constants
kas"~-kas=kabdrkt~s=ka~-~kab=O (mod 1)
(2.4)
are satisfied, ensuring that the closed string Fock space is symmetric between left- and right-movers. This allows for the truncation to unoriented closed superstrings which can then be coupled to unoriented open superstrings to form a type I superstring model. The open string models are specified by boundary condition vectors with half the number of entries as closed 176
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string vectors because the left- and right-moving components of the fermions are no longer independent, but are reflected into each other at the boundaries of the world sheet. The set of boundary conditions in the open string determining the GSO projectors corresponds to the subset of the LRS sectors of the closed string ( UI U) whose LRS states are not projected. The U's come in two varieties: from (UslUs) and from W a + W a = ( w k + ff'ka I Wka + ff'~ ) =- ( Ua I Ua). The set of boundary conditions determining the open string sectors are Va = ( Wka -- ffgka) = (Vka) which corresponds to the left-right asymmetric terms in the closed string GSO projectors. In general, these two sets of open string boundary conditions need not be the same. However, only for the case where the two sets of open string boundary conditions are identical, so that the model is "square", will there be a one-to-one correspondence between the sectors of the Jacobi transformed planar and Mrbius amplitudes. This allows for cancellation of the string one-loop potential infinities in general amplitudes. As shown in ref. [ 6 ], for det kss, ~ 0 the Us do not contribute to the open string GSO projections so the open string is square and the open string states kept under the GSO projection are those satisfying U a "N # ~ - ~ o = p a ( fl ) ~ ~.~ ( k ab -'l- k ab ) flb -I- .~a - U a " ( fl U -I- Uo - U o ) , b
(2.5)
where the statistical factor ga is the first entry in Ua. The annulus contribution to the open string partition function is then given by
,-.#U+Uo ._x
zann = N2
- - Y~ exp(2~ifls) exp[2ai~p(fl) ] r~--OV~t~ ) , M {ma}
(2.6)
where FV(r)=
k=l fi
( exp{2~iz[(Vk)2-Vk+~]/2}
× f i {1 - e x p [ 2 z f i z ( n + V k - 1 ) ] exp( --2niUk)}{1 - e x p [ 2 r d z ( n - V k) ] exp(2rdUk)}] / n=l
(2.7)
is the partition function for space boundary conditions Vwith exp [2zfi( U - Uo)"Nu] inserted into the trace over oscillators and z is the imaginary Teichmiiller parameter for the annulus. We have included a factor o f N 2 from the Chan-Paton trace over both boundaries and the normalisation factor M = 2 × (number of sectors), so that every unprojected state contributes with a coefficient of unity. (The extra 2 arises because we are considering unoriented theories where the annulus and M6bius amplitudes combine to give the spectrum.) This annulus vacuum amplitude, as well as more general amplitudes, has the property that when transformed into a closed string exchange [ 13,6,8 ] the terms appearing correspond to physical states of the closed string as guaranteed by our construction. To obtain a M6bius amplitude which contains the same set of closed string poles as the annulus amplitude it is necessary to impose the further conditions on the Ua [ 6 ]
Ua'Ua=O,
2Ua'Ub=O
(rood 1 ) .
(2.8)
When we start with a set of closed string basis vectors satisfying the modular conditions then the second of these conditions is always satisfied although the first is non-trivial. Conversely, if we have a set of Ua satisfying eq. (2.8) then we can always construct a suitable closed string model by taking the closed string basis vectors to be Wa = ( Ua + Uo I Uo), Wa = ( Uo I Ua + Uo ) plus I47o= ( Uo I Uo) if not already present in the set of closed string boundary condition vectors. It is not difficult to check that with conditions (2.8) and the world-sheet supersymmetry triplet condition [ 3 ] such a closed model is consistent. It is important to note that Uo need not be amongst the Ua and indeed in D = 4 Uo" Uo#0 and therefore does not satisfy the M6bius condition (2.8). (Equivalently, W= (Uol0) does not satisfy the consistency conditions 177
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for the closed string. ) This means that the purely R a m o n d boundary conditions are not present in four-dimensional open strings (and, in fact, may only appear in D = 10, 6, 2). The lack o f a Uo GSO projector in four-dimensional open string causes technical difficulties for the standard twist operator = (phase) × exp ( i n H ) ,
(2.9)
where H is the open string hamiltonian. Since only the Uo GSO projector [ 1 - e x p ( 2 n i U o . h r) ] can completely remove the half integer mass levels from the Neveu-Schwarz sector of the superstring (which is always present), without this projector ½( 1 + ~ ) will not act as a sensible projector since it is not possible to choose a phase so that ~ 2 = + 1. Thus, it appears that we have hit an obstacle to the construction of four-dimensional open strings. The solution of this problem is actually quite simple. The idea is to modify the twist operator to include GSO projectors so as to split the Fock space into integer and half-integer mass states before applying the twist. The physical restrictions on the twist operator are that ~ 2 = 1, that it actually flips the string over so that for the vertex operators g2 V ( a = 0)£2*= V ( a = n), and most importantly that it results in a consistent coupling to the closed string. It should also reverse the order of vertices in a tree amplitude,/2 VI... VNI 0 ) ~ VN...VI I0 ) [ 18 ]. In the U sector, the modified twist operator satisfying these requirements takes the form I')v = exp( - niECe) ½r/u{ [ 1 + exp(2niU.Nu) ] exp (ni/(tv) - i [ 1 - e x p ( 2 n i U ' N v ) ] e x p ( n i / t u ) }.
(2.10)
The phase exp( - n i E ~ c) is included to cancel the contribution t o / ~ from the zero-point energy E ~ ¢ = - ~ 4 d + ~dR where dR is the number of complex R a m o n d fermions in the Usector. At first sight, the factor o f i associated with the second term of the twist operator (2.10) may seem peculiar; however, its purpose is to cancel a factor of i which is hidden in exp(ni/~u) when acting on states with odd fermion numbers. It is not difficult to check that this choice of ~ v satisfies the necessary consistency condition above (see for example ref. [ 18 ] ). The ~/v control the C h a n - P a t o n gauge group representations at the various mass levels. For the Neveu-Schwarz Uo sector the twist operator is equivalent to the one given 16 years ago by Clavelli and Shapiro [ 18 ]. Using this form of the twist operator the contribution to the MSbius amplitude is z m o b - N E exp(2nifls) M {a,#} [ 117,,OU+Uo .a_t2flU+Uo
exp[2niap(fl)]exp(-niE~)½rlpv+vo t~_y,oU+Uo f;,,oU+Uo
× t ~-- ~u+ vo - - - ~v--O-W~) - i ~, ~v+vo - ,
,v--o-7~) ] ,
(2.11)
where the argument of the F ' s is (;r+ ½) and the factor N arises from the Chan-Paton trace over a single boundary. Both the annulus and MSbius amplitudes contain infinities which are most easily identified after being transformed into closed string exchanges via the Jacobi transformation z' = - 1/z. The transformation rules for the partition functions are
FV(z) = e x p [2nit U - Uo)" ( V - Uo) ] F V ( z ' ) ,
(2.12)
and
FW(z+½)=exp[4xi(U-Uo).(V-Uo)] exp[2~ri(V.V-~d)] exp[ni(U.U-{d)] -~.u - v - u t, ~~- , + ½ ) •
(2.13)
Thus, under the Jacobi transformation the annulus amplitude becomes v+vo. ,,) , - - ~ exp(2nifls) exp[2z~iotp(/~)] ~~av--ow~tz M {~,a}
zBB= N 2
(2.14)
where the superscript BB is to remind us that this amplitude can be represented as a closed string exchange between two boundary states [ 19,20 ]. By relabelling fl with ot and vice versa this takes on the same form as Z ~ " given in eq. (2.6) except the argument is now z' [6]. (The invariance of the form of the model under Jacobi 178
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transformation, o f course, only holds for the square models under consideration. ) Using eq. (2.13) the MSbius amplitude (2.11 ) after the Jacobi transformation is (following ref. [ 6 ] ) ZBc=
N vac ~ exp (2zrifls) exp [ 2niap (fl) ] exp ( - h i- e ~--O;~o ). ½pvr/TbW~o lvi {a#}
× [-iexp(-niflU.flU) tr,v+u o~'aU-r~-v-+~v)~au+v° +U° . .-cxp~'-inflU.flU) ... t~aU+U°~-~-v~O~.-~)-raU+U° ~lj ,
(2.15)
where the BC superscript is to remind us that this amplitude can be represented as a closed string exchange between a boundary and crosscap state [ 19,20,6] and the argument o f the F ' s is ( ~i z , + ~1 ). To obtain this result we have used various simplifications arising from eq. (2.8) which ensure that the sum over {aa} gives the same closed string states as in the transformed annulus after the standard rescaling ¼z ' ~ r' [ 16 ]. The potential infinities in the v a c u u m amplitudes can be obtained from eqs. (2.14) and (2.15 ) by including the contribution from the bosonic partition function, which yields a relative 2 °/2 between the Jacobi transformed M6bius and annulus contributions (see refs. [ 6,1,21 ] ), and an integration over the Teichmiiller parameter z' = In q/i~. The contribution from a particular state is proportional to f~dq qm,2_ 1, where ms is the mass o f the state (in units o f ~ ) , so that potential infinities occur for states which are tachyonic or are massless in the q--, 0 limit. A proper discussion o f infinity cancellation requires a study of general amplitudes. However, if infinities in the v a c u u m amplitudes cancel term-by-term between the M6bius strip and annulus, infinity cancellation of general amplitudes follows [ 6 ], so here we focus only on the vacuum amplitudes. (Although the vacuum amplitude will vanish because of supersymmetry, the requirement of term-by-term cancellation means that it is sensible to work with these amplitudes. ) Models which are space-time supersymmetric are non-tachyonic (such as the models o f the following section) and so that only potential infinities for these models correspond to massless states. In the Neveu-Schwarz Uo sector o f the cylinders, extracting the leading f~ dq/q from eqs. (2.14) and (2.15 ), including the relative 2 °/2 and summing, yields l
(N2+2°/2rluoN) f d-~qq
(2.16)
0
as the potential infinity. To cancel this infinity we must choose ~/vo= - 1 and N=2 °/2. Infinities may also be found in other sectors, but again these cancel by choosing the ~/'s suitably. The source o f the potential infinities in these sectors corresponds to the LRS states o f the corresponding closed string sectors, through the boundary and crosscap states. However, in some o f the closed string sectors the Klein bottle removes the massless LRS states so there are no physical states for these open string singularities to correspond to. This situation is completely analogous to case o f the standard D = 10 type I superstring, where the closed string (010) sector which couples to the R a m o n d sector o f the cylinder has its left-right symmetric states projected by the addition o f the Klein bottle. (This sector contains the anti-symmetric tensor, which is left-right antisymmetric, necessary to fill out the graviton supergravity multiplet. ) The presence of these unphysical states in the open string indicates the inconsistency of the non-finite strings, even if a renormalization scheme could be constructed to remove the infinities. Indeed these types of unphysical states are the source o f the anomaly in the standard D = 10 superstring with gauge group other than SO (32) [ 20 ]. Since the massless vectors must lie in the adjoint representation of the gauge group, we can determine the gauge group by counting the number o f gauge bosons kept after truncation to unoriented strings under the action of the twist operator. The gauge bosons which lie in the Neveu-Schwarz U0 sector are o f the form (d ~_1/2 or b ~_i/2 ) Ia, b ) where a and b are C h a n - P a t o n vector labels 1.... , N attached to the two open string endpoints. For t/vo = + 1 and t/vo = - 1 the a, b indices are respectively truncated to the symmetric and anti-symmetric components, after summing the annulus and Mrbius contributions, so that the resultant gauge groups are respectively S p ( N ) and S O ( N ) [ 1 ]. Taking the values o f N a n d r/uo required for infinity cancellation we find the low energy 179
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gauge group Sp (4). The total gauge group of the models is the product of the Chan-Paton gauge group cross the gauge group arising from the closed string internal degrees of freedom. For the models we consider here the gauge bosons from the closed string are those necessary to complete the supergravity multiplet. However, when more general superconformal field theories are used for the internal degrees of freedom (with c = 9) we expect a richer structure, although the closed string gauge group will always be much smaller than the gauge groups obtained from heterotic string theories (which typically have rank 22). A step towards constructing more general models has been taken in ref. [ 22 ].
3. Examples of models We now present some one-loop finite four-dimensional models whose massless spectrums are that of N = 4, 2, 1 supergravity coupled to super-Yang-Mills, in order to demonstrate the variety of models possible in our construction as well as to illustrate some of their features. The simplest possible open string model [ 14 ] consists of only the Neveu-Schwarz boundary condition Uo, while the supersymmetric models are obtained by adding additional basis vectors to both the open and closed models which respect the open-closed string coupling. The string theories we consider as examples have closed string sectors determined by the basis vectors Wo=((½)'°l(½)'°),
Wl=(O(0½½)3l(½)l°)
W: = (0(0½ ½) (½0½)21 (½)lO) ,
,
W3 = ( O ( 2"2J ! n ! ~ 2,t~a 2! !2~, 1 ( 1 ) ,0) ,
(3.1)
and Wi, W~ and W~ which respectively are identical to Wb W2 and W3 except that left- and right-movers are interchanged. After following the procedure explained in the previous section and choosing koo= ~ (which projects the LRS states in the 0 sector) so that the models are square, the corresponding open superstrings are determined by the open string basis vectors Uo+ flU where the Ui are U,=(½(½00)3),
U2=(½(I00)(0½0)2),
U3=(½(0½0)2(½00)).
(3.2)
The similar structure of the set of open and closed boundary condition basis vectors is, of course, a reflection of the open-closed coupling. It not difficult to check that these closed string boundary conditions satisfy the required consistency conditions of ref. [ 3 ], while the open string boundary conditions satisfy the M6bius consistency conditions (2.8). The Wi and U~have been chosen so as to introduce space-time fermions into the model as well as to truncate the spectrum to N < 4 supergravity. We denote the models of open and closed strings under consideration as closed
open
M l: 141o, Wl, Wi
UI
M2: Wo, W,, Wi, W2, W~
U~, U2
M3:Wo, W1, W~, W2, W2, W3, W~ U~, U2,/-13.
(3.3)
The spectrum of each of these models, of course, depends on the choice of GSO projectors, which in turn depends on the choice of k a. The first model M 1, whose massless spectrum corresponds to N = 4 supergravity coupled to super-Yang-Mills (for appropriate ka), has already been presented in ref. [ 14 ], so here we only briefly review its features. In the closed string, four gravitinos will lie in both the W, and HI, sectors, which after truncation to an unoriented closed string leaves a total of four gravitinos indicating that this model has N = 4 supergravity. The open string massless spectrum consists of one vector and two scalars in the Uo sector and four fermions in the Uo + U~ sector, 180
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all of which lie in the adjoint representation of the Chan-Paton gauge group indicating that the massless spectrum of the open string is N = 4 super-Yang-Mills. The closed string sector of the next model M2 has four sectors which potentially contain gravitinos, WI, W i, W2 and W~. After the truncation to unoriented models the Wi and Wr sectors are identified so we need only consider W 1 and WE. The W2 projection upon W1 and vice versa cuts the number ofgravitinos in each sector to two. However, as we shall see, there are additional projections which may eliminate all gravitinos from a sector leading to a spectrum containing four gravitinos, two gravitinos, or no gravitinos. This results in three choices for the space-time supergravity N = 4, 2, 0. The interesting new case is when N = 2 which we now describe in some detail. After imposing the closed string consistency [ 3 ] and truncation [ 6 ] conditions (2.4), the "structure constant" matrix which determines the GSO projection (2.2) is given by
I
½
k,,
k,,
k22 k22"]
kll
kit
kli
kl2
k~T
k~
k12 k ~ 2 | ,
k,~
k12+½
[ko]=|k~
kk2~
k22
kl~| ko~{0,½},
(3.4)
k22d
where we have arranged the rows and columns according to the ordering 0, 1, 1, 2,2. We have also set koo= ½, in order to satisfy the det ks,, # 0 squareness condition discussed in the previous section. The spectrum is determined from the closed projection conditions (2.2) which for the W~ sector are
Wo'Nw, =~o = k l l + ½ ,
w~ .Nw, =0, =½, W~ .Nw, =aT =k,T +k,, +½, W2 "Nw, =02 =kl2 +k22 dr ½ , W,) .Nwl =¢2 =k12+k22 + ½ •
(3.5)
(The Wt sector is non-LRS so there are no additional projection conditions arising from the Klein bottle contributions.) Since the gravitino states are 15)® (b]}2 or d]~2)10), where 10) is the degenerate Ramond vacuum, they are unprojected iff ~o + 0T = ½, and ~0o+ qh = ½, so we require k,t = ½ ,
kll +k,2+k22=½
(3.6)
to ensure that gravitinos are found in the W~ sector. The structure of the WE sector is quite similar, with the condition for eliminating the gravitinos from this sector k22 - - 0 .
(3.7)
With these choices of structure constants, after truncation to an unoriented string, the supergravity will be N = 2. For the above closed string, the corresponding contribution of the annulus to the open string partition function, from eq. (2.6), is zann
I ~ T 2 1 [ K'Uo
=v,
[~"Uo+UI
- - I,'t U0
t~2~t
.A..~:'Uo
K'Uo+U2 ,g K ' U o + U 2 U0+U2 Uo+UI --v3a Uo+U 2 --(~2(~3guo+ul+u2)
Jf_[l~eO+Ul+V2.{_(~l..~ I'~t U0
15"Uo
,f~ ~ I ~ U o + U I ,f~ ~ ' U o + U I - l - , . ~ .~ L T ' U 0 + U I --oil U0+UI --°'2zU0+U2 ~L'It"2"~ Uo+UI+U2)
[I~'Uo+U2..J_,.g
- - X.t UO
UO
t ~ , vo--Fvo+vl --~uo-t-u21Jt uo+ui-Fu2]
~,uo+el+v2
O'2A U o + U I
,J~ ..~ l ~ V o + V l + V 2 ..~ ,~ i ~ , u o + e l + v 2 " l --LI2LISXUo+U2 --WlW3~tUo+Ul+U21]
,
(3.8)
where the argument for the F ' s is r and 8 ~ = e x p [ 2 z r i ( k , , + k ~ ) ] ,82=exp[2ui(k~2+k,2)] and 83= exp [ 2~ri (k22 + k2~) ]. For simplicity, we have suppressed the overbars on the boundary condition vectors. 181
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Using the modified twist operator (2.10), the corresponding MObius contribution to the open string partition function is zm°b=~Nexp(~Tri)
[ ~t l v o ( F +
vo - x F•_ uo) +other sectors],
(3.9)
where F u+O_ ,.z vo
,.: uo
--IUo--IUo+UI
FU- ° =F~
w uo
--XUo-FU2
~ ,.: vo
TIUo-I-UIJwU2
,:: vo ,.: vo .a_ r: vo ° -vo+v, -vo+~2 - - vo+v,+v~
_,_ ,.: uo TJt
O
w uo
--Jt
,.: uo .a_ ,.z uo
UI - - ~ I
U21~t
UIq-U2 ~
,.: ~o .a. ,~ vo ~ ,.z vo ,.;, Vo --o - - tq - - v~ - v,+v2
,
(3.10)
with ( r + ½) as the argument of the F ' s . The form of the other sectors is similar to the explicitly given NeveuSchwarz sector. For simplicity, we only will explicitly show the finiteness conditions for the Neveu-Schwarz sector of the cylinders. The other sectors are quite similar. Furthermore, we will only demonstrate the term-by-term infinity cancellation in the vacuum amplitude. For more general amplitudes this is technically more involved but will follow the discussion in ref. [ 6 ]. Under the Jacobi transformation the annulus amplitude goes into itself but with the argument of the F ' s replaced by r' as may be explicitly checked from the transformation rule (2.12). Similarly, the MObius amplitude becomes ZSC=~Nexp(~ni)[_½qvoF~+O
_ ]lquo_l'.,wv0 s e +other ctors]_
,
(3.11)
where the argument of the F ' s is (r' + ½) (after the standard rescaling [ 1 ] ~z' ~ r ' ). The cancellation of the infinities between the annulus and MObius contributions can be observed by direct Taylor expansion of the two partition functions using the definition of the F's. By direct computation it is not difficult to verify the Neveu-Schwarz sector finiteness condition (2.16 ) for this particular model, after including the relative 2 D/2 from the world-sheet bosonic degrees of freedom. The spectrum of this model is determined by the GSO projection conditions in the various sectors Us' N~b~o =P~(fl), i= 1, 2, and by using the supergravity conditions (3.6) and (3.7). The two projection conditions in the Uo sector reduce the number of adjoint massless scalars from four to two, while leaving the vector untouched. The two projection conditions in the Uo + UI sector reduce the number of massless fermions from four to two. These two sectors combine to form an irreducible representation of N = 2 supersymmetry, the Uo + U2 sector under the two projections also contains two fermions, which together with the four scalars in the U o + U~ + U 2 sector again form an irreducible representation of N = 2 supersymmetry. One of the curious features of this model is that the partition function always vanishes independent of the choice o f k o, as may be checked using the Riemann theta function identity [23 ]. This occurs even when the total model (i.e. open plus closed) is not space-time supersymmetric. The last model, M3, is quite similar to the previous M2 model. Again a variety of supergravity spectra are possible. There are four gravitino containing sectors after truncation to an unoriented model, WI, W z , W 3 and W~ + W2 + W3. Each sector has more projections acting on it than in model M2. The additional sectors W3 and W3 serve to reduce the number of potential gravitinos in the W~ sector from two to one. Hence there is a possible maximum N = 4 supergravity, with one gravitino coming from each of the four sectors. Analogous to model M2, it is possible to select the k o so that four, two, one, or none of the sectors contribute gravitinos. (The projections in the four sectors are not entirely independent so it is not possible to arrange for only three gravitinos to survive). The case of interest is when only one gravitino survives projection yielding N = 1 supergravity. Following a similar analysis as for model M2, it is not difficult to check for this case that open string massless states fall into representations of N = 1 supersymmetry. However, this model does not contain Weyl fermions, as expected from the presence of adjoint representation scalars, analogous to the situation with heterotic and type II strings [24,25 ]. Possibly, models with Weyl fermions can be constructed by using higher level Kac-Moody algebras [251.
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4. Conclusions A l t h o u g h we have succeeded i n c o n s t r u c t i n g a n u m b e r o f e x a m p l e s o f o n e - l o o p finite f o u r - d i m e n s i o n a l o p e n superstrings, m u c h w o r k r e m a i n s to be d o n e before o p e n strings are as well d e v e l o p e d as the heterotic string. Besides the straightforward e x t e n s i o n o f o u r work to the other s t a n d a r d f o r m a l i s m s for the i n t e r n a l degrees o f f r e e d o m o f string theories such as lattices a n d orbifolds [ 2,26 ], there are a n u m b e r o f a v e n u e s which s h o u l d be investigated. F o r example, there has b e e n a suggestion to have projectors act n o n - t r i v i a l l y o n the C h a n - P a t o n factors which p o t e n t i a l l y w o u l d lead to a m u c h richer structure for the gauge group [ 12,8 ]. T h e r e is also the q u e s t i o n o f t r y i n g to c o n s t r u c t chiral models, p e r h a p s t h r o u g h the use o f higher level K a c - M o o d y algebras [ 25 ]. A n o t h e r i m p o r t a n t topic is the i n v e s t i g a t i o n o f higher-loop a m p l i t u d e s [ 11 ]. Finally, w i t h i n a n y p a r t i c u l a r f o r m a l i s m , it m a y be possible to classify the set o f all o n e - l o o p finite o p e n string m o d e l s since the n u m b e r o f i n t e r n a l degrees o f f r e e d o m is significantly smaller t h a n for the f o u r - d i m e n s i o n a l heterotic string. In c o n c l u s i o n , we have s h o w n that it is possible to c o n s t r u c t a variety o f f o u r - d i m e n s i o n a l o n e - l o o p finite o p e n superstring theories. In p a r t i c u l a r we have c o n s t r u c t e d m o d e l s which are s p a c e - t i m e s u p e r s y m m e t r i c a n d where the s u p e r g r a v i t y o f the massless states is N = 4 , 2 or 1.
Acknowledgement We w o u l d like to t h a n k L. Clavelli, D . R . T . Jones, D. Lewellen a n d F. Q u e v e d o for v a l u a b l e c o n v e r s a t i o n s . T h i s work was s u p p o r t e d in part by S E R C a n d in part by the U S D e p a r t m e n t o f Energy.
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