R2 corrections and non-perturbative dualities of N = 4 string ground states

Nuclear Physics B 510 (1998) 423-476

R2 corrections and non-perturbative dualities of N = 4 string ground states

Received 15 August 1997; accepted 24 September 1997

Abstract

We corn~~tc and analyse a variety of four~e~v~tive g~vi~tio~a~ teas in the elective action of six- and four-dimensional type II string ground states with N = 4 s~~~yrnme~. In six dimensions, we compute the relevant perturbative corrections for the type II string compactified on K3. In four dimensions we do analogous computations for severai models with (4,O) and (2,2) supersymmetry. Such ground states are related by heterotic-type II duality or type II-type II U-duality. Perturbative computations in one member of a dual pair give a non-perturbative result in the other member. In particular, the exact CP-even R2 coupling on the (2,Z) side reproduces the tree-level term plus NS 5-brane instanton con~butions on the (4,0> side. On the other hand, the exact CP-odd coupling yields the one-Ioop axionic inte~ction aR A R together with a similar i~~ta~to~ sum. in a subset of models, the expected breaking of the X(2,2 )S S-duality symme~ to a r(2)s subgroup is observed on the non-perturbative threshoids. Moreover, we present a duality chain that provides evidence for the existence of heterotic N = 4 models in which N = 8 supersymmetry appears at strong coupling. @ 1998 Elsevier Science B.V. PACS: 11.25.-w; 11.25.Mj; ll.JO.Pb; 12.6O.J~

Keywords: String theory; Gravitational corrections; String duality; Non-perturbative effects

’ E-mail: [email protected],ch 2 E-mail: [email protected] 3 On Leavefrom Laboratoire de Physique ThrSoriquede Wcole Normale Sup&ieure, CNRS, 24 rue Lhomond, 75231 Paris Cedex 05, France. E-mail: [email protected] 4 E-mail: o~rs~rn~i.c~~.ch ’ On leave from Centte de Physique ~~o~qu~, EC&e Polytecbnique, CNRS, 91128 p~~seau Cedex, France. E-mail: ~e~~rn~~.~~rn.~h d E-mail: bpi oline~rn~~.cc~.ch OXTO-3213/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII 50550-32 13 (97)00435-4

424

A. Cregori et aLlNuclear Physics B 510 (1998) 423-476

1. I~t~uctio~ There has been intriguing evidence that different perturbative string theories might be non-perturbatively equivalent [ l-41. In six dimensions, there is a conjectured duality between the heterotic string compactified

on T4 and the type IIA string compactified

K3 [ I,2]. Both theories have N = 2 supersymmetry in six dimensions.

Several arguments

(i) The tree-level

two-derivative

and 20 massless

on

vector multiplets

support this duality. actions of the two theories (in the Einstein

are related by a duality transfo~atio~. In ~~ic~l~, antisymmetric tensor (with the gauge Cbern-Simons

frame)

the field strength of the beterotic form irlcluded) is dual to that of

the type II string. (ii) The relation described above implies that the heterotic string is a magnetic or solitonic string of the type II theory and vice versa. This is also supported by the following facts [ 51. There is a singular string solution of the heterotie theory, electricaliy charged under the a~tisymmetric tensor, which can be identi~ed with the ~e~urbativ~ heterotic string [d . There is atso a magnetically charged solitonic (regular at the core) string solution, which has the correct zero-mode structure to be identified with the type II string. Upon the duality map, their role is interchanged in the type II theory 151. (iii) Anomaly cancellation of the heterotic string implies that there should be a oneloop R2 term in the type II theory. Such a term was found by direct calculation in [ ‘73. Its one-loop threshold correction upon compactification to four dimensions ES] implies insta~ton co~ectio~s on the heterotic side due to 5-branes wrapped on the six-torus, (iv) Upon toroidal compacti~cation to four dimensions, heterot~c-type II duality translates into S f--t II”interchange [ 11. As a cons~uence, perturbative IT-duality of the type 11 string implies S-duality [9] of the heterotic string (and vice versa). Electrically charged states are interchanged with magnetically charged states. (v) The six-dimensional ment [ lo] non-perturbative acting orbifolds

heterotic-type II duality implies by the adiabatic argudualities in lower-dimensional models obtained as freely

of the original pair.

(vi) More general {non-fry) symmetric orbifolds still give rise to N = 2 heterotictype II dual pairs in four Dimensions [ 1I-13]. On the heterotic side they can be viewed as compactifications on K3, while on the type IIA side they correspond to compactifications

on K3-fibred

Calabi-Yau

manifolds.

On the hererotic side the dilaton

is in a vector multiplet and the vector moduli space receives both perturbative and nonperturbative corrections. On the other hand, the hypermultiplet moduli space does not receive pert~rbativ~ corrections, or non-~~rt~rbati~e ones, if N = 2 is assumed to be unbroken. On the type II side the dilaton is in a by~e~ultiplet and the ~repotentia~ for the vector multiplets comes only from the tree level. This fact provides a quantitative test of duality; this was shown in Refs. [ Ii,13 J, where the tree-level type II prepotential was computed and shown to give the correct one-loop heterotic result and to predict the non-perturbative corrections on the heterotic side. This quantitative test is not applicable to N = 4 string duality. There is another class of non-~erturbativ~ duality symmetries known as U-duality I.21 y

A. Gregori

et al. /Nuclear Physics B 510 (1998) 423-476

425

relates type II vacua with maximal s~~~ymme~. Tbey are obtund from the of the SL(2,Z) symmetry of type III3 in ten dimensions and O(d,d, Z) duality upon compa~~fieatio~. Using freely acting orb~folds, the s~~y~et~ can be reduced but there should still be a ~-du~ity symmetry [ 101 s In Ref. [ 141 a class of models with N = 4 and N = 2 supersymmetry was discussed; these are related by U-duality. Again, there are several arguments in favour of U-duality, but no quantitative test to our k~ow~~ge~ In this paper we shall focus on the implications of heteroti~-type IIA and ~-dualities for higher-derivative gravitational terms in the effective action, namely R2 couplings and v~ations thereof. These terms have the prope~y that in vacua with 16 supercb~ges (N = 4 in four dimensions) they only receive contribution from short representations of the supersymmetry algebra, the so-called BPS multiplets. This property becomes obvious once these terms are written in terms of helicity supertraces [ 151, which are known to count only BPS states. Therefore, R2 terms in N = 4 vacua are very simile to the terms in the tw~~d~r~vat~v~ action for vacua with 8 su~er~h~ges (N = 2 in four dimensions), In fact, the two-derivative action can be shown to be uncorrected botb pe~~bative~y and non-~~t~~bativel~ in N = 4 vacua, so that these couplings are the first terms where quantum co~e~tio~s manifest themselves, in a still con~ollable way, though. The F4 [ 151 and ti terms in vacua with (4,0) supersymmetry also belong to this class of BPS-saturated couplings, together with higher-derivative terms constructed out of the Riema~n tensor and the gravipboton field strengths ( 141. contributions to R2 couplings depend on the type of N = 4 vacua we are ~o~sidering~ (2,2) vacua, where two s~persy~etries come from the left-muvers and two from the right-movers, or (4,O) vacua, where all four supersy~e~ies come from the left-movers only. All beteroti~ ground states with N = 4 su~rsymmetry are of the (4,O) type, but (4,O) type II vacua can also be constructed [2,17]. In that case, the axion-dilaton corresponds to the complex scalar in the gravitational multiplet in four dimensions and, as such, takes values in an SU( 1,1) /U( 1) eoset space, while the other scalars form an ~~~~,~~~~(~~(4~ x ~~(~v~~ rna~~~~~d~where NV is the comber of vector multi in four dimensions. On the other hand, (2,2) models only exist in type II and a different structure: the dilaton is now part of the ~~(~,~~~/(~~(~~ x SO(A$)j manifold, while the SU( 1,l) /U( 1) coset is spann~ by a perturbative modulus. reality always maps a (2,2) ground state to a (4,0) ground state [2]. We shall argue that R2 cauplings are exactly given by their one-loop result in (2,2) vacua. Translated into the dual (4,O) theory, the exact R2 coupling now appears to ise from non-pe~~rbative effects, identify with NS 5-brane i~sta~to~s in Ref. f8 FIere we shah carry the work of [S] further, and extend it to more general gravitational couplings and to other N = 4 models, obtund as freely acting orbifulds of the usual type IIA on K3 x T2 and heterotic on T6 theories. These exotic ground states Possess a number of vector multiplets smaller than that of their parents (NV = 22), and we shall generically refer to them as reduced-rank N = 4 mod&. They reduce to standard N = 4 or N = 8 models in proper d~ompactifi~atjon limits, and are invariant under ~ed~~e~ groups of T- or S-dualities. convolution

To be specific, we will consider

the hollowing N = 4 models:

(a) Type II theory compactified on K3 x T2 with (2,2) supersymmetry mu~ti~lets. We will denote this ground state by II:?*). It is ~o~jectnred the beterotic iu~e~~~~ge

stri [2f,

~urnpaet~fi~d on T6 ~d~n~~~dhenceforth e R2 ~a~~~~~~has already been ~o~s~der~

and 22 vector to be dual to

by I-I in th

re~o~si

it here in order to ~orn~ute also the thresholds of other four-derivative

as we11

to ~~rnp~e

it with the six-dimensional

thresholds

terms,

once we d~~rnpa~t~fy

the

P. (b) Type II theory compactified on a six-dimensional manifold with SU(2) holonomy, which is locally but not ~~u~~l~y K3 x 7’2. The supersymmetry is still {Z, 2). We present examples with NV - 6,10,14. Tke class of models with i~~~~a~~y constructed in Ref. f 171) using a ~ermioaic constr~c~i,o~ [ 18, constmct them by stating with the K3 x T’ model, going to a subs~ac~ of K3 with a

22 (Nan-freely active) symmetry and orbif~lding with this symmet~, a~~ompani~ by a lattice shift w on the two-torus (w is a ~our~dimensio~al vector of mod(2) integers with w2 = 0). In prac t i ce, we consider the T4/Zz orbifold limit of K3. The Z2 symmetry we use is a subgroup of the (D4)4 symmetry of T4/Zz. A~~ro~~ately choosing this 22 subgroup allows the co~s~uctio~ of (2,2) ground states with A$ = 6,fO, 14 vector mu~ti~~ets. Such a Z2 syrnrn~t~y has the pro~rty that if we orbif~~d by it, witbont a r2 shift, it reproduces the K3 x T2 models at a different point in the K3 moduli space, moreover, because of the shift on the T2, the SL(Z, Z>, duality symmetry is broken to a r(2)~ subgroup. We will denote these ground states by II$;*‘( w). Since the orbifold acts freely, by the adiabatic argument, the new model should be dual to a co~es~o~di~g ~rbifold of the beterotic string on T” with reduced rank, which we will denote by I-ETN~ (w) _ Duality will then imply that such N = 4 ground states have a reduced ~-d~a~~ty group, F(2)s c SL(~,Z>S. This property is reflected in the ~o~-i~v~ia~ce of the R2 threshold under the fuIl SL(2, Z)S roup. Wben the shift vector w involves ~~ojectio~s on momenta only on the type II ide, then its action is e on the heterotie

side, since it acts again on momenta.

If, cm the

ns projections on the winding numbers on the type II side, then in beterotic language the projection is on con-p~rturbative states c ng magnetic charges. (c) A (2,2) type II: model obtained by o~bifo~ding the t string ~ompa~ti~ed on Th ~rn~~rn~ IV = 8 s~~ersymmetry)~ We split T6 = T2 x and the ZZ ~rbifold action is an i~ve~siu~ on T4 and a shift w on T*, 7%is is a crowed state, where N - 8 s~persymmet~ is ~~~~ru~~~~sly broken to W = 4. It has NV = 6 v~tor mul~ipiets and will be denoted by $‘2)(w) = (d) A (4,O) type II model, constructed by freely orbifolding by (- 1)fi. times a Z2 ttice shift w on T6 ( (- 1)fi- is the left~moving fermion number). Such a ground state s NV = 6 and we will denote it by 11~4,~~(w). Here again N = 8 s~persymm~try is s~~~taneo~s~y broker ta N ground state of the previous two-torus electric and magnetic

to the ~~*~~(~~} It was argued in [14] to be U is a map of the agraph via S +-+ T interchange. to the case of ~tr~~g-st~i~g duality. charges si

A. Gregori et al. /Nuclear

String-string

duality and U-duality

421

Physics B 510 (1998) 423-476

imply that the aforementioned

models are related

through HET,,,,(w)(S,T)

(l.la)

=II$;2’(w’)(T,S),

11;4,e’ ( w) ( S, T) = @”

( w’) (r, S) )

where the lattice shift w’ is obtained

(l.lb)

from w through the duality map. For the particular

case of a shift vector w* acting on the momenta Moreover,

only, w*’ = w*.

we shall prove that, at least in the weak-coupling

models $2’2) (w) and IIi2*2’ (w) are actually identical, moduli.

regime S2 -+ 00, the two

up to relabelling

of perturbative

for w* = WI 5 (0, 0, 1 , 0) :

In particular,

II~,2)(wr)(s,zU)

=II~2,2’(wI)(s,-2/T,-1/2u).

We also have the following

decompactification

(1.2) limits, at least in the perturbative

regime:

IIi2*2’ (wt) (T2 -+ co) = $‘62,2)(WI) (T2 --) 0) = type IIA on K3 ,

(1.3)

IIi2,2’( wt) (T2 --) 0) = II, --(2’2) (WI) (T2 ---) co) = type IIA on p .

(1.4)

Now making use of the string-string Zurge-radius limit, HETb(

w*) (S2

duality

( l.la)

we obtain

that, at least in the

-+ 00) = heterotic on p ,

(1.5)

HETe( w*) (S2 -+ 0) = type IIA on p.

(1.6)

Thus we find that at weak coupling the HE’T6 (w*) has N = 4 supersymmetry while at strong coupling N = 8 supersymmetry is restored. It is known that the heterotic string can be viewed as a (non-freely acting) 22 orbifold of M-theory [ 201. Here, however, we find a ground state of the heterotic string in which N = 8 supersymmetry is spontaneously broken to N = 4. The extra gravitinos are magnetic solitons, with masses scaling as the inverse of the heterotic

coupling

constant.

Therefore,

in the strong-coupling limit where these gravitinos the situation described in [ 21-231. The structure

of this paper

is as follows:

N = 8 supersymmetry

become massless.

in Section

is restored

This is similar to

2 we describe

the potential

perturbative and non-perturbative contributions to the R2 couplings in the various string models we analyse further. In Section 3 we consider the type IIA, B string compactified to six dimensions on K3 and compute the perturbative corrections to the four-derivative couplings involving the metric, the NS-NS antisymmetric tensor and the dilaton. In Section 4 we describe the calculation of the one-loop R2 threshold in generic type II orbifold models with N = 4 supersymmetry. In Section 5 we reconsider the R2 thresholds of type II string on K3 x T2 and its dual heterotic theory. In Section 6 we analyse the BPS spectrum and R2 thresholds of the various models with N = 4 supersymmetry and reduced rank. Section 7 contains our conclusions. In Appendix A we describe the kinematics of on-shell string vertices relevant for our threshold calculations, and

in Appendix

B the calculation

of helicity

supertraces

that count BPS states in string

ground states with N = 4 supersymmetry and some associated a-function identities. In Appendix C we calculate the relevant fundamental-domain integrals appearing in the one-loop

calculation

Appendix

of the thresholds.

Details of one-loop

string calculations

are left to

D.

2. Perturbative

and instanton

corrections

to R* couplings

In this paper we shall be interested in four-derivative gravitationa couplings in the low-energy effective action of superstring vacua with 16 supercharges (N = 4 in four Dimensions,

or N = 2 in six Dimensions)*

The prototype

of these tees

is R* z

R~~~~~~~~~, but we shall also consider co~~~~~gs involving the NS a~tisymmetric tensor B,, and the dilaton CD. At tree level, such terms can be obtained directly from the relevant ten-dimensional calculations (see Ref, [24] ) upon compactification on the appropriate manifold, K3, K3 x T2 or T6. They turn out to be non-zero in (4,O) ground states (heterotic or type II) and zero for (2,2) ground states. They may a priori also receive higher-loop perturbative corrections, but (4,O) ground states appe~ to have no perturbative ~o~ectio~s at all, while the pert~rbativ~ corrections in (2,2) vacua are expected to come only from one loop owing to the presence of extended supersymmetry. These terms are related by supersymmetry to eight-fermion couplings. As such they may receive non-perturbative corrections from instantons having not more than 8 fermionic zero-modes. This rules out generic instanton configurations, which break all of the 16 supersymmetric charges and therefore possess at least 16 zero-modes. However, there exist p~t~c~lar ~on~gurations that preserve one half of the s~~ersymm~tries (this is the only possibility in six Dimensions where the supersymmet~ is N = Z), thereby possessing 8 fennionic zero-modes. 7 These configurations correspond to the various p-brane configurations of the original ten-dimensional theory: a Euclidean p-brane can generate an instanton when its (p + 1 )-dimensional world-volume wraps around some appropriate

submanifold

dimensions

have in common

of the compactification the NS 5-brane

manifold

(K3).

All superstrings

in ten

that couples to the dual of the NS-NS

a~tisymmetric tensor and breaks half of the tee-dimensional supersymmetry. Type II superstrings also have D p-branes that are charged under the various R-R forms and their for type lIl3. Obviously, duals: p = 0,2,4,&g for type IIA theory, p =z - 1,1,3,5,7 D-branes are absent from heterotic ground states, but also from the (4,0) type II model we shall consider, since the latter has no massless R-R fields. We conjecture that this is in fact true for any (4,O) vacuum. The only instanton configuration for such vacua is therefore the NS 5brane, which only starts to contribute for dimensions less than or equal to four. 7 tnstantons with Less than 8 zero-modes two-derivative or four-fermion action.

do not exis&, in agreement

with the absence of corrections

to the

In (2,2)

models the situation

IIA, B string compactified

is a bit more involved.

on K3 to six dimensions,

Let us consider

first the type

Since K3 is four dimensional, only

branes with p + 1 < 4 need be considered as instantons. Wrapped in a generic fashion around submanifolds of K3 they break all supersymmetries and thus do not contribute, in our calculation,

There are, however, supersymme~ic

relevant

will

insta~to~s

IIA theory we do not

0,2

and 4 cycles in X3. The

en have p + 1 = 0,2,4, found only in type ect any instanton corrections. In type IIB

fields span an S0(5,21)/(SO{S) x SO(21)) coset space. The perturbative T-duality symmetry 0(4,20,2) combines with the SL(2, Z) symmetry in ten dimensions into an O( 5,21, Z) U-duality symmetry group. The exact non-perturbative threshold should therefore be an 0( 5,21,Z)-invariant function of the moduli and, as argued in [ 251, it can be written as linear combinations of the Eisenstein-Poi~~~~ series. However, alf such series have distinct and non-zero ~~urbative terms when expels in terms of any modulus, in disagreement with the fact that all perturbative corrections should vanish. We thus conclude that the R* threshold is non-perturbatively zero also in type IIB on K3. There is an independent argument pointing to the same result. Consider compactifying type IIA, B on K3 x S’ . Then IIA and IIB are related by inverting the circle radius. From the type DA point of view there are now potential

instanton

corrections

from the p =

0,2,4-branes wrappi~~ around a 0,2,4 K3 cycle times S’. However, on the heterotic side we are still in a djme~sio~ larger than four so we still have no pertu~bative or nap-pe~urbative corrections. This implies that the contribution of the IIA instantons still vanishes, as it does for the IIB instantons, which are just the same as the six-dimensional ones. The instanton contributions in six dimensions thus also have to vanish. Compactifying further to four dimensions on an extra circle, the scalar manifold becomes SU(l, 1)/U(l) xS0(6,22)/(SO(6) xSO(22)) and theduality group SL(2.2) x0(6,22, Z) _ The inst~to~ cont~butio~s can come from 5-branes wrapped around K? x T2 as well as, in type IIB, from D3-branes wrapped around T2 times a K3 2-cycle, and (p,q) Dl-branes related via SL(2,Z)

wrapped around T*. The 1-brane contribution is zero since it is duality to that of the fundamental string world-sheet instantons,

which vanish from the one-loop result. * All other instanton corrections depend nontrivially on the O( 6,22) moduli. Again, it is expected that an O( 6,22)-invariant result would imply perturbative corrections depending on the O( 6,22) mod&i, which are absent, as we will show. re, we again obtain that the non-pert~rbative ~o~~tio~s vanish in IEB, and also This can also be argued via type II-heteroti~-type I triafity. On the heterotic and type I side these corrections come from the 5-brane wrapped on T6, The world-volume action of the DS-brane in type II theory is known and wrapping it around T6 and translating to heterotic variables produces a result depending only on the S field. Thus on the heterotic side we do not expect 0( 6,22)-dependent corrections, and therefore no instanton contributions in type II.

xThis is ~qui~~l~ntto the st~~~~~~~ that in 118the one-loop ~~~~holdonly de~nds on the complex structure U of the torus. This no longer holds for other thresholds such as VHVH and, accordingly, we shall lind that those xe non-perturbatively corrected even in type II.

430

A. Gregori et al./Nuclear

The upshot of the above discussion that on the type II side instanton dimensions.

Similar arguments

is that, in (2,2)

apply to the other reduced-rank corrections

computation

3. One-loop corrections In this section,

in six-dimensional the one-loop

(2,2)

models considered

We can therefore restrict ourselves

on both type IIA and type IIB (2,2)

we compute

imply

are related to the ones above by applying

rules on the possible brane wrappings.

to a one-loop

models, various dualities

to R2 terms are absent in six, five and four

corrections

in this paper, since their instanton some selection

Physics B 510 (1998) 423-476

models.

type IIA and IIB theories four-derivative

terms in the effective action

for type IIA and IIB theory compactified to six dimensions on the K3 manifold. We will work in the 22 orbifold limit of K3 in order to be explicit but, as we will show, the result will be valid for all values of the K3 moduli. 3.1. Type II super-string on K3: a reminder The one-loop

partition

function

of type IIA, B theory compactified

on the T4/22

orbifold is

where the T4 orbifold blocks rd.4 [i] are given by

[I 0

= r4,4

[I h

0

r4,4

7

r4,4

g

= 16

I%v2

I+h Ia [ If8 II

4,

(h,g)

+ (090)

9

(3.2)

where r4,4 is the (4,4) lattice sum. The parameter ,U takes the value 0 or 1 for type IIA or IIB superstrings, respectively, as it determines the sign of the antiholomorphic Ramond sector and hence the space-time chirality of the fermions. We shall also use the notations E = ( - 1) p and denote by left and right the holomorphic and antiholomorphic side, respectively. On each side the sum over spin structures splits into three even an d one odd (a, b) = ( 1,l) according to the structures (a,b) E {(0,0),(0,1),(1,0)} number of fermionic zero-modes on the world-sheet. To compute the massless spectrum we need the following geometric data of K3: the Einstein metric on K3 is parametrized by 58 scalars, and the non-zero Betti numbers are bc = b4 = 1 and b2 = 22. Out of the 22 two-forms, 3 are self-dual, while the remaining 19 are anti-self-dual. At the T4/Z2 orbifold point of K3, those correspond to the 3 + 3 Zz-even two-forms dx’ A dxj and to 16 anti-self-dual two-forms supported by

the two-sphere

that blows up each of 16 fixed points. With this in mind, it is easy to

derive the massless

spectrum:

Type HA. The ten-dimensional

bosonic

massless

fields GMN, BMN, @ and of the R-R t~~-fo~ Cum~a~tifi~atio~

spectrum

and one-form

on K3 then gives in the NS-NS

sector G,,

scalars, and the dilaton @; in the R-R sector we have A,,

consists ~tenti~s

of the NS-NS A~~~ and AM.

and 58 scala.rs, B,, and 22 and 22 vectors in addition

to A,. In six dimensions, APvp can be dualized into a vector, so all in all the bosonic fields comprise a graviton, 1 antisymmetric two-form tensor, 24 UC 1) vectors and 8 1 scalars. Hence, we end up with the following (non-chiral)

supermultiplets

of six-dimensional

( 1,l)

supersymmetry:

1 supergravity

multipIet , 20 vector m~lti~Iets,

where we recall that - the ( 1,l) supergravity

multiplet

comprises

(3.3)

a graviton,

2 Weyl gravitinos

of opposite

chirality, 4 vectors, 4 Weyl spinors of opposite chirality, 1 antisymmetric tensor, 1 real scalar; - a vector multiplet comprises 1 vector, 2 Weyl spinors of opposite chirality, 4 scalars. The scalars p~~etrize R x SU(4,20)/~~~~4) responds to the djla~o~ up to a global 0(4,2O,Z)

x ~U~20)), where the first factor cor~-duality id~nt~~~ation.

Type IZB. The ten-dimensional massless bosonic spectrum consists of the NS-NS fields GMN, BMN, Cp, and the self-dual four-form AhNRs, the two-form AMN and the zero-form A from the R-R sector. Compactification on K3 then gives in the NS-NS sector the same as for type IIA. In the R-R sector, we obtain respectively A&,po (which is not physical), 22 BZf;” (of which 19 anti-self-dual and 3 self-dual) and 1 scalar, A,, and 22 scalars, and the scalar A itself. If we decompose both B,, and A,, into a self-dual and an ~ti-~lf~dual part, tbe bosonic content comprises a gravito~, 5 self-dual and 21 anti-seff-dual antisymmetric tensors and 105 scalars. Hence, we end up with the following

six-dimensional

1 supergravity

(2,0)

(chiral)

supermultiplets:

multipket , 21 tensor multiplets ,

where we recall that - the (2,O) s~pe~~~avity multipIet

comprises

(3.4)

a gravito~~ 5 self-dual

~ti~~mrnet~ic

tensors, 2 left Weyl gravitinos, 2 Weyl fermions; - a (2,O) tensor multiplet comprises one anti-self-dual antisymmetric tensor, 5 scalars, 2 Weyl fermions of chirality opposite to that of the gravitinos. The scalars including the dilaton parametrize the coset space SO( 5,21) /( SO( 5) x SO(21) ), and the low-energy supergravity has a global O( 5,21, R) symmetry [ 261.

We must consider

the piece quartic in momenta

of the one-loop

three-point

function:

indices run over F = 0, I , . t 5 (see Appendix in the O-picture are

Here the space-time

A for conventions),

and the vertex operators

Vc”“(p, .Z, z) = (8X4(%

Z) + @ - Ic;(Z)$wZ))

x (dX”(Z,z)

+ip

.1Cl(z)flr”(~))e’~‘~(~l’),

(3.6)

where the polarization

tensor ~~~ is symmetric traceless for a graviton ant~symmetr~c for an a~t~symmetr~c two-form gauge field (p s - I >. A~toge~b~r the pbysic~ Efi,U =

pvp,

conditions

p@qw =0,

(p = I ) and

are

pq+L = 0,

Pl

+I% i-J-2

=o.

(3.7)

Note that they imply pi*pj = 0 for all i, j. Were the pi’s real and the metric Minkowskian, this would indicate that the momenta are in fact collinear, and all three-point amplitudes would vanish due to kjnematics. This can be evaded by going to complex momenta In Euclidean space. Tbe expression (3.6) gives the form for all the vertex operators when we take tbe even spin structure both on the left and the right. When one spin structure (say left) is odd, though, the presence of a conformal Killing spinor together with a world-sheet gravitino zero-mode requires one of the vertex operators to the -I -picture on the left

be inserted at an arbitrary

point on the world-sheet

(say the last one) be converted

1271.

There are four possible spin-structure combinations to consider, which can be grouped in two pairs accordion to whether they describe CP-even or CP-odd couplings,

where we denote e (0) the even (odd) spin structure on the left and the barred analogues for those on the right, Because of the physical conditions (3.7), the only kinematic structures that can appear at four-derivative order are, in an index-free notation f see Appendix A):

up to permutations

of ( 1,2,3).

The low-energy

action can then be determined

yield the same vertices on shell. Depending the string amplitude can be reproduced (see Appendix A for more details):

In these expressions, antisymmetric = a$?, H LGyP

h(pi)

and b(pi)

by finding Lorentz-invariant

on the polarization by the following

of the incoming

terms that particles,

terms in the effective action

denote the Fourier components

of the graviton and

tensor, which we identify with the polarizations pi in the string calculation, + cY,S, + &B, is the field strength of the two-form potential, and the

left-hod side defines a short-hard no~tio~ for the co~es~ndi~~ term (in agr~ment with stand~d rotating up to factors of fi). The precise meaning to be at~ibut~ to Eq. (3.12) is, for instance,

s

&~RpvpCFRWW’

Onell

Note that other four-derivative

s

d6pl d6p2 d6p3 #‘5) cpr + p2 + p3 >

(27T)‘Z

terms such as squared

Ricci tensor or squared

scalar

curvature do not contribute at thr~-grav~to~ scattering in traceless gauge, so that their coefficient cannot be fixed at this order. That this remains true at four-graviton scattering was proved in [28];

it can be seen as a consequence

of the field redefinition

freedom

which generates R2 and R,,R”” couplings from the g,u + g,v + a&w + b&v, variation of the Einstein term. Similarly, the coupling of two antisymmetric tensors and one graviton could as well be reproduced by a v~ety of RHH terms, equivalent under field rede~nitio~s. We now defer the interested reader to A~~~dix D for the actual detail of the string amplitude, and merely state the salient results. (i) The Z-e sector manifestly receives O(p4) contributions from contractions of four fermions on both sides, and the resulting terms in the effective action are (3.14) (ii)

In the 6-o sector we find the same result, but with an overall minus sign depending on whether we consider type IIA or IIB:

434

(3.15)

Therefore,

one-loop

string corrections

tive action of type IIA su~rs~ng

generate R2 and VHVH

terms in the effec-

on K3, while no such terms appear in the type

IIB supers&ring. (iii)

The CP-odd sectors GO and G-e again lead to the same vertices up to a sign depending on type IIA, B but also on the nature of the particles involved. This leaves

p-odd

eff, IIA

J

= 32t3

p%xld

eff, IIB =

-327r3

(3.16a) d’xfi;HAHAR.

(3.16b)

Summarizing, we can put the results (3.14), (3.15) for the CP-even terms and (3.16) for the CP-odd terms together, and we record the one-loop four-derivative terms in the six-dimensional effective action for type IIA and PIB:

&ff,IIA = 32r3&

J

d6xfi

(

2R2+$7HVH++IA(RAR+VHAVH}

, (3.17a) (3.17b)

where we introduced an overall normalization constant Nh. As a check note that the type IIA theory should be invariant

under a combined

space-

time (P) and world-sheet parity (0). Since the Levi-Civita E tensor changes sign under P while the B field changes sign under 0, we verify the correct invariance under PG. On the other hand, the type IIB theory is correctly invariant under the world-sheet parity a, since the interactions contain an even ~urnbe~ of a~tisymmetric tensor fields. We should stress here that these thresholds~ although they were computed at the p/Z2 orbifold point of K3 are valid for any value of the K3 moduli. The reason is that the threshold is proportional to the elliptic genus of K3 (which in this case is equal to the K3 Euler number) and thus is moduli-independent. It can also be seen directly in the T”/& calculation as follows. The result is obviously independent of the (4,4) orbifold moduli. AlI the other moduli have vertex operators that are proportional to the twist fields of the orbifo~d. The correlator of three grav~tons or ant~sym~etric tensors and one of the extra moduli is ident~caIIy zero, since the symmetry changes the sign of twist fields. Thus, the derivatives of the threshold with respect to the extra moduli are zero.

435

4. One-loop gravitational

corrections

in four-dimensional

type II models

Further compactification of six-dimensional N = 2 type IIA, I3 string theory on a two-torus yields N = 4 string theories in four dimensions. Six-dimensional duality between beterotic string on r”” and type IIA string on K3 is expected to descend to uality between the ~o~es~nding four-Dimensions N = 4 ~orn~~cti~ed theories. Moreover, the two compact flat dimensions make it possible to construct more exotic compactifications, preserving N = 4 in four dimensions [ 17,221) via the fermionic construction or constructions based on freely acting asymmetric orbifolds. As we will see in Section 6, the models obtained in this way may have h&erotic S-duals or type II U-duals. In the following we shall be interested in computing the four-dimensional ~~~n~e~~s of the sax-dimensional four-derivative gravitational terms for generic N = 4 ground states. Before that, however, we shall beefy recall some features of N = 4 supersymmetry. 4.3. Four-dimensional N = 4 super-symmetry and its BPS states Massless multiplets of N = 4 four-dimensiou~ supersymmet~, with helicity less than or equal to 2, are the gravity multip~et (1 gra~ito~, 4 graviti~os~ 6 graviphoto~s, 4 fe~io~s, 1 complex scalar) and the vector mult~plet f 1 photon, 4 fe~ions and 6 real scalars). In particular, the six-dimensional N = ( 1,1) gravity multiplet decomposes under reduction into the four-dimensional N = 4 gravity multiplet plus two N = 4 vector multiplets, while the six-dimensional chiral iV = (2,O) gravity multiplet yields one four-dimensional N =: 4 gravity multiplet plus one N = 4 vector multiplet (upon dualization of four-dimensional two-form potentials into scalars). On the other hand, both the six-dimensional N = ( 1,l) vector and N = (2, Of tensor multiFlets reduce to one N = 4 vector mult~~let each. The generic massive Li representation of N = 4 supe~ymmetry contains 128 bosonic plus 128 fermionic states generated by the action of eight fermionic raising operators on a spin j E Z/2 vacuum (j denotes the representation of the SO( 3) little group of massive representations). However, when the central charge matrix degenerates, only 6 or 4 of the raising operators survive, respectively yielding intermediate BPS representations lj of dimeusio~ 64 or short BPS representatives 9 Sj of dimension 16. Such BPS states can he traced by using helicity supertracesl. which behave as “indices” counting ~n~~r~ BPS multiplets [ 151. More details about the actual computation of helicity supertraces can be found in Appendix B. 4.2. Gravitational thresholds in four dimensions The Tao-derivative low-energy effective action for N = 4 thmries is believed to be exact at tree level, but hither-derivative terms can receive ~~~rbative and non-pe~~rbative 9 The massless

are always short representations.

one-loop

corrections.

four-derivative generally

We will be interested

terms involving

called gravitational

in computing

the graviton,

thresholds.

the moduli dependence

antisymmetric

tensor

and dilaton,

of the more

The terms of interest are therefore

(431) Again, we shall use a short-hand

notation

for each term appetite

in the above expres-

sion: R’, R /1 R, VHVH, VI-ZA VH, VV~VV~, VV~ A VH, R A VH. Note that there is no non-va~isbing on-shell RH coupling between one graviton and one two-form, nor any VV@ A VVQ, or VV@ A R couplings. The various terms in Eq. (4.1) will turn out to be expressible

in terms of helicity supertraces

and, as such, will receive contributions

from BPS states only. They therefore offer a reliable window into the strong-coupling regime. We will now concentrate on the derivation of general formulas for gravitational thresholds in four-dimensional type II models desponding from type II six-dimensional vacua compactified on X3. At first, one might think that such thresholds could be evaluated by computing two-graviton scattering. Such an amplitude, however, vanishes on shell, and is potentially infrared-divergent. A rigorous and unambiguous way to deal with this problem was described in [29,30] and further analysed in [ 311; this amounts to regularizing the infrared by turning on background fields that provide the theory with a mass gap. This method preserves some of the original supersymmetries of the theory: up to N = 2 for heterotic ground states, and up to (p, 4) = (2,2) for type II ground states, where we depots by p and 4 the number of s~~ersymm~tries Corning from the left and the right. IIowever, this procedure does not allow us to discriminate the various interaction operators

terms appearing

Here, however, the four-derivative s~f~cient

in (4.1),

by lack of a sufficient

that could be turned on as background

number

of marginal

fields.

we shall only be interested in the (T, U) moduli dependence gravitational couplings in the effective action; it will therefore

to compute the scattering

solitude

between two gravito~s

of be

(or two two-fobs

or two dilators) and mod~li fields. This will give access to $44 and a+@, which are infrared-unite. The same comments as in the six-dimensional case apply to the choice of vertices in Eq. (4.1) for describing the string amplitude. In particular, one may add to this expression terms such as R,,R Pv or R* without changing the S-matrix, and for instance the Gauss-Bonnet combination ( RCcvpaRpvPg-4R,, R@”4choose instead of R@‘PCIRPPPvpcr R*), which has the advantage of being a total derivative at second order in h and therefore does not correct the gravito~ propagator. lo This would be useful if one were “‘The Gauss-Bonnet

combination

in four dimensions

is a total derivative to any order in h, so one might

to look at four-particle scattering, where field theory subtraction enters into play [24]. Also, it will turn out that the naive R A R,

VH A BH and VV@ A VJFj!terms, chosen to represent the CT-odd interaction of gravitons with moduli, are inad~uate and have to be supplemented by ~~~r~-~~rnuns co~~~~ngs. With these provisos, the kinematical structures

contributing

to gravitational

thresholds read

and it is readily checked that these expressions are consistent with gauge invariance r--+6+pCS3k~Pk~ witbk+p=O. The second equation in (4.2) shows that the CAVE co~p~~~g carrot be revea by a three-particle amplitude, This forces us to look at scattering at least two gra~ito~s (or two two-fogs or two dilatons~ and two m~duli. Xn fact, the insertion of any number of rnodu~~ remains datable as long as two simplex-conjugated moduli are not simultaneously present, and we shall therefore keep with the general case of N moduli.

The class of (2,2)

s~~~symrnetri~ models descending

string un K3 can be generically partition Zfour II

from ~~x-d~rn~~sio~a~ type 11

described at the Z2 orbifold point of KJ by the fo~lowiug

function: dim

(4.3)

where Z,5,6[:] are generic ‘I orbifold blocks whose structure way the Z2 group acts on the various states of the spectrum. For such a vacuum, we shall need to extract the four-momenta

depends

on the specific

part from the following

amplitude:

(4.4) containing two gravitons or antisymmetric tensor fields {depending on the polarization tensors E+) and N two-torus rnQdu~~fields. In contrast to Section 3, the space-time indices now run over Al,= 0,. . . , 3, but the vertex operators of the space-time fields are (3.6) for the ~-~ieture, Eq. (3.8) for the -l-picture on identical to those given in the left, etc.. In close analogy, in the ~-picture the vertex operators of the mod~li fields are given by

&~r~(~)=~~(Gf~~~r~), In particular,

in the standard

I,J=

1,2.

14.6)

(r U) parametrization

recalled in Appendix

C, we have

with X = X4 += UX5, x” = X4 + uX5 (and similarly ?P = $4 -l- U@), while the vertices for T, g are obtains by complex conjugation. Nofe that chiral mad~li (K U) have a.? -as left-moving part, while the anticbiral ones (T, U) have 3.X instead. The modifications for - l-picture on the right and/or left are as described in Section 3, so that for example for the -l-picture on the left we have

together with an insertion

of the heft-moving

s~~ercu~ent

where we omitted the K3 internal part of CF. We will again defer the details of the compwtati~n to Appendix D, and simply outline the ca~c~~atio~ here, A drastic simplification occurs thanks to a sedition rule that forbids contractions not conserving the U( 1) charge of the T2 superconformal theory:

A, Gregori et al. /Nuclear Physics B 510 (1998) 423-476 MAXI = (Xri)

f: ~~~

= ~~~~

= 0

439

f

(4.10)

Except when a pair of complex-conjn~ated moduli occurs, only the zero-mode of the bosonic part of the moduli vertices contributes and generates for each insertion a derivative with respect

to the corresponding

a sign depending

on the nature of the last modulus).

modulus

(together,

that the fermionic

part of the two gravitons

in the odd structure,

Supersymmetry

be contracted

with

then demands

together, yielding

the four

powers of momenta as desired. The e-e and B-a kinematics turn out to be equal in the two-graviton case and op~site in the two-a~tisymmetric-tensor case (zero in the ~ravito~-two-fog case). Our final result for the one-loop moduli dependence couplings in Eq. (4.1) is summarized by

of the four-derivative

gravitational

(4.11a)

(4.1 lb)

(4.1 le)

(4.1 ld)

(4.1 le)

(4.1 lf)

(4.1 lg)

where A& is a normalization

constant

that we will fix later, The derivative +, stands for

the product n,“=, d#J. The K-‘J are numerical coefficients type II string as well as on the choice of moduli:

that depend on the choice of

where (~4, g,+) specifies the nature of the last modulus (see Eqs. (D.32a,b) ) and the conformal blocks Zij are expressed in terms of the blocks Z E 3 appearing in the four-dimensional petition function (4.3 >:

i,j

= Z, e

T,j=&a

i,j=a,e.

(4.13) En the previous expression,

a prime on the left indoor

the right stands for the operative

in Bq. (D.3).

4.4. Gravitational

thresholds and helicity supertraces

Using Riemann identity and (2,2) s~persymmetry, it is readily seen that the four blocks Z’j are equal to 2?. Moreover, identity (B.16) allows us to convert the 3, derivative in 2”” into a secon -order d~rivat~~~ with respect to the variable u conjugate to the left h&city At, as described in Appen ix B. A similar ~tat~rn~nt applies to the right side, yielding (4.14) Substituting

in Eq. (4.11),

we obtain for instance (4.15) 3

and similar relations

for the other thresholds.

This makes it obvious that only short BPS

states coutr~b~te to the one-loop four-derivative ~r~~~tatjo~al ~o~~t~o~s. it will be convenient to fix the normalization constarn to

JJ+ We note that the four different spin structures contribute but for signs depending on the type A or B of superstring considering, As a result of this in~crfer~~ce:

From now on,

(4.16) in the same way to d$~$,, and the modulus C# we are

A. Gregori et al/Nuclear

Physics B 510 (1998) 423-476

441

&Agr = 0 (4.17b)

type IIB: &A,,

We recover moduli

= SF $J[/B4.

in this way the well-known

is true in type IIB [32]. following

1~:

result that A,, only depends

T and not on the complex-structure Similar

moduli

interferences

on the K%hler

U in type IIA, while the reverse

occur for all thresholds

and yield the

moduli dependences:

A,,(T)

9

As(U)

9

&I(U)

3

@g(T)

7

@as(T)

3

@g-as(U), @d&a(u), (4.18a)

IIB : A,,(U)

9 A,(T)

9 Adil(T) 7 @g(U) 9 @as(U),

Ogreas(

Odil_s(T) . (4.18b)

The dependence of Agr(T) is consistent with our argument that the R* term does not get corrections beyond one loop. However, there exists a subgroup of SO( 6, NV, 2) that exchanges the (type IIA) U-modulus with the dilaton S-modulus, so that SO( 6, NV, Z) duality implies that A,, Adil, Ogreas,@d&as are also S-dependent, i.e. are perturbatively and non-perturbatively corrected. The loophole in the argument of Section 2 is that, for these couplings, the world-sheet instantons of the type IIB string are non-zero (since they depend on the type IIB T-modulus), and therefore the (p, q) Dl-branes do contribute to instanton corrections. From now on we shall restrict ourselves which the type II one-loop result is exact [ 2533,341.

5. Gravitational

thresholds

to R2 thresholds,

for

in ordinary type II on K3 x T*

We now apply the previous formalism to the trivial reduction to four dimensions of type II string theory on K3. Using the considerations in Subsection 4.1 and the six-dimensional spectrum, it follows that both type IIA and type IIB on K3 x T2 have

1 supergravity

multiplet , 22 vector multiplets .

(5.1)

The two theories are indeed exchanged by T-duality on one circle of T*, which corresponds to the exchange of T and U moduli. The scalars therefore span SU( 1,l) /U( 1) x SO(6,22)/(SO(6) x SO(22)), where the SU( 1, l)/U( 1) factor corresponds to the complex scalar in the gravitational multiplet (T for IIA, U for IIB) [ 171. Type II string theory on K3 x T2 at the T4/Z2 orbifold point is described by the following partition function:

442

where we use the same Ji,4[:]

blocks as in (3.2).

The he~jcity s~p~~t~~~e B4 entering in the threshold (4.15) can be readily commutes from (5.2) using the methods of Appendix 3, with the result B4

=

36r2*2.

(5.3)

It is easy to check the ~2 ---f 00 limit, where only short BPS massless states contribute, with the result

w

massless=Ix3+22x;=36,

where we used the contributions

(5.4) in Eq. (B.2) for the supergravity

and vector multiplets.

The expression in Eq. (5.3) further shows that the rest of the contributions to B4 come from the tower of massive short BPS multiplets whose vertex operators are those of the massless states plus momenta and windings of the two-torus. The matching condition implies that we should have FM = 0 for these states and they are in N = 4 supe~u~tiplets similar to the massless ones. This result is expected, since we know that a left-moving state breaks half of the two left-moving supersymmetries. Thus states that are ground states both on the left and right (plus momentum of the two-torus) are expected to break half out of the total of four supersymmetries in agreement with the helicity supertrace. Using Eq. (B.20),

not much more work is required to extract the 176 supertrace

whose 72 t 00 limit again agrees with the massless (short BPS) spect~m since lxYt22x7 = 90. However, although we know that ~n~~~ed~ate multiplets~ corresponding to states that are ground states on the left only, but with arbitrary oscillator excitations on the right (or reversed) could contribute to Bg, they turn out to cancel as a consequence of identity (B.22). We therefore conclude that intermediate BPS multiplets come in combinations that can aiways be paired into long massive multiplets and thus do not contribute to B6. Their multiplicities and mass formulae are therefore not protected from quantum corrections. is example indicates that one has to be careful when invoking non-reno~a~ization theorems for BPS states. Only BPS states having non-zero “‘index” are protected from quantum corrections.

A. Gregori et al. /Nuclear Physics B 510 (1998) 423-476

We now insert B4 into Eq. (4.15)

and use the fundamental-domain

443

integral

(C.8) to

obtain the R* thresholds: = -36 log (T2 Jv( T) I”) + const. ,

type IIA:

A,,(T)

type IIB:

Agr( U) = -36 log (Hz (q(U) 14) + const. ,

where the constant

is undetermined

(5.6a) (56b)

in our scheme. The above result is in agreement

with [ 81. Note that the one-loop thresholds are respectively invariant under SL( 2,Z)r and SL(2, Z),, as they should. Moreover, since only the twisted sectors (h,g) # (0,O) of T4/Z2 contribute to B4, A,, as well as the other thresholds are independent of the untwisted moduli of K3, and therefore of all K3 moduli. Consequently, the result obtained at the orbifold point T4/Z2 is valid everywhere in the moduli space of K3. 5.2. Decompactijcation

limit of CP-even couplings:

a pu&e

It is important to confront this result to our six-dimensional result (3.17a), which should be retrieved in the decompactification limit of the two-torus, T2 = & --f cc: II’2.2’ . Agl(T)~za 22 . in the type IIA situation.

- 36logT2 + 12rT2 + 0 (e-r’) This agrees with Eq. (3.17a)

(5.7)

provided we set

N+.

(5.8)

On the other hand, taking the large-volume limit in the type IIB theory does not affect the U-dependent threshold. However, only terms of order T2 (the volume of the torus) can be seen in the decompactification limit, so this agrees with the vanishing of R* coupling in six dimensions (3.17b). We can repeat the same discussion has the same behaviour

for the four-dimensional

up to T H U interchange,

VHVH

threshold, which

and predict that the six-dimensional

coupling VHVH should occur only in type IIB and not in type IIA, in contrast to R*. This is in disagreement with our six-dimensional result, which showed that cancellation between z-e and b-o spin structures had to occur in the same way for both R* and VHVH. Note that we could also have performed the three-graviton-two-form scattering calculation directly in four dimensions, finding the same result for E-e as in six dimensions, but a vanishing b-o contribution. We would have concluded that R* and VHVH have to occur with the same (T, U)-dependent coupling, in both types IIA and IIB. This shows that the three-particle amplitude has to be interpreted with great care. 5.3. CP-odd couplings

and holomorphic

Moving on to the CP-odd couplings Eq. (4.1 Id) yields

anomalies and focusing

on the IIA case for definiteness,

A. Gregari et al. /~u~ie~lr Pft~&.s B 510 (1998) 423-476

444 a@,,

=

-18idrlog

(Tz [v(T)14)

Would the non-harmonic

$@ar = 18i+log

,

.

(Tz lq(T)14)

T2 term be absent, those two equations

(5.9)

could be easily inte-

grated and would give ~~~~T) = 18Im logq4fT)

_

(5.10)

However, in the presence of the T2 term the notation

&-0 and +O for CP-odd couplings

between two gravitons and one modulus no longer makes sense, This non-integrability of CP-odd couplings has already been encountered before [ 351. This problem can be evaded simply by rewriting

the CP-odd coupling

as (5.1 I)

where fi is the gravitational Chern-Simons three-form, such that dJ2 = R I\ R. In the special case Zr = &+?J(T T>, 2~ = +0(7: T>, one retrieves by partial integration the usual integrable CP-odd coupling. In the case at hand, zr = -18iJrlog

(7i I?l(T)I4)

We can take advantage FP-Odd = 1877 as

Im (logT4(T>)

six dim

four dim +

and rewrite Eq. (5.11) .

OAdTi

as

(5.13)

limit T2 + 00, only the first term survives and we obtain RA R+0(ljT2))

16n3.A& 6~

(5.12)

.

>

the six-dimensional =

IlA

RA R - i

0

fir-odd = 18 ;7j gr SC

z

18ic+Jog (72 irl(Ti(4)

of the special structure of E$_ (5.12)

In the decompactification

This reproduces

z,=

,

J

(5.14)

.

type IIA result f3.17a):

d%J-g6

&UKAP%7~~R,.,&$~

s (5.15 j

d4+J-g?T2EfJB,J~K’poRxn~~Rpa(lp

since E’~BIJ = 2q/T2 and ~$6 = 4213. Exactly the same ~~at~~~arises for the &&. @sr_aSand @dtr_ascases, for which, in the type IIA case, the correct coupling should instead be written as

as 1877 s(

FP-Odd = c;;zdd

,

Im (log q4( T)) VH A OH - ;HAVHAdT,

= 36~

Im (10g~~(U))RA!7Iir0

(5.16a)

> _$RAHAdTf

,

(5.16b)

>

Gr;;dd = 247T (5.16~)

Note also that VHVH

correctly decompactifies

to the B A VH A VH of six-dimensional

type IJA theory in just the same way as R A R, while in type IIB R A VH gives the -?B A R A OH six-dimensional coupling. The VV@ A VH coupling correct HAHAR= cannot be checked here since we did not consider six-dimensional dilaton scattering.

Coming now to duality, it is well known that heterotic on T4 - type IIA on K3 duality in six dimensions implies, after compactification, the duality of the corresponding fourdimensional theories under exchange of S and T, where S is the axion-dilaton multiple& sitting in the gravitational multiplet on the heterotic side. For definiten~s we recall the p~ition function of heterotic string on T6:

HET22

:

2 =

1 T2 7p$P

1’

(5.17)

C-1) 5 c a,b=O

where r6,22( G, B, A) depends on the six-dimensional metric G, the antisymmetric tensor B and the Wilson lines A. At generic points of the moduli space (i.e. with gauge group broken to U( 1) factors), the massless bosonic s~~trurn is 1 supergravity

multiplet , 22 vector m~ltiplets,

(5.18)

in agreement with (5. l), as expected by duality. Contrary to the type II case, the heterotic string theory possesses a tree-level R A R coupling required for anomaly cancellation through the Green-Schwarz mechanism, together with an R2 coupling required for supersymmetry. The world-sheet fermions now have 10 zero-modes, so that the oneloop thr~-p~ti~le amplitude v~isbes (in even spin structures one would need four fe~io~i~ contraptions to have a non-vanishing result after spin-structure summation) . In particular, Following heterotic

we conclude that there is no one-loop correction to tree-level R* coupling. Ref. [S], we can therefore translate the type IIA resuit (5.6a) for the

string on T6:

HET22 : A,,( W = -361og (s* j?&s)fJ $---++- 36logS2 +- 127rSz + 0 (e-S’> . 2-+

(5.19) The 52 -+ 00 heterotic weak-coupling limit exhibits the tree-level R2 coupling together with a non-perturbatively seen logarithmic divergence. The latter was omitted in Ref. [ 81, where only the Wilsonian effective action was investigated, but is also present in other instances [25]. The full threshold is manifestly invariant under SL(2, Z)S, and could in fact be inferred from SL( 2,Z)s completion of the tree-level result. The exponentially suppr~ss~ terms in EZq. (5.19) were identified in [8] with the insta~ton ~o~~ib~tio~s of the neutral ~eteroti~ NS 5-brane wrapped on r”), the only i~sta~to~ ~on~guratio~ that can possibly occur in four-dimensional heterotic string. The same mapping can be executed for the CP-odd R A R coupling from Eq. (5.16):

FP-odd

gr

=

1 g

Im (logq4(S))

gence, we find a coupling ~uupling

-

$ OndS,

,

(5.20)

>

There, however, in addition realizing

RAR

to the tree-level between

term and instead of the logarithmic

the axion and the gravitationaf

the axion into a two-form and keeping S,, this translates

diver-

Chern-Simons

form.

track of the powers of Fhe heterotic

into a tine-Zut?p coupling

~~~~~~~~ between one two-form

and two gravitons, precluded by a one-loop h&erotic calculation. Happily enough, Chern-Simons form is co-closed, so that this coupling is a total derivative,

6. Reduced-rank

N = 4 models and baaing

the

of S-duality

Aithough the most studied N = 4 dual string pair is the stand~d heterotic on T6 type IIA on K3 x T2 pair with generic gauge group U( 1)28, more exotic models with a lower gauge-group rank do exist. Since all p\r = 4 matter multiplets have to transform into the adjoint representation of the gauge group, their expectation values cannot break it to a group with lower rank, and those theories therefore have to live in disconnected moduIi spaces On the type II side, such models can be easily obFain~

by ~~rn~a~tifying

the six-

dimensional IIA on KJ theory at orbifold points of K3 using a generalized ScherkSchwarz mechanism [ 36-381 to give a (moduli-dependent) mass [ 21-231 to part of the vector multiple& originating from the twisted sectors of K3. This can be implemented by orbifolding

the IIA on K3 x T2 theory by a translation

an action on the twisted sector. On the heterotic side, such models have been constructed

on the torus accompanied

by

in Ref. [39J with fermionic

~~ara~t~~s, but it is dif~c~lt to identify them with models dual to the above type II, since that would require identifying the point in heterotic moduli space ~o~espondin~ to the orbifold points of K3. Nevertheless, if one trusts six-dimensional IIA duality, such heterotic duals are guaranteed to exist. The construction on the type II side makes it clear that T-duality

heterotic-type is broken

to a

subgroup by the precise translation vector on the r2,2 lattice, which translates in heterotic variables into a ~~~~~~~g of S-duality. This breaking rnod~~es the eon-pertnrbativ~ insta~ton ~o~~~ions in lower-rank heterotic or type II theories djs~us~~d below. In the foIIowing, we shall examine the four-derivative perturbative gravitational corrections in various type II models, and translate them in terms of non-perturbative effects on the heterotic side.

IIere we consider a v~iation of the type II over p/Z2 x T” ~~m~actification described above (see model II,,(2$2),(5 .2) ) . The 22 will now act both as a twist on the T4 and as a shift on the two-torus. This model is a spontaneously broken N = 8 --P 4 theory with

(2,2)

supersymmetry,

The partition

function

as will become clear shortly, and will be denoted by @2’2) (w). reads

where r4.4 [t] are the twisted (4,4) lattice sums (see Eq. (3.2)) and RJ2[i] are the shifted f2,Zf lattice sums given in Appendix C. nodule inv~ia~c~ requires the shift vector w to satisfy w2 = 0. The 16 twisted vector rnu~t~~~et~ from the p/Z* x T2 model now acquire a mass of the order of the inverse radii of T2, so that the massless spectrum becomes 1 supergravity

multiplet

, 6 vector multiplets .

The scalars of the 6 vector multiplets the complex scalar in the gravitational type IIA (B) theories, It is now straightforward to compute using (B.19), (B-20). They read I2

(6.2)

p~ametri~e ~~~6,4~/~~~~6) x W(6)), rn~~t~~let co~espo~ds to the T modulus helicity supertraces

while (U> in

directly from Eq. (6.1) and

(6.3a)

(6.3b) where H[:]

are given in Eq. (B-21).

the contributions

of the massless

In Eq, (6.3), we have indicated after the N sign states, obtained by using the fact that only the r?,, [‘;‘I

block contains massless states, as well as the leading behaviour H[:] = 2 + U(q). check, we observe the correct values for the contributions of the massless states,

As a

(6.4a) (6.4b)

where we used the elementary contributions (B.2) and (B.3). Moreover, we observe that in contrast to the ordinary type IIA theory on K3 x T2 (model (52) ), the inte~ediate m~lti~lets do contribute to &. Inserting the result (6.3a) in Eq. (4.15) that we recaIl here, ‘2Theprimedsummationover(h,g)

standsfor(h,g)

E {(U,l),(t,O),(l,l)).

(6.5) allows us to determine integrals

involving

the gravitational

thresholds

f2.2 la] are computed

in terms of 84. ~~~darnental-domain

in A~~ndix

C, and yield, for the type IIA

case.

(WI : EL*,*)

AgrV) = -121og (T2 l&(7.)1’)

+ const.,

(6.6)

where i = 2,3,4, depending on the shift vector w (see Appendix C). An important consequence is that the resulting corrections break the SL(2,Z)r duality group to a 1’(2)r

subgroup.

The precise subgroup

depends on i as indicated

in Appendix

C.

This model was ogled in I141 to be U-dual to a (4,O) s~~rs~mrnetri~ type II model, to which we now turn. This model is obtained as a 22 orbifold of type II on rc’, where the Z2 acts as (-I >fi. together with a translation on 7”6. Again, this model exhibits spontaneously broken N = 8 -+ 4 supersymmetry and we will denote it by IIk4”) ( w) _ The resulting partition function reads

To compute the massless spectrum, we first recall that for N = 8 type II, obtained by compactifying on p, the spectrum is as follows: NS-NS gives G,,, B,,, 9 and E2 vectors as (6,6) and 6 x 6 = 36 scafars; R-R gives 16 vectors as {O, 16) and 32 scalars. Because of the (-I )‘L orbifold~ the R-R sector is projected out so we are left with the NS-NS

states only, which combine

into the following

four-dimensional

N =4

multiplets: 1 supergravity in agreement

multiplet,

with the massless

6 vector multiplets, spectrum

(6.2)

in the gravitational multip~et now ~o~es~o~ds For completeness, the helicity supertraces (B.16) and (B.17):

(6.8)

of the dual theory, The complex

scalar

to the axion-disavow field. for this model can be computed

using

(6.9a)

(6.9b)

where

from which we see that B4 again receives ~~~tri~ut~onsor@ from massless and massive short BPS multiplex,

while EQ, also gets ~o~tr~b~tio~s from ~~te~~iate

ones,

However, for (4,O) supersymmetric models, a four-graviton scattering calculation shows that the one-loop corrections to R2 terms do not involve h&city supertraces. Instead, the one-loop corrections simply vanish, and the only contributions to R2 couplings, as argued in the introduction, obtain

the non-perturbative

are non-perturbative. (4,O)

Now, U-duality

result from the one-loop

can be invoked to

result (6.6)

of the (2,2)

dual, by iden~fyi~g the ~-modulus of the (2,2) theory with the ~-modulus of the ory. There is however an ~rn~~a~~ subtlety involved in identifying the lattice shifts on both sides. We recall that in the full non-perturbative spectrum, states have not only electrical charges M;, ni under the Kaluza-Klein gauge fields of T2, but also have magnetic charges tii, fi”. Under S t+ 7’ interchange, mapped to each other according to [22] {~~,~~,~~,~~~

+

electric and magnetic

charges

are

(6,ll)

~~~,~~j~j,-~jj~,~~~.

In prickly, a (-1)“’ proj~tio~ on states with even electric winding FZIon the (2,2) side translates into a (-l>“z projection on the (4,O) side, of no effect in perturbation theory. A f-l)“* projection on the other hand in the (2,2) theory translates into a perturbative ( - 1) ml in the dual (4,O) theory. These two projections have a geometrical interpretation of doubling one radius of T2, in contrast to the ( - 1 )“I one. However, (2,2f perturbative modular invariance requires at the same time half-integer FZ’charges in the twisted sector. This implies also half-integer &z charges in the twisted sector of the dual (2,2) theory, which should presumably be a~comp~i~ by a (-I)“’ under some “non-perturbative modular invariance” requirement. This in turn would imply that the correct

projection

on the (2,2)

side is (-l)m1+fi2,

which reduces

to (-1)“’

in

the perturbative spectrum. This ambiguity does not affect the perturbative evaluation of thresholds. As for non-perturbative corrections, the relevant instantons are a subset of the original

ones, which have been shown to not contribute

to R2 couplings.

Restricting

to a pr~~~t~~~ on the electrical m~rne~ta only (cases I, II, III in Table Cl), from I$ (6.6) the result

II:470’ (w,11,111):

= -121og .6.1,,(S)

(S2 j&+(S)14) + const.

we find

(6.12)

This exhibits the expected feature [7] that the S-duality symmetry is broken to a r(2)~ subgroup of SL( 2, Z),, namely the subgroup that leaves 84(S) invariant. The two theories are weakly coupled in the regime T2, $2 --+ CQ. The T2 --+ 00 d~ompacti~catio~ limit of shifted (2,2) lattice sums was investigate in [Zl], with the result r;,II.rII

[

h]

g

-

Tz-+~

’

72

Sh,Oag.0

(6.13)

S-T

145 l#II.Il (2.2)

Fig.

t. The cube of duality, decompactification

up to exponentially

suppressed

corrections.

and strong-weak

coupling

This selects the untwisted

relations.

u~proje~ted sector

of the two models (6.1) * (6.7), thereby restoring N = 8 su~~rsymrn~t~ for both of them, in agreement with U-duality conjecture. Expanding &. (6.12) in the weak (4,O) coupling limit, we find II;4,0’ ( w1,n,rr1>: L&,(S)~~,

- 12logS2 t 0 (e+)

.

(6.14)

The result exhibits the correct vanishing of perturbative 0 (3;) corrections, together with the already encountered non-pe~urbative ~og~~rni~ divergence. Let us now turn to the strong-coupling ~~~avio~~ of the (4,0f ground state. The $2 --f 0 limit of (4, O} is mapped under duality to the T2 -+ 0 timit of the (2,2) ground state, for which we can again use the results of Ref. [ 211, (6.15) up to ex~o~~ntiall~ $u~~ressed ~o~~tio~s, The orbifold action does not affect the T2 part any longer, thereby yielding the standar type II on K3 x T” model of Section 5 at small radius. This is strictly true only in the perturbative regime of type II, because of the non-perturbative ambiguities mentioned before. This is further mapped to the HET22 model at large coupling & --f 0 and large radius T2 + 00. We therefore conclude that the (4,O) model and the standard heterotic model on T6 are equivalent in the strongcoupling large-radius limit, I3 This can also be checked on the explicit R* coupling I3 One may ask whether the two limits commute. The correct ~~sc~pt~on is to first take Tz --) 00 and then nniy 52 --t 0 in (4,O) variables, since we needed the (2,2) dual to be weakly coupled before we could conclude anything about its small-radius limit.

A. Gregori et al. /Nuclear Physics B 510 (1998) 423-476

IIC,O) ( W,II,III)

:

Agr(S)szo

- 12logS2 + 127rS2 + 0 (e-s’)

4.51

,

(6.16)

which reproduces the correct heterotic on T4 tree-level coupling (5.19). relations is depicted on the upper and rear faces of the cube in Fig. 1. free orbifold of type II on K3 and its heterotic dual HET6

6.2. IIr’2)

We now turn to another example descends

This set of

from six-dimensional

of N = 4 four-dimensional

string-string

duality, which this time

duality by a freely acting orbifold,

namely

a half-lattice shift on T2 together with a minus sign on the twisted sector of K3. The adiabatic argument [lo] guarantees that the heterotic model obtained by translating this action in heterotic string on T4 is still dual to the type II orbifold. To be explicit, the resulting partition function for this type II model, denoted by IIi2*2)( w), is given by

(6.17) Again, the shift vector w has to satisfy w2 = 0 for modular invariance. The (h,g) projections are associated with the T4/Z2 orbifold, while the freely acting transformations correspond to the (h’, g’) projection. The massless spectrum is most easily obtained from the results at the beginning of Section 5, by noting that the ( -l)h orbifold projects out the twisted states, so that we are left with the following 1 supergravity The relevant helicity

B4=6

untwisted

four-dimensional

N = 4 multiplets:

multiplet , 6 vector multiplets. supertraces

(6.18)

are (we use again the results (B.19)

(3r2,2 -~h,$])- 12,

&=15

(3r2,2-~‘(l-BeH[~])r~2[~])

where the functions thresholds II;2,2)(w)

H [i] are given in Eq. (B.21).

: A&T)

= -1210g

and (B.20) )

(6.19a)

-609

(6.19b)

We deduce the type IIA gravitational

+ const’ ’

(6.20)

where i = 2,3,4, depending on the shift vector w (see Table C, 1). As advocated in the previous section, we shall restrict our discussion to shift vectors leading to i = 4, for which the resulting T-duality group is r+( 2)~. We now want to discuss the beterotic dual for this model. From six-dimensional duality, the 22 symmetry

acting as -1

has to have an equivaIe~t

in the dual heterotic string for the co~es~ond~ng

SO(6,22)

heterotic

string

on all twisted states of K3 at the or&fold point

mod~li. At present, there remans

values of the

a puzzle as to what these values

are 1401. Nevertheless, this symmetry can in principle be used to construct a freely acting orbifold of heterotic string on T4, and the adiabatic argument guarantees that the resulting model wilI be dual to the present 11i2’” (w) model. Henceforth we shall refer to this model as HET6( w}. The heterot~c co~~l~~~ is given by the area of the type II torus, which, owing to the free action, is T/2. We t educe the non-perturbativ~ threshold for HETg ( us) :

HET6C

W,II,I~I>

:

A,rW

=

In particular, we observe that the tree-level contribution matches the one of the HETs2 model (5.19), as it s ould, since the tree-level effective action is universe for ah heterotic ground states. The cases co~~s~o~ ng to i = 2,3 in the thresbo~d ~~.2~~ are + 2s - 1, side, yielding T --+ -1J2SJ obtained by applying T-duality on the type respectively. The (large-radius} weak-coupling limit of HETa( w) is mapped to the (weak-coupling) large-radius Iimit of II, hi& by the same techniques as in the previous section turns out to be the standard I gn2’ model. The latter being dual. to the standard HET22, we conclude that ~~T4(~) and I-IETzz are the same in the {gage-radius~ p~rt~rbative regime. The relation between the quartet of theories that we have been discussing can be seen on the front side of the cube in Fig. 1, The (large-radius) strong-coupling limit of HET6( w) can be discussed in the same small-radius limit way as for the II,(4So)(no) model: it corresponds to the f weak-coupling) of II;“*” ( +v), which from the partition

function

(6.17)

and from Eq. (6.15) appears

to restore N = 8 supersymmetry. In fact, II~*“” { w} and $“” tra~sfo~atio~ of the moduli, thanks to the relation (C. 18),

Ir’,g’=O C-1)

hg’+gh’

pv

2.2

(,v) are identical

(T,W.

under

(6.22)

The precise rnap~in~ (YCU> + (T’, U’) is shown in Table Cl for the various lattice shifts, T --+ -2,/T for the cases I, IX,III at hand, leading to i = 4 in the above foi-rn~~a (6.20). The N = $ (weak-coupling) abbe-radius limit of II~*,~~(~) therefore coincides

with the N = 8 (weak-co~~l~~~)

small-radons

limit of $‘2’(

w), and is dual

ta the N = 8 (large-radius)

strong-coupling

limit of HETb( w). Furthermore,

this implies

that HETh( w) and IIF’O’ (w) are mapped to each other under S -+ -2/S. The various relations among the octet of theories that has been discussed in this and the previous section are summarized by the duality cube in Fig. 1. In this figure, the horizontal

c~~~ect~~~~ ~o~espo~d

to S * T d~~ity

and the various connections

on the

sides of the cube are limits.

6.3, IIj?‘) free orbifold of type II on K3 and its heterotic dual H&T14 We now turn to another example

of N = 4 four-dimensional

duality, which this time

descends from six-dirn~~sio~al string-string duality. We now wish to construct models with an inte~~iate adage-grout rank. To achieve that we need to project out part of the 16 twisted states of p/22_ This can be done by using a 22 subgroup of the (D4)4 discrete symmetry of the orbifold T4/Z2 [41]) generated by

I-->, p&n> - w>“‘Imn}q

D :

I+)

0:

I+> --+ --l-t-),

t-)

I-}

+ 1-1,

(6.23a) (6.238)

@,n> 4 C-l>“~w)

on each circle, where I&.> denote the two twisted states and Im, n> the untwisted momentum-winding states corresponding to the chosen circle. The operation D can be interpreted as the remnant of a Zz translation on the original circle, carrying one fixed point onto the other. As a first step we will examine the possibility of projecting out one half of the twisted states and obtain an 50(6,14) model. Stying from the T”/& x r2 orbifold blocks, r4,4[;] Q*//27112, we mod out a further Z2, which acts as a shift on the two-torus, and as the D-operation the partition

described

function

above on the l‘“/Zz. The (6,6)

conformal

blocks entering

(4”3) now read

(6.24)

In this expression, (h, g) refer to the original twist while ( h’, g’) refer to the D-shift. According to the definition of the latter (see Eq. (6.23) ), the (4,4) orbifold blocks possess the following properties: for (h, g) + (O,O), I’d.4 [$a = r4,4 [$!I = r4,4[!] (ordinary winding finally, g f g’ insertion -1.

twist);

r4,4 [$

is a (4,4)

lattice

sum with one shifted

momentum

(or

if B is used instead of D), analogous to the (2,2) constructions of Appendix C; (4,4) orbifold blocks with (?z,g) + (O,O), (~‘~~~) f (~~0) and Fz # A’ or vanish because the trace is perform over the origins twisted states with the of an operator under which half of the states have eigenvalue +1 and the others

454

We can now proceed (0,O)

our orbifold

and (B.20)

to the computation

of the helicity

blocks are of the form (B.18).

supertraces.

For (h,g)

#

We therefore use the results (B.19)

and find

(6.25a)

(6.25b)

We note again that the infrared behaviors of B4 and Bg are in agreement with the massless content of the model, namely I s~~ergravity mult~plet and 14 vector m~ltiplets (of which 8 are twisted), The gravitational threshold corrections follow from Eq. (6.5), ITi$2)(~)

: d,,(T)

= -2410g

(T2 12qi(T)l /q(T)i’)

+ const. ;

(6.26)

here the index i depends on the choice of shift vector w (see Table Cl ). As in Subsection 62, this model is guaranteed to have a heterotic dual obtained

by

translating the (D4)4 action on the beter~tic side, at the co~es~ondi~g point in rnodu~~ space. This symmetry is likely to be non-pert~rbative again. However, in this case it is possible to construct a perturbative heterotic dual with the correct rank 14, which we will denote by HET14( w). We consider

the decomposition

of the Z1~,22lattice according

to J-6,22 = J-53

G3 1”1,1 @ f0.x

@ h3

(4.27)

?

where the last two terms give Ep,x &. The o~rat~on that reduces the rank acts as an exchange of the two ro.8 lattices coupled with a translation in 1”1*1,thereby reducing EE: x &3 to its diagonal

level-2 subgroup. I4

Again, the heterotic non-perturbative threshold is obtained by exchanging (for lattice shift corresponding to i = 4 in Eq, (6.26) ), and reads HETI~{~~,I~,III} : ~~~(~~ = -24 log (2& j&(ZS)/

/71(2S) 1’) -+-const.

T with 2s

(6.28)

The above expression exhibits the correct tree-level heterotic ~o~tri~~t~on and the breaking of S-duality by instanton effects. 6.4. IIi:*2’ free orb&old uf type II ou K3 and its heterotic dual NET~o The method presented in the previous section can be slightly modified so that the original twisted sector of the p/Z? is left with one quarter of the states only. The model obtained in this way will have rank 16 and 50(6, ~~~~~~~~6) x SO{ IO]) moduli space. I4 A rank-14 heterotic model has also been constructed

in

I 391 I

Starting from the orbifold blocks (6.24) of the X)(6,14) model, we perform an extra 22, which acts on the (4,4) part as a D-operation along another circle (see Eq_ (6.23) ), while it amounts to a further shift on the (2,2) with respect to some momentum-winding direction. In other words, we perform a 2; x Z2 on the original p/Z2 x T2 const~~tj~~. Tbe result for the (6,6) blocks is

where F2wl,liw2 [“,:pij are the 22 x 22 freely acting constructions explained in Appendix C and ~~,~[~~~~~~~I are ~rbifold blocks whose Lou-v~ishing components are the following: for (h, g) Y (~,O) ~~,~[~~~ = ~~,~~~~~~] = ~4,4~~~~] = r4,4 shifted lattice sum co~esponding to twist) ; f4,4 [yg: :z] , which is an ordinary (4,4) a freely acting Z2 x Z2, analogous to the ones studied in Appendix C for the (2,2) lattices. The precise structure of the latter plays no role for the computation of helicity supertraces, since only the (h, g) # (0,O) blocks contribute to gravitational thresholds. By using the results (C.23), these blocks are recast as

(6.30)

for (kg)

f

(O,O).

This expression, combined with Eqs. (B. 18)) (B. 19) and (B.20)) therefore leads to the following helicity super&aces:

(6.31)

c PVE(Wl,W2,WI2}

x’(l+ (fk?)

;ReH

[I) [I) ,”

rT2 i

N F.

(6.32)

The leading i~fr~ed b~baviou~s reflect the presence of 10 vector m~~tiplets, 6 untwisted and 4 twisted, as ex acted by construction. The gravitational thresholds are determined as usual: p’

10

(W(i),(ii),(iii)

)

: &r(T)

= -1810g (T2 (fii(r)j2 1q(7’)i2) +const.,

(6.33)

where i = 4,2,3 respectively for the 22 x 22 shifted models (i) , (ii) and (iii} in Table C.2. yodels (iv), (v) and (vi) lead, on the other hand, to the result p’ 10

(w(iv),Cv>,(vil

):

A,rU)

=

-18 log (T2 Iv(T) I”> +

CXNUt.

(6.34)

It is remarkable

that this threshold

one should refrain from concluding

is invariant

under the full 5X(2, Z),

that the SL(2, Z),

symmetry

duality,

is restored,

but

since the

breaking may appear in quantities other than R* thresholds. In order to construct the heterotic dual with rank IO, we consider the SO( 8) x SO( 8) decom~Qsitio~ of eae EB and the decomposition of r,,6 into r,,, 83 r[t/ This lattice has an enhanced SO(S) x So’ x SUM’ x S~(8)“~ symmetry point,“’ from which we can switch on two discrete Wilson lines, which act iode~eudeut~y with exchange and shift as for rank 14. We then perform two 22 orbifolds, the first one exchanging SO(8) x SO(S)” with SO(S)” x X)(8>” while shifting the r{:1), and the second one excban~ing J-l:;. The rernai~i~~

(8) x SO($)‘”

with Sag’

x So”’

while shifting

the

gau

ymmetry is SO(S) at level 4. Again, identifying the precise bete~utic dual wo~i~ r~~~lre knowing the point in ~eterotic muduii space eo~esp~~di~~ a piece of insolation that is lacking at the duality maps T to 4X The other cases i = 2,3 are obtained by applying T-duality on the type IT side, yielding T --+ - l/4$ T -+ 4S - 1, respectively. We therefore conclude that the exact gravitational threshold in h&erotic variables reads

These results indeed are in agreement with the fact that the heterotic dilaton should co~~s~ond to the volume form of the base of the ~3-~bratio~, which in this case is T2/(&

x

22). 9 in

the case of models (iv),

(v)

(vi) biding

to (6.34),

the co~~ct

heterotic side is only obtained

contradiction

y substituting 7” -+ 2S, in ~pp~e~t with tbe fact that we have a Z2 x 22 ~rbifo~d. This is due to the ~~tie~~a~

translation

on T2 used to obtain this models: one 22 acts as a translation

momenta,

which are mapped

under type II-heterotic

duality

on the electric

to the electric momenta

on the beterotic side 6221. The second 22 acts instead on the electric windings, which are mapped to the ma netic momenta on tbe heterolic side, so it is not visible in the beter~t~c ~ert~rbativ~ tbeory~ from the beter~tic point of view there is only one Zz. The correct map is therefore T --+ 25, and we obtain the threshold

HETIo

(W(iv).(vi.(vi)) : &r(s)

= -1810g

(2&

lq(2S)j”)

+ const.

(636)

A. Gregori et al./Nuclear

Physics B 510 (1998) 423-476

451

7. Conclusions We have considered the threshold corrections to low-energy R2 and other fourderivative couplings in heterotic and type II ground states with 16 unbroken supercharges. In particular,

we have discussed

of type II vacua that have spontaneously sive gravitinos

in the perturbative

models with spontaneously

the ordinary

K3 compactification

broken N = 8 -+ 4 supersymmetry

spectrum.

and a family and 4 mas-

Those are special cases of more general

broken supersymmetry

studied in [ 21-231.

We have argued that there are no perturbative or non-perturbative corrections to the R2 couplings in heterotic ground states in dimension higher than four. In four dimensions, instanton

corrections

are expected from the heterotic Euclidean

5-brane, and they depend

on the S field only. In type II ground states with (2,2) supersymmetry we have argued that there are no non-perturbative corrections to the R2 couplings in four dimensions or more. The full result arises from one loop. We have first analysed this threshold in six dimensions, which provides a guide on what to expect in lower dimensions. We have subsequently evaluated this one-loop threshold for several (2,2) four-dimensional models with various numbers of massless vector multiplets. All such ground states have heterotic duals, and the type II result translates into 5brane instanton corrections from the heterotic point of view. Most reduced-rank models have an Olive-Montonen duality group that is a subgroup of SL( 2, Z)S, namely r(2)~, which is reflected in the behaviour of the non-perturbative corrections. The above non-perturbative results should provide a guideline towards the determination of the rules for calculating instanton corrections in string theory. Several steps in this direction were recently taken [8,25,33,34,43-481. The ultimate goal is to be able to handle non-perturbative effects with less supersymmetry or in its absence. We have also analysed the CP-odd R2 four-dimensional couplings, and resolved an apparent puzzle: the type II result implies, via duality, a CP-even coupling at one loop on the heterotic side between the antisymmetric tensor and the gravitational Chern-Simons form. We have shown that this is compatible with heterotic perturbation theory since such a coupling is invisible in on-shell amplitudes. Finally we have considered type II dual pairs with 16 supercharges and (2,2) or (4,O) supersymmetry. The situation there is analogous to the type II-heterotic case. By using some additional perturbative relationships, we find quartets of dual models, one of which is a heterotic ground state, with N = 4 in four dimensions, and which, at strong coupling, exhibits enhanced N = 8 supersymmetry! The interpretation of this ground state is a spontaneously broken N = 8 --f 4 theory, with 4 solitonic massive gravitinos that become massless at strong coupling, enhancing the supersymmetry to N = 8. We believe this possibility to be valuable for constructing interesting models with less supersymmetry and an N = 8 high-energy behaviour.

A. Gregori et al. /Nuclear Physics B 510 (1998) 423-476

458

Acknowledgements We would like to thank I. Antoniadis and R. Woodard for helpful discussions. N.A.O. acknowledges the Niels Bohr Institute for hospitality. The work of C.K. was supported by the TMR contract ERB-4061 -PL-95-0789,

and that of E.K. and P.M.P. by the contract

TMR-ERBFMRXCT96-0090.

Appendix A. Kinematics

and on-shell field theory vertices

Throughout this paper, we use a d-dimensional metric gcL,, with signature ( +, -, -, . . . ). We evaluate the leading fourth order in momenta scattering amplitudes of gravitational particles in six dimensions, together with moduli in four dimensions. Particles are characterized by their light-like momentum pi and (except for the moduli) their transverse (i.e. piei = 0) polarization tensors Ei. The latter are symmetric for gravitons h, antisymmetric

for antisymmetric tensors b and pure trace for dilatons @. By the latter we mean a polarization efiv = (g,, - pPk, - k,p, ) , where k is an auxiliary vector such that k . p = 1. We let pi = k 1 according to whether Ei is symmetric (h, 0) or antisymmetric

(b). All amplitudes

pi --+ li + pi @ li + pi li @ pi,

exhibit the gauge invariance

where gi is the transverse (i.e. pi . li = 0) infinitesimal gauge transformation parameter in momentum space (different for each particle). These gauge symmetries correspond to general covariance for gravitons, gauge invariance for antisymmetric tensors, and k-arbitrariness for dilatons. Therefore, k drops out of all amplitudes involving dilatons, and can safely be set to zero so long as one imposes the correct Tr E = 2 for the dilaton polarization tensor (as is obvious in light-cone gauge). Whenever possible, we omit Lorentz indices and implicitly contract indices from left to right, for example (A.la) PI AP2 A Pl 62 AP2El AC3 =

lAfivpmPI

where we define the CP-odd antisymmetric

P2 PIKE2

Levi-Civita

and Eci2345 = +J-g in four and six dimensions, forms is such that A = ApV..., dXp A dX” A . . . A dA = &A,..., dX”AdXpA...AdXP. First quantized string perturbation theory forces and we systematically impose, in the three-particle Pi’P.j=PiE~=Pl

This drastically PlE2P3 =

+P2+p3

of independent

(A.lb)

tensor E such that ~0123= +fi

respectively. Our convention for IIdXP. The exterior derivative acts as us to restrict to on-shell scattering case:

amplitudes,

(A.2)

CO.

reduces the number -Pl

KPp2[e,%3cT )

kinematic

e2PI = -P3E2P3 =P3EZPl =P2PlE2P3.

structures,

for instance (-4.3)

A. Gregori et d/Nuclear

In several inst~c~s

Physics 3 510 (1998) 423-476

459

we reduce the product of two CP-odd Levi-Civita

terms using the ~~~kowski~

tensors in CP-even

identity

where the sum runs over the d! permutations

cr of d elements with signature T(U) = 3~1.

When some indices are already contracted on the left-hand side, one can significantly reduce the number of terms in the sum by using the ~i~kowskia~ I6 identities:

The four-derivative low-energy effective action is obtained by finding Zlorentz invariants that will induce the same interactions of the massless spectrum as those given by the string amplitude. Three-particle interactions have the simplification that no fieldtheory subtraction is required, and the field-theory vertex has to match the precise string amplitudes This is also the case in the four-p~ticle inter~tio~s we are considering. The field theory vertices are obtained by expanding the Lorentz i~v~ia~t around flat backgrounds variables

g,, = qc,, + hpy, B,, = 0 + bpu, @ = &j + SQi, going to momentum

&d-a = and imposing

.Id”p* (21r)4

on-spell

(p)e4J”

(A.61

P

conditions.

To order h2, the Riemann

-(a

space

tensor with cova~iant indices

t--f p> - (Y

H

becomes

8) + U%Pl +-+(Ydm

(A.7)

so that

x (PI~~_FPZ)P~)

” The minus sign has to be omitted in Euclidean

space.

(pzh(PI

MP3)P2)

.

(A.81

The first term vanishes

on shell because

evant for two-graviton-one-modulus dependent

as in Eq. (4.2).

r~~roduc~~~ Bq_ (D.8). Bqs. (3.12)

H&city

of momentum

scattering

conservation,

when the coefficient

The second term induces a three-~raviton

The same kind of manipulations

but becomes

rel-

of R2 is moduliamplitude:

yield the other vertices

in

and (4.2).

sup~rtra~es are defined as

B&$-Strh

2s

)

(B.1)

where A stands for the physical four-dimensional helicity. In models with N = 4 supersymmetry, B2 vanishes (this is responsible for the vanishing of the ane-loop corrections to two-derivative terms in the effective actjon)~ B4 receives contributions from short representations only, w ile & receives also contributions from intermediate ones. This Fro~erty can be proved by computing B4

(supergravity)

=3,

supertra~~s for individ~~

&~(vectw)

~~(S’)=~~2j+1)(-1)2j, I34 (L-q

supe~ultiplets:

= -I_,

(3.2a)

B4 (1,‘)

= 0 )

=o;

(R2b)

(B.3a) Bfj

(si) = -f(Zj+

&5 (Lj) =o,

l)‘(-l)*j,

86 (lj)

= 7(2j-+.

l)(-1)2.i”

) (B.3b)

where Si, f.i, LJ are the short, i~terrn~d~~t~ and long ~epr~se~tat~~~s, respectively. In the framework of string theory, the physical four-dimensional helicity is h = ht -I- AR, where AL,~ are the contributions to the helicity from the left- (right-)movers, We introduce the helicity-generating function as

where the prime over the trace excludes the zero-modes related to the space-time coordinates ~co~seq~ent~~ Z(U, ii)j,.+~ = 722). At t e ~~rturbativ~ level, helicity supertraces are ohtained by taking appropriate derivatives of (B.4), using

A. Gregori et al./Nuclear

461

Physics B 510 (1998) 423-476

(B.5) In this paper we are mostly interested Zz-orbifold

type, for which the partition

in N = 4 type II four-dimensional function

models of the

(4.3) results into a helicity-generating

function ”

where (B.7)

counts the helicity contributions of the space-time bosonic oscillators. Owing to the (2,2) supersymmetry of our models, t* the first non-trivial supertraces B, =

can be computed

by using the following

( (A,+A,)4) =6(A[A;)

In the rest of this appendix,

=&

helicity

formulas:

&$Z(u,D)I+, ,

we collect some of the identities

(B.Sa)

involving

6-functions,

which are useful for these computations. Our conventions for the 8-functions are cu)

=

Ce~i,(p+;)‘+2,r(u+4)

(p+q)

(B.9)

PEZ so that I7we use the short-hand notation 6 [z] (L’) for 19 [g] (~17). ‘s In situations where N = 4 supersymmetry is realized as (4,0) (see e.g. the model IIT) amplitude given in (6.7) ), formulas (B.8) get modified as follows:

with vacuum

m

I

lsbd

t

31c-4

)c( ds /---.

463

A. Gregori ef u~.~~uc~e~r Physics B 510 (1998) 423-476

Here r4,4 [i] are the ordinary are generic

blocks.

used together

&-twisted

(4,4)

For these models,

with the definitions

lattice sums (see (3.2) >, whereas 22,~ [:I]

the above identities

(B.8)

(B. 16) and (B.17)

and the helicity-generating

function

can be (B.6)

to

obtain finally

where the functions

E?[i] are given by

iB.21)

Notice also the property (B-22)

Appendix

C. r&z lattice sums and fundamental-domain

In this appendix

we give our notation and conventions

integrals for the usual (2,2)

and shifted

(22) lattice sums used in the text. We also give the explicit results for the relevant ~~~dame~tal-domain integrals of these lattice sums. The (2,2) lattice sum is given by r2,2v?

w

=

=-

&J:

41 Pi

l!ci hPR)Eh T2

72

-2riTdet

eXP lz AEGLCZZ)

CC.11

A -

(

where Il/nai - m2 +Tn’ PZ =

27.2 U2

-t-iTh~~~~

9

pt -pi

(mn stands for mln’). In terms of the background momenta can be written as

=2mn

(C.2)

fields GJJ and B~J, the left and right

464

N’ = GiJ

(P~J- BJK#)

,

(C.4)

so that

The matrix identities

{D.32a),

tion of G,J and

in terms of the moduli 7’ and U: I9

The relation

BJJ

(D.32~)

(D.32b)

follow, after some algebra, using the parametrize-

follows from the defection

which may be derived from (C.5).

Finally,

of the fattice sum and the identity,

the relevant

fundamental-domain

integral

is [49]

The subtraction of the massless-states regularizing the logarithmic divergence, 7: u. The Zz-shifted

(2,2)

contribution in this integral is necessary and results in a non-harmonic dependence

for on

g take the values 0 or 1. Were, w denotes

the

lattice sums are

where the shifts h and projections

shift vector with components (at, ~22,b”, b”), and C introduced the inner product ** .e.w=mb+an,

w2=2ab9

(ml, m2, d, n2). We have also

(C.10)

so that a! generates a w~~d~~g shift in the J d~~ectio~~ where~ 6’ shifts the Ith moment turn. The vector -!?is associated with the rz.2 lattice and therefore the vector associated with the shifted lattice will be

p=l+W+ 2’ With these conventions,

(C.11) the left and right momenta

read

ly When T1 = Ut = 0, the usual ~aram~t~zati~~ is T2 = RI&, U2 = R2/Rg7 where I$ ;L~Zthe radii of compactification. 2”For ~1 = (~~1,211) and ~‘2 = (az,bz), the inner product is defined as ~1 +w2= al& + azbl.

p&+2

( mI =+-qh 2)(n1+~~~).

It is easy to cheek the p~riodicity

(C.12b)

properties

(h, g integers) (C.13)

as well as the modular

transformations

that the expression (6.14)

obeys

7--t--: The relevant parameter A-

W2 2

vi

+e

$! !$

11 h

z2+r2

(C.15a)

h+g’

’

(C.15b)

for these transformations

is

=ab.

(C.16)

From expressions (G.9) we learn that the integers ai and b’ are defined mod&o 2, in the sense that adding 2 to any one of them amounts at most to a change of sign in Z’& [:I. Such a modi~catio~

is ~e~essarily

~ompensat~

by an a~~~oFriate one in the

rest of the partition function in order to ensure modular invariance: we are thus left with the same model. On the other hand, adding 2 to aI or b’ translates into adding a multiple

of 2 to A. Therefore,

although

A. can be any integer, only A = 0 and A = 1

correspond to truly different situations. We now would like to discuss the issue of target-space

duality in these models, where

the 22 orb~fold acts as a translation in one complex plane. The rnodu~~ d~pende~ce of the two-torus shifted sectors (see Eq, (C.9) ) reduces in general the d~~ity group to some subgroup21 of SL(2, Z), x SL(2, Z), x .Zz rMU. Transformations that do not belong to this subgroup map a model w to some other model w’ leaving invariant, however, h =d 2=$. This means in particular that for a given model, decompactification limits that are related by transformations that do not belong to the actual duality group are no longer equivalent. 2i The sa~gr~~Fs

of SL(“L.Z)

that will actually

appear

in the sequel are F’*(2)

and T(2).

a c

If (

represents an element of the modular group, f+(Z) is defined by CL,d odd and b even, while for fhave (1, d odd and c even. Their intersection is r(2).

h d

> (2) we

466

To be more specific, by using expression properties

(6.9))

we can determine

of f,& [i] under the full group SL(2, Z),

x SL(2, Z),

the transformation x ZT”“:

:

SL(2,Z)r

(C.17a)

SL(2, Z),

:

(3)4(-i

-;

;;

;j

(~),

aldr-b’cf=l

(C.17b) and

(C.17c)

Thus, we can determine components

the duality

group for a given model by demanding

that the

of the vectors a and b remain

A = 0 situation defined turns out to be r’(2)r

invariant module 2. For example, in the by a = (0,O) and b = ( l,O), the target-space duality group x r-(2)11, whereas for the case with h = 1 and a = (l,O),

b = (l,O), we find 1”(2)r x f(2)u x Zr”“‘. At this point, we would like to mention a remarkable identity, which plays a role in the computation of fundamental-domain integrals, as well as in the identi~cation of several type II const~~ct~o~s (see subsection shifted lattice sums, one checks easily that 1

c’

2 h',g'=OC-1)

6.2). stating

hg’-f&?h’ p

from the definition

CT,u)

22

(C.9) of

(C.18)

for any shift vector such that & = 0. The precise relation between (T’, U’) and (z Ur) depends on the specific shift vector W, and is presented in the Table C.1 for all distinct A = 0 situations. After Poisson resummation in ml, m2, the shifted lattice sum (C.9) takes the alternative form h

qg9

w

[1 z-

g

T2

” x

exp -ni

E

W2

-4-_hg - a (gn - hm) >

A

-2&T

exp

det A - ~lil

u)A(;)12

i where the summation

is performed

over the set of matrices of the form

~C.19)

Table Cl The nine physically distinct models with A = 0 Case

Q

I 11 III

IV V VE

b

T’

u’

-- 2

--

--

-c

7 ii

1

w fi

-_ 2 T 1

-zF

e _- 2 u

I

1

i

j

4

2

4

4

4

3

2

4

2

2

-??:

-57

-+

WL I-U

2

3

3

4

VII

&g

I-T

-- 2 u

VIII

E

-57

3

2

.&IL 1-U

3

3

IX

u 1-T

1

(C.20) Modular-invariant d~ent~ domain

combinations of blocks S&(X U) [z] can be integrated over the funby dec~m~sing the set of matrices A with respect to orbits of the

rno~~l~ group. In this paper, we are rn~n~y interests in the case A = (2,22 for which the relevant integrals can be obtained from (C.8) by using (C.18) together with

As a result,

where the relation between the shift vector w = (a,b) and the gairs (i*j) is taken from Table C.1. In the co~st~~tio~ of ~~uc~-r~k models of Section 6, we i~~~d~~ shifted (2,2) lattices where the free action is of the type 22 x 22. Each of the 22’s acts according to the above analysis on a given set of momenta and windings. Consistency of the 22 x & action demands that the intersection of these two sets be empty. In other words, the co~es~nding shift vectors wr and w2 must satisfy wr - w2 = 0. Notice that the union of these sets corresponds to the action of the diagonal 22. The lattice sum will be denoted f ] and we have in particular 22Hcterotic constructions with A = 1 can be found in [ 231.

Table C.2 The six physically

distinct models with wi

wj = 0 Vi,j = I,2 WI

w2

(i)

f&O, I,Q9

(Q,O,O,1)

(ii) (iii)

~I,Q~~,~~ t1,o,o, 13 (1,0,&O) to,o, 190) (O,O, 191)

(0%i,o,s): (0,--l, 1,O) (Q,O,O, 11 (0,1,0,0) il,-l,O,O)

CZlSi2

(iv) (VI (vi)

where ~12 zz ~1 + w2 reflects the action of the diagonal Zz. As an example, consider the situation where wr = (O,O,l,O),

w2 = (O,O,O,l)

and

therefore wr2 = (O,O, 1) 1). In that case, the first (resp. second) 22 shifts the momenta of the first (resp. second) plane (insertion of (- I )“I’ (resp. ( - 1)“” )), while the diagonal 22 amounts to insertion (-I) mifnrz_The lattice sum now reads

from which Eqs. (C,23) are immediately checked. In the framework of ~~bs~tion 6.4, the requirement

of module

inv~iance

implies

that w: = w$ = wf2 = 0. This reduces the number of distinct possibilities to the six listed in Table C.2. The first of these corresponds to the example whose lattice sum is given in Eq. (C.24).

In this section

we compute

in great detail the stringy scattering

amplitude

(3.5)

of

three gravitons (or two-forms) for type II superstring on K3, and subsequently the scattering amplitude C4.4) of two gravitons (or two-forms or dilatons) with moduli of T2 for type II on K3 x T’.

For these computations

we use the following

contraction

formulae:

(X~(~,Z)X~(O))

=gp’A(Z,z)

where the Greek space-time

= -g@“log

(D.la)

indices run from 0 to 5 (3) in the six- (four-) dimensional

case, and the indices I, J run on the two compactified directions of T2. As in Appendix C, PL,R denote the left- and right-moving momenta of T2. A few remarks about these equations are in order. Eq. (D.la) gives the propagator of a non-compact boson on the space of non-zero-modes. Eq. (D. lb) omits a delta function singul~ity, which has to be subtracted for tree-level fa~tor~~t~on. The first term in (D.ld) is the co~trjb~tio~ of the winding zero-modes of a compact boson written in Hamiltonian representation, to be added to the non-zero-mode contribution {D. lb}. Eq. (D.le) holds only for even spin structures where the world-sheet fermions do not have any zero-modes. In the odd spin structure, there is one zero-mode for each space-time or T* fermion, a total of six in both the six- and four-dimensional cases. These zero-modes have to be saturated in order to give a non-vanishing result, and we no~ali~e them as ~*~~~~K~~~~~~~

= 8LYKAPuin six Dimensions,

(D.2a)

(@‘@‘~,VI+$”$I@)

= P’KAe’J

(D.2b)

where gt2 = -E*’ = I/&. replacement

in the partition

in four dimensions,

Saturation

of the zero-modes

at the same time induces the

function

We also need the integrated

propagators

on the torus: (D.4a) (D.4b)

where (a, b) is an even spin structure.

We normalize

the measure

of integration

on

vertex positions as s d2z/r2 = 1. Expressions analogous to (II. 1f,c) and (D.4b) for the left side follow by complex conjugation. Useful Riemann identities for the summing of spin structures

are assembled

in Appendix

B.

Here we wish to evaluate the amplitude (35) and derive the corresponding fourderivative terms in the effective action We need to distinguish according to the spin structures

on both sides.

D.2, 1. CP-even z-e In this sector we need to compute A+‘={

(&X/2(f~iz,)

+@I

the ~u~elation

function

,~(~~~~~~~,})

The Riemann identity (B.13) shows that at least two pairs of fermions must be contracted together on both sides, since ~o~tr~b~tio~s with less fe~~o~~c contraptions vanish after a sum on even spin structures. Each fermion pair comes wit one power of momentum; we therefore

have to be chosen

need precisely

two such ~o~tractions.23

in two different

The two pairs of fermions

vertices on both sides, since the polarizations

are

traceless: A:~;:

deriv = Csy&X%XV)

(1’2 .J(~22,~"(~~,)(P~.~(~~~~~t~*~)

x (pi *9(ii)S"lz:))C~2."t(Z2)31;~(~lf)+perm.

ID.6)

Maying use of Eq. (D. I f} and of the ~~ern~~~ identities in Ap endix B, it can be shown that the ~~~~~~~d~ after summation over spin stru~tures~ no longer depends on the position of the vertices, so that we can apply Eq. (D.4) to obtain

‘? It is known that those singularities arising when two vertices come together can yield poles U2( I/{pipj 11 that can cancel against s~~-~~~v~tiv~ terms to yield 0(p4) terms [SO] _ We evaluated these co~~bat~ons and found a precise caace~~at~on of the coKespoadja~ terms, in ag~e~e~t with the ~~~ectat~on that these term reproduce field-theory subtractions that are absent in the case at. band.

471

Here 211stands for the unintegrated kinematic

partition

function

in Eq. (3.1).

We also defined the

structure

?+'=

(plEZp3)(p241%p2)

+ 5 pem.

It can easily be shown that

so that the amplitude is non-vanishing only for three gravitons or for two antisymmetric tensors and one graviton, Identity (B.14) shows that the untwisted sector (h,g) = (0,O) does not contribute, and we use the explicit expression (3.2) for the twisted f4.4 [t] to obtain (D.10)

where we also used the standard modular-invariant integral JF 6%/r: = 7r/3. Comparing Eq. (D.8) with Eq. (3.12) we find that the three-gravitonand onegraviton-two-two-forms

in the Z-e

spin structure

can be described

by the following

vertex in the effective action: fD.11) O-2.2. CP-even S-0 In this sector the correlation #i--o =

function

~~X~(~~,~~~ + @i ,~~~~)~~(~~~)

0

x~~X~(~~,~,~

+@I

x (J2XKGT2,z2) +@2 x

we have to compute is modified to

(&X”(&,

.Jl(~i)~~(~~~)e~~‘.~(‘l,~~) ~~~~2~~K(&~)

~2) + ip2 - t+4(z*)+‘“( z~))ciL’2’x(tz3z2)

x p(~~:3)#fQ(Z&

~~~~~~“~-~,,x~(o)~~(o}~x~(o}~~(o)),

(D.12)

where we have used for the third vertex operator the -l-picture on both the left and the right sides, and inserted the left- and right-moving superc~~ents. All fe~ions have to be contracted

in order to saturate zero-modes,

and the remaining

two supercurrents does not yield any singular contribution of Levi-Civita tensors, We are therefore left with the following term:

dXCYaXP from the

because of the antisymmetry

(D.13) obtained by using Bq @.2a>. Then using Bq. (D.3) on both the left and the right sides, the integrated three-point amplitude becomes, after some algebra,

p-0

_

p-0

s

d% 7; 1 1

72yr3---

F

x (-4~‘)

x (--1)/“4rr2

2

= -&&&-“-”

x E x 16x 3 72

2

(D.14)

1

where we have defined t

Expanding

the product

of the two CP-odd

Levi-Civita

tensors in terms of the metric

in Eq. (A. I ) and comparing Eqs. (D.8) and (D. 15), we can show that, without any assumption on the symmetry properties of the polarization tensors, I’“-”

Therefore,

(D.16)

= - Cpr E@3) (p2EI E3pz> + 5 perm.

the O-O spin structure

yields exactly the same interactions

as the Z-e

one,

but with a sign depending on whether we are in type IIA or IIB. Hence, we record for the corresponding term in the effective action (D-17)

D.2.3. CP-odd We first work out the result in the sector GO,

in which we need to compute

the

cot-relator A+-“=

0

&XILc(&,zl)

+iPg .Ic;(z”&@‘(Zt))

x ~~~X~{~~,~~~+ ipI *~fzl)~“(z,))c”~‘.X(“‘,“‘) x (&XKE2g 22) + in .~(~~~~~~~~~) x (&X’(&,

22) + ip2 ~$(z~)@“(z~)j

ei~2.x~i2~Z2)

x (&XPC&, 23) + @3 * $~a)rl;v3)) ~~~~(Z3~~““.X(“,~~)~~~~~~~(0))

where

we have used the -l-picture

inserted the right-rnoyi~g sup~rcu~eat the relevant fuur-derivative term is

,

(0.18)

on the right for the third vertex operator (3.9).

and

Again, no contact terms are involved and

A. Gregori et al. /Nuclear Physics B 510 (1998) 423-476

where in the second step we used Eq. (D.lf) correlators

will contribute

473

and the fact that each of the three fermion

the same amount thanks to translation

invariance.

Using the result (D.19) in (3.5) along with the partition function (3.1), the integrated cot-relator (D.4b) and the replacement (D.3) on the right side, we obtain for the integrated

three-point

function

in this sector

d2r r32 1 1 72 7T3g r2 x (-47~~)

Z?‘-” = 7-F-O s 3

2

x 4?r2 x ;

x 16 x 3 = -32&+‘,

2

(D.20) where we have defined the tensor structure I’-”

= E1fiVE2K~E3po •~apAappl Kp2LL+ EaPPAuKplPp3’ + EapYAu’p2Pp3K plap2p >

(

(D.21)

=~p1Ap2Apl~2Ap2elAe3+perm.

Following

the same steps, and using Eq. (3.7),

it is not difficult to show that in the

other CP-odd sector the result is z”-’

)

= -ep,p2p3z”-0

(D.22)

so that the total result for the CP-odd part of the three-point

function

(3.5)

(D.23)

flP-Odd = - ( 1 - &~t~@s) 32%+-‘-“ . This implies

that we need, for the non-zero

type IIA :

~1~2~3

=

is

couplings:

-1,

(D.24a)

type IIB : pIp2p3 = 1~

(D.24b)

so that in type IIA we need an odd number of antisymmetric tensors and in type IIB an even number. Note also that one cannot construct any CP-odd four-derivative on-shell coupling between three gravitons. Comparing Eq. (D.21) with (3.12), we conclude CP-odd four-derivative gravitational couplings is

0.3.

Two-graviton-N-mod&i

scattering

that the one-loop

correction

to

in four dimensions

Here we evaluate the (leading) four-momentum piece of the ( N + 2)-point amplitude in Eq. (4.4). We first define a set of signs specifying the nature of the moduli:

A. Gregori ef ~~.~~u~~~~r P~ysj&s B SIO ~199~~ 423-476

474

x4 =

1,

-1,

With these notations,

qb=T,U -qb=T,U

O’=

the selection

1, -1, {

QS=T,T i$=u,U.

(D.26)

rules read

Let us first focus on the Z-e case, and first on the left side. If the moduli are chiral, they all have the same vertex dX+ip . @P on the left side; therefore, they can only contribute through the zero-mode pi of dX. If on the other hand modulus i is chiral and modulus j is antichiral, there can a priori be a contraction $Xfz;-, z_~)c??(~,, zj), but this will be a total derivative with respect to pi, unless there is also a con~a~ti~n $X{ Zi, zi)r?X( Z;i, zj) on the right side. But this can only occur if & and & have also opposite vertices on the right side, that is @i = &, a case that we excluded. Therefore, only the zero-modes p&p; of t?X’i?X” contribute.

Moreover,

we must contract

the fermionic

parts of the

graviton-two-form vertices together, since other contractions vanish after the sum over even spin structures, thereby providing four powers of momenta, All in all,

A similar reasoning applies when one of the spin structures is odd and shows that the 2 fermionic zero-modes on Tz have to come from the vertex in the - l-picture together with the T2 piece of the supercurrent, while all other vertices contribute through the bosonic zero-modes. The space-time fe~io~ic zero-modes are then provided by the ~ravi~on or two-fog vertex operators. We find

The Riema~n identity (B.13) allows us to carry out the spin structure summation and shows that the integrand is in fact ~ndepend~~~ of the position of the vertices. In the odd spin structure, the saturation of zero-modes induces the replacement (D.3). We can simplify the kinematic structures by making use of Eq. (A-4), and rewrite them as (recall that p1 . p2 is not restricted to vanish anymore)

A. Gregori et al. /Nuclear I”-’

= E2Pl A El P2 Jl Pl A P2 + (P,P2)

If particle 7”-’

Physics B 510 (1998) 423-476

1 is a dilaton,

= I”-”

tensor (CP-odd)

(D.30)

PI A P2 A El G.

this can be further reduced to ,

fjt-2-o = __Q+”

only couples

to another dilaton

= (p,p2)2k2>

so that one dilaton

475

(D.31)

=-(PIP~)PIAP~AE~,

(CP-even)

at this order. One also notes that the d-o

or to an antisymmetric

contribution

is opposite

to

the 2-e contribution in the two-graviton case, equal in the b2 and a2 cases. We can then make use of the identities (D.32a) (D.32b) (D.32~)

and find the general result (D.33)

where i, j run over the even and odd spin structures. The quantities

t?j and Z’j are defined

in Eqs. (4.12) and (4.13). In the last equation, J& stands for the product of derivatives with respect to the moduli di. For N > 1, these derivatives are actually promoted to modular covariant derivatives due to reducible diagrams that we disregarded. The signs Kij make reference to the modulus in the -l-ghost picture. The various coefficients in Eq. (4.11) are then obtained by comparing the kinematical factors (D.30) to the vertices

(4.2)

and taking proper account of symmetry

weights.

References ] 1) [2] [ 31 141 [ 51 161 [7] [8] [ 91

M.J. Duff and R. Khuri, Nucl. Phys. B 411 (1994) 473, hep-th19305142. C. Hull and P Townsend, Nucl. Phys. B 438 (1995) 109, hep-thI9410167. E. Witten, Nucl. Phys. B 443 (1995) 85, hep-th/9503124. J. Polchinski and E. Witten, Nucl. Phys. B 460 (1995) 525, hep-thI9510169. A. Sen, Nucl. Phys. B 450 (1995) 103, hep-th/9504027. J. Harvey and A. Strominger, Nucl. Phys. B 449 (1995) 535, hep-th19504047.B 458 (1996) 456 (E). C. Vafa and E. Witten, Nucl. Phys. B 447 (1995) 261, hep-th/9505053. J. Harvey and G. Moore, hep-th19610237. A. Font, L. Ibaifez, D. Lust and F. Quevedo, Phys. Lett. B 249 (1990) 35; A. Sen, Int. J. Mod. Phys. A 9 (1994) 3707, hep-th/9402002; Phys. Lett. B 329 (1994) 217, hepth/9402032. ] IO] C. Vafa and E. Witten, hep-thI9507050. [ 1 I] S. Kachru and C. Vafa, Nucl. Phys. B 450 (1995) 69, hep-th/9505105. [ 121 S. Ferrara, J. Harvey, A. Strominger and C. Vafa, Phys. Lett. B 361 (1995) 59, hep-th19505162.

[ 131 A. Klemm, W. Lerche and P Mayr, Phys. Lett. % 357 (1996) 313, hep-th/9506112; S. Kachru, A. Klemm, W. Lerche, F? Mayr and C. Wfa, hep-th/9508155; G. Aldazabal. A. Font, L. Ibafiez and E Quevedo, Nucl. Phys. B 461 ( 1996) 85, hep-th/9510093; Phys. Lett. B 380 ( 1996) 33, hep-th/9602097. [ I4] A. Sen and C. Vafa, NucI. Phys. B 455 (1995) 165, bep-th~95~~~. ] tS] C. Bachas and E. Riritsis, Nucl. Phys. B (Proc. Suppl.) 55 (1997) 194, h~~-th/96I~ZO5” t fS1 1.A~ton~ad~s,E. C&a, KS. Narain and T.R. Taylor, Nucl. Pbys. % 476 ( 1996) 133, he~-th~~6~~77. f 171S. Ferrara and C. Kounnas, Nucl. Phys. B 328 ( 1989) 406. [ IS] 1.Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87. ] 191 H. Kawai, D. Lewellen and S.H.H. Tye, Nucl. Phys. % 288 (1987) 1. 1Xl] P.Horava and E. Witten Nuci. Pbys. B 460 (1996) 506, hep-th/9510209. 12I I E. Kiritsis, C. Kounnas, P.M. Petropoulos and J. Rizos, Phys. Lett. B 385 ( 1996) 87, hep_th/96006087, [ 221 E. Kiritsis and C. Kounnas, hep-th/9703059. 1231 E. Kiritsis, C. Kounnas, PM. Petropoulos and J. Rizos, CERN-TH/97-44 preprint, to appear. 1241 D.J. Gross and J. Sloan, Nucl. Phys. % 29E (1987) 41. [ 251 E. Kiritsis and B. Pioline, he~-tb/97Q?~I~. [ 261 L. Remans, Nucl. Phys. % 276 f 1986) 71. [ 27) D.J. Gross and P.F. Mende, Nucl. Phys. B 291 ( 1987) 653. (281 K. Forger, B.A. Ovrut, S.J. Theisen and D. Waldram, Phys. Len. 3 388 (1996) 512, hep-th1960514S. 1291 E. Kiritsis and C. Kounnas, Nucl. Phys. B 442 (1995) 472, hep-th/9501020; Nucl. Phys. B (Proc. SuppI.) 41 (1995) 331, hep-th/9410212. I 301 E. Kiritsis and C. Kounnas, in Proc. of STRINGS 95, Future Perspectives in String Theory, Los Angeles, CA, 13-18 Mar 1995, hep-th19507051; E. Kiritsis, C. Kounnas, PM. Pe~poulos and J. Rizos, its Proc. of the 5th HeIlenic School and Workshops on Elements Particle Physics, Corfu, Greece, 3-24 Septem~~ 1995, bep-tb/96~501 I. ] 31) P.M.Pe~o~o~Ios and J. Rizos, Phys. Lett. B 374 ( 1996) 49, hep-th~9601~3~; PM. Petropoulos, in Froc. of the 5th Hellenic School and Workshops on Elementary Particle Physics, Corm, Greece, 3-24 September 1995, hep-tb/9605012; E. Kiritsis, C. Kounnas, PM. Petropoulos and J. Rizos, Nud. Phys. 3 483 ( 1997) 141, hep-th/9608034. 1321 M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Nucl. Phys. B 405 (1993) 279; Commun. Math. Phys. 165 (1994) 311. 1331 C. Bachas, C. Fabre, E. Kit&is, N. Obers and P Vanhave, hepth/9707 126. (341 1. Antoniadis, B. Pioline and TX Taylor, hep-tb/97~7222. 13.51 1.Anto~iadis, E. Gaya, KS. Narain and T.R. Taylor, Nucl, Phys. % 407 ( 1993) 706. /36] C. Koun~as and M. Potato, Nucl. Fbys, B 310 ( 1988) 355; S. Ferrara, C. Kounnas, M. Potrati and F. Zwimer, Nucl. Phys. B 318 ( 1989) 75. [h] 1.Antoniadis, Phys. L&t. B 244 ( 1990) 377; I. Antoniadis and C. Kounnas, Phys. Lett. % 261 (1991) 369. 1381 C. Kounnas and B. Rostand, Nucl. Phys. B 341 (1990) 641. I391 S. Chaudhuri, G. Hackney and J.D. Lykken, Phys. Rev. Lett. 75 ( 1995) 2264. ]4O] P.S. Aspinwall, hep-th/9507012. 141J R. Dijkgraaf, E. Verlinde and H. Verhnde, ~omrnu~, Math. Phys. I 15 f 1988) 649. ]42 ] J. Harvey and 6. Moore, hep-th~96~ 1176. 1431 E. W~tten, Nucl. phys. B 474 (11996) 343, be~-t~/96~030. [ 441 H. Ooguri and C. Vafa, Phys. Rev. Lett. 77 ( 1996) 3296, hep-th/960~079. 145 J M. Green and M. Gutperle, Phys. L&t. B 398 (1997) 69, hep-th/9612127; hep_th/9701093; M. Gutperle, hep-th/9705023. 1461 M. Green and P Vanhove, hep-th/9704145; M. Green, M. Gutperle and P. Vanhove, hep-th/9706175. 1471 1. Antoniadis, S. Ferrara, R. Minasian and KS. Narain, hep-th/9707013. 148 1J. Russo and A. Tseythn, he~-~/~707134. [ 49 J L.J. Dixon, VS. Ka~Ianovsky and J. Louis, Nucl. Phys. B 355 f 1991f 649, ] 501 J.A. Mi~aban, Nucl. Phys. B 298 ( 1988) 36.