CERNTH/96153
hepth/9611205
TERMS IN STRING VACUA
C. Bachas and E. Kiritsis
Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau, FRANCE
email:
Theory Division, CERN, CH1211, Geneva 23, SWITZERLAND
email:
ABSTRACT
We discuss terms in torroidal compactifications of typeI and heterotic string theory. We give a simple argument why only short BPS multiplets contribute to these terms at one loop, and verify heterotictypeI duality to this order. Assuming exact duality, we exhibit in the heterotic calculation nonzero terms that are twoloop, threeloop and nonperturbative on the typeI side.
CERNTH/96153
November 1996
To appear in the Proceedings of the Trieste Spring
School and
Workshop in String Theory, April 1996
Introduction. BPS states play a special role in theories with extended () supersymmetry. The fact that they form multiplets which are shorter than the generic representation of the supersymmetry algebra implies relations between their mass, charges and values of moduli which are valid in the exact quantum theory. For these relations are furthermore purely classical, and they ensure that BPS states are either stable or, at worse, marginallyunstable. Stable BPS states can thus be traced all the way to strong coupling, and their existence with appropriate multiplicities has constituted the main test of the various duality conjectures.
Another remarkable feature of BPS states is that they saturate certain oneloop terms in the effective lowenergy action. This fact has been articulated clearly by Harvey and Moore [1] in the context of heterotic thresholds, though it was implicit in much of the earlier work, such as for instance refs. [2]. BPSsaturated terms are furthermore typically related, by supersymmetry, to anomalies, and are thus expected to obey nonrenormalization theorems. This makes them a precious tool for checking duality conjectures. Tseytlin [3] has in particular used such terms, in order to test the conjectured duality between the typeI and heteroticstring theories in ten dimensions [4, 5]. In this paper we will extend Tseytlin’s analysis to torroidal compactifications.
The effective gaugefield action of openstring theory is closely related to the phaseshift or velocitydependent forces between Dbranes [6, 7, 8, 9, 10]. BPS saturation and a nonrenormalization assumption of the leading interaction are, furthermore, a crucial ingredient in the recent interesting conjecture by Banks et al [11] concerning Mtheory in the infinitemomentum frame. Despite their close relation the two calculations differ however in some significant ways. For instance in the effectiveaction calculation one subtracts diagrams with massless closed strings in the intermediate channels. These diagrams must be kept in the Dbrane calculation, where they are regulated effectively by the worldvolume dimensional reduction. Our analysis does not therefore translate into the Dbrane context immediately, but it raises by analogy some interesting questions.
Supertrace formulae. BPS saturation at one loop follows from supertrace formulae [12] involving powers of helicity and Rsymmetry charges. These are easier to discuss in terms of generating functionals. Define for instance
where the supertrace stands for a sum over bosonic minus fermionic states of the representation, and is the eigenvalue of a generator of the little group: or in the massive, respectively massless case in four dimensions. For a particle of spin we have
When tensoring representations the generating functionals get multiplied,
The supertrace of the th power of helicity can be extracted from the generating functional through
Consider now multiplets. The supersymmetry algebra contains four fermionic charges that may act independently: two of them raise the helicity by one half unit, while the other two lower it by the same amount. For the generic massive (long) multiplet all charges act nontrivially on some “ground state” of spin and one finds
For a massless or a short massive multiplet half of the supercharges have a trivial action so that one finds instead
where the factor 2 is required in the massive case, since short massive multiplets carry charge and are thus necessarily complex. Familiar examples of short massive multiplets include the monopoles () and charged gauge bosons () of pure N=2 YangMills theories. An immediate consequence of eqs. (5) is that only for short (BPS) multiplets is .
Let us turn next to the algebra. This has four raising and four lowering fermionic charges, all of which can act independently in a generic massive (long) representation,
Short representations, which include all the massless as well as some massive multiplets, annihilate half the supercharges so that
This is precisely the content of a long representation. has also intermediate (or semilong) multiplets, which annihilate onequarter of supercharges, and for which
The factors of two take again into account that massive short and intermediate multiplets have charge and are thus necessarily complex. It follows trivially from the above expressions that always, only for short multiplets, and in both the short and the intermediate case.
The discussion can be extended easily to take into account Rsymmetry charges. These are simply helicities in some (implicit) internal dimensions: there is a single Rcharge for , and three independent charges, corresponding to the Cartan generators of , in the case. To get a nonzero result for short, intermediate or long multiplets in the latter case, one must insert in the supertrace at least four, six, respectively eight powers of helicity and/or Rcharges.
TypeI effective action. Let us turn now to the oneloop calculation of the effective gaugefield action in typeI theory. In the backgroundfield method the oneloop free energy in noncompact dimensions reads [13]
where is a background magnetic field pointing in some direction in group space, is the corresponding charge distributed between the two string endpoints, and is a nonlinear function of the charges and the field that vanishes linearly with the latter
In the weakfield limit and for low spins this is a familiar fieldtheory expression: it follows directly from the fact that elementary charged particles have gyromagnetic ratio 2 and a spectrum given by equallyspaced Landau levels. The effects of nonminimal coupling for an open string are captured essentially by the replacement .
The supertrace in eq. (7) runs over all charged string states. For any given supermultiplet the mass and charges are however common, so that its contribution is proportional to
where we have used the fact that odd powers of helicity trace out automatically to zero. Since the expansion is an expansion in weakfield, the various nonrenormalization statements at one loop follow directly from the properties of helicity supertraces and eqs. (7,9). Thus in theories the first nonzero term, proportional to , is the oneloop gauge kinetic function: it only receives contributions from short (BPS) multiplets, as has been noted previously by using identities of functions [15, 14]. In theories the gauge coupling constant is not corrected at one loop. The first nonzero term, proportional to is quartic in the background field, and only receives contributions from short multiplets. This was noted again through function identities in the Dbrane context in refs. [10, 8]. The following term of order is also determined, incidentally, by short BPS states. This is because long multiplets do not contribute to , and there are no intermediate multiplets in the perturbative typeI spectrum.
Let us take now a closer look at the quartic term arising in N=4 (torroidal) compactifications. The only perturbative charged BPS states are the multiplets of the gauge bosons, together with all their KaluzaKlein descendants. For these states the mass is equal to the internal momentum, so that after some straightforward algebra one finds
where stands for the dimensional lattice of KaluzaKlein momenta, which must be shifted from the origin in the presence of nonvanishing Wilson lines. Each endpoint charge takes 32 values, but the sum runs only over antisymmetric states. For ease of notation we will from now on suppress the terms when writing effective actions.
The above expression is strictlyspeaking formal, since it diverges at the limit of integration. This is an openstring ultraviolet divergence, but can be also interpreted as coming from an onshell dilaton or graviton that propagates between two nonvanishing tadpoles. We are interested in the effective (Ẅilsonian)̈ action, so this divergence due to exchange of massless particles must be subtracted away. The right procedure is to change variables to the closedstring proper time , which is related to differently for the annulus and Möbiusstrip contributions,
Separating the two topologies amounts to writing the sum over ChanPatton states as an unconstrained sum over all left and all right endpoints, minus the diagonal. After performing also a Poisson resummation the result reads
where is now the compactification lattice. Our conventions are such that for a circle of radius .
The divergence in the above expression comes from the piece, as all other terms are exponentiallysmall in the region. Thanks to the factor that multiplies the Möbiusstrip contribution, this divergence is proportional to . It corresponds precisely to the tadpole masslesspropagator tadpole diagram, that must be removed in the effective action [14]. Switchingoff the Wilsonlines for simplicity, and changing integration variable once again for the Möbius contribution, we thus obtain our final expression
Since in the decompactification limit all terms disappear, we have just shown in particular that the 10d effective typeI Lagrangian has no oneloop corrections. This is in agreement with 10d heterotictypeI duality [3] as we will discuss in detail in the following section. The fact that only open BPS states contribute to the amplitude is in this respect crucial: it ensures that the stringscale does not enter in the expression for , which must thus cancel entirely when passing to the effective action in ten dimensions. More generally, after compactification, the fact that does not depend on implies that all corrections to the effective Lagrangian at one loop come from integrating out the KaluzaKlein modes of massless 10d string states.
HeterotictypeI duality. The predictions of this stringstring duality [4, 5] for the effective action in 10d, have been worked out and checked against earlier calculations by Tseytlin [3]. In summary, there exist two superinvariants quartic in the gaugefield strength [16], which are only distinguished by the groupindex contractions:
and
Here is the antisymmetric 2form, the LeviCivita tensor, and the covariant extension of the wellknown lightconegauge zeromode tensor. The parityeven part of appears however also independently, in the supersymmetric completion of the ChernSimmonsmodified twoderivative action. Since all these superinvariants have anomalycancelling pieces one may expect that they appear at one given order in the loop expansion. In heterotic theory in ten dimensions the twoderivative action comes from the sphere, does not appear at all, while appears at one loop only. Duality maps the model metrics and string couplings as follows [4]:
Simple power counting then shows that the parityeven parts of and should arise in typeI theory from surfaces of Euler number, respectively, minus one (disk, projective plane) and one (disk with two holes etc). This is compatible with the absence of all quartic terms at Euler number zero, as we have concluded.
Consider now torroidal compactifications. The gaugefielddependent oneloop free energy in heterotic theory reads [2]
Here stands for the usual sum over the Lorentzian Narain lattice, which factorizes in the integrand because we assumed zero Wilson lines, and
with the th Eisenstein series:
The ’s are modular forms of weight except for which must be modified to
The powers of in front of expression (16) come from the fact that we used typeI normalizations for the metric: of these powers are due to the spacetime volume, and the other four to the tensor . As for the fact that all holomorphic dependence in the integral appears through the sum over the Lorentzian lattice, this is a result of BPS saturation [1]. It can be derived by an argument similar to the one for the open string, except that the background field now only couples to the helicity coming from the left (supersymmetric) sector.
The Lorentzian lattice involves a sum over both momenta and windings on the dimensional torus. Setting to zero the antisymmetric tensor background, which is a RamondRamond field in typeI theory, and using again typeI normalizations for the compactification torus, we have
Now since inside the fundamental domain, is bounded away from the origin, all terms with nonzero winding are nonperturbatively small at weak . This is consistent with the fact that winding heterotic strings are solitonic Dbranes on the typeI side [5]. The remaining momentum lattice can be Poissonresummed back and written as follows:
We will now plug the above expression into eq. (16), and perform the modular integration. The term can be integrated explicitly, using the formulae
where we defined
subject to the modularinvariance condition . In what concerns the terms, we may extend their integration regime to the entire strip integration straightforward, since only terms without exponential dependence in survive: , modulo a nonperturbatively small error. This makes the
Putting all this together, redefining , and doing some tedious algebra leads to our final expression for the heterotic oneloop free energy at weak typeI coupling::
where here
The leading, term in this expression corresponds to the typeI diskdiagram [3]. The constant piece should be compared to the sum of the Möbiusstrip and annulus, given by eq. (13). These are indeed identical, if one notes that for a simple magnetic field . The remaining terms, as well as the moduliindependent contribution of the heterotic spherediagram [3]
correspond to two and threeloop diagrams on the typeI side. If we assume exact duality and no further corrections on the heterotic side, we conclude that beyond three loops there are only instanton corrections on the typeI side.
Afterword. The effective expansion parameter in eq. (23) is , where is the heterotic Regge slope, and is a typical radius of the compactification torus. Stretched heterotic strings are (nonperturbative) charged BPS states on the typeI side, so it is not surprising that they should control at least part of the terms in . The role of analogous degrees of freedom, as well as of the twoloop renormalization, eq. (24), in the Dbrane context must be elucidated further. The study of fundamentalstring scattering [17] may shed some different light on these issues.
Acknowledgements
We thank the organizers of the Spring Workshop on String Theory for the invitation. C.B. aknowledges support from EEC contract CHRXCT930340, and thanks M. Green, S. Shenker, A. Tseytlin and P. Vanhove for conversations on some related issues.
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