An Alternative for Moduli Stabilisation

The one-loop vacuum energy is explicitly computed for a class of perturbative string vacua where supersymmetry is spontaneously broken by a T-duality invariant asymmetric Scherk-Schwarz deformation. The low-lying spectrum is tachyon-free for any value of the compactification radii and thus no Hagedorn-like phase-transition takes place. Indeed, the induced effective potential is free of divergence, and has a global anti de Sitter minimum where geometric moduli are naturally stabilised.

The true vacuum state in a quantum field theory is given by the configuration with the lowest vacuum energy. In general, superstrings yield a vanishing vacuum energy for toroidal or orbifold compactifications if supersymmetry is intact. As a consequence, superstring compactifications are usually characterised by a moduli space of supersymmetric vacua. This moduli space is spanned by certain dynamical moduli with vanishing potential, whose undetermined vacuum expectation values fix the shape and size of compact internal spaces, and set the strength of gravitational and gauge interactions. This is clearly an embarrassment for any attempt at phenomenological study in String Theory since experiments set severe bounds on Brans-Dicke-like forces and on time variation of coupling constants. It is then clear that the search for mechanisms for moduli stabilisation in string compactifications is of utmost importance. Recently, considerable progress has been made along this direction by allowing for non-trivial fluxes. In this class of compactifications, the internal manifold is permeated by constant fluxes for the field strengths of some Neveu-Schwarz-Neveu-Schwarz and Ramond-Ramond fields. In this way, a non-trivial potential is generated whose extrema fix the vacuum expectation values of complex structure moduli as well as the dilaton field [1]. Moreover, the inclusion of non-perturbative effects (like gaugino condensation) and/or the emergence of perturbative string-loop and α ′ corrections leads also to the stabilisation of Kähler class moduli [2][3][4][5] * . Despite the indiscussed importance of these results, any attempt to lift the moduli space of supersymmetric vacua via geometric fluxes and non-perturbative effects relies on a low-energy supergravity analysis, while a full-fledged perturbative string description is by definition missing. Alternative approaches to the moduli stabilisation problem involve the introduction of open-string magnetic backgrounds [7][8][9][10][11]. Although a perturbative string description is now in principle available (at least for Abelian backgrounds), no consistent string-theory vacua with stabilised moduli have been constructed so far.
An alternative way of stabilising moduli might instead rely on non-supersymmetric string compactifications where corrections to the moduli space of supersymmetric vacua are generated in perturbation theory. This approach has the great advantage of allowing a perturbative string theory description, and can be democratically realised in heterotic strings, type II superstrings and their orientifolds, both on toroidal and orbifold backgrounds. In this case, the first correction to the flat potential for moduli fields is determined by the non-vanishing one-loop vacuum energy. Despite non-supersymmetric strings are typically plagued by the presence of tachyons in the twisted sector, thus inducing a divergent vacuum energy, tachyon-free heterotic models have been constructed in the past and it has also been shown that the associated one-loop vacuum energies are finite and have extrema at symmetry-enhancement points [12,13]. In this letter, we extend this analysis to superstrings with spontaneously broken supersymmetry and show how, in models without tachyons, symmetries of the (deformed) Narain lattice determine the minima of the one-loop cosmological constant. Constructions with a similar target have been carried out recently also in the context of non-critical string theory [14,15].
Among the various mechanisms for breaking supersymmetry, the Scherk-Schwarz deformation provides an elegant realisation based on compactification [16,17]. In the simplest case of circle compactification, it amounts in Field Theory to allowing the higher dimensional fields to be periodic around the circle up to an R-symmetry transformation. The Kaluza-Klein momenta of the various fields are correspondingly shifted proportionally to their R charges, and modular invariance dictates the extension of this mechanism to the full perturbative spectrum in models of oriented closed strings [18][19][20][21]. Actually, it is a known fact that Scherk-Schwarz deformations can be conveniently realised in String Theory as freely acting orbifolds [22]. Typically, non-supersymmetric projections are accompanied by shifts of internal coordinates, so that the moduli describing their size set the supersymmetry-breaking scale. The simplest instance of a Scherk-Schwarz deformation involves the Z 2 orbifold generated by g = (−1) F δ, where (−1) F is the space-time fermion index and δ shifts the compact coordinate y ∈ S 1 (R) by half of the length of the circle, δ : y → y + πR. As usual, the resulting string spectrum is encoded in the one-loop partition function where V 9 is the (infinite) volume of the non-compact dimensions, (O 8 , V 8 , S 8 , C 8 ) are the characters associated with the SO(8) little group, and and thus the tachyonic ground state is actually massive for large values of R, while it is massless and then really tachyonic The absence of tachyonic excitations in the large-radius regime suggests that in this range the theory is perturbatively under control. Using standard techniques [23][24][25][26][27][28][29], one where d N counts the degeneracy of states at each mass level, The contribution of massless-states reproduces the standard field theory result, while massivestate contributions are exponentially suppressed for large values of R, but yield a divergent contribution in the tachyonic region R ≤ √ 8α ′ . This divergence is a consequence of the exponential growth of string states and is responsible for the well-known first-order Hagedorn phase transition [30], after we reinterpret the compact radius in terms of the finite temperature, β ∼ R.
Although this representation of the Scherk-Schwarz deformation has a natural field theory limit (after the compactification radius is properly halved), where the anti-periodic fermions have half-integer Kaluza-Klein excitations and the vacuum energy has the typical R −n behaviour, for n non-compact dimensions, String Theory can afford more possibilities: one can actually deform only the momenta, as in eq. (1), only the windings or both. These † Actually, to compare with the standard Scherk-Schwarz deformation one has to halve the compactification radius, so that bosons and fermions have indeed integer and half-odd-integer KK momenta, and the twisted tachyon appears for R < √ 2α ′ .
correspond in turn to the freely acting orbifolds g δ i , where g is any non-supersymmetric generator and the δ i act as on a circle of radius R.
Clearly the δ 2 shift does not introduce new physics, since it is directly related to the δ 1 shift via T-duality: R → α ′ /R. As a result, it yields a tachyon-free spectrum in the smallradius regime, the associated effective potential behaves like R n , while a Hagedorn-like transition occurs now at R = α ′ /8.
More interesting is instead the δ 3 shift. It preserves T-duality and thus one can expect to have a sensible theory both in the small-radius and in the large-radius regimes ⋆ . In this case, a generic vertex operator V m,n = e ip L X L +ip R X R , with p L,R = m R ± nR α ′ , gets the phase (−1) m+n under the action of δ 3 , and, as a consequence, both momenta and windings are half-odd-integers in the twisted sector. In this sector the mass formulae read where we have consider the more general case of a higher-dimensional internal manifold consisting of the product of various circles, each of radius R i . Evidently, level-matching ⋆ Actually, this asymmetric δ 3 (−1) F freely acting orbifold with diagonal metric is equivalent to a more conventional symmetric Scherk-Schwarz deformation but with a non-vanishing B-field background, as shown in [31]. However, in the following we concentrate on the asymmetric shift since it is clearly T-duality invariant and thus some results are easier to prove.
is not satisfied if δ 3 acts on a single coordinate. Actually this deformation is only allowed when acting on coordinates of an even 2d-dimensional torus. In this case, the lightest states have masses and the twisted spectrum is clearly free of tachyons for d = 2, 3 ‡. Henceforth, one expects to have a finite and well-behaved one-loop result for any values of R. This is in contrast to the previous case, where a divergence induced by the emergence of tachyonic excitations triggers a first-order phase transition. In fact, in the case at hand the partition function schematically reads and thus the absence of tachyons prevents from IR divergence while, as usual, modular invariance excludes the dangerous UV region from the integration domain F.
To be more precise, the complete partition function now reads where ǫ = (1, 1, . . . , 1) is a 2d-dimensional unit vector, and the second line clearly spells-out the "non-canonical" deformation of the Narain lattice in the twisted sector.
Also in this case, one can use standard unfolding techniques to convert the integral over the fundamental domain into the integral over the half-infinite strip. Typically, this procedure involves a Poisson re-summation in order to disentangle the contributions to the integral of different orbits of SL(2, Z). Re-summing over windings or momenta clearly spells-out the small or large radius behaviour. For this reason, in the case of δ 1 and δ 2 shifts the very consistency of the perturbative string expansion suggests to re-sum over momenta and windings, respectively, while in the more interesting case of a δ 3 shift either ways are ‡ It is amusing to note similarities with mass-formulae of other tachyon-free non-supersymmetric orientifold models [32,33] meaningful, since the spectrum is free of tachyons and no first-order phase transitions are expected to occur. In particular, from acts as left multiplication on the (2 × 2d)-dimensional integral matrices As a result, both M andM can be arranged into orbits of Γ 0 [2] and picking-up a single representative for each orbit allows one to disentangle the τ 1 and τ 2 integrals [34]. In particular, one has to distinguish between the degenerate orbit M = 0 . . . 0 2ℓ 1 + 1 . . . 2ℓ 2d + 1 , and the non-degenerate one and similarly forM . Let us consider, for instance, the contribution of the degenerate orbit to V large . One finds The above expression, however, cannot be computed analytically for arbitrary radius, since for values of the radius of the order of the string scale the N -series is not uniformly convergent and thus one cannot exchange the integration with the summation. However, one can easily extract the large-radius behaviour where, for simplicity we assumed the radii to be equal. The R → ∞ behaviour of the nondegenerate orbit can be also computed to find the exponentially suppressed contributions of massive states. Similarly, had we started from V small the series would have been convergent only for small values of R, and for R → 0 , whose behaviour could have been anticipated by T-duality arguments. What about the behaviour of the potential for finite values of R? In the absence of tachyonic excitations no divergences are expected, while from the asymptotic behaviours something special is expected to occur at the self-dual radius. Indeed, the gradient of the potential Clearly, it would be interesting to generalise this construction to more general supersymmetry-breaking set-ups. Extrema of the potential should then occur at points of enhanced symmetry and eventually yield a more complex landscape of (meta-)stable vacua.
It might also be rewarding to combine this mechanism with brane-supersymmetry-breaking [35,36,37,38] where NS-NS tadpoles are typically uncancelled, thus introducing a new positive source for the potential. Barring the stabilisation of the dilaton, the combination of these two effects might uplift the AdS minimum to a de Sitter metastable vacuum, as in [2]. Of course, a different question is whether the dilaton (which cannot be stabilised by our method) stays small in the AdS minimum, and whether higher-loop corrections might destroy the extrema of V, but a definite answer is out of reach from present technology.