We now study a CY orientifold with flux. In such a model, one has the 'tadpole' consistency condition:

2 a' is the coupling in the Nanbu—Goto—Polyakov world-sheet action of the string and gs is the string coupling itself. These two parameters control the two types of quantum corrections in string theory. The first is related to the world-sheet corrections; these correspond to higher derivative terms in the effective gravitational theory and are calculated via loop diagrams in the sigma model. The second is related to the vacuum expectation value (vev) of the dilaton which controls the corrections due to string loops; these are due to higher genus Riemann surfaces on which the string propagates.

2 a' is the coupling in the Nanbu—Goto—Polyakov world-sheet action of the string and gs is the string coupling itself. These two parameters control the two types of quantum corrections in string theory. The first is related to the world-sheet corrections; these correspond to higher derivative terms in the effective gravitational theory and are calculated via loop diagrams in the sigma model. The second is related to the vacuum expectation value (vev) of the dilaton which controls the corrections due to string loops; these are due to higher genus Riemann surfaces on which the string propagates.

Here T3 is the tension of a D3 brane, Nd3 is the net number of (D3 — D3) branes one has inserted to fill the non-compact dimensions, and H3 and F3 are the 3-form fluxes which arise in the NS and RR sectors, respectively. We assume that we are working with a model with only one Kahler modulus, so that h1'1(M) = 1. (In taking the F-theory limit, where one shrinks the elliptic fibre, one has hl,1(X) = 2 and one modulus is frozen.) Such models can be explicitly constructed, for example by using the examples of CY 4-folds or by explicitly constructing orientifolds of known CY 3-folds with h1'1 = 1.

In the presence of the non-zero fluxes, one generates a superpotential for the CY moduli,

where G3 = F3 — tH3 and t is the IIB axiodilaton. Combining this with the tree-level Kahler potential,

K = — 3 ln[—i(p — p)] — ln[—i(T — r)], (12.12)

where p is the single volume modulus given by p = b/V2 + ie4u-^, (12.13)

and using the standard N = 1 supergravity formula for the potential, one obtains

V = eK ^^gabDaWDbW — 3\W|2j ^ eK gijDiWDjW^. (12.14)

Here a and b run over all moduli fields, while i and j run over all moduli fields except p; we see that, because p does not appear in Eq. (12.11), it cancels out of the potential energy given by Eq. (12.14), leaving the positive semi-definite potential characteristic of no-scale models. These models are not satisfactory, as they lead to the cosmological decompactification of the internal space during the cosmological evolution.

One can use two known corrections to the no-scale models, both parametrizing possible corrections to the superpotential.

(i) In type IIB compactifications of this type, there can be corrections to the superpotential coming from Euclidean D3 branes:

where T(zi) is a complex structure-dependent one-loop determinant and the leading exponential dependence comes from the action of a Euclidean D3 brane wrapping a 4-cycle in M. (ii) In general models of this sort, one finds non-Abelian gauge groups arising from geometric singularities in X, or (in type IIB language) from stacks of D7 branes wrapping 4-cycles in M. This theory undergoes gluino condensation, which results in a non-perturbative superpotential. This leads to an exponential superpotential for p similar to the one above (but with a fractional multiple of p in the exponent, since the gaugino condensate looks like a fractional instant on effect in W):

So effects (i) and (ii) have rather similar consequences for our analysis; we will simply assume that there is an exponential superpotential for p at large volumes. There are some interesting possibilities for cosmology if there are multiple non-Abelian gauge factors. Using the 4-folds, it is easy to construct examples which could yield gauge groups of total rank up to — 30. However, much larger ranks should be possible.

The corrections to the superpotential discussed above can stabilize the volume modulus, leading to a supersymmetry Supersymmetry-preserving AdS minimum. We analyze the vacuum structure, just keeping the tree-level Kahler potential, and a superpotential,

where W0 is a tree-level contribution which arises from the fluxes. The exponential term arises from either of the two sources above and the coefficient a can be determined accordingly. At a supersymmetric vacuum, we have DpW = 0. We simplify the situation by setting the axion in the p modulus to zero and letting p = ia. In addition, we take A, a and W0 to be real and W0 to be negative. The condition DW = 0 then implies that the minimum lies at

The potential at the minimum is negative and equal to a2 A2

OCTcr

We see that we have stabilized the volume modulus, while preserving super-symmetry. It is important to note that the AdS minimum is quite generic. For example, if Wo = —10_4, A = 1 and a = 0.1, the minimum is at acr ~ 113, as shown in Fig. 12.7.

Another possibility to obtain a minimum for large volumes is to consider a situation where the fluxes preserve supersymmetry and the superpotential involves multiple exponential terms, i.e. 'racetrack potentials' for the stabilization of p. Such a superpotential could arise from multiple stacks of 7-branes wrapping 4-cycles, which cannot be deformed into each other in a supersymmetry-preserving manner. In this case, by tuning the ranks of the gauge groups appropriately, one can obtain a parametrically large value of a at the minimum.

Now we lift the supersymmetric AdS vacua to obtain the dS vacua of string theory. In the consistency condition given by Eq. (12.10), there are contributions from both localized D3 branes and fluxes. To find the AdS vacua with no moduli, of the kind discussed in Section 12.2, we assumed that the condition was saturated by turning on fluxes in the compact manifold.

Next we assume that, in fact, we turn on too much flux, so that Eq. (12.10) can only be satisfied by introducing one D3 brane. The consistency equation is now satisfied due to the presence of the anti-D3 brane, but there is an extra bit of energy density from the 'extra' flux and D3 brane. In general, we obtain a term in the potential which takes the following form:

12 M/string theory and anthropic reasoning

Fig. 12.8. The potential (multiplied by 1015) for the case of an exponential superpotential, including a D/a3 correction with D = 3 x 10~9, which uplifts the AdS minimum to a dS minimum.

Fig. 12.8. The potential (multiplied by 1015) for the case of an exponential superpotential, including a D/a3 correction with D = 3 x 10~9, which uplifts the AdS minimum to a dS minimum.

where the factor of 8 is added for later convenience. The coefficient D depends on the number of D3 branes and on the warp factor at the end of the throat. These parameters can be altered by discretely changing the fluxes. This allows us to vary the coefficient D and the supersymmetry-breaking in the system, while still keeping them small. (Strictly speaking, since the flux can only be discretely tuned, D cannot be varied with arbitrary precision.) We will see that, by tuning the choice of D, one can perturb the AdS vacua to produce dS vacua with a tunable cosmological constant.

We now add to the potential a term of the form D/a3, as explained above. For suitable choices of D, the AdS minimum will become a dS minimum, but the rest of the potential does not change too much. However, there is one new important feature: a dS maximum separating the dS minimum from the vanishing potential at infinity. The potential is given by nAe-aa /1 \ D

By fine-tuning D, it is easy to have the dS minimum very close to zero. For the model W0 = —10-4, A = 1, a = 0-1 and D = 3 x 10-9, the potential is as indicated in Fig. 12.8.

Note that, if one does not require the minimum to be so close to zero, D does not have to be so fine-tuned. A dS minimum is obtained as long as D lies within certain bounds, eventually disappearing for large enough D. If one does fine-tune to bring the minimum very close to zero, the resulting potentials are quite steep around the dS minimum. In this circumstance, the new term effectively uplifts the potential without changing the shape too much around the minimum, so the p field acquires a surprisingly large mass (relative to the final value of the cosmological constant).

If one wants to use this potential to describe the present stage of acceleration of the Universe, one needs to fine-tune the value of the potential in the dS minimum to be V0 — 10"120 in units of the Planck density. In principle, one could achieve this, for example, by fine-tuning D. However, the tuning achievable by varying the fluxes in microscopic string theory is limited, though it may be possible to tune well if there are enough 3-cycles in M.

12.4 Discussion of the anthropic landscape of string theory

It is difficult to construct realistic cosmologies in string theory if the moduli fields are not frozen. We have found that it is possible to stabilize all moduli in a controlled manner in the general setting of compactification with flux. This opens up a promising arena for the construction of realistic cosmolog-ical models based on string theory. More specifically, we have seen that it is possible to construct metastable dS vacua by including anti-branes and incorporating non-perturbative corrections to the superpotential from D3 instantons or low-energy gauge dynamics.

In the simplest possible case, our examples require knowledge of at least six parameters: two to specify the distinct electric and magnetic fluxes required to fix the dilaton; three to specify the non-perturbative corrections to the superpotential; and one to specify the anti-brane contribution. Moduli stabilization in more complicated models may depend on many more parameters, which means there are many ways to realize these vacua.

One may hope that the number of vacua in string theory is very large, at least of the order N > 10120. In this case, it may be possible that some of these vacua have a positive cosmological constant of order A — M4/N, so the selection of a vacuum with A — 10_120Mp could then be anthropic. The basic estimate for the number of flux vacua, satisfying the tadpole consistency condition of Eq. (12.10), is given by Douglas [5, 6] as

Here K is the number of distinct fluxes and L = x/'24 is the 'tadpole charge' on the left-hand side of Eq. (12.10). The estimates are K —100—400 and L — 500—5000, which lead to Nvac — 10500. This number is extremely large, even larger than the number 10120 required for the anthropic solution of the cosmological constant problem. Each of these vacua will have a different vacuum energy density and each part of the Universe with a particular positive cosmological constant will be exponentially large. Particles living in the different vacua will have dramatically different properties.

It is interesting that all of these conclusions have been reached after the recent discovery that the Universe is accelerating. Attempts to describe this acceleration in string theory forced us to invent a way to describe dS vacua. As a result, we have found that the solution of this problem is not unique and the same string theory there could have an incredibly large number of different vacua. This explains the sudden increased attention of cosmologists and string theorists towards the concept of the multiverse and anthropic reasoning.

It is a pleasure to thank T. Banks, M. Dine, N. Kaloper, A. Klypin, L. Kofman, L. Susskind, A. Vilenkin and S. Weinberg and all my collaborators for useful discussions. This work was supported by NSF grant PHY-9870115.

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