Saturday, November 19, 2005

Non-Kähler compactifications

Li-Sheng Tseng from University of Utah gave a Duality Seminar about
  • Flux Compactifications and Moduli

The talk focused on non-Kähler compactifications with fluxes which is a very interesting topic that however remains controversial. Let me try to explain why.

For the sake of simplicity, Li-Sheng discussed the case of a non-zero H-field only; other fluxes are set to zero. The condition of unbroken supersymmetry implies that the three-form H must actually be expressed in terms of a derivative of the differential two-form that gives you a Hermitean metric. Why? Recall that the H-field enters as a kind of torsion to the overall connection - and a spinor should be covariantly constant under this connection.

Because this exterior derivative of the metric is not zero, you can't really call it a Kähler form. And you can't call the manifold a Kähler manifold either. However, when we deal with heterotic string theory, there is also the usual Bianchi identity for H

  • dH = alpha' Tr (R /\ R - F /\ F)

When we consider the classical compactifications with "H=0", we relate the curvature of the geometry "R" and the curvature of the gauge bundle "F". For example, we can cancel them trivially by embedding the spin connection to the gauge connection - but that's definitely not the most general solution. Both connections are nevertheless of the same order. The size of the manifold is not determined. The Kähler moduli are not fixed and "alpha'/a^2" is a good expansion parameter.

That's both a good news as well as bad news. It is good news because reliable calculations can be done even perturbatively. It is bad news because the moduli are not stabilized which disagrees with basic properties of the real world around us, among many other worlds.

However, when we choose a non-zero value of "H", the equation above makes it clear that terms with different powers of alpha' are getting mixed. Another equation tells you that "H" and the metric are of the same order, and therefore you can't assign unique powers of alpha' to all your fields; dimensional analysis breaks down. This implies that the size of the manifold is comparable to the string length (i.e. "of order one") if the equations are to be solved. For a finite value of the fluxes, the expansion in "alpha'/a^2" is strongly coupled. Li-Sheng actually argued that this conclusion is only correct for a T2-fiber in his main example while the K3-base may be arbitrarily large. But at any rate, there are some dimensions whose size is stringy which forces us to include infinitely many terms at all orders in alpha'. Any truncation seems unreliable.

While the Hermitean metric is not a closed form, its square continues to be co-closed, and Li-Sheng told us about some examples of generalized geometry - conformal Calabi-Yau geometry, for example. These are nice abstract notions, ideas, and formulae, but the question whether an actual CFT that describes such a perturbative string theory exists remains unanswered. No such a CFT has been explicitly found - and not even proved by an existence theorem. Unlike the Calabi-Yau case, no explicit orbifold point in the moduli space is known either. At the level of geometry only, the existence of some particular examples has been shown, but I doubt that these proofs exceed some low-energy geometric approximations and imply the existence of the full backgrounds of string theory.

Note that you must solve the equations for "H"; Einstein's equations with the appropriate right-hand side; and it turns out that the dilaton is non-trivial which also forces you to solve a non-trivial Laplace-like equation for the dilaton.

All of these equations are pretty hard to satisfy. A proof that such a combined solution for all these fields exists amounts to a rather extensive generalization of Yau's theorem (such a description is especially appropriate for the metric and its Einstein's equations with a source). There are some details that seem hard and potentially incompatible with the perturbative approach to the question. For example, a static dilaton must satisfy something like

  • nabla^2 (exp(-phi)) = F^2 + H^2 + ...

whose right-hand side actually turns out to be positively definite at the leading order. This would prevent the dilaton from being smooth but non-constant everywhere, as required in Li-Sheng's construction: the Laplacian has opposite signs at the maxima and minima of the dilaton. However, he argues that the higher-order terms in alpha' appear in the equation above and these extra terms may be negative and compensate the leading-order positive terms.

Quite generally, I remain skeptical about the constructions that seem impossible or inconsistent at a leading order in a certain perturbative expansion and whose consistency is being explained by a mere existence of some higher-order (or even non-perturbative) terms. Don't get me wrong: constructions that are not accessible perturbatively are all but guaranteed to be important in many situations. All moduli in the real world are stabilized, for example, which makes any expansion about their extreme values problematic.

Even though the backgrounds that do not solve the equations of motion order by order in a certain perturbative expansion are important and will be important, the importance does not prove that they actually exist. Cancelling a first-order problem by pointing out a possible second-order cure may be a hint that a consistent theory exists - but it is definitely not a proof. If we cancel terms with a different parametric dependence on any parameter G, it certainly means that the perturbative expansion in G breaks down. That is not enough to prove that the theory is doomed, but it is not enough to prove that it is consistent either.

For example, the second-order terms may be negative but the third-order terms could be positive and such that the sum of the first three corrections remains positively definite. Is it up to the collective decision of the terms at all orders whether the qualitative conclusion based on the first-order approximation (the theory does not exist) is correct; or whether the conclusion based on the second-order approximation (the theory does exist) is right.

Note that the existence of simpler vacua in string theory has been firmly established. In my opinion, the people thinking about the non-Kähler issues may want to focus on proving the existence of their conjectured generalized theories - for example the existence of CFTs in the cases in which the dilaton may be kept near minus infinity. When the perturbative expansions break down, it is only the exact result that is relevant. One should try to decide whether a truncation of some expansions leads to qualitatively correct results. And the answer can be either Yes or No.

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