Topological constraints on stabilized flux vacua

We study the influence of four-form fluxes on the stabilization of the Kahler moduli in M-theory compactified on a Calabi-Yau four-fold. We find that, under certain non-degeneracy condition on the flux, M5-instantons of a new topological type generate a superpotential. The existence of such an instanton restricts possible four-folds for which the stabilization by this mechanism is expected. These topological constraints on the background are different from the previously known constraints, derived from the flux-free analysis of the nonperturbative effects.


Introduction
Whether or not we wish to accept the anthropic philosophy [1], [2], a necessary condition for a plausible phenomenologically realistic background is the stabilization of all of its moduli. In the context of the orientifold type IIB models [3], [4], [5], [6], [7] it is now clear that the complex structure moduli and the axion-dilaton modulus are fixed by a perturbative superpotential proportional to the fluxes [8]. On the other hand, the stabilization of the Kähler moduli relies on the generation of the nonperturbative superpotential. The nonperturbative effects originate from gaugino condensation on coincident D7-branes [9], [10] present in the background and from the D3-brane instantons [10], [11], [12], [13].
The KKLT paper [6] qualitatively discussed the nonperturbative superpotential deriving its intuition from the flux-free compactifications. The subsequent successful search for the realistic backgrounds [10], [14], [15] was also based on the flux-free analysis [11], [12] of the nonperturbative effects.
However, recently it was realized [13], [16], [17] that the presence of background fluxes may actually modify the conditions for the generation of an instanton-induced superpotential.
In this note we study the effect of the background flux on the generation of a nonperturbative superpotential for the Kähler moduli. We find that, under certain restrictions on the background flux, instantons of a new topological type generate a superpotential.
We investigate these new instantons in M-theory compactified on a Calabi-Yau fourfold CY 4 with four-form fluxes [18]. The effective 3D theory has four supercharges. Moreover, if the four-fold is elliptically fibered and the area of the elliptic fiber is sent to zero, a new fourth dimension appears and the background is described as a flux compactification of type IIB string theory on a Calabi-Yau orientifold [4]. In the framework of M-theory the D7-branes are described as singular fibers of the elliptic fibration, while the D3-brane instantons become the M5-brane instantons wrapped on the "vertical" divisors 1 of CY 4 .
An M5-brane wrapped on a divisor in the four-fold generates a superpotential required for the stabilization of Kähler moduli if there are exactly two fermionic zero modes on its world-volume. The relevant analysis of the generalized Dirac equation [16] has not yet been done in the presence of fluxes. The purpose of this note is to fill in this gap.
We find exactly two fermionic zero modes by restricting the choice of fluxes and global properties of the divisors. We consider divisors with Hodge numbers We would like to emphasize that it is not trivial to reduce the number of the fermionic zero modes of the instanton to two. For example, [17] have counted the number of the fermion zero modes in the context of a type IIB compactification on the orientifold T 6 /Z 2 in the presence of fluxes and found four zero modes. In their case, no instanton-induced superpotential is generated. The 11D metric is a warped product where η µν is the metric on the three-dimensional Minkowski space and the internal metric has the form: Here g Our goal in Section 3 will be to recast the equations of motion for fermions living on the M5-instanton as a set of equations on differential forms on the divisor D. We will further use this in Section 4 to find the case with exactly two fermion zero modes.
We have introduced complex coordinates z i , i = 1, 2, 3 along the divisor and the complex coordinate w normal to the divisor inside CY 4 . In (3.1) T w , Tw are SO(2) Dirac matrices:

Note that
The six dimensional chiral(anti-chiral) gamma matricesγ αβ i ,γ αβ j (γ i αβ , γj αβ ) have the properties where g ij is Kähler metric on the divisor D. Note that nothing in the equation (3.1) acts on the index A = 1, 2 of a spinor in R 3 . In what follows we will not write this index explicitly but we will keep it in mind in the future counting of the number of zero modes.
The covariant derivatives ∇ j , ∇j include the connection on the bundle of chiral Spin (6) spinors as well as connection on the spin bundle derived from the normal bundle N . Now we use the known fact (see for example [11]) that the bundle S + of chiral spinors on a Kähler manifold of complex dimension three is isomorphic to the bundle Here Ω (0,p) stands for the bundle of (0, p) forms. We will further use that the normal bundle on the divisor in CY 4 is isomorphic to the canonical bundle K. Recalling that θ is a section of the bundle 4 S + ⊗ K 1 2 ⊕ S + ⊗ K − 1 2 , we find the following degrees of freedom. A (0,2) form a w (2) taking values in the canonical bundle K, a section of K a w (0) as well as a (0,2) form b (2) and a scalar b (0) .

The M5-instanton with two fermion zero modes.
Here we study the set of equations (3.6)-(3.9) for the fermionic degrees of freedom on an M5-brane wrapped on a divisor D in a Calabi-Yau 4-fold. We consider divisors with Hodge numbers where h (0,p) stands for the number of harmonic (0, p) forms on D. The goal of this section is to show that for generic background fluxes the M5-branes wrapped on the divisors of this topological type generate a superpotential.
In the absence of fluxes there would be four fermion zero modes for the divisors with these Hodge numbers. Two zero modes 6 would be coming from harmonic (0,2) form and the other two from (0,0) form. It is natural to expect that choosing flux appropriately one can lift zero modes associated with (0,2) form. Below we realize this expectation.
The equation ( Recalling that all the fields carry hidden index A = 1, 2 of a spinor in R 3 (see discussion below (3.3)), we conclude that we found exactly two fermion zero modes. Therefore, the M5-instanton of the topology (4.1) in the presence of generic fluxes (4.6) generates nonperturbative superpotential for the Kähler moduli.

Conclusion
In this note we studied how the conditions for the stabilization of the Divisors with these Hodge numbers appeared before 8 in the discussion of the gaugino condensation on coincident D7-btanes [13]. We found that in the presence of the special background fluxes such divisors are relevant for the generation of nonperturbative superpotential induced by the M5-instantons.
The condition for the existence of such a divisor restricts possible four-folds for which the stabilization of the Kähler moduli by this mechanism is expected. These topological constraints on the background are different from the previously known constraints derived from the flux-free analysis of the nonperturbative effects.
It would be interesting to find other choices of fluxes which can make the M5-branes wrapped on more general divisors to contribute to the nonperturbative superpotential.
Another interesting question is to find an explicit example of a Calabi-Yau four-fold, other than K 3 × K 3 , that contains a divisor with the desired properties (4.1) and admits the appropriate non-degenerate flux (4.6).