Example 2: Five-Brane Superpotential in Heterotic/F-Theory

Let us now discuss a second example for which the F-theory flux superpotential can be computed explicitly since the F-theory fourfold admits only few complex structure moduli.

Clearly, using mirror symmetry such fourfolds can be obtained as mirror manifolds of examples with few K¨ahler moduli.

To start with, let us consider heterotic string theory on the mirror of the Calabi-Yau threefold ˜Z₃ which we studied in the last section 6.2.2. This mirror is the heterotic manifold Z₃ that we studied already in section 6.1.2 in the context of mirror symmetry on fourfolds.

As we noted there using the methods of [260] Z₃ is also elliptically fibered, such that it has an F-theory dual description and that bundles can at least in principle be constructed explicitly using the spectral cover construction of section 4.1.4. The polyhedron ofZ₃ is the dual polyhedron to (6.9) presented in (6.11) and we recall its Weierstrass form,

p₀ =x³+y²+xyza˜ ₀u₁u₂u₃+ ˜z⁶(a₁u¹⁸₁ +a₂u¹⁸₂ +a₃u¹⁸₃ +a₄u⁶₁u⁶₂u⁶₃). (6.82) The coordinatesuare the homogeneous coordinates of the twofold baseB₂and (x, y,z) denote˜

14If one considers exactly the mirror of ˆX4, as we will in fact do in section 6.2.3, it might be possible to embed this reduced deformation problem into the complicated deformation problem of ˆX4constructed in this section.

6.2. HETEROTIC/F-THEORY DUALITY: MODULI AND SUPERPOTENTIALS 163 the homogeneous coordinate ofP²(1,2,3). Note that one finds that the elliptic fibration ofZ₃ is highly degenerate over B₂. The threefold is nevertheless non-singular since the singularities are blown up by many divisors in the toric ambient space of Z₃. However, in writing (6.82) many of the coordinates parameterizing these additional divisors have been set to one¹⁵. Turning to the perturbative gauge bundleE₁⊕E₂we restrict in the following to the simplest bundleSU(1)×SU(1) which thus preserves the full perturbative E₈×E₈ gauge symmetry in four dimensions.

To nevertheless satisfy the anomaly condition (4.9) one has to include five-branes. In particular, we consider a five-brane inZ3 given by the equations

h₁:=b₁u¹⁸₁ +b₂u⁶₁u⁶₂u⁶₃= 0 , h₂ := ˜z= 0 . (6.83) The curve Σ wrapped by the five-brane is thus in the baseB₂ ofZ₃ and horizontal. Unfortu-nately, it is hard to check the heterotic anomaly (4.9) explicitly as in the example of section 6.2.2 since there are too many K¨ahler classes in Z₃.

However, one can proceed to construct the associated Calabi-Yau fourfold X₄ which en-codes a consistent completion of the setup by duality. The associated fourfold X₄ cannot be constructed as it was done in section 6.2.2. However, one can follow the strategy of [103]

summarized in diagram (6.1) to construct heterotic/F-theory dual geometries by employing mirror symmetry to first obtain the mirror fourfold ˜X₄ of X₄ as the Calabi-Yau threefold fibration with generic fiber ˜Z₃ being the mirror of the heterotic threefoldZ₃. This then natu-rally leads us to identifyX4 as the mirror to the fourfold (6.79) in section 6.2.2. This fourfold is also the main example discussed in detail in section 6.1 and in [79].

In the following we check that this is indeed the correct identification by using the formal-ism of [103, 105]. The Weierstrass form of X₄ can be computed using the polyhedron (6.31) that is dual to (6.79) yielding

P =y²+x³+m₁(u, w, k)xyz+m₆(u, w, k)z⁶ = 0, (6.84) where we abbreviated

m1(w, u, k) = a0u1u2u3w1w2w3w4w5w6k1k2, (6.85) m₆(w, u, k) = a₁(k₁k₂)⁶u¹⁸₁ w¹⁸₁ w¹⁸₂ w₅⁶w⁶₆+a₂(k₁k₂)⁶u¹⁸₂ w₃¹⁸w¹²₅

+a₃(k₁k₂)⁶u¹⁸₃ w¹⁸₄ w¹²₆ +a₄(k₁k₂)⁶(u₁u₂u₃w₁w₂w₃w₄w₅w₆)⁶ +b₁k₂¹²u¹⁸₁ w²⁴₁ w₂¹²w⁶₃w₄⁶+b₂k₂¹²(u₁u₂u₃)⁶(w₁w₃w₄)¹²

+c₁k₁¹²(u₁u₂u₃)⁶(w₂w₅w₆)¹².

The coordinates u are the coordinates of the twofold base B₂ as before and w, k₁, k₂ are the additional coordinates of the threefold base B₃¹⁶. Again, note that we have set many

15The blow-down of these divisors induces a large non-perturbative gauge group in the heterotic string.

16We note that we slightly changed our labeling of the coordinates compared to (6.32), (6.33).

coordinates to one to displayP. The chosen coordinates correspond to divisors which include the vertices of ∆^X₅ and hence determine the polyhedron completely. In particular, one finds that (k₁, k₂) are the homogeneous coordinates of theP¹-fiber overB₂. The coefficients a, b, c₁ encode the complex structure deformations of X₄. However, since h^(3,1)( ˆX₄) = 4, there are only four complex structure parameters rendering six of them redundant.

As the first check that X4 is indeed the correct dual Calabi-Yau fourfold, we follow the discussion of section 4.3.2 and use the stable degeneration limit [182] and writeP in a local patch with appropriate coordinate redefinition as [103]

P =p₀+p₊+p₋, (6.86)

where

p₀ = x³+y²+xyza˜ ₀u₁u₂u₃+ ˜z⁶ a₁u¹⁸₁ +a₂u¹⁸₂ +a₃u¹⁸₃ +a₄u⁶₁u⁶₂u⁶₃

, (6.87) p₊ = vz˜⁶ b₁u¹⁸₁ +b₂u⁶₁u⁶₂u⁶₃

, p₋ = v⁻¹z˜⁶c1u⁶₁u⁶₂u⁶₃ .

The coordinate v is the affine coordinate of the fiber P¹. In the stable degeneration limit {p₀ = 0} describes the Calabi-Yau threefold of the heterotic string. In this case p₀ coincides with the constraint (6.12) of the threefold geometry Z₃ as discussed in section 6.1.2 which directly identifies it as the dual heterotic Calabi-Yau threefold toX₄. This shows in particular that the geometric moduli of the heterotic compactification on Z₃ are correctly embedded in X₄. The polynomialsp_±encode the perturbative bundles, and the explicit form (6.87) shows that we are dealing with a trivial SU(1)×SU(1) bundle, which is also directly checked by analyzing the polyhedron of ˆX₄ using the methods of [101, 210]. Indeed, one shows explicitly that over each divisor k_i = 0 in B₃ a full E₈ gauge group is realized. Since the full E₈×E₈ gauge symmetry is preserved we are precisely in the situation of section 4.3.3, where we recalled from [105] that a smooth X₄ has to contain a blow-up corresponding to a horizontal heterotic five-brane. We now check that this allows us to identify the heterotic five-brane and its moduli explicitly in the duality to F-theory.

Let us begin by making contact to the formalism of section 4.3.3. First, to make the perturbativeE₈×E₈gauge group visible in the Weierstrass equation (6.84), we have to include new coordinates (˜k₁,k˜₂) replacing (k₁, k₂). This can be again understood by analyzing the toric data using the methods of [101, 210]. We denote by (3,2, ~µ) the toric coordinates of the divisor corresponding to ˜k₁ in the Weierstrass model. Then the resolved E₈ singularity corresponds to the points¹⁷

(3,2, n~µ), n= 1, ...,6 , (2,1, n~µ), n= 1, ...,4 , (6.88) (1,1, n~µ), n= 1,2,3 , (1,0, n~µ), n= 1,2 , (0,0, ~µ)

17Note that we have chosen the vertices in theP_1,2,3[6] to be (−1,0),(0,−1),(3,2) to match the discussion in refs. [101, 210]. However, if one explicitly analyses the polyhedron ofX4 one finds that one has to apply a Gl(2,Z) transformation to find a perfect match. This is due to the fact thatX4, in comparison to its mirror X˜4, actually contains the dual torus as elliptic fiber.

6.2. HETEROTIC/F-THEORY DUALITY: MODULI AND SUPERPOTENTIALS 165 While (3,2,6~µ) corresponding to k₁ is a vertex of the polyhedron, (3,2, ~µ) corresponding to k˜₁ is an inner point. Using the inner point for ˜k₁, the Weierstrass form P changes slightly, while the polynomials p₀, p₊ and p₋ can still be identified in the stable degeneration limit.

Next, to determine g₅ in (4.58), we compute g of the Weierstrass form in a local patch where ˜k₂ = 1

g= ˜k⁵₁

b1u¹⁸₁ +b2u⁶₁u⁶₂u⁶₃+ ˜k1 a1u¹⁸₁ +a2u¹⁸₂ +. . .

. (6.89)

The dots contain only terms of order zero or higher in ˜k₁. Comparing this with (4.61), we note that the Calabi-Yau fourfold X₄ can be understood as a blow-up¹⁸ along the curve k˜₁ =g₅= 0 in the base of X₄, where g₅ is given by

g5 =b1u¹⁸₁ +b2u⁶₁u⁶₂u⁶₃. (6.90) This identifies {g₅ = 0} with the curve of the five-brane Σ in the base B₂ of Z₃, which is in perfect accord with (6.83). Thus, one concludes thatX4is indeed a correct fourfold associated to Z₃ with the given horizontal five-brane. As we can see from (6.90), the five-brane has one modulus. If we compareg₅ withp₊, we see that p₊=vz˜⁶g₅. This nicely fits with the bundle description of a five-brane as a small instanton. In fact, in our configuration, p₊ and p₋ should describe SU(1)-bundle since we have the full unbroken perturbativeE₈ ×E₈-bundle as described above. The SU(1)-bundles do not have any moduli, such that the moduli space corresponds to just one point [182]. In the explicit discussion of the Weierstrass form in our setting, p₊ has one modulus which corresponds to the modulus of the five-brane.

Finally, we consider the computation of the F-theory flux superpotential that has been carried out already in 6.1.3 but in a different context, see also [79]. Here, we do not need to recall all the details. As was discussed there, the different triangulations of ˜X4 correspond to different five-brane phases. It was proposed that the four-form flux, for the two possible five-brane phases, is given by the basis element

ˆ

γ₁⁽²⁾= (−θ₁²+¹₂θ₃(θ₁+θ₃) +¹₆θ₄(θ₁+θ₃))Ω₄|z=0 , (6.91) and the element (6.65) for the other phase, where theθ_i=z_idz^d

i are the logarithmic derivatives as introduced in (5.65). The moduli z₁, z₂ could be identified as the deformations of the complex structure of the heterotic threefold Z₃, whilez₃ corresponded to the deformation of the heterotic five-brane.¹⁹ Indeed, a non-trivial check of this identification was provided in section 6.1.3 and in [79], where it was shown that the F-theory flux superpotential (5.98) in the directions (6.91) matches with the superpotential for a five-brane configuration in a local Calabi-Yau threefold KP² → P² obtained by decompactifying Z3, where the non-compact five-brane is described as a point on a Riemann surface in the base B₂ of Z₃ as discussed in section 6.1.2. Using heterotic F-theory duality as in section 4.3.3 then implies that the flux (6.91) actually describes a compact heterotic five-brane setup.

18This blow-up can be equivalently described as a complete intersection as we will discuss in sections 7.2. A simple example of such a construction will be presented in section 8.4.2.

19The deformationz4 describes the change inp−.

Part III

Blow Up Geometries and

SU(3)-Structure Manifolds

Chapter 7

Five-Branes and Blow-Up Geometries

In this chapter we provide, following [60,81,100], a novel geometric description of the dynamics of five-branes via a dual non-Calabi-Yau threefold ˆZ₃. Starting with a spacetime-filling five-brane on a curve Σ in a Calabi-Yau threefoldZ3, as it may occur in Type IIB and heterotic M-theory compactifications, the basic motivation of [60] was to find a natural description, where both the bulk and brane fields associated to deformations of Z₃ respectively Σ are treated equally in an unified framework. To obtain such a description a very canonical procedure was proposed in [60], where the pair (Z₃,Σ) is replaced by a new threefold ˆZ₃with a distinguished divisorE, that is obtained by blowing up along Σ inZ₃. It was further argued, that both the geometric deformations of (Z₃,Σ) are canonically unified as complex structure deformations of Zˆ₃and it was suggested to use Picard-Fuchs equations for the complex structure deformations of ˆZ3 to a study the open-closed deformations of (Z3,Σ). Ultimately, both the flux and five-brane superpotential should be given as solutions to these Picard-Fuchs equations.

Indeed the details of this proposal and concrete examples of five-branes in Calabi-Yau threefolds Z₃, their open-closed Picard-Fuchs equations and the superpotentials have been worked out in [100]. Furthermore it has been argued to view the geometry of the blow-up Zˆ3 not only as a tool for calculations but as being dynamically generated by the five-brane backreaction. Evidence for its use to define a flux compactification of the Type IIB and heterotic string that is dual to the original five-brane setup was provided by the definition of an SU(3)-structure on ˆZ₃. It is the purpose of this chapter to introduce the physical and mathematical ideas that lead to the blow-up threefold ˆZ₃ as the natural geometry to analyze five-brane dynamics. Simultaneously this serves as a preparation both for the calculations of the Picard-Fuchs system on ˆZ₃ as well as for the physical interpretation of the blow-up. The details of the calculations of the open-closed superpotential for two concrete examples and the definition of the SU(3)-structure are presented in chapter 8 respectively in chapter 9.

We begin our discussion of five-brane dynamics in N = 1 Calabi-Yau compactifications in section 7.1. We emphasize that the presence of five-brane sources for the flux is more thoroughly treated using the notion of currents and naturally leads to the consideration of the open manifold Z₃ −Σ. In addition we discuss geometric deformations of the five-brane

curve Σ, where we distinguish between deformations leading, even at first order, to massive or light fields in the four dimensional effective action. We comment on the use of the five-brane superpotential for determining higher order obstructions of the light fields. Next in section 7.2 we are naturally led to the blow-up ˆZ₃, that we construct explicitly both locally and globally as a complete intersection. We give a detailed explanation for the unification of the bulk and brane geometrical deformations ofZ₃ respectively Σand their obstructions as pure complex structure deformations on ˆZ3 and a specific flux element. Then in section 7.3 we analyze the open-closed deformation space by studying the variation of the pullback ˆΩ of the Calabi-Yau three-form Ω of Z₃ to ˆZ₃ under a change of complex structure on ˆZ₃. There we also discuss the general structure of the open-closed Picard-Fuchs equations as obtained from a residue integral for ˆΩ. Finally in section 7.4 we present the lift of the flux and five-brane superpotential of (Z3,Σ) to the blow-up ˆZ3 and argue that both are obtained as solutions to the same open-closed Picard-Fuchs equations on ˆZ₃.

7.1 Five-Brane N = 1 Effective Dynamics

In this section we discuss basic aspects of five-brane dynamics. Our point of view will be geometrical and appropriate to formulate the blow-up proposal in section 7.2. We begin our discussion in section 7.1.1 with a brief summary of heterotic and Type IIB string compact-ifications with five-branes focusing on global consistency conditions and the use of currents to describe the localized brane sources. We are naturally led to work on the open manifold Z3−Σ, where all fields in the theory are well-behaved, even in the presence of the singular brane source. The geometrical deformations of the five-brane around the supersymmetric con-figuration specified by a holomorphic curve Σ are discussed in section 7.1.2, where we present a physically motivated discussion of both light and massive fields and their behavior under complex structure deformations of Z₃. A purely mathematical analysis of analytic families of holomorphic curves provides the necessary background to appreciate the use of the brane superpotential to determine higher order obstructions, as discussed in section 7.1.3.