Special Relations on the N = 1 Moduli Space

In this section we discuss a subtlety in the decomposition (3.19). The notion of ζA being a complex scalar field depends on the background complex structure chosen on the ambient Calabi-Yau Z3, i.e. on the split (3.19), NRZ3Σ+⊗ C = NZ3Σ+⊕ NZ3Σ+, into holomorphic

and anti-holomorphic parts. To explore this dependence further it is natural to consider the contractions of the sA with the holomorphic (3, 0)-form Ω, the (2, 1)-forms χκ introduced in

(3.13) and their complex conjugates. In the background complex structure defined at z0 we

find, in the cohomology of Z3 as well as in the cohomology of Σ, that

sAyΩ(z₀) = 0 , s_Ayχ¯_κ(z₀) = 0 , s_Ay ¯Ω(z₀) = 0 . (3.20)

These contractions vanish on Z3 since there are no non-trivial (2, 0)-forms in H2(Z3). More-

over, they also vanish on Σ for a supersymmetrically embedded D5-brane. In addition, since every two-form pulled back to Σ has to be proportional to the (1, 1)-K¨ahler form J, only sAyχ_κ can be a non-trivial (1, 1)-form on Σ. Note, however, that also s_Ayχ_κ is trivial in the cohomology of Z3 due to the primitivity of H(2,1)(Z3).

However, in the four-dimensional effective theory we also have to allow for possible fluctua- tions around the supersymmetric background configuration, including those corresponding to complex structure deformations of Z3. The holomorphic three-form Ω as well as the complex

scalars ζ are then functions of the complex structure parameters zκ_{. In a different complex}

structure on Z3, the notion of holomorphic and anti-holomorphic coordinates on Z3, encoded

by the type of Ω(z), has not to be aligned with the splitting into complex scalars (3.19) in general. To measure this discrepancy, we consider the pullback ι∗_(sAyΩ(z)) on Σ. For z = z0+ δz near a background complex structure z0 we expand Ω(z) to linear order in δz as

ι∗(sAyΩ(z)) = (1− K_κδzκ)ι∗(s_AyΩ(z₀)) + ι∗(s_Ayχ_κ(z₀))δzκ = ι∗(s_Ayχ_κ(z₀))δzκ, (3.21) where we used (2.18) and (3.20). In other words, the form sAyΩ is a (2, 0)-form on Σ in the complex structure z but a (1, 1)-form on Σ in the complex structure z0 to linear order in the

complex structure variation δz. However, a similar argument shows that

(sAy ¯Ω)(z) = (s_Ayχ)(z¯ κ) = 0 , (3.22)

even to linear order in δzκ. These forms only appear at higher order in the complex structure variations.

The above considerations allow us to describe the metric deformations of the induced metric ι∗_{g on the two-cycle Σ+. In general, both the complex structure deformations of}

Z3 and the fluctuations of the embedding map ι contribute. Here, we discuss only those

variations δ(ι∗g) originating from complex structure deformations and postpone the analysis of all possible metric variations to section 3.2.3. Analogously to (3.13) the complex structure

deformations on Σ+ are encoded in the purely holomorphic metric variation ι∗(δg)uu= 2ivΣ R Ω∧ ¯Ωι ∗_(s AyΩ)_uu(ι∗g)u¯uι∗(¯s_¯ Byχ¯_κ¯)_uu¯ GA ¯B δ¯z¯κ . (3.23) Here we have introduced the volume of the holomorphic two-cycle Σ+ as

vΣ= Z Σ+ d2u√g = Z Σ+ ι∗J (3.24)

and a natural hermitian metric_{GA ¯B} given by GA ¯B=− i V Z Σ+ sAy¯s_¯ By(J)ι∗J. (3.25)

We will show later on that GA ¯B can be obtained by dimensional reduction, cf. section 3.2.3.

It is identified with the metric for the moduli ζ on the open string moduli space and is independent of the coordinates u, ¯u on Σ.

The metric variation (3.23) can be explained by application of some useful formulas for the open string moduli space. First, we use the fact that H(1,1)_(Σ

+) is spanned by the pullback

ι∗J. This is exploited to rewrite the pullback of any closed (1, 1)-form ω to Σ+in cohomology,

cf. (3.45). In particular we obtain ι∗(sAyχ_κ) = ι∗J vΣ Z Σ+ ι∗(sAyχ_κ) , (3.26)

which can be written after multiplication with_V−1_GA ¯B_g(s

C, ¯s_B¯) and by using (3.25) as Z Σ+ ι∗(sAyχ_κ) =− vΣ V Z Σ+ g(sA, ¯s_B¯)GBC¯ ι∗(sCyχ_κ). (3.27) We evaluate this for every choice of sA and compare the coefficients on both sides to relate

the metric on the normal bundle NZ3Σ and the metric GA ¯B.

Thus, the identity (3.27) allows us to infer the metric variations (3.23) from the complex structure deformations on Z3. First, we consider the pullback to Σ+ of the metric variations

δgij in (3.13) of the ambient Calabi-Yau Z3

ι∗(δg)uu = R iV

Ω_{∧ ¯}Ω Ω

¯ı¯

u ( ¯χ¯κ)¯ı¯uδ¯z¯κ. (3.28)

Then we replace, motivated by (3.27), the inverse metric gi¯ _{occurring in the contraction of}

¯

χ¯κ and Ω by si_As_B¯¯GA ¯B to obtain our ansatz for the induced metric deformation on Σ+ given

in (3.23).

However, there are some remarks in order. Since there are no (2, 0)-forms on Σ+ in the

background complex structure z0, the form ι∗(sAyΩ) should vanish identically. Thus, in order to make sense of the metric variation (3.23) we have to evaluate it, following the logic of (3.21),

3.2. DYNAMICS OF D5-BRANES IN CALABI-YAU ORIENTIFOLDS 47 in the complex structure z = z0+ δz. Applying this to (3.23) we expand δ(ι∗g) to linear order

in δz, i.e. ι∗(δg)uu(z) = ι∗(δg)uu(z0) + ι∗(δg)u¯u(z0)· δz, to obtain

ι∗(δg)u¯u(z0) = 2ivΣ R Ω∧ ¯Ωι ∗_(s Ayχ_κ)_uu¯ (ι∗g)u¯uι∗(¯s_¯ Byχ¯_κ¯)_uu¯ GA ¯B δzκδ¯zκ¯ . (3.29) Here we emphasize the change in type from purely holomorphic indices δguu at z to mixed

type δgu¯u at z0. It is important to note that there are neither any metric deformations linear

in the complex structure parameter δz nor any of pure type.

These calculations stress that the analysis of the open string moduli space crucially de- pends on the chosen background complex structure encoded by the moduli zκ. It is hence natural that the complex structure parameters zκ _{of Z}

3 and the open string moduli ζAshould

be treated on an equal footing to characterize the structure of theN = 1 field space. This has let to the the introduction of the blow-up proposal in [60] that was further exploited in [81] and ultimately in [100]. It will be used in chapter 7 to unify the open-closed fields and to understand and derive the effective brane superpotential. In the next sections we derive the general four-dimensional effective D5-brane action and show that the brane superpotential is naturally encoded by the forms sAyΩ and s_Ayχ_κ that are sensible both to the complex structure on Z3 and the open string deformations.

We conclude by noting that the analysis of δ(ι∗g) and the induced variation (3.29) can also be derived in two alternative ways, one based on the consideration of all vector fields in C∞_{(Σ, NZ}

3Σ) and a different by analyzing the variation of vol(Σ) under a change of complex

structure. For the first way we introduce a basis sa of the infinite dimensional Hilbert space

of all sections in _C∞(Σ, NZ3Σ). The natural metricGa¯b on these sections is formally identical

to the integral in (3.25). Then we introduce holomorphic vielbeins en of the normal bundle

NZ3Σ obeying the elementary relation

gi¯eine¯¯m¯ = ηn ¯m, gi¯= ηn ¯meine¯m¯¯ , (3.30)

where gi¯denotes the hermitian metric on NZ3Σ and ηn ¯mthe flat hermitian metric. We readily

expand the holomorphic vielbeins with respect to the basis of sections sa as ein =

P ca

nsia.

Upon integrating the first relation in (3.30) over Σ+, we obtain

ηn ¯m= 1 vΣ Z Σ+ ηn ¯mι∗(J) = 1 vΣ Z Σ+ gi¯eine¯m¯¯ ι∗(J) = i_vV ΣGa¯b ca_n¯c¯b_m¯ , (3.31)

which is readily inserted into the second equation in (3.30) to obtain gi¯= vΣ

V G

a¯b_sn

asm¯b¯ . (3.32)

This is then again used in (3.28) to replace, to quadratic order in sa, the metric gi¯. By

expanding in the complex structures to second order, we reproduce (3.29) as before, however, with the basis of sections sa of C∞(Σ, NZ3Σ) instead of sA. Nonetheless, the relation (3.29)

is still a good approximation for the purpose of deriving the effective action since most of the modes associated to the saare massive, cf. section 3.4, and are consistently integrated out in

the effective Lagrangian maintaining only the light fields, that are precisely counted by the finite number of sections sA in H0(Σ, NZ3Σ).

For brevity, we describe the second method only in words. Since a variation δ(ι∗_{g) violates}

the calibrations (3.8) and thus the volume minimization property of Σ, the volume of Σ will increase by δ vol(Σ). Analyzing this variation to appropriate order in the fields it is possible to extract (3.29) or to directly obtain the corresponding potential term in the dimensional reduction of Dirac-Born-Infeld action (3.40). Indeed we follow a similar logic in section 3.4 to determine the potential on the infinite dimensional field space of massive deformations of Σ.