The N = 1 Couplings and Coordinates

Its inverse is given by

G^αβ =−2

3KK^αβ + 2v^αv^β , G^ab=−2

3KK^ab , (2.36)

whereK^αβ andK^ab are the formal inverses ofKαβ and Kab, respectively.

Having defined the couplings of the reduced quaternionic sector we note that the corre-sponding orientifold actions (2.25), (2.27) for theO5/O9- and theO3/O7-setup differ since the massless spectrum from the R–R-sector differs, which in particular yields different couplings of the B-field zero modesb^a.

2.2. EFFECTIVE ACTION OF TYPE IIB CALABI-YAU ORIENTIFOLDS 33

The K¨ahler potential and N = 1 coordinates

As a first step to identify the N = 1 characteristic data we have to find the appropriate complex coordinates for which the scalar metrics are manifestly K¨ahler. First, we note that the complex structure moduliz^κ, respectively z^k, already define K¨ahler coordinates for both orientifolds since the scalar metric in the actions (2.25), (2.27) is already K¨ahler with K¨ahler potential (2.30). The other K¨ahler coordinates are different for the two setups and are dis-cussed next.

ForO5/O9-orientifolds the additional complex fields in the chiral multiplets read [128,130]

t^α=e^−φv^α−ic^α , Pa= Θ_abb^b+iρa , S=e^−φV+i˜h−¹₄(ReΘ)^abPa(P+ ¯P)_b , (2.39) where v^α, b^a, c^α, ρ_a and ˜h = h− ¹₂ρ_ab^a are given in (2.15), (2.19) and (2.21). The complex symmetric tensor Θ appearing in (2.39) is given by Θ_ab =Kabαt^α and (ReΘ)^ab denotes the inverse of ReΘ_ab. More conceptually, the coordinates (2.39) can be obtained by probing the cycles in Z₃/σ by D-branes and are then formally extracted from [128, 130, 149–151]

Im(ϕ^ev)−iA=t^αω_α−P_bω˜^b−Sm₆. (2.40) Here we introduced the polyforms of even formsϕ^ev,Ain the NS–NS- respectively R–R-sector,

ϕ^ev =e^−φe^−B2+iJ , A=e^−B2 ∧ X

q=0,2,4,6,8

Cq , (2.41)

which have to be interpreted, as (2.40), as formal polynomials of forms in the even cohomology H^ev(Z₃) of different degrees. Then the reduction ans¨atze (2.15), (2.19) and (2.21) have to be inserted to obtain the coordinates (2.39) as the coefficients of a basis expansion of (2.40).

Roughly, the forms ϕ^ev and A are the Dirac-Born-Infeld and Chern-Simons actions of Dp-instantons, p= 1,3,5, wrapping the even-dimensional cycles and the coordinates (2.39) are the complexified volumes of the corresponding cycles in the Type IIBO5/O9-background.

For O3/O7-orientifolds besides the complex structure moduli z^k the additional K¨ahler coordinates read [128, 130],

τ =l+ie^−φ, G^a=c^a−τ b^a, T_α =i(ρ_α−¹₂Kαabc^ab^b)+¹₂e^−φKα+ i

2(τ−τ¯)KαbcG^b(G−G)¯ ^c , (2.42) whereKα is introduced in (2.35). This data is again summarized as

Re(ϕ^ev)−iA=−iτ−iG^aω_a−T_αω˜^α, (2.43) which is interpreted as the complexified volume of even dimensional cycles wrapped by Dp-instantons,p=−1,1,3, in the Type IIB O3/O7-background.

The definition of the appropriateN = 1 coordinates prepares us for the determination of theN = 1 coupling functions. As mentioned before, we do not obtain superpotential a W

due to the absence of fluxes and D-branes,W = 0. We first determine the K¨ahler potential K. In general, the fullN = 1 K¨ahler potential is determined by integrating the kinetic terms of the complex scalarsM^I.

ForO5/O9-orientifolds it takes the form [128, 130]

K = K_cs(z,z) +¯ K^q(S, t, P) , K_cs=−lnh

−i Z

Ω∧Ω¯i

(2.44) where we introduce the K¨ahler potential

K^q = −lnh

1

48Kαβγ(t+ ¯t)^α(t+ ¯t)^β(t+ ¯t)^γi

−lnh

S+ ¯S+¹₄(P + ¯P)a(ReΘ⁻¹)^ab(P+ ¯P)_bi

= −2 ln√

2e^−2φV

. (2.45)

We note that the scalar metric corresponding to K is block-diagonal in the fields z and (S, P , t). Thus, we infer that the scalar manifold of the O5/O9-orientifold effective theory is locally a direct productM^cs× M^q. The first factor is the special K¨ahler manifold with local coordinates z^κ. It is realized as a submanifold of the complex structure moduli space of Z₃, which itself is special K¨ahler. The second factor M^q is a K¨ahler submanifold of dimension h^(1,1)+ 1 in the quaternionic manifold of theN = 2 hypermultiplets. Locally it is a fibration of the scalar manifold of P,S over the K¨ahler moduli space of the orientifold Z₃/σ.

ForO3/O7-orientifolds the K¨ahler potential reads [128, 130]

K = K_cs(z,z) +¯ K^q(τ, T, G) , K_cs=−lnh

−i Z

Ω∧Ω¯i

(2.46) where we introduce the K¨ahler potential

K^q = −lnh

−i(τ−τ¯)i

−2 lnh1

6e^−32φKαβγv^αv^βv^γi

=−2 ln√

2e^−2φV

. (2.47) We note that the scalar manifold of the O3/O7-orientifold effective theory is again locally a direct product M^cs× M^q. However, the first factor is a different special K¨ahler submanifold with local coordinates z^k of the complex structure moduli space of Z3 than in the O5/O9 case. The second factor M^q is a K¨ahler submanifold of the quaternionic manifold of the N = 2 hypermultiplets. Locally it is a fibration of the scalar manifold of T, G over the dilaton moduli space. We note that the full K¨ahler potentialK is invariant, up to a K¨ahler transformation, under the action of SL(2,R) on τ as required by SL(2,R)-duality of the ten-dimensional Type IIB theory.

In general a first check of the consistency of the K¨ahler coordinates (2.39), (2.39) with the corresponding K¨ahler potentials (2.44), (2.46) is provided by a comparison with the expected result from Weyl rescaling,

K =−2 ln√ 2e^−2φV

Z

Ω∧Ω¯

. (2.48)

2.2. EFFECTIVE ACTION OF TYPE IIB CALABI-YAU ORIENTIFOLDS 35 This is in perfect agreement with the above results. However, we emphasize that the real difficulty in the determination of the N = 1 characteristic data is to express the general expression (2.48) as a function of appropriate coordinates so that all scalar kinetic terms, as well as some mixing terms, in the action are reproduced.

We conclude by rephrasing the N = 1 K¨ahler potential in a more formal and compact form for both orientifolds. We note that both K¨ahler potentials (2.45), (2.47) can be written in a unified way as [128, 130]

K^q = −2 ln Φ_B , Φ_B :=ihϕ^ev,ϕ¯^evi (2.49) up to a K¨ahler transformation where we introduced the pairing [152]

ϕ, ψ

= Z

Z3

X

m

(−1)^mϕ_2m∧ψ_6−2m (2.50) on the space of polyforms. The only difference between the two orientifold setups then reduces to the choice of variables on which (2.49) depends. For the case of O5/O9-orientifolds K^q is a function of Im(ϕ^ev) only, in particular the real part has to be understood as a function of Im(ϕ^ev). For O3/O7-orientifolds, however, (2.49) is a function of Re(ϕ^ev) and the imaginary parts is determined by Re(ϕ^ev).

The Gauge Kinetic Function

We conclude with the determination of the gauge kinetic couplingf_κλwhich is a holomorphic function of the chiral superfields. By comparison of the actions (2.25), (2.27) with the general N = 1 action (2.37) we obtain [128, 130]

O5/O9 : f_kl=−₂ⁱM¯kl

z^k=0=¯z^l , O3/O7 : f_κλ=−₂ⁱ M¯κλ

z^κ=0=¯z^λ (2.51) whereMis introduced in appendix A.1 and a submatrix of theN = 2 gauge kinetic coupling matrix. It has to be evaluated on the submanifoldz^k = ¯z^l = 0 or z^κ= ¯z^λ = 0, respectively, in the complex structure moduli space of Z₃. Its holomorphicity in the complex structure deformations z^κ, z^k follows since f can be rewritten for both orientifolds in terms of the holomorphicN = 2 prepotentialF(z) evaluated on the submanifold M^cs [128, 130],

O5/O9 : f_kl(z^κ) =−₂ⁱFkl

z^k=0=¯z^l , O3/O7 : f_κλ(z^k) =−₂ⁱFκλ

z^κ=0=¯z^λ . (2.52)

Chapter 3

The D5-Brane Effective Action

In this chapter we begin with the analysis of brane dynamics for the example of D5-branes in generic O5/O9-Calabi-Yau orientifolds. We present a detailed computation of the N = 1 effective action of a spacetime-filling D5-brane wrapping a curve Σ, mainly following the original work [60]. First in section 3.1 we start with a general review of supersymmetric D-branes in Calabi-Yau manifolds, where we focus on their low-energy effective dynamics as encoded by the Dirac-Born-Infeld and Chern-Simons action and on the geometric calibration conditions on the internal cycles supporting a BPS D-brane. Then the actual calculation of the D5-brane effective action is performed in section 3.2 by a purely bosonic reduction.

This is divided into four parts, in which we first count the bosonic four-dimensional massless spectrum both of the closed and open string sector, then present some special identities on the space of open-closed geometric moduli before we actually derive the D5-brane effective action, where we put special emphasis on the determination of the scalar potential. The calculation of the scalar potential reveals the ad-hoc surprising relevance of keeping non-dynamical three-forms in the four-dimensional action in order to recover the complete F-term potential in a purely bosonic reduction. Next in section 3.3 we work out theN = 1 characteristic data of the D5-brane action, in particular in comparison withO5/O9-orientifold result of chapter 2. We determine new N = 1 chiral coordinates and the K¨ahler potential that is derived explicitly at large volume of Z3. Then the effective superpotential is read off, which is a projection of the familiar Type IIB flux superpotential and the D5-brane superpotential, that is given as a chain integral. The N = 1 data is completed by the determination of the D5-brane gauge kinetic function, the gaugings of chiral fields and the corresponding D-terms. Finally in section 3.4 we extend, following [100], the reduction of [60] to the full geometric deformation space of the D5-brane which is infinite dimensional and does include massive fields. This is proposed to be the natural domain of the D5-brane superpotential which encodes the general deformation theory of the curve Σ in Calabi-Yau threefoldZ₃. We concluded by a calculation of the corresponding F-term scalar potential as a functional of the infinite dimensional space of massive modes. Further details on some calculations performed in the context of the derivation of the D5-brane action can be found in appendix A.

3.1 BPS D-Branes in Calabi-Yau Manifolds

In general the existence of D-branes as dynamical objects in string theory can be inferred from first principles, namely the spectrum of BPS-particles and string dualities. It is a basic fact found in textbooks, see e.g. [5], that the fundamental string state with two oscillator modes, winding numbermand no momentum along a compact spacetime direction, here an S¹, has a mass proportional to m. The winding number m is the quantized charge with respect to the NS–NS B-field and indeed this state is identified as a BPS-state in the theory [132]. In combination with the non-perturbativeSL(2,Z)-symmetry of Type IIB theory, that acts on the B-fieldB₂ and the R–R-formC₂ as a doublet, this predicts a whole family of BPS-states with charge (p, q) under (B₂, C₂) for any pair of relative primes p, q [52]. In particular one expects fundamental states with charge (0,1) that are charged under C₂. Since C₂ couples electrically to string-like objects the fundamental object corresponding to these states is called a D-string. Analogously, the strings associated to BPS-states with charge (p, q) are denoted (p, q)-strings. Then by T-duality one expects BPS-states charged under all R–R-forms C_p in the theory which correspond to quantized p+ 1-dimensional membranes. However, these states do not have a perturbative description in terms of quantized fundamental string exci-tations since all perturbative states do not couple to the R–R-forms but their field strengths.

Consequently, these states are invisible in the spectrum of the perturbative string. However, they have convenient descriptions in various limits of string theory.

On the one hand, at low-energies membranes with non-zero electric R–R-charge are found in the effective supergravity theory as solitons, i.e. stable finite energy solutions of the classical equations of motions with a conserved (topological) charge called p-branes, see e.g. [5] for a summary. Also the fundamental string occurs on an equal footing as a solitonic solution with non-zero NS–NS-charge. It can further be shown that these solutions indeed obey a BPS-bound. On the other hand, membranes are visible at small coupling in the perturbative string as 9−p Dirichlet boundary conditions on the string endpoints, denoted by Dp-branes.

Both descriptions of membranes have different regimes of validity, however. The require-ment of the supergravity description is low curvature, which in more technical terms means, that the Schwarzschild radius r_S of the p-brane solution is big compared to the string scale

√α^′. This requirement yields [40]

n_pg_S>>1 (3.1)

wheren_p is the integral number of fundamentalp-brane charge of the solution. The require-ment for the validity of the perturbative string is of course small string coupling g_S so that we obtain

n_pg_S <<1, (3.2)

which is the opposite to (3.1). Despite their different validity, however, it is believed that both thep-brane and the Dp-brane describe one and the same fundamental object simply denoted

3.1. BPS D-BRANES IN CALABI-YAU MANIFOLDS 39 as a Dp-brane. This conjecture is well established by various checks invoking techniques from different corners of string theory starting with the CFT analysis in the groundbreaking work [37], see [16, 17, 28, 29] for reviews.

For our purposes we treat spacetime-filling D-branes and their dynamics as a BPS-background state in Type IIB string theory with small fluctuations around it. Here we will mainly be restricted to low energies which means that we are working with Type IIB super-gravity and to scales larger than the thickness of the D-brane where the description in terms of the D-brane effective action is valid. Then the BPS condition on the D-brane is translated into BPS calibration conditions [36] on the embedding of the D-brane worldvolume Wp+1

and the D-brane worldvolume fields into the Type IIB background under consideration. By additional consistency conditions of the setup imposed on the one hand byN = 1 supersym-metry in extended four-dimensional Minkowski space and on the other hand by cancellations of tadpoles we will consider Calabi-Yau threefoldsZ₃ with O5/O9-orientifolds.

Following this program we start by introducing the effective action of a single D-brane.

The spectrum of light fields on the D-brane can in general be thought of as a reduction of the N = 1 Super-Yang-Mills theory in ten dimensions to the worldvolumeWp+1 [52]. It consists of anN = 1 supersymmetricU(1)-gauge theory on the submanifoldWp+1. In addition there are adjoint-valued scalars corresponding to fluctuationsδιof the embeddingι : Wp+1 →M₁₀ into ten-dimensional spacetime. The effective action of these fields consists of two parts, one part being the Dirac-Born-Infeld action [4, 38, 39]

S_DBI^SF =−Tp

Z

Wp+1

e^−φp

detι^∗(g+B2)−ℓF (3.3) in string-frame units, whereT_p denotes the Dp-brane tension and ℓ= 2πα^′. The second part of the effective action is the Chern-Simons action encoding the coupling of the D-brane to the R–R-sector,

S_CS =−µ_p Z

Wp+1

X

q

C_q∧e^ℓF−B2, (3.4)

where µ_p denotes the D-brane charge. The formal sum over all R–R-potentials indicates a coupling of the D-brane to forms of degree lower than C_p+1 in a background of worldvolume flux/instantons [153]. We note that the presence of the B-field B₂ in the combination B :=

B₂−ℓF can be understood from gauge invariance in the bulk. The string sigma-model action for world-sheets W2 with boundary contains the terms

S_σ ⊃ 1 ℓ

Z

W2

B₂+ Z

∂W2

A , (3.5)

where A denotes the gauge field on the D-brane. Since this is not invariant under the gauge transformation δB₂ = dΛ alone, one has to demand δA = ¹_ℓΛ. Thus, B is the only gauge invariant combination [52].

Next we formulate the BPS conditions a D-brane in a Calabi-Yau manifold has to meet to define a supersymmetric background [36, 134]. The most immediate condition is of course

µ_p =T_pwhich is enforced by the SUSY algebra for every BPS state. For vanishing background fields the BPS-conditions further imply for a spacetime-filling D6-brane in Type IIA to wrap a special Lagrangian submanifold L, which can roughly be thought of as a real locus of real dimensions d in a complex manifold of complex dimension d. In terms of the complex structureI onZ₃ this is expressed byI(T L)⊂N L. A trivial example is a special Lagrangian L inC which is just e.g. the real axis, whereI is the usual multiplication by i. Equivalently L is defined as the volume-minimizing representative within its homology class such that the volume form vol_L on L is calibrated as [134]

ι^∗(J) = 0, e^−iθD6ι^∗(Ω) =

pdet(g+B)

pdet(g) vol_L . (3.6)

The constant angle θD6 is a priori free and determines the covariantly constant spinor corre-sponding to the supercharge conserved by the BPS D-brane. More precisely θ_D6 determines the linear combination of the two covariantly constant spinors of the Type II compactifica-tion onZ₃ [36] that is unbroken by the D-brane background and thus yields a supercharge in four dimensions generating N = 1 spacetime supersymmetry. All additional D-branes in the setup have to be calibrated with respect to the same calibration forme^−iθDpι^∗(Ω) in order to preserve the same covariantly constant spinor and thus preserve N = 1 spacetime supersym-metry. The identical argumentation applies to orientifold planes, which are O6-planes in the Type IIA case, that have to be included for tadpole cancellation discussed at the end of this section. Then, the free angle θ_D6 has to equal the angle θin the orientifold calibration (2.8).

In Type IIB a spacetime-filling BPS D-brane is supported along holomorphic submanifolds inZ₃ [36]. This in particular yields a natural choice of complex structure on Σ by aligning it with the ambient complex structure using the embeddingιwhich then defines a holomorphic map obeying ∂¯z^¯ı(u^a)/∂u=∂zⁱ(u^a)/∂u¯= 0, where we introduced complex coordinates u on the worldvolume Wp+1. In other words, D-branes in Type IIB test the even dimensional homology of Z₃, that are D3-branes on points , D5-branes on holomorphic curves Σ, D7-branes on holomorphic divisors D or D9-branes on the entire Calabi-Yau threefold Z₃. A holomorphic submanifold is volume minimizing since its volume form is just proportional to (powers of) the pullback ofJ, a well-known mathematical fact for complex submanifolds of K¨ahler manifolds [154]. In the presence of background fields this is replaced by the more general calibration conditions [134]

ι^∗(Ω) = 0, e^−iθDpι^∗e^J+iB|Dp=

pdet(g+B)

pdet(g) vol_Dp, p= 3,5,7,9. (3.7) where vol_Dp denotes the volume form on the analytic cycle wrapped by the Dp-brane. Again we note that in the presence ofOp-orientifold planes, a necessary ingredient to cancel tadpoles, the angle θ_Dp has to equal the angle in the orientifold calibration (2.8), which is θ_Dp = 0.

Then the calibrations (3.7) are even more restrictive. Explicitly, we evaluate the real and

3.2. DYNAMICS OF D5-BRANES IN CALABI-YAU ORIENTIFOLDS 41

Im Dokument Holomorphic Couplings In Non-Perturbative String Compactifications (Seite 44-53)