D5-Brane N = 1 Effective Couplings and Coordinates

3.3. D5-BRANE N = 1 EFFECTIVE COUPLINGS AND COORDINATES 55

denotes the inverse of ReΘ_ab. The function L^α_AB¯ is defined in (3.42). Note that we recover the N = 1 coordinates (2.39) of the bulk O5/O9 setup discussed in section 2.2.3 if we set ζ =a= 0. The completion (3.64) by the open string fields is inferred from the couplings in the D5-brane action (3.39) and (3.48).

The full N = 1 K¨ahler potential is determined by integrating the kinetic terms of the complex scalars M^I = (S, t, P , z, ζ, a). It takes the form

K=−ln

−i Z

Ω∧Ω¯i

+K_q , K_q =−2 ln√

2e^−2φVi

, (3.65)

where K_q has to be evaluated in terms of the coordinates (3.64). In contrast to general compactifications withO3/O7-orientifold planes, cf. section 2.2.3, this can be done explicitly for O5-orientifolds yielding

K_q =−ln₁

48KαβγΞ^αΞ^βΞ^γ

−ln

S+ ¯S+¹₄(ReΘ)^ab(P+ ¯P)_a(P+ ¯P)_b−2µ₅ℓ²C^IJ¯a_Ia¯J¯

, (3.66) where we write

Ξ^α=t^α+ ¯t^α−µ₅L^α_AB¯ζ^Aζ¯^B¯ . (3.67) Note that the expression (3.65) for K can already be inferred from general Weyl rescaling arguments, e.g. from the factore^Kin front of theN = 1 potential (2.38). However, the explicit form (3.66) displaying the field dependence of K has to be derived by taking derivatives ofK and comparing the result with the bulk and D5-brane effective action. Let us also note that the expression (3.66) reduces to the results found in [160, 161] in the orbifold limit.

3.3.2 The Superpotential

Having defined the N = 1 chiral coordinates as well as the K¨ahler potential we are prepared to deduce the effective superpotential W. Using the general supergravity formula (2.38) for the scalar potential expressed in terms of W we are able, as presented below, to deduce the superpotential W entirely by comparison to the scalar potential V_F in (3.61) as derived from dimensional reduction. This indeed identifiesVFas an F-term potential of theN = 1 effective theory as indicated by the notation.

The superpotential W yielding V_F consists of two parts, a truncation of the familiar flux superpotential for the closed string moduli [14] and a contribution encoding the dependence on the open string moduli of the wrapped D5-brane,

W = Z

Z3

F₃∧Ω +µ₅ Z

Σ+

ζyΩ, (3.68)

where we introduced the R–R-flux F₃. Now, it is a straightforward but lengthy calculation summarized in appendix A.3 to obtain the F-term contribution of the scalar potential (2.38).

3.3. D5-BRANE N = 1 EFFECTIVE COUPLINGS AND COORDINATES 57 The detailed calculations yield the positive definite F-term potential

V = ie^4φ 2V²R

Ω∧Ω¯

|W|²+D_z^κW D_z¯^¯κW G¯ ^κ¯κ+µ₅ G^AB¯e^−φ Z

Σ+

s_AyΩ Z

Σ+

¯ sB¯yΩ¯

. (3.69) Here the covariant derivatives with respect to the complex structure coordinatesz^κ and the open string moduliζ^A read

D_z^κW = Z

F₃∧χ_κ+µ₅ Z

ζyχ_κ , D_ζ^AW =µ₅ Z

s_AyΩ + ˆK_ζ^AW . (3.70) Furthermore, we have to use the first order expansion ofs_AyΩ discussed in (3.21) to obtain a form of type (1,1) that can be integrated over Σ₊ yielding a potentially non-vanishing result,

Z

Σ+

s_AyΩ = Z

Σ+

s_Ayχ_κδz^κ. (3.71)

Inserting this into (3.69), the F-term potential perfectly matches the scalar potential V_F of (3.61) obtained by dimensional reduction of the D5-brane as well as the bulk supergravity action.

The superpotential (3.68) is the perturbative superpotential of the Type IIB compactifi-cation. However, in the form (3.68) it is just the leading term in the expansion of the chain integral⁵ [65, 67, 106, 107]

W_brane= Z

Γ

Ω, (3.72)

where Γ is a three-chain with boundary given as ∂Γ = Σ−Σ0, where Σ0 is a fixed refer-ence curve in the same homology class as Σ. W_brane depends on the closed string complex structure moduli through the holomorphic three-form Ω and on the open string fields through the deformation parameters of the curve Σ. Using the general power series expansion of a functional about a reference function, we recover our result for the superpotential (3.68) to linear order.⁶ It is one central aim of this thesis to study and exactly calculate this brane superpotential in various setups and invoking different physical and mathematical techniques.

We conclude with a discussion of the derivation and the special structure of the F-term potential. We first note that the potential (3.69) is positive definite unlike the generic F-term potential of supergravity. This is due to the no-scale structure [162–164] of the underlying N = 1 data. Indeed, the superpotential (3.68) only depends on z and ζ and is independent of the chiral fieldsS,P,aandt. Consequently, theN = 1 covariant derivativeD_M^IW of the superpotential simplifies to K_MIW when applied with respect to the fields M^I= (S, P , a, t).

The K¨ahler potential (3.65) for these fields has the schematic form

K =−m ln(t+ ¯t+f(ζ,ζ))¯ −nln(S+ ¯S+g(P + ¯P , t+ ¯t) +h(a,¯a)) (3.73)

5In this section we setµ5= 1.

6The general Taylor expansion is given byF[g] =P∞ k=0

R dx1· · ·dxk1 k!

δ^kF[g]

δg(x1)···δg(x_k)

˛

˛

˛_g=˜

g

δg(x1)· · ·δg(xk).

ForW as a functional of the embeddingιandδι≡ζ as well as ˜g=ιwe to first order derive the second term of (3.68)).

withm= 3 andn= 1, where we concentrate on the one-modulus case for each chiral multiplet in order to clarify our exposition. The generalization to an arbitrary number of moduli is straightforward, cf. appendix A.3, where also the functionsf, g and h can be found. Then the contributions of the fieldsM^I = (S, P, a, t, ζ) to the scalar potentialV are found to take the characteristic form given by

K^IJ¯D_MIW D_M¯J¯W¯ =|∂_ζW|²K^ζζ¯+ (n+m)|W|² (3.74) as familiar from the basic no-scale type models of supergravity.⁷ Consequently, this turns the negative term−3|W|² in (2.38) into the positive definite term |W|² of (3.69) for the case n= 1 andm= 3. A similar structure for the underlyingN = 1 data has been found for D3-and D7-branes as shown in [61–63, 165–167].⁸ In particular, this form for the scalar potential V on the complex structure and D-brane deformation space implies that a generic vacuum is de Sitter, i.e. has a positive cosmological constant, while in a supersymmetric vacuum bothV andW vanish. However, the potential depends on the K¨ahler moduli only through an overall factor of the volume and thus drives the internal space to decompactify.

3.3.3 The Gauge-Kinetic Function, Gaugings and D-term Potential

In the following we discuss the terms of the four-dimensional effective action arising due to the U(1) vector multiplets in the spectrum. Firstly, there are the kinetic terms of the D5-brane vectorAand the vectorsV arising from the expansion (3.14) of the R–R formC₄. The gauge-kinetic function is determined from the actions (3.39) and (3.48) and reads

f_ΣΣ(t^Σ) = ¹₂µ₅ℓ²t^Σ , f_kl(z^κ) =−₂ⁱM¯kl=−₂ⁱFkl

z^k=0=¯z^¯k , (3.75) where the complex matrix M is defined in appendix A.1. Here f_ΣΣ is the gauge-coupling function for the D5-brane vector Aandf_klis the gauge-coupling function for the bulk vectors V discussed in (2.51). As reviewed in section 2.2.3, we note that the latter can be expressed via Fkl =∂_zk∂_zlF as the second derivative of the N = 2 prepotential F with respect to the N = 2 coordinatesz^k which have then to be set to zero in the orientifold set-up. This ensures that the gauge-coupling function is holomorphic in the coordinates z^κ which would not be the case for the full N = 2 matrix ¯MKL given in (A.2).

There are some remarks in order. Firstly, we note that the gauge-kinetic function encod-ing the mixencod-ing between the D5-brane vector and the bulk vectors is discussed in appendix A.2. Secondly, we observe that the quadratic dependence of fΣΣ on the open string moduli ζ through the coordinatet in (3.64) is not visible on the level of the effective action. These

7This no-scale structure will be clarified further, extending the example of [128], in appendix A.3 using the dual description ofS+ ¯S in terms of a linear multipletL.

8See [168] for a similar discussion in heterotic M-theory.

3.3. D5-BRANE N = 1 EFFECTIVE COUPLINGS AND COORDINATES 59 corrections as well as further mixing with the open string moduli are due to one-loop correc-tions of the sigma model and thus not covered by our bulk supergravity approximation nor the Dirac-Born-Infeld or Chern-Simons actions of the D5-brane.

Let us now turn to the terms in the scalar potential induced by the gauging of global shift symmetries and compare to the potentialV_D in (3.63). There are two sources for such gaugings. The first gauging arises due to the source term proportional to d(˜ρ^Σ− C(2)B^Σ)∧A in (3.48). It enforces a gauging of the scalars dual to the two-forms ˜ρ^Σ and C(2). In fact, eliminating d˜ρ^Σ and dC(2) by their equations of motion, the kinetic terms of the dual scalars ρa and h contain the covariant derivatives

Dρ_a=dρ_a+µ₅ℓδ_a^ΣA , Dh=dh+µ₅ℓB^ΣA , (3.76) where A is the U(1) vector on the D5-brane. Rearranging this into N = 1 coordinates we observe that the signs in the covariant derivative of h and ρ arrange⁹ to ensure that the complex scalar S defined in (3.64) remains neutral under A. However, the gaugings (3.76) imply a charge for the chiral field P_Σ. It is gauged by the D5-brane vector A. Its covariant derivative is given by

DP_Σ=dP_Σ+iµ₅ℓA . (3.77)

The second gauging arises in the presence of electric NS–NS three-form flux ˜e_K introduced in (3.62). It was shown in [128], that the scalarhis gauged by the bulkU(1) vectorsV arising in the expansion (3.14) ofC₄. This forces us to introduce the covariant derivative

DS =dS−i˜e_K˜V^K˜ . (3.78)

The introduction of magnetic NS–NS three-form flux is more involved and leads to a gauged linear multiplet (φ, C(2)) as described in [128].

Having determined the covariant derivatives (3.77) and (3.78) it is straightforward to evaluate the D-term potential. Recall the general formula for the D-term [59]

K_IJ¯X¯_k^J¯=i∂_ID_k , (3.79) where X^I is the Killing vector of the U(1) transformations defined as δM^I = Λ^k₀X_k^J∂_JM^I. For the gaugings (3.77) and (3.78) we find the Killing vectorsX^PΣ =iµ₅ℓ and X^S_˜

K =−i˜e_K˜ which are both constant. Integrating (3.79) one evaluates the D-terms usingK_PΣ = ˜K_PΣ and K_S given in (A.14) respectively above (A.13) in appendix A.3 as

D=−¹₄µ₅ℓe^φB^ΣV⁻¹ , D_K˜ = ¹₂˜e_K˜e^φV⁻¹ . (3.80) Inserting these D-terms into the N = 1 scalar potential (2.38) and using the gauge-kinetic functions (3.75), we precisely recover the D-term potential V_D in (3.63) found by dimensional reduction.

9The plus sign in the covariant derivative ofharises due to the minus sign in the duality conditions (2.10).

Im Dokument Holomorphic Couplings In Non-Perturbative String Compactifications (Seite 67-72)