Moduli Stabilization, Large-Volume dS Minimum Without anti-D3-Branes, (Non-)Supersymmetric Black Hole Attractors and Two-Parameter Swiss Cheese Calabi-Yau's
aa r X i v : . [ h e p - t h ] M a r Moduli Stabilization, Large-Volume dS Minimum Without D -Branes,(Non-)Supersymmetric Black Hole Attractors and Two-ParameterSwiss Cheese Calabi-Yau’s Aalok Misra ( a ) , ( b )1 and Pramod Shukla ( a ) 2 (a) Department of Physics, Indian Institute of Technology, Roorkee - 247 667, Uttaranchal, India(b) Physics Department, Theory Unit, CERN, CH1211, Geneva 23, Switzerland Abstract
We consider two sets of issues in this paper. The first has to do with moduli stabilization, existence of“area codes” [1] and the possibility of getting a non-supersymmetric dS minimum without the additionof D WCP [1 , , , ,
9] in the “large-volume-scenario” limit [5]. The main result of our paperis that we show that by including non-perturbative α ′ and instanton corrections in the K¨ahler potentialand superpotential [6], it may be possible to obtain a large-volume non-supersymmetric dS minimum without the addition of anti-D3 branes a la KKLT. The chosen Calabi-Yau has been of relevance alsofrom the point of other studies of K¨ahler moduli stabilization via nonperturbative instanton contributions[7] and non-supersymmetric AdS vacua (and their subsequent dS-uplifts) using ( α ′ ) corrections to theK¨ahler potential [5, 8, 9, 10]. e-mail: [email protected] email: [email protected] Introduction
Flux compactifications have been extensively studied from the point of view of moduli stabilization (See [11]and references therein). Though, generically only the complex structure moduli get stabilized by turningon fluxes and one needs to consider non-perturbative moduli stabilization for the K¨ahler moduli[12]. Inthe context of type II compactifications, it is naturally interesting to look for examples wherein it may bepossible to stabilize the complex structure moduli (and the axion-dilaton modulus) at different points ofthe moduli space that are finitely separated, for the same value of the fluxes. This phenomenon is referredto as “area codes” that leads to formation of domain walls. Another extremely important issue related tomoduli stabilization is the problem of getting a non-supersymmetric de Sitter vacuum in string theory. TheKKLT scenario which even though does precisely that, has the problem of addition of an uplift term tothe the potential, corresponding to addition of D N = 1 SUGRAformalism. It would be interesting to be able to get a de Sitter vacuum without the addition of such D α ′ and instanton contributions) aspects of (non)supersymmetric flux vacua andblack holes in the context of type II compactifications on (orientifold) of compact Calabi-Yau’s of a projectivevariety with multiple singular conifold loci in their moduli space. The compact Calabi-Yau we work withis of the “Swiss cheese” type. The paper is planned as follows. In section , based on [16], we performa detailed analysis of the periods of the Calabi-Yau three-fold considered in this paper, working out theirforms in the symplectic basis for points away and close to the two singular conifold loci. The results ofsection get used in the subsequent section ( ). We then discuss, in section , stabilization of the complexstructure moduli including the axion-dilaton modulus by extremizing the flux superpotential for points nearand close to the two conifold loci, arguing the existence of “area codes” and domain walls. In section , we show that by the inclusion of non-perturbative α ′ -corrections to the K¨ahler potential that surviveorientifolding and instanton contributions to the superpotential, one can, analogous to [5], get a large-volume non-supersymmetric dS vacuum without the addition of D -branes . We consider this to be the mostsignificant result of this paper. In section , we explicitly solve the “inverse problem” using the techniquesof [2]. In section , using the techniques of [3] we show the existence of multiple superpotentials (includingtherefore “fake superpotentials”). Section has the conclusions. In this section, based on results in [16], we look at different regions in the moduli space of a two-parameterCalabi-Yau three fold of a projective variety expressed as a hypersurface in a weighted complex projectivespace, and write out the explicit expressions for the periods. The explicit expressions, though cumbersome,will be extremely useful when studying complex structure moduli stabilization and existence of “area codes”in section , solving explicitly the “inverse problem” in section and showing explicitly the existence of “fakesuperpotentials” in section in the context of non-supersymmetric black hole attractors. More precisely,1ased on [16], we will consider the periods of the “Swiss cheese” Calabi-Yau three-fold obtained as aresolution of the degree-18 hypersurface in
WCP [1 , , , , x + x + x + x + x − ψ Y i =1 x i − φx x x = 0 . (1)Similar to the explanation given in [17], it is understood that only two complex structure moduli ψ and φ are retained in (1) which are invariant under the group G of footnote 3, setting the other invariant complexstructure moduli appearing at a higher order (due to invariance under G ) at their values at the origin.Defining ρ ≡ (3 . ψ , the singular loci of (1) are in WCP [3 , ,
1] with homogenous coordinates [1 , ρ , φ ]and are given as under:1. Conifold Locus1 : { ( ρ, φ ) | ( ρ + φ ) = 1 } Conifold Locus2 : { ( ρ, φ ) | φ = 1 } Boundary : ( ρ, φ ) → ∞ Fixed point of quotienting : The fixed point ρ = 0 of A where A : ( ρ, φ ) → ( αρ, α φ ), where α ≡ e πi .We will be considering the following sectors in the ( ρ, φ ) moduli space: • | φ | > , < argφ < π , large ψ The fundamental period ̟ , obtained by directly integrating the holomorphic three-form over the“fundamental cycle” (See [16]), is given by: ̟ = ∞ X k =0 (6 k )! k !(2 k )!(3 k )! (cid:18) − ψ (cid:19) U k ( φ )= ∞ X k =0 ( − ) k Γ( k + )Γ( k + )( k !) (cid:18) ρ k (cid:19) U k ( φ ) , (2)where U ν ( φ ) ≡ φ ν F ( − ν , − ν , − ν ; 1 , φ ); the other components of the period vector are given by: ̟ i = ̟ ( α i ψ, α i φ ) where α ≡ e πi , i = 1 , , , , • | φ | < , large ψ The fundamental period is given by: ̟ = ∞ X m =0 ∞ X n =0 (18 n + 6 m )!( − φ ) m (9 n + 3 m )!(6 n + 2 m )!( n !) m !(18 ψ ) n +6 m , (3) The term “Swiss cheese” (See [4]) is used to denote those Calabi-Yau’s whose volume can be written as: V = ( τ B + P i = B a i τ Si ) − ( P j = B b j τ Sj ) − ... , where τ B is the volume of the big divisor and τ Si are the volumes of the h , − h , − This is induced by the group action: ( x , x , x , x , x ; ψ, φ ) → ( α A x , α A x , α A x , α A x , α A x ; α − a ψ, α − a φ ) where a = A + A + A + 6 A + 9 A and ( A , A , A , A , A ) are related to the coefficients of the most general degree-18 polynomialin ( x , x , x , x , x ) invariant under G = Z × Z ( Z : (0 , , , , , Z : (1 , − , , , p = x + x + x + x + x = 0, according to the Greene-Plesser construction, is given by { p = 0 } /G . The G − invariant polynomial is givenby: A ′ Q i =1 x i + A ′ x x x x + A ′ x x x x + A ′ x x x x + A ′ x x x + A ′ x + A ′ x + A ′ x + A ′ x + A ′ x ; some of thedeformations can be redefined away by suitable automorphisms. ρ = ρ and φ = φ : ̟ ̟ ̟ ̟ ̟ ̟ = (cid:16) P P P (cid:17) φ − φ )( ρ − ρ ) , (4)where P , , are given in appendix A. • | ρ φ − ω , − , − | < ̟ a + σ = 13 π X r =1 , α ar sin (cid:18) πr (cid:19) ξ σr ( ψ, φ )[ a = 0 , σ = 0 , , , (5)where ξ σr ( ψ, φ ) = P ∞ k =0 (Γ( k + r ) k !Γ( k + r ) ρ k + r U σ − ( k + r ) ( φ ) , U σν ( φ ) = ω − νσ U ν ( ω σ φ ) , ω ≡ e πi ; for small φ , U ν ( φ ) = 3 − − ν Γ( − ν ) ∞ X m =0 Γ( m − ν )(3 ωφ ) m (Γ(1 − m − ν ) m ! . (6)Expanding about a suitable φ = φ and ρ = ρ , one can show: ̟ ̟ ̟ ̟ ̟ ̟ = (cid:16) M M M (cid:17) ρ − ρ )( φ − φ ) , (7)where M , , are given in appendix A. • Near the conifold locus : ρ + φ = 1The periods are given by: ̟ i = C i g ( ρ, φ ) ln ( ρ + φ −
1) + f i ( ρ, φ ) , (8)where C i = (1 , , − , , , g ( ρ, φ ) = i π ( ̟ − ̟ ) ∼ a ( ρ + φ −
1) near ρ + φ − ∼ a is aconstant and f i are analytic in ρ and ψ . The analytic functions near the conifold locus are given by: f a + σ = 12 π X r =1 , e iπar sin (cid:18) πr (cid:19) ξ σr ( ρ, φ ) , a = 0 , σ = 0 , , . (9)Defining x ≡ ( ρ + φ − ξ σr = ∞ X k =0 ∞ X m =0 − k + r + m e iπ ( σ +1)3 + − iπ ( k + r ( − ) k (Γ( k + r )) Γ( k + 1)Γ( k + r )Γ( k + r )(Γ(1 − m + k + r )) m ! ( x − φ + 1) k + r . (10) The three values of σ correspond to the three solutions to (1 − φ ) U ′′′ ν ( φ )+3( ν − φ U ′′ ν ( φ ) − (3 ν − ν +1) φU ′ ν ( φ )+ ν U ν ( φ ) =0; the Wronskian of the three solutions is given by: − i π e − iπν sin ( πν )(1 − φ ) ν − - the solutions are hence linearly independentexcept when ν ∈ Z f f f f f f = (cid:16) N N N (cid:17) xφ , (11)where N , , are given in appendix A. • Near φ = 1 , Large ρ From asymptotic analysis of the coefficients, one can argue: U ν ( φ ) ∼ − √ π ( ν + 1) h ( φ − ν +1 − ω ( φ − ω − ) + ω ( φ − ω − ) i , ≡ y ν − y ν + y ν , (12)where ω ≡ e iπ . Defining U σν ( φ ) = P τ =0 γ σ,τν y τν ( φ ), where γ σ,τν = − e − iπν − − e − iπν e − iπν , onecan show that U ν , (cid:18) P σ =0 U σν ( φ )1 − e − iπν = (cid:19) y ν ( φ ) − y ν ( φ ) ≡ V ν ( φ ) , (cid:16) V ν ( φ ) − U ν ( φ ) − U ν ( φ )1 − e − iπν = (cid:17) y ν ( φ ) ≡ W ν ( φ ) arelinearly independent even for ν ∈ Z .For small ρ , ξ σr = Z Γ dµ isin ( π ( µ + r )) (Γ( − µ )) Γ( − µ + )Γ( − µ + ) ρ − µ U σµ ( φ ) , (13)where the contour Γ goes around the Im( µ ) < ′ goingaround the Im( µ ) > σ = 0 but not for σ = 1 ,
2. For the latter,one modifies U σµ ( φ ) by adding a function which does not contribute to the poles and has simple zerosat integers as follows: U σµ ( φ ) → ˜ U σµ,r ( φ ) ≡ U σµ ( φ ) − e iπr sin ( π ( µ + r )) sin ( πµ ) f σµ ( φ ) , (14)where f µ ( φ ) = 0 ,f µ ( φ ) = − (1 − e − iπν ) y µ ( φ ) ,f µ ( φ ) = (1 − e − iπν ) V ν ( φ ) + (1 − e − iπν ) W µ ( φ ) . (15)One can then deform the contour Γ to the contour Γ ′ to evaluate the periods. This is done in appendixA. The Wronskian of these three solutions is given by i (2 π ) e iπν (1 − φ ) ν − = 0 , ν ∈ Z . φ = ω − , and a large ρ = ρ , one gets the following periods: ̟ ̟ ̟ ̟ ̟ ̟ = A ′ + B ′ x + C ′ ( ρ − ρ ) A ′ + B ′ x + C ′ ( ρ − ρ ) A ′ + B ′ x + C ′ ( ρ − ρ ) A ′ + B ′ x + B ′ x lnx + C ′ ( ρ − ρ ) A ′ + B ′ x + B ′ x lnx + C ′ ( ρ − ρ ) A ′ + B ′ x + B ′ x lnx + C ′ ( ρ − ρ ) , (16)where x ≡ ( φ − ω − ).The equations (A10) and (16) will get used to arrive at (20) and finally (22) and (23).The Picard-Fuchs basis of periods evaluated above can be transformed to a symplectic basis as under(See [16]): Π = F F F X X X = M ̟ ̟ ̟ ̟ ̟ ̟ , (17)where M = − − − . (18)In the next section, we use information about the periods evaluated in this section, in looking for “areacodes”. In this section, we argue the existence of area codes, i.e., points in the moduli space close to and away fromthe two singular conifold loci that are finitely separated where for the same large values (and hence notnecessarily integral) of RR and NS-NS fluxes, one can extremize the (complex structure and axion-dilaton)superpotential (for different values of the complex structure and axion-dilaton moduli) .The axion-dilaton modulus τ gets stabilized (from D τ W c.s. = 0, W c.s. being the Gukov-Vafa-Wittencomplex structure superpotential R ( F − τ H ) ∧ Ω = (2 π ) α ′ ( f − τ h ) · Π, F and H being respectivelythe NS-NS and RR three-form field strengths, and are given by: F = (2 π ) α ′ P a =1 ( f a β a + f a +3 α a ) and H = (2 π ) α ′ P a =1 ( h a β a + h a +3 α a ); α a , β a , a = 1 , ,
3, form an integral cohomology basis) at a value givenby: τ = f T . ¯Π h T . ¯Π , (19)where f and h are the fluxes corresponding to the NS-NS and RR fluxes; it is understood that the complexstructure moduli appearing in (19) are already fixed from D i W = 0 , i = 1 , For techniques in special geometry relevant to this work, see [18] for a recent review; see [19] for moduli-stablizationcalculations as well. Near the conifold locus : φ = 1 , Large ψ The period vector in the symplectic basis, is given by:Π = − A ′ + A ′ + ( C ′ − C ′ )( ρ − ρ ) + ( B ′ − B ′ ) x ( A ′ + A ′ + 3 A ′ + 3 A ′ + 2 A ′ ) + ( C ′ + C ′ )( ρ − ρ ) + ( B ′ + B ′ + 3 B ′ + 3 B ′ + 2 B ′ ) x + (3 B ′ + 2 B ′ ) xlnxA ′ + A ′ + A ′ + ( B ′ + B ′ + B ′ ) x + ( C ′ + C ′ + C ′ )( ρ − ρ ) + B ′ xlnxA ′ + C ′ x + C ′ ( ρ − ρ ) − A ′ + C ′ x + C ′ ( ρ − ρ ) − A ′ + A ′ + ( B ′ − B ′ ) x + ( C ′ − C ′ )( ρ − ρ ) + B ′ xlnx A ′ − A ′ + A ′ + A ′ + (2 B ′ − B ′ + B ′ + B ′ ) x + (2 C ′ − C ′ + C ′ + C ′ )( ρ − ρ ) + ( − B ′ + B ′ + B ′ ) xlnx ≡ A + B x + C ( ρ − ρ ) A + B x + B xlnx + C ( ρ − ρ ) A + B x + B xlnx + C ( ρ − ρ ) A + B x + C ( ρ − ρ ) A + B x + B xlnx + C ( ρ − ρ ) A + B x + B xlnx + C ( ρ − ρ ) . (20)The tree-level K¨ahler potential is given by: K = − ln ( − i ( τ − ¯ τ )) − ln (cid:16) − i Π † ΣΠ (cid:17) , (21)where the symplectic metric Σ = − ! . Near x = 0, one can evaluate ∂ x K, τ and ∂ x W c.s. -this is done in appendix B.Using (20) - (21) and (B1)-(B5), one gets the following (near x = 0 , ρ − ρ = 0): D x W c.s. ≈ lnx A + B x + C xlnx + D ( ρ − ρ ) + B ′ ¯ x + C ′ ¯ xln ¯ x + D ′ (¯ ρ − ¯ ρ ) ! = 0 . (22)Similarly, D ρ − ρ W c.s. ≈ A + B x + C xlnx + D ( ρ − ρ ) + B ′ ¯ x + C ′ ¯ xln ¯ x + D ′ (¯ ρ − ¯ ρ ) = 0 . (23) • Near ρ + φ = 1Near y ≡ ρ + ρ − ρ = ρ ′ , one can follow a similar analysis as (20 - (23) and arriveat similar equations: D y W c.s. ≈ lny A + B y + C ylny + D ( ρ − ρ ′ ) + B ′ ¯ y + C ′ ¯ yln ¯ y + D ′ (¯ ρ − ¯ ρ ′ ) ! = 0 ,D ρ − ρ ′ W c.s. ≈ A + B y + C ylny + D ( ρ − ρ ′ ) + B ′ ¯ y + C ′ ¯ yln ¯ y + D ′ (¯ ρ − ¯ ρ ′ ) = 0 . (24) • Points away from both conifold lociIt can be shown, again following an analysis similar to the one carried out in (20) - (24), that one getsthe following set of equations from extremization of the complex-structure moduli superpotential: A i + B i ψ + C i φ + B ′ i ¯ ψ + C ′ i ¯ φ = 0 , (25)where i indexes the different regions in the moduli space away from the two conifold loci, as discussedin section earlier. 6herefore, to summarize,Near φ = ω − : A + B ( φ − ω − ) + C ( φ − ω − ) ln ( φ − ω − ) + D ( ρ − ρ )+ B ′ ( ¯ φ − ω ) + C ′ ( ¯ φ − ω ) ln ( ¯ φ − ω ) + D ′ (¯ ρ − ¯ ρ ) = 0 , A + B ( φ − ω − ) + C ( φ − ω − ) ln ( φ − ω − ) + D ( ρ − ρ )+ B ′ ( ¯ φ − ω ) + C ′ ( ¯ φ − ω ) ln ( ¯ φ − ω ) + D ′ (¯ ρ − ¯ ρ ) = 0 ,τ = Ξ[ f i ; ¯ φ − ω, ρ − ρ ] P i =0 h i ¯ A i " − Ξ[ h i ; ¯ φ − ω, ρ − ρ ] P j =0 h i ¯ A i ;Near ρ + φ − A + B ( ρ + φ −
1) + C ( ρ + φ − ln ( ρ + φ −
1) + D φ + B ′ (¯ ρ + ¯ φ − C ′ (¯ ρ + ¯ φ − ln (¯ ρ + ¯ φ −
1) + D ′ ¯ φ = 0 , A + B ( ρ + φ −
1) + C ( ρ + φ − ln ( ρ + φ −
1) + D φ + B ′ (¯ ρ − ¯ φ −
1) + C ′ (¯ ρ + ¯ φ − ln (¯ ρ + ¯ φ −
1) + D ′ ¯ φ = 0 ,τ = Ξ[ f i ; ¯ ρ + ¯ φ − , φ ] P i =0 h i ¯ A ′ i " − Ξ[ h i ; ¯ ρ + ¯ φ − , φ ] P j =0 h i ¯ A ′ i | φ | < , Large ψ : A + B ( φ − φ ′′ ) + C ( ρ − ρ ′′ ) B ′ ( ¯ φ − ¯ φ ′′ ) + C ′ (¯ ρ − ¯ ρ ′′ ) = 0 , A + B ( φ − φ ′′ ) + C ( ρ − ρ ′′ ) + B ′ ( ¯ φ − ¯ φ ′′ ) + C ′ (¯ ρ − ¯ ρ ′′ ) = 0 ,τ = ˜Ξ[ f i ; ¯ φ , ρ ] P i =0 h i ¯ A ′′ i " − ˜Ξ[ h i ; ¯ φ , ρ ] P j =0 h i ¯ A ′′ i ; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ φ − ω , − , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < A + B ( φ − φ ′′′ ) + C ( ρ − ρ ′′′ ) + B ′ ( ¯ φ − ¯ φ ′′′ ) + C ′ (¯ ρ − ¯ ρ ′′′ ) = 0 , A + B ( φ − φ ′′′ ) + C ( ρ − ρ ′′′ ) + + B ′ ( ¯ φ − ¯ φ ′′′ ) + C ′ (¯ ρ − ¯ ρ ′′′ ) = 0 ,τ = ˜Ξ[ f i ; ¯ φ , ρ ] P i =0 h i ¯ A ′′′ i " − ˜Ξ[ h i ; ¯ φ , ρ ] P j =0 h i ¯ A ′′′ i , (26)where on deleting the ln terms in Ξ one gets the form of ˜Ξ in (26). Given that the Euler characteristic of theelliptically-fibered Calabi-Yau four-fold to which, according to the Sen’s construction [20], the orientifoldof the Calabi-Yau three-fold of (1) corresponds to, will be very large , and further assuming the absenceof D f T . Σ .h , and hence the fluxes -therefore, similar to the philosophy of [2], we would disregard the integrality of fluxes. Without doing thenumerics, we will now give a plausibility argument about the existence of solution to any one of the foursets of equations in (26). As one can drop x as compared to xlnx for x ∼
0, the equations in (26) pair offeither as: See [7] - χ ( CY ) = 6552 where the CY for the WCP [1 , , , , D and E singularities along two sections, with the three-fold a CP -fibration over CP with the two divisorscontributing to the instanton superpotential a la Witten being sections thereof. Near either of the two conifold loci: A i + ( B i cosα i + B ′ i sinα i ) ǫ i lnǫ i + C i β i + C ′ i ¯ β i = 0 , ˜ a i + (˜ b i cosα i + ˜ b ′ i sinα i ) ǫ i lnǫ i + ˜ c i β i + ˜ c ′ i ¯ β i = 0 , (27)or • Away from both the conifold loci: A i + B i γ i + C i δ i + B ′ i ¯ γ i + C ′ i ¯ δ i = 0 , ˜ A i + ˜ B i γ i + ˜ C i δ i + ˜ B ′ i ¯ γ i + ˜ C ′ i ¯ δ i = 0 , (28)where ǫ i , α i correspond to the magnitude and phase of the extremum values of either φ − ω − or ρ + φ −
1, and γ i , δ i are different (functions of) extremum values of φ, ψ near and away, respectively,from the two conifold loci, and both sets are understood to be “close to zero” each.From the point of view of practical calculations, let us rewrite, e.g., (27) as the equivalent four real equations: A i + B i cosα i ǫ i lnǫ i + B ′ i sinα i ǫ i lnǫ i + C i Re ( β i ) + C ′ i Im ( β i ) = 0 , f A i + f B i cosα i ǫ i lnǫ i + f B i ′ sinα i ǫ i lnǫ i + e C i Re ( β i ) + e C ′ i Im ( β i ) = 0 ,ν i + χ i cosα i ǫ i lnǫ i + χ ′ i sinα i ǫ i lnǫ i + ϑ i Re ( β i ) + ϑ ′ Im ( β i ) = 0 , e ν i + f χ i cosα i ǫ i lnǫ i + f χ i ′ sinα i ǫ i lnǫ i + e ϑ i Re ( β i ) + e ϑ ′ i Im ( β i ) = 0 . (29)In (26), by “close to zero”, what we would be admitting are, e.g., ǫ i , | β i | ∼ e − ≈ × − implying that ǫ i lnǫ i ≈ − . Let us choose the moduli-independent constants in (29), after suitable rationalization, to be7 × O (1), the coefficients of the ǫ i lnǫ i -terms to be 7 × and the coefficients of Re ( β i ) and Im ( β i ) to be ∼ . On similar lines, for (28), we could take the moduli to be ∼ e − and the moduli-independent andmoduli-dependent constants to be 7 × O (1) and ∼ respectively. Now, the constants appearing in (29)(and therefore (26)) are cubic in the fluxes (more precisely, they are of the type h f in obvious notations),which for (1) would be ∼ (See [7]). In other words, for the same choice of the NS-NS and RR fluxes - 12 innumber - one gets 6 or 9 or 12 complex (inhomogenous [in ψ, φ ] algebraic/transcendetal) constraints (comingfrom (26)) on the 6 or 9 or 12 extremum values of the complex structure moduli ( φ i , ψ i , τ i ; i = 1 , , , f T . Σ .h fixed,one should be able to tune the fluxes f i , h i ; i = 0 , ..., , are reasonable implyingthe possibility of existence of “area codes”, and the interpolating domain walls [21]. Of course, completenumerical calculations, which will be quite involved, will be needed to see explicitly everything working out. α ′ - and In-stanton Corrections In this section, using the results of [6], we show that after inclusion of non-perturbative α ′ -corrections to theK¨ahler potential, in addition to the perturbative α ′ corrections of [8], as well as the non-perturbative instan-ton contributions to the superpotential, it may be possible to obtain a large volume non-supersymmetric dSminimum (analogous to [5] for the non-supersymmetric AdS minimum) without the addition of D -branes -see also [9]. 8et us begin with a summary of the inclusion of perturbative α ′ -corrections to the K¨ahler potential intype IIB string theory compactified on Calabi-Yau three-folds with NS-NS and RR fluxes turned on, asdiscussed in [8]. The ( α ′ ) − corrections contributing to the K¨ahler moduli space metric are contained in Z d x √ ge − φ (cid:18) R + ( ∂φ ) + ( α ′ ) ζ (3) J . + ( α ′ ) ( ▽ φ ) Q (cid:19) , (30)where J ≡ t M N ...M N t M ′ N ′ ....M ′ N ′ R M N M ′ N ′ ...R M N M ′ N ′ + 14 ǫ ABM N ...M N ǫ ABM ′ N ′ ...M ′ N ′ R M ′ N ′ M N ...R M ′ N ′ M N , (31)the second term in (31) being the ten-dimensional generalization of the eight-dimensional Euler density, and t IJKLMNP Q ≡ − ǫ IJKLMNP Q − (cid:20) ( δ IK δ JL − δ IL δ JK )( δ MP δ NQ − δ MQ δ NP )+( δ KM δ LN − δ KN δ LM )( δ P I δ QJ − δ P J δ QI ) + ( δ IM δ JN − δ IN δ JM )( δ KP δ LQ − δ KQ δ LP ) (cid:21) + 12 h δ JK δ KM δ NP δ QI + δ JM δ NK δ LP δ QI + δ JM δ NP δ QK δ KI i + 45 terms obtained by antisymmetrization w . r . t . ( ij ) , ( kl ) , ( mn ) , ( pq ) , (32)and Q ≡ π ) (cid:18) R IJ R MNKL R IJMN − R K LI K R M NK L R I JM N (cid:19) . (33)The perturbative world-sheet corrections to the hypermultiplet moduli space of Calabi-Yau three-fold com-pactifications of type II theories are captured by the prepotential: F ( X ) = i κ abc X a X b X c X + ( X ) ξ, (34)where the ( α ′ ) -corrections are contained in ξ ≡ − ( α ′ ) χ ( CY ) ζ (3)2 , κ abc being the classical CY intersectionnumbers. Substituting (34) in K = − ln (cid:2) X i ¯ F i + ¯ X i F i (cid:3) gives: K = − ln (cid:20) − i z a − ¯ z a )( z b − ¯ z b )( z c − ¯ z c ) + 4 ξ (cid:21) . (35)Truncation of N = 2 to N = 1, implying reduction of the quaternionic geometry to K¨ahler geometry,corresponds to a K¨ahler metric which becomes manifest in K¨ahler coordinates: T a = g a + i ˆ V a , τ = l + ie − φ ,the hat denoting the Einstein’s frame in which, e.g., ˆ V a = e φ (cid:16) κ abc v b v c (cid:17) , v a being the K¨ahler moduli, andthe K¨ahler potential is given by: K = − ln ( − ( τ − ¯ τ )) − ln (cid:18) ˆ V + 12 ξe − φ (cid:19) − ln (cid:18) − i Z CY Ω ∧ ¯Ω (cid:19) , (36)substituting which into the N = 1 potential V = e K (cid:16) g i ¯ j D i W ¯ D ¯ j ¯ W − | W | (cid:17) (one sums over all the moduli),one gets: V = e K " ( G − ) α ¯ β D α W D ¯ β ¯ W + ( G − ) τ ¯ τ D τ W D ¯ τ ¯ W − ξ ˆ V e − φ ( ˆ ξ − ˆ V )( ˆ ξ + 2 ˆ V ) ( W ¯ D ¯ τ ¯ W + ¯ W D τ W ) − ξ (( ˆ ξ ) + 7 ˆ ξ ˆ V + ( ˆ V ) )( ˆ ξ − ˆ V )( ˆ ξ + 2 ˆ V ) | W | , (37)9he hats being indicative of the Einstein frame - in our subsequent discussion, we will drop the hats fornotational convenience. The structure of the α ′ -corrected potential shows that the no-scale structure is nolonger preserved due to explicit dependence of V on ˆ V and the | W | term is not cancelled. In what follows,we will be setting 2 πα ′ = 1.The type IIB Calabi-Yau orientifolds containing O3/O7-planes considered involve modding out by( − ) F L Ω σ where N = 1 supersymmetry requires σ to be a holomorphic and isometric involution: σ ∗ ( J ) = J, σ ∗ (Ω) = − Ω. Writing the complexified K¨ahler form − B + iJ = t A ω = − b a ω a + iv α ω α where ( ω a , ω α ) formcanonical bases for ( H − ( CY , Z ) , H ( CY , Z )), the ± subscript indicative of being odd under σ , one sees thatin the large volume limit of CY /σ , contributions from large t α = v α are exponentially suppressed, howeverthe contributions from t a = − B a are not. Note that it is understood that a indexes the real subspace of real dimensionality h , − = 2; the complexified K¨ahler moduli correspond to H , ( CY ) with complex dimensionality h , = 2 or equivalently real dimensionality equal to 4. So, even though G a = c a − τ b a (forreal c a and b a and complex τ ) is complex, the number of G a ’s is indexed by a which runs over the realsubspace h , − ( CY ) As shown in [6], based on the R -correction to the D = 10 type IIB supergravity action[22] and the modular completion of N = 2 quaternionic geometry by summation over all SL (2 , Z ) imagesof world sheet corrections as discussed in [23], the non-perturbative large-volume α ′ -corrections that survivethe process of orientifolding of type IIB theories (to yield N = 1) to the K¨ahler potential is given by (in theEinstein’s frame): K = − ln ( − i ( τ − ¯ τ )) − ln " V + χ X m,n ∈ Z / (0 , (¯ τ − τ ) (2 i ) | m + nτ | − X β ∈ H − ( CY , Z ) n β X m,n ∈ Z / (0 , (¯ τ − τ ) (2 i ) | m + nτ | cos ( n + mτ ) k a ( G a − ¯ G a ) τ − ¯ τ − mk a G a ! , (40)where n β are the genus-0 Gopakumar-Vafa invariants for the curve β and k a = R β ω a , , and G a = c a − τ b a ,the real RR two-form potential C = C a ω a and the real NS-NS two-form potential B = B a ω a . As pointedout in [6], in (40), one should probably sum over the orbits of the discrete subgroup to which the symmetrygroup SL (2 , Z ) reduces. Its more natural to write out the K¨ahler potential and the superpotential in terms To make the idea more explicit, the involution σ under which the NS-NS two-form B and the RR two-form C are odd canbe implemented as follows. Let z i , ¯ z i , i = 1 , , σ be defined as: z ↔ z , z → z ;in terms of the x i figuring in the defining hypersurface in equation (1) on page 2, one could take z , = x , x , etc. in the x = 0coordinate patch. One can construct the following bases ω ( ± ) of real two-forms of H even/odd under the involution σ : ω ( − ) = { X ( dz ∧ d ¯ z ¯2 − dz ∧ d ¯ z ¯1 ) , i ( dz ∧ d ¯ z ¯1 − dz ∧ d ¯ z ¯2 ) } ≡ { ω ( − )1 , ω ( − )2 } ,ω (+) = { X i ( dz ∧ d ¯ z ¯2 + dz ∧ d ¯ z ¯1 ) , X idz ∧ d ¯ z ¯1 } ≡ { ω (+)1 , ω (+)2 } . (38)This implies that h , ( CY ) = h , − ( CY ) = 2 - the two add up to give 4 which is the real dimensionality of H ( CY ) for thegiven Swiss Cheese Calabi-Yau. As an example, let us write down B ∈ R as B = B dz ∧ d ¯ z ¯2 + B dz ∧ d ¯ z ¯3 + B dz ∧ d ¯ z ¯1 + B dz ∧ d ¯ z ¯1 + B dz ∧ d ¯ z ¯2 + B dz ∧ d ¯ z ¯3 + B dz ∧ d ¯ z ¯1 + B dz ∧ d ¯ z ¯2 + B dz ∧ d ¯ z ¯3 . (39)Now, using (38), one sees that by assuming B = B = B = b , and B = − B = ib , B = 0, one can write B = b ω ( − )1 + b ω ( − )2 ≡ P h , − =2 a =1 b a ω ( − ) a .
10f the N = 1 coordinates τ, G a and T α where T α = i e − φ κ αβγ v β v γ − (˜ ρ α − κ αab c a b b ) − τ − ¯ τ ) κ αab G a ( G b − ¯ G b ) , (41)where ˜ ρ α being defined via C (the RR four-form potential)= ˜ ρ α ˜ ω α , ˜ ω α ∈ H ( CY , Z ).Based on the action for the Euclidean D ) action iT D R Σ e − φ √ g − B + F + T D R Σ e C ∧ e − B + F , the nonperturbative superpotential coming from a D ∈ H ( CY /σ, Z ) such that the unit arithmetic genus condition of Witten [24] issatisfied, will be proportional to (See [6]) e R Σ e − φ ( − B + iJ ) − i R Σ ( C − C ∧ B + C B = e iT α R Σ ˜ ω α ≡ e in α Σ T α , (42)where C , , are the RR potentials. The prefactor multiplying (42) is assumeto factorize into a functionof the N = 1 coordinates τ, G a and a function of the other moduli. Based on appropriate transformationproperties of the superpotential under the shift symmetry and Γ S ⊂ SL (2 , Z ):( i ) τ → aτ + bcτ + d , C B ! → a bc d ! C B ! ,G a → G a ( cτ + d ) ,T α → T α + c κ αab G a G b ( cτ + d ) ;( ii ) b a → b a + 2 πn a ,G a → G a − πτ n a ,T α → T α − πκ αab n a G b + 2 π τ κ αab n a n b , (43)the non-perturbative instanton-corrected superpotential was shown in [6] to be: W = Z CY G ∧ Ω + X n α θ n α ( τ, G ) f ( η ( τ )) e in α T α , (44)where the theta function is given as: θ n α ( τ, G ) = X m a e iτm e in α G a m a . (45)In (45), m = C ab m a m b , C ab = − κ α ′ ab , α = α ′ corresponding to that T α = T α ′ (for simplicity) that isinvariant under (43).Now, for (1), as shown in [7], there are two divisors which when uplifted to an elliptically-fibered Calabi-Yau, have a unit arithmetic genus ([24]): τ ≡ ∂ t V = t , τ ≡ ∂ t V = ( t +6 t ) . In (41), ρ = ˜ ρ − iτ and ρ = ˜ ρ − iτ .To set the notations, the metric corresponding to the K¨ahler potential in (40), will be given as: G A ¯ B = ∂ ρ ¯ ∂ ¯ ρ K ∂ ρ ¯ ∂ ¯ ρ K ∂ ρ ¯ ∂ ¯ G K ∂ ρ ¯ ∂ ¯ G K∂ ρ ¯ ∂ ¯ ρ K ∂ ρ ¯ ∂ ¯ ρ K ∂ ρ ¯ ∂ ¯ G K ∂ ρ ¯ ∂ ¯ G K∂ G ¯ ∂ ¯ ρ K ∂ G ¯ ∂ ¯ ρ K ∂ G ¯ ∂ ¯ G K ∂ G ¯ ∂ ¯ G K∂ G ¯ ∂ ¯ ρ K ∂ G ¯ ∂ ¯ ρ K ∂ G ¯ ∂ ¯ G K ∂ G ¯ ∂ ¯ G K , (46)11here A ≡ ρ , , G , . We have taken the involution to be such that h , − = 2. From the K¨ahler potentialgiven in (40), one can show that the corresponding K¨ahler metric of (46) is given by: G A ¯ B = (cid:16) √ √ ¯ ρ − ρ Y +
118 (¯ ρ − ρ ) Y (cid:17) (cid:18) √ (¯ ρ − ρ )(¯ ρ − ρ ) Y (cid:19) − ie − φ √ ¯ ρ − ρ Z ( τ )6 √ Y − ie − φ √ ¯ ρ − ρ Z ( τ )6 √ Y (cid:18) √ (¯ ρ − ρ )(¯ ρ − ρ ) Y (cid:19) (cid:16) √ √ ¯ ρ − ρ Y + √ ¯ ρ − ρ Y (cid:17) − ie − φ √ ¯ ρ − ρ Z ( τ )6 √ Y − ie − φ √ ¯ ρ − ρ Z ( τ )6 √ Y ik e − φ √ ¯ ρ − ρ Z (¯ τ )6 √ Y ik e − φ √ ¯ ρ − ρ Z (¯ τ )6 √ Y k X k k X ik e − φ √ ¯ ρ − ρ Z (¯ τ )6 √ Y ik e − φ √ ¯ ρ − ρ Z (¯ τ )6 √ Y k k X k X , (47)where Z ( τ ) ≡ X c X m,n A n,m,n kc ( τ ) sin ( nk.b + mk.c ) , Y ≡ V E + χ X m,n ∈ Z / (0 , ( τ − ¯ τ ) (2 i ) | m + nτ | − X β ∈ H ( CY , Z ) n β X m,n ∈ Z / (0 , ( τ − ¯ τ ) (2 i ) | m + nτ | cos ( n + mτ ) k a ( G a − ¯ G a ) τ − ¯ τ − mk a G a ! , X ≡ P c P m,n ∈ Z / (0 , e − φ | n + mτ | | A n,m,n kc ( τ ) | cos ( nk.b + mk.c ) Y + | P c P m,n ∈ Z / (0 , e − φ | n + mτ | A n,m,n kc ( τ ) sin ( nk.b + mk.c ) | Y ,A n,m,n kc ( τ ) ≡ ( n + mτ ) n k c | n + mτ | . (48)The inverse metric is given as: G − = ( G − ) ρ ¯ ρ ( G − ) ρ ¯ ρ ( G − ) ρ ¯ G G − ) ρ ¯ ρ ( G − ) ρ ¯ ρ ( G − ) ρ ¯ G G − ) ρ ¯ G ( G − ) ρ ¯ G k − k ) X k ( k k − k ) X k ( k k − k ) X k − k ) X , (49)where the non-zero elements are given in appendix C.Now, analogous to [5], we will work in the large volume limit: V → ∞ , τ ∼ ln V , τ ∼ V . In this limit,the inverse metric (49) simplifies to (we will not be careful about the magnitudes of the numerical factorsin the following): G − ∼ −V√ ln V V ln V − i X ln VX V ln V V i X V k X i X ln VX − i X V k X k − k ) X k ( k k − k ) X k ( k k − k ) X k − k ) X , (50)12here X ≡ X c X m,n ∈ Z / (0 , | n + mτ | | A n,m,n kc ( τ ) | cos ( nk.b + mk.c ) . Refer to [5] for discussion on the minus sign in the (cid:0) G − (cid:1) ρ ¯ ρ . Having extremized the superpotential w.r.t.the complex structure moduli and the axion-dilaton modulus, the N = 1 potential will be given by: V = e K " X A,B = ρ α ,G a ( ( G − ) A ¯ B ∂ A W np ¯ ∂ ¯ B ¯ W np + (cid:16) ( G − ) A ¯ B ( ∂ A K ) ¯ ∂ ¯ B ¯ W np W + c.c. (cid:17)) + X A,B = ρ α ,G a ( G − ) A ¯ B ∂ A K ¯ ∂ ¯ B K − | W | + X α, ¯ β ∈ c . s . ( G − ) α ¯ β ∂ α K c.s. ¯ ∂ ¯ β K c.s. | W np | , (51)where the total superpotential W is the sum of the complex structure moduli Gukov-Vafa-Witten super-potential W c.s. and the non-perturbative superpotential W np arising because of instantons (obtained bywrapping of D τ and τ ).Now, using: ∂ ρ α W = θ n α ( τ, G a ) f ( η ( τ )) e in α T α in α − e − φ ! ,∂ G a W = X n α e i τm f ( η ( τ )) e im a G a n α e in α T α im a n α − in α κ αab | τ − ¯ τ | " ¯ τ ( G b − ¯ G b ) + (¯ τ G b − τ ¯ G b ) + (2 G b − ¯ G b )( τ − ¯ τ ) , (52)in the large-volume limit, one forms tables 1, 2 and 3.One therefore sees from table 1 that the dominant term in ( G − ) A ¯ B ∂ A W np ¯ ∂ ¯ B ¯ W np is ( G − ) ρ ¯ ρ | ∂ ρ W np | ,given by: Y√ ln VV n e − φ e − φ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) θ n ( τ, G ) f ( η ( τ )) (cid:12)(cid:12)(cid:12)(cid:12) e − n Im ( T ) . (53)From table 2 we see that the dominant term in ( G − ) A ¯ B ( ∂ A K ) ¯ ∂ ¯ B ¯ W np W is ( G − ) ρ ¯ ρ ∂ ρ K ¯ ∂ ¯ ρ ¯ W np W + c.c. ,which gives: W c.s. ln VV n e − φ θ n (¯ τ , ¯ G ) f ( η (¯ τ )) ! e − in ( − ˜ ρ + κ ab ¯ τGa − τ ¯ Ga (¯ τ − τ ) ( Gb − ¯ Gb )(¯ τ − τ ) − κ ab Ga ( Gb − ¯ Gb )( τ − ¯ τ ) ) + c.c. (54)Note, from table 3, the dominant and the (sub) sub-dominant terms in ( G − ) A ¯ B ∂ A K ¯ ∂ ¯ B K | W | , givenrespectively by ( G − ) ρ ¯ ρ | ∂ ρ K | | W | and (cid:2) ( G − ) ρ ¯ ρ | ∂ ρ K | + ( G − ) ρ ¯ ρ ∂ ρ ¯ ∂ ¯ ρ K + c.c. (cid:3) | W | are actuallyof the form: h VV + ξ ≈ − ξ V + O (cid:16) V (cid:17)i | W | and (cid:20) ( ln V ) ( V + ξ ) + ( ln V ) ( V + ξ ) (cid:21) | W | ≈ (cid:20) ( ln V ) V − ξ ( ln V ) V + ( ln V ) V − ξ ( ln V ) V + O (cid:16) V (cid:17)(cid:21) | W | respectively and the ξ -independent terms together cancel the “-3” in (50). This is just arederivation of the last term in (37). One notes that there are additional terms of O (cid:16) V (cid:17) that one gets from " ( G − ) G ¯ G | ∂ G K | + ( G − ) G ¯ G | ∂ G K | + ( G − ) G ¯ G ∂ G K ¯ ∂ ¯ G K | W | , which is given by: | W | V k + k k − k ! (cid:12)(cid:12)(cid:12)P c P n,m ∈ Z / (0 , e − φ A n,m,n kc ( τ ) sin ( nk.b + mk.c ) (cid:12)(cid:12)(cid:12) P c ′ P m ′ ,n ′ ∈ Z / (0 , e − φ | n + mτ | | A n ′ ,m ′ ,n kc ′ ( τ ) | cos ( n ′ k.b + m ′ k.c ) , (55)13hich one sees can be either positive or negative .To summarize, from (52) - (55), one gets the following potential: V ∼ Y√ ln VV n +2 e − φ ( n ) P m a e − m gs + maban gs + n κ abbabb gs ! | f ( η ( τ )) | + ln VV n +2 θ n (¯ τ , ¯ G ) f ( η (¯ τ )) ! e − in ( − ˜ ρ + κ ab ¯ τGa − τ ¯ Ga (¯ τ − τ ) ( Gb − ¯ Gb )(¯ τ − τ ) − κ ab Ga ( Gb − ¯ Gb )( τ − ¯ τ ) ) + c.c. + | W | V k + k k − k ! (cid:12)(cid:12)(cid:12)P c P n,m ∈ Z / (0 , e − φ A n,m,n kc ( τ ) sin ( nk.b + mk.c ) (cid:12)(cid:12)(cid:12) P c ′ P m ′ ,n ′ ∈ Z / (0 , e − φ | n + mτ | | A n ′ ,m ′ ,n kc ′ ( τ ) | cos ( n ′ k.b + m ′ k.c ) + ξ | W | V . (56)On comparing (56) with the analysis of [5], one sees that for generic values of the moduli ρ α , G a , k , and O (1) W c.s. , and n = 1, analogous to [5], the second term dominates; the third term is a new term.However, as in KKLT scenarios (See [12]), W c.s. <<
1; we would henceforth assume that the fluxes andcomplex structure moduli have been so fine tuned/fixed that W ∼ W n.p. . Further, from studies related tostudy of axionic slow roll inflation in Swiss Cheese models [27], it becomes necessary to take n >
2. Weassume that the fundamental-domain-valued b a ’s satisfy: | b a | π << . This implies that the first term in(56) - | ∂ ρ W np | - a positive definite term and denoted henceforth by V I , is the most dominant. Hence, if aminimum exists, it will be positive. To evaluate the extremum of the potential, one sees that: ∂ c a V I ∼ − √ ln VV n +2 X β ∈ H − ( CY , Z ) n β X m,n ∈ Z / (0 , mk a (¯ τ − τ ) (2 i ) | m + nτ | sin ( nk.b + mk.c ) P m a e − m gs + maban gs + n κ abbabb gs ! | f ( η ( τ )) | = 0 ⇔ nk.b + mk.c = N π ; ∂ b a V I | nk.b + mk.c = Nπ ∼ V√ ln VV n +1 e − m gs + ma ′ ba ′ n gs + n κ a ′ b ′ ba ′ bb ′ gs P m a e − m gs + maban gs + n κ abbabb gs | f ( η ( τ )) | n m a g s + n κ ab b b g s ! = 0 . (57)Now, given the O (1) triple-intersection numbers and super sub-Planckian NS-NS axions, we see that poten-tial V I gets automatically extremized for D m a >>
1. Note that if the NS-NS axionsget stabilized as per n m a g s + n κ ab b b g s = 0, satisfying ∂ b a V = 0, this would imply that the NS-NS axions getstabilized at a rational number, and in particular, a value which is not a rational multiple of π , the samebeing in conflict with the requirement nk.b + mk.c = N π . It turns out that the locus nk.b + mk.c = N π for | b a | << π and | c a | << π corresponds to a flat saddle point with the NS-NS axions providing a flat direction- See [27].Analogous to [5], for all directions in the moduli space with O (1) W c.s. and away from D i W cs = D τ W =0 = ∂ c a V = ∂ b a V = 0, the O ( V ) contribution of P α, ¯ β ∈ c.s. ( G − ) α ¯ β D α W cs ¯ D ¯ β ¯ W cs dominates over (56), If one puts in appropriate powers of the Planck mass M p , | b a | π << | b a | << M p , i.e., NS-NS axions aresuper sub-Planckian. V I ,this will be a dS minimum. There has been no need to add any D nk.b + mk.c = N π gurantees that the slow rollparameters “ ǫ ” and “ η ” are much smaller than one for slow roll inflation beginning from the saddle pointand proceeding along an NS-NS axionic flat direction towards the nearest dS minimum (See [27]). We now switch gears and address two issues in this and the subsequent sections, related to supersymmetricand non-supersymmetric black hole attractors . In this section, using the techniques discussed in [2], weexplicitly solve the “inverse problem” for extremal black holes in type II compactifications on (the mirror of)(1) - given a point in the moduli space, to find the charges ( p I , q I ) that would satisfy ∂ i V BH = 0, V BH beingthe black-hole potential. In the next section, we address the issue of existence of “fake superpotentials” inthe same context.We will now summarize the “inverse problem” as discussed in [2]). Consider D = 4 , N = 2 supergravitycoupled to n V vector multiplets in the absence of higher derivative terms. The black-hole potential can bewritten as [14]: V BH = −
12 ( q I − N IK p K ) (cid:16) ( Im N ) − (cid:17) IJ ( q J − ¯ N p L ) , (58)where the ( n V + 1) × ( n V + 1) symmetric complex matrix, N IJ , the vector multiplet moduli space metric,is defined as: N IJ ≡ ¯ F IJ + 2 iIm ( F IK ) X K Im ( F IL ) X L Im ( F MN ) X M X N , (59) X I , F J being the symplectic sections and F IJ ≡ ∂ I F J = ∂ J F I . The black-hole potential (58) can be rewritten(See [2]) as: ˜ V BH = 12 P I Im ( N IJ ) ¯ P J − i P I ( q I − N IJ p J ) + i P I ( q I − ¯ N IJ p J ) . (60)The variation of (60) w.r.t. P I gives: P I = − i (cid:16) ( Im N ) − ) IJ (cid:17) ( q J − N IJ p J ) , (61)which when substituted back into (60), gives (58). From (61), one gets: p I = Re ( P I ) ,q I = Re ( N IJ P J ) . (62)Extremizing ˜ V BH gives: P I ¯ P J ∂ i Im ( N IJ ) + i ( P I ∂ i N IJ − ¯ P J ∂ i ¯ N IJ ) p J = 0 , (63)which using (62) yields: ∂ i Im ( P I N IJ P J ) = 0 . (64)Similar to what was done in section , one uses the semi-classical approximation and disregards the inte-grality of the electric and magnetic charges taking them to be large.The inverse problem is not straight forward to define as all sets of charges ( p I , q I ) which are related toeach other by an Sp (2 n V + 2 , Z )-transformation, correspond to the same point in the moduli space. This is See [18] for a nice review of special geometry relevant to sections and . V BH ) (and ∂ i V BH ) is (are) symplectic invariants. Further, ∂ i V BH = 0 give 2 n V real equationsin 2 n V + 2 real variables ( p I , q I ). To fix these two problems, one looks at critical values of V BH in a fixedgauge W = w ∈ C . In other words, W = Z M Ω ∧ H = q I X I − p I F I = X I ( q I − N IJ p J ) = w, (65)which using (62), gives: X I Im ( N IJ ) ¯ P J = w. (66)Thus, the inverse problem boils down to solving: p I = Re ( P I ) , q I = Re ( N IJ P J ); ∂ i ( P I N IJ P J ) = 0 , X I N IJ ¯ P J = iw. (67)One solves for P I s from the last two equations in (67) and substitutes the result into the first two equationsof (67).We will now solve the last two equations of (67) for (1). As an example, we work with points in the modulispace close to one of the two conifold loci: φ = 1. We need to work out the matrix F IJ so that one canwork out the matrix N IJ . From the symmetry of F IJ w.r.t. I and J , one sees that the constants appearingin (16) must satisfy some constraints (which must be borne out by actual numerical computations). Tosummarize, near x = 0 and using (A10)-(16): F = F ⇔ lnx B B + C C = B B + B ( lnx + 1) + C C ⇒ B = 0 , C C = C C ; F = F ⇔ lnx B B + C C = B B + B ( lnx + 1) + C C ⇒ B = 0 , C C = C C ; F = F ⇔ B B + C C = B B + B ( lnx + 1) + C C ⇒ C C = C C . (68)In (68), the constants A i , B ij , C k are related to the constants A i , B ij , C k via matrix elements of M of (18).Therefore, one gets the following form of F IJ : F IJ = B C + C C C C C C C C C C C C C C C C C C (69)Using (69), one can evaluate X I Im ( F IJ ) X J - this is done in appendix D.Using (69), (C1) - (C2), one gets: N = a + b (1)00 x + b (2)00 xlnx + c ( ρ − ρ ) a + b (1)01 x + b (2)01 xlnx + c ( ρ − ρ ) a + b (1)02 x + b (2)02 xlnx + c ( ρ − ρ ) a + b (1)01 x + b (2)01 xlnx + c ( ρ − ρ ) a + b (1)11 x + b (2)11 xlnx + c ( ρ − ρ ) a + b (1)12 x + b (2)12 xlnx + c ( ρ − ρ ) a + b (1)02 x + b (2)02 xlnx + c ( ρ − ρ ) a + b (1)12 x + b (2)12 xlnx + c ( ρ − ρ ) a + b (1)22 x + b (2)22 xlnx + c ( ρ − ρ ) . (70)The constants a ij , b (1) , (2) jk , c lm are constrained by relations, e.g., F I = N IJ X J , (71)16hich, e.g., for I = 0 would imply: a A + a A + a A = A a B + b (1)00 A + a B + A b (1)01 + a B + b (1)02 A = B b (2)00 A + a B + b (2)01 A + a B + A b (2)02 = 0 a C + c A + a C + c A + a C + c A = C . (72)So, substituting (70) into the last two equations of (67), one gets: ∂ x ( P I N IJ P J ) = 0 ⇒ lnx h ( P ) b (2)00 + ( P ) b (2)11 + ( P ) b (2)22 + 2 P P b (2)01 + 2 P P b (2)02 + 2 P P b (2)12 i = 0; ∂ ρ − ρ ( P I N IJ P J ) = 0 ⇒ ( P ) c (2)00 + ( P ) c + ( P ) c + 2 P P c + 2 P P c + 2 P P c = 0 , (73)and ¯ X I Im ( N IJ ) P J = − iw implies:¯ A I ( a IJ − ¯ a IJ ) P J + ¯ x [ ¯ B I ( a IJ − ¯ a IJ ) P J − ¯ b (1) IJ ¯ A I P J ] + x [ b (1) IJ ¯ A I P J ] + xlnx [ ¯ A I b (2) IJ P J ] + ( ρ − ρ )[ ¯ A I c IJ P J ]+(¯ ρ − ¯ ρ )[ ¯ C I ( a IJ − ¯ a IJ ) P J − ¯ c IJ A I P J ] + ¯ xln ¯ x [ B I a IJ P J ] = − w or X I =0 Υ I ( x, ¯ x, xlnx, ¯ xln ¯ x ; ρ − ρ , ¯ ρ − ¯ ρ ) P I = ¯ w. (74)Using (74), we eliminate P from (73) to get: α ( P ) + β ( P ) + γ P P = λ ,α ( P ) + β ( P ) + γ P P = λ . (75)The equations (75) can be solved and yield four solutions which are: P = 12 √ α λ − α λ ! γ λ − γ λ + √ Y ! √ X P = − √ X √ P = − √ α λ − α λ ! γ λ − γ λ + √ Y ! √ X P = √ X √ P = 12 √ α λ − α λ ! γ λ − γ λ − √ Y ! √ X P = − √ X √ = − √ α λ − α λ ! γ λ − γ λ − √ Y ! √ X P = √ X √ X ≡ α β + α " − α β β + γ β γ − β γ ! + α " α β + γ − β γ + β γ ! × " α β λ + α γ λ + 2 α β λ − γ γ λ + p X !! ,Y ≡ γ λ − γ γ λ λ + 4 α λ − β λ + β λ ! + λ α β λ − α β λ + γ λ ! , (77)and X ≡ Y + α " − α β λ + β λ ! + γ − γ λ + γ λ + vuut γ λ − γ γ λ λ + 4 α λ − β λ + β λ ! + λ α β λ − α β λ + γ λ ! ! . (78)One can show that one does get P I ∼ X I as one of the solutions - this corresponds to a supersymmetricblack hole, and the other solutions correspond to non-supersymmetric black holes. In this section, using the results of [3], we show the existence of “fake superpotentials” corresponding toblack-hole solutions for type II compactification on (1).As argued in [3], dS-curved domain wall solutions in gauged supergravity and non-extremal black holesolutions in Maxwell-Einstein theory have the same effective action. In the context of domain wall solutions,if there exists a W ( z i , ¯ z i ) ∈ R : V DW ( ≡ Domain Wall Potential) = −W + γ g i ¯ j ∂ i W ∂ ¯ j W , z i being complexscalar fields, then the solution to the second-order equations for domain walls, can also be derived from thefollowing first-order flow equations: U ′ = ± e U γ ( r ) W ; ( z i ) ′ = ∓ e U γ g i ¯ j ∂ ¯ j W , where γ ≡ q e − U Λ W .Now, spherically symmetric, charged, static and asymptotically flat black hole solutions of Einstein-Maxwell theory coupled to complex scalar fields have the form: dz = − e U ( r ) dt + e − U ( r ) (cid:20) c sinh ( cr ) dr + c sinh ( cr ) ( dθ + sin θdφ ) (cid:21) , where the non-extremality parameter c gets related to the positive cosmologicalconstant Λ > c = 0 that corresponds to extremalblack holes, one can write down first-order flow equations in terms of a W ( z i , ¯ z i ) ∈ R : U ′ = ± e U W ; ( z i ) ′ = ± e U g i ¯ j ∂ ¯ j W , and the potential ˜ V BH ≡ W + 4 g i ¯ j ∂ i W ∂ ¯ j W can be compared with the N = 2 supergravityblack-hole potential V BH = | Z | + g i ¯ j D i ZD ¯ j ¯ Z by identifying W ≡ | Z | . For non-supersymmetric theories18r supersymmetric theories where the black-hole constraint equation admits multiple solutions which mayhappen because several W s may correspond to the same ˜ V BH of which only one choice of W would correspondto the true central charge, one hence talks about “fake superpotential” or “fake supersymmetry” - a W : ∂ i W = 0 would correspond to a stable non-BPS black hole. Defining V ≡ e U V ( z i , ¯ z i ) , W ≡ e U W ( z i , ¯ z i ),one sees that V ( x A ≡ U, z i , ¯ z i ) = g AB ∂ A W ( x ) ∂ B W ( x ), where g UU = 1 and g Ui = 0. This illustrates thefact that one gets the same potential V ( x ) for all vectors ∂ A W with the same norm. In other words, W and ˜ W defined via: ∂ A W = R BA ( z, ¯ z ) ∂ B ˜ W correspond to the same V provided: R T gR = g .For N = 2 supergravity, the black hole potential V BH = Q T M Q where Q = ( p Λ , q Λ ) is an Sp (2 n v +2 , Z )-valued vector ( n V being the number of vector multiplets) and M ∈ Sp (2 n V + 2) is given by: M = A BC D ! , (79)where A ≡ Re N ( Im N ) − B ≡ − Im N − Re N ( Im N ) − Re N C ≡ ( Im N ) − D = − A T = − ( Im N − ) T ( Re N ) T . (80)Defining M : M = I M where M ≡ D CB A ! , I ≡ − n V +1 n V +1 ! . (81)The central charge Z = e K ( q Λ X Λ − p Λ F λ ), a symplectic invariant is expressed as a symplectic dot productof Q and covariantly holomorphic sections: V ≡ e K ( X Λ , F Λ ) = ( L Λ , M Λ )( M Λ = N ΛΣ L Σ ), and hence can bewritten as Z = Q T IV = L Λ q Λ − M λ p Λ . (82)Now, the black-hole potential V BH = Q T M Q (being a symplectic invariant) is invariant under: Q → SQ,S T M S = M . (83)As S is a symplectic matrix, S T I = I S − , which when substituted in (83) yields:[ S, M ] = 0 . (84)In other words, if there exists a constant symplectic matrix S : [ S, M ] = 0, then there exists a fakesuperpotential Q T S T IV whose critical points, if they exist, describe non-supersymmetric black holes.We now construct an explicit form of S . For concreteness, we work at the point in the moduli space for(1): φ = 1 and large ψ near x = 0 and ρ = ρ . Given the form of N IJ in (73), one sees that: N − = ˜ a + ˜ b (1)00 x + ˜ b (2)00 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)01 x + ˜ b (2)01 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)02 x + ˜ b (2)02 xlnx + ˜ c ( ρ − ρ )˜ a + ˜ b (1)01 x + ˜ b (2)01 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)11 x + ˜ b (2)11 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)12 x + ˜ b (2)12 xlnx + ˜ c ( ρ − ρ )˜ a + ˜ b (1)02 x + ˜ b (2)02 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)12 x + ˜ b (2)12 xlnx + ˜ c ( ρ − ρ ) ˜ a + ˜ b (1)22 x + ˜ b (2)22 xlnx + ˜ c ( ρ − ρ ) , (85)19hich as expected is symmetric (and hence so will Re N and ( Im N ) − ). One can therefore write M ≡ U VX − U T ! , (86)where V T = V, X T = X and U, V, X are 3 × Re N and ( Im N ) − . Writing S = A BC D ! , (87)( A , B , C , D are 3 × S ∈ Sp (6), implying: A T C T B T D T ! − ! A BC D ! = − ! , (88)which in turn implies the following matrix equations: −A T C + C T A = 0 , −B T D + D T B = 0 , −A T D + C T B = − , −B T C + D T A = . (89)Now, [ S, M ] = 0 implies: A U + B X A V − B U T C U + D X C V − D U T ! = U A + V C U B + V D X A − U T C X B − U T D ! . (90)The system of equations (89) can be satisfied, e.g., by the following choice of A , B , C , D : B = C = 0; D = ( A − ) T . (91)To simplify matters further, let us assume that A ∈ O (3) implying that ( A − ) T = A . Then (90) wouldimply: [ A , V ] = 0 , [ A , X ] = 0 , [ A − , U ] = 0 , [ A , U ] = 0 . (92)For points near the conifold locus φ = ω − , ρ = ρ , using (A10)-(16) and (69) and dropping the moduli-dependent terms in (20), one can show: (cid:16) Im N − (cid:17) I ( Re N ) IK = 0 , K = 1 , (cid:16) Im N − (cid:17) K = 0 , K = 1 , Im N ) K + ( Re N ) I (cid:16) Im N − (cid:17) IJ ( Re N ) JK = 0 , K = 1 , . (93)20his is equivalent to saying that the first two and the last equations in (92) can be satisfied by: A = − − . (94)The form of A chosen in (94) also satisfies the third equation in (92) - similar solutions were also consideredin [3]. Hence, S = − − − − . (95)We therefore see that the non-supersymmetric black-hole corresponding to the fake superpotential Q T S T IV , S being given by (95), corresponds to the change of sign of two of the three electric and magetic chargesas compared to a supersymmetric black hole. The symmetry properties of the elements of M and hence M may make it generically possible to find a constant S like the one in (95) for two-paramater Calabi-Yaucompactifications. We looked at several aspects of complex structure moduli stabilization and inclusion, in the large volumelimit, of perturbative and specially non-perturbative α ′ -corrections and instanton contributions in the K¨ahlerpotential and superpotential in the context of K¨ahler moduli, for a two-parameter “Swiss cheese” Calabi-Yau three-fold of a projective variety expressed as a (resolution of a) hypersurface in a complex weightedprojective space, with mutliple conifold loci in its moduli space. As regards N = 1 type IIB compactificationson orientifold of the aforementioned Calabi-Yau, we argued the existence of (extended) “area codes” whereinfor the same values of the RR and NS-NS fluxes, one is able to stabilize the complex structure and axion-dilaton moduli at points away from and close to the two singular conifold loci. It would be nice to explicitlywork out the numerics and find the set of fluxes corresponding to the aforementioned area codes (whoseexistence we argued), as well as the flow of the moduli corresponding to the domain walls arising as aconsequence of such area codes. Further, in the large volume limit of the orientifold, we show that with theinclusion of non-perturbative α ′ -corrections that survive the orientifolding alongwith the nonperturbativecontributions from instantons, it is possible to get a non-supersymmetric dS minimum without the inclusionof anti-D3 branes . It would interesting to investigate the effect of string loop corrections in the context oforientifolds of compact Calabi-Yau of the type considered in this work (See [25]). As regards supersymmetricand non-supersymmetric black-hole attractors in N = 2 type II compactifications on the same Calabi-Yauthree-fold, we explicitly solve the “inverse problem” of determining the electric and magnetic charges of anextremal black hole given the extremum values of the moduli. In the same context, we also show explicitlythe existence of “fake superpotentials” as a consequence of non-unique superpotentials for the same black-hole potential corresponding to reversal of signs of some of the electric and magnetic charges. There maybe interesting connection between the existence of such fake superpotentials and works like [26] AM thanks S.Mathur for bringing [26] to our attention. cknowledgements PS is supported by a C.S.I.R., Government of India, (junior) research fellowship and the work of AM ispartially supported by D.A.E. (Government of India) Young Scientist Award project grant. AM also thanksthe CERN theory group for hosptiality and the theoretical high energy physics groups at McGill University(specially Keshav Dasgupta), University of Pennsylvania (specially Vijay Balasubramanian), Ohio StateUniversity (specially Samir Mathur) and Columbia University (specially D.Kabat), for hospitality wherepart of this work was done and also where preliminary versions of this work were presented. AM also thanksSamir Mathur and Jeremy Michelson for interesting discussions, and Rajesh Gopakumar and Ashoke Senfor interesting questions (that helped improve the arguments of section 4) during AM’s presentation of partof the material of this paper at ISM07, Harish-Chandra Research Institute, Allahabad.
A Periods
In this appendix, we fill in the details relevant to evaluation of periods in different portions of the complexstructure moduli space of section . | φ | < , large ψ The expressions for P , , relevant to (4) in section are given as under: P ≡ P ∞ m =0 P ∞ n =0 A m,n φ m ρ n +6 m P ∞ m =0 P ∞ n =0 e iπ ( − m +128 n )9 A m,n φ m ρ n +6 m P ∞ m =0 P ∞ n =0 e iπ ( − m +128 n )9 A m,n φ m ρ n +6 m P ∞ m =0 P ∞ n =0 e iπ ( − m +128 n )3 A m,n φ m ρ n +6 m P ∞ m =0 P ∞ n =0 e iπ ( − m +128 n )9 A m,n φ m ρ n +6 m P ∞ m =0 P ∞ n =0 e iπ ( − m +128 n )9 A m,n φ m ρ n +6 m , (A1) P ≡ P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m e iπ ( − m +128 n )9 P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m e iπ ( − m +128 n )9 P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m e iπ ( − m +128 n )3 P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m e iπ ( − m +128 n )9 P ∞ m =0 P ∞ n =0 A m,n mφ m − ρ n +6 m e iπ ( − m +128 n )9 (A2)22nd P ≡ − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 e iπ ( − m +128 n )9 − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 e iπ ( − m +128 n )9 − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 e iπ ( − m +128 n )3 − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 e iπ ( − m +128 n )9 − P ∞ m =0 P ∞ n =0 A m,n (18 n +6 m ) φ m ρ n +6 m +10 e iπ ( − m +128 n )9 . (A3)The coefficients A m,n appearing in (A1)-(A3) are given by: A m,n ≡ (18 n + 6 m )!( − φ ) m (3 . n +6 m (9 n + 3 m )!(6 n + 2 m )!( n !) m !18 n +6 m . The equations (A1)-(A3) will be used in obtaining (25) and the third set of equations in (26). | ρ φ − ω , − , − | < M , , relevant to (7) in section are given as under: M ≡ P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m e iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m e iπ ( k + r + iπm + iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r ρ k + r φ m e iπ ( k + r + iπm + iπr , (A4) M ≡ P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m − e iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m e iπ ( k + r + iπm + iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r (6 k + r ) ρ k + r − φ m e iπ ( k + r + iπm + iπr (A5)and M ≡ P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − e iπ ( k + r + iπm P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − e iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − e iπ ( k + r + iπm + iπr P r =1 , P ∞ k =0 P ∞ m =0 A k,m,r mρ k + r φ m − e iπ ( k + r + iπm + iπr . (A6)23n equations (A4)-(A6), the coefficients A k,m,r are given by: A k,m,r ≡ e iπar sin (cid:18) πr (cid:19) ( − ) k − k + r + m e − iπ ( k + r + imπ Γ( m + k + r )(Γ( k + r )) (Γ(1 − m + k + r )) m ! . The equations (A4)-(A6) will be used in obtaining (25) and the fourth set of equations in (26).Near the conifold locus : ρ + φ = 1The expressions for N , , relevant for evaluation of (11) in section , are given as under: N ≡ P r =1 , P ∞ k =0 A k, ,r e − iπ ( k + r P r =1 , P ∞ k =0 A k, ,r e iπ ( k + r P r =1 , P ∞ k =0 A k, ,r e iπ ( k + r P r =1 , P ∞ k =0 A k, ,r e iπ ( − k +5 r P r =1 , P ∞ k =0 A k, ,r e − iπ ( k +7 r P r =1 , P ∞ k =0 A k, ,r e − iπ (3 k +3 r , (A7) N ≡ P r =1 , P ∞ k =0 ( k + r ) A k, ,r e − iπ ( k + r P r =1 , P ∞ k =0 ( k + r ) A k, ,r e iπ ( k + r P r =1 , P ∞ k =0 ( k + r ) A k, ,r e iπ ( k + r P r =1 , P ∞ k =0 ( k + r ) A k, ,r e iπ ( − k +5 r P r =1 , P ∞ k =0 ( k + r ) A k, ,r e − iπ ( k +7 r P r =1 , P ∞ k =0 ( k + r ) A k, ,r e − iπ (3 k +3 r (A8)and N ≡ P r =1 , P ∞ k =0 ( A k, ,r e iπ − A k, ,r ( k + r )) e − iπ ( k + r P r =1 , P ∞ k =0 ( A k, ,r e iπ − A k, ,r ( k + r )) e iπ ( k + r P r =1 , P ∞ k =0 ( A k, ,r − A k, ,r ( k + r )) e iπ ( k + r P r =1 , P ∞ k =0 ( A k, ,r e iπ − A k, ,r ( k + r )) e iπ ( − k +5 r P r =1 , P ∞ k =0 ( A k, ,r e iπ − A k, ,r ( k + r )) e − iπ ( k +7 r P r =1 , P ∞ k =0 ( A k, ,r − A k, ,r ( k + r )) e − iπ (3 k +3 r . (A9)The coefficients A k,m,r figuring in (A7)-(A9) are given by: A k,m,r ≡ − k + r + m e iπ ( σ +1)3 + − iπ ( k + r ( − ) k (Γ( k + r )) Γ( k + 1)Γ( k + r )Γ( k + r )(Γ(1 − m + k + r )) m ! sin (cid:18) πr (cid:19) . Near φ = 1 , Large ρ The expressions for ̟ ,..., relevant for evaluation of (16) in section , are given as under:( i ) ̟ ∼ − Z Γ ′ dµ π i Γ( − µ )Γ( µ + )Γ(1 + µ ) ρ − µ √ P τ =0 γ ,τµ ω τ ( φ − ω − τ ) µ +1 π ( µ + 1)24 A , h ( φ − − ω ( φ − ω − ) + ω ( φ − ω ) i + A , ρ h ( φ − − ω ( φ − ω − ) + ω ( φ − ω − ) i , (A10)where A , = − √ π i Γ( )Γ( ) and A , = √ π i Γ( )Γ( ).( ii ) ̟ ∼ Z Γ ′ dµ π (Γ( − µ )) Γ( µ + 16 )Γ( µ + 56 ) ρ − µ × √ π ( µ + 1) (cid:20) isin ( πµ ) (cid:16) e − iπµ ( φ − µ +1 + ω ( φ − ω − ) µ +1 − ω ( φ − ω − ) µ +1 + ( φ − µ +1 (cid:17) + e − iπµ ( φ − µ +1 (cid:21) ∼ A , h ( φ − A + ln ( ρ − )) + iπ (( φ −
1) + 2 ω ( φ − ω − − ω ( φ − ω − ) + ( φ − ln ( φ − i + A , ρ h ( φ − ( A + ln ( ρ − ) + iπ (cid:16) ( φ −
1) + 2 ω ( φ − ω − ) − ω ( φ − ω − ) (cid:17) + ( φ − ln ( φ − i , (A11)where A ≡ − − ) + Ψ( ) and A ≡ − − − ) + ψ ( ).( iii ) ̟ ∼ − Z Γ ′ dµ π (Γ( − µ )) Γ( µ + 16 )Γ( µ + 56 ) ρ − µ × √ π ( µ + 1) (cid:20) isin ( πµ ) (cid:16) − e − iπµ ( φ − µ +1 + e − iπµ ω ( φ − ω − ) µ +1 + ω ( φ − ω − ) µ +1 (cid:17) − (cid:16) ( φ − µ +1 − ω ( φ − ω − ) µ +1 (cid:17) − ( φ − µ +1 (cid:21) ∼ A , (cid:20)(cid:16) ω ( φ − ω − ) − φ − (cid:17) ( A + ln ( ρ − )) + 2 iπ (cid:16) − φ −
1) + ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) − φ − ln ( φ − (cid:21) + A , ρ (cid:20)(cid:16) ω ( φ − ω − ) − φ − (cid:17) ( A + ln ( ρ − )) − iπ (cid:18) − φ − + ω ( φ − ω − ) + ω ( φ − ω − ) (cid:19) − φ − ln ( φ − (cid:21) (A12)( iv ) ̟ ∼ Z Γ ′ dµ π (Γ( − µ )) Γ( µ + 16 )Γ( µ + 56 ) ρ − µ e iπµ × − √ π ( µ + 1) h ( φ − µ +1 − ω ( φ − ω − ) µ +1 + ω ( φ − ω − ) µ +1 i ∼ A , (cid:20)(cid:16) ( φ − − ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − ))+( φ − ln ( φ − − ω ( φ − ω − ) ln ( φ − ω − ) + ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) + A , ρ (cid:20)(cid:16) ( φ − − ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − ))+( φ − ln ( φ − − ω ( φ − ω − ) ln ( φ − ω − ) + ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) (A13)25 v ) ̟ ∼ Z Γ ′ dµ π (Γ( − µ )) Γ( µ + 16 )Γ( µ + 56 ) ρ − µ e iπµ × − √ π ( µ + 1) h e − iπµ ( φ − µ +1 + ωω ( φ − ω − ) µ +1 − ω ( φ − ω − ) µ +1 i ∼ A , (cid:20)(cid:16) ( φ − − ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − )) − iπ ( φ −
1) + ( φ − ln ( φ −
1) + ω ( φ − ω − ) ln ( φ − ω − ) − ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) + A , ρ (cid:20)(cid:16) ( φ − − ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − )) − iπ ( φ − + ( φ − ln ( φ −
1) + ω ( φ − ω − ) ln ( φ − ω − ) − ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) (A14)( vi ) ̟ ∼ Z Γ ′ dµ π (Γ( − µ )) Γ( µ + 16 )Γ( µ + 56 ) ρ − µ e iπµ × − √ π ( µ + 1) h − e − iπµ ( φ − µ +1 + e − iπµ ωω ( φ − ω − ) µ +1 + ω ( φ − ω − ) µ +1 i ∼ A , (cid:20)(cid:16) − φ −
1) + ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − ))+4 iπ ( φ − − φ − ln ( φ − − iπω ( φ − ω − ) + ω ( φ − ω − ) ln ( φ − ω − ) + ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) + A , ρ (cid:20)(cid:16) − φ − + ω ( φ − ω − ) + ω ( φ − ω − ) (cid:17) ( A + iπ + ln ( ρ − ))4 iπ ( φ − − φ − ln ( φ − − iπω ( φ − ω − ) + ω ( φ − ω − ) ln ( φ − ω − )+ ω ( φ − ω − ) ln ( φ − ω − ) (cid:21) . (A15) B Complex Structure Superpotential Extremization
In this appendix, the details pertaining to evaluation of the covariant derivative of the complex structuresuperpotential in (22), are given.One can see from (21): ∂ x K ∼ − lnx (cid:0) ¯ A B − ¯ B ¯ A + ¯ A B − ¯ A B (cid:1) K , (B1)where K ≡ i Im " ( ¯ A A + ¯ A A + ¯ A A ) + ( ¯ B A + ¯ A B + ¯ A B + ¯ B A + ¯ A B + ¯ B A ) x +( ¯ A B + ¯ B A + ¯ A B + ¯ B A ) xlnx + ( ¯ A C + ¯ C A + ¯ A C + ¯ C A + ¯ A C + ¯ C A )( ρ − ρ ) . (B2)26t the extremum values of the complex structure moduli ( x, ρ − ρ ), τ = f T . ¯Π h T . ¯Π ≈ P i =0 h i ¯ A i ) " f (cid:0) ¯ A + ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) + f (cid:0) ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) + f (cid:0) ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) + f (cid:0) ( ¯ A + ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) + f (cid:0) ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) + f (cid:0) ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ) (cid:1) × − P i =0 h i ¯ A i ) " h ( ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ h ( ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ A + ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ h ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) ! . (B3)Hence, ∂ x W c.s. ≈ lnx " ¯ B f − h Ξ[ f i ; ¯ x, (¯ ρ − ¯ ρ )] P i =0 h i ¯ A i + h ( P j =0 f i ¯ A i )Ξ[ h i ; ¯ x, (¯ ρ − ¯ ρ )]( P i =0 h i ¯ A i ) ! + ¯ B f − h Ξ[ f i ; ¯ x, (¯ ρ − ¯ ρ )] P i =0 h i ¯ A i + h ( P j =0 f i ¯ A i )Ξ[ h i ; ¯ x, (¯ ρ − ¯ ρ )]( P i =0 h i ¯ A i ) ! + ¯ B f − h Ξ[ f i ; ¯ x, (¯ ρ − ¯ ρ )] P i =0 h i ¯ A i + h ( P j =0 f i ¯ A i )Ξ[ h i ; ¯ x, (¯ ρ − ¯ ρ )]( P i =0 h i ¯ A i ) ! + ¯ B f − h Ξ[ f i ; ¯ x, (¯ ρ − ¯ ρ )] P i =0 h i ¯ A i + h ( P j =0 f i ¯ A i )Ξ[ h i ; ¯ x, (¯ ρ − ¯ ρ )]( P i =0 h i ¯ A i ) ! , (B4)where Ξ[ f i ; ¯ x, (¯ ρ − ¯ ρ )] ≡ f ( ¯ A + ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ )) + f ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ f ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ) + f ( ¯ A + ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ f ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) + f ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) , ≡ f T ¯Π(¯ x, ¯ ρ − ¯ ρ )Ξ[ h i ; ¯ x, (¯ ρ − ¯ ρ )] = h ( ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ h ( ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ B ¯ x + ¯ C (¯ ρ − ¯ ρ ))+ h ( ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) + h ( ¯ A + ¯ B ¯ x + ¯ B ¯ xln ¯ x + ¯ C (¯ ρ − ¯ ρ )) ≡ h T (cid:2) ¯Π(¯ x, ¯ ρ − ¯ ρ ) − ¯Π( x = 0 , ρ = ρ ) (cid:3) . (B5) C Inverse Metric Components
The components of the inverse of the metric (46), relevant to almost all equations in section starting from(49) are given as under: 27 G − ) ρ ¯ ρ =1∆ " Y √− ρ + ¯ ρ (cid:18) ρ X √− ρ + ¯ ρ − (cid:16) X + e φ X Y (cid:17) ¯ ρ √− ρ + ¯ ρ e φ + X Y (cid:16) √ Y + ρ √− ρ + ¯ ρ (cid:17)(cid:19) , ( G − ) ρ ¯ ρ = 1∆ " Y (cid:16) − X + e φ X Y (cid:17) ( ρ − ¯ ρ ) ( ρ − ¯ ρ ) , ( G − ) ρ ¯ G = 1∆ 24 i e φ X Y ( ρ − ¯ ρ ) (cid:16) Y + √ ρ √− ρ + ¯ ρ − √ ρ √− ρ + ¯ ρ (cid:17) ( G − ) ρ ¯ ρ =1∆ 144 Y (cid:20) − ρ X √− ρ + ¯ ρ + (cid:16) X + e φ X Y (cid:17) ¯ ρ √− ρ + ¯ ρ + e φ X Y (cid:16) √ Y − ρ √− ρ + ¯ ρ (cid:17)(cid:21) √− ρ + ¯ ρ , ( G − ) ρ ¯ G = 1∆ " − i e φ X Y (cid:16) Y − √ ρ √− ρ + ¯ ρ + √ ρ √− ρ + ¯ ρ (cid:17) ( ρ − ¯ ρ ) , ( G − ) G ¯ G = 1∆ " e φ k X Y − √ k ρ X Y √− ρ + ¯ ρ − √ e φ k ρ X Y √− ρ + ¯ ρ +6 √ k ρ X Y √− ρ + ¯ ρ + 3 √ e φ k ρ X Y √− ρ + ¯ ρ − k ρ ρ X √− ρ + ¯ ρ √− ρ + ¯ ρ − (cid:16) √ e φ k X Y + 2 k X (cid:16) √ Y − ρ √− ρ + ¯ ρ (cid:17)(cid:17) ¯ ρ √− ρ + ¯ ρ + ¯ ρ √− ρ + ¯ ρ (cid:16) √ e φ k X Y − k X ¯ ρ √− ρ + ¯ ρ + 2 k X (cid:16) √ Y + 4 ρ √− ρ + ¯ ρ (cid:17)(cid:17) , with: ∆ = − e φ X Y + 6 √ ρ X Y √− ρ + ¯ ρ + 3 √ e φ ρ X Y √− ρ + ¯ ρ − √ ρ X Y √− ρ + ¯ ρ − √ e φ ρ X Y √− ρ + ¯ ρ + 8 ρ ρ X √− ρ + ¯ ρ √− ρ + ¯ ρ + (cid:16) √ e φ X Y + X (cid:16) √ Y − ρ √− ρ + ¯ ρ (cid:17)(cid:17) ¯ ρ √− ρ + ¯ ρ − ¯ ρ √− ρ + ¯ ρ (cid:16) √ e φ X Y − X ¯ ρ √− ρ + ¯ ρ + X (cid:16) √ Y + 8 ρ √− ρ + ¯ ρ (cid:17)(cid:17) ; X ≡ X c X ( n,m ) ∈ Z / (0 , A n,m,n kc ( τ ) sin ( nk.b + mk.c ) . D Ingredients for Evaluation of N IJ In this appendix we fill in the details relevant to evaluation of X I Im ( F IJ ) X J to arrive at (70).First, using (69), one arrives at; Im ( F I ) X I = − i " B B + C C − ¯ B ¯ B − ¯ C ¯ C ! ( A + B x + C ( ρ − ρ )28 C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) + C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) ; Im ( F I ) X I = − i " C C − ¯ C ¯ C ! ( A + B x + C ( ρ − ρ )+ C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) + C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) ; Im ( F I ) X I = − i " C C − ¯ C ¯ C ! ( A + B x + C ( ρ − ρ )+ C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) + C C − ¯ C ¯ C ! ( A + B x + B xlnx + C ( ρ − ρ )) . (C1)This hence yields X I Im ( F IJ X J = ( X ) Im ( F ) + ( X ) Im ( F ) + ( X ) Im ( F )+2 x X Im ( F ) + 2 X X Im ( F ) + 2 X X Im ( F ) ≈ − i " A + 2 A B x + 2 A C ( ρ − ρ ) ! B B + C C − ¯ B ¯ B − ¯ C ¯ C ! + A + 2 A B x + 2 A B xlnx + 2 A C ( ρ − ρ ) ! C C − ¯ C ¯ C ! + A + 2 A B x + 2 A B xlnx + 2 A C ( ρ − ρ ) ! C C − ¯ C ¯ C ! + A A + [ A B + A B ] x + A B xlnx + [ A C + A C ]( ρ − ρ ) ! C C − ¯ C ¯ C ! + A A + [ A B + A B ] x + A B xlnx + [ A C + A C ]( ρ − ρ ) ! C C − ¯ C ¯ C ! + A A + [ A B + A B ] x + A B xlnx + [ A C + A C ]( ρ − ρ ) ! C C − ¯ C ¯ C ! . (C2) References [1] A. Giryavets,
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