A symplectic rearrangement of the four dimensional non-geometric scalar potential

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We present a symplectic rearrangement of the effective four-dimensional non-geometric scalar potential resulting from type IIB superstring compactification on Calabi Yau orientifolds. The strategy has two main steps. In the first step, we rewrite the four dimensional scalar potential utilizing some interesting flux combinations which we call new generalized flux orbits. After invoking a couple of non-trivial symplectic relations, in the second step, we further rearrange all the pieces of scalar potential into a completely 'symplectic-formulation' which involves only the symplectic ingredients (such as period matrix etc.) without the need of knowing Calabi Yau metric. Moreover, the scalar potential under consideration is induced by a generic tree level Kähler potential and (non-geometric) flux superpotential for arbitrary numbers of complex structure moduli, Kähler moduli and odd-axions. Finally, we exemplify our symplectic formulation for the two well known toroidal examples based on type IIB superstring compactification on ......-orientifold and ......-orientifold.

Published for SISSA by Springer

Received: August 10, 2015

Accepted: November 9, 2015 Published: November 24, 2015

Pramod Shukla1

Universit a di Torino, Dipartimento di Fisica and INFN | sezione di Torino,

Via P. Giuria 1, I-10125 Torino, Italy

E-mail: mailto:[email protected]

Web End [email protected]

Abstract: We present a symplectic rearrangement of the e ective four-dimensional non-geometric scalar potential resulting from type IIB superstring compacti cation on Calabi Yau orientifolds. The strategy has two main steps. In the rst step, we rewrite the four dimensional scalar potential utilizing some interesting ux combinations which we call new generalized ux orbits. After invoking a couple of non-trivial symplectic relations, in the second step, we further rearrange all the pieces of scalar potential into a completely symplectic-formulation which involves only the symplectic ingredients (such as period matrix etc.) without the need of knowing Calabi Yau metric. Moreover, the scalar potential under consideration is induced by a generic tree level Kahler potential and (non-geometric) ux superpotential for arbitrary numbers of complex structure moduli, Kahler moduli and odd-axions. Finally, we exemplify our symplectic formulation for the two well known toroidal examples based on type IIB superstring compacti cation on

T6=(Z2 [notdef] Z2)-orientifold and T6=Z4-orientifold.

Keywords: Flux compacti cations, Supergravity Models, Superstring Vacua

ArXiv ePrint: 1508.01197

1From October 1, 2015, the address has been changed to: ICTP, Strada Costiera 11, Trieste 34014, Italy; E-mail: [email protected]

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2015)162

Web End =10.1007/JHEP11(2015)162

A symplectic rearrangement of the four dimensional non-geometric scalar potential

JHEP11(2015)162

Contents

1 Introduction 1

2 Preliminaries 32.1 Splitting of various cohomologies under orientifold action 32.2 Four dimensional e ective scalar potential 52.3 New generalized ux orbits 9

3 Rearrangement of scalar potential: step 1 103.1 Using rst set of symplectic relations 103.2 Using new generalized ux orbits 123.3 Summary of rst rearrangement 14

4 Rearrangement of scalar potential: step 2 154.1 Invoking a set of important symplectic identities 154.2 Symplectic rearrangements 164.3 Adding D-term contributions 184.4 Summary of nal symplectic form 194.5 Towards the ten-dimensional uplift of the symplectic rearrangement of the scalar potential 20

5 Explicit examples for checking the proposal 235.1 Example A: type IIB ,! T6=(Z2 [notdef] Z2)-orientifold 23

5.2 Example B: type IIB ,! T6=Z4-orientifold 26

6 Conclusions and future directions 28

A Useful symplectic relations 30

1 Introduction

For more than a decade, moduli stabilization has been among the most challenging goals of realistic model building attempts in superstring compacti cations. In this regard, a lot of attractive progress has been made in type II orientifold compacti cation in recent years [1{7]. Turing on various possible uxes on the internal background induces e ective potentials for the moduli and hence create the possibility of uxes being utilized for moduli stabilization and in search of string vacua [8{15]. Moreover, interesting connections between the toolkits of superstring ux-compacti cations and the gauged supergravities have given the platform for approaching phenomenology based goals from two directions [16{26]. A consistent incorporation of various kinds of possible uxes makes the compacti cation

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JHEP11(2015)162

background richer and more exible for model building. For example, inclusion of non-geometric ux breaks the no-scale structure of low energy 4D type IIB supergravity, and opens the possibility of stabilizing all moduli at tree level. However, the task does not remain as simple as many technical challenges are inevitable, and the same have led to enormous amount of progress in recent years [16{18, 20{25, 27{37]. For example the resulting 4D scalar potential are very often so huge in concrete examples (say in Type IIB on T6=(Z2 [notdef] Z2) orientifold) that even it gets hard to analytically solve the extremization

conditions, and one has to look either for simpli ed ansatz by switching o certain ux components at a time, or else one has to opt for an involved numerical analysis.

There have been close connections between the symplectic geometry and e ective potentials of type II supergravity theories [38, 39], and the role of symplectic geometry gets crucially important while dealing with Calabi Yau orientifolds. The reason for the same being the fact that unlike toroidal orientifold examples, one does not know the explicit analytic representation of Calabi Yau metric needed to express the e ective potential. However, in the context of type IIB orietifolds with the presence of standard NS-NS three-form ux (H3) and RR three-form ux (F3), it has been shown that the complete four dimensional scalar potential (derived from F=D-term contributions) could be expressed via merely using the period matrices and without the need of CY metric [40, 41]. A chain of successive T -duality operations on H- ux of type II orientifold theories lead to various geometric and non-geometric uxes, namely !; Q and R- uxes. Moreover, S-duality invariance of type IIB superstring compacti cation demands for including additional P - uxes S-dual to non-geometric Q- ux [30, 42, 42{46]. Now the question arises if it could be possible to take the next step to include generalized (non-geometric) uxes in symplectic formalism of [40, 41].

Moreover, in the context of non-geometric ux compacti cations, there have been great amount of studies via considering the 4D e ective potential merely derived by knowing the Kahler and super-potentials [9{12, 26, 34, 47{49], and without having a good understanding of their ten dimensional origin. Some signi cant steps have been taken towards exploring the form of non-geometric 10D action via Double Field Theory (DFT)1 [33, 35] as well as supergravity [26, 34, 49, 51, 52]. In this regard, toroidal orientifolds have been always in the center of attraction because of their relatively simpler structure to perform explicit computations, and so toroidal setups have served as promising toolkits. For example the knowledge of metric has helped in anticipating the ten-dimensional origin of the geometric ux dependent [26] as well as the non-geometric ux dependent potentials [34] via a dimensional oxidation process in the T6=(Z2 [notdef] Z2) toroidal orientifolds of type IIA and

its T-dual type IIB model. Later on, this dimensional oxidation process has been further extended with the inclusion of P- ux, the S-dual to non-geometric Q- ux in [49] as well as with the inclusion of odd axions B2=C2 and geometric ux (!) as well as non-geometric ux (R) in [51, 52] leading to the appearance of peculiar ux combinations which are called as new generalized ux orbits.

In this article, we aim to extend the symplectic formalism of [40, 41] by rewriting the four dimensional non-geometric scalar potential in terms of symplectic ingredients. To be

1A much better understanding of the ten dimensional origin of the 4D non-geometric scalar potential has been proposed very recently in a nice work [50] via considering dimensional reduction of DFT.

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speci c, in the context of type IIB non-geometric Calabi Yau orientifold compacti cation, we will consider generic tree level Kahler and (non-geometric) ux super-potentials for arbitrary number of moduli/axions, and rearrange the F=D term contributions using new generalized ux orbits and some symplectic relations.

The article is organized as follows: section 2 provides a very brief review of type IIB non-geometric ux compacti cation relevant for the present work. In section 3, we compute the four dimensional non-geometric scalar potential resulting from the generic tree level expressions of Kahler - and super-potentials valid for arbitrary numbers of complex structure moduli, Kahler moduli and odd-axions. Subsequently as a rst step, we rewrite the scalar potential into a compact manner via using interesting ux combinations what we call new generalize ux orbits. In section 4, we invoke a couple of symplectic relations to further rewrite the rst rearrangement of section 3 into an entirely symplectic and very compact fashion. In section 5, we illustrate the utility of the same for rewriting the 4D scalar potentials of two concrete well-known examples of Type IIB superstring compacti cations

T6=(Z2 [notdef] Z2) and T6=Z4 orientfolds. Finally, in section 6, we provide an overall conclusion

followed by an appendix of additional useful symplectic relations.

2 Preliminaries

Let us consider Type IIB superstring theory compacti ed on an orientifold of a Calabi-Yau threefold X.

2.1 Splitting of various cohomologies under orientifold action

The admissible orientifold projections can be classi ed by their action on the Kahler form J and the holomorphic three-form 3 of the Calabi-Yau, given as under [53]:

O =

8

<

:

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p : (J) = J ; ( 3) = 3 ;

()FL p : (J) = J ; ( 3) = 3 ;

(2.1)

where p is the world-sheet parity, FL is the left-moving space-time fermion number, and is a holomorphic, isometric involution. The rst choice leads to orientifold with O5=O9-planes whereas the second choice to O3=O7-planes. The massless states in the four dimensional e ective theory are in one-to-one correspondence with harmonic forms which are either even or odd under the action of , and these do generate the equivariant cohomology groups Hp;q(X). Let us x our conventions as those of [47], and

denote the bases of even/odd two-forms as ( ; a) while four-forms as (~

; ~

a) where

2 h1;1+(X); a 2 h1;1(X).2 Also, we denote the zero- and six- even forms as 1 and 6

respectively. The de nitions of integration over the intersection of various cohomology

2For explicit construction of some type-IIB toroidal/CY orientifold examples with odd-axions, see [54{59].

{ 3 {

bases are,

ZX ^ ^ = k ; [integraldisplay]X ^ a ^ b =^ k ab

Note that if four-form basis is appropriately chosen to be dual of the two-form basis, one will of course have ^

d = ^

and d ba = ba. However for the present work, we follow the conventions of [47], and take the generic case. Considering the bases for the even/odd cohomologies H3(X) of three-forms as symplectic pairs (aK; bJ) and (A ; B ) respectively,

we x the normalization as under,

ZX aK ^ bJ = KJ; [integraldisplay]X A ^ B = (2.3)

Here, for the orientifold choice with O3=O7-planes, K 2 [notdef]1; : : : ; h2;1+[notdef] and 2 [notdef]0; : : : ; h2;1[notdef]

while for O5=O9-planes, one has K 2 [notdef]0; : : : ; h2;1+[notdef] and 2 [notdef]1; : : : ; h2;1[notdef].

Now, the various eld ingredients can be expanded in appropriate bases of the equivariant cohomologies. For example, the Kahler form J, the two-forms B2, C2 and the R-R four-form C4 can be expanded as [53]

J = t ; B2 = ba a; C2 = ca a (2.4)

C4 = D 2 ^ + V K ^ aK + UK ^ bK + ~

where t is string-frame two-cycle volume moduli, while ba; ca and are various axions. Further, (V K, UK) forms a dual pair of space-time one-forms and D 2 is a space-time two-form dual to the scalar eld . Also, since re ects the holomorphic three-form 3, we have h2;1(X) complex structure moduli z appearing as complex scalars. Now, we consider

a complex multi-form of even degree evenc de ned as [60],

evenc = eB2 ^ CRR + i eRe(eB2+iJ) (2.5)

+ Ga a + T ~

;

which suggests the following forms for the Einstein-frame chiral variables appearing in N = 1 4D-e ective theory,

= C0 + i e ; Ga = ca + ba ; (2.6)

T =

+ ^ abcabb + 12 ^ abba bb[parenrightbigg] i2 t t ;

where = ( ^

d1) k and ^

ab = ( ^

d1) ^

k ab. It is worth to mention that as compared

to chiral variables de ned in [53], we have rescaled our T by a factor of 2=(3i) along with a sign ip in NS-NS axion ba.

{ 4 {

ZX 6 = f; [integraldisplay]X ^ ~ = ^d ;

ZX a ^ ~ b = d ba (2.2)

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2.2 Four dimensional e ective scalar potential

The dynamics of low energy e ective supergravity action is encoded in three building blocks; namely a Kahler potential (K), a holomorphic superpotential (W ) and a holomorphic gauge kinetic function ( ^

G) written in terms of appropriate chiral variables. Subsequently, the total N = 1 scalar potential can be computed from various F=D-term contributions via

V = eK Ki|DiW D|W 3 [notdef]W [notdef]2[parenrightBig]+ 12(Re^

G)1JK DJDK :

Let us provide some details on the basic ingredients needed to generate the scalar potential V .

Kahler potential (K) and moduli space metrices. Using appropriate chiral variables, a generic form of the tree level Kahler potential can be written as a sum of two pieces motivated from their underlying N = 2 special Kahler and quaternionic structure, and the same is give as under,

K := Kcs + Kq; where (2.7)

Kcs = ln

16 lijk zi zj zk +

X0 are used, and lijk are triple intersection numbers on the Mirror Calabi Yau. Further, the quantities lij; li are real numbers while l0 is a pure imaginary number [61, 62]. In general, f(zi) will have an in nite series of non-perturbative contributions (say Finst:(zi)), however for the current purpose, we are assuming the large

complex structure limit to suppress the same. Now, the overall internal volume VE in the

Einstein frame can be generically written in terms of two-cycle volume moduli as below,

VE = 16 k t t t (2.10)

The well known fact which can be seen from the structure of the Kahler potential (2.7) is that the total moduli space metric is block diagonal with one block corresponding to the complex structure moduli while the other one involving Kahler moduli, odd-axions and axion-dilaton. So the scalar potential computations via utilizing Kahler derivatives and metrices gets much simpli ed.

One should represent VE in terms of chiral variables (; Ga; T ) as de end in eq. (2.6)

for computing the moduli space Kahler metrices, and for doing this, one needs to invert

{ 5 {

; Kq = ln (i( )) 2 ln VE :

Here, the involutively-odd holomorphic three-form 3 generically depends on the complex structure moduli (zk) and can be written out in terms of period vectors,

3 X A F B (2.8)

via using a genetic tree level pre-potential as under,

F = (X0)2 f(zi) ; f(zi) =

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i

ZX 3 ^ 3

1

2 lij zi zj + li zi +

1

2 l0 (2.9)

where special coordinates zi =

i X

the expression of T . Though it is not possible to do it for a general CY orientifold compacti cation, nevertheless one can still represent the Kahler matrix components into another suitable form involving two cycle volume moduli (t ), the dilaton (s), NS-NS B2 axion (ba) and triple intersection numbers ( and ^

ab) [53]. For the present work,

let us just recollect the relevant Kahler derivatives along with the inverse Kahler metric components as under,

K = i 2 s

1 + 2 s Gab ba bb[parenrightBig]= K (2.11)

KGa = 2 i Gab bb = KG

a ; KT =

3 i ^

d t

k0 = KT ;

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and

K = 4 s2; KGa = 4 s2 ba; KT = 2 s2 ^

abbabb; (2.12)

KGa G

= s Gab + 4 s2 babb; KT G

a

= s Gab ^

bcbc + 2 s2^

bcbbbc ba;

b

KT T = 49 k20

+ s Gab ^

acbc ^

bdbd + s2 ^

abbabb ^

cdbcbd;

where = [parenleftBig]

( ^

d1) [prime] G [prime] [prime] (

^

d1) [prime]

. Also, for writing Kahler metric we have used

^

ab = ( ^

d1) ^

k ab along with the following short hand notations for G and G1 compo-

nents,

G =

3 2

k k0 32k k k20

; Gab = 23 k0^kab (2.13)

G =

2

3 k0 k + 2 t t ; Gab =

3 2

^

kab

k0

Moreover we have introduced k0 = 6 VE = k t , k = k t , k = k t and

^

kab =

^

k ab t . Let us mention that apart from a slight di erence in the de nition of chiral variables (2.6), due to the presence of ^

d and dab matrices, there is a further slight change in the expressions of various components of inverse Kahler metric as compared to the ones given in [53]. One can show that the dependence on these d-matrices can be picked up via considering the fact that @VE

@T scales with ^

d @VE@T as can be anticipated directly from

6 VE := k t t t = (t

^

d )( t t ). Taking these into account, one nds that,

49 k20

= (

^

d1) [prime]

k [prime] k [prime] 23 k0 k [prime] [prime]

[parenrightbigg]( ^d1) [prime] (2.14)

= 4 VE (^d1) [prime] k [prime] [prime] ( ^ d1) [prime] [parenrightBig]

Note that one often uses orientifold constructions such that ^

d = and dab = ab, and so one will not need to take care of these extra normalizations, however in cases otherwise, e.g. in the second example Example B, the same is important as we will see later.

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Non-geometric ux superpotential (W ). Turning on various uxes on the internal background induces a non-trivial ux superpotential [40]. To construct a generic form of the superpotential, one has to understand the splitting of various geometric as well as non-geometric uxes into the suitable orientifold even/odd bases. Moreover, it is important to note that in a given setup, all ux-components will not be generically allowed under the full orietifold action O = p()FL. For example, only geometric ux ! and non-

geometric ux R remain invariant under ( p()FL), while the standard uxes (F; H) and

non-geometric ux (Q) are anti-invariant [15, 47]. Therefore, under the full orientifold action, we can only have the following ux-components

F F ; F [parenrightbig]

; H H ; H

[parenrightbig]

; ! !a ; !a ; ^

! K; ^

! K

[parenrightbig]

;

QaK; QaK; ^

Q ; ^

Q

[parenrightBig]; (2.15)

For writing a general ux-superpotential, one needs to de ne a twisted di erential operator,

D involving the actions from all the NS-NS (non-)geometric uxes as [47],

D = d + H ^ : + ! / : + Q . : + R : (2.16)

The action of operator /; . and on a p-form changes it into a (p+1), (p1) and (p3)-form

respectively. Considering various ux-actions on the di erent even/odd bases to result in even/odd three-forms, we have [47],

H = H A + H B ; F = F A + F B ;

!a (! / a) = !a A + !a B ; ^

Q (Q . ~

) = ^

Q A +

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R RK; RK

[parenrightbig]

; Q

^

Q B (2.17)

^

! (! / ) = ^

! KaK + ^

! KbK;

Qa (Q . ~

a) = QaK aK + QaKbK;

R = RKaK + RKbK :The rst three lines involve ux components counted via odd-index 2 h2;1(X) while the later three have even-index K 2 h2;1+(X). Using de nitions in (2.17), we have the

following additional useful non-trivial actions of uxes on various 3-form even/odd basis elements [47],

H ^ A = f1H 6; H ^ B =f1H 6 (2.18) ! / A = d1

ab !b ~ a; ! / B =

d1

ab !b ~ a

Q . A =

[parenleftBig]

^

d1

^

Q ; Q . B = [parenleftBig]^ d1

^

Q ;

and

R aK = f1 RK 1; R bK =f1 RK 1 ! / aK =

[parenleftBig]

^

d1

^! K ~ ; ! / bK =

[parenleftBig]^ d1

^! K ~

Q . aK = d1

abQaK b; Q . bK =

d1

ab QaK b :

{ 7 {

With these ingredients in hand, a generic form of ux superpotential is as under,

W =

ZX

F + D evenc[bracketrightbigg]3 ^ 3 =

ZX

F + H + !aGa + ^

Q T

3^ 3: (2.19)

This generic ux superpotential W can be equivalently written as,

W = e X + m F ; (2.20)

where

e = F + H + !a Ga + ^

Q T ; (2.21)

m = F + H + !a Ga + ^

Q T :

Using the superpotential (2.20), one can compute the various derivatives with respect to chiral variables, ; Ga and T as followings,

W =H X + H F ; W =H X + H F

WGa =!a X + !a F ; W Ga =!a X + !a F

WT = ^

Q X +

^

Q F ; W T =

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Q X +

^

Q F : (2.22)

^

Note that, only !a and ^

Q components are allowed by the choice of involution to contribute into the superpotential, and in order to turn-on the non-geometric R- uxes, one has to induce D-terms via implementing a non-trivial even sector of H2;1(X)-cohomology [47, 50{ 52]. However, for the cases with homomorphic involutions with h2;1(CY=O) = 0, which one often adopts in moduli stabilization and subsequent phenomenological purposes, no such D-terms involving non-geometric R- ux will be induced.

The D-terms (DK[arrowhookleft] DK). In the presence of a non-trivial sector of even (2,1)-cohomology, i.e. for h2;1+(X) [negationslash]= 0, there are additional D-term contributions to the four

dimensional scalar potential. Following the strategy of [47], the same can be determined via considering the following gauge transformations of RR potentials CRR = C0 + C2 + C4,

CRR ! CRR + D( KaK + KbK) (2.23)

C0 f1RK K + f1RK K

[parenrightbig]

+ cb (d1)abQaK K + (d1)abQaK K[parenrightBig] b

+ (^d1) ^! K K + ( ^d1) ^! K K[parenrightBig]~

Recollection of various pieces suggests the following two D-terms being generated by the gauge transformations,

DK = i

f1RK (@K) + (d1)baQbK (@aK) + ( ^d1) ^! K (@ K)

[bracketrightbigg](2.24)

DK = i

f1RK (@K) + (d1)baQbK (@aK) + ( ^d1) ^! K (@ K) [bracketrightbigg]

Now using the expressions for tree level Kahler derivatives (2.11), one nds

DK = 1

2 s VE

RK f

VE s2^k abt babb[parenrightBig]+ s (d1)baQbK ^k act bc s t ^ ! K

[bracketrightbigg]

(2.25)

DK =

1

2 s VE

RK f

VE s2^k abt babb[parenrightBig]+ s (d1)baQbK ^k act bc s t ^ ! K

[bracketrightbigg]

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2.3 New generalized ux orbits

A closer investigation of the symplectic vectors (e ; m ) and (DK; DK), which are responsible for generating F -term and D-term contributions to the scalar potential, suggests for de ning peculiar ux combination as new generalized ux orbits [51, 52]. The ux orbits in NS-NS sector with orientifold odd-indices k 2 h2;1(X) are given as,

H = H + !a ba + ^

Q

12 ^ ab babb [parenrightbigg]

H = H + !a ba + ^

Q

12 ^ ab babb [parenrightbigg]

(2.26)

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fa = !a + ^ Q

^ ab bb[parenrightBig]; [Omegainv]a = !a + ^ Q

^ ab bb [parenrightBig]

^

Q = ^

Q ; ^

Q = ^

Q

while the ux components of even-index K 2 h2;1+(X) are given as,

^

f K = ^! K (d1)ba QbK [parenleftBig]^k ac bc

[parenrightBig]

+ f1 RK

12^k ab ba bb

[parenrightbigg]

f K = ^! K (d1)ba QbK [parenleftBig]^k ac bc

[parenrightBig]

^

+ f1 RK

12^k ab ba bb [parenrightbigg]

(2.27)

QaK = QaK + f1 dba (RK bb); QaK = QaK + f1 dba (RK bb);

RK = RK; RK = RK :

In the rst set of orbits (2.26), we have used ^

ab = ( ^

d1) ^

k ab. Now, the RR three-form

ux orbits are generalized in the following form,

F = F + !a ca + ^

Q

+ ^ abcabb[parenrightBig]+ c0H ; (2.28)

F = F + !a ca + ^

Q

+ ^ abcabb[parenrightBig]+ c0 H :

Using these ux orbits along with the de nitions of chiral variables in eq. (2.6), the symplectic vectors (e ; m ) and (DK; DK) are compactly written as under,

e = F + i (s H ) i

^

Q

[parenrightBig];

m = F + i s H

[parenrightbig]

i

^

Q

[parenrightBig]; (2.29)

and

DK = 1

2 s VE

f1RK VE s t ^

f K

[bracketrightbigg]

;

DK =

1

2 s VE

f1RK VE s t ^

f K

[bracketrightbigg]

; (2.30)

where the symbol represents Einstein-frame four-cycle volume given as: =

1

2 t t .

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3 Rearrangement of scalar potential: step 1

Here we provide a detailed computation of the N = 1 four dimensional e ective scalar potential for the type IIB superstring theory compacti ed on a Calabi Yau orientifold. Our aim is to perform the most genetric tree level analysis with arbitrary number of moduli and axions, i.e. for h1;1+(CY=O) number of complexi ed Kahler moduli T , h1;1(CY=O) number

of complexi ed odd-axions Ga as well as h2;1(CY=O) number of complex structure moduli

zi. In the process of doing the taxonomy of various pieces of F-term scalar potential, we will utilize new generalized ux orbits (2.26){(2.28) which has been proposed in [34, 49, 51, 52] in a series of iterative attempts.

3.1 Using rst set of symplectic relations

Let us start with the scalar potential analysis via considering the following splitting of pieces coming from generic N = 1 F-term contribution,

eK VF = KAB (DAW ) (DBW ) 3[notdef]W [notdef]2 Vcs + Vk : (3.1) Here we have separated the complex structure and rest of the moduli dependent piece as facilitated by the block diagonal form of Kahler metric in these two sectors, and

Vcs = Kij (DiW ) (DjW ); Vk = KAB (DAW ) (DBW ) 3[notdef]W [notdef]2 (3.2) The indices (i; j) corresponds to complex structure moduli zis while the other indices (A; B) are counted in rest of the chiral variables [notdef]; Ga; T [notdef]. Now the plan is to rewrite the

F-term scalar potential into various pieces which could be expressed in terms of components of period-matrix, and we will do it in three parts.

Part 1. In the rst step of simpli cation, we use the following symplectic identity [38],

Kij (DiX ) (DjX ) = X X

(3.4)

Using period matrix components, one can introduce the following de nitions of the new-matrices (M) for computing the hodge star of various odd-three forms [38],

? A = M

= ReN ImN (3.6)

M = M

T

M = ImN ReN ImN ReN

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1

2 eKcs ImN (3.3)

where the period matrix N for the involutively odd (2,1)-cohomology sector is de ned

as under,

N = F + 2 i

Im(F ) X X (ImF ) Im(F )X X

A + M B ; and (3.5) ? B = M A + M B

where we also de ne the following useful components to be heavily utilized in the present work,

M = ImN

M

Now, we can split the piece Vcs into two parts as follows

Vcs = Kij (DiW ) (DjW ) = Vcs1 + Vcs2 ; (3.7)

where the two pieces are further simpli ed as,

Vcs1 =

1

2 eKcs e + m N

[parenrightbig]

ImN e + m N

[parenrightbig]

(3.8)

=

1

2 eKcs e M e e M m + e M m m M m

[parenrightbig]

+ i

2 eKcs e m e m

[parenrightbig]

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:= V (1)cs1 + V (2)cs1 :

and

Vcs2 = e + m N

[parenrightbig] [parenleftBig]

X X [parenrightBig][parenleftBigg][parenleftBigg]e

+ m N

[parenrightbig]

(3.9)

= e (X X ) e m (F X ) e e (X F ) m m (F F ) m

=

e X + m F [parenrightBig][parenleftBigg][parenleftBigg]e

X + m F

[parenrightbig]

where in the last line of Vcs2, an exchange of indices $ has been utilized. The reason

for this splitting is the fact that the second piece of Vcs1 is nulli ed via using a set of tadpole conditions and NS-NS Binachi identities. This point will be detailed later on when we will see the explicit expressions of e and m written in terms of NS-NS and RR generalized ux orbits.

Part 2. Now, we take the piece Vk of the scalar potential (3.2), and consider a taxonomy of pieces recollected as under

Vk =

KAB KA KB[notdef]W [notdef]2 3[notdef]W [notdef]2[parenrightBig]+ KAB WA W B

[parenrightBig]

+KAB (KA W ) W B + WA (KBW )

[parenrightbig]

(3.10)

Using the derivatives of Kahler potential (2.11) and inverse Kahler metric (2.12), one nds the following useful relations,

KA KA = ( ) = KB KBKA KAGa = (Ga Ga) = KGaB KB (3.11)

KA KAT = (T T ) = KT B KB

using which, one gets the well known no-scale relation,

KAB KA KB = 4 : (3.12)

This simpli es the piece Vk as under

Vk = [notdef]W [notdef]2 + KAB (KA W ) W B + WA (KBW )

[parenrightbig]

+ KAB WA W B[parenrightBig](3.13)

{ 11 {

A closer investigation of the second pieces shows that,

KAB (KA W ) W B + WA (KBW )

[parenrightbig]

= ( ) W W W W [parenrightbig]

+ (Ga Ga) W W Ga W WG

a

[parenrightbig]

+(T T ) W W T W WT

[parenrightbig]

= 2 [notdef]W [notdef]2 + W [parenleftBig]

e X + m F [parenrightBig]

(3.14)

Here in simplifying the last step, we have used the fact that superpotential (2.20) is a linear function in chiral variables ; Ga and T which results in following relations,

W + Ga WGa + T WT = W F X + F F [parenrightbig]

(3.15)

+ W e X + m F

[parenrightbig]

W + Ga W Ga + T W T = W [parenleftBig]

F X + F F [parenrightBig]

This follows directly from derivatives of superpotential given in eq. (2.22).

Part 3. After observing eq. (3.14), we nd that the rst two (of the three) pieces of Vk given in eq. (3.13) can be recombined with Vcs2. Now we can have a new rearrangement of

the total F -term scalar potential into three pieces as under,

eK VF = V1 + V2 + V3 (3.16)

where we consider a new collection of pieces given as under,

V1 := Vcs1 (3.17)

V2 := Vcs2 + KAB KA KB[notdef]W [notdef]2 3[notdef]W [notdef]2[parenrightBig]+ KAB (KA W ) W B + WA (KBW )

[parenrightbig]

V3 := KAB WA W B

Now, using the simpli cation results from Part: 1 and Part: 2, we try to rewrite these three pieces V1; V2 and V3 of the scalar potential in terms of new generalized ux combinations. The reason for such a collection will be clearer as we proceed in this section.

3.2 Using new generalized ux orbits

Rewriting V1 using generalized ux orbits: after a detailed investigation of pieces within V1 Vcs1 = V (1)cs1 + V (2)cs1 given in eq. (3.8) and using new generalized ux or

bits (2.26){(2.28), we nd that rst piece of simpli ed Vcs1 takes the form,

V (1)cs1 =

1

2 eKcs

JHEP11(2015)162

[bracketleftBig][parenleftBigg]

F

M F F M F + F M F F M F

[parenrightbig]

(3.18)

+s2 H M H H M H + H M H H M H +

^ M

^

Q

^ M

^

Q + ^

Q M

^

Q

^

Q M

^

Q

[parenrightBig]

2 s

H M ^

Q

H M

^

Q + H M

^ H M

^

Q

[parenrightBig] [bracketrightBig]

{ 12 {

and the second piece V (2)cs1 can be further rearranged as under,

V (2)cs1 = i

2 eKcs e m e m

[parenrightbig]

(3.19)

=

1

2 eKcs [bracketleftbigg]

2 s F H F H

[parenrightbig]

2

F ^

Q F ^

Q

[parenrightBig][bracketrightbigg]

where ^

Q := ^

Q and ^

Q := ^

Q have been utilized in these expressions. The

piece (3.19) combines various NS-NS and RR-Bianchi identities and can be considered as generalized RR Tadpoles [49, 52], and these have to vanish by adding local sources.

Rewriting V2 using generalized ux orbits: as a next step, we consider the second piece V2 which using the results of Part 2 simpli es as under

V2 := Vcs2 + KAB KA KB[notdef]W [notdef]2 3[notdef]W [notdef]2[parenrightBig]+ KAB (KA W ) W B + WA (KBW )

[parenrightbig]

= (e e )Re(X X )(e e ) + (e e )Re(X F )(m m )

+(m m )Re(F X )(e e ) + (m m )Re(F F )(m m ) (3.20) Now, the eq. (3.20) can be re-expressed using generalized ux orbits as,

V2 = 4 Re(X X )[bracketleftBig]

s2 H H + s H ^

Q + s H ^

Q

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^

Q ^

Q

[bracketrightBig]

+4 Re(X F )[bracketleftBig]

s2 H H + s H ^

Q + s H ^

Q

^

Q ^

Q

[bracketrightBig]

(3.21)

+4 Re(F X )[bracketleftBig]

s2 H H + s H ^

Q + s H ^

Q

^

Q ^

Q

[bracketrightBig]

hs2 H H + s H ^

Q + s H ^

Q ^

Q ^ Q [bracketrightBig]

Rewriting V3 using generalized ux orbits: we consider the third collection of piece V3 which is given as:

V3 := KAB WA W B = X X [bracketleftBig]

H K H + !a KGaG

+4 Re(F F )

b

!b + ^

Q KT T ^

Q

+H KG

a

!a + !a KGa H + H KT ^

Q + ^

Q KT H

a

!a + !a KGaT ^

Q

+ ^

Q KT G

[bracketrightBig]

(3.22)

+X F [ ] + F X [ ] + F F [ ]

Using generalized ux combinations and Kahler metric components, we nd the following rearrangement in V3,

V3 = 4 Re(X X )[bracketleftBig]

s2 H H + s

4 [Omegainv] a Gab [Omegainv]b +

k20 9

^

Q

^

Q

[bracketrightBig]

+4 Re(X F )[bracketleftBig]

s2 H H + s

4 [Omegainv] a Gab [Omegainv]b +

k20 9

^

Q

^

Q

[bracketrightBig]

(3.23)

+4 Re(F X )[bracketleftBig]

s2 H H + s

4 [Omegainv] a Gab [Omegainv]b +

k20 9

^

Q

^

Q

[bracketrightBig]

+4 Re(F F )

hs2 H H + s4 [Omegainv] a Gab [Omegainv]b +k209^

Q ^Q [bracketrightBig]

{ 13 {

One should observe that in the case when only H3 and F3 uxes are present, the whole contribution of [notdef]H[notdef]2 type is already embedded into Vcs1 [40, 41], and hence a cancellation

of pure H- ux pieces of V2 and V3 is anticipated, and we have

V2+V3 = 4 Re(X X )[bracketleftbigg]

2 sH ^

Q + s

4 [Omegainv] a Gab [Omegainv]b +

1 4

^ Q

4 k209 4 [parenrightbigg][bracketrightbigg]

+4 Re(X F )[bracketleftbigg]

2 s H ^

Q + s

4 [Omegainv] a Gab [Omegainv]b +

1 4

^ Q

4 k209 4 [parenrightbigg][bracketrightbigg]

+4 Re(F X )[bracketleftbigg]

2 s H ^

Q + s

4 [Omegainv] a Gab [Omegainv]b +

1 4

Q ^

^ Q

4 k209 4 [parenrightbigg][bracketrightbigg]

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+4 Re(F F )

2 s H ^

Q + s4 [Omegainv] a Gab [Omegainv]b +14^

Q ^ Q

4 k209 4 [parenrightbigg][bracketrightbigg]

(3.24)

3.3 Summary of rst rearrangement

As a summary at this stage, we have the following scalar potential rearrangement,

VFF =

1

2 eKKcs [bracketleftbigg]

F

M F F M F + F M F F M F [bracketrightbigg]

(3.25)

VHH =

1

2 eKKcs [bracketleftbigg]

s2 H M H H M H + H M H H M H

[parenrightbig][bracketrightbigg]

VFH =

1

2 eKKcs [bracketleftbigg]

(2) s F H F H

[parenrightbig][bracketrightbigg]

VFQ =

1

2 eKKcs [bracketleftbigg]

(+2)

F ^

Q F ^

Q

[parenrightBig][bracketrightbigg]

VHQ =

1

2eKKcs [bracketleftbigg]

(2) s

[parenleftBig]

H

M

^ H M

^

Q + H M

^ H M

^

Q

[parenrightBig]

+ 8 eKcs

H Re(X X )^

Q + H Re(X F )

^

Q

+ H Re(F X )

^

Q + H Re(F F )

^

Q

[parenrightbigg][bracketrightbigg]

V[Omegainv][Omegainv] =

1

2 eKKcs [bracketleftbigg]

(2) s eKcs Gab

f a Re(X X ) [Omegainv]b + [Omegainv] a Re(X F ) [Omegainv]b

+ [Omegainv] a Re(F X ) [Omegainv]b + [Omegainv] a Re(F F ) [Omegainv]b [bracketrightbigg]

VQQ =

1

2 eKKcs [bracketleftbigg][parenleftBig]

^ M

^

Q

^ M

^

Q + ^

Q M

^

Q

^

Q M

^

Q

[parenrightBig]

2eKcs

4 k209 4 [parenrightbigg] ^

Q Re(X X )^

Q + ^

Q Re(X F )^Q

+ ^

Q Re(F X )

^

Q + ^

Q Re(F F )

^

Q

[bracketrightbigg]

:

{ 14 {

4 Rearrangement of scalar potential: step 2

In this section we will provide three sets of equivalent symplectic representations of the F -term scalar potential taking a next step to our rst rearrangement in eq. (3.25). For that purpose, let us rst present a couple of very important symplectic identities.

4.1 Invoking a set of important symplectic identities

We nd that the following interesting and very analogous relation as compared to the de nition of period matrix (3.4) holds,

F = N + 2 i

Im(N ) X X (ImN ) Im(N )X X

(4.1)

Moreover, similar to the de nition of the period matrices (3.6), one can de ne another set of symplectic quantities as under,

L = ImF

L

= ReF ImF (4.2)

L = L

JHEP11(2015)162

T

L = ImF ReF ImF ReF

Now we will use these two sets of matrices M and L as building blocks and will de ne some

new combinations of the same which will be useful for our scalar potential rearrangement purpose. In this context, we de ne three new sets of symplectic quantities M1, M2 and

M3 as under,

M1 = M + L

M1 = M

+ L

(4.3)

M1 = M + L M1 = M + L

M2 = M + 2 L

[parenrightbig]

M2 = M

+ 2 L

[parenrightbig]

(4.4)

M2 = M + 2 L

[parenrightbig]

M2 = (M + 2 L )

and

M3 = + M L + M L

[parenrightbig]

M3 = + M L + M L

[parenrightbig]

M3 = M L + M L

[parenrightbig]

(4.5)

M3 = M L + M L

[parenrightbig]

Apart from these de ning equations (4.3){(4.5), there are some more relations among

M1; M2 and M3 which we will present in the appendix A. Now the most important

{ 15 {

relation which will serve as a bridging segment for the symplectic rearrangement of the

scalar potential is given as under,3

4 eKcs Re(X X ) = M1

4 eKcs Re(F X ) = M1 (4.6)

4 eKcs Re(X F ) = + M1

4 eKcs Re(F F ) = + M1

We have checked eq. (4.6) for h2;1(CY ) = 0; 1; 2 and 3 using pre-potential given in

eq. (2.9). As the computations involve inverting complicated matrices of order (h2;1 + 1),

for h2;1(CY ) 4, it gets too huge to verify the identities, however we expect the same to

be generically true for an arbitrary number of complex structure moduli.

4.2 Symplectic rearrangements

As we have many symplectic identities with many quantities such as M; L and Mis, this

will result in more than one equivalent rearrangements of the scalar potential. In order to prefer one over the other, let us try to gure out some points as guidelines for our rearrangement,

Considering the moduli space metrices given in eq. (2.13) we nd that,

Gab =

2

3 k0 ^

kab (4.7)

4 k209 4 [parenrightbigg]= 23 k0 (^d1) [prime] k [prime] [prime] ( ^ d1) [prime]

= 4 VE (

4 k209 4 [parenrightBig]factor can be clubbed with the other one to

M + 8 eKcs Re(X X )[parenrightBig]etc. which is similar to the only piece of

Q type. This appears to be a better one as then, one piece of both of QQ and [Omegainv][Omegainv] can be written with h1;1 ux-indices being contracted by the even/odd sector metrics and Gab.

Here we note that the rst four pieces of collection (3.25) are already in desired form as those are already written in the suitable symplectic forms as we will see later. Now using relations (3.6), (4.3){(4.5), and considering the points above we can rearrange the VHQ,

V[Omegainv][Omegainv] and VQQ pieces of eq. (3.25) in the following three representations,

3The rst equation of (4.6) can be also obtained by comparing eqs. (11) and (27) of [38], and this has motivated us to de ne what we call our L matrices and invoke for its three other components.

{ 16 {

JHEP11(2015)162

ab = 4 VE

^

d1) [prime] k [prime] [prime] ( ^

^

d1) [prime]

where k0 = 6 VE has been used. From eq. (3.25), this shows that coe cient of the

pieces with Re(X X ), Re(X F ) etc. in both V[Omegainv][Omegainv] as well as VQQ are similar, and

so may be clubbed in a similar manner in the rearrangement.

Apart from the rst choice mention above, we also observe that ( ) contributions

in QQ piece via

look similar as

H ^

Representation-I: using M and M1 matrices.

VHQ =

1

2 eKKcs [bracketleftbigg]

(2) s

H M 2 M1

[parenrightbig]

^ H M 2 M1

[parenrightbig]

^

Q

+H M 2 M1

[parenrightbig]

^ H (M 2 M1 )

^

Q

[parenrightbigg][bracketrightbigg]

^ [Omegainv]a M1 [Omegainv]b [Omegainv]a M1 [Omegainv]b (4.8)

+[Omegainv]a M1 [Omegainv]b [Omegainv]a M1 [Omegainv]b [parenrightbigg][bracketrightbigg]

VQQ =

V[Omegainv][Omegainv] =

1

2 eKKcs [bracketleftbigg]

(2 VE s)

JHEP11(2015)162

1

2 eKKcs [bracketleftbigg][parenleftBig]

2 VE (

^

d1) [prime] k [prime] [prime] ( ^

d1) [prime]

[parenrightBig] [parenleftbigg]

^

Q M1

^

Q

^

Q M1

^

Q

+ ^

Q M1

^

Q

^

Q M1

^

Q

[parenrightbigg][bracketrightbigg]

1

2 eKKcs [bracketleftbigg]

^ M

^

Q

^ M

^

Q + ^

Q M

^

Q

^

Q M

^

Q

[bracketrightbigg]

:

Representation-II: using M and M2 matrices.

VHQ =

1

2 eKKcs [bracketleftbigg]

(2) s

H M2 ^

Q

H M2

^

Q

+H M2

^ H M2

^

Q

[parenrightbigg][bracketrightbigg]

V[Omegainv][Omegainv] =

1

2 eKKcs [bracketleftbigg]

14 s [notdef] Gab [parenleftbigg]

fa

M [Omegainv]b [Omegainv]a M [Omegainv]b

+[Omegainv]a M [Omegainv]b [Omegainv]a M [Omegainv]b [parenrightbigg][bracketrightbigg]

1

2 eKKcs [bracketleftbigg]

(s VE

kab) [notdef] [parenleftbigg]

fa

^

M2 [Omegainv]b [Omegainv]a M2 [Omegainv]b (4.9)

+[Omegainv]a M2 [Omegainv]b [Omegainv]a M2 [Omegainv]b [parenrightbigg][bracketrightbigg]

VQQ =

1

2 eKKcs [bracketleftbigg]

1 4

4 k209 [parenrightbigg] [notdef] [parenleftbigg]^

Q M ^

Q ^

Q M ^Q

+ ^

Q M

^

Q

^

Q M

^

Q

[parenrightbigg][bracketrightbigg]

1

2 eKKcs [bracketleftbigg][parenleftBig]VE

( ^

d1) [prime] k [prime] [prime] ( ^

d1) [prime]

[parenrightBig] [parenleftbigg]

^ M2 ^

Q

^ M2 ^ Q

+ ^

Q M2

^

Q

^

Q M2

^

Q

[parenrightbigg][bracketrightbigg]

:

Representation-III: using M and M3 matrices.

VHQ =

1

2 eKKcs [bracketleftbigg]

(2) s

[parenleftBig]

H

M

^ H M

^

Q +H M

^ H M

^

Q

[parenrightBig]

+ 2

H M ~Q H M ~Q + H M ~Q H M ~Q [parenrightBig][bracketrightbigg]

{ 17 {

V[Omegainv][Omegainv] =

1

2 eKKcs [bracketleftbigg]

s4 Gab [parenleftbigg]

~[Omegainv]a M

~[Omegainv]b

~[Omegainv]a M

~[Omegainv]b

+~[Omegainv]a M

~[Omegainv]b

~[Omegainv]a M

~[Omegainv]b

[parenrightbigg][bracketrightbigg]

VQQ =

1

2 eKKcs [bracketleftbigg][parenleftBig]

^ M

^

Q

^ M

^

Q + ^

Q M

^

Q

^

Q M

^

Q

[parenrightBig]

+14 [parenleftbigg]

4 k20

9 4 [parenrightbigg] [parenleftbigg]

~Q M

~Q

~Q M

~Q

; (4.10)

where eq. (A.4) has been utilized for this representation via de ning ~[Omegainv]a and ~Q as under,~[Omegainv]a = M3 [Omegainv]a + M3 [Omegainv]a

[parenrightbig]

+ M3 [Omegainv]a + M3 [Omegainv]a

+ ~Q M

~Q

~Q M

~Q [parenrightbigg][bracketrightbigg]

JHEP11(2015)162

[parenrightbig]

(4.11)

M3 ^

Q + M3 ^

Q

[parenrightBig]

There is a bit of abuse of notation as di erent quantities are denoted with similar (however not the same) notations; e.g. [Omegainv]; ^

[Omegainv]; ~[Omegainv] as well as ~Q and

^

Q are di erent.

~Q = [parenleftBig]M3

^

Q + M3

^

Q

[parenrightBig]

+

4.3 Adding D-term contributions

As we have mentioned earlier, if the choice of homolorphic involution is such that one can have h2;1+(CY ) [negationslash]= 0, then additional contributions to the e ective scalar potential are

introduced via D-terms written in new generalized ux orbits as under [50, 52],

DK = 1

2 s VE

f1RK VE s t ^

f K

[bracketrightbigg]

; DK =

1

2 s VE

f1RK VE s t ^

f K

[bracketrightbigg]

:

Now similar to the period matrices M of involutively odd (2,1)-cohomology sector, following

from the underlying N = 2 symplectic structure, one can introduce similar matrices for the disjoint even sector as under,

^

MJK = Im

^

N JK;

^

M

KJ = Re ^

NJI Im ^

N IK (4.12)

^

MJK = [parenleftBig]

^

M

K J

T

^

MJK = Im

^

NJK Re

NJI Im ^

N IL Re

^

NLK

^

Also, as the gauge kinetic function ^

G are given as [60],

^

GJK =

i2^

NJK

at (zK =0=zK); (4.13)

where ^

N is the period matrix on the even (2,1)-cohomology sector similar to (3.4). Using

these ingredients one nds the D-term contributions to the four dimensional scalar potential as under,

V (1)D = V[Omegainv][Omegainv] + VR[Omegainv] + VRR (4.14)

{ 18 {

where

V[Omegainv][Omegainv] =

1

2 eKKcs [bracketleftbigg]

fJ ^

^ MJK

^

fK

fJ ^

^ MJK

^

[Omegainv]K + ^

[Omegainv]J ^

M KJ

^

fK

[Omegainv]J ^

^ MJK ^ [Omegainv]K

[bracketrightbigg]

(4.15)

VR[Omegainv] =

1

2 eKKcs [bracketleftbigg]

(2VE)

RJ ^

MJK^

fK

RJ

^

^ [Omegainv]K +RJ ^

M KJ

^

fK

RJ

MJK ^ [Omegainv]K

[parenrightbigg][bracketrightbigg]

^

VRR =

1

2 eKKcs [bracketleftbigg]

V2Es [notdef]

RJ ^

M RK RJ^

MJK RK + RJ^

M KJ RK RJ^

MJK RK

[parenrightBig][bracketrightbigg]

Here ^

fK = t ^

f K and ^

[Omegainv]K = t ^

f K. We also mention that total D-term is positive de nite and can be written as

1 4V2E

RCY3 D ^ ^ D for D = DKaK + DKbK.

4.4 Summary of nal symplectic form

The good thing about presenting several symplectic arrangements via introducing symplectic matrices (M;

^

M and Mis) is the fact that now one can express various pieces either

as O1 ^ O2 or O1 ^ O2 form. For example, considering the third representation, we can

express the full scalar potential as,

Ve =

1

2 eKKcs [integraldisplay]CY3 [bracketleftbigg]

F ^ F + s2 H ^ H +

^

Q ^

^

Q 2 s H ^

^

Q (4.16)

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4 s H ^

~Q +

s4 Gab

~[Omegainv]a ^

~[Omegainv]b + 14 [parenleftbigg]

4 k20

9 4 [parenrightbigg]

~Q ^

~Q

+V2Es R ^ ^ R + s

^

[Omegainv] ^ ^

^

[Omegainv] 2 VE R ^ ^

^

[Omegainv] + 2 s F ^ H 2 F ^

^

Q

[bracketrightbigg]

:

where ^ and denote the Hodge star operations in the even/odd (2,1)-cohomology sector.

Here while introducing the integral sign, we have assumed that uxes are constant parameters, and so M matrices have been simply replaced by their respective integral forms. The

last two pieces with F^H and F^

^

Q terms are nulli ed via a combination of NS-NS and RR Bianchi identities. In order words, the same can be nulli ed by adding contributions from local sources such as branes/orientifold planes. The same can be expressed as additional generalized D3=D7 contributions given as under

V (2)D = VFH VFQ (4.17)

Note that in addition to the actual RR tadpole constraints, setting the above V (2)D to zero will need the following (subset of) NS-NS Bianchi identities obtained via demanding the nillpotency of the twisted di erential operator as D2 = 0 [47],

H ^

Q H

^

Q =0; H !a H !ak =0 (4.18)

Q ^

^ Q

Q ^

^ Q =0; !a !b !b !ak =0; !a

^

Q !a

^

Q =0

RK ^

! K RK ^

! K =0; RK QaK RK QaK =0

! K ^

^ ! K ^

! K ^

! K =0; QaKQbK QbKQaK =0; QaK ^

! K QaK ^

!K =0

{ 19 {

Finally, using generic tree level Kahler potential in eq. (2.7), we get eKKcs = 1

2 s V2E , and

subsequently the total scalar potential takes a nal form as under,

Ve =

14 s V2E [integraldisplay]CY3 [bracketleftbigg]

F ^ F + s2 H ^ H +

^

Q ^

^

Q 2 s H ^

^

Q

4 s H ^

~Q +

s4 Gab

~[Omegainv]a ^

~[Omegainv]b + 14 [parenleftbigg]

4 k20

9 4 [parenrightbigg]

~Q ^

~Q

+V2Es R ^ ^ R + s

^

[Omegainv] ^ ^

^

[Omegainv] 2 VE R ^ ^

[bracketrightbigg]

^ : (4.19)

It appears to be quite remarkable that the total F/D-term scalar potential of arbitrary number of complex structure moduli, Kahler moduli and odd-axions has been written out so compactly in terms of symplectic ingredients along with the moduli space metrices, and also without the need of knowing the Calabi Yau metric.

4.5 Towards the ten-dimensional uplift of the symplectic rearrangement of the scalar potential

So far our aim has been only to rearrange the 4D e ective scalar potential which has been derived using a generalized version of the GVW ux superpotential. As an evidence that the collection of various 4D scalar potential pieces, written in terms of new generalized ux orbits, can indeed be derived from the dimensional reduction of a ten-dimensional theory, now we connect the various pieces of our symplectic formulation with those obtained from the reduction of Double Field Theory (DFT) on a Calabi Yau orientifold4 [50]. Although for details on the later, we refer the readers to [50], we hereby collect the relevant results needed to establish the connection with our approach. The DFT Lagrangian on a Calabi Yau threefold can be given as the sum of following two pieces,

LRR =

1

2 G ^ G

LNS NS = e2[bracketleftbigg]

1

2 ~ ^ ~ +

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1

2 ^

1 4

^ ~[parenrightBig] ^ [parenleftBig] ^ ~ [parenrightBig]

1 4

^ ~[parenrightBig] ^ [parenleftBig] ^ ~ [parenrightBig][bracketrightbigg]

where

The N = 2 string-frame de nition of the ux combination ~ (in our conventions) can

be given as,

~ D eiJ = H + [Omegainv] / (iJ) + Q . [parenleftbigg]

(iJ) ^ (iJ)

2

(iJ) ^ (iJ) ^ (iJ)6[parenrightbigg](4.20)

where similar to the previously de ned twisted di erential operator D, a new operator

D = d + H ^ : + [Omegainv] / : + Q . : + R : have been introduced to incorporate the e ects

of B2- eld such that,

H = H + ! / B2 + Q .

B2 ^ B22[parenrightbigg]+ R

[parenrightbigg]

+ R

B2 ^ B2 ^ B2 6

; etc: (4.21)

4We thank the referee for suggesting us to comment on the ten-dimensional origin of our symplectic rearrangement of the 4D scalar potential.

{ 20 {

The second ux combination is de ned as,

D = H ^ + [Omegainv] / + Q . + R (4.22)

The generalized RR three form eld strength G is given as,

G F + D C = F + H ^ C(0) + [Omegainv] / C(2) + Q . C(4) + R C(6) (4.23)

where RR-form C = C(0) + C(2) + C(4) + C(6) + : : :

Now, we will investigate the four types of terms in LNS NS and LRR to connect with those of ours. Note that, as we have already converted the total scalar potential into real moduli/axions (and the nal collection does not use the chiral super elds), so we can directly check the connection by simply considering the orientifold projected version of various terms in LNS NS and LRR.

Matching the RR sector: as a very rst observation, let us recall that in our approach the generalized RR eld strength is as given in eq. (2.28),

F = F + !a ca + ^

Q

+ ^ abcabb[parenrightBig]+ c0 H ;

F = F + !a ca + ^

Q

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+ ^ abcabb[parenrightBig]+ c0 H :

Thus noting from orbit collection in eq. (2.26) that [Omegainv]a = !a + ^

Q ^

abcabb and [Omegainv]a =

!a + ^

Q ^

abbb, we have the rst identi cation from eq. (4.23) and eq. (2.28),

G F : (4.24)

This identi es the RR sectors of the 4D scalar potentials in the two approaches as,

(I) :=

14 s V2E [integraldisplay]CY3=O

F ^ F (4.25)

Matching the NS-NS sector: under the orientifold involution, for the ux combination ~ being de ned in eq. (4.20), we get the following splitting into the even/odd (2,1)-cohomology bases,

~ ~ A + ~ B (4.26)

[bracketleftbigg][parenleftbigg]

= H

G ^ G =

14 s V2E [integraldisplay]CY3=O

1 s

^

Q

A +

H 1s^

Q

[parenrightbigg] B [bracketrightbigg]

+i 1

ps

[bracketleftbigg][parenleftbigg]

^

f K t

1s VE RK[parenrightbigg]

aK +

^

f K t

1s VE RK

bK [bracketrightbigg]

Note that some factors of \s" are introduced as we have changed various orientifold projected pieces of ~ into Einstein-frame. Using the collection of ~ in terms of our generalized

{ 21 {

ux combinations, we nd the following identi cation of pieces in Einstein-frame,

(II) :=

14 s V2E [integraldisplay]CY3=O

s2 ~ ^ ~ (4.27)

=

14 s V2E [integraldisplay]CY3=O [bracketleftbigg]

s2 H ^ H +

^

Q ^

^

Q 2 s H ^

^

Q

+V2Es R ^ ^ R + s

^

[Omegainv] ^ ^

^

[Omegainv] 2 VE R ^ ^

^

[Omegainv]

[bracketrightbigg]

Comparing above with our symplectic collection given in eq. (4.19), we nd that the pieces in the rst line are from F -term superpotential contribution while those in the last line are induced via D-terms.

Further, as the holomorphic three-form is odd under orientifold involution, the multi-degree form , as de ned in eq. (4.22), will have three components appearing as 6-form, 4-form and 2-form respectively. Therefore Einstein-frame expression of ( ^ ) piece can

be expanded as,

(III) :=

14 s V2E [integraldisplay]CY3=O

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s2 ^

=

14 s V2E [integraldisplay]CY3=O [bracketleftbigg]

s2 (H ^ ) ^ (H ^ )

+s ([Omegainv] / ) ^ ([Omegainv] / ) + (Q . ) ^ (Q . )

[bracketrightbigg]

(4.28)

Now let us consider the two cross-pieces of LNS NS which are given as under,

(IV ) := 1

8 s V2E [integraldisplay]CY3=O [bracketleftbigg][parenleftBig]

^ ~

[parenrightBig] ^ [parenleftBig]

^ ~

[parenrightBig]

+ ^ ~[parenrightBig] ^ [parenleftBig] ^ ~ [parenrightBig][bracketrightbigg]

14 s V2E [integraldisplay]CY3=O [parenleftBig]

^ Re(~)

[parenrightBig] ^ [parenleftBig]

^ Re(~)

[parenrightBig]

= 1

4 s V2E [integraldisplay]CY3=O[bracketleftbigg]

s2 (H ^ ) ^ (H ^ ) + (

^

Q ^ ) ^ (

^

Q ^ )

s (H ^ ) ^ (

^

Q ^ ) s (

^

Q ^ ) ^ (H ^ )

[bracketrightbigg]

(4.29)

Q with appropriate indices. Now, notice that the rst piece in eqs. (4.28) and (4.29) cancel each other. Recall that this is the same cancellation which we have observed in eqs. (3.21) and (3.23) while considering V2 + V3 in the analysis of previous section. Now using the symplectic relations (A.1){(A.3)

we nd that (III) + (IV ) results in the remaining following pieces of our collection,

(III) + (IV )

14 s V2E [integraldisplay]CY3=O [bracketleftbigg]

where in the last equality, we have used Re(~) = H 1s

^

s4 Gab

~[Omegainv]a ^

~[Omegainv]b (4.30)

4 s H ^

~Q +

1 4

4 k209 4 [parenrightbigg]~Q ^ ~Q [bracketrightbigg]:

{ 22 {

In this way, we are able to ensure that the various pieces of the scalar potential rearrangement collected in our symplectic formalism can indeed be derived from a ten-dimensional theory (namely DFT) after compactifying the same on a Calabi Yau orientifold. Now we will examine the proposal in two concrete examples.

5 Explicit examples for checking the proposal

Here we will present two explicit examples to illustrate the insights of our symplectic formulation of the four dimensional scalar potential.

5.1 Example A: type IIB [arrowhookleft]! T6/(Z2 Z2)-orientifoldLet us brie y revisit the relevant features of a setup within type IIB superstring theory compacti ed on T6= (Z2 [notdef] Z2) orientifold. The complex coordinates zis on each of the tori

in T6 = T2 [notdef] T2 [notdef] T2 are de ned as

z1 = x1 + U1 x2; z2 = x3 + U2 x4; z3 = x5 + U3 x6; (5.1)

where the three complex structure moduli Uis can be written as Ui = vi + i ui; i = 1; 2; 3. Further, the two Z2 orbifold actions are being de ned as

: (z1; z2; z3) ! (z1; z2; z3) (5.2) : (z1; z2; z3) ! (z1; z2; z3) :

Moreover, the full orientifold action is: O ( p (1)FL ) has the holomorphic involution

being de ned as

: (z1; z2; z3) ! (z1; z2; z3) ; (5.3)

resulting in a setup with the presence of O3=O7-plane. The complex structure moduli dependent pre-potential is given as,

F = X 1 X 2 X 3X 0

which results in the following period-vectors,

X 0 = 1 ; X 1 = U1 ; X 2 = U2 ; X 3 = U3 ; (5.5)

F0 = U1 U2 U3 ; F1 = U2 U3 ; F2 = U3 U1 ; F3 = U1 U2 Now, the holomorphic three-form 3 = dz1 ^ dz2 ^ dz3 can be expanded as,

3 = 0 + U1 1 + U2 2 + U3 3

+ U1 U2 U3 0 U2 U3 1 U1 U3 2 U1 U2 3

where we have choosen the following basis of closed three-forms

0 = dx1 ^ dx3 ^ dx5 ; 0 = dx2 ^ dx4 ^ dx6 ;

1 = dx2 ^ dx3 ^ dx5 ; 1 = dx1 ^ dx4 ^ dx6 ; (5.6)

{ 23 {

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= U1 U2 U3 (5.4)

2 = dx1 ^ dx4 ^ dx5 ; 2 = dx2 ^ dx3 ^ dx6 ;

3 = dx1 ^ dx3 ^ dx6 ; 3 = dx2 ^ dx4 ^ dx5 Subsequently we nd that

Kcs = ln

i

X F X F [parenrightBig][bracketrightbigg]= 3

Xj=1ln i(Uj Uj)

[parenrightbig]

: (5.7)

This also demands that Im(Ui) < 0 which is rooted from the condition of physical domain to be de ned via period matrix (3.4) condition Im(N ) < 0 [38, 41]. This condition

Im(Ui) < 0 is equally important as to demand Im() > 0 and Im(T ) < 0 which are related to string coupling and volume moduli to take positive values, or in other words positive de niteness of moduli space metrices. Now, the basis of orientifold even two-forms and four-forms are as under,

1 = dx1 ^ dx2; 2 = dx3 ^ dx4; 3 = dx5 ^ dx6 (5.8)

~

1 = dx3 ^ dx4 ^ dx5 ^ dx6; ~

2 = dx1 ^ dx2 ^ dx5 ^ dx6; ~

3 = dx1 ^ dx2 ^ dx3 ^ dx4

implying that ^

d = . The only non-trivial triple intersection number (k ) is given as 123 = k123 = 1 which implies the volume form of the sixfold to be VE = t1 t2t3 and so

the four cycle volume moduli are given as, 1 = t2 t3; 2 = t3 t1; 3 = t1 t2: implying that

t1 =

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r2 31 ; t2 = [radicalbigg]1 32 ; t3 = [radicalbigg]1 23 =) VE = p123 (5.9)

Let us mention that for this example there are no two-forms anti-invariant under the orientifold projection, i.e. h1;1(X6) = 0, and therefore no B2 and C2 moduli as well as no

geometric ux components such as !a ; !a are present. Moreover, as h2;1+(X6) = 0, so no geometric as well as non-geometric ux components with index K 2 h2;1+(X6) are present, and this implies that respective D-terms will not be induced. The only D-term can arise from the local sources such as branes and orientifold planes to cancel the RR tadpoles. Now, the expressions for Kahler potential and the generalized ux-induced superpotential take the following forms,

K = ln (i( ))

3

Xj=1ln i(Uj Uj)

[parenrightbig]

3

X =1ln

i (T T )2[parenrightbigg]

(5.10)

F + H + ^

Q T

[parenrightBig] F [bracketrightbigg]; (5.11)

where = 0; 1; 2; 3 and = 1; 2; 3 implying the presence of 8 components for each of three form uxes H3 and F3 given as,

H0; H1; H2; H3; H0; H1; H2; H3

F0; F1; F2; F3; F 0; F 1; F 2; F 3

and similarly 24 Q- ux components can be written for ^

Q and ^

Q . Now to analyze and express the total F-term scalar potential in our symplectic formulation, we do the followings,

{ 24 {

W =

[bracketleftbigg][parenleftBig]

F + H + ^

Q T

[parenrightBig] X

+

First, we utilize the Kahler potential (5.10) and superpotential (5.11) which results

in 2422 terms in total.

Subsequently, using new generalized ux orbits and the relevant symplectic relations

given in appendix A, we enumerate terms in each of the three rearrangements, and nd that the counting of terms can be distributed into the various pieces of our symplectic formulation given in eq. (4.19) as under,

VFF =

14 s V2E [integraldisplay]CY3

F ^ F; #(VFF) = 1630

VHH =

s4 V2E [integraldisplay]CY3

H ^ H; #(VHH) = 76

VQQ =

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14 s V2E [integraldisplay]CY3 [parenleftBig]

^

Q ^

^

Q (VE ^

) ~Q ^

~Q [parenrightBig]

; #(VQQ) = 408

VHQ =

14 V2E [integraldisplay]CY3 [parenleftBig]

2 H ^

^

Q 4 H ^

~Q[parenrightBig]

; #(VHQ) = 180

VHF =

14 V2E [integraldisplay]CY3

(2 F ^ H) ; #(VHF) = 32

VFQ =

14 s V2E [integraldisplay]CY3

(2 F ^ Q) ; #(VFQ) = 96

Here, for checking the symplectic formulation, we have used the following relations,

49 k20 [parenrightbigg] 4 4VE k =

0

B

@

4 21 0 0

0 4 22 0 0 0 4 23

1

C

A

(5.12)

and

VE k =

0

B

@

0 1 2 1 3

1 2 0 2 3

1 3 2 3 0

1

C

A

(5.13)

Thus we are able to rewrite the total F-term scalar potential in terms of symplectic ingredients and without using internal background metric. As mentioned earlier, the last two pieces VHF and VFQ correspond to generalized tadpole contributions and these have to be canceled by local sources plus satisfying a subset of NS-NS Bianchi identities given in (4.18).

Note that our example A is too simple to illustrate all the features of proposed symplectic formalism in eq. (4.19) basically in two sense; rst it does not have odd axions due to h1;1(X6) = 0 and so use of generalized ux orbits corrected via odd-axions B2=C2 have

not been demonstrated. Second, this example could not introduce non-geometric R- ux due to a trivial sector of even (2,1)-cohomology as h2;1+(X6) = 0. For that purpose, we now come to our example B in the next subsection.

{ 25 {

5.2 Example B: type IIB [arrowhookleft]! T6/Z4-orientifoldIn this case, we consider a type IIB compacti cation setup on the orientifold of T6=Z4 orbifold and analyze the scalar potential for the untwisted sector moduli/axions. This setup has h2;1(X) = 1+ + 0, and h1;1(X) = 3+ + 2, i.e. there are three complexi ed

Kahler moduli (T ), two complexi ed odd axions (Ga) and no complex structure moduli. The only non-zero intersection numbers are: k311 = 1=2; k322 = 1 which results in overall volume form VE = 14(t21 2 t22) t3. In addition, one has odd intersection numbers as

^

k311 =

d = diag[notdef]1=2; 1; 1=4[notdef] and dab = diag[notdef]1; 1=2[notdef]. Also,

given that h2;1(X) = 0, complex structure moduli dependent piece of the Kahler potential

is just a constant piece. Here we x our conventions by considering X 0 = 1; F0 = i, which results in eKcs = 1=2. While we leave additional orientifold construction related details to be directly referred from [47, 52], here we simply provide the explicit expressions of Kahlerand super-potentials for analyzing F-term scalar potential. The Kahler potential is given as under,

K = ln 2 ln (i( )) 2 ln VE(T ; ; Ga; T ; ; Ga) (5.14)

where the Einstein frame volume is given as,

VE VE(T ; S; Ga) =

1; ^k322 = 1=2 along with

^

i(T3 T 3)2 i4( )^ 3ab (Ga Ga)(Gb Gb)[parenrightbigg]1=2

[notdef]

[bracketleftbigg][parenleftbigg]

i(T1 T 1) 2

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1 4

2

1=2

2 2

i(T2 T 2) 2

(5.15)

Further, the generic form of the tree level ux superpotential with all allowed uxes being included is given as,

W =

[bracketleftBig][parenleftBig]

F0 + H0 + !0a Ga + ^

Q 0 T

F 0 + H0 + !a0 Ga + ^

Q 0 T

[parenrightBig][bracketrightBig]; (5.16)

where a = [notdef]1; 2[notdef] and = [notdef]1; 2; 3[notdef]. Now, one can compute the full F-term scalar potential

from these explicit expressions of K and W . For this toroidal setup, the new generalized ux orbits given in eqs. (2.26), (2.27) and (2.28) are simpli ed. The ones with odd-indexed uxes are given as under,

H0 = H0 + (!01 b1 + !02 b2) + ^

Q30

12 ^ 311(b1)2 + 12 ^ 322(b2)2 [parenrightbigg]

F0 = c0 H0 +

[parenrightBig]

i

hF0 + (!01 c1 + !02 c2) + ^

Q10 1

+ ^

Q20 2 + ^

Q30 3 + ^

311c1b1 + ^

322c2b2

[parenrightbig][bracketrightBig]

(5.17)

f01 =

h!01 + ^

Q30 ^

311 b1

[parenrightbig] [bracketrightBig]

; [Omegainv]02 =

h!02 + ^

Q30 ^

322 b2

[parenrightbig][bracketrightBig]

^

Q10 = ^

Q10; ^

Q20 = ^

Q20; ^

Q30 = ^

Q30

while the ones with even-indexed uxes are given as,

R1 = f1R1; ^ f11 = ^!11; ^ f21 = ^!21; ^ f31 = ^!31 Q11b1 + Q12b2

[parenrightbig]

R1(2b21 + b22) (5.18)

{ 26 {

and similarly ux components with upper index = 0 and K = 1 can be analogously written. Using these ux orbits one gets a total of 382 terms in F-term contribution while 72 terms in V (1)D. The F-term pieces can be rearranged as,

VFF = 1

4 s V2E [bracketleftbigg]

F20 + (F0)2

[bracketrightbigg]

#(VFF) = 178

VHH = s

4 V2E [bracketleftbigg]

H20 + (H0)2

[bracketrightbigg]

#(VHH) = 30

VFH = 1

4 V2E [notdef]

2 H0F0 F0H0[bracketrightbigg]

#(VFH) = 60

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VFQ =

14 s V2E [notdef]

2 F0

^

Q0

[parenrightBig] [parenleftBig]^

Q0

[parenrightBig]

F0

[bracketrightbigg]

#(VFQ) = 64

V[Omegainv][Omegainv] = 1

4 V2E [bracketleftbigg]

3 [Omegainv]01[Omegainv]01 + [Omegainv]10[Omegainv]10

[parenrightbig]

+ 2 3 [Omegainv]02[Omegainv]02 + [Omegainv]20[Omegainv]20

[parenrightbig][bracketrightbigg]

VHQ =

14 V2E [notdef]

(+2)

3 H0

^

Q0

[parenrightBig]+ 3

^

Q0

[parenrightBig]

H0

[bracketrightbigg]

#(V[Omegainv][Omegainv] + VHQ) = 34

VQQ =

14 s V2E [bracketleftbigg][parenleftBigg]

422 21

[parenrightbig]

Q01 ^

^ Q01 + 21 22

[parenrightbig]

Q02 ^

^ Q02 + 23 ^

Q03 ^

Q03

+ 2 12 ^

Q01 ^

Q02 6 23

Q02 ^

^ Q03 6 13

Q01 ^

^ Q03

[bracketrightbigg]

(5.19)

+ 1

4 s V2E [bracketleftbigg][parenleftBigg]

422 21

[parenrightbig]

^ Q01 + 21 22

[parenrightbig]

^ Q02 + 23 ^

Q03 ^

Q03

+ 2 12 ^

Q01 ^

Q02 6 23

^ Q03 6 13

^

Q01 ^

Q03

[bracketrightbigg]

#(VQQ) = 16 :

while the pieces coming from D-term V (1)D are as under,

VRR =R21 + (R1)2

4 s2 #(VRR) = 2;

V[Omegainv][Omegainv] =

t ^

f 1

2+ (t ^ f 1)24 V2E

#(V[Omegainv][Omegainv]) = 56; (5.20)

t ^

f 1

[parenrightBig]

+ R1

t ^

f 1

[parenrightBig]

#(VR[Omegainv]) = 14 ;

This example also illustrates how a huge scalar potential can be so compactly rewritten using the new generalized ux orbits. Moreover, this rearrangement of the total scalar potential can be easily seen from our eq. (4.19) and any of the three symplectic representations in eqs. (4.8){(4.10) after supplementing the following symplectic ingredients,

M00 = 1; M00 = 0; M00 = 0; M00 = 1

L00 = 1; L00 = 0; L00 = 0; L00 = 1(M1)00 = 2; (M1)00 = 0; (M1)00 = 0; (M1)00 = 2 (5.21)

(M2)00 = 3; (M2)00 = 0; (M2)00 = 0; (M2)00 = 3

{ 27 {

VR[Omegainv] = 2 [notdef]

R1

4 s V2E

(M3)00 = 0; (M3)00 = 2; (M3)00 = 2; (M3)00 = 0^

M11 = 1;

^

M11 = 0;

^

M11 = 0; ^

M11 = 1

For example, V[Omegainv][Omegainv] can be known simply by considering Gab = 4 VE

^

kab. Now ^

kab = ^

k abt ,

so one has the only non-zero components given as G11 = 3 and G22 = 2 3 as can be

seen from collection in eq. (5.19). Similarly, for the largest piece VQQ, let us consider

the followings,

49 k20 4 [parenrightbigg]:= 4 VE (^d1) [prime] k [prime] [prime] ( ^d1) [prime] (5.22)

=

0

B

@

422 221 0 41 3

0 21 222 42 3

41 3 42 3 0

1

C

A

Now one can immediately read o the precise sum of two ^

Q ^

Q pieces as given in collec-

tion (5.19) from the following coe cient matrix,

49 k20 4 [parenrightbigg]+ =

0

B

B

B

@

422 21 1 2 31 3

1 2 21 22 32 3

31 3 32 3 23

1

C

C

C

A

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(5.23)

Here we recall that = 12 t t t and results in 1 = t1 t3; 2 = t2 t3 and 3 = (t21 2 t22). Thus we have illustrated our generic proposal of symplectic rearrangement in

two Toroidal examples.

6 Conclusions and future directions

In [40], the four dimensional e ective potentials obtained in the context of type IIB super-string compacti caion with superpotentials induced by the standard NS-NS and RR three form uxes (H3 and F3) have been expressed in terms of symplectic ingredients using [38].

In this article, we have extended that symplectic formulation for a superpotential induced by generalized uxes turned-on on generic Calabi Yau orientifold backgrounds. This has been done in a two-step strategy. First we have rewritten the total scalar potential into suitable pieces using a set of new generalized ux orbits, and subsequently after invoking some non-trivial symplectic relations we have further rearranged various pieces into a symplectic formulation.

As a check of our proposal, we have considered two concrete examples of type IIB superstring compacti cation on the orientifolds of T6=(Z2 [notdef] Z2) and T6=Z4. Both of these

simple examples have their own advantages and limitations. For example, the rst example with T6=(Z2 [notdef] Z2)-orientifold illustrates the utility of period matrix part in the symplectic

rearrangement as it has 3 complex structure moduli, however this example does neither

support involutively odd-axions nor has involutively even (2; 1)-cohomology sector to illus

trate the appearance of D-term involving R- ux. On the other hand, the second example

{ 28 {

with T6=Z4-orientifold has two odd axions as h1;1(CY ) = 2, and moreover h2;1+(CY ) = 1

which help in demonstrating the crucial use of the new generalized ux combinations we have, and also in the embedding of D-terms. However, the second example does not have any complex structure moduli and so the information within period matrix sector has been indeed trivial. Thus we can say that the two examples considered in this article compliment each other quite well, and at the same time remain simple enough to perform explicit analytic computations needed to check the proposal.

The symplectic rearrangement of the 4D scalar potential proposed in this article has many possible advantages and applications; for example,

The total symplectic rearrangement is very compact, and helps in rewriting the scalar

potential consisting of thousands of terms into a few lines. Moreover, we do not need to know the Calabi Yau metric as the desired relevant pieces of information for rewriting the total scalar potential can be extracted via the moduli space matrices and the period matrices.

The symplectic rearrangement is what we call suitable for dimensional oxidation

purpose (on the lines of [34, 49, 51, 52]), and at least for the scenarios when the uxes are treated as constant parameters, one could naively guess the ten-dimensional uplift of the four dimensional scalar potential. In fact, we have connected the various pieces of our rearrangement with those of a scalar potential obtained by dimensional reduction of Double Field Theory on a CY orientifold [50].

Moreover, the scalar potential under consideration is valid for an arbitrary Calabi

Yau orientifold compacti cation, and so is equipped with arbitrary numbers of complex structure moduli, Kahler moduli and odd-axions. In addition, the symplectic rearrangement generically consists of all kinds of (non-)geometric uxes along with the standard H3 and F3 uxes, however which of those can be consistently turned-on on a given background still needs an answer.

In the light of the aforesaid points, the present analysis should be helpful in the model

independent studies of phenomenological aspects, e.g. moduli stabilization, searching de-Sitter solutions etc.

For example, to elaborate on the point of moduli stabilization, let us consider an orientifold setup with h2;1+(CY3) = 0 which has been very common in the setups of previous moduli stabilization studies and let us say that we want to focus on the stabilization of universal axion (c0) and dilaton (s), then using eq. (4.19) the total e ective potential can be rewritten as,

V (c0; s; : : : : : :) =

l1s + l2 + s l3[parenrightbigg]+ l4s c0 +l5s c20 (6.1)

where lis depend on all the moduli/axions except the dilaton (s) and RR axion c0. Note that it has been possible to extract the dilaton dependence from all the pieces as we

{ 29 {

JHEP11(2015)162

have already expressed the symplectic collection into Einstein-frame. Further, the explicit expressions of lis can be collected as under,

l1 =

14 V2E [integraldisplay]CY3 [bracketleftbigg]

G ^ G +

^

Q ^

Q + 14 [parenleftbigg]

4 k20

^

9 4 [parenrightbigg]

~Q ^

~Q [bracketrightbigg]

l2 =

14 V2E [integraldisplay]CY3 [bracketleftbigg]

2 H ^

^

Q 4 H ^

~Q +

14 Gab

~[Omegainv]a ^

~[Omegainv]b

;

l3 =

14 V2E [integraldisplay]CY3

H ^ H l5; (6.2)

l4 =

14 V2E [integraldisplay]CY3

(G ^ H + H ^ G) ;

JHEP11(2015)162

where G = F + [Omegainv]a ca + ^

Q while other ux combinations are as de ned in eq. (2.26).

Now extremizing the potential (6.1) w.r.t. universal axion and dilaton, one nds that

c0 =

l42 l5 ; s = [radicalbig]4

l1l5 l242 pl3 l5 (6.3)

Moreover, the two- eld analysis shows that Hessian at the above critical point leads to,

Vc0c0 = 4 pl

3 l3=25

p4 l1l5 l24; Vc0s = 0 = Vsc0 ; Vss = 4 pl

5 l3=23

p4 l1l5 l24

(6.4)

By this two- eld analysis we have shown some indications how the symplectic rearrangement could be useful for performing a model independent moduli stabilization. Finally we may agree that many things work quite nicely, however there are several issues to be settled in order to have a complete understanding of the setups with non-geometric uxes. Moreover, which and how many uxes can be truly and consistently turned on simultaneously remains an open issue which is essential for studying the moduli stabilization and subsequent phenomenology, and we hope to get back on some of these issues in future.

Acknowledgments

I am very grateful to Ralph Blumenhagen for useful discussions and encouragements throughout. Moreover, I am thankful to Ralph Blumenhagen, Anamaria Font, Xin Gao, Daniela Herschmann, Oscar Loaiza-Brito and Erik Plauschinn for useful discussions during earlier collaboration. This work was supported by the Compagnia di San Paolo contract \Modern Application of String Theory" (MAST) TO-Call3-2012-0088.

A Useful symplectic relations

Using the symplectic matrices M; M1; M2 and M3 de ned in eqs. (3.6), (4.3), (4.4)

and (4.5) respectively, one nds that,

Re(X X ) =

14 eKcs M + L

[parenrightbig]

=

14 eKcs M1 (A.1)

Re(F X ) =

14 eKcs M

+ L

[parenrightbig]

=

14 eKcs M1

{ 30 {

Re(X F ) = +

14 eKcs M + L

[parenrightbig]

= +14 eKcs M1

Re(F F ) = +

14 eKcs (M + L ) = +

14 eKcs M1

and

M + 8 eKcs Re(X X )[parenrightBig]= M2 = M 2M1

[parenrightbig]

(A.2)

M + 8 eKcs Re(F X )[parenrightBig]= M2 = M

2M1

[parenrightbig]

M + 8 eKcs Re(X F )

[parenrightbig]

= M2 = M 2M1

[parenrightbig]

JHEP11(2015)162

= M2 = (M 2 M1 )

and so equivalently we have another set of relations as under,

M1 = 12 M M2

[parenrightbig]

; (A.3)

M1 = 12 M

M + 8 eKcs Re(F F )

[parenrightbig]

M2

[parenrightbig]

;

M1 = 12 M M2

[parenrightbig]

;

M1 = 12 (M M2 )

M1 = 12

hM3 M M3

+ M M3

[parenrightbig]

+M3 M M3

+ M M3

[parenrightbig][bracketrightBig]

M1 = 12

hM3 M M3

+ M M3

[parenrightbig]

+M3 M M3

+ M M3

[parenrightbig][bracketrightBig]

M1 =

1

2

hM3

M M3 + M M3

[parenrightbig]

+M3 M M3 + M M3

[parenrightbig][bracketrightBig]

M1 =

1

2

hM3 M M3 + M M3

[parenrightbig]

(A.4)

Being directly related to produce Hodge star of three-forms as in eq. (3.5), we consider that M should be present in all the rearrangement of the scalar potential pieces, and so

we choose either of M1; M2 and M3 along with M for rewriting the various pieces. This

leads to three rearrangements of the scalar potential.

Verifying the non-trivial symplectic identities. Though verifying these symplectic identities is quite non-trivial for generic h2;1(CY ) case, let us present some veri cation of

the same by considering particular cases in the limit of not presenting too huge expressions.

{ 31 {

+M3 M M3 + M M3

[parenrightbig][bracketrightBig]

Case 1: h2,1(CY ) = 0. For the case of frozen complex structure moduli (e.g. models

studied in [14, 15, 51, 52]), we can have all the lijk; lij and li to be zero while choosing the

pure imaginary number l0 as l0 = i, and so we have

X 0 = 1; F0 = i (A.5)

implying that Kcs := ln i (X F X F )[parenrightBig]= ln 2, and subsequently from the re-

spective de nitions, one has

M00 = 1; M00 = 0; M00 = 0; M00 = 1

L00 = 1; L00 = 0; L00 = 0; L00 = 1 (M1)00 = 2; (M1)00 = 0; (M1)00 = 0; (M1)00 = 2 (M2)00 = 3; (M2)00 = 0; (M2)00 = 0; (M2)00 = 3 (M3)00 = 0; (M3)00 = 2; (M3)00 = 2; (M3)00 = 0

^

M11 = 1;

^

M11 = 0;

M11 = 1

Using these ingredients, we nd that identities (A.1), (A.2), (A.3) and (A.4) follow quite immediately.

Case 1: h2,1(CY ) = 1. The pre-potential for this case can be written as,

F(X 0; X 1) =

16 X 0

JHEP11(2015)162

^

M11 = 0; ^

3 l0 (X 0)3 + 6 l1 (X 0)2 X 1 + 3 l11 (X 1)2 X 0 + l111 (X 1)3[bracketrightbigg](A.6)

Even for this simple pre-potential, the period matrix N as well as other symplectic matrices

are quite huge to represent, so just for the sake of simple illustration, let us assume that l l111 [negationslash]= 0 and other triple intersection numbers to be zero. Subsequently, setting X 0 = 1

and X 1 = v + i u, the various symplectic matrices are simpli ed as under,

M =[parenleftBigg][parenleftBigg]

6 lu3

6v lu3

!; M =[parenleftBigg]v3u33v2u3 v2(u2+v2)

u3

2u2v+3v3u3 [parenrightBigg]

(A.7)

6v lu3

2(u2+3v2)

lu3

M = v3u3 v

2(u2+v2)

u3

3v2 u3

0

@

l

(u2+v2)36u3

lv(u2+v2)2 2u3

lv(u2+v2)2 2u3

2u2v+3v3 u3

!; M =

l

(u4+4u2v2+3v4)2u3

1

A

L =[parenleftBigg][parenleftBigg]

3lu3 3vlu3

3vlu3

u23v2

lu3

!; L = 0

@

3u2v+v32u3

1

A

3(u2+v2) 2u3

(u2+v2)2 2u3

v(u2+3v2) 2u3

(A.8)

l(u2+v2)3 12u3

lv

(u2+v2)2

4u3

L =

3u2v+v3 2u3

(u2+v2)2

2u3

3(u

2+v2) 2u3

v

(u2+3v2)

2u3

!; L =

0

@

lv

(u2+v2)2

4u3

l

(u42u2v23v4)

4u3

1

A

{ 32 {

M1 =

3 lu3

3v lu3

!; M1 =[parenleftBigg]

v33u2v2u3u4v4 2u3

3

(u2v2)

2u3

(A.9)

3v lu3

3(u2+v2)

lu3

3v(u2+v2) 2u3

[parenrightBigg]

M1 =

0

@

v33u2v

2u3

3(u2v2)

2u3

v4u4

2u3

1

A; M1 =

3v

(u2+v2)2u3

0

@

l

(u2+v2)312u3

lv(u2+v2)2 4u3

lv(u2+v2)2 4u3

3l

(u2+v2)24u3

1

M2 = 0 0

0 4lu [parenrightBigg]

; M2 =[parenleftBigg][parenleftBigg]

3vu u2+v2u

3 u

vu [parenrightBigg]

(A.10)

JHEP11(2015)162

M2 =

3v u

3u

v2u + u

v u

!; M2 =

0 00 l(u2+v2)

u

M3 = 0

3 lu2

3lu2 0 [parenrightBigg]

; M3 =[parenleftBigg][parenleftBigg]

3v22u2 + 12

v3u2 + v

3vu2 3v22u2 12 [parenrightBigg]

; (A.11)

M3 = 3v22u2 12

3v u2

l(u2+v2)2 4u2

l

3v22u2 + 12

!; M3 =

0

@

0

v

(u2+v2)

u2

(u2+v2)2

4u2 0

1

A

Now using eKcs = 3

4lu3 and F0 = 16l(v + iu)3; F1 = 12l(v + iu)2 for the simpli ed ansatz,

one can verify that

eKcs Re(X X ) =[parenleftBigg][parenleftBigg]

3

4lu3

3v4lu3

3v4lu3 3

(u2+v2)

4lu3

[parenrightBigg]

(A.12)

eKcs Re(F X ) =[parenleftBigg][parenleftBigg]

v33u2v

8u3

v4u4

8u3

3(u2v2)

8u3

3v

(u2+v2)

8u3

[parenrightBigg]

eKcs Re(X F ) = 0

@

v33u2v

8u3

3(u2v2)

8u3

v4u4

8u3

3v

(u2+v2)

8u3

1

A

eKcs Re(F F ) = 0

@

l

(u2+v2)3

48u3

lv(u2+v2)2 16u3

lv(u2+v2)2 16u3

3l

(u2+v2)2

16u3

1

A

which are precisely (14M1) matrices, and hence we veri ed identities in eq. (A.1) though

for a simpli ed ansatz to show analytic form of intermediate matrices involved. Following similar procedure, and using generic pre-potential (2.9), we can verify these identities for h2;1(CY ) = 0; 1; 2; 3 and we conjecture the same to be generically true.

Useful symplectic expressions for Example A

Considering the pre-potential (5.4), we get,

F =

0

B

B

B

@

2U1U2U3 U2U3 U1U3 U1U2

U2U3 0 U3 U2

U1U3 U3 0 U1

U1U2 U2 U1 0

1

C

C

C

A

(A.13)

{ 33 {

Using which one can compute the real and imaginary parts of period matrix N which are

given as,

Re N =

0

B

B

B

@

2v1 v2 v3 v2 v3 v1 v3 v1 v2

v2 v3 0 v3 v2

v1 v3 v3 0 v1

v1 v2 v2 v1 0

1

C

C

C

A

(A.14)

and

u2 u3 v21u1 + u1 u3 v

2 2

u2 + u1 u2 v

2

3

u3 + u1 u2 u3 u2 u3 v1u1 u1 u3 v2u2 u1 u2 v3u3

u2 u3 v1u1

u2 u3u1 0 0

u1 u3 v2u2 0

u1 u3u2 0

u1 u2 v3u3 0 0

u1 u2 u3

(A.15)

Recall that condition for physical domain is ImN < 0 which is ensured by (u1 u2 u3) < 0. Using these ingredients, we get the four sets of period matrices M de ned in eq. (3.4) to

get expressions given as under,

M =

Im N =

0

B

B

B

@

1

C

C

C

A

0

B

B

B

B

B

@

v1v2v3u1u2u3 v2v3

JHEP11(2015)162

(u21+v21)

(u22+v22)

(u23+v23)

u1u2u3

v1v3

u1u2u3

v1v2

u1u2u3

v2v3 u1u2u3

v1v2v3 u1u2u3

v3(u22+v22)

u1u2u3

v2(u23+v23)

u1u2u3

1

C

C

C

C

C

A

(A.16)

v1v3 u1u2u3

v3(u21+v21)

u1u2u3

v1v2v3 u1u2u3

v1(u23+v23)

u1u2u3

v1v2 u1u2u3

v2(u21+v21)

u1u2u3

v1(u22+v22)

u1u2u3

v1v2v3 u1u2u3

(u21+v21)(u22+v22)(u23+v23)

u1u2u3

v1(u22+v22)(u23+v23)

u1u2u3

v2(u21+v21)(u23+v23)

u1u2u3

v3(u21+v21)(u22+v22)

u1u2u3

v1(u22+v22)(u23+v23)

u1u2u3

(u22+v22)(u23+v23)

u1u2u3

v1v2

(u23+v23)

M =

0

B

B

B

B

B

@

u1u2u3

v1v3

(u22+v22)

u1u2u3

v2(u21+v21)(u23+v23)

u1u2u3

v1v2

(u23+v23)

(u21+v21)(u23+v23)

u1u2u3

v2v3

(u21+v21)

1

C

C

C

C

C

A

(A.17)

u1u2u3

u1u2u3

v3(u21+v21)(u22+v22)

u1u2u3

v1v3

(u22+v22)

u1u2u3

v2v3

(u21+v21)

(u21+v21)(u22+v22)

u1u2u3

u1u2u3

1u1 u2 u3

v1 u1u2u3

v2u1 u2 u3

v3u1 u2 u3

v1 u1 u2u3

u21+v21 u1u2u3

v1v2 u1u2u3

v1v3 u1u2u3

M =

0

B

B

B

B

@

u22+v22 u1u2u3

v2v3 u1u2u3

(A.18)

v2 u1u2u3

v1v2 u1u2u3

v3 u1u2u3

v1v3 u1u2u3

v2v3 u1u2u3

u23+v23 u1u2u3

1

C

C

C

C

A

v1v2v3 u1u2u3

v2v3 u1u2u3

v1v3 u1u2u3

v1v2 u1u2u3

v2v3(u21+v21)

u1u2u3

v1v2v3u1u2u3 v3

(u21+v21)

u1u2u3

v2

(u21+v21)

u1u2u3

M =

0

B

B

B

B

B

@

(A.19)

v1v3(u22+v22)

u1u2u3

v3

(u22+v22)

u1u2u3

v1v2v3u1u2u3 v1

(u22+v22)

u1u2u3

1

C

C

C

C

C

A

v1v2(u23+v23)

u1u2u3

v2

(u23+v23)

u1u2u3

v1

(u23+v23)

u1u2u3

v1v2v3u1u2u3

Now, we provide the M2-matrices are given as under,

M2 =

0

B

B

B

@

0 0 0 0 0 0 1u3 1u2 0 1u3 0 1u1 0 1u2 1u1 0

1

C

C

C

A

(A.20)

{ 34 {

v1u1 v2u2 v3u3 u

21+v21 u1

u

22+v22 u2

u

23+v23 u3

v2u2 v3u3 0 0

1u2 0 v1u1 + v2u2 v3u3 0

1u3 0 0 v1u1 v2u2 + v3u3

1

C

C

C

A

M2 =

0

B

B

B

@

1 u1

v1 u1

(A.21)

M2 =

0

B

B

B

B

@

v1u1 + v2u2 + v3u3 1u1 1u2 1u3

v21u1 + u1 v1u1 + v2u2 + v3u3 0 0

v22u2 + u2 0

v1 u1

(A.22)

v2u2 + v3u3 0

v23u3 + u3 0 0

v1u1 + v2u2 v3u3

1

C

C

C

C

A

0 0 0 0 0 0 v23

u3 + u3

v22u2 + u2

(A.23)

Other matrices are quite large to present here, however using M and M2 all of those can

be determined.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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