The Generalised Geometry of Type II Non-Geometric Fluxes Under T and S Dualities
TThe Generalised Geometry of Type II Non-Geometric FluxesUnder T and S Dualities
George James Weatherill ∗ School of Physics and Astronomy,University of Southampton,Highfield, Southampton SO17 1BJ, UK
Abstract
We examine the flux structures defined by NS-NS superpotentials of Type IIA and Type IIBstring theories compactified on a particular class of internal spaces which include non-geometricflux contributions due to T duality or mirror symmetry. This is then extended to the TypeIIB R-R sector through the use of S duality and then finally to its mirror dual Type IIA R-Rsector, with note of how this sector breaks S duality invariance in Type IIA. The nilpotency andtadpole constraints associated with the fluxes induced by both dualities are derived, explicitlydemonstrated to be mirror invariant and classified in terms of S duality multiplets. These resultsare then used to motivate the postulation of an additional symmetry for internal spaces which aretheir own mirror duals and an analysis is done of the resultant constraints for such a construction.
Contents p -forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Complex structure moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 K¨ahler moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Additional definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 p -form defined flux components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Cohomology defined flux components . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Matrix defined superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Type IIB Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Flux Interdependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ∗ email : [email protected] a r X i v : . [ h e p - t h ] D ec Flux constraints 27 (cid:48) duality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 Alternate flux matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3.1 T (cid:48) duality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3.2 S duality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Reduced superpotential expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Z × Z orientifold 49 B.1 H (3) basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.2 (cid:101) H (3) basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 C Flux matrix identities 56
C.1 Type IIB SL(2 , Z ) S action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.2 Flux matrix constraints on H (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.3 Alternative flux relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Calabi-Yau manifolds have been an area of considerable interest in string theory due to their partialbreaking of supersymmetry in theories with internal spaces because of their SU(3) holonomy. However,due to their complexity few explicit cases are known in sufficient detail to allow for detailed analysis ofthe dynamics of the moduli of the manifold. Furthermore, a larger class of spaces with the same partialsupersymmetry breaking properties is obtained from the generalisation of SU(3) holonomy to SU(3)structure, which arise when some of the simplifying assumptions of Calabi-Yau spaces are relaxed,with such particular cases as ‘half flat’ spaces being a well investigated area [1, 2]. This relaxationcan be viewed in terms of torsion classes which are in turn expressible in terms of fluxes such as theinclusion of fluxes like H = d B . It was realised in the second string theory revolution that thesefluxes can be used to stablise the moduli of the internal space, who gain vacuum expectation valuesdue to the non-trivial potential induced by the fluxes [4].Considerable work [5] has been done usingorbifolds and orientifolds, whose construction via Kaluza-Klein reduction onto a six dimensional torusfollowed by quotienting by discrete groups allows for the behaviour of fluxes and their effect on modulito be examined explicitly.The flux structure differs between Type IIA and Type IIB theories, but in both cases there is anNS-NS 3-form flux H and R-R fluxes which live on the branes appropriate to each theory, which in the2ype IIB case is only F because F = 0 = F due to H (1) and H (5) being empty. This is related to thecohomology condition B ∧
Ω = 0 which is closely tied to the non-total breaking of the supersymmetryand the compatibility of SU(3) spin structures for the space [3]. In order for there to be invariance inthe moduli effective theory these fluxes must be augmented by additional terms, such as the NS-NSgeometric flux ω which in the case of Type IIA is sufficient to provide stable vacua [6, 7, 8, 9, 10, 11] butnot in Type IIB as the moduli potential is independent of the K¨ahler moduli and thus those moduli arenot stablised. This too can be addressed by the inclusion of further NS-NS fluxes [12, 13, 14] which aresuch that the geometric interpretation of the space is no longer straight forward, making the new fluxes‘non-geometric’ in nature [15], at which point the space can no longer be categorised by torsion classes.The properties of possible orbifolds, such as the number of moduli and their complex structure,have been fully classified [16, 17], with the Z × Z case being one of the most examined cases in termsof non-geometric fluxes [12, 13, 14], T duality invariance [19, 20] and S duality invariance [21, 22]. Oneof the reasons for this is that the orbifold discrete group actions can be such that the six dimensionaltorus is factorised into three two dimensional tori and this can be further restricted by isotropy tomake the three tori equal, reducing the number of moduli of each type to one, reducing the number ofpossible independent flux terms and thus making it one of the simplest non-trivial orbifolds.In this paper we shall consider the superpotential from which the moduli potential can be con-structed in its most general form when all possible T and S duality induced fluxes are included. Thepolynomial form of the superpotential is reexpressed in terms of fluxes, K¨ahler forms and holomorphicforms and then converted into expressions involving covariant derivatives. The formulation of the su-perpotential in terms of covariant derivatives has been seen in the case of twisted derivatives, where theinclusion of the H flux causes the exterior derivative to gain a torsion-like term, d → d H = d + H ∧ ,with the associated internal space being a twisted Calabi-Yau. With the inclusion of geometric fluxes p -forms which were originally closed under d are no longer as d λ = ω · λ . This can be further extendedto also include the non-geometric fluxes, as considered explicitly in the Z orientifold [18], with suchspaces being generalised twisted Calabi-Yaus [3]. In the standard Calabi-Yau case the properties ofd allow for the definition of the cohomologies H ( p ) but with the inclusion of geometric fluxes this isno longer the case. However, due to p -form truncations the bases for the untwisted H ( p ) can still beused as bases for the generalised twisted space [3]. For twisted Calabi-Yaus we have d H = 0 butin generalised twisted Calabi-Yaus the altered action of d is such that d H = ω · H but the 3-formexpansion of H remains unchanged provided the new Bianchi constraint d H = ω · H = 0 is satisfied.The inclusion of additional fluxes makes the Bianchi constraints non-trivial but we may still use thefluxless and closed H ( p ) as our bases. [3]It is this approach we will consider, first reexpressing the polynomial form of a Type IIB superpo-tential in terms of a pair of covariant derivatives, one for each flux sector, and the two holomorphicforms associated to the complex structure and K¨ahler moduli. The covariant derivatives, D and D (cid:48) ,are then expressed in two different ways; the first of which uses the tensor structure of the fluxes suchas H ∼ H abc and ω ∼ ω abc and the second of which defines the fluxes by their actions on the twistedgeneralised H ( p ) basis elements. The Bianchi constraints are then obtained by requiring the nilpotencyof the derivatives, which in the former case requires us to apply D to the H ( p ) basis elements andin the latter case the matrix representation must be nilpotency, D = 0 and likewise for D (cid:48) . We thenconsider the Type IIA superpotential, expressing it in terms of two covariant derivatives, D and D (cid:48) , andthe holomorphic forms but noting how it treats the NS-NS and R-R flux sectors differently and how theinternal space’s degrees of freedom are labelled in a different manner to the Type IIB case, before thenconsidering the relationship between the two different flux formulations of Type IIA and Type IIB anddemonstrating the equivalence of the Bianchi constraints. Using results obtained in these two sectionswe then consider S duality in Type IIB using the matrix representation of the covariant derivatives,obtaining sets of SL(2 , Z ) S triplets and singlets which are either Bianchi constraints induced by T andS duality or tadpoles which can couple to the D-branes or O-planes wrapping particular cycles of theinternal space. Finally we discuss the general structure of the superpotential and note the existenceof two T duality inequivalent formulations of the superpotential. Based on this we postulate an addi-3ional symmetry between the two moduli spaces for internal spaces which are their own mirror dual.Throughout the paper the Z × Z orientifold is used as an explicit example to illustrate results interms of flux entries, however because it is often more illuminating to maintain the index structureonly in the Appendix is the notation commonly found in the literature [14, 20, 22] used. Motivation : The Z × Z orientifold The Z × Z orientifold, which we will refer to as M Z , is extensively examined in the literature dueto it possessing several important features. • With h , ( M Z ) = 3 the space possesses dynamical complex structure moduli, as well as K¨ahlerand dilaton moduli, unlike such cases as the Z orbifold whose complex structure is determinedentirely by the orbifold group. [16, 17] • The orbifold group action can be set such that the six dimensional torus from which M Z is builtfactorises into three two dimensional tori. Each torus possesses a complex structure modulusand a K¨ahler modulus and these two moduli possess equivalent qualitative dynamics. [12, 21] • The degrees of freedom can be reduced in a straight forward manner, by requiring ‘isotropy’,while maintaining the existence of a modulus of each type. This is acheived by restricting thethree tori to being equal to one another. [12, 20, 22] • It is its own mirror dual, which is inherited from the self mirror dual nature of two dimensionaltori. [21]In their discussion of non-geometric fluxes induced by T duality [12, 13, 14] use the isotropic caseas their explict example for constructing non-geometric fluxes and examining the modifications theymake to the space and this has been extended to include the S duality induced constraints [21]. Theseformed the basis of the explcit examples used in work done on solving these T [20] and S [22] dualityconstraints, which are the precursors to this work. As described in [21] an IIB/O3 superpotential onthe Z × Z orientifold which is seperately invariant under S and T duality transformations can beconstructed from four fluxes, two of which are non-geometric in nature. O ( T ) O ( T ) O ( S ) F = F abc Q = Q abc O ( S ) H = H abc P = P abc The Type IIB/O3 non-geometric fluxes Q and P must be contracted with a 4-form but it is linear inthe K¨ahler moduli T a [21] and so this is taken to be B c = T a (cid:101) ω a , where (cid:101) ω a are the H (4) versions of the ω a and their specific relationship discussed later. From this we can construct a superpotential linearin the K¨ahler moduli and cubic in the complex structure moduli. W = (cid:90) M Ω ∧ (cid:16) J (0) · ( F − SH ) + J c · ( Q − SP ) (cid:17) (1.1)The lack of higher order K¨ahler moduli terms in the superpotential suggests not all fluxes have beenincluded, as the Hodge numbers of the orientifold are h , = h , = 3. Type IIB possesses self Sduality symmetry and it is this which motivates the existence of the R-R non-geometric flux P in[21, 22] so that by construction SL(2 , Z ) S transformation to leave (1.1) invariant, but Type IIA doesnot possess this symmetry, its S dual theory is eleven dimensional supergravity. The superpotentialwhich is exactly mirrror dual to (1.1) will have SL(2 , Z ) S invariance but this is not generically true,it requires certain Type IIA fluxes to be turned on which in turn induce further fluxes in the Type Under the orientifold action used in the literature [19, 21, 22] h , = h , , there are no 2 n -forms odd under theprojection. O ( T ) O ( T ) O ( S ) w = w abc V = V abc O ( S ) f = f abc R = R abc The existence of w and V is motivated [19] by the effect S duality transformations have on Type IIAtheories and their tensor structure on M Z can be obtained by modular transformations, they are thecomplement or duals of the 3-form fluxes and the non-geometric fluxes. However, to account for theK¨ahler moduli contributions to the superpotential they are not contracted with terms proportional to J and J , as this would produce terms of O ( T ) and O ( T ). This, as well as the issue with the K¨ahlerdependence of the non-geometric contribution, can be resolved by defining a new set of K¨ahler forms (cid:101) J ( n ) , obtained by modular transformations on the K¨ahler moduli. (cid:101) J ( n ) ≡ G T ( J ( n ) ) , G T : T a → − T a ∀ a = 1 , . . . , h , The order of K¨ahler moduli dependency in K¨ahler forms is J ( n ) ∼ O ( T n ), but the G T transformationchanges this to (cid:101) J ( n ) ∼ O ( T − n ) and the contracted tensor Q · (cid:101) J (2) is therefore linear in the K¨ahlermoduli, as is known to be required. However, if we are to write the superpotential integrand so that it’sK¨ahler dependence is only constructed from (cid:101) J ( n ) K¨ahler forms then the object which gives the O ( T )contribution must contract with the 6-form (cid:101) J (3) and so F is not viewed in terms of components F abc but rather (cid:101) F abc , the Hodge dual ∗ of F abc . Similarly the tensor structure of the term contributing thecubic K¨ahler moduli dependency is not viewed as V abc but rather it’s Hodge dual, (cid:101) V abc . The spacealso possesses structure which allows modular invariance in both the K¨ahler and complex structuremoduli [21] and through this analysis precisely this kind of alternative tensor structure for the fluxesarises. Using tildes to represent the dual tensor the superpotential then can be written in a way whichmakes its K¨ahler moduli dependence much clearer. W = (cid:90) M Ω ∧ (cid:16) (cid:101) F · (cid:101) J (3) + Q · (cid:101) J (2) + w · (cid:101) J (1) + (cid:101) V · (cid:101) J (0) (cid:17) ++( − S ) (cid:90) M Ω ∧ (cid:16) (cid:101) H · (cid:101) J (3) + P · (cid:101) J (2) + f · (cid:101) J (1) + (cid:101) R · (cid:101) J (0) (cid:17) (1.2)Because of Bianchi constrants and possible orientifold projections not all of these fluxes can be non-zeroat the same time but in principle this is the most general superpotential which can arise from considerT and S duality induced fluxes. From this point onwards we will not consider such apriori effects asthe orientifold projection, rather we will assume all of these fluxes can in principle be turned on atonce and construct various flux structures from them, with the assumption that once these structuresare found they can be simplified down by such things as which fluxes or forms are even or odd underthe projection. As will be observed, the symmetry between the two moduli spaces is more easily seenby the inclusion of all of the terms in (1.2) and the alteration of the index structure of such fluxes as Q or H by the Hodge star does not alter the constraints on the fluxes. Before beginning our analysis of the superpotential in terms of fluxes we need to define how we representthe fluxes, moduli, p -forms, H ( p ) bases in general. Much of the notation is choosen to match [23]. Thespecific case of the Z × Z orientifold is done in Section 7. The definitions of G U and G S follow in the same manner as G T . .1 Cohomologies and p -forms Working on a connected orientated six dimensional manifold M we wish to define a set of bases forthe p -form spaces Λ ( p ) ( M ), whose direct sum form the ring of p -forms Λ ( ∗ ) ( M ) = ⊕ p =0 Λ ( p ) ( M ).Assuming that all fluxes are set to zero we can also straightforwardly define the cohomologies H ( p ) of M and their sum H ( ∗ ) ( M ) = ⊕ p =0 H ( p ) ( M ). By orientated connectedness M has a unique 6-form µ vol which satisfies (cid:82) M µ vol = 1 and so given the set of tangent 1-forms η σ which span Λ (1) ( M ) we alsohave their Hodge duals ˇ η τ which form the basis of Λ (5) ( M ), defined by η σ ∧ ∗ η τ = δ στ µ vol = η σ ∧ ˇ η τ . p basis dim Elements1 η σ η , · · · , η η σ η , · · · , ˇ η (2.1)Though we construct ( p > η σ , we are making the stipulation that H (1) ( M ), and thus by duality H (5) ( M ), is empty. We will also make use of interior forms ι σ , whichsatisfy ι σ ( η τ ) = δ τσ and so belong to the space dual to Λ (1) which we denote as Λ (1) ∗ . Elements of theother Λ ( p ) ∗ are defined as exterior products of these forms such as ι abc ≡ ι a ι b ι c , with the restriction to H ( p ) ∗ being straightforward though we will later consider them in more detail. A generic element of Λ (3) ( M ) can be constructed from wedge products of three of the tangent 1-form, η abc ∈ Λ (3) ( M ) and fluxes can be expanded in this form. F = 13! F abc η abc (2.2)The properties of M determine the structure of the coefficients F abc but it is more convenient towork with the basis of H (3) ( M ) for fluxes. On M we can define h , + 1 pairs of 3-cycles A I , B J , I, J = 0 , · · · , h , with a symplectic structure in their intersection numbers. A I ∩ B J = − B J ∩ A I = δ IJ A I ∩ A J = B I ∩ B J = 0The ( α I , β J ) basis of H (3) is defined as the Poincare dual of the ( A I , B J ) homology 3-cycles. (cid:90) A J α I = (cid:90) M α I ∧ β J = δ IJ (cid:90) B J β I = (cid:90) M β I ∧ α J = − δ IJ (2.3)The set of h , + 1 coordinates U I are defined by the 3-cycles and the holomorphic 3-form Ω, which inturn define h , projective coordinates, the complex structure moduli U i . U I = (cid:90) A I Ω ⇒ U i = U i U (2.4)Therefore the holomorphic 3-form can then be expanded in terms of these coordinates, with the factorof − abc η abc = U I α I − U J β J The U I are defined as the U I derivatives of the complex structure prepotential F and all functions andtensors dependent upon U are then evaluated at U = 1. U I = (cid:90) B J Ω ≡ ∂ F ∂ U I F = 13! 1 U F ijk U i U j U k (2.5)The K¨ahler potential for the complex structure moduli is also constructable from the holomorphic3-form. K , = − ln (cid:18) i (cid:90) M Ω ∧ Ω (cid:19) ≡ − ln i (cid:16) U I U I − U J U J (cid:17) = − ln i (cid:18) U I ∂ F ∂ U I − U J ∂ F ∂ U J (cid:19)
6e will find it more convenient to work in a slightly ammended sympletic basis ( a I , b J ) which isobtained from ( α I , β J ) by a sympletic transformation and which forces us to define a set of moduli U such that Ω is left schematically invariant.( a , a i , b , b j ) ≡ ( α , β i , β , − α j )( U , U i , U , U j ) ≡ ( U , −U i , U , U j ) ⇒ Ω = (cid:110) U I α I − U J β J U I a I − U J b J (2.6)Though the definition of this second set of sympletic 3-forms and related moduli has no apriori mo-tivation the reason for this exchange of α i and β i will be given in our discussion of Type IIB fluxes,as well as the action of T duality on R-R fluxes. Also for later convenience we define two bilinearforms associated to the intersection numbers of the cohomology bases, the first of which is g , definedon φ ∈ H ( p ) ( M ) and ϕ ∈ H (6 − p ) ( M ) mapping them to Z . g ( φ, ϕ ) ≡ (cid:90) M φ ∧ ϕ (2.7)If φ and ϕ are vectors of forms then g ( φ, ϕ ) is a matrix with entries defined by g ( φ m , ϕ n ). The basisvector for the complex structure moduli space we take to be e ( a ) = (cid:0) a I , b J (cid:1) so that g takes on theform of the canonical symplectic form. g a ≡ g ( e ( a ) , e ( a ) ) = (cid:18) g ( a I , a J ) g ( a I , b J ) g ( b I , a J ) g ( b I , b J ) (cid:19) = (cid:18) g IJ g JI g IJ g IJ (cid:19) = (cid:18) − (cid:19) ⊗ I h , +1 By construction this choice of basis leads to a formulation which is invariant under transformations e ( a ) → L · e ( a ) where L ∈ Sp( h , +1) as L (cid:62) · g a · L = g a . The second bilinear form is h ( φ, ϕ ) ≡ g ( φ, σ · ϕ )where σ = (cid:18) (cid:19) ⊗ I n and 2 n is the dimension of the ϕ . h a ≡ g ( e ( a ) , σ · e ( a ) ) = (cid:18) g JI g IJ g IJ g IJ (cid:19) = (cid:18) − (cid:19) ⊗ I h , +1 Henceforth unless the dimensions are ambigious we shall drop the subscript on I . With these twomatrices defined by e ( a ) we can express superpotential-like integrals in terms of matrices and vectors,such as two elements C and D of H (3) with the expansion in terms of the symplectic basis. C = C I a I − C J b J = (cid:0) C I C J (cid:1) (cid:18) I − I (cid:19) (cid:18) a I b J (cid:19) ≡ C (cid:62) · h a · e ( a ) The expansion of D in the symplectic basis follows in the same manner and using these vector expres-sions we can represent the inner product between these two 3-forms in a particularly straight forwardmanner. g ( C, D ) = (cid:90) M C ∧ D = D I C I − D J C J = C (cid:62) · h a · g a · h (cid:62) a · D = D (cid:62) · g a · C Expanding D (cid:62) · g a · C in terms of the components we note that the structure of the expression ispreserved, there is a factor of − H (3) basis leave g ( C, D ) invariant. The vector associated to the holomorphic form is the modulivector, Ω = U (cid:62) · h a · e ( a ) and so we can obtain an expression for a generic Type IIB superpotentialcontribution due to some 3-form flux G . (cid:90) M Ω ∧ G = (cid:90) M ( U I a I − U J b J ) ∧ ( G I a I − G J b J ) = U J G J − U I G I = g (cid:16) Ω , G (cid:17) The d defined deRham cohomology H (3) is decomposable in terms of Dolbeault operator definedcohomologies H (3 − n,n ) , which are spanned by forms related to the holomorphic 3-form, its complexconjugate and their K¨ahler derivatives D U i ≡ ∂ U i + ∂ U i K (2 , , due to the manner in which Ω is definedas a holomorphic 3-form. H (3) ≡ H (3 , ⊕ H (2 , ⊕ H (1 , ⊕ H (0 , ≡ (cid:104) Ω (cid:105) ⊕ (cid:104) D U i Ω (cid:105) ⊕ (cid:104) D U i Ω (cid:105) ⊕ (cid:104) Ω (cid:105) α I , β J ) sympleticbasis elements, which we denote by H (3 − n,n ) . H (3) = (cid:104) α (cid:105) ⊕ (cid:104) α i (cid:105) ⊕ (cid:104) β j (cid:105) ⊕ (cid:104) β (cid:105)≡ H (3 , ⊕ H (2 , ⊕ H (1 , ⊕ H (0 , = (cid:104) a (cid:105) ⊕ (cid:104) b i (cid:105) ⊕ (cid:104) a j (cid:105) ⊕ (cid:104) b (cid:105) We now consider the K¨ahler modui space, constructing it from even dimensional cycles as opposed tothe odd dimensional cycles of the complex structure moduli space, but with the guiding principle thatthe two moduli spaces must be as close in structure as possible. The set of complex K¨ahler moduli aredefined as the coefficients of the expansion of the complex K¨ahler 2-form
J ≡ B + iJ , in terms of the(1 ,
1) forms ω a , J = T a ω a , although when we consider the K¨ahler moduli holomorphic form we willfind it more convenient to work with a set of projective coordinates T A defined in the same manneras the complex structure moduli space, with T a ≡ T a T . The (1 , h , A a with a, b = 1 , · · · , h , , which in turn have a set of 4-cycles B b partners whichdefine the set of h , (2 , (cid:101) ω b . These two sets of forms are partnered in the same manner inwhich a I and b J are partnered but have non-trivial intersection numbers g ba and g ab defined in thesame manner as the sympletic interaction numbers previously. (cid:90) A b ω a = (cid:90) M ω a ∧ (cid:101) ω b = g ba (cid:90) B b (cid:101) ω a = (cid:90) M (cid:101) ω a ∧ ω b = g ab We can expand this basis to include the (0 ,
0) and (3 ,
3) forms, ω ≡ (cid:101) ω ≡ µ vol . The 6-form (cid:101) ω is associated with B ≡ M itself, which is the only 6-cycle if M is connected. In the case of ω wehave to associate it with a 0-cycle point A , which can be any point other than orbifold singularities. (cid:90) A ω = (cid:90) M ω ∧ (cid:101) ω = 1 (cid:90) B (cid:101) ω = (cid:90) M (cid:101) ω ∧ ω = 1For the purposes of clarity we shall sometimes regard expressions such as ω a ∧ (cid:101) ω a (no sum) not as (cid:101) ω but µ vol , so that the pair of forms ω and (cid:101) ω satisfy the same notation. As in the complex structure casewe can define a vector of these even forms e ( ω ) = (cid:0) ω ω a (cid:101) ω (cid:101) ω b (cid:1) , from which the general intersectionmatrix g of the K¨ahler moduli space is built and the associated second bilinear form h , where we onceagain have written the elements such that (cid:101) ω comes before the (cid:101) ω a , allowing us to extend our indexlabelling to A, B = 0 , , . . . , h , in the same manner as the sympletic form indices. g ω ≡ g ( e ( ω ) , e ( ω ) ) = (cid:18) g ( ω A , ω B ) g ( ω A , (cid:101) ω B ) g ( (cid:101) ω A , ω B ) g ( (cid:101) ω A , (cid:101) ω B ) (cid:19) = (cid:18) g BA g AB (cid:19) h ω ≡ h ( e ( ω ) , e ( ω ) ) = (cid:18) g ( ω A , (cid:101) ω B ) g ( ω A , ω B ) g ( (cid:101) ω A , (cid:101) ω B ) g ( (cid:101) ω A , ω B ) (cid:19) = (cid:18) g BA g AB (cid:19) Unlike the complex structure case the intersection matrix g ω is symmetric due to the commutativeproperties of the even forms but we have not assumed any specific form of g BA = (cid:0) g BA (cid:1) (cid:62) and a generalchange of basis on either the (1 ,
1) or (2 , g ω .We can explicitly construct a convenient set of intersection numbers by considering the K¨ahler moduliversion of the holomorphic 3-form, obtained by exponentiating the complexified K¨ahler form. e J ≡ (cid:102) = ∞ (cid:88) n =0 n ! J n = ∞ (cid:88) n =0 J ( n ) = J (0) + J (1) + J (2) + J (3) = T ω + T a ω a + 12! T a T b ω a ∧ ω b + 13! T a T b T c ω a ∧ ω b ∧ ω c (2.8)The first and second terms in (cid:102) have a simple expansion in terms of the H (0 , and H (1 , bases andbear a close resemblence to the a I terms in Ω. However, the latter two terms are not expressed in8erms of the H (2 , and H (3 , forms we have already seen, but combinations of the (1 , ω a ∧ ω b ∧ ω c = κ abc µ vol . The (2 , , (cid:101) ω a = f abc ω b ∧ ω c , which relate to the intersection numbers κ abc by g ab = f acd κ bcd . In order to make the expansion of (cid:102) as close to the structure of the expansionof Ω we make an ansatz for (cid:102) so that the latter two terms are written in the more natural basis of the (cid:101) ω A . Ω = U a + U i a i − U j b j − U b , (cid:102) = T ω + T a ω a + T b (cid:101) ω b + T (cid:101) ω The lack of minus signs is due to the h ω being positive definite, compared to h a , and because of thisthere is no pre-potential associated to (cid:102) , which requires a sympletic structure to the forms. Sucha structure can be given to these forms by the use of Grassmannian variables [23] but this wouldintroduce the notion of Grassmannian fluxes, an unwanted complication. If we treat the A, B = 0cases as seperate from the
A, B = 1 , . . . , h , cases then we can define a reduced pre-potential, in that T is not the T derivative of a pre-potential but infact plays the role of the reduced pre-potential. Toobtain this we compare the coefficients of each of the seperate 2 n -form terms and express T a and T in terms of intersection numbers and T A . Comparing coefficients we can express T a in terms of thederivatives of T , in an analogous way to the complex structure moduli space. T = 13! κ abc T a T b T c ⇒ T b g ba = 12! κ abc T b T c = ∂ T ∂ T a (2.9)Therefore, if we set T a = ∂ T ∂ T a , inline with the complex structure case of (2.5), then the intersectionmatrix g ba reduces to δ ba and the associated h ω to the identity. (cid:102) = T A ω A + T B (cid:101) ω B = ω + T a ω a + ∂ G ∂T b (cid:101) ω b + G (cid:101) ω G = 13! κ abc T a T b T c With g ω = σ ⊗ I h , +1 the symmetry group associated to e ( ω ) is not the sympletic group but isisomorphic to O( h , + 1 , h , + 1) instead because of the invariant under transformations e ( ω ) → L · L · e ( a ) where L ∈ O( h , + 1 , h , + 1). This follows from the fact L (cid:62) · g ω · L = η where η is theLorentzian metric with h , + 1 positive and negative entries. As a result of this the K¨ahler potentialfor the moduli can be expressed generically in terms of the moduli only. K , = − ln (cid:18)(cid:90) M (cid:102) ∧ (cid:102) (cid:19) = − ln (cid:16) T A T A + T B T B (cid:17) The factor of i has been dropped because T A T A + T B T B is real, unlike U I U I − U J U J , which is purelyimaginary. Given X ∈ (cid:101) H (3) we define its components by the expansion X = X A ω A + X B (cid:101) ω B , andlikewise for Y ∈ (cid:101) H (3) , choosen so that integration over M preserves the structure of the expansion. g ( X, Y ) = g ω ( X, Y ) = (cid:90) M X ∧ Y = X A Y A + X B Y B As in the complex structure case we can associated a vector to any element of (cid:101) H (3) by its coefficientsin the e ( ω ) basis. X = X A ω A + X B (cid:101) ω B = (cid:0) X A X B (cid:1) (cid:18) I I (cid:19) (cid:18) ω A (cid:101) ω B (cid:19) ≡ X (cid:62) · h ω · e ( ω ) We again express the inner product between two elements of a cohomology in terms of vectors. g ( X, Y ) = (cid:90) M X ∧ Y = X A Y A + X B Y B = X (cid:62) · h ω · g ω · h (cid:62) ω · Y = Y (cid:62) · g ω · X In the definition of the vector associated to Ω the explicit h a factorisation was required for the entriesof the vector to be the complex structure moduli and while we have defined the (cid:101) H (3) basis to make h ω h ω in the definition of the (cid:101) H (3) vectors on grounds of symmetry. As a result inboth moduli spaces the holomorphic form has a vector associated to it whose entries are the moduli, (cid:102) = T (cid:62) · h ω · e ( ω ) . As with H (3) the cohomology (cid:101) H (3) decomposes into subspaces of H ( ∗ ) which arespanned by forms constructed from (cid:102) in the same manner as in the H (3) case by the use of a K¨ahlerderivative of the K¨ahler moduli, D T a = ∂ T a + ∂ T a K , . (cid:101) H (3) ≡ (cid:104) (cid:102) (cid:105) ⊕ (cid:104) D T a (cid:102) (cid:105) ⊕ (cid:104) D T a (cid:102) (cid:105) ⊕ (cid:104) (cid:102) (cid:105) As in the case of H (3) it is more convenient to work with the Dolbeault operator defined cohomologiesbut unlike the complex structure case they are synonymous with deRham cohomologies due to theproperties of spaces with SU(3) structure such as h , = 0, resulting in such identities as b n = h n,n where b p is the p ’th Betti number. (cid:101) H (3) = (cid:104) ω (cid:105) ⊕ (cid:104) ω a (cid:105) ⊕ (cid:104) (cid:101) ω b (cid:105) ⊕ (cid:104) (cid:101) ω (cid:105) = H (0) ⊕ H (2) ⊕ H (4) ⊕ H (6) ≡ H (0 , ⊕ H (1 , ⊕ H (2 , ⊕ H (3 , The two moduli spaces split H ( ∗ ) into H (3) and (cid:101) H (3) and so we define the Λ ( ∗ ) version of this splitting,but due to the non-emptiness of Λ (1) and Λ (5) we denote them as Λ ( − ) and Λ (+) respectively.Λ ( − ) = Λ (1) ⊕ Λ (3) ⊕ Λ (5) = (cid:104) η τ (cid:105) ⊕ (cid:104) η τσρ (cid:105) ⊕ (cid:104) η τσρλυ (cid:105) Λ (+) = Λ (0) ⊕ Λ (2) ⊕ Λ (4) ⊕ Λ (6) = (cid:104) (cid:105) ⊕ (cid:104) η τσ (cid:105) ⊕ (cid:104) η τσρλ (cid:105) ⊕ (cid:104) η τσρλυχ (cid:105) The compatibility condition
J ∧
Ω = 0 is automatically satisfied if H (5) is empty and can be reexpressedin terms of the H (3) and (cid:101) H (3) bases.d ω a = 0d a I = 0d b J = 0 ⇔ d( ω a ∧ a I ) = 0d( ω a ∧ b J ) = 0 ⇔ ω a ∧ a I ω a ∧ b J (cid:111) ∈ H (5) ⇒ ω a ∧ a I = 0 ω a ∧ b J = 0 (2.10)The general expansion χ = χ · h · e is used to define the vector associated to any element of H ( ∗ ) ,irrespective of the cohomologies χ has support in. The general elements of H ( ∗ ) can be written as C + X and D + Y and since g ≡ g ( e , e ) is the direct sum of g a and g ω we have that g ( C + X, D + Y ) = g a ( C, D ) + g ω ( X, Y ). The natural, positive definite, inner product on Λ ( ∗ ) follows from g by use ofthe Hodge star ∗ ≡ ∗ , whose action on the cohomologies can be written as a matrix ∗ by defining ∗ ( χ ) = χ (cid:62) · h · ∗ · e . (cid:10)(cid:10) φ, χ (cid:11)(cid:11) ≡ g (cid:16) φ, ∗ ( χ ) (cid:17) = (cid:90) M (cid:16) φ (cid:62) · h · e (cid:17) ∧ (cid:16) χ (cid:62) · h · ∗ · e (cid:17) = (cid:90) M (cid:16) φ (cid:62) · h · e (cid:17) ∧ (cid:16) χ (cid:62) · Ad( ∗ ) · h · e (cid:17) = χ (cid:62) · Ad( ∗ ) · g · φ = χ (cid:62) · φ With Ad( ∗ ) ≡ h ·∗· h (cid:62) = g (cid:62) we can make use of the definitions of of g ω and g α to obtain the result ∗ = g .In Type IIB theories the superpotential integrand is written with the holomorphic 3-form Ω, whilein the NS-NS sector of Type IIA theories the complexified holomorphic 3-form Ω c is used, a point wewill return to shortly. Ω c is dilaton independent and Ω not, with the specific dependency being easilyexpressible in terms of the symplectic basis of H (3) .Ω = U a + U i a i − U j b j − U b Ω c = − S U a + U i a i − U j b j + S U b Using this as a guide we define four complexified holomorphic forms, two for each moduli type.Ω c = − S U a + U j a j − U j b j + S U b Ω (cid:48) c = U a − S U j a j + S U j b j − U b (cid:102) c = − S T ω + T a ω a + T b (cid:101) ω b − S T (cid:101) ω (cid:102) (cid:48) c = T ω − S T a ω a − S T b (cid:101) ω b + T (cid:101) ω Neveu-Schwarz-Neveu-Schwarz Superpotentials
With the explicit example of the Z × Z orientifold superpotential (1.2) for motivation we wish toconsider the most general superpotential of this form on a general internal space M . However, it ispreferable to first begin with the Type IIA NS-NS sector where the subtlies of taking duals of thefluxes are not required but the schematic form of the superpotential is of the same form. In orderto systematize our analysis of the fluxes we drop the notation commonly used to denote the variousfluxes [12, 13, 14] and instead label the fluxes which couple to the dilaton as (cid:98) F n and those which donot as F m . W = (cid:90) M Ω c ∧ (cid:16) F · J (0) + F · J (1) + F · J (2) + F · J (3) (cid:17) ≡ (cid:90) M Ω c ∧ G Since G contributes to the superpotential via exterior multiplication with the complexified holomorphic3-form only 3-form terms in G will contribute, all others will be integrated out. By allowing thisadditional freedom in how we view G we are able to factorise it into a pair of terms, one of whominvolves the moduli in J ( n ) and the other of which is dependent entirely upon the fluxes only. Thisfactorisation is not unique as there is a degeneracy in terms of the signs of the fluxes and the termsthey contract with but we set the signs such that the factorisation is written in terms of (cid:102) . G = (cid:16) F + F + F + F (cid:17) . (cid:16) J (0) + J (1) + J (2) + J (3) (cid:17) ≡ G · (cid:102) Without fluxes the action of G is trivialised to zero and this is synonymous with the fact that all basiselements of the H ( p ) are closed and though fluxes break the closure we shall still refer to such spaces ascohomologies. Therefore we can view the action of G as equivalent to that of the exterior derivative, d.With the inclusion of the 3-form field strength H G is no longer trivial and the exterior derivative istwisted by the inclusion of a torsion term defined by H , d → d H = d + H ∧ , further suggesting a linkbetween the flux actions and exterior derivatives. This can be extended [3, 18] to include the T dualityimages of the NS-NS field strength, making the twisted Calabi-Yau into a generalised non-geometricone with covariant exterior derivative D . By considering G ∼ D the action of d on Λ (2) , Λ (4) and Λ (6) is F , F and F respectively. D ≡ F + F + F + F ⇒ G = D ( (cid:102) )This is only a schematic factorisation, the precise forms of D as a differential operator can be foundby considering its individual flux terms. As a result of the definition of the superpotential’s integrandin terms of the 3-form Ω c it acts on the subspaces of (cid:101) H (3) as F n : H ( n,n ) → H (3) . W = (cid:90) M Ω c ∧ G = (cid:90) M Ω c ∧ D ( (cid:102) ) p -form defined flux components In this section the indices a, b range from 1 to dim( M ) = 6, not from 1 to h , . We can define thecomponents of the various fluxes using the bases of Λ ( ∗ ) and Λ ( ∗ ) ∗ and from here we drop the j indexon F j when working with such components since it is clear to which flux is being refered to by theindex structure. Doing the same for the J ( n ) then allows us to express G in terms of the componentsof the J ( n ) and the fluxes. G = D ( (cid:102) ) = (cid:16) λ F abc + λ F dab J dc + λ F dea J debc + λ F def J defabc (cid:17) η abc We will refer to this way of expressiong the components of fluxes as the Λ ( ∗ ) components defined inthe (Λ ( ∗ ) , Λ ( ∗ ) ∗ ) basis. The λ i are combinatorical factors which are determined by the specific index11tructure of the fluxes and K¨ahler forms. Denoting a T duality in the η a direction as T a we can obtainthe T duality sequence of fluxes. [12, 13, 14] F abc ←→ F abc ←→ F abc ←→ F abc (3.1)The covariant derivative can be expressed explicitly as a differential operator by making use of theinterior forms ι σ which allow us to factorise contracted indices by ι a ( η b ) = δ ba . The specific coefficientsof each terms in D are determined by the requirement that the Bianchi constraints of D are equivalentto the Jacobi constraints of the twelve dimensional gauge algebra whose structure constants are thefluxes. D = 13! F abc η abc + 12! F abc η bc ι a + 12! F abc η c ι b ι a + 13! F abc ι c ι b ι a (3.2)By construction we have defined the derivative action as D : Λ (+) → Λ (3) ⊂ Λ ( − ) as these are the onlycontributions to the superpotential if the integrand involves Ω c ∈ H (3) . However, it can be see thatthe expressions in (3.2) also act as D : Λ (3) → Λ (+) , the two different kinds of action of the derivativesare not independent. With this in mind we consider a general element of Λ (3) , ϕ = ϕ abc η abc andapply D so as to get a more explicit understanding of this different action of the derivatives. It isclear that the action of the fluxes is F − n : Λ (3) → Λ (2 n ) and so the action of the derivatives on Λ (3) naturally splits into four cases relating to the decomposition of Λ (+) . • Case 1 : F : Λ (3) → Λ (0)
13! 13! F abc ϕ ijk ι c ι b ι a ( η ijk ) = 13! 13! F abc ϕ ijk (cid:0) P δ ai δ bj δ ck (cid:1) = 13! F abc ϕ abc • Case 2 : F : Λ (3) → Λ (2)
13! 12! F abc ϕ ijk η c ι b ι a ( η ijk ) = 13! 13! F abc ϕ ijk η c (cid:0) P δ ai δ bj η k (cid:1) = 12! F abi ϕ abj η ij • Case 3 : F : Λ (3) → Λ (4)
13! 12! F abc ϕ ijk η bc ι a ( η ijk ) = 13! 12! F abc ϕ ijk η bc (cid:0) P δ ai η jk (cid:1) = 12! 12! F aij ϕ akl η ijkl • Case 4 : F : Λ (3) → Λ (6)
13! 13! F abc ϕ ijk η abc ( η ijk ) = 13! 13! F abc ϕ ijk η abc ( η ijk ) (cid:0) P (cid:1) = 13! 13! F abc ϕ ijk η abcijk This can be summarised by noting the defining action of F n , F n : Λ (2 n ) → Λ (3) → Λ (6 − n ) canbe rewritten as F n : Λ (2 n ) → Λ (3) → ∗ Λ (2 n ) , a result which will be of importance later on. This isnot the complete action of D as the image of some flux terms acting on elements of Λ (+) reside inΛ (1) or Λ (5) but as pointed out in our factorisation of the superpotential they do not contribute to thesuperpotential because such terms are projected out from the integrand by Ω c ∈ H (3) ⊂ Λ (3) . F : Λ (2) → Λ (5) , F : Λ (4) → Λ (5) Λ (0) → Λ (1) , F : Λ (6) → Λ (5) Λ (2) → Λ (1) , F : Λ (4) → Λ (1) (3.3)These actions are not relevant to constructing the superpotential but are to the Bianchi constraintsof D . In the fluxless case the exterior derivative is trivially nilpotent, d = 0, and defines an exactsequence on the even and odd p -form bundles. The same is true for the covariant derivatives in thecase of non-zero fluxes but in a less straightforward way. This is because the action d : Λ ( p ) → Λ ( p +1) is no longer true for D , giving rise to non-trivial Bianchi constraints, as done in full generality in theappendix of [18]. · · · D −−→ Λ (+) D −−→ Λ ( − ) D −−→ Λ (+) D −−→ · · · ( p ) doesnot lend itself to a straightforward analysis of the flux obtained from the superpotential. An exampleof this is how F can be defined in terms of the geometry of the space by d η τ ∝ F τσρ η σρ and theproperties of M determine the number and location of non-zero fluxes in F τσρ . This formulation, whilegeneral to any internal space, is not manifest in its independent fluxes or how they contribute to thesuperpotential. In fluxless cases the moduli are massless so moduli dependent forms are expanded inthe H ( p ) basis and those p -forms not in the cohomologies can be ignored in the effective theory becausethey are associated to ‘heavy modes’. Though the fluxes break this massless/heavy splitting of thedegrees of freedom of M through the non-closure of elements originally in H ( p ) giving the modulimasses, it is possible to use the original H ( p ) bases to expand flux contributions to the superpotentialintegrand [3]. The SU(3) structure is that which reduces to massless field modes in the fluxless case,while the ‘heavy modes’ are those which have mass even in the absence of fluxes. We therefore havetwo approaches we might wish to consider; analysing the ‘light’ modes using the SU(3) structure oranalysing all modes using all p -forms as our basis. The latter case has the advantage of not excludinganything from our considerations, removing the issue of just how light is light, how heavy is heavyor which p -forms or expressions we neglect. However, as we will show, the cohomologies approachis considerably more ‘natural’ when it comes to describing the superpotential. This will allow usto construct general superpotentials and see the effect T and S duality have on them, rather thanrestricting ourselves to a single particular M . We cannot completely ignore the fact an analysisin terms of SU(3) structure will not be a complete description of the fluxes of M but assuming themoduli are sufficiently lighter than other fields we will mostly restrict our discussion to the cohomologyconstruction only. With our attention restricted to the H ( p ) the p -forms ( p odd) of interest are only those in H (3) andthe generic exact sequence for D previously considered simplifies down to being in terms of H (3) and (cid:101) H (3) . · · · D −−→ H (3) D −−→ (cid:101) H (3) D −−→ H (3) D −−→ · · · To facilitate an analysis of the flux actions on the cohomologies we reformulate the fluxes and derivativessuch that they can be viewed as matrices (or components of matrices) acting upon vectors of basiselements of cohomologies. To that end we define a 2 h , + 2 h , + 4 dimensional vector of p -forms e by combining e ( a ) and e ( ω ) . e ≡ (cid:0) e ( ω ) e ( a ) (cid:1) ≡ (cid:0) ω ω a (cid:101) ω (cid:101) ω b a a i b b j (cid:1) With the entries of e forming the basis for any differential form in M we can express the D imageof any given form as a linear combination of other cohomology basis elements, thus giving a matrixrepresentation to D . We choose to write this action in such a manner that the h matrices are explicitfactors, for later convenience. D (cid:18) e ( ω ) e ( a ) (cid:19) = (cid:18) MN (cid:19) (cid:18) h ω h a (cid:19) (cid:18) e ( ω ) e ( a ) (cid:19) D ≡ (cid:18) MN (cid:19) As a result of the action of the derivatives exchanging the two cohomologies, H (3) ↔ (cid:101) H (3) , we havethat D must be block-off-diagonal, hence only the M and N sub-matrices are non-zero and and wecan express the exact sequence in terms of these flux matrices. · · · M · h ω −−−−→ H (3) N · h a −−−−→ (cid:101) H (3) M · h ω −−−−→ H (3) N · h a −−−−→ · · · We also observed in the previous section that the action of the fluxes on (cid:101) H (3) defines the action ofthe fluxes on H (3) and vice versa, implying that the entries of M define those of N and it is thisdependence which we aim to derive. In the NS-NS superpotential the flux multiplets F n are definedby the action of D on the basis elements of (cid:101) H (3) by F n : H ( n,n ) → H (3) . As a result the action of13 is used to define the flux entries of the multiplets by D ( e ( ω ) ) = M · h a · e ( a ) . As in the previoussubsection we consider the F n in order of increasing n , beginning with F : H (0 , → H (3) , whoseaction is equivalent to multiplying by a 3-form, expanded in the sympletic basis. D ( ω ) = F = ( F ) I a I − ( F ) J b J = ( M · h a · e ( a ) ) Therefore the first row of M is written in terms of the coefficients of F , providing us with a vector offlux entries. ( M ) · = (cid:0) ( F ) ( F ) i ( F ) ( F ) j (cid:1) The next case, D ( ω a ), defines F as it is the part of D whose action is of the form H (1 , → H (3) andwe define the flux entries of F such that its expansion in terms of the H (3) basis is of the same formatas the F case. F ( ω a ) = ( F ) ( a ) I a I − ( F ) J ( a ) b J ≡ ( M · h a · e ( a ) ) a +1 Each of the ω a has a corresponding vector of flux entries and so overall F can be associated with a h , × h , + 1) submatrix of M , specifically the 2nd through to the ( h , + 1)’th rows of M .( M ) a +1 · = (cid:16) ( F ) ( a )0 ( F ) ( a ) i ( F ) a ) ( F ) j ( a ) (cid:17) By considering coefficients of the moduli in the superpotential it is clear there are no other independentflux components in F , a result which will be demonstrated explicitly for the Z × Z orientifold.Continuing with this method for F and F we can define all of the entries for M . Due to our choiceof ordering in the entries of e ( ω ) the components of F are before those of F . M = ( F ) ( F ) i ( F ) ( F ) j ( F ) ( a )0 ( F ) ( a ) i ( F ) a ) ( F ) j ( a ) ( F ) ( F ) i ( F ) ( F ) j ( F ) ( b )0 ( F ) ( b ) i ( F ) ( b )0 ( F ) ( b ) j = ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J (3.4)Using this definition of the components of each of the fluxes and this matrix we can construct a morenatural differential operator representation for D than that given in (3.2). This can be done by usinga set of interior forms defined in terms of the basis elements of H (3) and (cid:101) H (3) and their intersectionnumbers, which in turn we take to define the basis elements of H ( p ) ∗ such as H (2) ∗ = (cid:104) ι ω a (cid:105) . ι ω a ( ω b ) = g ab = δ ab = ι e ω b ( (cid:101) ω a ) , ι a i ( a j ) = g ij = δ ij = ι b j ( b i )The volume form is factorisable into pairs of Hodge dual forms, such as µ vol = a ∧ b , which we denotegenerally as µ vol = ξ ∧ ζ . Therefore an interior form applied to µ vol is non-zero. µ vol = ξ ∧ ζ ⇒ ι ξ ( µ vol ) = ζ All other combinations do not need to be specified because they lead to forms of degree higher thansix when combined with the other parts of the superpotential integrand and so do not contribute. Itis also worth noting that
J ∧
Ω = 0 excludes any H (5) contributions of the form ω a ∧ a I and ω a ∧ b J .This is in contrast to the action of F on Λ (2) in (3.3), where a non-zero 5-form might exist but simplynot contribute to the superpotential. F = F στρ η στρ = ( F ) I a I ι ω − ( F ) J b J ι ω F = F στρ η τρ ι σ = ( F ) ( a ) I a I ι ω a − ( F ) J ( a ) b J ι ω a F = F στρ η ρ ι τσ = ( F ) ( b ) I a I ι e ω b − ( F ) ( a ) J b J ι e ω b F = F στρ ι ρτσ = ( F ) I a I ι e ω − ( F ) J b J ι e ω (3.5)14his reformulation of D , with its action on the basis elements of the (cid:101) H (3) cohomologies manifest, canbe written in a more compact form once we define ι e to be a vector of interior forms, with the n ’thentry being the dual of the n ’th entry of e and likewise for e ( a ) and e ( ω ) . D = F + F + F + F ≡ e (cid:62) ( a ) · h (cid:62) a · M (cid:62) · ι e ( ω ) (3.6)This manner of expressing D has a clear action on (cid:101) H (3) but the D of (3.2) can also be applied to H (3) elements, with the action being related in some way to N . On the grounds of symmetry betweenthe two cohomologies and the definition of D on e we can postulate the alternative form of D withmanifest action on H (3) . D = F + F + F + F ≡ e (cid:62) ( ω ) · h (cid:62) ω · N (cid:62) · ι e ( a ) (3.7)In order to find the entries of N in terms of the entries of M we can reformulate each of the fluxesin (3.5) individually. We first consider F and F which have the simplest actions, reducing to eitherremoving or adding 3-forms. F : H (0 , → H (3) → H (3 , F : H (0 , ← H (3) ← H (3 , In the case of F the differential operator action is simply F ∧ , whose action on both H (3) and (cid:101) H (3) is easily found and we use this as a guide in reformulating the expressions. F : H (0 , → H (3) , F = ( F ) I a I ι ω − ( F ) J b J ι ω F : H (3) → H (3 , , F = ( F ) J (cid:101) ω ι a J + ( F ) I (cid:101) ω ι b I Conversely the removal of 3-forms by the action of F on (cid:101) H (3) is easily converted into an action on H (3) . F : H (3 , → H (3) , F = ( F ) I a I ι e ω − ( F ) J b J ι e ω F : H (3) → H (0 , , F = − ( F ) J ω ι a J − ( F ) I ω ι b I Comparing the two different actions of F and F we have the following expressions : a I ι ω (cid:39) (cid:101) ω ι b I b J ι ω (cid:39) − (cid:101) ω ι a J a I ι e ω (cid:39) − ω ι b I b J ι e ω (cid:39) ω ι a J (3.8)In the cases of F and F this simple addition or removal of 3-forms does not occur so we try toreexpress the p -form and interior form components of F , F in (3.5) into something resembling theright hand sides of (3.8). F arises from the non-closed nature of the basis forms and due to itsgeometric nature it can be expressed in terms of the exterior derivative d. We have defined the entriesof F by its action on (cid:101) H (3) and using integration by parts [18] the alternative action on H (3) can befound. d( ω a ) = F ( ω a ) ≡ F ( a ) I a I − F J ( a ) b J (3.9)The two sets of coefficients can be extracted from this expression by integrating over the approriate3-cycles and then converting these to integrals over the entire space. (cid:90) A J d( ω a ) = (cid:90) A J F ( a ) I a I ⇒ (cid:90) M d( ω a ) ∧ b J = (cid:90) M F ( a ) I a I ∧ b J By Stokes theorem and H (5) being empty the left hand side integral with integrand d( ω a ) ∧ b J convertsto an integral over M with integrand − ω a ∧ d( b J ) but this can be reexpressed as an intergral over a4-cycle, that which is associated to the ω a b J . F ( a ) J = − (cid:90) M ω a ∧ d( b J ) = (cid:90) M d( − b J ) ∧ ω a = (cid:90) B a d( − b J )15y writing d b J in terms of the (cid:101) H (3) basis we obtain the contribution to the non-closure of b J spannedby the (2 , (cid:101) ω b in terms of the fluxes defining the non-closure of the ω a and the (cid:101) H (3) intersectionnumbers. However, due to our definition of the (2 , δ ab ,simplify the end result. F ( a ) J = F ( b ) J δ ba = F ( b ) J (cid:90) M ω a ∧ (cid:101) ω b = (cid:90) M ω a ∧ ( F ( b ) J (cid:101) ω b ) = (cid:90) M ω a ∧ d( − b J )Therefore the action of F on the b J , F ( b J ), or equivalently the non-closure of b J , has a contributionin the (2 , −F ( b ) J (cid:101) ω b . Repeating this method but integrating over the B I a I , d a I ∼ F ( a I ) in the (2 , F I ( b ) (cid:101) ω b and we can summarise this in the same way as (3.8). a I ι ω a (cid:39) − (cid:101) ω a ι b I b J ι ω a (cid:39) (cid:101) ω a ι a J (3.10)In terms of their explicit action on different cohomologies we can represent F in two different ways. F : H (1 , → H (3) , F = ( F ) ( a ) I a I ι ω a − ( F ) J ( a ) b J ι ω a F : H (3) → H (2 , , F = − ( F ) J ( a ) (cid:101) ω a ι a J − ( F ) ( a ) I (cid:101) ω a ι b I (3.11)The remaining case of F does not immediately lend itself to the same methodology since the schematicaction of the flux is F : H ( p ) → H ( p − , the opposite behaviour of F and the exterior derivative d,a reflection of its non-geometric nature. However, such action is seen in adjoint derivatives and so wecan rephrase F in terms of the adjoint action of an exterior derivative. To that end we define d andits (cid:10)(cid:10) , (cid:11)(cid:11) adjoint d † by the action of F on H (2 , . d † ( (cid:101) ω b ) ≡ F ( (cid:101) ω b ) ≡ F ( a ) I a I − F ( a ) J b J As in the F case we can project out particular coefficients of F ( (cid:101) ω b ) by taking its inner product withparticular H (3) basis elements, allowing us to then make use of the adjoint properties of the innerproduct. F ( a ) I = (cid:10)(cid:10) F ( (cid:101) ω a ) , a I (cid:11)(cid:11) = (cid:10)(cid:10) d † (cid:101) ω a , a I (cid:11)(cid:11) ≡ (cid:10)(cid:10) (cid:101) ω a , da I (cid:11)(cid:11) Making use of Stokes theorem again we can change which form the derivative d acts upon, which isnot possible to do when working with d † .0 = (cid:90) M d ( a I ∧ ω a ) = (cid:90) M da I ∧ ω a − (cid:90) M a I ∧ d ω a = (cid:10)(cid:10) da I , (cid:101) ω a (cid:11)(cid:11) − (cid:90) M a I ∧ ∗ ( ∗ − d ∗ ) (cid:101) ω b Having obtained an expression for d acting on an element of H (3) we need to revert back to expressingderivatives as d † . This is done by using the definition of adjoint derivatives in terms of Hodge stars andderivatives, taking note that the action of the derivatives on the symplectic basis elements acquires anadditional factor of − d † = (cid:110) ∗ − d ∗ d : (cid:101) H (3) → H (3) − ∗ − d ∗ d : H (3) → (cid:101) H (3) Inverting this relationship, to express d in terms of d † , we obtain the alternative action of d † on H (3) bynoting that ∗ = ∗ − on (cid:101) H (3) due to the intersection numbers of the basis elements being the Kroneckerdelta. F ( a ) I = (cid:10)(cid:10) d † (cid:101) ω a , a I (cid:11)(cid:11) = (cid:10)(cid:10) (cid:101) ω a , da I (cid:11)(cid:11) = (cid:10)(cid:10) (cid:101) ω a , − ∗ d † ∗ − a I (cid:11)(cid:11) = (cid:10)(cid:10) ω a , d † b I (cid:11)(cid:11) F gives terms related to the F image of the a I . F ( a ) J = (cid:10)(cid:10) d † (cid:101) ω a , b J (cid:11)(cid:11) = (cid:10)(cid:10) (cid:101) ω a , db J (cid:11)(cid:11) = (cid:10)(cid:10) (cid:101) ω a , − ∗ d † ∗ − b J (cid:11)(cid:11) = (cid:10)(cid:10) ω a , d † ( − a J ) (cid:11)(cid:11) Putting these results together we obtain the remaining set of relations, in line with (3.8) and (3.10). a I ι e ω a (cid:39) ω a ι b I b J ι e ω a (cid:39) − ω a ι a J (3.12)In terms of their explicit action on different cohomologies we can represent F in two different ways. F : H (2 , → H (3) , F = ( F ) ( b ) I a I ι e ω b − ( F ) ( b ) J b J ι e ω b F : H (3) → H (1 , , F = ( F ) ( b ) J ω b ι a J + ( F ) ( b ) I ω b ι b I Despite the derivation of the alternative actions of each of the F n using a different method, they allshare the feature that if a particular term f ∈ F n has an action f : (cid:104) ξ (cid:105) → (cid:104) ζ (cid:105) then it will also have anaction f : (cid:104)∗ ζ (cid:105) → (cid:104)∗ ξ (cid:105) , where (cid:104) χ (cid:105) is the space spanned by the form χ . To illustrate this and the resultsthemselves we consider the explicit case of the Z × Z orientifold. Using these results the expansionof D in (3.6) is converted into the expansion postulated in (3.7), giving us the manifest action of D on H (3) and the entries of N in terms of those of M . D ≡ e (cid:62) ( ω ) · h (cid:62) ω · N (cid:62) · ι e ( a ) ⇒ N = (cid:32) −F (0) J F ( b ) J F J (0) −F J ( a ) −F (0) I F ( b ) I F (0) I −F ( a ) I (cid:33) (3.13)We are abusing notation somewhat, as N is related to the transpose of M and so all the flux coefficientmatrices in (3.13) should be transposed, such as −F (0) J defining part of a row in M but part of acolumn in N . However, due to their explicit index labels it is unambiguous how they combine withother terms once matrix expressions are written out in terms of their entries. Before moving onto howthe superpotential might be expressed in terms of these flux matrices it is convenient to define a morecompact notation for the flux components. In (3.4) we are able to express M in a more convenientmanner by running the complex structure indices over the range { , . . . , h , } , where the a and b contributions are easily combined with the other sympletic basis terms. In the case of the elements of (cid:101) H (3) there was an obstruction to being able to run K¨ahler indices over the ranges A, B ∈ { , , . . . , h , } as the expressions for ω in (3.8) do not follow the pattern given in (3.10) for the ω a and likewise for (cid:101) ω in (3.8) compared to the (cid:101) ω b of (3.12).Any change of basis e → e (cid:48) which allows these expressions to be combined must preserve theassociated bilinear forms g and h and ideally not alter how the complexified holomorphic forms areexpanded in terms of the dilaton. It is clear that because the differences in (3.8), (3.10) and (3.12)occur in terms of the K¨ahler indices the change of basis must be in the e ( ω ) section of e . Thereare two ways in which to do this which preserve the schematic layout of the K¨ahler moduli dependentcomplexified holomorphic forms. In the case of the non-complexified holomorphic form (cid:102) = T · h ω · e ( ω ) any non-singular transformation in e ( ω ) can be countered by a corresponding transformation in T . • Exchange ω and (cid:101) ω and therefore T and T ω ω a (cid:101) ω (cid:101) ω b → ν ν a (cid:101) ν (cid:101) ν b = (cid:101) ω ω a ω (cid:101) ω b , T T a T T b → T T a T T b = T T a T T b (3.14) • Exchange ω a and (cid:101) ω a and therefore T a and T a . ω ω a (cid:101) ω (cid:101) ω b → ν ν a (cid:101) ν (cid:101) ν b = ω (cid:101) ω a (cid:101) ω ω b , T T a T T b → T T a T T b = T T a T T b (3.15)17t is noteworthy that this kind of exchanging of moduli T ↔ T or T a ↔ T a is precisely that whichwas used in Section 1 to argue that the Type IIB superpotential could be written in a particular way.The difference between these two choices in how we might change the e ( ω ) basis reduces to a differencein sign, with the change of basis in (3.14) giving the + sign in the ± of the following expressions. ± a I ι ν A (cid:39) − (cid:101) ν A ι b I ± b J ι ν A (cid:39) (cid:101) ν A ι a J ± a I ι e ν B (cid:39) ν B ι b I ± b J ι e ν B (cid:39) − ν B ι a J (3.16)Though this change in basis is prompted by our examination of N it also transforms M , which is themore convenient flux matrix in which to examine the flux components. As we can now run the K¨ahlerindices over the range { , . . . , h , } we can view the fluxes of F and F as part of the fluxes of F and F , though which belongs to which depends upon which change of basis we choose. The changeof basis in (3.14) leads us to view F as part of F and F as part of F . M = ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J → ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J (3.17)Conversely, in the case of the change of basis in (3.15) F is part of F and F part of F and theindex structure of the components of M are altered accordingly. M = ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J → ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) (3.18)With the compact 2 × M it is possible to conveniently express the action of D on (cid:101) H (3) usinga set of matrices C i , rather than having to refer to how the flux components arrange themselves intoT duality multiplets. M = (cid:18) C C C C (cid:19) ⇒ D ( ν ) = C · α − C · β D ( (cid:101) ν ) = C · α − C · β (3.19)Using (3.16) we convert (3.19) from the action D : (cid:101) H (3) → H (3) to D : H (3) → (cid:101) H (3) , where the sign isas previously stated. ±D ( α ) = C (cid:62) · ν − C (cid:62) · (cid:101) ν ±D ( β ) = C (cid:62) · ν − C (cid:62) · (cid:101) ν ⇒ D (cid:18) αβ (cid:19) = ± (cid:18) C (cid:62) − C (cid:62) C (cid:62) − C (cid:62) (cid:19) (cid:18) ν (cid:101) ν (cid:19) Using the definition of N , D ( e ( a ) ) = N · h ν · e ( ν ) we obtain the expression for N in terms of the C i . N = ± (cid:18) C (cid:62) − C (cid:62) C (cid:62) − C (cid:62) (cid:19) (3.20)Using the ansatz that the expression for D in terms of N will be of the same form as the expressionfor D ’s natural action on (cid:101) H (3) in (3.6). D = F + F + F + F = ± (cid:0) ν (cid:101) ν (cid:1) · (cid:18) C C − C − C (cid:19) · (cid:18) ι α ι β (cid:19) (3.21)From this point we will work with the explicit case of the second basis, that of (3.15), as convertingto the first case is simply a matter of relabelling and altering index structures on the fluxes, and thesecond redefinition will be more convenient in our analysis of Type IIB. M = ( F ) I ( F ) J ( F ) ( b ) I ( F ) ( b ) J ( F ) I ( F ) J ( F ) ( a ) I ( F ) J ( a ) ≡ (cid:32) F ( A ) I F J ( A ) F ( B ) I F ( B ) J (cid:33) , a I ι ν A (cid:39) (cid:101) ν A ι b I b J ι ν A (cid:39) − (cid:101) ν A ι a J a I ι e ν B (cid:39) − ν B ι b I b J ι e ν B (cid:39) ν B ι a J (3.22)18ence we have the following two alternative actions for D , in terms of the flux components. D ( ν A ) = F ( A ) I a I − F J ( A ) b J D ( (cid:101) ν B ) = F ( B ) I a I − F ( B ) J b J ⇔ D ( a I ) = −F ( A ) I ν A + F I ( B ) (cid:101) ν B D ( b J ) = −F ( A ) J ν A + F ( B ) J (cid:101) ν B (3.23)The contribution to the superpotential integrand D ( (cid:102) ) splits into four parts, each associated to oneof the flux multiplets F n . F · J (0) = T (cid:16) F (0) I a I − F J (0) b J (cid:17) = F pqr η pqr F · J (1) = T b (cid:16) F ( b ) I a I − F ( b ) J b J (cid:17) = F ipq J ir η pqr F · J (2) = T a (cid:16) F ( a ) I a I − F J ( a ) b J (cid:17) = F ijp J ijqr η pqr F · J (3) = T (cid:16) F (0) I a I − F (0) J b J (cid:17) =
13! 13! F ijk J ijkpqr η pqr (3.24)With these we can construct the general Type IIA superpotential in this formulation. (cid:90) M Ω c ∧ D ( (cid:102) ) = T (cid:16) − S F (0)0 U + F (0) j U j + S F U − F i (0) U i (cid:17) ++ T b (cid:16) − S F ( b )0 U + F ( b ) J U J + S F ( b )0 U − F ( b ) i U i (cid:17) ++ T (cid:16) − S F (0)0 U + F (0) j U j + S F (0)0 U − F (0) i U i (cid:17) ++ T a (cid:16) − S F ( a )0 U + F ( a ) j U j + S F a ) U − F i ( a ) U i (cid:17) Though we did not originally give a motive for our redefining of the sympletic basis and associatedmoduli ( α I , β J , U ) → ( a I , b J , U ) we have now derived precisely the same type of exchange in theK¨ahler moduli. While a symplectic transformation on the defining H (3) basis of (3.16) does not alterthe identities the K¨ahler moduli space transformation does and it has motivated e ( ω ) → e ( ν ) . Byconsidering mirror symmetry we can view this Type IIA H (3) redefinition in the context of a TypeIIB (cid:101) H (3) redefinition, where the cohomologies are defined on the mirror pair ( M , W ), with Type IIAon M and we will denote the map associated to mirror symmetry by Υ. Under mirror symmetry thedegrees of freedom of the Type IIA theory on M are labelled differently to those of the Type IIB on W . The complex structure of M is defined to be equivalent to the K¨ahler structure of W and so the T of M relate to the U of W . Due to properties of the H (3) ( M ) and (cid:101) H (3) ( M ) basis elements we havefound it convenient to exchange ω a and (cid:101) ω a so as to obtain (3.22). This results in a redefinition T → T and the effect such changes have on W is to prompt its complex structure moduli and H (3) ( W ) basiselements to be altered. Conversely, in order to obtain the results of (3.22) on H ( p ) ( W ) we would berequired to exchange the ω a and (cid:101) ω a of (cid:101) H (3) ( W ), whose Υ image is the redefinition of the H (3) ( M )elements α a and β a . Thus the choice we made of how to denote the complex structure moduli, U → U ,was in anticipation of this redefinition of the K¨ahler moduli in Type IIA, allowing us to forego furtherredefinitions of flux components due to the exchange of such terms as F ( A ) i and F i ( A ) . This kind ofexchange in the basis of (cid:101) H (3) will be seen to arise in Type IIB due to the manner in which the fluxescouple to the moduli. Before that we find it useful to further develop the way of expressing integrals interms of matrices and vectors in anticipation of comparing the superpotentials of the Type II theories.We previously noted that the superpotential can be written in terms of g and the vectors associatedto the elements of H (3) which make up the integrand. This section, unless otherwise stated, is done in Type IIA. The superpotential can be expressed interms of the flux matrices by the use of an inner product involving a vector of moduli.Φ (cid:62) ≡ (cid:0) T (cid:62) U (cid:62) (cid:1) ≡ (cid:0) T T a T T b U U i U U j (cid:1) (3.25)Given any form χ we recall that the associated vector is defined via the general factorisation χ = χ · h · e ,though this is a basis dependent expression, as was seen in the case of (cid:102) , whose moduli vector was19edefined by the change of basis e ( ω ) → e ( ν ) . Hence we defined our moduli by T ≡ (cid:102) and U ≡ ΩΩ = U I a I − U J b J = (cid:0) U I U J (cid:1) · h a · (cid:18) a I b J (cid:19) = Ω (cid:62) · h a · e ( a ) = U (cid:62) · h a · e ( a ) (cid:102) = T A ν A + T B (cid:101) ν B = (cid:0) T A T B (cid:1) · h ν · (cid:18) ν A (cid:101) ν B (cid:19) = (cid:102) (cid:62) · h ν · e ( ν ) = T (cid:62) · h ν · e ( ν ) Before considering how the dilaton couples to particular fluxes, which would complicated our analysisthrough the use of complexified holomorphic forms, we first consider the toy model of a superpotentialwhose integrand is of the form Ω ∧ D ( (cid:102) ) so as to neglect dilaton dependence. We restrict our attentionto the NS-NS fluxes and for comparision also consider the superpotential which would be obtainedfrom the integrand (cid:102) ∧ D (Ω) as this will arise later. To this end we recall the two alternative actionsof D on the basis elements of (cid:101) H (3) and H (3) . D (cid:18) ν (cid:101) ν (cid:19) ≡ M · h a · e ( a ) , D (cid:18) αβ (cid:19) ≡ N · h ν · e ( ν ) By writing the holomorphic forms in terms of their vector factorisations and using the above expressionsfor the images of (cid:101) H (3) and H (3) basis elements under D , we can construct the vectors associated to D ( (cid:102) ) and D (Ω). D ( (cid:102) ) = D ( T (cid:62) · h ν · e ( ν ) )= T (cid:62) · h ν · D ( e ( ν ) )= T (cid:62) · h ν · M · h a · e ( a ) = D ( (cid:102) ) (cid:62) · h a · e ( a ) D (Ω) = D ( U (cid:62) · h a · e ( a ) )= U (cid:62) · h a · D ( e ( a ) )= U (cid:62) · h a · N · h ν · e ( ν ) = D (Ω) (cid:62) · h ν · e ( ν ) (3.26)Given the vectors associated to any two wedge-paired elements in H ( ∗ ) their integral can be writtenin terms of g and in the cases of integrands Ω ∧ D ( (cid:102) ) and (cid:102) ∧ D (Ω) the expressions reduce to beingin terms of g a or g ν only. (cid:90) Ω ∧ D ( (cid:102) ) = g (cid:16) Ω , D ( (cid:102) ) (cid:17) = D ( (cid:102) ) (cid:62) · g a · Ω = T (cid:62) · h ν · M · g a · U (3.27) (cid:90) (cid:102) ∧ D (Ω) = g (cid:16) (cid:102) , D (Ω) (cid:17) = D (Ω) (cid:62) · g ν · (cid:102) = U (cid:62) · h a · N · g ν · T (3.28)It is worth noting that the sum of these expressions can be expressed in a very natural way in terms ofΦ and D and the bilinear forms defined on e , putting the two moduli spaces into a single expression,because of the identity (cid:102) + Ω = Φ (cid:62) · h · e . (cid:90) (cid:102) ∧ D (Ω) + (cid:90) Ω ∧ D ( (cid:102) ) = g ν ( (cid:102) , D (Ω)) + g a (Ω , D ( (cid:102) ))= g ( (cid:102) + Ω , D (Ω) + D ( (cid:102) ))= Φ (cid:62) · h · D · g · Φ (3.29)This expression for superpotential-like terms treats the two moduli spaces in exactly the same mannerbut this symmetry is broken when we consider the complexified holomorphic forms, as T dualityrelated fluxes do not all couple to the dilaton in the same manner. In spaces where the fluxes areobtained by requiring modular invariance in the complex structure and K¨ahler moduli it is possibleto express the superpotential in such a way that all those fluxes which define a covariant derivativecouple to the dilaton in the same manner, as discussed for the Z × Z orientifold in [19]. We shallreturn to this result later when considering a particular set of internal spaces where such a symmetricformalism is possible even for the complexified holomorphic forms. To examine precisely how theinclusion of dilaton couplings in the complexified holomorphic forms breaks this symmetry we definea set of matrices associated to the holomorphic forms in the e ( a ) and e ( ν ) bases.Ω = Ω (cid:62) · h a · e ( a ) = U (cid:62) · Ω (cid:58) · e ( a ) , (cid:102) = (cid:102) (cid:62) · h ν · e ( ν ) = T (cid:62) · (cid:102) (cid:58) · e ( ν )
20 second set of matrices, of the same dimensions as D , can be defined for the holomorphic forms bytaking the basis to be e and the moduli vector Φ. In this way both holomorphic forms can be writtenin the same manner, Ω = Φ (cid:62) · Ω · e and (cid:102) = Φ (cid:62) · (cid:102) · e . These matrices can be expressed in terms ofΩ (cid:58) and (cid:102) (cid:58) through the use of projection matrices P ± = ( I ± Γ ). We note how Ω (cid:58) and (cid:102) (cid:58) relate to h a and h ν . Ω = P − ⊗ Ω (cid:58) = P − ⊗ h a , (cid:102) = P + ⊗ (cid:102) (cid:58) = P + ⊗ h ω Though much of our analysis will be done in terms of Ω (cid:58) , (cid:102) (cid:58) and their complexified forms the resultsand methods for Ω, (cid:102) and their complexified forms follow in the same way and will be used in latersections so we consider both matrix types in tandem. Although the Type IIA integrand has thecomplex structure moduli complexified we consider the complexification of (cid:102) instead, with the Ω casesfollowing the same general method, so as to have a simpler sign convention in our algebra. We defineits vector/matrix factorisation in the same way as (cid:102) , (cid:102) c ≡ T (cid:62) · (cid:102) c (cid:58)(cid:58) · e ( ν ) . (cid:102) c = T T a T T b (cid:62) − S I I h , − S I I h , ν ν a (cid:101) ν (cid:101) ν b This can be factorised so that the complexification is due to a single matrix, C , which modifies theoriginal expressions for the holomorphic forms. (cid:102) = T (cid:62) · (cid:102) (cid:58) · e ( ν ) → (cid:102) c = T (cid:62) · (cid:102) c (cid:58)(cid:58) · e ( ν ) ≡ T (cid:62) · C · (cid:102) (cid:58) · e ( ν ) (3.30)We can extract the expression for C from this factorisation but it will be useful to write it, and C (cid:48) , asa linear combination of a set of projection operators for examining S duality. The projection operatorsare such that they seperate out the H (0 , and H (3 , basis elements from the H (1 , and H (2 , basesand are built from SO ( n, m ) metrics with signature (+ , · · · , − , · · · ), which we shall denote as η ( n,m ) . A n ≡ I ⊗ (cid:16) η ( n +1 , − η (1 ,n ) (cid:17) = I ⊗ (cid:18) I n (cid:19) = I ⊗ A n B n ≡ I ⊗ (cid:16) η ( n +1 , + η (1 ,n ) (cid:17) = I ⊗ (cid:18) n (cid:19) = I ⊗ B n Of note are the following set of identities for combining A m and B n and since the dimensionalities ofthe matrices are unambigious we suppress the indices. A · A = A , B · B = B , A · B = 0 = B · A (3.31)The A and B inherit the same set of identities due to their definitions in terms of A and B and it isthese matrices which define the two K¨ahler complexified holomorphic forms. (cid:102) c (cid:58)(cid:58) = A h , − S B h , (cid:102) (cid:48) c (cid:58)(cid:58) = B h , − S A h , In the case of the non-complexified holomorphic form matrices they can be written as a tensor productof matrices of the form η ( n,m ) and a similar tensor product formulation can be done for the complexifiedforms but with projection operators P ( (cid:48) ) n . (cid:102) c (cid:58)(cid:58) = I ⊗ (cid:16) A h , − S B h , (cid:17) ≡ I ⊗ P h , (cid:102) (cid:48) c (cid:58)(cid:58) = I ⊗ (cid:16) B h , − S A h , (cid:17) ≡ I ⊗ P (cid:48) h , Infact there are two complexified holomorphic forms, (cid:102) c and (cid:102) (cid:48) c and so we distinguish their complexification matriceswith a prime, C and C (cid:48) respectively. (cid:102) c ≡ P − ⊗ (cid:102) c (cid:58)(cid:58) = P − ⊗ I ⊗ P h , , (cid:102) (cid:48) c ≡ P + ⊗ (cid:102) (cid:48) c (cid:58)(cid:58) = P + ⊗ I ⊗ P (cid:48) h , By our definitions of the complexified holomorphic forms, the matrix representations are factorisableinto complexification matrices and the original expressions for (cid:102) (cid:58) etc . (cid:102) c (cid:58)(cid:58) = I ⊗ P h , = (cid:16) I ⊗ P h , (cid:17) · (cid:16) I ⊗ I h , +1 (cid:17) = C h , · (cid:102) (cid:58) = C h , · h ω Ω c (cid:58)(cid:58) = η (1 , ⊗ P h , = (cid:16) I ⊗ P h , (cid:17) · (cid:16) η (1 , ⊗ I h , +1 (cid:17) = C h , · Ω (cid:58) = C h , · h a As the dimensions of the matrices are unambigious due to their context we will henceforth drop thesubscripts unless required. This is extended to the full 2 h , +2 h , +4 dimensional case by defining thecombination combining C ( (cid:48) ) ≡ C ( (cid:48) ) h , ⊕ C ( (cid:48) ) h , , resulting in factorisations such as Ω (cid:48) c = C (cid:48) · Ω or (cid:102) c = C · (cid:102) .With complexification having the effect of h ν → C ( (cid:48) ) · h ν on the expansion of (cid:102) the inner productexpression for the T duality induced superpotential is obtained by altering the toy model expressionpreviously found in (3.27), with the R-R fluxes following from the fact Type IIB treats them in thesame manner as NS-NS fluxes. W = (cid:90) M Ω c ∧ D ( (cid:102) ) = T (cid:62) · h ν · M · g a · C · U (3.32) Type IIA and Type IIB theories are related by T duality, with the particular case of three T dualities(in different directions) being equivalent to the mirror map Υ. In general the compact space obtainedfrom M under this map, which we shall denote as W , is not the same as M which can be most readilyseen from the Hodge numbers of the two spaces. h , ( M ) = h , ( W ) , h , ( M ) = h , ( W )This exchange of Hodge numbers is closely related to the fact the K¨ahler and complex structure modulispaces of the two theories are exchanged, though for the purposes of clarity we will continue to use h , to mean the Type IIB Hodge number, rather than using additional A and B labels to distinguishwhich Type II theory we are working in. The Type IIB complex structure moduli space therefore has h , dimensions and in keeping with previous index notation the index labels a , b etc will range over1 to h , . Conversely, the dimension of the Type IIB K¨ahler moduli space is h , and the indices i , j etc will range over 1 to h , .Type IIA Type IIB U I , U J I, J = 0 , · · · , h , T I , T J T A , T B A, B = 0 , · · · , h , U A , U B This will be of particular convenience when comparing flux expansions or matrix dimensions betweenthe two theories and we therefore have a new p -form vector f moduli vector due to the new modulivector Ψ. f (cid:62) ≡ (cid:16) f (cid:62) ( a ) f (cid:62) ( ν ) (cid:17) ≡ (cid:0) a a a b b b ν ν i (cid:101) ν (cid:101) ν j (cid:1) Ψ (cid:62) ≡ (cid:0) U (cid:62) T (cid:62) (cid:1) ≡ (cid:0) U U a U U b T T i T T j (cid:1) The Type II theories represent the degrees of freedom of the internal space in different manners, so thedegrees of freedom denoted by the Type IIA U are represented by the T in Type IIB. In anticipation As [ C , h ν ] = 0 this could alternatively be written as h ν · C and likewise for Ω → Ω c and other complexificationmatrices.
22f having to construct the two alternative actions of the covariant derivatives we have taken our (cid:101) H (3) basis f ( ν ) to be analogous to the second Type IIA change of basis e ( ω ) → e ( ν ) and work with the T J K¨ahler moduli. With this redefinition of the form vector e → f we also have a new bilinear form g = g ( e ) → g ( f ) ≡ g . Due to our choice of the f ( ν ) basis we can use the mirror of (3.16), obtained byrelabelling a I → ν I etc and setting the ± sign choice to − and overall this gives a set of expressionswhich are schemicatically unchanged. ν I ι a A (cid:39) b A ι e ν I (cid:101) ν J ι a A (cid:39) − b A ι ν J ν I ι b B (cid:39) − a B ι e ν I (cid:101) ν J ι b B (cid:39) a B ι ν J (3.33)Type IIA and Type IIB have the NS-NS sector ‘in common’ and in terms of the superpotential thisamounts to the natural flux multiplets being defined as covariant derivative images of the (cid:101) H (3) sub-cohomologies [18, 3]. In the case of the 3-form flux H → (cid:98) F the superpotential contribution isproportional to T = T and in the same methodology as the Type IIA theory the extension of theNS-NS sector by T duality we would expect to be of a particular form.IIB : W NS = (cid:90) W Ω ∧ D ( (cid:102) c ) , IIA : W NS = (cid:90) M Ω c ∧ D ( (cid:102) )The fact both NS-NS sectors are defined by the derivatives acting on elements of their respective (cid:101) H (3) suggests we are able to make the same Λ ( ∗ ) component expansion of D as for D in terms of four fluxesinduced by T duality; (cid:98) F , F , F and (cid:98) F , where the hatted fluxes couple to the dilaton in their entirity. (cid:98) F = 13! (cid:98) F pqr η pqr , F = 12! (cid:98) F rpq η pqr ι r , F = 12! (cid:98) F pqr η r i qp , (cid:98) F = 13! (cid:98) F pqr ι rqp (3.34)These form precisely the same kind of gauge Lie algebra as in Type IIA and so the natural choice forthe Type IIB NS-NS derivative is of the same form as D in Type IIA but for the time being we shalldenote the sum of the fluxes as G , in line with the G of Type IIA. G = 13! (cid:98) F pqr η pqr + 12! F rpq η pqr ι r + 12! F pqr η r ι qp + 13! (cid:98) F pqr ι rqp = (cid:98) F + F + F + (cid:98) F (3.35)However, it is known that the non-geometric flux F (commonly denoted as Q in the literature)contributes a linear K¨ahler moduli dependency to the superpotential, in contrast to F which gavequadratic K¨ahler dependence in Type IIA. Therefore the integrand of the superpotential cannot bewritten as G ( (cid:102) c ), the fluxes of F and F couple to the K¨ahler moduli in the wrong manner. To rectifythis in a manner which leaves the Bianchi constraints invariant we consider two holomorphic forms ˇΩand ˇ (cid:102) , which are modifications of the standard expressions over and above simple relabellings. (cid:102) = T (cid:62) · h ν · e ( ν ) → T (cid:62) · L (cid:62) · h ν · e ( ν ) = ˇ (cid:102) Ω = U (cid:62) · h a · e ( a ) → U (cid:62) · K (cid:62) · h a · e ( ν ) = ˇΩ (3.36)A Type IIB superpotential whose F contributes quadratic K¨ahler moduli and F complex structuremoduli is then easily constructed from these new holomorphic forms. W NS = (cid:90) W ˇΩ ∧ G ( ˇ (cid:102) c ) = T (cid:62) · L (cid:62) · h ν · C · G · g a · K · U (3.37)We make the assumption that L commutes with C , which will be justified later, as allows us torefactorise the scalar product expression for the superpotential. W NS = (cid:90) W ˇΩ ∧ G ( ˇ (cid:102) c ) = T (cid:62) · h ν · C · (cid:16) L (cid:62) · G · g a · K · g (cid:62) a (cid:17) · g a · U ≡ (cid:90) W Ω ∧ D ( (cid:102) c ) (3.38)This provides us with the matrix representation of D acting on (cid:101) H (3) elements in terms of the entriesof G , where we define the matrix representation in the same manner as Type IIA. D ( f ) = D (cid:18) f ( a ) f ( ν ) (cid:19) = (cid:18) NM (cid:19) (cid:18) h a h ν (cid:19) (cid:18) f ( a ) f ( ν ) (cid:19) ≡ D · h · f (3.39)23he components of G are defined by (3.35), in that they have the same relationship that the fluxes of F have with the components of M . However, it is more convenient to define flux components in termsof M than to define them from G and then construct how they define M . M ≡ L (cid:62) · G · g a · K · g (cid:62) a ⇒ G = ( L (cid:62) ) − · M · g a · K − · g (cid:62) a (3.40)The relationship between N and M is the same as that between N and M , up to an exchange of Hodgenumbers. Having defined a set of bilinear forms, g and h , in addition to the Type IIA g and h , we areable to express N and N in terms of M and M and these bilinear forms. N = g ν · M (cid:62) · g a , N = g ν · M (cid:62) · g a (3.41)Using these identities we can construct the action of D on H (3) from its defining action on (cid:101) H (3) . D ( ν I ) = F ( I ) A a A − F B ( I ) b B D ( (cid:101) ν J ) = F ( J ) A a A − F ( J ) B b B ⇔ D ( a A ) = − F ( I ) A ν I + F A ( J ) (cid:101) ν J D ( b B ) = − F ( I ) B ν I + F ( J ) B (cid:101) ν J (3.42)In order to associate these components with the F n and (cid:98) F m of the NS-NS sector we need to determinethe specific form of both L and K but schematically it can be seen that F defines the fluxes of D ( ν i )and F the fluxes of D ( (cid:101) ν j ). As will be done explicitly later the requirement that both D and G giverise to the same Bianchi constraints determines the entries of K and L . Having constructed scalarproduct expressions for both Type II NS-NS superpotentials we can obtain the relationship betweenthe fluxes by comparing these scalar products. We now turn our attention to finding the relationship between the entries of M and N , which is done bycomparing the Type IIA superpotential with that of the Type IIB superpotential, though presently onlyin the NS-NS sector. In preparation for the R-R sector we consider an additional superpotential-liketerm in Type IIA, whose form is motivated by the moduli space exchange symmetry ζ : (cid:102) ↔ Ω. • Type IIB NS-NS superpotential. (cid:90) W Ω ∧ D ( (cid:102) c ) = g (cid:16) Ω , D ( (cid:102) c ) (cid:17) = T (cid:62) · h ν · C · M · g a · U (3.43) • Type IIA NS-NS superpotential moduli dual to the Type IIB NS-NS superpotential. (cid:90) M (cid:102) ∧ D (Ω c ) = g (cid:16) (cid:102) , D (Ω c ) (cid:17) = U (cid:62) · h a · C · N · g ν · T (3.44) • Type IIA NS-NS superpotential mirror dual to Type IIB NS-NS superpotential. (cid:90) M Ω c ∧ D ( (cid:102) ) = g (cid:16) Ω c , D ( (cid:102) ) (cid:17) = T (cid:62) · h ν · M · g a · C · U (3.45)In each case the dimension of the complexification matrix is the same, a point which we willrefer back to later. If the Type II superpotentials were related by a simple exchange of their modulispaces then (3.43) is equal, up to a relabelling of the moduli, to (3.44) and by comparing the matrixexpressions we can determine N in terms of M . h ν · M · g a = h a · N · g ν ⇒ M = h a · N · h a (3.46)However, as the Type IIA superpotential is defined as the mirror dual of the Type IIB superpotentialwe must equate (3.43) with (3.45) instead, once we relabel the moduli and take the transpose of thematrix expression in (3.45). h ν · M · g a = (cid:16) h ν · M · g a (cid:17) (cid:62) ⇒ M = g a · M (cid:62) · g a (3.47)24ecalling the general expression for M in terms of the F flux we now have the explicit dependence of N in terms of these fluxes. M ≡ (cid:32) F ( A ) I F J ( A ) F ( B ) I F ( B ) J (cid:33) ⇒ M = (cid:32) F ( I ) A F B ( I ) F ( J ) A F ( J ) B (cid:33) = (cid:32) −F ( A ) I F I ( B ) F ( A ) J −F ( B ) J (cid:33) Putting these results into (3.23) we have the Type IIA NS-NS derivative action given the Type IIBNS-NS derivative action. The choose of expressing Type IIA fluxes in terms of Type IIB fluxes, ratherthan vice versa, is convenient for our later analysis of tadpoles. D ( ν I ) = F ( I ) A a A − F B ( I ) b B D ( a A ) = − F ( I ) A ν I + F A ( J ) (cid:101) ν J D ( (cid:101) ν J ) = F ( J ) A a A − F ( J ) B b B D ( b B ) = − F ( I ) B ν I + F ( J ) B (cid:101) ν J D ( ν A ) = − F ( I ) A a I − F A ( J ) b J D ( a I ) = F ( I ) A ν A + F B ( I ) (cid:101) ν B D ( (cid:101) ν B ) = F ( I ) B a I + F ( J ) B b J D ( b J ) = − F ( J ) A ν A − F ( J ) B (cid:101) ν B (3.48)This interdependence between the fluxes is not what would have been obtained if the relationshipbetween the different superpotentials was that of ζ , the exchange of moduli spaces. Under such atransformation we would have expected one of the Type II theories to have their flux multiplets definedby the covariant derivatives acting a particular H (3 − n,n ) cohomology. To refer to these two differentkinds of ways of reformulating the superpotential we define a pair of operators ζ and ζ by their actionon integrands, where D A and D B are derivatives defined in Type IIB and Type IIA respectively. ζ : (cid:90) W Ω ∧ D A ( (cid:102) c ) ↔ (cid:90) M Ω c ∧ D B ( (cid:102) ) ζ : (cid:90) W Ω ∧ D A ( (cid:102) c ) ↔ (cid:90) M (cid:102) ∧ D B (Ω c )Therefore, on the NS-NS sector we have that Υ = ζ . We refer to the Υ image of a superpotentialas its mirror dual and we distinguish between that and the moduli space exchanging nature of ζ bycalling the ζ image of a superpotential as its moduli dual. Since Type IIB is self S dual its R-R sector is constructed by from the NS-NS sector by the dilatoninversion S → − S . (cid:90) W Ω ∧ D ( (cid:102) c ) → (cid:90) W Ω ∧ D (cid:48) ( (cid:102) (cid:48) c ) = (cid:90) W ˇΩ ∧ G (cid:48) ( ˇ (cid:102) (cid:48) c ) (4.1)The flux components of G (cid:48) follow the same structure as in the G expansion but with fluxes couplingto the dilaton in a different manner and so the hatted and unhatted fluxes are exchanged. G (cid:48) = F + (cid:98) F + (cid:98) F + F = 13! F abc η abc + 12! (cid:98) F abc η bc ι a + 12! (cid:98) F abc η c ι b ι a + 13! F abc ι c ι b ι a (4.2)These components are related to the entries of the pair of flux matrices which determine the action of D on the cohomology bases in the same way G and M were. D (cid:48) (cid:18) f ( a ) f ( ν ) (cid:19) = (cid:18) N (cid:48) M (cid:48) (cid:19) (cid:18) h a h ν (cid:19) (cid:18) f ( a ) f ( ν ) (cid:19) ≡ D (cid:48) · h · f , M (cid:48) ≡ (cid:32)(cid:98) F ( I ) A (cid:98) F B ( I ) (cid:98) F ( J ) A (cid:98) F ( J ) B (cid:33) (4.3)With the relationship between M (cid:48) and N (cid:48) being the same as that between M and N the constructionof the alternative action of D (cid:48) on H (3) follows in a straightforward manner. D (cid:48) ( ν I ) = (cid:98) F ( I ) A a A − (cid:98) F B ( I ) b B D (cid:48) ( (cid:101) ν J ) = (cid:98) F ( J ) A a A − (cid:98) F ( J ) B b B ⇔ D (cid:48) ( a A ) = − (cid:98) F ( I ) A ν I + (cid:98) F A ( J ) (cid:101) ν J D (cid:48) ( b B ) = − (cid:98) F ( I ) B ν I + (cid:98) F ( J ) B (cid:101) ν J (4.4)25 .2 Type IIA Type IIA does not possess a self SL(2 , Z ) S symmetry and so we cannot construct the R-R flux sector bysimply taking the SL(2 , Z ) S partner of the NS-NS sector. The NS-NS superpotential of (3.45) involvesΩ c and if the Type IIA superpotential as a whole is to follow a schematically similar form to the TypeIIB superpotential then the R-R superpotential would be expected to involve Ω (cid:48) c in some way. Since F RR is independent of the complex structure moduli we deduce that in terms of the U it would beassociated to the U modulus, which is not a degree of freedom due to its projective definition. The3-form associated to this complex structure modulus in the expansion of Ω (cid:48) c is α = a and we havethat F RR = U F RR ≡ F · ( U α ), which suggests that F RR is the first term in a more general fluxdependent expression D (cid:48) (Ω (cid:48) c ). (cid:90) M e J ∧ F RR = (cid:90) M (cid:102) ∧ F RR → (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c )This demonstrates that the R-R sector has Υ = ζ , the moduli spaces are exchanged in a manifestway. As in the NS-NS case, we generalise this to include fluxes relating to the action of D (cid:48) on theother H (3) basis elements and it is clear that this expression for the Type IIA R-R superpotential isthe mirror of the Type IIB R-R superpotential (cid:90) M (cid:102) ∧ F RR → (cid:90) M (cid:102) ∧ (cid:16) U F · α − S U i (cid:98) F · α i + S U j (cid:98) F · β j − U F · β (cid:17) (4.5)We have defined the F n and (cid:98) F m by the way in which they couple to the U i in the same way in which F n · J ( n ) contribute O ( T na ) terms to the superpotential in the Type IIB case. This labelling is entirelya choice of notation and has no consequence in terms of the components of the flux matrices whichdefine the action of D (cid:48) . Even when considering an orientifold projection it is possible to work entirelyin terms of which moduli coefficients are included or projected out, rather than determining the actionof the projection on the fluxes themselves. As will be done in a more explicit manner all T and Sduality transformations or induced structures can be written entirely in terms of the flux matrices,allowing us to forgo the issue of considering precisely how to regard these new R-R contributions interms of physical constructions such as branes. This extended R-R superpotential prompts us to definea complex structure counterpart to the K¨ahler forms J ( n ) so that Ω has a simplified expansion similarto (cid:102) , but we use the original bases of the cohomologies. (cid:102) = T ω + T a ω a + T b (cid:101) ω b + T (cid:101) ω = J (0) + J (1) + J (2) + J (3) ⇔ Ω = U α + U i α i − U j β j − U β = J (0) + J (1) − J (2) − J (3) (4.6)We have defined the Type IIA version here but the Type IIB version follows in precisely the samemanner and is something we will make use of in a later section. Using these we are able to reexpressthe Type IIA R-R superpotential so as to make the equality of mirror duality with moduli dualitymore manifest. (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c ) = (cid:90) M (cid:102) ∧ (cid:16) F · + (cid:98) F · + (cid:98) F · + F · (cid:17)(cid:16) J (0) − S J (1) + S J (2) − J (3) (cid:17) (4.7)We define the entries of N (cid:48) in terms of the components of the F by the action of D (cid:48) on the basis elementof H (3) but it must be done in the ( a I , b J , U ) symplectic basis in order to conform to the Type IIBresults. (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c ) = (cid:90) M (cid:102) ∧ (cid:16) U F · a − S U i (cid:98) F · a i + S U j (cid:98) F · b j − U F · b (cid:17) (4.8)Given that the F multiplets are defined as D (cid:48) images of H (3) basis elements they define the rows of N (cid:48) and as in previous cases we absorb F and F into F and F . N (cid:48) ≡ (cid:32)(cid:98) F ( I ) A (cid:98) F B ( I ) (cid:98) F ( J ) A (cid:98) F ( J ) B (cid:33) D (cid:48) images of H (3) elements it is not possible to express D (cid:48) in a componentexpansion of the form previously considered for the NS-NS sector and the Type IIB R-R sector. Thisalso negates being able to easily construct a gauge algebra whose structure constants are the Type IIAR-R fluxes. D (cid:48) = F · + (cid:98) F · + (cid:98) F · + F · (cid:54) = 13! F abc η abc + 13! (cid:98) F abc η bc ι a + 13! (cid:98) F abc η c ι b ι a + 13! F abc ι c ι b ι a (4.9)While the R-R flux multiplets do not lend themselves to a concise manner of expression involving the η τ and ι σ we can use the matrix representation of the derivative to construct an expression for D (cid:48) interms of ι e ( a ) and e ( ν ) . D (cid:48) ( e ( a ) ) = N (cid:48) · h a · e ( ν ) ⇒ D (cid:48) = e (cid:62) ( ν ) · h (cid:62) a · N (cid:48)(cid:62) · ι e ( a ) (4.10)The action of D (cid:48) on elements of (cid:101) H (3) can be constructed from its action on elements of H (3) . D (cid:48) ( a I ) = (cid:98) F ( I ) A ν A + (cid:98) F B ( I ) (cid:101) ν B D (cid:48) ( b J ) = (cid:98) F ( J ) A ν A + (cid:98) F ( J ) B (cid:101) ν B ⇔ D (cid:48) ( ν A ) = (cid:98) F ( I ) A a I − (cid:98) F A ( J ) b J D (cid:48) ( (cid:101) ν B ) = − (cid:98) F ( I ) B a I + (cid:98) F ( J ) B b J (4.11) The explicit relationship between the (cid:98) F and the (cid:98) F , or more specifically M (cid:48) and N (cid:48) , is found in the sameway as the NS-NS sector case, except this time we can use moduli duality directly, without having tochange the argument of the derivative. • Type IIB R-R superpotential. (cid:90) W Ω ∧ D (cid:48) ( (cid:102) (cid:48) c ) = g (cid:16) Ω , D (cid:48) ( (cid:102) (cid:48) c ) (cid:17) = T (cid:62) · h ν · C (cid:48) · M (cid:48) · g a · U (4.12) • Type IIA mirror and moduli dual to the Type IIB R-R superpotential. (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c ) = g (cid:16) (cid:102) , D (cid:48) (Ω (cid:48) c ) (cid:17) = U (cid:62) · h a · C (cid:48) · N (cid:48) · g ν · T (4.13)Comparing these two expressions we can determine M (cid:48) in terms of N (cid:48) . h ν · M (cid:48) · g a = h a · N (cid:48) · g ν ⇒ N (cid:48) = h a · M (cid:48) · h a (4.14)As a result of this we can express (cid:98) F in terms of the (cid:98) F flux matrices. N (cid:48) ≡ (cid:32)(cid:98) F ( I ) A (cid:98) F B ( I ) (cid:98) F ( J ) A (cid:98) F ( J ) B (cid:33) = (cid:32) (cid:98) F ( I ) A − (cid:98) F B ( I ) − (cid:98) F ( J ) A (cid:98) F ( J ) B (cid:33) Putting these results together we obtain the R-R version of (3.48) but in this case it is more convenientto express all fluxes in terms of Type IIB fluxes. D (cid:48) ( ν I ) = (cid:98) F ( I ) A a A − (cid:98) F B ( I ) b B D (cid:48) ( a A ) = − (cid:98) F ( I ) A ν I + (cid:98) F A ( J ) (cid:101) ν J D (cid:48) ( (cid:101) ν J ) = (cid:98) F ( J ) A a A − (cid:98) F ( J ) B b B D (cid:48) ( b B ) = − (cid:98) F ( I ) B ν I + (cid:98) F ( J ) B (cid:101) ν J D (cid:48) ( a I ) = (cid:98) F ( I ) A ν A − (cid:98) F B ( I ) (cid:101) ν B D (cid:48) ( ν A ) = (cid:98) F ( I ) B a I + (cid:98) F A ( J ) b J D (cid:48) ( b J ) = − (cid:98) F ( J ) A ν A + (cid:98) F ( J ) B (cid:101) ν B D (cid:48) ( (cid:101) ν B ) = (cid:98) F ( I ) B a I + (cid:98) F ( J ) B b J (4.15) Thus far we have only considered what fluxes might in principle be required for a flux compactificationwhich is invariant under T and S duality transformations, we have not addressed the issue of theirconsistency constraints. 27 .1 T Duality Constraints
Though T duality alone does not induce additional fluxes in the Type IIA R-R sector we assume thatall fluxes which might be required for full T and S duality invariance are turned on. However, weinitially restrict our analysis to the conditions which T duality induces. This approach allows us toregard the NS-NS sector and the R-R sector as disjoint from one another, even though the R-R sectoris considered to include fluxes which require S duality transformations to be induced. Once we haveexamined the T duality induced structures in terms of these fluxes we shall extend our analysis toinclude SL(2 , Z ) S transformations, thus providing the proper description of the fluxes. D must satisfy Bianchi constraints in order for consistency, which in absense of any fluxes are equivalentto d = 0 and so is trivially satisfied. Turning on the fluxes induced by T duality is equivalent toremoving the closure properties of the cohomology base. As previously discussed the action of F isclosely related to d itself, as they both map p -forms to p + 1-forms and while the non-geometric fluxesdo not have the same equivalent behaviour they too contribute to D . D = (cid:16) d + F ∧ (cid:17) ∼ (cid:16) F ∧ + F · + F · + F · (cid:17) = 0 (5.1)As previously commented, using the component expressions in (3.2) the constraints arising from actingsuch a D on elements of each Λ ( p ) ( M ) are derived in full in [18]. Considering only H ( ∗ ) definedfluxes is insufficient for full nilpotency but it has a number of advantages over the Λ ( ∗ ) analysis; theequivalence of the Bianchi constraints due to G and D in Type IIB is manifest, SL(2 , Z ) S multipletsare constructable in both Type IIB and Type IIA and the analysis of tadpoles can be subsumed intothe analysis of Bianchi constraints. All of these will be derived and examined in this section. If sucha restricted analysis is to be valid then there needs to be a clearcut distinction between light andheavy modes. The light modes are those whose mass is zero in the case where all fluxes are turnedoff, while the heavy are the converse, they are not harmonic even in the fluxless limit. The expressionfor the Laplacian in terms of the fluxes is given in Appendix Section A and since there are two fluxsectors there are two Laplacians, ignoring the issue of S duality induced mixing. The construction ofthe Laplacians follows the same method as the construction of the Bianchi constraints and we firstconsider the Type IIA NS-NS constraints. These can be expressed in two ways, in terms of the fluxcomponents or in terms of flux matrices, the latter of which we consider first. D ( e ) = D · h · D · h · e = (cid:18) M · h a · N · h ν N · h ν · M · h a (cid:19) (cid:18) e ( ν ) e ( a ) (cid:19) (5.2)Using the flux component definition of M and its relationship to N we can expand these two expressionsinto four sets of flux polynomials. D ( ν A ) = (cid:16) F ( A ) I F ( B ) I − F J ( A ) F ( B ) J (cid:17) ν B + (cid:16) F J ( A ) F ( B ) J − F ( A ) I F I ( B ) (cid:17)(cid:101) ν B D ( (cid:101) ν B ) = (cid:16) F ( B ) I F ( A ) I − F ( B ) J F ( A ) J (cid:17) ν A + (cid:16) F ( B ) J F ( A ) J − F ( B ) I F I ( A ) (cid:17)(cid:101) ν A D ( a I ) = (cid:16) F ( A ) I F ( A ) J − F I ( B ) F ( B ) J (cid:17) a J + (cid:16) F I ( B ) F ( B ) J − F ( B ) I F J ( B ) (cid:17) b J D ( b J ) = (cid:16) F ( A ) J F ( A ) I − F ( B ) J F ( B ) I (cid:17) a I + (cid:16) F ( B ) J F ( B ) I − F ( A ) J F I ( A ) (cid:17) b I (5.3)The nilpotency of D is therefore expressible in terms of an ideal whose generating functions are thesecomponents of ( D · h ) . (cid:104)D (cid:105) = (cid:104) M · h a · N, N · h ν · M (cid:105) We have neglected the ‘external’ factor of h as it is non-degenerate and therefore does not alter theideal the constraints generate. The Type IIB case for D follows in the same manner, defining four sets28f flux component constraints. D ( ν I ) = (cid:16) F ( I ) A F ( J ) A − F B ( I ) F ( J ) B (cid:17) ν J + (cid:16) F B ( I ) F ( J ) B − F ( I ) A F A ( J ) (cid:17)(cid:101) ν J D ( (cid:101) ν J ) = (cid:16) F ( J ) A F ( I ) A − F ( J ) B F ( I ) B (cid:17) ν I + (cid:16) F ( J ) B F ( I ) B − F ( J ) A F A ( I ) (cid:17)(cid:101) ν I D ( a A ) = (cid:16) F ( I ) A F ( I ) B − F A ( J ) F ( J ) B (cid:17) a B + (cid:16) F A ( J ) F ( J ) B − F ( J ) A F B ( J ) (cid:17) b B D ( b B ) = (cid:16) F ( I ) B F ( I ) A − F ( J ) B F ( J ) A (cid:17) a A + (cid:16) F ( J ) B F ( J ) A − F ( I ) B F A ( I ) (cid:17) b A (5.4)Likewise we have a pair of flux matrix dependent expressions. (cid:104) D (cid:105) = (cid:104) M · h a · N , N · h ν · M (cid:105) Due to the manner in which the Type IIB fluxes couple to the K¨ahler moduli in a way different tothat of the Type IIA fluxes we had to define two different flux dependent expressions, G and D . Byconstruation G ’s nilpotency constraints were equal to the constraints of the Type IIB gauge algebrabut this is not automatically true for D . By comparing the superpotentials of (3.38), which followedfrom these two different expressions, we obtained a flux matrix of D in terms of a flux matrix of G . M = L (cid:62) · G · g a · K · g (cid:62) a (5.5)Rather than construct their flux matrix partners it is more convenient to express the Bianchi constraintsin terms of single flux matrix by using (3.41), thus allowing us to construct the ideals of (cid:104) D (cid:105) and (cid:104) G (cid:105) in terms of M and G respectively. (cid:104) D (cid:105) = (cid:104) M · g a · M (cid:62) , M (cid:62) · g a · M (cid:105) , (cid:104) G (cid:105) = (cid:104) G · g a · G (cid:62) , G (cid:62) · g a · G (cid:105) We have once again droped irrelevant factors of non-degenerate matrices since they do not changethe ideal. Considering each case in turn and not yet making any assumption about the specific formof L and K we can derive the constraints they must satisfy for D and G to have equivalent Bianchiconstraints on the H ( ∗ ) . (cid:104) M · g a · M (cid:62) (cid:105) = (cid:104) G · g a · K · g a · K (cid:62) · g a · G (cid:62) (cid:105) ⇒ K · g a · K (cid:62) = g a (cid:104) M (cid:62) · g a · M (cid:105) = (cid:104) G (cid:62) · L · g a · L (cid:62) · G (cid:105) ⇒ L · g a · L (cid:62) = g a (5.6)Therefore both L and K are sympletic matrices as they preserve the canonical sympletic form. This isfurther restricted by recalling that we required C to commute with L and in order to treat the modulispaces in the same manner we require K to commute with (cid:101) C . Since the existence of L is motivated byexchanging the T i and T j moduli in the construction of the superpotential we set the action L to notalter the superpotential’s T and T terms. This is further justified by the fact C treats those modulidifferently, coupling them to the dilaton. Thus we can make a general ansatz for L and K . L = L L , K = K K (5.7)The sympletic constraints now reduce to skew-orthogonality of the submatrixes, L · L (cid:62) = − I and K · K (cid:62) = − I . The simplest specific solution is L = − L = I which has the effect J (1) = T i ω i →−T j ω j = J c , the standard definition used for the K¨ahler 4-form. Provided we frame our analysis ofthe fluxes and their duality induced structures in terms of the flux matrices of D we are not required touse specific L i or K j as we can construct the polynomial expressions for the Type IIB superpotentialvia D in the same manner as we did for the Type IIA case. This greatly simplifies the analysis of theType IIB theory, as everything can be written in terms of M rather than G . With this knowledge wenow turn to comparing the nilpotency constraints of the fluxes of Type IIA to those of Type IIB.29he generic problem of comparing the equivalence of the Type IIA and Type IIB nilpotencyconstraints reduces to comparing the ideals defined by the associated polynomials.Type IIA : I A,N = (cid:104)D (cid:105) I A,R = (cid:104)D (cid:48) (cid:105) Type IIB : I B,N = (cid:104) D (cid:105) I B,R = (cid:104) D (cid:48) (cid:105) (5.8)To confirm that the constraints are equivalent we use (3.47) and (4.14) to convert the flux matrixexpressions between the Type II constructions. I B,N = (cid:68) N · h ν · MM · h a · N (cid:69) → (cid:68) h a · (cid:16) M · h a · N (cid:17) · h ν − h a · (cid:16) N · h ν · M (cid:17) · h ν (cid:69) = I A,N I B,R = (cid:68) N (cid:48) · h ν · M (cid:48) M (cid:48) · h a · N (cid:48) (cid:69) → (cid:68) h a · (cid:16) M (cid:48) · h a · N (cid:48) (cid:17) · h ν − h ν · (cid:16) N (cid:48) · h ν · M (cid:48) (cid:17) · h a (cid:69) = I A,R (5.9)In (5.6) we used the relationship between N and M to express the Bianchi constraints in terms of asingle flux matrix and this can be done for the other flux sectors and in both Type IIA or Type IIB. (cid:104) D (cid:105) ⇔ (cid:40) (cid:104) M · h a · N (cid:105) ⇒ (cid:110) (cid:104) M · g a · M (cid:62) (cid:105) (cid:104) N (cid:62) · g a · N (cid:105)(cid:104) N · g a · N (cid:62) (cid:105) (cid:104) M (cid:62) · g a · M (cid:105)(cid:104) N · h ν · M (cid:105) ⇒ (cid:110) (cid:104) M (cid:62) · g a · M (cid:105) (cid:104) N · g a · N (cid:62) (cid:105)(cid:104) N (cid:62) · g a · N (cid:105) (cid:104) M · g a · M (cid:62) (cid:105) (5.10)The expression M · g a · M (cid:62) suggests the nilpotency constraints can be rephrased to be in terms ofan integrand defined by a pair of exact 3-forms, with g a implying the forms are defined in the H (3) of Type IIA. To examine this we use a pair of vectors χ and ϕ of dimension 2 h , + 2 and a pair ofvectors φ and ψ of dimension 2 h , + 2, which then allow us to define sets of forms in either Type IItheory. IIA (cid:110) χ ≡ χ (cid:62) · h ν · e ( ν ) ↔ (cid:101) χ ≡ χ (cid:62) · h a · f ( a ) φ ≡ φ (cid:62) · h a · e ( a ) ↔ (cid:101) φ ≡ φ (cid:62) · h ν · f ( ν ) (cid:111) IIB (5.11)In order to obtain expressions which are quadratic in M we consider the pair of exact 3-forms D ( φ )and D ( ϕ ) and their skew-inner product in Type IIA. (cid:90) M D ( χ ) ∧ D ( ϕ ) = g a (cid:16) D ( χ ) , D ( ϕ ) (cid:17) = ϕ (cid:62) · h ν · M · g a · M (cid:62) · h ν · χ (5.12)Therefore the first set of constraints on the fluxes can be reexpressed as the vanishing of the skew-innerproduct of any two D exact elements of the Type IIA H (3) , with the R-R sector following the samemethod, but with D (cid:48) in the place of D . g ν (cid:16) χ, D (cid:0) D ( ϕ ) (cid:1)(cid:17) = 0 ⇔ g a (cid:16) D ( χ ) , D ( ϕ ) (cid:17) = 0Since the bilinear form in each equation of (5.10) is g a or g a it is clear that the constraints can onlybe rephrased in terms of exact forms belonging to H (3) in either Type IIA or Type IIB. Therefore theonly other combination we need consider is that of the exact forms defined by the D images of theType IIB (cid:101) H (3) elements (cid:101) φ and (cid:101) ψ . (cid:90) W D ( (cid:101) φ ) ∧ D ( (cid:101) ψ ) = g a (cid:16) D ( (cid:101) φ ) , D ( (cid:101) ψ ) (cid:17) = (cid:101) ψ (cid:62) · h ν · M · g a · M (cid:62) · h ν · (cid:101) φ (5.13)Therefore the first set of constraints on the fluxes can be reexpressed as the vanishing of the skew-innerproduct of any two D exact elements of the Type IIB H (3) , with the R-R sector following the samemethod, but with D (cid:48) in the place of D . g ν (cid:16) (cid:101) φ, D (cid:0) D ( (cid:101) ψ ) (cid:1)(cid:17) = 0 ⇔ g a (cid:16) D ( (cid:101) φ ) , D ( (cid:101) ψ ) (cid:17) = 0 (5.14)30herefore we can write nilpotency defined ideals in terms of orthogonal exact form defined ideals. (cid:104)D (cid:105) = (cid:68) g a (cid:16) D ( · ) , D ( · ) (cid:17) , g a (cid:16) D ( · ) , D ( · ) (cid:17)(cid:69) = (cid:104) D (cid:105)(cid:104)D (cid:48) (cid:105) = (cid:68) g a (cid:16) D (cid:48) ( · ) , D (cid:48) ( · ) (cid:17) , g a (cid:16) D (cid:48) ( · ) , D (cid:48) ( · ) (cid:17)(cid:69) = (cid:104) D (cid:48) (cid:105) (5.15) Given M is six dimensional the Type IIA models can have non-zero potentials of the form C , C and C but due to the emptiness of H (1) and H (5) for M the only non-trivial case can be C , whose tadpolecontributions arise from 3-form flux combinations. It is known [3, 18] that the tadpole expression forspaces with only F RR R-R fluxes is, up to proportionality factors, measured by D ( F RR ). In our analysisof the Type IIA flux sector we noted that F RR could be written as an exact derivative in terms of D (cid:48) and so the tadpole term can be written as a quadratic derivative. F RR ≡ F · α = D (cid:48) ( α ) = D (cid:48) ( a ) ⇒ D ( F RR ) = DD (cid:48) ( a ) (5.16)The fact F RR could be written in this way provided a natural extension of the sector to include other F fluxes and as a result we are able to form additional 3-forms which have the schematic form of beinga D derivative of a D (cid:48) exact form. D ( F · α ) = DD (cid:48) ( α ) = + DD (cid:48) ( a ) , D ( (cid:98) F · α a ) = DD (cid:48) ( α a ) = −DD (cid:48) ( b a ) D ( F · β ) = DD (cid:48) ( β ) = + DD (cid:48) ( b ) , D ( (cid:98) F · β b ) = DD (cid:48) ( β b ) = + DD (cid:48) ( a b ) (5.17)Using (3.48) and (4.15) we can determine the component form of these expressions in terms of theType IIB fluxes. DD (cid:48) ( a I ) = − (cid:16) (cid:98) F ( I ) A F ( J ) A + (cid:98) F B ( I ) F ( J ) B (cid:17) a J − (cid:16) (cid:98) F ( I ) A F A ( J ) + (cid:98) F B ( I ) F ( J ) B (cid:17) b J DD (cid:48) ( b I ) = (cid:16) (cid:98) F ( I ) A F ( J ) A + (cid:98) F ( I ) A F ( J ) A (cid:17) a J + (cid:16) (cid:98) F ( I ) A F ( J ) A + (cid:98) F ( I ) B F B ( J ) (cid:17) b J (5.18)All of these expressions are constructed from applying DD (cid:48) to 3-forms and since they are also expandedin terms of the H (3) basis it follows that all of these expressions can contribute tadpoles. The fluxes of F RR have a straightforward definition in terms of the D-brane content of Type IIA but the remainingcases, DD (cid:48) ( a i ) and DD (cid:48) ( b J ), do not. The F RR tadpole contributions can be decomposed as a I ∧DD (cid:48) ( a )and b J ∧ DD (cid:48) ( a ) and the physical interpretation is the D6-brane wrapping particular 3-cycles in W .In order to justify considering these expressions as new tadpole contributions we also construct theequivalent expressions for D (cid:48) D . D (cid:48) D ( a I ) = (cid:16) F ( I ) A (cid:98) F ( J ) A + F B ( I ) (cid:98) F ( J ) B (cid:17) a J + (cid:16) F ( I ) A (cid:98) F A ( J ) + F B ( I ) (cid:98) F ( J ) B (cid:17) b J D (cid:48) D ( b I ) = − (cid:16) F ( I ) A (cid:98) F ( J ) A + F ( I ) A (cid:98) F ( J ) A (cid:17) a J − (cid:16) F ( I ) A (cid:98) F ( J ) A + F ( I ) B (cid:98) F B ( J ) (cid:17) b J (5.19)By considering (5.18) and (5.19) we observe a number of identities relating the individual coefficientswhich allows us to link this standard tadpole to the new expressions. We use the relabellings D = D and D (cid:48) = D for simplicity. a I ∧ D n D m ( a J ) = − a J ∧ D m D n ( a I ) a I ∧ D n D m ( b J ) = − b J ∧ D m D n ( a I ) b I ∧ D n D m ( b J ) = − b J ∧ D m D n ( b I ) (5.20)In order to examine the D ( F RR ) expression further it is convenient to use the projection-like propertiesof ι a I and ι b J because they project out a 0-form rather than a 6-form. a I ∧ D n D m ( a J ) = − a J ∧ D m D n ( a I ) → ι b I D n D m ( a J ) = − ι b J D m D n ( a I ) a I ∧ D n D m ( b J ) = − b J ∧ D m D n ( a I ) → ι b I D n D m ( b J ) = + ι a J D m D n ( a I ) b I ∧ D n D m ( b J ) = − b J ∧ D m D n ( b I ) → ι a I D n D m ( b J ) = − ι a J D m D n ( b I ) (5.21)31sing these and the fact that for 3-forms 1 = a I ι a I + b J ι b J we can reexpress the F RR tadpole in a newway. D ( F RR ) = DD (cid:48) ( a ) = a I ι a I DD (cid:48) ( a ) + b J ι b J DD (cid:48) ( a ) = − a I ι a D (cid:48) D ( a I ) + b J ι a D (cid:48) D ( b J ) (5.22)Though we are not explicitly considering S duality transformations yet it follows from the dilatoninversion that D and D (cid:48) should be treated in the same manner, though not the flux multiplets whichdefine their components. With the fluxes of DD (cid:48) ( a ) also contributing to particular D (cid:48) D images ofall H (3) basis forms it follows that all coefficients of the H (3) basis elements in (5.18) are tadpoles,not just those corresponding to DD (cid:48) ( a ). The physical interpretation of these tadpoles is not clearas the fluxes which they are defined in terms of are induced by the inclusion of SL(2 , Z ) S symmetryto the Type IIA superpotential. Furthermore, we have not considered a fully consistent picture ofthese tadpoles since we are assuming the non-zero nature of fluxes which are not required for onlyT duality invariance; under modular transformations of the dilaton we would expect the tadpoles toform SL(2 , Z ) S multiplets, mixing DD (cid:48) and D (cid:48) D terms. We defer such an analysis for the time beingand instead consider the Type IIB tadpoles.The Type IIB tadpoles are those flux combinations which couple to the C n potentials living onthe D-branes of Type IIB. Following the same principle as the Type IIA case possible contributions tosuch tadpoles can be found by considering how a 10 − n form can be contructed to couple to C n . C has a tadpole contribution H ∧ F → (cid:98) F ∧ F which we immediately note can be written in a numberof ways. (cid:98) F ∧ F = (cid:26) D ( F ) = DD (cid:48) ( ν ) − D (cid:48) ( (cid:98) F ) = − D (cid:48) D ( ν ) (cid:27) = 12 ( DD (cid:48) − D (cid:48) D )( ν ) (5.23)The flux component version of this tadpole follows from the known actions of D and D (cid:48) . DD (cid:48) ( ν I ) = (cid:16) (cid:98) F ( I ) A F ( J ) A − (cid:98) F B ( I ) F ( J ) B (cid:17) ν J + (cid:16) (cid:98) F B ( I ) F ( J ) B − (cid:98) F ( I ) A F A ( J ) (cid:17)(cid:101) ν J DD (cid:48) ( (cid:101) ν J ) = (cid:16) (cid:98) F ( J ) A F ( I ) A − (cid:98) F ( J ) B F ( I ) B (cid:17) ν I + (cid:16) (cid:98) F ( J ) B F ( I ) B − (cid:98) F ( J ) A F A ( I ) (cid:17)(cid:101) ν I (5.24)The C tadpole’s flux polynomial expression is the (cid:101) ν component of DD (cid:48) ( ν I ), which can be projectedout by ν ∧ or ι e ν . C : (cid:98) F B (0) F (0) B − (cid:98) F (0) A F A (0) (5.25)The C case requires a 4-form, which is F · F for the T duality only case. Since the fluxes of F coupleto T j = T i the F B ( i ) and F ( i ) A are the components of F and so the corresponding terms in (5.24) areseen to be the coefficients of (cid:101) ν j in DD (cid:48) ( ν ). C : (cid:98) F B (0) F ( j ) B − (cid:98) F (0) A F A ( j ) (5.26)Due to the different way in which F couples in a tadpole compared too a Bianchi constraint theseterms are not the coefficients of a 4-form, but a 2-form. However, the physical interpretation canstill be viewed in terms of D5-branes wrapping cycles; the D5 wraps a 2-cycle A i and the associatedtadpole is obtained by integrating DD (cid:48) ( ν ) over its dual 4-cycle B i . The C and C follow in the samemanner, with C tadpoles being defined on the dual cycles to the wrapped D7.As in the Type IIA case these expressions only account for a small number of the possible fluxpolynomials of (5.24). Each of the C n we have considered has had tadpoles of the form D ( F )but the inclusion of other R-R fluxes allows for other possibilities. They do not naturally form p -forms alone, such as F · (cid:98) F · not being a p -form unless it acts on an element of Λ ( p ≥ . Such a fluxcombination appears in (5.24), formed by acting DD (cid:48) on (cid:101) ν i . The physical interpretation of these fluxesand the objects they live on we shall not address, instead we restrict our discussion to how these fluxpolynomials are constructed and transform under the dualities we are considering. In that regard by32onstructing the D (cid:48) D expressions and comparing with those of DD (cid:48) we obtain a set of identities whichare the Type IIB version of (5.28). D (cid:48) D ( ν I ) = (cid:16) F ( I ) A (cid:98) F ( J ) A − F B ( I ) (cid:98) F ( J ) B (cid:17) ν J + (cid:16) F B ( I ) (cid:98) F ( J ) B − F ( I ) A (cid:98) F A ( J ) (cid:17)(cid:101) ν J D (cid:48) D ( (cid:101) ν J ) = (cid:16) F ( J ) A (cid:98) F ( I ) A − F ( J ) B (cid:98) F ( I ) B (cid:17) ν I + (cid:16) F ( J ) B (cid:98) F ( I ) B − F ( J ) A (cid:98) F A ( I ) (cid:17)(cid:101) ν I (5.27)Relabelling the derivatives as D = D and D (cid:48) = D we can generalise the results to consider the D i D j actions on all elements of (cid:101) H (3) . ν I ∧ D n D m ( ν J ) = − ν J ∧ D m D n ( ν I ) → ι e ν I D n D m ( ν J ) = − ι e ν J D m D n ( ν I ) ν I ∧ D n D m ( (cid:101) ν J ) = + (cid:101) ν J ∧ D m D n ( ν I ) → ι e ν I D n D m ( (cid:101) ν J ) = + ι ν J D m D n ( ν I ) (cid:101) ν I ∧ D n D m ( (cid:101) ν J ) = − (cid:101) ν J ∧ D m D n ( (cid:101) ν I ) → ι ν I D n D m ( (cid:101) ν J ) = − ι ν J D m D n ( (cid:101) ν I ) (5.28) Type IIB supergravity is known to have a self dual SL(2 , R ) symmetry at the ten dimensional levelwhich is broken by compactification down to the quantised SL(2 , Z ) subgroup but this is not sharedby Type IIA, as we have seen in its asymmetric treatment of the two flux sectors. However, we areable to do much of our analysis of S duality in the Type IIB side and then convert the results, usingthe relationships between the IIA and IIB fluxes summarised in ( ?? ), into the equivalent Type IIAresults. As a result, unless otherwise stated this section is entirely within Type IIB. We shall denotea general SL(2 , Z ) S transformation on S by Γ S , which dependent on the context we will take to beeither a matrix on some flux doublet or a M¨obius transformation on S itself. S → S (cid:48) = Γ S ( S ) ≡ aS + bcS + d ⇔ Γ S ≡ (cid:18) a bc d (cid:19) (5.29)SL(2 , Z ) has two generators, which we choose to be those related to the moduli transformations Γ : S → − S and Γ : S → S + 1. Γ = (cid:18) −
11 0 (cid:19) , Γ = (cid:18) (cid:19) To examine the flux structures induced by S duality we define a pair of matrices, F (cid:58) and (cid:98) F (cid:58) , in termsof matrices A n and B m previously defined in relation to complexification matrices. F (cid:58) = A · M + B · M (cid:48) (cid:98) F (cid:58) = A · M (cid:48) + B · M (5.30)These definitions can be written in terms of flux matrix doublets. (cid:32) F (cid:58) (cid:98) F (cid:58) (cid:33) = (cid:18) A BB A (cid:19) (cid:18) MM (cid:48) (cid:19) ⇒ (cid:18) MM (cid:48) (cid:19) = (cid:18) A BB A (cid:19) (cid:32) F (cid:58) (cid:98) F (cid:58) (cid:33) (5.31)The superpotential can then be rewritten in a way such that its dilaton dependence is manifest. W = T (cid:62) · (cid:102) (cid:58) · (cid:16) C · M + C (cid:48) · M (cid:48) (cid:17) · g · Ω (cid:58) · U = T (cid:62) · (cid:102) (cid:58) · (cid:16) F (cid:58) − S (cid:98) F (cid:58) (cid:17) · g · Ω (cid:58) · U If the superpotential is to be invariant then F (cid:58) − S (cid:98) F (cid:58) must be invariant, up to a gauge choice, and aswith the individual flux multiplets F n and (cid:98) F n these two matrices transform as an SL(2 , Z ) S doublet. S → aS + bcS + d ⇒ (cid:32) F (cid:58) (cid:98) F (cid:58) (cid:33) → (cid:18) a bc d (cid:19) (cid:32) F (cid:58) (cid:98) F (cid:58) (cid:33) , Z ) S transformation properties of the D flux matrices follow from this. (cid:18) MM (cid:48) (cid:19) → (cid:18) A BB A (cid:19) (cid:18) a bc d (cid:19) (cid:18)
A BB A (cid:19) (cid:18) MM (cid:48) (cid:19) (5.32)= (cid:20)(cid:18) a bc d (cid:19) ⊗ A + (cid:18) d cb a (cid:19) ⊗ B (cid:21) (cid:18) MM (cid:48) (cid:19) = (cid:16) Γ S ⊗ A + (cid:0) σ · Γ S · σ (cid:1) ⊗ B (cid:17) (cid:18) MM (cid:48) (cid:19) If M transforms as M → m · M , where m is a matrix that commutes with both g a and g ν then thecorresponding transformation on N is N → N · m (cid:62) . Both A and B satisfy this and are also symmetric,allowing us to express the SL(2 , Z ) S transformations on each flux matrix in similar ways. (cid:0) N N (cid:48) (cid:1) → (cid:0) N N (cid:48) (cid:1) (cid:16) Γ (cid:62) S ⊗ A + (cid:0) σ · Γ (cid:62) S · σ (cid:1) ⊗ B (cid:17) = (cid:0) N N (cid:48) (cid:1) (cid:18)
A BB A (cid:19) (cid:18) a bc d (cid:19) (cid:62) (cid:18)
A BB A (cid:19) (5.33)Thus far we have only considered the constraints which arise from the nilpotency of the two flux sectorsseperately, as without S duality the two sectors can be viewed as disjoint. Under S duality transforma-tions the flux matrices of the derivatives mix and thus the four quadratic derivative expressions usedto define Bianchi and tadpole constraints also mix.
The analysis of the two Type II theories is much the same when working on the level of the fluxmatrices, as opposed to working with the flux multiplets, and so we shall restrict our discussion to theType IIB case, D and D (cid:48) . (cid:104) D (cid:105) = (cid:104) M · h a · N , N · h ν · M (cid:105) (cid:104) DD (cid:48) (cid:105) = (cid:104) M (cid:48) · h a · N , N (cid:48) · h ν · M (cid:105)(cid:104) D (cid:48) (cid:105) = (cid:104) M (cid:48) · h a · N (cid:48) , N (cid:48) · h ν · M (cid:48) (cid:105) (cid:104) D (cid:48) D (cid:105) = (cid:104) M · h a · N (cid:48) , N · h ν · M (cid:48) (cid:105) (5.34)It is noteworthy that due to their linear independence, relationship A + B = I and projection-likemultiplicative action we can use A and B to decompose any matrix into four disjoint submatrices. X = ( A + B ) · X · ( A + B ) = A · X · A + A · X · B + B · X · A + B · X · B (5.35)We first consider quadratic derivative actions of the same schematic form as D : (cid:101) H (3) → (cid:101) H (3) andas in previous sections we are using the Z × Z orientifold as an explicit example for our analysis.Previous work [21, 22] on the behaviour of the fluxes in this space under S duality has discussedcombinations of geometric and non-geometric fluxes, in both sectors, which transform as either tripletsor singlets of SL(2 , Z ) S . An example of a triplet is seen in the case of the non-geometric fluxes, Q abc ∼ Q and P abc ∼ P , by considering the Bianchi constraints due to T duality under SL(2 , Z ) S . Under T dualityonly, the NS-NS non-geometric flux satisfies the Lie algebra inspired Jacobi constraint Q · Q = 0 and thecorresponding R-R flux P satisfies the same kind of constraint P · P = 0. Separately these expressionsform ideals (cid:104) Q · Q (cid:105) and (cid:104) P · P (cid:105) but since these two fluxes transform as an SL(2 , Z ) S doublet theseideals are not closed under S duality. Under the inversion S → − S the generating functions of thesetwo ideals are exchanged and therefore under S duality they must belong to the same ideal, along witha third flux combination Q · P + P · Q in order to make the ideal closed under SL(2 , Z ) S . (cid:18) QP (cid:19) → (cid:18) a bc d (cid:19) (cid:18) QP (cid:19) = (cid:18) aQ + bPcQ + dP (cid:19) ⇒ (cid:104) Q · Q (cid:105)(cid:104) P · P (cid:105) → (cid:28) Q · Q , P · PQ · P + P · Q (cid:29) (5.36)To begin we consider the flux matrix expressions associated to Q · Q ∼ F · F ; the NS-NS sectorconstraints M · h a · N . M · h a · N = (cid:0) M M (cid:48) (cid:1) (cid:18) I
00 0 (cid:19) (cid:18) h a h a (cid:19) (cid:18) NN (cid:48) (cid:19) (5.37)34his scalar product notation is not required to consider the SL(2 , Z ) S image of M · h a · N but itis convenient for more general expressions encountered later. The SL(2 , Z ) S image we arrange inaccordance with the A , B inspired decomposition of (5.35) and for less cluttered notation use (cid:5) ≡ · h a · . M · h a · N → A · ( a M + b M (cid:48) ) (cid:5) ( a N + b N (cid:48) ) · A + A · ( a M + b M (cid:48) ) (cid:5) ( d N + c N (cid:48) ) · B + B · ( d M + c M (cid:48) ) (cid:5) ( a N + b N (cid:48) ) · A + B · ( d M + c M (cid:48) ) (cid:5) ( d N + c N (cid:48) ) · B (5.38)The corresponding SL(2 , Z ) S image of M (cid:48) (cid:5) N (cid:48) , M (cid:5) N (cid:48) and M (cid:48) (cid:5) N is given in the Appendix as (C.3-C.5).Two transformations of particular note are Γ S = I and Γ S = Γ . The former implies that the NS-NSderivative is still nilpotent and the latter implies that the R-R derivative is also nilpotent.Γ ( M (cid:5) N ) = ( A − B ) · M (cid:48) (cid:5) N (cid:48) · ( A − B )Though Γ has not mapped the NS-NS constraint matrix exactly into the R-R version due to thechange of sign on B the linear independence of the submatrices make this irrelevant and with thesetwo conditions (5.38) can be reduced down to only terms which mix the two sectors. M · h a · N → A · (cid:16) ab M (cid:5) N (cid:48) + ab M (cid:48) (cid:5) N (cid:17) · A + A · (cid:16) ac M (cid:5) N (cid:48) + bd M (cid:48) (cid:5) N (cid:17) · B + B · (cid:16) bd M (cid:5) N (cid:48) + ac M (cid:48) (cid:5) N (cid:17) · A + B · (cid:16) cd M (cid:5) N (cid:48) + cd M (cid:48) (cid:5) N (cid:17) · B (5.39)Comparing this expression with the similarly reduced form of (C.3) we note that in each case theSL(2 , Z ) S integers factorise out as overall factors in both the A · X · A and B · X · B terms. Using thispre- and post-multiplication by A or B we can project out particular parts of (5.38) to form SL(2 , Z ) S multiplets. AA ≡ (cid:104) A · M (cid:5) N · A , A · M (cid:48) (cid:5) N (cid:48) · A , A · ( M (cid:48) (cid:5) N + M (cid:5) N (cid:48) ) · A (cid:105) BB ≡ (cid:104) B · M (cid:5) N · B , B · M (cid:48) (cid:5) N (cid:48) · B , B · ( M (cid:48) (cid:5) N + M (cid:5) N (cid:48) ) · B (cid:105) (5.40)Schematically the action of A is to project out the components associated with F and F , while B does the opposite, projecting out the (cid:98) F and (cid:98) F fluxes. As such we would expect the triplet of (5.36)to be associated to a triplet of the form AA . The components of (5.38) yet to be put into a multipletare the B · X · A and B · X · A terms and by considering the SL(2 , Z ) S integers for these parts we canconstruct another pair of triplets associated to these components. AB ≡ (cid:104) A · M (cid:48) (cid:5) N · B , A · M (cid:5) N (cid:48) · B , A · ( M (cid:5) N + M (cid:48) (cid:5) N (cid:48) ) · B (cid:105) BA ≡ (cid:104) B · M (cid:48) (cid:5) N · A , B · M (cid:5) N (cid:48) · A , B · ( M (cid:5) N + M (cid:48) (cid:5) N (cid:48) ) · A (cid:105) (5.41)This is not sufficient for full SL(2 , Z ) S invariance, for each of the terms in (5.35) we have constructedour triplets from the components of four terms, M (cid:5) N , M (cid:48) (cid:5) N , M (cid:5) N (cid:48) , M (cid:48) (cid:5) N (cid:48) . We have seen how alinear combination of these terms makes a triplet and so we would expect it to be possible to build asinglet to go along with each of these triplets. An explicit example in the form of an SL(2 , Z ) S singletof the Z × Z orientifold discussed in [21, 22] has the schematic form Q · H − P · F . This we wouldassociate it with the terms of the form A · X · B . By considering the triplets of (5.40) and (5.41) andusing (C.2-C.5) we can straightforwardly construct the four singlets associated to the terms of (5.35). AA ≡ (cid:104) A · ( M (cid:5) N (cid:48) − M (cid:48) (cid:5) N ) · A (cid:105) BB ≡ (cid:104) B · ( M (cid:5) N (cid:48) − M (cid:48) (cid:5) N ) · B (cid:105) AB ≡ (cid:104) A · ( M (cid:5) N − M (cid:48) (cid:5) N (cid:48) ) · B (cid:105) BA ≡ (cid:104) B · ( M (cid:5) N − M (cid:48) (cid:5) N (cid:48) ) · A (cid:105) (5.42)Given the four possible combinations of the flux matrices and the four terms arising from the A , B induced decomposition the multiplets of (5.40-5.42) make up all possible SL(2 , Z ) S multiplets. How-ever, not all of these expressions are independent nor are they all Bianchi constraints. Only those35xpressions which are an SL(2 , Z ) S image of a T duality Bianchi constraint are S duality constraintsand not all expressions defining the multiplets of (5.40-5.42) are of this form. To use a more explicitexample, the Z × Z triplet of (5.36) is formed from particular linear combinations of the four terms Q · Q , P · Q , Q · P and P · P and so there exists an SL(2 , Z ) S singlet which is the complement of thistriplet and which does not arise as a Bianchi constraint, namely Q · P − P · Q . In the ( H ( ∗ ) , H ( ∗ ) ∗ ) basisthis would belong to the singlet AA . The triplet can be regarded as being the SL(2 , Z ) S image of theterm Q · Q , a pairing of two NS-NS fluxes and only pairwise combinations of two NS-NS fluxes will giveT duality constraints if the R-R sector is turned off. In the flux matrix definition of AA there is noterm which can be decomposed into a pair of NS-NS flux terms, all of the terms are a mixture of NS-NSand R-R fluxes. Therefore AA does not arise as an SL(2 , Z ) S image of a T duality Bianchi constraintand we have no apriori reason to expect it to be zero. The case of the singlet Q · H − P · F ∈ AB does not follow in precisely the same manner, as AB has a term with M (cid:5) N in it also. In this casewe note that QF ∈ A · M (cid:5) N (cid:48) · B , a known tadpole condition in the T duality only case and thus AB cannot be set to zero. Also following on from our previous discussion of the tadpole expressions wecan deduce that n AB ∼ = n BA and so we can split these multiplets into two different categories; thosewhose generating functions are set to zero by Bianchi contraints and those whose generating functionsare equal to integers which define the number and type of charged extended objects in the space.Bianchi Tadpole AA AA BB BB AB = BA BA = AB (5.43)We now turn our attention to the action of the derivatives of general form D : (cid:101) H (3) → (cid:101) H (3) whereunlike the quadratic action on H (3) the A and B are ‘internal’ to the flux matrix expressions, ratherthan projecting out linearly independent sections of the constraints. A result of this is that the inducedtransformations are not as straightforward and as in the M · h a · N case we find it convenient to expressthe flux matrix nilpotency expressions as scalar products. N · h ν · M = (cid:0) N N (cid:48) (cid:1) (cid:18) I
00 0 (cid:19) (cid:18) h ν h ν (cid:19) (cid:18) MM (cid:48) (cid:19) N (cid:48) · h ν · M (cid:48) = (cid:0) N N (cid:48) (cid:1) (cid:18) I (cid:19) (cid:18) h ν h ν (cid:19) (cid:18) MM (cid:48) (cid:19) (5.44)In order to simplify our expressions we note that h ν = I and so no longer explictly write it and in whatfollows factors of I are surpressed. For a general combination of pairs of flux matrices we can define arelated quadratic form, X , and we will consider how this transforms under SL(2 , Z ) S . p N · M + q N · M (cid:48) + r N (cid:48) · M + s N (cid:48) · M (cid:48) ≡ (cid:0) N N (cid:48) (cid:1) · X · (cid:18) MM (cid:48) (cid:19) (5.45)We can construct the transformation properties of X by using (5.33) and (5.33) and since the entriesare scalar multiples of I the A and B terms decouple. X ≡ (cid:18) p qr s (cid:19) → (cid:16) Γ (cid:62) S ⊗ A + (cid:0) σ · Γ (cid:62) S · σ (cid:1) ⊗ B (cid:17) (cid:18) p qr s (cid:19) (cid:16) Γ S ⊗ A + (cid:0) σ · Γ S · σ (cid:1) ⊗ B (cid:17) = (cid:18) Γ (cid:62) S (cid:18) p qr s (cid:19) Γ S (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Ξ A ⊗A + (cid:18)(cid:0) σ · Γ (cid:62) S · σ (cid:1) (cid:18) p qr s (cid:19) (cid:0) σ · Γ S · σ (cid:1)(cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Ξ B ⊗B (5.46)Proceeding as before we wish to obtain an SL(2 , Z ) S triplet by considering the image of the T dualityconstraints and also combination of terms which form a singlet. Due to the splitting of X by A and B the equations which are satisfied by a singlet reduce to Ξ A = X = Ξ B , so by using the fact thatany element of SL(2 , R ) is a symplectic matrix and both Ξ A and Ξ B are of the form m (cid:62) · X · m it36ollows that if X is the canonical symplectic form then the equations Ξ A = X = Ξ B are automaticallysatisfied for any SL(2 , Z ) S transformation and we obtain a singlet.Γ S (cid:16) N · M (cid:48) − N (cid:48) · M (cid:17) = N · M (cid:48) − N (cid:48) · M This can be taken a step further by noting that due to the linear independence of A and B X can bewritten as two independent terms, which transform separately. X = (cid:18) p qr s (cid:19) = (cid:18) a a a a (cid:19) ⊗ A + (cid:18) b b b b (cid:19) ⊗ B ≡ X A ⊗ A + X B ⊗ B (5.47)With the decomposition of the SL(2 , Z ) S image of X in (5.46) we have that Ξ A depends on the a i onlyand Ξ B depends on the b j only. Therefore the necessary and sufficient conditions for a singlet becomethe pair of conditions Ξ A = X A , Ξ B = X B and we can construct two seperate non-trivial singlets bysetting one of X A or X B to zero and the other to the canonical sympletic form. A ≡ (cid:104) N · A · M (cid:48) − N (cid:48) · A · M (cid:105) , B ≡ (cid:104) N · B · M (cid:48) − N (cid:48) · B · M (cid:105) (5.48)For the triplet we begin with the known NS-NS sector T duality constraint N · M = 0 and consider itsimages under particular elements of SL(2 , Z ) S , which in the case of Γ = I and Γ = Γ we obtain theT duality constraints of both the NS-NS sector and the R-R sector.Γ S = (cid:18) −
11 0 (cid:19) : (cid:18) p q (cid:19) → (cid:18) q p (cid:19) (5.49)Therefore we have that given S duality N · M = 0 implies N (cid:48) · M (cid:48) = 0. The other generator of SL(2 , Z ) S on a general linear combination of these two terms leads to different transformations in the A and B terms. Γ S = (cid:18) (cid:19) : (cid:18) p q (cid:19) → (cid:18) p q (cid:19) + p (cid:18) AA A (cid:19) + q (cid:18) B BB (cid:19) (5.50)Although setting the expressions associated with the second and third terms in the above expressionis a necessary condition for joint T and S duality invariance, it is not sufficient. This can be seen byconsidering another SL(2 , Z ) S transformation, those which is associated to the negative integer shift, S → S −
1. Γ S = (cid:18) −
10 1 (cid:19) : (cid:18) p q (cid:19) → (cid:18) p q (cid:19) + p (cid:18) −A−A A (cid:19) + q (cid:18) B −B−B (cid:19) (5.51)Using (5.49) the requirement that both (5.50) and (5.51) are also zero leads to stronger constraints, N · M = 0 is true by virtue of the I = A + B decomposition terms both vanishing seperately. Apriori wecould not assume that the A and B related terms form two separate, independent, systems but giventhe singlet structures we would expect the triplets to follow with the same splittings. A ≡ (cid:104) N · A · M , N (cid:48) · A · M (cid:48) , N · A · M (cid:48) + N (cid:48) · A · M (cid:105) B ≡ (cid:104) N · B · M , N (cid:48) · B · M (cid:48) , N · B · M (cid:48) + N (cid:48) · B · M (cid:105) (5.52)As with the quadratic derivative actions of the form D : (cid:101) H (3) → (cid:101) H (3) it is possible that not all ofthese expressions automatically give Bianchi constraints. However, unlike the previous case we cannotexpress these flux matrix expressions in terms of the natural Type IIB flux multiplets since they are notdefined by derivative actions on H (3) . The triplets contain expressions which arise in the T dualityonly case while the singlets do not, but they cannot be Type IIB tadpoles as they are expandedin the H (3) basis and so they can only couple to the Type IIA C form. However, the use of fluxmatrices to examine the effect of SL(2 , Z ) S transformations on the superpotential is independent ofhow we might label the flux multiplets of the particular Type II theory they are defined in. HenceSL(2 , Z ) S transformations in Type IIA will result in its flux matrices forming the same set of multiplets.Therefore while the s we have constructed here do not represent tadpole constraints in Type IIB theywould in Type IIA and conversely the tadpoles found in the (cid:101) H (3) case would not be Type IIA tadpoles.37 Moduli space exchange symmetry
Thus far we have observed a great deal of symmetry in how the different moduli spaces of M (and W )can be described, though distinct differences exist. For example, the manner in which the Type IIBsuperpotential is defined, with the derivatives acting upon the K¨ahler moduli dependent terms ratherthan the complex structure. Af we wish to make our descriptions of the moduli spaces as symmetricas possible we would postulate that the superpotential and fluxes can be reformulated such that theroles of the moduli spaces are exchanged even within the Type IIB framework. However, because thisimplies the two moduli spaces of M are equivalent we would not expect it to be possible for all M ,only a particular set of spaces.To lend some justification for this hypothesis we again turn to the explict case of the Z × Z orientifold. At has been shown [20, 22] that the Bacobi-like set of constraints of non-geometric fluxespossess an invariance under the subset of GL(6 , Z ) which satisfies the orbifold symmetries. An thecase of M Z this manifests itself as the complex structure U i moduli possessing modular invariancedue to the way the orientifold factorises into three similar sub-tori, each of which possess SL(2 , Z )symmetries. The modular invariance is only in the complex structure moduli, which can be seen tofollow from the fact that a redefinition of the complex structure moduli does not mix different TypeIIB fluxes and is related to a redefinition of the sympletic basis ( a A , b B ) and thus how the fluxes areexpanded in terms of components. The orientifold has also been examined [ ? ] from the point of viewof enforcing, by construction, such modular invariance on the flux Lie algebras and while it does notresult in the same constraints as T duality the methodology of their analysis is qualitatively the same.An [20, 22] the change of basis on the non-geometric fluxes was viewed in terms of the subset ofGL(6 , Z ) which satisfies the orbifold symmetries but this can be reinterpreted as the set of sympletictransformations for the H (3) basis of the orientifold. More generally this is a reflection of how sympletictransformations on H (3) are a symmetry if the complex structure moduli are appropriately redefined,precisely what we have observed in our analysis of the Type IIB theory. Conversely, if the K¨ahlermoduli can be redefined then we can regard this as being related to the redefinition of the (cid:101) H (3) basiselements and in the case of the M Z precisely the same kind of transformations occur for the K¨ahlermoduli as the complex structure moduli, another result of the properties of the three sub-tori. Overallwe have two kinds of symmetry associated with the complex structure moduli space; T duality andredefinitions of the sympletic basis of H (3) , but only one kind of symmetry associated to the K¨ahlermoduli space; redefinitions of the (cid:101) H (3) basis and the redefinition of the H (3) and (cid:101) H (3) is a symmetryinherent to M and W outside of their string theory contexts [23]. To illustrate this for the M Z moreexplicitly we recall its general polynomial form, where the moduli are grouped in terms of their K¨ahlermoduli dependence. W = (cid:90) W Ω ∧ (cid:16) D ( (cid:102) c ) + D (cid:48) ( (cid:102) (cid:48) c ) (cid:17) = T (cid:16) P ( U ) − S (cid:98) P ( U ) (cid:17) + T a (cid:16) P ( a )1 ( U ) − S (cid:98) P ( a )1 ( U ) (cid:17) ++ T (cid:16) P ( U ) − S (cid:98) P ( U ) (cid:17) + T b (cid:16) P ( b )2 ( U ) − S (cid:98) P ( b )2 ( U ) (cid:17) (6.1)Due to its factorisation in terms of three two dimensional sub-tori the moduli pair off, with ( T a , U a )being those moduli which describe the a ’th sub-torus. Under Υ the moduli are exchanged ( T a , U a ) → ( (cid:101) T a , (cid:101) U a ) = ( U a , T a ) but the underlying structure of the space remains a two dimensional torus, sug-gesting that the dynamics of the complex structure moduli space is equivalent to the dynamics of theK¨ahler moduli space, in line with the definition of mirror symmetry; the complex structure modulispace of M being related to the K¨ahler moduli space of W . We have seen this in the way the differentsuperpotentials are defined, but in cases where M = W this would imply an enhanced symmetry on M itself and the superpotential defined by its fluxes, with the two moduli spaces interchangable andstructures existing on one should appear in the other. To illustrate this on the M Z rather than use Tduality or mirror symmetry to alter the superpotential we instead can simply rearrange the polynomial38uperpotential to exchange the roles of the two moduli types. W → U (cid:16) P ( T ) − S (cid:98) P ( T ) (cid:17) + U a (cid:16) P ( a )1 ( T ) − S (cid:98) P ( a )1 ( T ) (cid:17) ++ U (cid:16) P ( T ) − S (cid:98) P ( T ) (cid:17) + U b (cid:16) P ( b )2 ( T ) − S (cid:98) P ( b )2 ( T ) (cid:17) (6.2)T duality defines a sequence of fluxes induced by the non-zero nature of (cid:98) F and for each of thesefluxes there is a corresponding polynomial in the complex structure moduli, which in the case of (cid:98) F we have denoted as (cid:98) P ( U ). With the reformulation of the superpotential in (6.2) due to the symmetryin the moduli we now have K¨ahler moduli dependent polynomials such as (cid:98) P . We can postulate thesepolynomials are built from some new set of fluxes but precisely what those might be we will considerin the next section. However, if the K¨ahler moduli space for the M Z is equivalent to the complexstructure moduli space of M Z then this leads us to argue that there is a duality which has the sameeffect on the K¨ahler moduli as T duality has on the complex structure moduli, which we will refer toas T (cid:48) duality. To examine this more quantitatively and for spaces other than M Z we are required toconsider the many different ways we have of constructing superpotential-like expressions from objectswe have been examining. Since we will be discussing how various derivatives and their matrix representations relate to oneanother we will dispense with the different D’s used for different derivatives in previous sections andsimply label them with an index, D i , and likewise with their associated matrix representations, whichin the case of Type IIB has M i representing the action on the (cid:101) H (3) basis and N j on the H (3) basis andthe superpotentials of Type IIA are distinguished from those of Type IIB by W compared to W . InType IIB we can construct objects which have a superpotential-like form in two different ways; one ofwhich is the Type IIB superpotential and the second resembles its ζ moduli dual form, except it isdefined upon W rather than M , which is a critical difference. W = (cid:90) W Ω ∧ (cid:16) D ( (cid:102) c ) + D (cid:48) ( (cid:102) (cid:48) c ) (cid:17) = T (cid:62) · h ν · (cid:16) C · M + C (cid:48) · M (cid:48) (cid:17) · g a · U (6.3) W = (cid:90) W (cid:102) ∧ (cid:16) D (Ω c ) + D (cid:48) (Ω (cid:48) c ) (cid:17) = U (cid:62) · h a · (cid:16)(cid:101) C · N + (cid:101) C (cid:48) · N (cid:48) (cid:17) · g ν · T (6.4)In general these are the only two expressions which can be formed of integrals and from pairs ofelements of either H (3) or (cid:101) H (3) . It is possible, however, to construct Type IIB scalar products whichare dependent upon the bilinear forms g and h defined in Type IIA. W = T (cid:62) · h a · (cid:16) C · N + C (cid:48) · N (cid:48) (cid:17) · g ν · U (6.5) W = U (cid:62) · h ν · (cid:16)(cid:101) C · M + (cid:101) C (cid:48) · M (cid:48) (cid:17) · g a · T (6.6)These two expressions are constructable using matrices because the dimensions of such pairs as h ν and h a are equal by h , ( M ) = h , ( W ). This allows us to build forms such as T (cid:62) · h a · f ( ν ) , hybrids ofterms defined in different spaces and in different Type II theories but this fact means that generallysuch constructs are ill defined. The expression T (cid:62) · h a · f ( ν ) can be built in W if h , = h , and it ispossible to choose H (3) bases in M and W such that h a = h a . This is a reflection of the link betweenthe K¨ahler moduli space of W and the complex structure moduli space of M , T I ↔ U I . Af the linkis to be between the two moduli spaces of W itself then we instead wish to consider the equivalence T I ↔ U I which is possible if, given h , = h , , T A ↔ U A and as a result we narrow our considerationsto those spaces which satisfy M = W , those spaces which are self mirror dual. Such a restrictionautomatically allows us to make the equivalence g a = g a and likewise with the other bilinear formsbecause of the equality of the Hodge numbers and that a redefinition of the basis of H (3) ( M ) is also We do not consider Ω ∧ D ( (cid:102) c ) and D (Ω) ∧ (cid:102) c as different for reasons which will shortly be given. At should be noted that although the IIA complex structure indices
I, J, . . . and the IIA K¨ahler indices
A, B, . . . range over the same values we retain their distinction for the purposes of clarity.
39 redefinition of the basis of H (3) ( W ) since they are one and the same. Is a result it is possible toconstruct the Type IIB form (cid:101) Ω ≡ Ω (cid:12)(cid:12)(cid:12) U A → T A = T (cid:62) · h a · f ( a ) = T (cid:62) · h a · f ( a ) on W . With this equalitybetween the Type IIA and Type IIB bilinear forms on M = W both (6.5) and (6.6) therefore obtainan integral representation, in terms of (cid:101) Ω and (cid:101) (cid:102) ≡ (cid:102) (cid:12)(cid:12)(cid:12) T I → U I . W = (cid:90) W (cid:101) (cid:102) ∧ (cid:16) D ( (cid:101) Ω c ) + D (cid:48) ( (cid:101) Ω (cid:48) c ) (cid:17) W = (cid:90) W (cid:101) Ω ∧ (cid:16) D ( (cid:101) (cid:102) c ) + D (cid:48) ( (cid:101) (cid:102) (cid:48) c ) (cid:17) (6.7)To illustrate this more explicitly we consider an integral similar to that of (6.3), namely using non-complexified holomorphic forms and use the properties of the symplectic basis and the equality ofthe Hodge number to convert it into something similar to (6.7). Thus illustrating a rearrangement ofthe superpotential akin to that between (6.2) and (6.1). For the purpose of simplification we neglectdilaton complexifications. (cid:90) W Ω ∧ D ( (cid:102) ) = (cid:90) W (cid:16) U A a A − U B b B (cid:17) ∧ T I (cid:16) F ( I ) A a A − F B ( I ) b B (cid:17) − T J (cid:16) F ( I ) A a A − F ( J ) B b B (cid:17) = (cid:90) W (cid:16) T I a I − T J b J (cid:17) ∧ U A (cid:16) F ( I ) A a I − F B ( J ) b J (cid:17) − U B (cid:16) F ( I ) A a I − F ( J ) B b J (cid:17) = (cid:90) W (cid:101) Ω ∧ D ( (cid:101) (cid:102) ) (6.8)We have had to make the assumption that A, B and
I, J range over the same indices and that thesympletic structure of M is equivalent to that of W , as such expressions as T I a I − T J b J are the TypeIIB holomorphic 3-form Ω but with the moduli labelled in the Type IIA manner. The general factthat these expressions bear a striking resemblence to the Type IIA superpotential integrals promptsus to now turn our attention to those superpotential-like integrals defined in Type IIA on a generic M . As with Type IIB, there are two expressions which can be written as integrals and two which, ingeneral, cannot. We label the Type IIA derivatives with a tilde and an index, D i and likewise withtheir associated matrix representations, which in the case of Type IIA has M i representing the actionon the (cid:101) H (3) basis and N j on the H (3) basis. W = (cid:90) M (cid:102) ∧ (cid:16) D (Ω c ) + D (cid:48) (Ω (cid:48) c ) (cid:17) = U (cid:62) · h a · (cid:16) C · N + C (cid:48) · N (cid:48) (cid:17) · g ν · T (6.9) W = (cid:90) M Ω ∧ (cid:16) D ( (cid:102) c ) + D (cid:48) ( (cid:102) (cid:48) c ) (cid:17) = T (cid:62) · h ν · (cid:16)(cid:101) C · M + (cid:101) C (cid:48) · M (cid:48) (cid:17) · g a · U (6.10)As in Type IIB we can construct a pair of superpotential-like scalar products which are a mixture ofType IIA and Type IIB defined objects. W = U (cid:62) · h ν · (cid:16) C · M + C (cid:48) · M (cid:48) (cid:17) · g a · T (6.11) W = T (cid:62) · h a · (cid:16)(cid:101) C · N + (cid:101) C (cid:48) · N (cid:48) (cid:17) · g ν · U (6.12)For the case of M = W it is possible to construct integral representations in the same manner as theType IIB case and we again take (cid:101) (cid:102) and (cid:101) Ω to represent the holomorphic forms which have had theirmoduli spaces exchanged. W = (cid:90) M (cid:101) Ω ∧ (cid:16) D ( (cid:101) (cid:102) c ) + (cid:102) D (cid:48) ( (cid:101) (cid:102) (cid:48) c ) (cid:17) W = (cid:90) M (cid:101) (cid:102) ∧ (cid:16) D ( (cid:101) Ω c ) + (cid:102) D (cid:48) ( (cid:101) Ω (cid:48) c ) (cid:17) (6.13)These eight ways of constructing a superpotential can split into two subsets by considering how themoduli are coupled to the dilaton by the complexification matrices. The standard Type IIB superpo-tential has its dilaton dependence defined in the same manner as W , it is the T which are complexified.40he Type IIA superpotential has terms of the form seen in both W and W but in both cases it is the U which are complexified. In these three cases, as well as W it is the same degrees of freedom whichis dilaton complexified, the T and U are different labellings for the same moduli.Type IIA : Ω c = − S U a + U i a i − U j b j + S U b Type IIB : (cid:102) c = − S T ν + T i ν i + T j (cid:101) ν j − S T (cid:101) ν (6.14)When comparing the Type IIA and IIB fluxes for each flux sector we noted that the complexificationmatrices could be neglected due to each Type II superpotential being dependent on the same complex-ification matrices. By considering all of these possible superpotential constructions it can be seen thatall elements of W − = { W , W , W , W } and, seperately, W + = { W , W , W , W } have dilaton com-plexification of the same form. The elements of W ± are related to one another by moduli relabellingand ζ and these lead to simple expressions for the interdependency of their associated flux matrices.Comparing W with W gives the flux matrix relations previously observed in the NS-NS sector in(3.46), but now in both sectors. h ν · M · g a = h a · N · g ν ⇒ M = h a · N · h a N = − h ν · M · h ν Provided M = W the expressions for W and W are well defined and their flux matrices are relatedin a similar way. Their interdependency with the flux matrices of W and W are trivial since W is W under a moduli relabelling and likewise for W and W . h ν · M · g a = h a · N · g ν ⇒ M = h a · N · h a = M N = − h ν · M · h ν = N We previously saw that the Type IIA nilpotency constraints are equal to the Type IIB nilpotencyconstraints, in either flux sector, and it therefore follows that the nilpotency constraints of the deriva-tives used to construct the superpotentials of W − are all equivalent. Repeating this analysis for W + = { W , W , W , W } we observe they too are linked by moduli relabelling and moduli duality.The dilaton couples to the same degrees of freedom in each case and therefore the complexificationmatrices can be factorised out when comparing the expressions. From this it is straightforward toobtain the relationship between the different flux matrices and to show the nilpotency conditions tobe equal. As an example we consider W and W where the complexification matrices combine witheither T or U , as is the case for any other pairwise comparision of W + elements. h a · N · g ν = h ν · M · g a ⇒ M = h a · N · h a = M N = − h ν · M · h ν = N ⇐ h a · M · g ν = h ν · N · g a By the same reasoning as W − the Bianchi constraints for each flux sector of each superpotential areequivalent to one another. However, if we are to equate an element of W − with an element of W + itis no longer the case that the Bianchi constraints are equivalent. The Bianchi constraints of D and D are equivalent and those of D and D are equivalent but D and D are inequivalent, the changein dilaton couplings between the expressions is a non-trivial effect. To examine this more indepthwe shall consider the specific cases of the Bianchi constraints of W and W , a choice motivated bythe results of (3.29). We shall denote the map which converts the standard Type IIB fluxes andderivatives of W into those of W by M . In terms of the polynomial form of the superpotential this isequivalent to converting (6.1) into (6.2) and it is this which we wish to express in terms of derivativesand holomorphic forms. The K¨ahler moduli in (6.3) arise due to the K¨ahler forms J ( n ) and we have previously defined theircomplex structure counterparts J ( n ) . The W of (6.4) can be broken down into simpler expressionsby defining a set of flux multiplets, F n and (cid:98) F m , as the images of these J under the derivatives. The41atted and unhatted fluxes choosen to follow the same layout as the F and (cid:98) F fluxes. (cid:90) M (cid:102) ∧ D (Ω c ) = (cid:90) M (cid:102) ∧ D (cid:16) − S J (0) + J (1) − J (2) + S J (3) (cid:17) = (cid:90) M (cid:102) ∧ (cid:16) − S (cid:98) F · J (0) + F · J (1) − F · J (2) + S (cid:98) F · J (3) (cid:17) (6.15) (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c ) = (cid:90) M (cid:102) ∧ (cid:16) F · J (0) − S (cid:98) F · J (1) + S (cid:98) F · J (2) − F · J (3) (cid:17) (6.16)We can define the components of the F fluxes in the same manner as we have for the F but due totheir definition in mapping H (3 − n,n ) to (cid:101) H (3) their indices will run over different ranges. (cid:98) F : (cid:16) F (0) I ν I + F J (0) (cid:101) ν J (cid:17) ι a : H (3 , → (cid:101) H (3) F : (cid:16) F ( a ) I ν I + F J ( i ) (cid:101) ν J (cid:17) ι a a : H (2 , → (cid:101) H (3) (cid:98) F : (cid:16) F (0) I ν I + F (0) J (cid:101) ν J (cid:17) ι b : H (0 , → (cid:101) H (3) F : (cid:16) F ( b ) I ν I + F ( j ) J (cid:101) ν J (cid:17) ι b b : H (1 , → (cid:101) H (3) (6.17)The superpotential is then straightforward to express in terms of these fluxes, in the same manner as(3.25). (cid:90) M (cid:102) ∧ D (Ω c ) = − S U (cid:16) F (0) I T I + F J (0) T J (cid:17) + U a (cid:16) F ( a ) I T I + F J ( i ) T J (cid:17) + S U (cid:16) F (0) I T I + F (0) J T J (cid:17) − U b (cid:16) F ( b ) I T I + F ( j ) J T J (cid:17) (6.18) (cid:90) M (cid:102) ∧ D (cid:48) (Ω (cid:48) c ) = U (cid:16) (cid:98) F (0) I T I + (cid:98) F J (0) T J (cid:17) − S U a (cid:16) (cid:98) F ( a ) I T I + (cid:98) F J ( i ) T J (cid:17) − U (cid:16) (cid:98) F (0) I T I + (cid:98) F (0) J T J (cid:17) + S U b (cid:16) (cid:98) F ( b ) I T I + (cid:98) F ( j ) J T J (cid:17) The action of M on the various objects of Type IIB theory can now be written in a more explicitmanner, one which bears close resemblence to the action of ζ in (3.49) but without M ↔ W or TypeIIA ↔ Type IIB. M : D ( (cid:48) )1 → D ( (cid:48) )2 , ( F , (cid:98) F ) → ( F , (cid:98) F ) M ( (cid:48) )1 → M ( (cid:48) )2 , Ω → (cid:102) N ( (cid:48) )1 → N ( (cid:48) )2 , (cid:102) → Ω (6.19)These actions are such that the superpotential is left invariant by M but the fluxes and derivatives areredefined. At is noteworthy also that M satisfies M = Ad, where Id is the identity map which leavesall objects in (6.19) unchanged. (cid:48) duality constraints By comparing these two ways of writing the superpotential we can obtain the components of the F and (cid:98) F in terms of the usual fluxes F and (cid:98) F , which are given in Table 1. The global factor of − H (3) defined W would pick up a sign change, but this is purely a matter of definition.As a result of different dilaton couplings the fluxes of D are a non-trivial mixture of the fluxes fromboth D and D (cid:48) . The I, J = 0 terms being in a different flux sector to those of the
I, J >
A, B = 0 terms are in a different flux sector to the
A, B > W has noclear distinction between the usual notion of NS-NS fluxes and R-R fluxes in W and the immediatecorrollary of this is that the Bianchi constraints due to D and D (cid:48) will not be the same as those dueto D and D (cid:48) . We can construct the Bianchi constraints for D in terms of the F components in the42 : F (0)0 F (0) i F F j (0) F (0)0 F (0) i F (0)0 F (0) j ∈ W : − S U T − S U T i − S U T − S U T j S U T S U T i S U T S U T j F : − F (0)0 − (cid:98) F ( i )0 − F − (cid:98) F j ) − F (0)0 − (cid:98) F ( i )0 − F (0)0 − (cid:98) F ( j )0 F : F ( a )0 F ( a ) i F a ) F j ( a ) F ( b )0 F ( b ) i F ( b )0 F ( b ) j ∈ W : U a T U a T i U a T U a T j − U b T − U b T i − U b T − U b T j F : − (cid:98) F (0) a − F ( i ) a − (cid:98) F a (0) − F a ( j ) − (cid:98) F (0) b − F ( i ) b − (cid:98) F (0) b − F ( j ) b (cid:98) F : (cid:98) F (0)0 (cid:98) F (0) i (cid:98) F (cid:98) F j (0) (cid:98) F (0)0 (cid:98) F (0) i (cid:98) F (0)0 (cid:98) F (0) j ∈ W : U T U T i U T U T j − U T − U T i − U T − U T j F : − (cid:98) F (0)0 − F ( i )0 − (cid:98) F − F j ) − (cid:98) F (0)0 − F ( i )0 − (cid:98) F (0)0 − F ( j )0 (cid:98) F : (cid:98) F ( a )0 (cid:98) F ( a ) i (cid:98) F a ) (cid:98) F j ( a ) (cid:98) F ( b )0 (cid:98) F ( b ) i (cid:98) F ( b )0 (cid:98) F ( b ) j ∈ W : − S U a T − S U a T i − S U a T − S U a T j U b T U b T i U b T U b T j F : − F (0) a − (cid:98) F ( i ) a − F a (0) − (cid:98) F a ( j ) − F (0) b − (cid:98) F ( i ) b − F (0) b − (cid:98) F ( j ) b Table 1: M defined components of F and (cid:98) F in terms of the components of F and (cid:98) F and associatedsuperpotential coefficients.same manner as was done for D ; converting the flux actions on H (3) into the equivalent actions on (cid:101) H (3) . (cid:98) F : b (cid:16) F (0) I ι e ν I − F J (0) ι ν J (cid:17) F : b a (cid:16) F ( a ) I ι e ν I − F J ( a ) ι ν J (cid:17)(cid:98) F : − a (cid:16) F (0) I ι e ν I − F (0) J ι ν J (cid:17) F : − a b (cid:16) F ( b ) I ι e ν I − F ( b ) J ι ν J (cid:17) (6.20)Therefore we have the two equivalent actions of D , in the same manner as (3.23), with the actions of (cid:98) F and (cid:98) F once again able to be subsumed into the other fluxes. D ( a A ) = F ( A ) I ν I + F J ( A ) (cid:101) ν J D ( b B ) = F ( B ) I ν I + F ( B ) J (cid:101) ν J ⇔ D ( ν I ) = F ( A ) I a A − F I ( B ) b B D ( (cid:101) ν J ) = − F ( A ) J a A + F ( B ) J b B (6.21)Combining these two actions of each flux we obtain their Bianchi constraints. D ( a A ) = (cid:16) F ( A ) J F ( B ) J − F I ( A ) F ( B ) I (cid:17) a B + (cid:16) F J ( A ) F ( B ) J − F ( A ) I F I ( B ) (cid:17) b B D ( b A ) = (cid:16) F ( A ) J F ( B ) J − F ( A ) I F ( B ) I (cid:17) a B + (cid:16) F ( A ) J F ( B ) J − F ( A ) I F I ( B ) (cid:17) b B D ( ν I ) = (cid:16) F ( B ) I F ( B ) J − F I ( A ) F ( A ) J (cid:17) ν J + (cid:16) F ( B ) I F J ( B ) − F I ( A ) F ( A ) J (cid:17)(cid:101) ν J D ( (cid:101) ν I ) = (cid:16) F ( B ) I F ( B ) J − F ( A ) I F ( A ) J (cid:17) ν J + (cid:16) F ( A ) I F ( A ) J − F ( B ) I F J ( B ) (cid:17)(cid:101) ν J (6.22)Although we can use Table 1 to convert these expressions into the F and (cid:98) F components, it is moreconvenient to work with flux matrices, as the generalisation to the S duality case is more forthcomingin that formulation; a fact we have previously seen in Section 5.2. Before considering the constraints induced on the fluxes of D ( (cid:48) )2 we shall derive the dependence of thosefluxes on the usual D ( (cid:48) )1 fluxes by equating the two ways of writing the superpotential in terms of fluxmatrices in (6.3) and (6.4). T (cid:62) · h ν · (cid:16) C · M + C (cid:48) · M (cid:48) (cid:17) · g a · U = U (cid:62) · h a · (cid:16)(cid:101) C · N + (cid:101) C (cid:48) · N (cid:48) (cid:17) · g ν · T (6.23)43ince the Type IIB superpotential is naturally written in terms of M and M (cid:48) we wish to express N and N (cid:48) in terms of them. Hence, because of the non-trivial dilaton coupling caused by the inability toneglect the complexification matrices we must consider the NS-NS and R-R sector simultaneously. C · M + C (cid:48) · M (cid:48) = h ν · g (cid:62) ν · (cid:16) N (cid:62) · (cid:101) C + ( N (cid:48) ) (cid:62) · (cid:101) C (cid:48) (cid:17) · h a · g (cid:62) a (6.24)The complexification matrices, tilded and not, are all diagonal and commute with the g and h bilinearforms in both Type IIA and Type IIB and though we are assuming h , = h , we retain the distinctionbetween C ( (cid:48) ) and (cid:101) C ( (cid:48) ) and their definition in terms of other matrices. C = A − S B = A h , − S B h , C (cid:48) = B − S A = B h , − S A h , (cid:101) C = (cid:101) A − S (cid:101) B = A h , − S B h , (cid:101) C (cid:48) = (cid:101) B − S (cid:101) A = B h , − S A h , (6.25)Commuting the C ( (cid:48) ) through the bilinear forms we can reexpress the N ( (cid:48) )2 in terms of M ( (cid:48) )2 so that all thetransposed matrices are removed and the result is an expression which can be written with manifestdilaton dependence and F (cid:58) and (cid:98) F (cid:58) matrices. C · M + C (cid:48) · M (cid:48) = − h a · (cid:16) M · (cid:101) C + M (cid:48) · (cid:101) C (cid:48) (cid:17) ⇒ F (cid:58)(cid:58) − S (cid:98) F (cid:58)(cid:58) = − h a · (cid:16) F (cid:58)(cid:58) − S (cid:98) F (cid:58)(cid:58) (cid:17) (6.26)By considering dilaton couplings this decomposes into a pair of equations, each involving all of theflux matrices, which can be written in terms of the F n (cid:58)(cid:58) and (cid:98) F m (cid:58)(cid:58)(cid:58) matrices. F (cid:58)(cid:58) = M · (cid:101) A + M (cid:48) · (cid:101) B = − h a · (cid:16) A · M + B · M (cid:48) (cid:17) = − h a · F (cid:58)(cid:58) (cid:98) F (cid:58)(cid:58) = M · (cid:101) B + M (cid:48) · (cid:101) A = − h a · (cid:16) B · M + A · M (cid:48) (cid:17) = − h a · (cid:98) F (cid:58)(cid:58) (6.27)Using the properties of (cid:101) A and (cid:101) B these simultaneous equations allow us to express M ( (cid:48) )2 entirely in termsof M ( (cid:48) )1 . M = − h a · (cid:16)(cid:16) A · M + B · M (cid:48) (cid:17) · (cid:101) A + (cid:16) B · M + A · M (cid:48) (cid:17) · (cid:101) B (cid:17) = − h a · (cid:16) F (cid:58)(cid:58) · (cid:101) A + (cid:98) F (cid:58)(cid:58) · (cid:101) B (cid:17) M (cid:48) = − h a · (cid:16)(cid:16) A · M + B · M (cid:48) (cid:17) · (cid:101) B + (cid:16) B · M + A · M (cid:48) (cid:17) · (cid:101) A (cid:17) = − h a · (cid:16) F (cid:58)(cid:58) · (cid:101) B + (cid:98) F (cid:58)(cid:58) · (cid:101) A (cid:17) It was previously noted that Table 1 shows the
A, B = 0 cases of the flux components of F and (cid:98) F aretreated differently to the A, B (cid:54) = 0 case, the reverse of the behaviour seen in our examination of thestandard Type IIB fluxes, where
I, J = 0 cases were different from those with
I, J (cid:54) = 0. An the case ofthe K¨ahler indices it is related to the left action of A and B on M ( (cid:48) )1 , with the dimensions of A and B being defined in terms of h , . Here, however, it is due to the right multiplication of (cid:101) A and (cid:101) B , whosedimensions are defined in terms of h , instead. The combined left and right actions of these matrices isthat which provides this alteration and so (6.28) is of the general form we might have apriori expected,given the results of Table 1. Previously, when discussing S duality transformations in Type IIB, itwas convenient to view the two flux matrices as doublet partners due to their relationship with theSL(2 , Z ) S doublets and the same is true here; we can express the relationship between the M ( (cid:48) )2 and the M ( (cid:48) )1 in terms of transformations on a two component vector using the same transformation matricesthat relate the S duality flux doublet with the flux matrix doublet, as in (5.31). (cid:18) M M (cid:48) (cid:19) = − h a (cid:18) A BB A (cid:19) L (cid:18) M M (cid:48) (cid:19) (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) R = − h a F (cid:58)(cid:58) (cid:98) F (cid:58)(cid:58) (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) R (6.28)The h a is understood to be a common factor, hence its factorisation out of the matrices in a waynot entirely consistent with its matrix definition. The L and R subscripts define the direction of44ultiplication. (cid:18) A BB A (cid:19) L (cid:18) XY (cid:19) ≡ (cid:18) A · X + B · Y A · X + B · Y (cid:19) , (cid:18) XY (cid:19) (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) R ≡ (cid:32) X · (cid:101) A + Y · (cid:101) B X · (cid:101) A + Y · (cid:101) B (cid:33) Rather than repeating the entire method just used, the N ( (cid:48) ) i forms of these expressions are straightfor-ward to construct from (6.28) but this time removing the M ( (cid:48) ) j by using the fact h a anticommutes withthe g a /ν and so picks up a factor of − N = (cid:16) (cid:101) A · (cid:16) N · A + N (cid:48) · B (cid:17) + (cid:101) B · (cid:16) N · B + N (cid:48) · A (cid:17)(cid:17) ( h a ) N (cid:48) = (cid:16) (cid:101) B · (cid:16) N · A + N (cid:48) · B (cid:17) + (cid:101) A · (cid:16) N · B + N (cid:48) · A (cid:17)(cid:17) ( h a ) (6.29)These form the same kind of tranformed doublet structure as in (6.28), but with the h a now an overallfactor on the right. (cid:18) N N (cid:48) (cid:19) = (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) L (cid:18) N N (cid:48) (cid:19) (cid:18) A BB A (cid:19) R ( h a ) (6.30) (cid:48) duality constraints The constraints on the fluxes as a result of the nilpotency of D are not equivalent to the D nilpotencyconstraints, due to the existence and placement of the projection-like matrices A and B . To examinethis we redefine our notation for each of the flux matrices such that the expressions relating to M i · h a · N i = 0 simplify and we again use · h a · = (cid:5) . (cid:18) M M (cid:48) (cid:19) = (cid:18) m m (cid:48) (cid:19) (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) R , (cid:18) N N (cid:48) (cid:19) = (cid:32) (cid:101) A (cid:101) B (cid:101) B (cid:101) A (cid:33) L (cid:18) n n (cid:48) (cid:19) (6.31)Due to the orthogonality of (cid:101) A and (cid:101) B half of the terms in the expansion of M ( (cid:48) )2 (cid:5) N ( (cid:48) )2 as linear com-binations of M ( (cid:48) )1 (cid:5) N ( (cid:48) )1 are identically zero, as was seen when considering S duality constraints. Using (cid:101) A = (cid:101) A , and likewise for (cid:101) B , the constraints can be put into a particular format which was seenpreviously in (5.37) except that the vectors of flux matrices are redefined. M (cid:5) N = (cid:0) m m (cid:48) (cid:1) (cid:32) (cid:101) A (cid:101) B (cid:33) (cid:18) h a h a (cid:19) (cid:18) n n (cid:48) (cid:19) M (cid:48) (cid:5) N (cid:48) = (cid:0) m m (cid:48) (cid:1) (cid:32) (cid:101) B (cid:101) A (cid:33) (cid:18) h a h a (cid:19) (cid:18) n n (cid:48) (cid:19) M (cid:5) N (cid:48) = (cid:0) m m (cid:48) (cid:1) (cid:32) (cid:101) A (cid:101) B (cid:33) (cid:18) h a h a (cid:19) (cid:18) n n (cid:48) (cid:19) M (cid:48) (cid:5) N = (cid:0) m m (cid:48) (cid:1) (cid:32) (cid:101) B (cid:101) A (cid:33) (cid:18) h a h a (cid:19) (cid:18) n n (cid:48) (cid:19) (6.32)With each of the four cases being of the same format, only differing by location and number of primedflux matrices, without much loss of generality we explicitly consider the first case. − h a · M (cid:5) N · h a = − h a · (cid:16) m · (cid:101) A (cid:5) (cid:101)
A · n + m (cid:48) · (cid:101) B (cid:5) (cid:101)
B · n (cid:48) (cid:17) · h a = A · (cid:16) M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) B (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) B (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) M · (cid:101) A (cid:5) N (cid:48) + M (cid:48) · (cid:101) B (cid:5) N (cid:17) · B + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N (cid:48) + M · (cid:101) B (cid:5) N (cid:17) · B (6.33) The case of N i · h ν · M i = 0 follows in the same manner if we did a different redefinition in which we factorised outthe matrices „ A BB A « . A and B factors between the two fluxmatrices as well as being external to each term. By using the projection properties of the external A and B we can compare the components of M (cid:5) N with those of M (cid:5) N and M (cid:48) (cid:5) N (cid:48) from (5.37), aswell as M (cid:48) (cid:5) N (cid:48) . In order to drop the non-degenerate factors of h a we consider the ideals generated bythe components of the flux matrices instead. (cid:68) A · M (cid:5) N · A (cid:69) = (cid:68) A · (cid:16) M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) B (cid:5) N (cid:48) (cid:17) · A (cid:69)(cid:68) A · M (cid:48) (cid:5) N (cid:48) · A (cid:69) = (cid:68) A · (cid:16) M · (cid:101) B (cid:5) N + M (cid:48) · (cid:101) A (cid:5) N (cid:48) (cid:17) · A (cid:69)(cid:68) A · M (cid:5) N · A (cid:69) = (cid:68) A · (cid:16) M · I (cid:5) N + M (cid:48) · (cid:5) N (cid:48) (cid:17) · A (cid:69)(cid:68) A · M (cid:48) (cid:5) N (cid:48) · A (cid:69) = (cid:68) A · (cid:16) M · (cid:5) N + M (cid:48) · I (cid:5) N (cid:48) (cid:17) · A (cid:69) (6.34)At is clear from the fact (cid:101) A and (cid:101) B are internal to the flux matrix pairings of M ( (cid:48) )2 (cid:5) N ( (cid:48) )2 that they cannotbe written as some linear combination of the M ( (cid:48) )1 (cid:5) N ( (cid:48) )1 and so the T (cid:48) constraints associated with thederivatives defining W in (6.4) provide different constraints to those of W in (6.3). However, it isclear from (6.34) that the constraints are equivalent on a slightly weaker level, in that the sum ofthe two terms associated with W is equal to the sum of the terms associated with W . This can beobtained by making use of (C.6-C.9). (cid:68) A · (cid:16) M (cid:5) N + M (cid:48) (cid:5) N (cid:48) (cid:17) · A (cid:69) = (cid:68) A · (cid:16) M (cid:5) N + M (cid:48) (cid:5) N (cid:48) (cid:17) · A (cid:69)(cid:68) A · (cid:16) M (cid:48) (cid:5) N + M (cid:48) (cid:5) N (cid:17) · A (cid:69) = (cid:68) A · (cid:16) M (cid:5) N (cid:48) + M (cid:48) (cid:5) N (cid:17) · A (cid:69) (6.35)These kinds of flux combinations have been previously seen in our analysis of S duality, forming termsin SL(2 , Z ) S multiplets. Since we have explicitly assumed both NS-NS and R-R fluxes are all potentiallynon-zero we have to consider what kind of flux structures are induced by S duality. We repeat the method used to examine the S duality of the Type IIB W superpotential but now welook at W , by expressing M · (cid:101) C + M (cid:48) · (cid:101) C (cid:48) as an inner product. M · (cid:101) C + M (cid:48) · (cid:102) C (cid:48) = (cid:0) M M (cid:48) (cid:1) · (cid:32) (cid:101) C (cid:102) C (cid:48) (cid:33) Using previous results for how the complexification matrices transform under SL(2 , Z ) S we have thetransformation properties of the doublet formed of the two flux matrices and the transformation onthe N ( (cid:48) )2 follow or can be obtained directly from the definition of W . (cid:32) (cid:101) C (cid:101) C (cid:48) (cid:33) → (cid:16)(cid:0) Γ (cid:62) S (cid:1) − ⊗ A + (cid:0) σ · Γ (cid:62) S · σ (cid:1) − ⊗ B (cid:17) (cid:32) (cid:101) C (cid:101) C (cid:48) (cid:33)(cid:0) M M (cid:48) (cid:1) → (cid:0) M M (cid:48) (cid:1) (cid:0) Γ (cid:62) S ⊗ A + (cid:0) σ · Γ (cid:62) S · σ (cid:1) ⊗ B (cid:1)(cid:18) N N (cid:48) (cid:19) → (cid:0) Γ S ⊗ A + (cid:0) σ · Γ S · σ (cid:1) ⊗ B (cid:1) (cid:18) N N (cid:48) (cid:19) (6.36)These are precisely those transformations seen in our previous analysis S duality in (5.33) and (5.33)but with the roles of M and N exchanged, a result which could be deduced apriori from our definition(6.23). The immediate implication of this fact and that A and B commute with h a /ν is that we candeduce all the SL(2 , Z ) S multiplets associated to W from the known SL(2 , Z ) S multiplets associatedto W by exchanging M , ↔ N , , M (cid:48) , ↔ N (cid:48) , and h a ↔ h ν . Applying this to A / B of (5.40) weobtain (cid:101) A / B and the pair of singlets (cid:101) A / B follow in the same manner from A / B in (5.42). (cid:101) A ≡ (cid:104) M · (cid:101) A (cid:5) N , M (cid:48) · (cid:101) A (cid:5) N (cid:48) , M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) A (cid:5) N (cid:48) (cid:105) (cid:101) B ≡ (cid:104) M · (cid:101) B (cid:5) N , M (cid:48) · (cid:101) B (cid:5) N (cid:48) , M (cid:48) · (cid:101) B (cid:5) N + M · (cid:101) B (cid:5) N (cid:48) (cid:105) (cid:101) A ≡ (cid:104) M (cid:48) · (cid:101) A (cid:5) N − M · (cid:101) A (cid:5) N (cid:48) (cid:105) (cid:101) B ≡ (cid:104) M (cid:48) · (cid:101) B (cid:5) N − M · (cid:101) B (cid:5) N (cid:48) (cid:105) (6.37)46he introduction of these A and B terms inside the flux matrix pairings allows us to make use of(C.6-C.9) to compare these W multiplets with the W multiplets. Due to the linearly independentdecomposition (5.35) (cid:101) A can be written as a union of ideals defined by this decomposition. (cid:101) A = (cid:104) A · M · (cid:101) A (cid:5) N · A , A · M (cid:48) · (cid:101) A (cid:5) N (cid:48) · A , A · ( M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) A (cid:5) N (cid:48) ) · A (cid:105)∪ (cid:104) B · M (cid:48) · (cid:101) A (cid:5) N · A , B · M · (cid:101) A (cid:5) N (cid:48) · A , B · ( M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) A (cid:5) N (cid:48) ) · A (cid:105)∪ (cid:104) A · M · (cid:101) A (cid:5) N (cid:48) · B , A · M (cid:48) · (cid:101) A (cid:5) N · B , A · ( M (cid:48) · (cid:101) A (cid:5) N (cid:48) + M · (cid:101) A (cid:5) N ) · B (cid:105)∪ (cid:104) B · M (cid:48) · (cid:101) A (cid:5) N (cid:48) · B , B · M · (cid:101) A (cid:5) N · B , B · ( M · (cid:101) A (cid:5) N (cid:48) + M (cid:48) · (cid:101) A (cid:5) N ) · B (cid:105) By considering the splittings and decompositions due to A , B , (cid:101) A and (cid:101) B it can be seen that the unionof all the SL(2 , Z ) S ideals of W is equal to the union of all the SL(2 , Z ) S ideals of W but individuallythe ideals are not equal to one another. (cid:101) A ∪ (cid:101) B ∪ (cid:101) A ∪ (cid:101) B = A ∪ B ∪ A ∪ B (6.38)The second set of SL(2 , Z ) S triplets on W follow (5.40) and (5.41) by the same relabelling. (cid:101) e A e A ≡ (cid:104) (cid:101) A · N · M · (cid:101) A , (cid:101) A · N (cid:48) · M (cid:48) · (cid:101) A , (cid:101) A · ( N (cid:48) · M + N · M (cid:48) ) · (cid:101) A (cid:105) (cid:101) e A e B ≡ (cid:104) (cid:101) A · N (cid:48) · M · (cid:101) B , (cid:101) A · N · M (cid:48) · (cid:101) B , (cid:101) A · ( N · M + N (cid:48) · M (cid:48) ) · (cid:101) B (cid:105) (cid:101) e B e A ≡ (cid:104) (cid:101) B · N · M (cid:48) · (cid:101) A , (cid:101) B · N (cid:48) · M · (cid:101) A , (cid:101) B · ( N (cid:48) · M (cid:48) + N · M ) · (cid:101) A (cid:105) (cid:101) e B e B ≡ (cid:104) (cid:101) B · N (cid:48) · M (cid:48) · (cid:101) B , (cid:101) B · N · M · (cid:101) B , (cid:101) B · ( N · M (cid:48) + N (cid:48) · M ) · (cid:101) B (cid:105) (6.39)These can then be written in terms of the W flux matrices using (6.28) and (6.29), though we only doso explicitly for (cid:101) e A e A due to the length of the expressions. The remaining multiplets follow the samegeneral structure but with appropriate (un)priming of the flux matrices. (cid:101) e A e A = (cid:104) (cid:101) A · ( N · A · M + N (cid:48) · A · M (cid:48) ) · (cid:101) A(cid:105) ∪∪ (cid:104) (cid:101)
A · ( N · B · M + N (cid:48) · A · M (cid:48) ) · (cid:101) A(cid:105) ∪∪ (cid:104) (cid:101)
A · ( N · M (cid:48) + N (cid:48) · M ) · (cid:101) A(cid:105) (6.40)As before the singlets are the third term of each triplet with a sign change. (cid:101) e A e A ≡ (cid:104) (cid:101) A · ( N (cid:48) · M − N · M (cid:48) ) · (cid:101) A (cid:105) (cid:101) e A e B ≡ (cid:104) (cid:101) A · ( N · M − N (cid:48) · M (cid:48) ) · (cid:101) B (cid:105) (cid:101) e B e A ≡ (cid:104) (cid:101) B · ( N (cid:48) · M (cid:48) − N · M ) · (cid:101) A (cid:105) (cid:101) e B e B ≡ (cid:104) (cid:101) B · ( N · M (cid:48) − N (cid:48) · M ) · (cid:101) B (cid:105) (6.41)In our examination of the standard formulation of the fluxes and superpotential we noted that not allof these SL(2 , Z ) S multiplets are Bianchi constraints, some of them are non-zero and measure tadpolecontributions due to branes and their S duality images. Which type of constraint a particular multipletfell into was given in (5.43) and we would expect a similar behaviour in these multiplets. The simplesttadpole considered was the C potential which coupled to the external space filling D3 branes whoseflux contribution H ∧ F ∼ (cid:98) F ∧ F could be written in terms of derivatives as being proportional to (cid:101) ν ∧ ( D D (cid:48) − D (cid:48) D )( (cid:101) ν ). The M image of this is obtained by replacing the W derivatives with thoseof W and the flux polynomials associated to that appear in the (cid:101) e B e B singlet and can be written interms of derivatives as (cid:101) ν ∧ ( D D (cid:48) − D (cid:48) D )( (cid:101) ν ). This construction does not have a straight forwarddefinition in terms of the F and (cid:98) F fluxes of W , in the same manner that SL(2 , Z ) S images of Type IIAfluxes do not have a straight forward expression in terms of fluxes due to the way they are defined.How the tadpole contributions are to be viewed in terms of the action of M on the branes of the TypeIIB theory is a question we shall not address other than to comment that (cid:101) ν ∧ ( D D (cid:48) − D (cid:48) D )( (cid:101) ν )contains the fluxes found on branes other than the D3s in the formulation of the W superpotentialincluding extended objects which are the NS-NS counterparts of the D branes.47 .4 Reduced superpotential expression If we assume that the formulation of W is as valid as that of W then we can express the superpotentialin a way which is symmetric in its treatment of the moduli spaces. By using (3.29) as a guide we haveobtained the relationship between the flux matrices of W in (6.3) and those of W in (6.4). Tomotivate this further we consider a superpotential-like expression W D which is defined as a scalarproduct involving Ψ, whose entries are the moduli vectors T and U , and a matrix D . We do not treat D as the matrix associated to a derivative, only a linear operator on the cohomology bases so thatthe associated flux matrices M D and N D are independent but we use notation which follows previoussuperpotential-like scalar products. W D ≡ Ψ (cid:62) · h · C · D · g · Ψ D = (cid:18) M D N D (cid:19) C = (cid:18) C (cid:101) C (cid:19) (6.42)With h , = h , the complexification matrices are equal, C = (cid:101) C , and so C = I ⊗ C . Expanding W D out in terms of the individual moduli sectors results in a pair of terms, one of the form seen in W andthe other of the form seen in W . W D = T (cid:62) · h ν · C · M D · g a · U + U (cid:62) · h a · C · N D · g ν · T (6.43)This is in contrast to previous superpotential expressions considered, where the matrices Ω and (cid:102) aredefined with a projection matrix P ± so that one of the two terms is projected out. In general thereare two contributions to the superpotential due to the different flux sectors so if the two moduli spacesare equivalent we would expect it to be possible to express the full superpotential in the same manneras ( ?? ). On the assumption that W ≡ M ( W ) = W the superpotential W , which is normally writtenas having the form of W , is proportional to W + W and the proportionality constant can be gaugedto 1. W = W + W = (cid:32) T (cid:62) · h ν · C · M · g a · U + U (cid:62) · h a · (cid:101) C · N · g ν · T + T (cid:62) · h ν · C (cid:48) · M (cid:48) · g a · U + U (cid:62) · h a · (cid:101) C (cid:48) · N (cid:48) · g ν · T (cid:33) (6.44)Given the fact M = Id by construction we have that W = W + M ( W ) is M invariant and thereforethe two moduli spaces are treated in the same manner. Comparing the scalar product expression for W with (6.43) we can see that the two pairs of terms, relating to primed and non-primed flux matrices,suggest we consider a pair of matrices where there is no mixing between the primed and non-primedflux matrices. D = (cid:18) M N (cid:19) = (cid:18) I
00 0 (cid:19) (cid:18) M N (cid:19) + (cid:18) I (cid:19) (cid:18) M N (cid:19) ≡ P + · D + P − · D D (cid:48) = (cid:18) M (cid:48) N (cid:48) (cid:19) = (cid:18) I
00 0 (cid:19) (cid:18) M (cid:48) N (cid:48) (cid:19) + (cid:18) I (cid:19) (cid:18) M (cid:48) N (cid:48) (cid:19) ≡ P + · D (cid:48) + P − · D (cid:48) Due to the non-trivial mixing between the NS-NS and R-R sectors in (6.28) and (6.29) the distinctionbetween the two flux sectors is no longer a simple one but with Ψ = Ψ (cid:62) · h · f = (cid:102) + Ω we are ableto express the superpotential in a way which treats the two moduli spaces in the same manner, using D ◦ C (Ψ) ≡ D (cid:0) C (Ψ) (cid:1) = Ψ (cid:62) · C · D · f . W = Ψ (cid:62) · h · (cid:0) C · D + C (cid:48) · D (cid:48) (cid:1) · g · Ψ= g (cid:16) Ψ , ( D ◦ C + D (cid:48)◦ C (cid:48) )(Ψ) (cid:17) = (cid:90) W (cid:16) (cid:102) + Ω (cid:17) ∧ (cid:16) D ◦ C + D (cid:48)◦ C (cid:48) (cid:17)(cid:16) (cid:102) + Ω (cid:17) (6.45)In our examination of S duality we found it convenient to consider the invariance of C · M + C (cid:48) · M (cid:48) = F (cid:58) − S (cid:98) F (cid:58) , from which we could deduce the S duality transformation properties of the fluxes and (5.30).Now that we have combined the two moduli spaces we can extend this further. C · D + C (cid:48) · D (cid:48) ≡ F − S (cid:98) F ⇒ (cid:18) CC (cid:48) (cid:19) · (cid:0) D D (cid:48) (cid:1) = (cid:18) − S (cid:19) · (cid:16) F (cid:98) F (cid:17) (6.46)48ith C = C ⊗ I and likewise for C (cid:48) the same relationship between the matrices as (5.30) occurs, whichcan be rephrased as (5.31) except that the dimensions of the matrices have increased so we denote A ⊗ I by A and likewise for B . (cid:18) F (cid:98) F (cid:19) = (cid:18) A BB A (cid:19) (cid:18) DD (cid:48) (cid:19) ⇒ (cid:18) DD (cid:48) (cid:19) = (cid:18) A BB A (cid:19) (cid:18) F (cid:98) F (cid:19) (6.47)This allows us to much more succinctly state the S duality transformation properties of the fluxes inthis moduli symmetric formulation under S → aS + bcS + d , in line with (5.32). (cid:18) DD (cid:48) (cid:19) → (cid:18) A BB A (cid:19) (cid:18) a bc d (cid:19) (cid:18)
A BB A (cid:19) (cid:18) DD (cid:48) (cid:19) (6.48)This result combined with the flux matrix definitions of D and D (cid:48) and the I term in A and B againillustrates that the M ( (cid:48) )1 and N ( (cid:48) )2 have equivalent SL(2 , Z ) S transformations, as noted in (6.36) andrequired by definition (6.23). With this definition of F and (cid:98) F we can reduce the superpotential downto a simple form. W = Ψ (cid:62) · h · (cid:16) F − S (cid:98) F (cid:17) · g · Ψ (6.49)This formulation makes S duality transformation properties and the symmetry in moduli treatmentmanifest. Z × Z orientifold In their discussion of non-geometric fluxes induced by T duality [12, 13, 14] use the isotropic caseof the Z × Z orientifold, henceforth M Z , as their explict example for constructing non-geometricvacua and this is extended to include S duality induced constraints in both Type IIA and Type IIBon the non-isotropic space in [21], with particular S and T duality invariant vacua constructed inthe isotropic case. The non-geometric fluxes have been analysed in terms of Lie algebras for casesinvolving only T duality [19] and cases where S duality is included [22], where the latter demonstrateshow non-geometric flux constraints can be examined through the use of integrability conditions. Thetwo Z orbifold groups factorise M Z into three two dimensional tori T → ⊗ n =1 T n and thus inheritsthe property of being its own mirror dual from the two dimensional tori. The orbifold symmetries are such that any 3-form must have an index on each torus and from this weconstruct our symplectic basis given in Table 2. From the definition of the H (3) basis the coefficientsof the holomorphic 3-form Ω can be written in terms of the period matrix entries.Ω = dz ∧ dz ∧ dz = ( η + τ η ) ∧ ( η + τ η ) ∧ ( η + τ η )= α + τ i α i + τ τ τ τ j β j − τ τ τ β = a − τ i b i + τ τ τ τ j b j − τ τ τ b The action of the orbifold groups are such that only those 2 n -forms with indices on n tori are allowedand we define the even dimensional cohomology bases in Table 3 to be as simple as possible. J ≡ B + iJ = T η + T η + T η = T ω + T ω + T ω = T (cid:101) ν + T (cid:101) ν + T (cid:101) ν The K¨ahler moduli holomorphic form can also be defined and takes the same general form as in thecomplex structure case, but for the sign changes in half the terms. (cid:102) = e J = ω + T a ω a + T T T T b (cid:101) ω b + T T T (cid:101) ω = ν + T a (cid:101) ν a + T T T T b ν b + T T T (cid:101) ν α α α − β β β β η η η η η η η η a − b − b − b − b a a a Table 2: Sympletic bases of H (3) . ω ω ω ω (cid:101) ω (cid:101) ω (cid:101) ω (cid:101) ω η η η η η η η ν (cid:101) ν (cid:101) ν (cid:101) ν (cid:101) ν ν ν ν Table 3: Commutitative bases of (cid:101) H (3) . Given an explicit basis we are able to consider the equivalence of the expressions in (3.16). In the caseof the terms responsible for the sequence H (1 , → H (3) → H (2 , we consider the case of those termswhich are responsible for ω → α and ω → β . η ι : ω = η → η = α − η ι : ω = η → η = β In general for terms whose action on (cid:101) H (3) is ω a → α I the corresponding action on H (3) will be to map β I to some element in H (2 , . Hence we apply − η ι to α and − η ι to β . η ι : β = η → − η = − (cid:101) ω − η ι : α = η → η = (cid:101) ω We therefore have two ways of expressing these η ab ι c operators in terms of the cohomology bases andTables 2 and 3 allow for conversion into the alternative basis. η ι = α ι ω = − (cid:101) ω ι β = − b ι e ν = − ν ι a − η ι = β ι ω = (cid:101) ω ι α = a ι e ν = − ν ι b The other cases for a i and b j follow the same pattern, it is straightforward to see that acting theoperators onto other elements of H (1 , and H (3) are zero and so we obtain half of the results givenin (3.16). The second half we obtain by considering those terms which are responsible for H (2 , → H (3) → H (1 , . − η ι ι : (cid:101) ω = η → η = β : α = η → − η = − ω η ι ι : (cid:101) ω = η → η = α : β = η → η = ω We therefore have two ways of expressing these η ab ι c operators in terms of the cohomology bases andTables 2 and 3 allow for conversion into the alternative basis. − η ι ι = β ι e ω = − ω ι α = a ι ν = (cid:101) ν ι b η ι ι = α ι e ω = ω ι β = − b ι ν = (cid:101) ν ι a The other cases for a i and b j follow the same pattern and give the second half of the results in (3.16).50 .3 Flux components There are a number of different flux structures we have considered. • The Type IIA NS-NS fluxes F given in Table 5 obtained from (3.24). • The Type IIA R-R flux F given in Table 4 obtained from (4.5). F (0)0 F (0)1 F (0)2 F (0)3 F F F F F (0) A −F −F −F −F F F F F F B (0) Table 4: Component labels for the fluxes of F RR • The Type IIB NS-NS fluxes F n and (cid:98) F m given in Table 6 from (3.38). We set L and K equalto the canonical sympletic forms of their respective dimensions and determine the fluxes of D interms of G by equating K¨ahler moduli coefficients. G ( T ω ) = D ( T ν ) G ( −T i (cid:101) ω i ) = D ( T i (cid:101) ν i ) G ( T (cid:101) ω ) = D ( T (cid:101) ν ) G ( T j ω j ) = D ( T j ν j ) (7.1)This choice results in those fluxes coupling to T i acting on J c = −T i (cid:101) ω i . The redefinition of thesympletic basis simplifies our analysis and to illustrate this we consider the first case G ( T ω ) = D ( T ν ) and what it reduces to.13! (cid:98) F pqr η pqr = F (0) A α A − F B (0) β B (7.2)The redefinition of the sympletic basis reverts our notation back to the ( α A , β B ) basis and therelationship between the (cid:98) F pqr and the F (0) A etc follow from comparing coefficients. • The Type IIB alternate fluxes in the D of W given in Table 7. The fluxes associated to W do not allow for them to be written in terms of coefficients in the same manner as the F and (cid:98) F . They lack a simple tensor formulation such as F abc η abc , a problem also encountered with theR-R fluxes of Type IIA in Table 4. In this work we have seen that using the cohomology bases for the construction of the fluxes simplifiesthe analysis of T and S dualities considerably and naturally takes into account the properties of theinternal space, rather than working with Λ ( ∗ ) and Λ ( ∗ ) ∗ . Using Λ ( ∗ ) and Λ ( ∗ ) ∗ as the bases for the fluxeshas the advantage that no additional work was required to find the action F n : Λ (3) → Λ (6 − n ) if given F n : Λ (2 n ) → Λ (3) and the resultant nilpotency conditions are complete. However, this has the disad-vantages that it does not make the number of independent fluxes on M manifest, the superpotentialcannot be stated in terms of the components easily and the Type IIA R-R flux sector does not havea simple factorisation for its covariant derivative. All of these issues were resolved by restricting ourattention to the cohomology bases, which were sufficient to entirely describe the independent fluxesand moduli coefficients in the superpotentials of both Type IIA and Type IIB. As seen in the literaturethe price paid for this was that not all Bianchi constraints are captured by the formulation and due tothe fact the fluxes broke the closure of the elements of H ( ∗ ) the distinction between the lightest modesand the massive modes was lost. The assumption that the cohomology bases are sufficient to describethe space could be justified if the H ( ∗ ) basis elements are eigenforms of the Laplacians of (A.3) withsmall eigenvalues. 51 (0)0 F (0)1 F (0)2 F (0)3 F F F F F (0) I F F F F F F F F F J (0) F ( a )0 F ( a )1 F ( a )2 F ( a )3 F ( a )0 F ( a )1 F ( a )2 F ( a )3 F (1) I −F −F + F + F + F + F −F −F F (1) J F (2) I −F + F −F + F + F −F + F −F F (2) J F (3) I −F + F + F −F + F −F −F + F F (3) J F (0)0 F (0)1 F (0)2 F (0)3 F (0)0 F (0)1 F (0)2 F (0)3 F (0) I F F F F F F F F F (0) J F ( a )0 F ( a )1 F ( a )2 F ( a )3 F a ) F a ) F a ) F a ) F (1) I −F −F + F + F −F −F + F + F F J (1) F (2) I −F + F −F + F −F + F −F + F F J (2) F (3) I −F + F + F −F −F + F + F −F F J (3) Table 5: Alternate component labels for fluxes F n F (0)0 F (0)1 F (0)2 F (0)3 F F F F F (0) A (cid:98) F (cid:98) F (cid:98) F (cid:98) F (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F F B (0) F ( i )0 F ( i )1 F ( i )2 F ( i )3 F i ) F i ) F i ) F i ) F (1) A − F + F − F − F + F + F − F − F F B (1) F (2) A − F − F + F − F + F − F + F − F F B (2) F (3) A − F − F − F + F + F − F − F + F F B (3) F (0)0 F (0)1 F (0)2 F (0)3 F (0)0 F (0)1 F (0)2 F (0)3 F (0) A (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F F (0) B F ( i )0 F ( i )1 F ( i )2 F ( i )3 F ( i )0 F ( i )1 F ( i )2 F ( i )3 F (1) A + F + F − F − F + F − F + F + F F (1) B F (2) A + F − F + F − F + F + F − F + F F (2) B F (3) A + F − F − F + F + F + F + F − F F (3) B Table 6: Alternate component labels for fluxes F n and (cid:98) F m .52 (0)0 F (0)1 F (0)2 F (0)3 F F F F F (0) I + (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F F J (0) F ( a )0 F ( a )1 F ( a )2 F ( a )3 F a ) F a ) F a ) F a ) F (1) I + F + F − F − F + F − F + F + F F J (1) F (2) I + F − F + F − F + F + F − F + F F J (2) F (3) I + F − F − F + F + F + F + F − F F J (3) F (0)0 F (0)1 F (0)2 F (0)3 F (0)0 F (0)1 F (0)2 F (0)3 F (0) I − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F − (cid:98) F + (cid:98) F + (cid:98) F + (cid:98) F F (0) J F ( b )0 F ( b )1 F ( b )2 F ( b )3 F ( b )0 F ( b )1 F ( b )2 F ( b )3 F (1) I + F − F + F + F − F − F + F + F F (1) J F (2) I + F + F − F + F − F + F − F + F F (2) J F (3) I + F + F + F − F − F + F + F − F F (3) J Table 7: Explicit M defined components for fluxes F n We observed the differences and similarities between the two Type II constructions. The deriva-tives act on the holomorphic forms of different moduli spaces in Type IIA, thus preventing Type IIAfrom being self S-dual. Type IIB possesses self S duality but the manner in which the fluxes coupleto the K¨ahler moduli is different from the Type IIA case. This fact required additional attention toensure that when the Type IIB superpotential is written in terms of the natural holomorphic formsthe Bianchi constraints of the resultant covariant derivatives are equal to those obtained from its TypeIIA mirror dual. The tadpole constraints were reformalised into a representation similar to Bianchiconstraints but with derivatives from each flux sector. In the case of Type IIB the inclusion of R-Rpartners to all T duality induced fluxes suggested an extension to the tadpole constraints which wasfurther justified in the Type IIA construction.Having used the cohomology bases to provide a convenient notation to examine Type II super-potentials under combinations T and S dualities we noted that for particular M it is possible forreformulate the superpotential such that the roles of the two moduli spaces are exchanged. Such asymmetry is not clear without including all superpotential contributions under T and S dualities. Thefluxes, constraints and multiplets of the alternative formulation have the same schematic form as theusual formulation and we have argued that for M = W these two formulations are equally valid. InType IIB the standard flux multiplets are defined as D ( J ( n ) ), with T duality inducing contributionsfor each value of n , but the alternative set of fluxes were defined as D ( J ( n ) ). Since the flux contribu-tions for each n are not synonymous with those of D ( J ( n ) ) the duality which induces the sequenceis not T duality, but one which we referred to as T (cid:48) duality. Since the dualities induce different fluxmultiplet sequences they have different covariant derivatives, D (cid:54) = D , and we demonstrated that as aresult their respective nilpotency constraints are not equivalent. This reformulation was not justifiedrigorously but rather based on arguments of symmetry. We considered the explicitly case of the Z × Z orientifold whose factorisation into three two dimensional sub-tori results in the complex structure andK¨ahler moduli having analogous roles. There are a number choices in how this work might be extendedand used : • We found a large number of additional terms which could in principle contribute to tadpole53onstraints but how they do this and what the physical interpretation in terms of D-branes, NS-branes and O-planes was not considered. Whether this can be done on the effective supergravitylevel or requires the contributions to descend from a full string theory is not clear and in the lattercase the fact some of the fluxes are non-geometric is a large obstruction as stringy constructionsof spaces with non-geometric fluxes is an open problem at present. • We have only used symmetries on the level of the effective moduli theory to motivate the existenceof M , rather than rigorously derived it from a string theory and a stringy proof to its existence(if it indeed exists) might lead to a greater understanding of particular internal spaces. • Assuming the existence of M the additional constraints restrict the number of independent fluxeson the space and thus narrows down the number of possible vacua a particular M = W mightbe able to have. The implications for this in terms of phenomenology and moduli stablisationwould be worth investigating. • The enhanced symmetry group of g = g a ⊕ g ν compared to g a and g ν individually would furtherconnect the NS-NS and R-R flux sectors and the implications this would have for the fluxes andpossible vacua constructed from resultant superpotentials are not immediately clear. Construc-tions of the (cid:101) H (3) basis which are manifestly symplectic [23] require Grassman valued bases andhow this affects flux compactifications is unknown.The fluxes of the Z × Z orientifold given in Section 7 allow for these unresolved problems to beexamined in more explicit detail and may lead to a more general understanding of flux compactifica-tions. Acknowledgements
The author would like to thank Anthony Ashton and James Gray for useful discussions, as well asSouthampton University for the support of a scholarship.
A Moduli masses
If the approximation to include only ‘light’ modes spanned by H ( ∗ ) is to be valid then the eigenvaluesof the H ( ∗ ) bases in terms of the Laplacian are to be small. This is complicated by the fact we haveconstructed a pair of derivatives and so can build two different Laplacians, which is further complicatedwhen S duality is included. To that end we construct the Laplacian operator in terms of the fluxes in D and D (cid:48) , ∆ D ≡ D D † + D † D using φ ∈ (cid:101) H (3) and χ ∈ H (3) in Type IIA on M . D ( φ ) = φ (cid:62) · h ν · M · h a · e ( a ) D ( χ ) = χ (cid:62) · h a · N · h ν · e ( ν ) D † ( φ ) = φ (cid:62) · h ν · M † · h a · e ( a ) D † ( χ ) = χ (cid:62) · h a · N † · h ν · e ( ν ) It should be noted that M † and N † are not the hermitian conjugates of M and N but the flux matricesassociated to D † instead. φ (cid:62) · h ν · M · χ = (cid:10)(cid:10) χ, D ( φ ) (cid:11)(cid:11) ≡ (cid:10)(cid:10) φ, D † ( χ ) (cid:11)(cid:11) = χ (cid:62) · h a · N † · φ ⇒ N † = h a · M (cid:62) · h ν χ (cid:62) · h a · N · φ = (cid:10)(cid:10) φ, D ( χ ) (cid:11)(cid:11) ≡ (cid:10)(cid:10) χ, D † ( φ ) (cid:11)(cid:11) = φ (cid:62) · h ν · M † · χ ⇒ M † = h ν · N (cid:62) · h a (A.1)From this the two terms of the Laplacian can be constructed for both H (3) and (cid:101) H (3) . D † D ( φ ) = φ (cid:62) · h ν · M · h a · N † · h ν · e ( ν ) = φ (cid:62) · h ν · M · M (cid:62) · e ( ν ) DD † ( φ ) = φ (cid:62) · h ν · M † · h a · N · h ν · e ( ν ) = φ (cid:62) · N (cid:62) · N · h ν · e ( ν ) D † D ( χ ) = χ (cid:62) · h a · N · h ν · M † · h a · e ( a ) = χ (cid:62) · h a · N · N (cid:62) · e ( a ) DD † ( χ ) = χ (cid:62) · h a · N † · h ν · M · h a · e ( a ) = χ (cid:62) · M (cid:62) · M · h a · e ( a ) (A.2)54sing (3.41) the Laplacian for each flux sector can be written entirely in terms of the (cid:101) H (3) → H (3) defined fluxes, in the same manner as the Bianchi constraints for T duality could. (cid:10)(cid:10) φ, ( DD † + D † D )( φ ) (cid:11)(cid:11) = φ (cid:62) · h ν · (cid:16) M · M (cid:62) + N (cid:62) · N (cid:17) · φ = φ (cid:62) · h ν · (cid:16) M · M (cid:62) + g a · M · M (cid:62) · g (cid:62) a (cid:17) · φ (cid:10)(cid:10) χ, ( DD † + D † D )( χ ) (cid:11)(cid:11) = χ (cid:62) · h a · (cid:16) M (cid:62) · M + N · N (cid:62) (cid:17) · χ = χ (cid:62) · h a · (cid:16) M (cid:62) · M + g ν · M (cid:62) · M · g (cid:62) ν (cid:17) · χ (A.3)Therefore if M · M (cid:62) = 0 = M (cid:62) · M the H ( p ) are D -harmonic in terms of the NS-NS fluxes of D beingnon-zero and the R-R sector has the same result but M → M (cid:48) . If both ∆ D and ∆ D (cid:48) have the H ( p ) are harmonic then we would expect the distinction between the moduli and the ‘heavy’ fields to beenough to make the analysis of the fluxes using the cohomology bases consistent. B Interior form identities
In order to derive the action of N given M we generalise the ι τ ( η σ ) = δ στ to the basis elements of H ( ∗ ) and its interior form dual. B.1 H (3) basis The basis of H (3) is defined to be sympletic, g ( a I , b J ) = δ JI and from this basis we define a set ofinterior forms. H (3) = (cid:104) a I , b J (cid:105) , (cid:90) M a I ∧ b J = δ JI , ι a J ( a I ) = δ JI = ι b I ( b J ) (B.1)The 3-forms can be rewritten in terms of the general space of 3-forms and a set of coefficient, a I =( a I ) abc η abc and b J = ( b J ) ijk η ijk , with the sympletic definition of the basis defining a set of constraintson these coefficients. δ JI = (cid:90) M a I ∧ b J = ( a I ) abc ( b J ) ijk (cid:90) M η abc η ijk = ( a I ) abc (cid:15) abcijk ( b J ) ijk (B.2)By the same methodology we can rewrite the interior forms in terms of ι abc and sets of coefficients, ι a I = ( A I ) abc ι cba and ι b J = ( B J ) ijk ι kji , which are also constrained in terms of the basis coefficientsby their relationship with the sympletic forms. δ JI = ι a J ( a I ) = ( A J ) abc ( a I ) ijk ι cba η ijk = 3!( A J ) abc ( a I ) abc δ JI = ι b I ( b J ) = ( B I ) abc ( b J ) ijk ι cba η ijk = 3!( B I ) abc ( b J ) abc (B.3)Comparing the three coefficient expansions for δ JI we obtain the ( A J ) abc and ( B I ) abc in terms of the( a I ) abc , ( b J ) abc and the antisymmetric (cid:15) .( A J ) abc = 13! (cid:15) abcijk ( b J ) ijk , ( B I ) abc = 13! (cid:15) abcijk ( a I ) ijk (B.4)With these explicit expressions for the coefficients we can construct ι a J ( b I ) and ι b I ( a J ) in terms ofthe a I and b J components. ι a I ( b J ) = (cid:15) abcijk ( b I ) ijk ι cba ( b J ) pqr η pqr = ( b I ) ijk (cid:15) abcijk ( b J ) abc ι b J ( a I ) = (cid:15) abcijk ( a J ) ijk ι cba ( a I ) pqr η pqr = ( a J ) ijk (cid:15) abcijk ( a I ) abc (B.5)Converting the antisymmetric tensor back into an integral over M of the 6-form η abcdef these expres-sions can be written entirely in terms of the symplectic basis. ι a I ( b J ) = ( b I ) ijk (cid:15) abcijk ( b J ) abc = (cid:90) M ( b J ) abc η abc ( b I ) ijk η ijk = (cid:90) M b J ∧ b I = 0 (B.6)55y the same method we obtain the second expression. ι b J ( a I ) = ( a J ) ijk (cid:15) abcijk ( a I ) abc = (cid:90) M ( a I ) abc η abc ( a J ) ijk η ijk = (cid:90) M a I ∧ a J = 0 (B.7)With these integral expressions we can see the relationship between the ι α and ι β and integrating overthe 3-cycles of M . B.2 (cid:101) H (3) basis In the case of (cid:101) H (3) the ω a and (cid:101) ω b are not in the same cohomologies and so such expressions as ι ω a ( (cid:101) ω b )do not reduce to scalar quantities. However particular expressions of interest can be obtained byexpressing the (cid:101) ω b in terms of the ω a as (cid:101) ω a ≡ f abc ω b ∧ ω c , with the f abc coefficients being related tothe K¨ahler intersection numbers. δ ba = g ba = (cid:90) M ω a ∧ (cid:101) ω b = (cid:90) M ω a ∧ ω c ∧ ω d f bcd = κ acd f bcd (B.8)Using this allows for ι ω a ( (cid:101) ω b ) and ι e ω b ( ω a ) to be simplified, once we make use of an additional identityrelating interior forms and p -forms. Given a general p -form λ we suppose it has factorisation λ = ξ ∧ ζ and consider their associated interior forms, under the map ϕ : λ → ι λ . ϕ : λ = ξ ∧ ζ → ι λ = ι ζ ι ξ ⇒ (cid:101) ω a = f abc ω b ∧ ω c → ι e ω a = f abc ι ω c ι ω b (B.9)With this factorisation of (cid:101) ω b and the associated interior forms we can easily express the remainingcombination of interior forms and p -forms, where we make use of the fact f abc = f acb . ι ω a (cid:101) ω b = ι ω a ( f bcd ω c ∧ ω d ) = 2 f bcd ω c ∧ ι ω a ( ω d ) = 2 f bca ω c ι e ω b ω a = ι ( f bcd ω c ∧ ω d ) ω a = f bcd ι ω d ι ω c ω a = f bad ι ω d (B.10) C Flux matrix identities
C.1 Type IIB SL (2 , Z ) S action Given an SL(2 , Z ) S transformation on the flux matrices of Type IIB they transform in the followingways : S → aS + bcS + d ⇒ (cid:40) (cid:18) MM (cid:48) (cid:19) → (cid:18) A · ( a M + b M (cid:48) ) + B · ( d M + c M (cid:48) ) A · ( c M + d M (cid:48) ) + B · ( b M + a M (cid:48) ) (cid:19)(cid:18) NN (cid:48) (cid:19) → (cid:18) ( a N + b N (cid:48) ) · A + ( d N + c N (cid:48) ) · B ( c N + d N (cid:48) ) · A + ( b N + a N (cid:48) ) · B (cid:19) (C.1) C.2 Flux matrix constraints on H (3) As used in the main body of work we compress notation by replacing · h a · by (cid:5) and consider how theflux matrix expressions associated to D i D j : H (3) → H (3) transform under (C.1). M · h a · N → (cid:16) A ( a M + b M (cid:48) ) + B ( d M + c M (cid:48) ) (cid:17) (cid:5) (cid:16) ( a N + b N (cid:48) ) A + ( d N + c N (cid:48) ) B (cid:17) = A · (cid:16) a M (cid:5) N + ab M (cid:5) N (cid:48) + ab M (cid:48) (cid:5) N + b M (cid:48) (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) ad M (cid:5) N + ac M (cid:5) N (cid:48) + bd M (cid:48) (cid:5) N + bc M (cid:48) (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) ad M (cid:5) N + bd M (cid:5) N (cid:48) + ac M (cid:48) (cid:5) N + bc M (cid:48) (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) d M (cid:5) N + cd M (cid:5) N (cid:48) + cd M (cid:48) (cid:5) N + c M (cid:48) (cid:5) N (cid:48) (cid:17) · B (C.2)56 · h a · N (cid:48) → (cid:16) A ( a M + b M (cid:48) ) + B ( d M + c M (cid:48) ) (cid:17) (cid:5) (cid:16) ( c N + d N (cid:48) ) A + ( b N + a N (cid:48) ) B (cid:17) = A · (cid:16) ac M (cid:5) N + ad M (cid:5) N (cid:48) + bc M (cid:48) (cid:5) N + bd M (cid:48) (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) ab M (cid:5) N + a M (cid:5) N (cid:48) + b M (cid:48) (cid:5) N + ab M (cid:48) (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) cd M (cid:5) N + d M (cid:5) N (cid:48) + c M (cid:48) (cid:5) N + cd M (cid:48) (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) bd M (cid:5) N + ad M (cid:5) N (cid:48) + bc M (cid:48) (cid:5) N + ac M (cid:48) (cid:5) N (cid:48) (cid:17) · B (C.3) M (cid:48) · h a · N → (cid:16) A ( c M + d M (cid:48) ) + B ( b M + a M (cid:48) ) (cid:17) (cid:5) (cid:16) ( a N + b N (cid:48) ) A + ( d N + c N (cid:48) ) B (cid:17) = A · (cid:16) ac M (cid:5) N + bc M (cid:5) N (cid:48) + ad M (cid:48) (cid:5) N + bd M (cid:48) (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) cd M (cid:5) N + c M (cid:5) N (cid:48) + d M (cid:48) (cid:5) N + cd M (cid:48) (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) ab M (cid:5) N + b M (cid:5) N (cid:48) + a M (cid:48) (cid:5) N + ab M (cid:48) (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) bd M (cid:5) N + bc M (cid:5) N (cid:48) + ad M (cid:48) (cid:5) N + ac M (cid:48) (cid:5) N (cid:48) (cid:17) · B (C.4) M (cid:48) · h a · N (cid:48) → (cid:16) A ( c M + d M (cid:48) ) + B ( b M + a M (cid:48) ) (cid:17) (cid:5) (cid:16) ( c N + d N (cid:48) ) A + ( b N + a N (cid:48) ) B (cid:17) = A · (cid:16) c M (cid:5) N + cd M (cid:5) N (cid:48) + cd M (cid:48) (cid:5) N + d M (cid:48) (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) bc M (cid:5) N + ac M (cid:5) N (cid:48) + bd M (cid:48) (cid:5) N + ad M (cid:48) (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) bc M (cid:5) N + bd M (cid:5) N (cid:48) + ac M (cid:48) (cid:5) N + ad M (cid:48) (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) b M (cid:5) N + ab M (cid:5) N (cid:48) + ab M (cid:48) (cid:5) N + a M (cid:48) (cid:5) N (cid:48) (cid:17) · B (C.5) C.3 Alternative flux relations
The M images of the D i D j : H (3) → H (3) expressions have the following expressions in terms of the D i D j flux matrices. M (cid:5) N = m · (cid:101) A (cid:5) (cid:101)
A · n + m (cid:48) · (cid:101) B (cid:5) (cid:101)
B · n (cid:48) = A · (cid:16) M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) B (cid:5) N (cid:48) (cid:17) · A + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) B (cid:5) N (cid:48) (cid:17) · A + A · (cid:16) M · (cid:101) A (cid:5) N (cid:48) + M (cid:48) · (cid:101) B (cid:5) N (cid:17) · B + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N (cid:48) + M · (cid:101) B (cid:5) N (cid:17) · B (C.6) M (cid:48) (cid:5) N (cid:48) = m · (cid:101) B (cid:5) (cid:101)
B · n + m (cid:48) · (cid:101) A (cid:5) (cid:101)
A · n (cid:48) = A · (cid:16) M (cid:48) · (cid:101) A (cid:5) N (cid:48) + M · (cid:101) B (cid:5) N (cid:17) · A + B · (cid:16) M · (cid:101) A (cid:5) N (cid:48) + M (cid:48) · (cid:101) B (cid:5) N (cid:17) · A + A · (cid:16) M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) B (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) B (cid:5) N (cid:48) (cid:17) · B (C.7) M (cid:5) N (cid:48) = m · (cid:101) A (cid:5) (cid:101)
A · n (cid:48) + m (cid:48) · (cid:101) B (cid:5) (cid:101)
B · n = A · (cid:16) M · (cid:101) A (cid:5) N (cid:48) + M (cid:48) · (cid:101) B (cid:5) N (cid:17) · A + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N (cid:48) + M · (cid:101) B (cid:5) N (cid:17) · A + A · (cid:16) M · (cid:101) A (cid:5) N + M (cid:48) · (cid:101) B (cid:5) N (cid:48) (cid:17) · B + B · (cid:16) M (cid:48) · (cid:101) A (cid:5) N + M · (cid:101) B (cid:5) N (cid:48) (cid:17) · B (C.8)57 (cid:48) (cid:5) N = m · (cid:101) B (cid:5) (cid:101)
B · n (cid:48) + m (cid:48) · (cid:101) A (cid:5) (cid:101)
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