M-theory on Calabi-Yau Five-Folds
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION
Imperial/TP/08/KSS/02; CERN-PH-TH/2008-201; arXiv:0810.2685 [hep-th]
M-theory on Calabi-Yau Five-Folds
Alexander S. Haupt a,b,c , Andre Lukas d , K. S. Stelle a,c,e a Theoretical Physics Group, Imperial College London,Prince Consort Road, London SW7 2AZ, U.K. b Institute for Mathematical Sciences, Imperial College London,53 Prince’s Gate, London SW7 2PG, U.K. c Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut),Am M¨uhlenberg 1, D-14476 Potsdam, Germany d Rudolf Peierls Centre for Theoretical Physics, Oxford University,1 Keble Road, Oxford, OX1 3NP, U.K. e Theory Division, Physics Department, CERN, CH-1211 Geneva 23, SwitzerlandE-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the compactification of M-theory on Calabi-Yau five-folds and theresulting N = 2 super-mechanics theories. By explicit reduction from 11 dimensions,including both bosonic and fermionic terms, we calculate the one-dimensional effectiveaction and show that it can be derived from an N = 2 super-space action. We findthat the K¨ahler and complex structure moduli of the five-fold reside in 2 a and 2 b super-multiplets, respectively. Constrained 2 a super-multiplets arise from zero-modes of theM-theory three-form and lead to cross-couplings between 2 a and 2 b multiplets. Fermioniczero modes which arise from the (1 ,
3) sector of the 11-dimensional gravitino do not havebosonic super-partners and have to be described by purely fermionic super-multiplets inone dimension. We also study the inclusion of flux and discuss the consistency of thescalar potential with one-dimensional N = 2 supersymmetry and how it can be describedin terms of a superpotential. This superpotential can also be obtained from a Gukov-type formula which we present. Supersymmetric vacua, obtained by solving the F-termequations, always have vanishing vacuum energy due to the form of this scalar potential.We show that such supersymmetric solutions exist for particular examples. Two substantialappendices develop the topology and geometry of Calabi-Yau five-folds and the structure ofone-dimensional N = 2 supersymmetry and supergravity to the level of generality requiredfor our purposes. Keywords:
M-Theory, Flux compactifications, Field Theories in Lower Dimensions,Superspaces. ontents
1. Introduction 22. The M-theory low energy effective action 53. Calabi-Yau five-folds 84. Compactification on Calabi-Yau five-folds 12
5. Supersymmetry and Calabi-Yau five-folds 23 N = 2 supersymmetry transformations and multiplets 255.2 The one-dimensional effective action in superspace 27
6. Flux and the one-dimensional scalar potential 30
7. Conclusion and Outlook 38A. Index conventions and spinors 42B. Calabi-Yau five-folds 44
B.1 Basic topological properties 44B.2 Examples of Calabi-Yau five-folds 47B.2.1 Complete intersection Calabi-Yau five-folds 47B.2.2 Torus quotients 54B.3 Some differential geometry on five-folds 54B.4 Five-fold moduli spaces 56B.4.1 Real vs. complex forms 60 C. N = 2 supersymmetry in one dimension 64 C.1 Global N = 2 supersymmetry 65C.2 Local N = 2 supersymmetry 69– 1 – . Introduction The technique of compactification has connected string- and M-theory to a wealth of super-gravity theories in diverse dimensions and has led to important insights into both theoreticaland phenomenological aspects of the theory. Ever since the seminal work [1], compactifi-cations on Calabi-Yau spaces and related constructions have played a central rˆole in thiscontext. While most of this work has concentrated on Calabi-Yau three-folds, primarilyin order to connect string theory to four-dimensional physics, Calabi-Yau four-folds havebeen used, for example in F-theory compactifications [2], and compactification on K3 hasplayed an important rˆole in uncovering elementary duality relations [3, 4]. Calabi-Yaufour-folds have also appeared in string-/M-theory compactifications to two and three di-mensions [5, 6]. To the best of our knowledge, the first time Calabi-Yau five-folds haveappeared in the physics literature was in Ref. [7] where subclasses of those manifolds featurein the discussion of certain vacuum constructions of F-theory and thirteen dimensional S-theory leading to supersymmetric two dimensional N = (1 ,
1) and three dimensional N = 2theories, respectively, and then again more detailed later in Ref. [8] in a similar but moregeneral context.The main purpose of the present paper is to close an apparent gap in the scheme ofM-theory compactifications by considering 11-dimensional supergravity on Calabi-Yau five-folds. Eleven-dimensional supergravity is the only one of the six “known” limits of M-theorywith a sufficient number of physical spatial dimensions to allow for such compactifications(although, Calabi-Yau five-folds can, of course, be used for F-theory compactifications totwo dimensions). M-theory backgrounds based on Calabi-Yau five-folds and their correc-tions induced by higher-order curvature terms have been considered in Ref. [9]. Here, wewill be concerned with the actual compactifications on such backgrounds and the resultingone-dimensional (super-)mechanics theories. Calabi-Yau five-folds reduce supersymmetryby a factor of 1 /
16 and, given the eleven-dimensional theory has 32 real supercharges, oneexpects one-dimensional theories with N = 2 supersymmetry from such reductions.Specifically, we will derive the general form of this one-dimensional N = 2 super-mechanics theory and analyse its relation to the underlying topology and moduli-spacegeometry of the five-folds. The necessary mathematical details regarding the topology andgeometry of five-folds are, to a large extend, analogous to the the well-established three-fold case, and will be systematically developed as a preparation for our reduction. Anothervital ingredient in our discussion is the structure of one-dimensional N = 2 supersymmetrictheories [10]. Although gravity is non-dynamical in one dimension, the component fields ofthe one-dimensional gravity supermultiplet (the lapse function and the gravitino) generateconstraint equations which cannot be ignored. Therefore, we have to consider local one-dimensional N = 2 supersymmetry. Moreover, it turns out that the structure of theone-dimensional theories obtained from M-theory reduction is more general than the super-mechanics theories usually considered in the literature. In the present paper, we, therefore,invest considerable work in order to develop one-dimensional N = 2 supergravity to asufficiently general level.Our work is motivated by a number of general considerations. Reductions of M-– 2 –heory to one dimension have played some rˆole in the attempts to understand quantumM-theory [11, 12] and we hope the results of the present paper may prove useful in thiscontext. Arguments from topological string theory suggest a mini-superspace descriptionof quantum string cosmology [13] along the lines of “traditional” quantum cosmology [14].Mini-superspace quantisation may be applied to the one-dimensional effective theories de-rived in this paper, hoping that this will describe some aspects of quantum M-theory onCalabi-Yau moduli spaces. In the present paper, we will not pursue this explicitly butpossible applications in this direction are currently under investigation. A further motiva-tion is related to the general problem of string vacuum selection and its possible interplaywith cosmology. One aspect of the string vacuum degeneracy, which is often overlooked, isthe ambiguous split of space-time into a number of internal, usually compact dimensionsand four external dimensions. One might speculate that a more plausible geometry for an“initial” state in the early universe is one where all spatial dimension are treated on anequal footing. In the context of M-theory, such “democratic” backgrounds are given by10-dimensional compact Ricci-flat spaces (neglecting flux for the time being) and, hence,Calabi-Yau five-folds provide a natural arena for this discussion. Assuming sufficientlyslow, adiabatic time evolution, the problem of how three large spatial dimensions emergefrom such a background can then be addressed by studying dynamics on the five-fold mod-uli space. This dynamics is, of course, described by the one-dimensional effective actionswe will be deriving in the present paper.As a low-energy effective description of M-theory, 11-dimensional supergravity is cor-rected by an infinite series of higher-order terms which are organised by their associatedpower of β ∼ κ / , where κ is the 11-dimensional Newton constant. Let us first considerthe situation at zeroth order in β , that is for 11-dimensional supergravity in its standardform. A background with vanishing flux, that is with zero anti-symmetric four-form ten-sor field G = dA , and an 11-dimensional metric which consists of a direct product of aRicci-flat Calabi-Yau metric and time, clearly solves the 11-dimensional equations of mo-tion at this lowest order. However, at linear order in β the anomaly cancellation term − β R A ∧ X , where X is the well-known quartic in the curvature two-form, has to beadded to the action. It has been observed in Ref. [9] that X can be non-zero when evalu-ated on Calabi-Yau five-folds backgrounds. In fact, here we will show that it is proportionalto c ( X ), the fourth Chern class of the five-fold X . At order β , the equation of motionfor G is accordingly corrected by a term βX and is, hence, no longer necessarily satisfiedfor G = 0. A further contribution to the A equation of motion can arise from membraneswrapping a holomorphic curve C with cohomology class W = [ C ] in the Calabi-Yau five-fold. Taking into account these contributions, we show the three-form equation of motionleads to a topological consistency condition, required for a solution at order β to exist.It states (modulo factors) that the cohomology class [ G ∧ G ] plus the membrane class W must be proportional to the fourth Chern class, c ( X ). Here, we will consider several waysof solving this consistency equation. First, for vanishing flux, G = 0, and no membranes,the five-folds X needs to have vanishing fourth Chern class c ( X ) and we will show thatsuch five-folds indeed exist. Alternatively, for five-folds with c ( X ) = 0 a compensatingnon-zero flux and/or membrane is required. By means of a number of simple examples we– 3 –ill demonstrate that this can indeed frequently be achieved. In particular, we show thatthe consistency condition can be satisfied for the Calabi-Yau five-fold defined by the zerolocus of a septic polynomial in P . The “septic” is arguably the simplest five-fold and theanalogue of the quintic three-fold in P .The one-dimensional effective action will be calculated as an expansion in powers of β .As a first step we consider the situation at zeroth order in β . Effects from flux or membranesonly come in at order β and are, therefore, not relevant at this stage. In particular, weclarify the relation between Calabi-Yau topology/geometry and the structure of the one-dimensional supermechanics induced by M-theory at this lowest order in β . Many aspectsof this relation are analogous to what happens for compactifications on lower-dimensionalCalabi-Yau manifolds, others, as we will see, are perhaps less expected. The topologyof a Calabi-Yau five-fold X is characterised by six a priori independent Hodge numbers,namely h , ( X ), h , ( X ), h , ( X ), h , ( X ), h , ( X ) and h , ( X ). In analogy with thefour-fold case [15], an index theorem calculation together with the Calabi-Yau condition c ( X ) = 0, leads to one relation between those six numbers. The moduli space of aCalabi-Yau manifold consists (locally) of a direct product of a K¨ahler and a complexstructure moduli space [16]. For Calabi-Yau five-folds, these two parts of the modulispace are associated with the (1 ,
1) and the (1 ,
4) sectors, respectively. As we will see,the associated K¨ahler and complex structure moduli are part of 2 a and 2 b multiplets [17]of one-dimensional N = 2 supersymmetry. A further set of bosonic zero modes originatesfrom the M-theory three form A in the (2 ,
1) sector. We will show that these modes becomepart of constrained 2 a multiplets. This exhausts the list of bosonic zero modes. Expandingthe 11-dimensional gravitino leads to fermionic zero modes in the sectors (1 , q ) where q = 1 , , ,
4. For q = 1 , , ,
3) fermions have no bosonic zero mode partners. We will show thatthis apparent contradiction can be resolved by the introduction of fermionic 2 b multiplets,that is 2 b multiplets with a fermion as their lowest component. With this assignment ofzero modes to super-multiplets, the one-dimensional effective theory is an N = 2 sigmamodel which we present both in its component and superspace form. Some of its featuresare worth mentioning. For example, the sigma model metric for the 2 a multiplets in the(1 ,
1) sector is not the standard Calabi-Yau K¨ahler moduli space metric [16], as is usuallythe case for three-fold compactifications. However, the physical sigma model metric andthe standard Calabi-Yau metric are related in a simple way. Also, it turns out that thesigma model metrics in the (2 ,
1) and (1 ,
3) sector depend inter alia on the K¨ahler moduli,so that we require a coupling of 2 a and 2 b multiplets. As far as we know such a couplingbetween 2 a and 2 b multiplets has not been studied in the context of one-dimensional N = 2supersymmetry before.Then, we proceed to include the order β effects from flux and membranes. We calculatethe scalar potential, including four-form flux, membrane effects and effects from the non-zero Calabi-Yau curvature tensor. The latter requires evaluating the non-topological R terms of M-theory on a five-fold background and we show that these terms can be expressedin terms of the fourth Chern class, c ( X ). Our results indicate that the part of the scalarpotential induced by the (1 , = 2 supersymmetry. Setting the (1 , , , , W . As we will show, this superpotential can beobtained from the Gukov-type formula W ∼ R X G flux ∧ J , where J is the K¨ahler form ofthe Calabi-Yau five-fold. We also present the explicit superpotential and scalar potentialfor a number of particular examples, including the septic in P , and discuss implicationsfor moduli stabilisation and dynamics.The plan of the paper is as follows. In Section 2 we review some basic facts about 11-dimensional supergravity. Some general results on the topology and moduli space geometryof Calabi-Yau five-folds are collected in Section 3. In this section, we also present severalexplicit examples of five-fold backgrounds which solve the M-theory consistency condition.More details on this and derivations of some of the results are given in Appendix B. InSection 4, we perform the reduction of M-theory on such backgrounds at zeroth order in β ,starting with the bosonic action and then including terms bilinear in fermions. Section 5shows that the one-dimensional effective action obtained in this way has indeed two localsupersymmetries and can be written in superspace form. Many of the necessary details andtechnical results on one-dimensional N = 2 supersymmetry and supergravity are collectedin Appendix C. In Section 6, we derive the order β corrections to the effective action andcalculate the scalar potential and superpotential. We conclude in Section 7. Conventionsand notation used throughout this paper are summarised in Appendix A.
2. The M-theory low energy effective action
In this section, we review a number of results on 11-dimensional supergravity and its higher-derivative corrections, focusing on the aspects that will be important for the reduction onCalabi-Yau five-folds. More detailed reviews on the subject can, for example, be found inRefs. [18, 19].The field content of 11-dimensional supergravity consists of the 11-dimensional space-time metric g MN , the anti-symmetric three form tensor field A MNP with field strength G = dA and the gravitino Ψ M , an 11-dimensional Majorana spinor. Here, we denote 11-dimensional curved indices by M, N, . . . = 0 , , . . . ,
10 and their flat, tangent-space coun-terparts by
M , N , . . . . Where possible, we will use differential forms to keep our notationconcise. Our conventions largely follow Ref. [18] and are summarised in Appendix A.We split the 11-dimensional action into four parts as S = S , B + S , F + S , GS + S ,R + . . . . (2.1)Here, the first and second terms are the bosonic and fermionic parts of 11-dimensionalsupergravity [20], respectively, S GS is the Green-Schwarz term related to the cancellation of– 5 –he M5-brane world-volume anomaly [21], S R are the non-topological R terms [22, 23, 24]and the dots indicate additional higher order contributions, which we will not need for ourpurposes.The bosonic part of the action reads [20] S , B = 12 κ Z M (cid:26) R ∗ − G ∧ ∗ G − G ∧ G ∧ A (cid:27) , (2.2)where κ is the 11-dimensional gravitational constant, R is the Ricci scalar of the 11-dimensional metric g and M is the space-time manifold. The equations of motion fromthis bosonic action are given by R MN = 112 G MM ...M G N M ...M − g MN G M ...M G M ...M , (2.3) d ∗ G = − G ∧ G . (2.4)The gravitino dynamics is encoded in the fermionic action S , F = − κ Z M d x √− g n ¯Ψ M Γ MNP D N ( ω )Ψ P + 196 (cid:0) ¯Ψ M Γ MNP QRS Ψ S + 12 ¯Ψ N Γ P Q Ψ R (cid:1) G NP QR + (fermi) o , (2.5)where ¯Ψ M = i Ψ † M Γ . Here and in much of what follows, we omit four-fermi terms. Thecovariant derivative D M is defined by D N ( ω )Ψ P = ( ∂ N + 14 ω N QR Γ QR )Ψ P , (2.6)with the spin connection ω N QR . The corresponding equation of motion for Ψ M readsΓ MNP D N ( ω )Ψ P + 196 (cid:0) Γ MNP QRS Ψ S + 12 g MN Γ P Q Ψ R (cid:1) G NP QR + (fermi) = 0 . (2.7)The action S , B + S , F for 11-dimensional supergravity is invariant under the supersym-metry transformations δ ǫ g MN = 2¯ ǫ Γ ( M Ψ N ) ,δ ǫ A MNP = − ǫ Γ [ MN Ψ P ] ,δ ǫ Ψ M = 2 D M ( ω ) ǫ + 1144 (Γ M NP QR − δ NM Γ P QR ) ǫG NP QR + (fermi) , (2.8)which are parameterised by an 11-dimensional Majorana spinor ǫ .In its rˆole as the low-energy effective theory of M-theory the action S , B + S , F receives an infinite series of higher-order derivative corrections which are organised byinteger powers of the quantity β = 1(2 π ) (cid:18) κ π (cid:19) / . (2.9)– 6 –ne such correction which appears at order β is the Green-Schwarz term S , GS = − (2 π ) β κ Z M A ∧ X , (2.10)where X is a quartic polynomial in the curvature two-form R . It can be convenientlyexpressed in terms of the first and second Pontrjagin classes p ( T M ) and p ( T M ) of thetangent bundle T M of M as X = 148 (cid:18)(cid:16) p (cid:17) − p (cid:19) ,p ( T M ) = − (cid:18) π (cid:19) tr R ,p ( T M ) = 18 (cid:18) π (cid:19) (cid:0) (tr R ) − R (cid:1) . (2.11)This Green-Schwarz term leads to a correction to the equation of motion (2.4) for A , whichnow reads d ∗ G = − G ∧ G − (2 π ) βX . (2.12)We note that the exactness of d ∗ G implies that the eight-form G ∧ G + (2 π ) βX must becohomologically trivial on M . This integrability condition will play an important rˆole forcompactifications on Calabi-Yau five-folds, as we will see. There is also a non-topological R term at order β which is related to the Green-Schwarz term (2.10) by supersymmetry.This term which we will need for our discussion of flux and scalar potentials in the one-dimensional effective theory is given by [22, 23, 24] S ,R = β κ · Z M d x √− g t M ...M t N ...N R M M N N . . . R M M N N , (2.13)with the famous rank eight tensor t which has been defined in Ref. [27].Equations of motion for anti-symmetric tensor fields can receive contributions fromelectrically charged objects and, for the case at hand, an additional term has to be addedto eq. (2.12) in the presence of M-theory membranes. Clearly, such a term can affect theintegrability of eq. (2.12) and should be taken into account.We start with the bosonic part of the membrane action S = − T Z M n d σ p − ˆ g + ˆ A o , (2.14)where ˆ g and ˆ A are the pullbacks of the 11-dimensional space-time metric g and three-form A under the embedding X M = X M ( σ ) of the membrane world-volume M into space-time Care has to be taken in order to obtain the correct sign of the Green-Schwarz term relative to the
GGA
Chern-Simons term in the action (2.2) and different versions exist in the literature [18, 25, 26]. Ingeneral, the sign of the Chern-Simons term is fixed by supersymmetry and the relative sign is fixed by theanomaly cancellation condition on the five-brane world volume [21]. In Ref. [18], several different argumentsare presented for why the relative sign must be positive (in our conventions) and we adopt this result inthe present paper. – 7 – . Here, σ = ( σ , σ , σ ) are coordinates on the membrane world volume. The membranetension T is given by T = 12 π √ β . (2.15)Adding this action to the bosonic one for 11-dimensional supergravity, eq. (2.2), and re-computing the equation of motion for A leads to d ∗ G = − G ∧ G − (2 π ) βX − κ T δ ( M ) . (2.16)Here, δ ( M ) is an eight-form current associated with the membrane world-volume. It ischaracterised by the property Z M w = Z M w ∧ δ ( M ) (2.17)for any three-form w .
3. Calabi-Yau five-folds
Our M-theory reduction depends on a range of results on Calabi-Yau five-folds, includingresults about their topology, their differential geometry and moduli spaces. Perhaps mostimportantly, to be sure we are not dealing with an empty set, we require some explicitexamples of Calabi-Yau five-folds on which consistent M-theory reductions can be carriedout. In this chapter, we provide a non-technical summary of the main facts and results forthe reader’s convenience. For the details we refer to Appendix B.We begin by defining what we mean by a Calabi-Yau five-fold X . As usual, we requirethat X be a compact, complex K¨ahler manifold with vanishing first Chern class, c ( X ) = 0.In addition, X should break supersymmetry by a factor of 1 /
16. This means that theholonomy group Hol( X ) ⊂ SU(5) is a sufficiently large subgroup of SU(5) such that in thedecomposition Spin(10) → [ + ¯ + ] SU(5) (3.1)of (chiral) spinors on X under SU(5) only the SU(5) singlet is invariant under the holonomygroup. An immediate consequence is that the Hodge numbers h p, ( X ) = h ,p ( X ) for p = 1 , , , h , ( X ) = h , ( X ) = h , ( X ) = h , ( X ) = 1. The reason forthis additional condition on supersymmetry breaking is to avoid “non-generic” cases whichlead to a larger number of preserved supersymmetries and additional zero modes (dueto h p, ( X ) = 0 for some p ∈ { , , , } ), such as 10-tori, products of lower-dimensionalCalabi-Yau manifolds (for example, a product of a three-fold with K3) or products oflower-dimensional Calabi-Yau manifolds with tori (for example, a four-fold times a two-torus).Given the restrictions on Hodge numbers discussed above, we remain with six, a pri-ori independent Hodge numbers, namely h , ( X ), h , ( X ), h , ( X ), h , ( X ), h , ( X ) and h , ( X ). For Calabi-Yau four-folds it is known [15] that one additional relation between the– 8 –odge numbers can be derived using the index theorem together with the Calabi-Yau con-dition c ( X ) = 0. In Appendix B, we show that the same is true for Calabi-Yau five-foldsand we derive the relation11 h , ( X ) − h , ( X ) − h , ( X ) + h , ( X ) + 10 h , ( X ) − h , ( X ) = 0 . (3.2)Hence, we are left with five apparently independent Hodge numbers. The precise rˆole of thecohomology groups in the reduction of M-theory will be explained in the following section.Here, we summarise the relation between cohomology groups, M-theory zero modes andflux (see Table 1). As usual, the moduli space of Ricci-flat metrics consists of K¨ahler cohomology bosonic zero modes fermionic zero modes flux H , ( X ) h , ( X ) real, K¨ahler moduli h , ( X ) complex,from gravitino − H , ( X ) h , ( X ) complex, from three-form h , ( X ) complex,from gravitino − H , ( X ) − h , ( X ) complex,from gravitino G -flux H , ( X ) − − G -flux H , ( X ) h , ( X ) complex structure moduli h , ( X ) complex,from gravitino − H , ( X ) − − − Table 1:
Cohomology groups of a Calabi-Yau five-fold X and their relation to zero modes and fluxin an M-theory reduction. and complex structure deformations. For Calabi-Yau five-folds they are associated withharmonic (1 ,
1) and (1 ,
4) forms, respectively. Another set of bosonic zero modes arisesfrom the M-theory three-form A and is related to the cohomology H , ( X ). As Table 1shows, for all these bosonic modes, we have fermionic zero modes counted by the sameHodge number. This suggests an obvious way of arranging modes into one-dimensionalsuper-multiplets. However, the (1 ,
3) sector is somewhat puzzling in that it gives rise toa set of fermionic but not bosonic zero modes. We will come back to this later and showhow this apparent mismatch of bosonic and fermionic degrees of freedom can be reconciledwith supersymmetry.As discussed before, the equation of motion for the M-theory three-form A leads toan important integrability condition which amounts to the right-hand side of eq. (2.16)being cohomologically trivial. Let us now consider this condition for the case of an 11-dimensional space-time of the form M = R × X , with a Calabi-Yau five-fold X . Thetotal Pontrjagin class of such a space-time is p ( M ) = p ( X ). In general, for a complexmanifold Z , the Pontrjagin and Chern classes are related by p ( Z ) = c ( Z ) − c ( Z ) and p ( Z ) = c ( Z ) − c ( Z ) c ( Z ) + 2 c ( Z ). Given that c ( X ) = 0 for a Calabi-Yau five-fold we have p ( X ) = − c ( X ) and p ( X ) = c ( X ) − c ( X ). Inserting this into the– 9 –efinition (2.11) of X , we find X = − c ( X ) . (3.3)In general, we also allow four-form flux G flux on X and it is convenient to introduce there-scaled version g = (cid:20) T π G flux (cid:21) , (3.4)where we recall that T is the membrane tension defined in eq. (2.15) and the squarebrackets indicate the cohomology class. As is clear from the Wess-Zumino term in themembrane action (2.14) this re-scaled flux is quantised in integer units, that is, it shouldbe an element of the fourth integer cohomology of X . More accurately, taking into accountthe subtlety explained in Ref. [28], the quantisation law reads g + 12 c ( X ) ∈ H ( X, Z ) . (3.5)Finally, we should allow for membranes which wrap a holomorphic cycle C ⊂ X of the five-fold, that is membranes with world volume M = R × C . The membrane current δ ( M )then takes the form δ ( M ) = δ ( C ). Inserting this current, together with eqs. (3.3) and(3.4) into the right-hand side of the G equation of motion (2.16) and taking the cohomologyclass of the resulting expression, one finds c ( X ) − g ∧ g = 24 W . (3.6)Here, W = [ C ] ∈ H ( X, Z ) is the second homology class of the curve C , wrapped by themembranes. Eq. (3.6) is a crucial condition which is clearly necessary for consistent M-theory backgrounds based on Calabi-Yau five-folds. When solving this condition, it mustbe kept in mind that the homology class W , having a holomorphic curve representative C ,must be an effective class in H ( X, Z ), that is, it must be an element in the Mori cone of X . Our task is now to establish the existence of Calabi-Yau five-fold backgrounds whichsatisfy the above consistency condition. Formally, this amounts to finding Calabi-Yau five-folds X , an element g in the fourth cohomology of X and an effective class W ∈ H ( X, Z )such that eqs. (3.5) and (3.6) are satisfied. In Appendix B.2 we analyse this problem indetail for a number of explicit examples. In particular, we consider torus quotients andcomplete intersection Calabi-Yau five-folds (CICY five-folds) [29].Let us briefly review some basic properties of CICY five-folds. CICY five-folds areembedded in an ambient space A = N mr =1 P n r , given by a product of m projective spaceswith dimensions n r . They are defined by the common zero locus of K = P mr =1 n r − p α in A . The polynomials p α are characterised by their degrees q rα in the coordinates of the r th projective factor of the ambient space. A short-handnotation for CICY manifolds is provided by the configuration matrix [ n | q ] = n ... n m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q . . . q K ... ... q m . . . q mK (3.7)– 10 –hich encodes the dimensions of the ambient projective spaces and the (multi)-degrees ofthe defining polynomials. Such configuration matrices are constrained by the Calabi-Yaucondition, c ( X ) = 0, which for CICY manifolds reads K X α =1 q rα = n r + 1 (3.8)for all r . The simplest CICY five-fold is given by the zero locus of a septic polynomialin P and is represented by the configuration matrix [6 | P is the directanalogue of the best-known example of a Calabi-Yau three-fold, the quintic hypersurfacein P . In total, there are 11 CICY five-folds which can be defined in a single projectivespace and these manifolds are listed in Table 5.The main results of Appendix B.2 can be summarised as follows. The simplest way ofsatisfying the integrability condition (3.6) is to turn off flux, g = 0, and have no membranesso that W = 0. In this case, a Calabi-Yau five-fold X with vanishing fourth Chern class, c ( X ) = 0, is required. It can be shown in general that CICY configurations with all q ra ≥ m = 1 or K = 1) always have c ( X ) = 0. In addition,we have verified that no configuration matrix with m ≤ K ≤ c ( X ) = 0.For larger configurations with m > K > q ra < c ( X ) = 0might still exist although we have been unable to find an explicit example. It is conceivablethat c ( X ) = 0 for all CICY five-folds. Given the lack of a viable CICY example, we haveturned to torus quotients of the form X = T / Z . We have shown that, for an appropriatechoice of shifts in the Z symmetries, Z is freely acting and, hence, X is a manifold. Each Z reduces supersymmetry by 1 /
2, so in total it is reduced by a factor of 1 /
16. This meansthat X , although its holonomy is merely Z , is a Calabi-Yau manifold in the sense definedearlier. Clearly, as X admits a flat metric, we have c ( X ) = 0. It remains an open questionwhether a Calabi-Yau five-fold with full SU(5) holonomy and c ( X ) = 0 exists. We are notaware of a general mathematical reason which excluded this and it would be interesting tosearch for such a manifold, for example among toric five-folds. In the present paper, wewill not pursue this explicitly.Next, we should consider the possibility of satisfying the integrability condition (3.6) inthe presence of non-vanishing flux but without membranes. The CICY manifolds definedin a single projective space, given in Table 5, all have b ( X ) = 1 and, hence, there is onlya single flux parameter. Eq. (3.6) then turns into a quadratic equation for this parameter.Unfortunately, there is no rational solution to this equation for any of the 11 cases. Thismeans that the quantisation condition (3.5) cannot be satisfied and, hence, that flux isnot sufficient to obtain viable examples for CICYs in a single projective space. Essentially,the reason is that there is only one flux parameter available which is too restrictive. Forsimple CICYs defined in a product of two projective space, where b ( X ) = 2 or b ( X ) = 3depending on the case, we run into a similar problem. The simplest viable example wehave found involves the space X ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.9)– 11 –efined in the ambient space A = P × P × P . In this case, we have b ( X ) = 5and flux can be parameterized as g = k , J J + k , J J + k , J + k , J J + k , J ,where J r are the three K¨ahler forms of the ambient projective spaces, normalised as ineq. (B.12). It turns out that both conditions (3.5) and (3.6) can be satisfied for the choice( k , , k , , k , , k , , k , ) = (1 , , / , , | W = 227 ˜ J , where ˜ J is the eight-form dual to the ambient spaceK¨ahler form J .Finally, by combining flux and membranes, the anomaly condition can frequently besatisfied. For example, for the septic, [6 | g = kJ we find the conditions (3.5)and (3.6) are solved for k = 15 / W = 6 J .In summary, we have demonstrated that the quantisation and integrability condi-tions (3.5) and (3.6), necessary for a consistent compactification of M-theory on Calabi-Yaufive-fold backgrounds, can be satisfied for a range of simple examples. Flux and membranesare usually necessary and even the septic in P leads to a viable background for appropriatenon-zero choices of both flux and membranes. We have also given an example, based ona torus quotient, with c ( X ) = 0 which is consistent without flux and membranes. Wehave not been able to find a Calabi-Yau manifold with c ( X ) = 0 and full SU(5) holonomyand it might be interesting to search for such a case, for example among toric Calabi-Yaumanifolds.
4. Compactification on Calabi-Yau five-folds
In this section, we consider the compactification of 11-dimensional supergravity on a space-time of the form M = R × X , where X is a Calabi-Yau five-fold. At zeroth order in β , westart with the background configuration ds = − dt + g mn dx m dx n , G = 0 , (4.1)where g mn = g mn ( x p ) is the Ricci-flat metric on X and m, n . . . = 1 , . . . ,
10. Clearly, thisbackground solves the leading order bosonic equations of motion (2.3) and (2.4). At order β , additional higher-derivative terms appear in the 11-dimensional equations of motion andcorrections of the same order will have to be added to the above background. It is nota priori clear that suitable corrections to the background exist in order for it to remaina solution at order β . We have seen that the integrability condition (3.6) is a necessarycondition for this to be the case. In the absence of flux and membranes, the integrabilitycondition is solved by Calabi-Yau five-folds with c ( X ) = 0 and, in the previous section,we have given an explicit example of such a five-fold. In Ref. [9], it has been shown thata full solution at order β does indeed exist in this case. For five-folds with c ( X ) = 0 fluxand/or membranes need to be included in order to satisfy the integrability condition andwe have seen that this can be achieved for a number of simple examples. In this case, theexistence of a full 11-dimensional solution at order β has not been analysed in detail. In– 12 –he presence of flux, one expects a scalar potential in the effective one-dimensional theory.Flux potentials frequently lead to some runaway direction in moduli space and in suchcases, one would not expect a static β , generalising the results of Ref. [9], isan interesting subject to which we intend to return in a future publication. In the presentpaper, we focus on deriving the one-dimensional effective theory for backgrounds wherethe integrability condition (3.6) is satisfied and under the assumption that a full order β background can be found. For now we will focus on the effective action at zeroth orderin β for which the above simple background is sufficient. Higher order corrections to theeffective action and, in particular, the scalar potential due to flux will be discussed later.We start with the reduction of the bosonic part of the action before we move on to thefermionic terms in the second part of this section. Our starting point is the bosonic part of 11-dimensional supergravity (2.2) which cor-responds to the leading, zeroth order terms in a β expansion together with the back-ground (4.1). The order β Green-Schwarz term (2.10) will also play a certain rˆole. We nowneed to identify the moduli of this background. As discussed in detail in Appendix B.4, theformalism to deal with Calabi-Yau five-fold moduli spaces is largely similar to the one de-veloped for Calabi-Yau three-folds [16]. Here, we only summarise the essential informationneeded for the dimensional reduction. As for Calabi-Yau three-folds, the moduli space ofRicci-flat metrics on Calabi-Yau five-folds is (locally) a product of a K¨ahler and a complexstructure moduli space which are associated to (1 ,
1) and (2 ,
0) deformations of the metric.They can be described in terms of harmonic (1 ,
1) forms for the K¨ahler moduli space andharmonic (1 ,
4) forms for the complex structure moduli space. We begin with the K¨ahlermoduli which we denote by t i = t i ( τ ), where i, j, . . . = 1 , . . . , h , ( X ) and τ is time (for asummary of our index conventions see Appendix A). They are real scalar fields and canbe defined by expanding the K¨ahler form J on X in terms of a basis { ω i } of H ( X ) as J = t i ω i . (4.2)The complex structure moduli are denoted by z a = z a ( τ ), where a, b, . . . = 1 , . . . , h , ( X ),and these are, of course, complex scalar fields.After this preparation, the ansatz for the 11-dimensional metric including moduli canbe written as ds = − N ( τ ) dτ + g mn ( t i , z a , ¯ z ¯ a ) dx m dx n (4.3)where N = N ( τ ) is the einbein or lapse function. The lapse function can, of course,be removed by a time reparameterization. However, its equation of motion in the one-dimensional effective theory is the usual zero-energy constraint (the equivalent of the Fried-man equation in four-dimensional cosmology). In order not to miss this constraint, we willkeep N explicitly in our metric ansatz.The zero modes of the M-theory three-form A are obtained by an expansion in harmonicforms, as usual. From the Hodge diamond (B.2) of Calabi-Yau five-folds, it is clear that– 13 –nly the harmonic two- and three-forms on X are relevant in this context. For the latterwe also introduce a basis { ν p } , where p, q, . . . = 1 , . . . , h , ( X ). The zero mode expansionfor A can then be written as A = ( ξ p ( τ ) ν p + c . c . ) + N µ i ( τ ) ω i ∧ dτ , (4.4)with h , ( X ) complex scalar fields ξ p and h , ( X ) real scalars µ i . It is clear that thelatter correspond to gauge degrees of freedom since N µ i ( τ ) ω i ∧ dτ = d ( f i ( τ ) ω i ) with thefunction f i being integrals of N µ i . Note that N enters here merely to ensure worldlinereparameterization covariance. Since the fields µ i do not represent physical degrees offreedom, the one-dimensional effective action should not depend on these modes. It is,therefore, safe to ignore them in the above ansatz for A . Nevertheless, we will find itinstructive to keep these modes for now to see explicitly how they drop out of the effectiveaction.Further zero-modes can arise from membranes if they are included in the compactifi-cation, such as moduli of the complex curve C which they wrap and their superpartners.Presently, we will not include these additional modes but rather focus on the modes frompure 11-dimensional supergravity.While the way we have parametrised the zero modes of A in eq. (4.4) appears to bethe most natural one, it is not actually the most well-suited ansatz for performing thedimensional reduction. This is due to the fact that we have split a three form into (2 , , , , ξ p ˙ z a etc.), whichwould in turn force us into attempting lengthy field re-definitions in order to diagonalisethe kinetic terms.It would, on the other hand, be much more economic to start out with a formulationin which no such mixing of kinetic terms arises in the first place. Indeed, it is possible tocircumvent, yet fully capture, this complication by using real harmonic 3-forms instead ofcomplex (2 , , A , we first need to introduce a basis { N P } P =1 ,...,b ( X ) of real harmonic 3-forms on X . Instead of eq. (4.4), we can then write A = X P ( τ ) N P + N µ i ( τ ) ω i ∧ dτ , (4.5)with b ( X ) = 2 h , ( X ) real scalar fields X P and h , ( X ) real scalars µ i . The two ans¨atzefor A are readily related by devising linear maps, denoted A and B , translating back andforth between real harmonic 3-forms and complex harmonic (2 , , A = A ( z, ¯ z ), B = B ( z, ¯ z ). Infixed bases, they possess a matrix representation: ν p = A p Q N Q (and: ¯ ν ¯ p = ¯ A ¯ p Q N Q ) , (4.6)– 14 – P = B P q ν q + ¯ B P ¯ q ¯ ν ¯ q . (4.7)Inserting eqs. (4.6)-(4.7) into eqs. (4.4)-(4.5), we learn how the two formulations are relatedat the level of zero mode fields ξ p = X Q B Q p (and c.c.) , (4.8) X P = ξ q A q P + ¯ ξ ¯ q ¯ A ¯ q P . (4.9)For the reasons outlined above, we henceforth adopt the 3-form formulation. At eachstep of the calculation, one may, of course, revert if desired to the complex (2 , A s and B s as∆ P Q := i ( B P q A q Q − ¯ B P ¯ q ¯ A ¯ q Q ) . (4.10)It is readily verified that ∆ satisfies the properties of a complex structure.The A and B matrices turn out to be an effective way to parametrize our ignoranceof the actual dependence of the (2 , R . As usual, for given values of the complex structure moduli, we introducelocal complex coordinates z µ and ¯ z ¯ µ , where µ, ν, . . . = 1 , . . . , µ, ¯ ν, . . . = ¯1 , . . . , ¯5, sothat the metric is purely (1 , g µ ¯ ν are the only non-vanishing ones.This leads to12 N R = 4 N ddτ (cid:0) N − g µ ¯ ν ˙ g µ ¯ ν (cid:1) + g µ ¯ ρ g ν ¯ σ ˙ g µ ¯ σ ˙ g ν ¯ ρ + g µ ¯ ρ g σ ¯ ν ˙ g µσ ˙ g ¯ ν ¯ ρ + 2 g µ ¯ ν ˙ g µ ¯ ν g σ ¯ ρ ˙ g σ ¯ ρ + 4 N − ˙ N g µ ¯ ν ˙ g µ ¯ ν . (4.11)where here and in the following the dot denotes the derivative with respect to τ . Into thisexpression, we have to insert the expansion of the metric (B.67) which can also be writtenas ˙ g µ ¯ ν = − iω i,µ ¯ ν ˙ t i , ˙ g µν = − || Ω || Ω µ ¯ µ ... ¯ µ χ a, ¯ µ ... ¯ µ ν ˙ z a , ˙ g ¯ µ ¯ ν = ( ˙ g µν ) ∗ . (4.12)Here { χ a } , where a, b, . . . = 1 , . . . , h , ( X ), is a basis of harmonic (1 ,
4) forms. Further weneed the field strength G = dA for the three-form ansatz (4.5) and its Hodge dual whichare given by G = ˙ X P dτ ∧ N P , ∗ G = − N − ˙ X P ∆ P Q N Q ∧ J . (4.13)To derive the second equation we have used the result (B.102) for the dual of a real 3-formon a Calabi-Yau five-fold. The ∆ appearing here has been defined in eq. (4.10) and isdiscussed further in Appendix B.4.1. – 15 –nserting the ansatz (4.3), (4.5) together with the last three equations into the bosonicaction (2.2) and integrating over the Calabi-Yau five-fold, one finds the bosonic part of theone-dimensional effective action S B , kin = l Z dτ N − (cid:26) G (1 , ij ( t ) ˙ t i ˙ t j + 12 G (3) PQ ( t, z, ¯ z ) ˙ X P ˙ X Q + 4 V ( t ) G (1 , a ¯ b ( z, ¯ z ) ˙ z a ˙¯ z ¯ b (cid:27) (4.14)at order zero in the β expansion. Here l = v/κ and v is an arbitrary reference volume ofthe Calabi-Yau five-fold . The moduli space metrics in the (1 , ,
4) and 3-form sectorsare given by G (1 , ij ( t ) = 4 Z X ω i ∧ ∗ ω j + 8 V ˜ w i ˜ w j , (4.15) G (1 , a ¯ b ( z, ¯ z ) = R X χ a ∧ ¯ χ ¯ b R X Ω ∧ ¯Ω , (4.16) G (3) PQ ( t, z, ¯ z ) = Z X N P ∧ ∗ N Q , (4.17)where ˜ ω i = g µ ¯ ν ω i,µ ¯ ν . Since h , ( X ) need not be even, G (1 , ij is a genuinely real metricthat cannot be complexified in general. This is compatible with the anticipated N = 2supersymmetry in one dimension, which only demands target spaces of sigma models tobe Riemannian manifolds [10]. Using the results of Appendix B.4, these metrics can becomputed as functions of the moduli. In the (1 ,
1) sector we have G (1 , ij ( t ) = 8 V h G (1 , ij ( t ) − κ i κ j κ i = − κ ij − κ i κ j κ , (4.18)where κ is a quintic polynomial in the K¨ahler moduli given by κ = 5! V = d i ...i t i . . . t i , d i ...i = Z X ω i ∧ · · · ∧ ω i , (4.19) d i ...i are intersection numbers and κ i = d ii ...i t i . . . t i , κ ij = d iji i i t i t i t i . The stan-dard moduli space metric G (1 , ij , as defined in Appendix B.4, can be obtained from theK¨ahler potential K (1 , = − ln κ as G (1 , ij = ∂ i ∂ j K (1 , . We note that the physical sigmamodel metric (4.18) differs from the standard moduli space metric G (1 , ij by a term pro-portional to κ i κ j and a rescaling by the volume. The latter is not really required at thisstage and can be removed by a redefinition of time τ but it will turn out to be a usefulconvention in the full supersymmetric version of the effective action. The additional term,however, cannot be removed, for example by a re-scaling of the fields t i . As a consequence,unlike the standard moduli space metric, the physical metric is not positive definite. Inthe direction u i ∼ t i we have G (1 , ij u i u j < u i ,defined by G (1 , ij t i u j = 0, we have G (1 , ij u i u j >
0. This means G (1 , has a Minkowski Related factors of 1 /v should be included in the definition of the moduli space metrics (4.15)–(4.17)but will be suppressed in order to avoid cluttering the notation. These factors can easily be reconstructedfrom dimensional arguments. – 16 –ignature ( − , +1 , . . . , +1). This is in contrast to, for example, M-theory compactifica-tions on Calabi-Yau three-folds [16, 30] where the sigma model metric in the (1 ,
1) sector isidentical to the standard moduli space metric and, in particular, is positive definite. In thepresent case, the appearance of a single negative direction is, of course, not a surprise. Oursigma model metric in the gravity sector can be though of as a “mini-superspace” versionof the de-Witt metric which is well-known to have precisely one negative eigenvalue [31].Here, we see that this negative direction lies in the (1 ,
1) sector. Another difference tothe Calabi-Yau three-fold case is the degree of the function κ . For three-folds κ is a cubicwhile, in the present case, it is a quintic polynomial.We now turn to the (1 ,
4) moduli space metric G (1 , a ¯ b which is, in fact, equal to thestandard moduli space metric in this sector and can, hence, be expressed as G (1 , a ¯ b = ∂ a ∂ ¯ b K (1 , , K (1 , = ln (cid:20) i Z X Ω ∧ ¯Ω (cid:21) (4.20)in terms of the K¨ahler potential K (1 , . This is very similar to the three-fold case. Inparticular, G (1 , a ¯ b is positive definite as it should be, given that the single negative directionarises in the (1 ,
1) sector.Finally, in the 3-form sector one finds from the results in Appendix B.4.1 that themetric can be written as G (3) PQ ( t, z, ¯ z ) = 12 ∆ ( P R d Q ) R ij t i t j , d PQ ij = Z X N P ∧ N Q ∧ ω i ∧ ω j , (4.21)where we have introduced the intersection numbers d PQ ij = − d QP ij , which are purelytopological. The metric (4.21) is Hermitian with respect to the complex structure ∆ (seeeq. (B.110)).This completes the definition of all objects which appear in the action (4.14).We see that this action does not depend on the gauge degrees of freedom µ i whichappear in the ansatz (4.4) for the three-form A , as should be the case. This demonstrates µ i independence at zeroth order in β but what happens at first order in β ? At this order,there are three terms in the 11-dimensional theory, all of them topological, which contributeto µ i dependent terms in one dimension. These are the Chern-Simons term A ∧ G ∧ G ineq. (2.2), the Green-Schwarz term (2.10) and the Wess-Zumino term in the membraneaction eq. (2.14). Evaluating these three terms leads to the one-dimensional contribution S B , gauge = − lβ Z dτ N [12 g ∧ g + 24 W − c ( X )] i µ i , (4.22)where β = (2 π ) β/v / is the one-dimensional version of the expansion parameter β . Thenotation [ . . . ] i indicates the components of the eight-form in brackets with respect to abasis { ˜ ω i } of harmonic eight-forms dual to the harmonic two-forms { ω i } . Hence, at order β the µ i dependent terms do not automatically vanish. However, the bracket in eq. (4.22)vanishes once the integrability condition (3.6) is imposed. Put in a different way, theequation of motion for µ i from eq. (4.22) is simply the integrability condition (3.6)12 g ∧ g + 24 W − c ( X ) = 0 . (4.23)– 17 –ence, the rˆole of the gauge modes µ i is to enforce the integrability condition at the levelof the equations of motion and, once the condition is imposed, the gauge modes disappearfrom the action as they should. The condition (3.6) can, therefore, also be interpreted asan anomaly cancellation condition which has to be satisfied in order to prevent a gaugeanomaly of the M-theory three-form A along the Calabi-Yau (1 ,
1) directions.
One may ask if an explicit dimensional reduction of the fermionic part of the 11-dimensionalaction (2.1) is really necessary, for in many other cases, once the bosonic terms in theeffective action are known the fermionic ones can be inferred from supersymmetry. In thepresent case, there are a number of reasons why reducing at least some of the fermionicterms might be useful. First of all, the structure of the bosonic action (4.14) points to somefeatures of one-dimensional N = 2 supersymmetry which have not been well-developed inthe literature. For example, the bosonic action (4.14) indicates a coupling between the twomain types of N = 2 supermultiplets, the 2 a and 2 b multiplets, which, to our knowledge,has not been worked out in the literature. Also, in the last section, we have seen that it isimportant to keep the lapse function as a degree of freedom in the one-dimensional theory,as it generates an important constraint. In the context of supersymmetry, the lapse is partof the one-dimensional supergravity multiplet which one expects to generate a multipletof constraints. Therefore, even though gravity is not dynamical in one dimension, we needto consider local one-dimensional N = 2 supersymmetry. Again, it appears this has notbeen developed in the literature to the extend required for our purposes. We will deal withthese problems in Appendix C where we systematically develop one-dimensional N = 2supersymmetry and supergravity both in component and superspace formalism. At anyrate, given that the relevant supersymmetry is not as well established as in some othercases, it seems appropriate to back up our results by reducing some of the 11-dimensionalfermionic terms as well. Finally, the list of M-theory zero modes on Calabi-Yau five-foldsin Table 1 contains (1 ,
3) fermionic zero modes but no matching bosons. This feature issomewhat puzzling from the point of view of supersymmetry and can certainly not beclarified from the bosonic effective action alone.In this section, we will, therefore, reduce the terms in the 11-dimensional actionquadratic in fermions. These results together with the bosonic action are sufficient tofix the one-dimensional action in superspace form uniquely and, in addition, provide uswith a number of independent checks. Four-fermi terms in the one-dimensional theory arethen obtained from the superspace action and we will not derive them by reduction from11 dimensions.We should start by writing down a zero mode expansion of the 11-dimensional gravitinoΨ M on the space-time M = R × X . The covariantly constant, positive chirality spinor on X is denoted by η and its negative chirality counterpart by η ⋆ (for a summary of our spinorconventions see Appendix A). The spinor η is characterised by the annihilation conditions γ ¯ µ η = 0. Further, by ω ( p,q ) i we denote the harmonic ( p, q ) forms on X . Then, followingthe known rules for writing down a fermionic zero mode ansatz (see for example Refs. [32]– 18 – § = ψ ( τ ) ⊗ η ⋆ + ¯ ψ ( τ ) ⊗ η, (4.24)Ψ ¯ µ = X p,q ζ ( i )( p,q ) ( τ ) ⊗ ( ω ( p,q )( i ) ,α ...α ( p ) ¯ β ... ¯ β ( q − ¯ µ γ α ...α ( p ) ¯ β ... ¯ β ( q − η )+ X p,q ζ ′ ( i )( p,q ) ( τ ) ⊗ ( ω ( p,q )( i ) ,α ...α ( p ) ¯ β ... ¯ β ( q − ¯ µ γ α ...α ( p ) ¯ β ... ¯ β ( q − η ⋆ ) , (4.25)Ψ µ = (Ψ ¯ µ ) ∗ . (4.26)Here, ζ ( i )( p,q ) and ζ ′ ( i )( p,q ) are one-dimensional complex fermions which represent the zero-modesin the ( p, q ) sector of the Calabi-Yau five-fold and ψ is the one-dimensional gravitino. Thesums over ( p, q ) in (4.25) run over all non-trivial cohomology groups of the five-fold. Letus discuss the various ( p, q ) sectors in the first sum in (4.25) in detail. For ( p, q ) = (1 , γ ¯ µ exceeds the number of creating ones, γ µ ,and, as a result, this term vanishes. Further, for all cases with q = p + 1 the numberof creation and annihilation gamma matrices is identical. Anti-commuting all γ ¯ µ to theright until they annihilate η one picks up inverse metrics g µ ¯ ν which ultimately contract theharmonic ( p, p + 1) forms ω ( p,p +1) i to harmonic (0 ,
1) forms. Since the latter do not exist onCalabi-Yau five-folds all terms with q = p + 1 vanish. This leaves us with the cases where p ≥ q . Among those, only the terms with ( p, q ) = (2 , , (3 ,
2) contain both creation andannihilation matrices. For ( p, q ) = (2 , ,
2) forms into harmonic (1 ,
1) forms. Therefore, the (2 , ,
1) term and does not need to be written downindependently. The same argument applies to the (3 ,
2) part which can be absorbed intothe (2 ,
1) contribution. By the gamma matrix structure and the annihilation property of η ⋆ all but the (5 ,
0) term in the second sum in (4.25) vanish. Using the Fierz identity(see eq. (B.51)) the (5 ,
0) term in the second sum can be converted into a term with the(1 ,
1) structure of the first sum and can, hence, be absorbed by the (1 ,
1) contribution.In summary, all we need to write down explicitly are the ( p, q ) terms with q = 1 and p = 1 , , , , , , , ,
1) and (2 ,
2) pieces using the complex structure of theCalabi-Yau five-fold X . Henceforth, we will restrict our attention to Calabi-Yau five-foldswhose (2 , , , , ,
3) and a (3 ,
1) piece as a ˆ4-form and given the restriction on h , ( X ),– 19 –his restriction is also purely topological. The ˆ4-forms are thus well-suited to describe the(1 ,
3) + (3 , { O X } X =1 ,...,b ( X ) ,such that the first 2 h , ( X ) 4-forms, denoted { O ˆ X } ˆ X =1 ,..., h , ( X ) , only contain (1 ,
3) and(3 , h , ( X ) 4-forms, denoted { O ˜ X } ˜ X =1 ,...,h , ( X ) , only contain(2 , { O ˆ X } . For ageneral Calabi-Yau five-fold, a more complicated intertwining of the K¨ahler and complexstructure moduli with the (1 , , = ψ ( τ ) ⊗ η ⋆ + ¯ ψ ( τ ) ⊗ η, (4.27)Ψ ¯ µ = ψ i ( τ ) ⊗ ( ω i,α ¯ µ γ α η ) + i P ( τ ) ⊗ ( N P ,α α ¯ µ γ α α η )+ 14 ¯Υ ˆ X ( τ ) ⊗ ( O ˆ X ,α ...α ¯ µ γ α ...α η ) −
14! ¯ κ ¯ a ( τ ) ⊗ ( || Ω || − ¯ χ ¯ a,α ...α ¯ µ γ α ...α η ) , (4.28)Ψ µ = (Ψ ¯ µ ) ∗ , (4.29)The four terms in eq. (4.28) correspond to the (1 , , ,
1) and (4 ,
1) sectors, respec-tively. The harmonic (1 ,
1) forms are denoted by ω i , where i, j, . . . = 1 , . . . , h , ( X ), the real3-forms are denoted by N P , where P , Q , . . . = 1 , . . . , b ( X ), the real ˆ4-forms by O ˆ X , whereˆ X , ˆ Y , . . . = 1 , . . . , h , ( X ) and the (1 ,
4) forms by χ a , where a, b, . . . = 1 , . . . , h , ( X ). Inthe same order, the associated zero modes, which are complex one-dimensional fermions,are denoted by ψ i , Λ P , Υ ˆ X and κ a . It is clear that the number of zero modes cannot bereduced any further and that these four types of modes are independent. Three of them,the (1 , ,
4) modes pair up with corresponding bosonic zero modes in thesame sectors. The ˆ4-form modes, however, have no bosonic zero mode partners, as men-tioned earlier and one of our tasks will be to understand how they can be incorporatedinto a supersymmetric one-dimensional effective theory.Had we written the second term in eq. (4.28) in (2 , ¯ µ = . . . − / λ p ( τ ) ⊗ ( ν p,α α ¯ µ γ α α η ) + . . . , we would have identified a set of h , ( X ) complex one-dimensionalfermions in this sector. From eq. (4.28) however, there appear to be b ( X ) = 2 h , ( X )complex one-dimensional fermions. This apparent factor of two discrepancy in the numberof degrees of freedom is resolved by observing that a successive insertion of eqs. (B.93)-(B.96) into the second term in eq. (4.28) leads to a constraint in the form of a projectioncondition on the 3-form fermions Λ P P + P Q Λ P = Λ Q , (and: P −P Q ¯Λ P = ¯Λ Q ) , (4.30)where P ±P Q were defined in eq. (B.104). This condition, which is equivalent to P −P Q Λ P =0, precisely halves the number of degrees of freedom so as to match the counting in (2 , / b ( X ) = h , ( X ) complex one-dimensional fermionsin this sector, as claimed in Table 1. It can be shown that this constraint also applies tothe time derivative and supersymmetry transformation of Λ P P + P Q ˙Λ P = ˙Λ Q , P + P Q ( δ ǫ Λ P ) = δ ǫ Λ Q , (4.31)implying in particular that the projection operators commute with both supersymmetryand time translation when acting on Λ P (cid:2) P ±P Q , ∂ (cid:3) Λ P = 0 , (cid:2) P ±P Q , δ ǫ (cid:3) Λ P = 0 . (4.32)The projection condition is thus preserved under both operations as is required by con-sistency. Eqs. (4.30)-(4.32) will play important rˆoles in finding the correct superspaceformulation for this sector later in section 5.By complete analogy, we learn that the ˆ4-form sector really only contains h , ( X )complex one-dimensional fermions (cf. Table 1) and not 2 h , ( X ) as is suggested by thethird term in eq. (4.28). By using eqs. (B.118) and (B.120) and the third term in eq. (4.28),we infer P + ˆ Y ˆ X Υ ˆ Y = Υ ˆ X , (and: P − ˆ Y ˆ X ¯Υ ˆ Y = ¯Υ ˆ X ) , (4.33)thereby halving the number of degrees of freedom. The projection operators P ± ˆ Y ˆ X weredefined in eq. (B.130). Eq. (4.33) implies P + ˆ Y ˆ X ˙Υ ˆ Y = ˙Υ ˆ X , h P ± ˆ Y ˆ X , ∂ i Υ ˆ Y = 0 , (4.34) P + ˆ Y ˆ X ( δ ǫ Υ ˆ Y ) = δ ǫ Υ ˆ X , h P ± ˆ Y ˆ X , δ ǫ i Υ ˆ Y = 0 (4.35)guaranteeing the preservation of the projection condition under time translation and su-persymmetry. The compatibility conditions (4.34)-(4.35) are, of course, required for con-sistency.In order to reduce the fermion terms, we also need explicit expressions for the vielbein,its time derivative and the spin connection. In particular, it should be kept in mind thatthe gravitino ansatz (4.27)–(4.29) implicitly depends on the vielbein since the curved indexgamma matrices γ µ that appear have to be replaced by flat index gamma matrices γ µ via γ µ = e µν γ ν . We begin with the vielbein. From the metric ansatz (4.3) with the 10-dimensional metric taken to be purely (1 ,
1) its non-zero components are e = − N/ e µν and e ¯ µ ¯ ν , so that g µ ¯ ν = e µρ e ¯ ν ¯ σ η ρ ¯ σ is the Ricci-flat metric on the Calabi-Yau five-fold. Ofcourse, the 10-dimensional part of the vielbein depends on the Calabi-Yau K¨ahler moduli t i = t i ( τ ) and the complex structure moduli z a = z a ( τ ) and, hence, its time-derivative isnon-zero. From the time derivative (4.12) for the metric one finds˙ e µν = − i ω i,µρ e ρν ˙ t i , (4.36)˙ e µ ¯ ν = − || Ω || Ω µ ¯ µ ... ¯ µ χ a, ¯ µ ... ¯ µ ρ e ρ ¯ ν ˙ z a , (4.37)– 21 –nd similarly for the complex conjugates. From the equations above and the covariantconstancy of the vielbein, we find expressions for the 11-dimensional spin-connection ω N QR .Its only non-zero components are given by ω µν = − iN − ω i,µρ e ρν ˙ t i , (4.38) ω µ ¯ ν = − || Ω || N − Ω µ ¯ µ ... ¯ µ χ a, ¯ µ ... ¯ µ ρ e ρ ¯ ν ˙ z a , (4.39)plus their complex conjugates and the components ω mnp of the Calabi-Yau spin connection,computed from the 10-dimensional vielbein e mn . The complex conjugates of the compo-nents listed above are, of course, also present. The components of the eleven dimensionalcovariant derivative, defined in eq. (2.6), then become D = ∂ , (4.40) D µ = ˜ D µ + i N − ω i,µ ¯ ν ˙ t i γ ¯ ν Γ + 112 || Ω || N − Ω µ ¯ µ ... ¯ µ χ a, ¯ µ ... ¯ µ ν ˙ z a γ ν Γ , (4.41) D ¯ µ = ( D µ ) ∗ , (4.42)where ˜ D µ is the covariant derivative on the Calabi-Yau five-fold.We are now ready to perform the reduction. Inserting the gravitino ansatz (4.27)-(4.29)into the fermionic action (2.5) produces a vast number of terms – even when restricting toterms quadratic in fermions. Each of these terms contains a product of a certain numberof gamma matrices sandwiched between two spinors η or η ⋆ . Luckily, on a Calabi-Yau five-fold there only exist a very limited number of non-vanishing such spinor bilinears, namely η † η , J µ ¯ ν , Ω µ ...µ and their complex conjugates (see Appendix B.3 for details). As a result,many terms in the reduction vanish immediately, due to their gamma matrix structure.The remaining terms can be split into two types. The first type leads to one-dimensionalfermion kinetic terms and such terms originate from the 11-dimensional Rarita-Schwingerterm in the action (2.5). The second type leads to one-dimensional Pauli terms, that iscouplings between two fermions and the time derivative of a boson, which descend from allthe remaining terms in the action (2.5), quadratic in fermions.After inserting the gravitino ansatz and integrating over the Calabi-Yau manifold, theRarita-Schwinger term gives rise to the following fermion kinetic terms S F , kin = − l Z dτ i n G (1 , ij ( t )( ψ i ˙¯ ψ j − ˙ ψ i ¯ ψ j ) + G (3) PQ ( t, z, ¯ z )(Λ P ˙¯Λ Q − ˙Λ P ¯Λ Q )+3 G (ˆ4)ˆ X ˆ Y ( t )(Υ ˆ X ˙¯Υ ˆ Y − ˙Υ ˆ X ¯Υ ˆ Y ) + 4 V ( t ) G (1 , a ¯ b ( z, ¯ z )( κ a ˙¯ κ ¯ b − ˙ κ a ¯ κ ¯ b ) o . (4.43)Here, G (1 , ij , G (3) PQ and G (1 , a ¯ b are the moduli space metrics for the (1 , , G (ˆ4)ˆ X ˆ Y . It is given by G (ˆ4)ˆ X ˆ Y ( t ) = Z X O ˆ X ∧ ∗ O ˆ Y = − d ˆ X ˆ Y i t i , d ˆ X ˆ Y i = Z X O ˆ X ∧ O ˆ Y ∧ ω i (4.44)– 22 –n terms of the intersection numbers d ˆ X ˆ Y i = d ˆ Y ˆ X i , which are purely topological for theclass of five-folds we are considering. To evaluate ∗ O ˆ Y in the above integral we have usedthe result for the Hodge dual of ˆ4-forms from eq. (B.126).Reducing the other fermion bilinear terms in the 11-dimensional action (2.5) we findfor the Pauli terms S F , Pauli = l Z dτ (cid:26) i N − G (1 , ij ( t )( ψ i ψ + ¯ ψ i ¯ ψ ) ˙ t j + i G (1 , ij,k ( t )( ψ k ¯ ψ i + ¯ ψ k ψ i ) ˙ t j + iN − G (3) PQ ( t, z, ¯ z )(Λ P ψ + ¯Λ P ¯ ψ ) ˙ X Q + iG (3) PQ ,i ( t, z, ¯ z )( ψ i ¯Λ P + ¯ ψ i Λ P ) ˙ X Q − i G (3) PQ ,a ( t, z, ¯ z )Λ P ¯Λ Q ˙ z a + G (3) PQ ,a ( t, z, ¯ z ) κ a ¯Λ P ˙ X Q + i G (3) PQ , ¯ a ( t, z, ¯ z )Λ P ¯Λ Q ˙¯ z ¯ a − G (3) PQ , ¯ a ( t, z, ¯ z )¯ κ ¯ a Λ P ˙ X Q +2 iV G (1 , a ¯ b,c ( z, ¯ z ) κ a ¯ κ ¯ b ˙ z c − iV G (1 , a ¯ b, ¯ c ( z, ¯ z ) κ a ¯ κ ¯ b ˙¯ z ¯ c − N − V G (1 , a ¯ b ( z, ¯ z )( ψ κ a ˙¯ z ¯ b − ¯ ψ ¯ κ ¯ b ˙ z a ) − K i G (1 , a ¯ b ( z, ¯ z )( ψ i ¯ κ ¯ b ˙ z a − ¯ ψ i κ a ˙¯ z ¯ b ) (cid:27) . (4.45)This completes the dimensional reduction of the fermionic part of the 11-dimensional actionat the level of terms quadratic in fermions. Our complete result for the one-dimensionaleffective action in components, four-fermi terms not included, is given by the sum of thebosonic action (4.14) and the two fermionic parts (4.43) and (4.45). Next, we have to verifythat this action is indeed invariant under one-dimensional N = 2 local supersymmetry, asit should be. In the following section, we will do this by writing down a superspace actionwhose associated component action coincides with our reduction result. This superspaceaction then also determines the four-fermion terms, which we have not explicitly computedfrom the dimensional reduction.
5. Supersymmetry and Calabi-Yau five-folds
Compactification on Calabi-Yau five-folds reduces the number of supersymmetries by afactor of 16, so the effective theory derived in the previous section should, in fact, haveone-dimensional N = 2 supersymmetry. We will now show that this is indeed the case. Ourfirst step is to identify how the five-fold zero modes have to be arranged in one-dimensional N = 2 supermultiplets. This is done by reducing the 11-dimensional supersymmetry trans-formations to one dimension and comparing the result with the known supersymmetrytransformations of the various types of one-dimensional multiplets. Then, we write downa superspace action and show that its associated component action, after integrating outauxiliary fields and neglecting four-fermi terms, is identical to the component action de-rived from reduction. As we have already mentioned, the required one-dimensional N = 2theories have not been worked out in sufficient detail and generality for our purposes. Wehave, therefore, included a systematic exposition of both globally and locally supersym-metric one-dimensional N = 2 theories, tailored to our needs, in Appendix C. Here, wewill briefly summarise the main results of this appendix, focusing on the structure of the– 23 –ultiplets and other information necessary to relate N = 2 superspace and componentactions.One-dimensional, N = 2 superspace (“supertime”) is labelled by coordinates ( τ, θ, ¯ θ )where θ is a complex one-dimensional spinor and ¯ θ its complex conjugate. General su-perfields are functions of these coordinates and can, as usual, be expanded in powers of θ and ¯ θ to obtain their component fields. Since θ = ¯ θ = 0, only four terms arise insuch an expansion, namely the theta-independent term and the ones proportional to θ , ¯ θ and θ ¯ θ . In order to develop the geometry of supertime one needs to introduce a super-vielbein, a superconnection and supertorsion and solve the Bianchi identities subject tocertain constraints on the torsion tensor. This is explicitly carried out in Appendix C andhere we simply cite the main results. The field content of the supergravity multiplied canmost easily be read off from the component expansion of the super-determinant E of thesupervielbein. It is given by E = − N − i θ ¯ ψ − i θψ , (5.1)where N is a real scalar, the einbein or lapse function and ψ is a complex fermion, theone-dimensional gravitino or lapsino.A 2 a multiplet is a real supermultiplet, that is a supermultiplet φ satisfying φ † = φ .Its component expansion is given by φ = ϕ + iθψ + i ¯ θ ¯ ψ + 12 θ ¯ θf , (5.2)and contains the real scalars ϕ and f and the complex fermion ψ . The highest component f turns out to be an auxiliary field so we remain with a real scalar and a complex fermionas the physical degrees of freedom.A 2 b multiplet, Z , is defined by the constraint ¯ D Z = 0, where D is the super-covariantderivative D = (cid:18) − i N − ¯ θψ − N − θ ¯ θψ ¯ ψ (cid:19) ∂ θ + (cid:18) i N − ¯ θ − N − θ ¯ θ ¯ ψ (cid:19) ∂ − i N − ¯ θ ¯ ψ ∂ ¯ θ (5.3)and ¯ D its conjugate. For the component expansion of a 2 b multiplet one finds Z = z + θκ + i N − θ ¯ θ ( ˙ z − ψ κ ) , (5.4)with a complex scalar z and a complex fermion κ . Unlike a 2 a multiplet, a 2 b multipletdoes not contain an auxiliary field so that its physical field content consists of a complexscalar and a complex fermion. This distinction in physical field content between 2 a and2 b multiplets will be useful in identifying the supermultiplet structure of the five-fold zeromodes.For both 2 a and 2 b multiplets fermionic versions exist, that is multiplets satisfying thesame constraint as their bosonic counterparts but with a fermion as the lowest component. For our spinor conventions see Appendix A. – 24 –ere, we only need the fermionic 2 b multiplet, R , defined by the constraint ¯ D R = 0. Itscomponent expansion R = ρ + θh + i N − θ ¯ θ ( ˙ ρ − ψ h ) (5.5)is analogous to that of an ordinary 2 b multiplet except that the lowest component, ρ , isnow a (complex) fermion, while h is a complex scalar. As we will see, for a suitable chosenaction, the scalar h is an auxiliary field so that the fermion ρ is the only physical degree offreedom.A superfield action can now be written as an integral R dτ d θ E over some functionof the above fields and their super-covariant derivatives, where d θ = dθ d ¯ θ . Explicitsuperfield actions and their component expansions as required for our purposes are givenin Appendix C. N = 2 supersymmetry transformations and multiplets We should now identify how the zero modes of M-theory on Calabi-Yau five-folds fall intosuper-multiplets of one-dimensional N = 2 supersymmetry. It is a plausible assumptionthat bosonic and fermionic zero modes that arise from the same sector of harmonic ( p, q )forms on the five-fold pair up into supermultiplets. For example, the h , ( X ) K¨ahlermoduli t i should combine with the same number of (1 ,
1) fermions ψ i . Since the K¨ahlermoduli t i are real scalars the resulting h , ( X ) supermultiplets must be of type 2 a . Inthe (1 ,
4) sector, on the other hand, we have h , ( X ) complex scalars z a (the complexstructure moduli) and the same number of complex fermions κ a so one expects h , ( X )supermultiplets of type 2 b . The 3-form sector is somewhat more peculiar. There are b ( X )real scalars X P and the same number of complex fermions Λ P fitting nicely into b ( X )2 a multiplets. However, we also need to take into account the constraint (4.30) on thefermions, which halves their number. The result is a set of constrained 2 a multiplets withthe same number of degrees of freedom as 1 / b ( X ) 2 b multiplets, reminding us of theiroriginal nature. This leaves us with the ˆ4-form fermions Υ ˆ X . They have no bosonic zeromode partners so cannot be part of either the standard 2 a or 2 b multiplets. The naturalguess is for them to form 2 h , ( X ) fermionic 2 b multiplets. As for the 3-form fermions,there is the constraint (4.33), which reduces their number to by a factor of two. That is, wehave h , ( X ) complex one-dimensional fermions in this sector. Finally, the lapse function N and the component ψ of the 11-dimensional gravitino should form the one-dimensionalgravity multiplet. We now verify this assignment of supermultiplets by a reduction of the11-dimensional supersymmetry transformations.Our task is to reduce the 11-dimensional supersymmetry transformations (2.8) for themetric ansatz (4.3), the associated spin connection (4.38)-(4.39), the three-form ansatz (4.4)and the gravitino ansatz (4.27)–(4.29). We denote the spinor parameterising 11-dimensionalsupersymmetry transformations by ǫ (11) in order to distinguish it from its one-dimensionalcounterpart ǫ . The 11-dimensional spinor can then be decomposed as ǫ (11) = i ǫ ⊗ η ⋆ − i ǫ ⊗ η , (5.6)– 25 –here η is the covariantly constant spinor on the Calabi-Yau five-fold. Inserting all thisinto the 11-dimensional supersymmetry transformations and collecting terms proportionalto the same harmonic Calabi-Yau forms we find the supersymmetry transformations of thevarious zero modes. For the lapse function N and the time component ψ of the gravitinothey are δ ǫ N = − ǫ ¯ ψ , δ ǫ ψ = i ˙ ǫ, δ ǫ ¯ ψ = 0 δ ¯ ǫ N = ¯ ǫψ , δ ¯ ǫ ψ = 0 , δ ¯ ǫ ¯ ψ = − i ˙¯ ǫ . (5.7)These transformations are identical to the one for a one-dimensional N = 2 supergravitymultiplet as can be seen by comparing with Appendix C.For the other zero modes we find the supersymmetry transformations(1 ,
1) : δ ǫ t i = − ǫψ i , δ ǫ ψ i = 0 , δ ǫ ¯ ψ i = i N − ǫ ˙ t i + . . . , (5.8)3-form : δ ǫ X P = − ǫ Λ P , δ ǫ Λ P = 0 , δ ǫ ¯Λ P = iN − ǫP −QP ˙ X Q + . . . , (5.9)ˆ4-form : δ ǫ Υ ˆ X = 0 + . . . , δ ǫ ¯Υ ˆ X = 0 , (5.10)(1 ,
4) : δ ǫ z a = iǫκ a , δ ǫ ¯ z ¯ a = 0 , δ ǫ κ a = 0 , δ ǫ ¯ κ ¯ a = N − ǫ ˙¯ z ¯ a + . . . , (5.11)and similarly for the ¯ ǫ -variation. The dots indicate terms cubic in fermions which wehave omitted . To arrive at the last equation in (5.9), we have performed a compensatingtransformation, making use of a local fermionic symmetry. Namely, the action (4.43)and (4.45) is invariant under δ Λ P = P −QP l Q , (and: δ ¯Λ P = P + QP ¯ l Q ) , (5.12)for a set of local complex fermionic parameters l Q , while all other fields do not transform.The constraint (4.30) on Λ P may be viewed as a gauge choice with respect to this symmetry.The form of the last equation in (5.9) then guarantees the preservation of this gauge choiceunder a supersymmetry transformation as required by eq. (4.31). Even though the ˆ4-form fermions Υ ˆ X are subject to the same kind of constraint (cf. eq. (4.33)), there is noassociated local symmetry. This is because the proof that (5.12) is a symmetry cruciallyhinges on the Hermiticity of the 3-form metric (cf. eq. (B.110)), but the ˆ4-form metric isnot Hermitian.Again, comparing with the results for the supersymmetry transformations of the vari-ous one-dimensional N = 2 multiplets given in Appendix C, we confirm the assignment ofzero modes into supermultiplets discussed above. In particular, the transformation of theˆ4-form fermions Υ ˆ X indicates that they should indeed be part of fermionic 2 b supermulti-plets.To summarise these results, we write down the explicit off-shell component expansionfor all superfields in terms of the Calabi-Yau five-fold zero modes and appropriate auxiliary It may be a bit surprising that the transformations above do not seem to mix fields of different types(that is (1 , , terms. That is, the sector-mixing terms in thetransformations are all of order (fermi) , which can be seen by taking the full, off-shell supersymmetrytransformations of Appendix C and eliminating the auxiliary fields. – 26 –elds. Taking into account the component structure of the various supermultiplets derivedin Appendix C, we haveSUGRA (2 a ) : E = − N − i θ ¯ ψ − i θψ , (5.13)(1 ,
1) (2 a ) : T i = t i + iθψ i + i ¯ θ ¯ ψ i + 12 θ ¯ θf i , (5.14)3-form (2 a ) : X P = X P + iθ Λ P + i ¯ θ ¯Λ P + 12 θ ¯ θg P , (5.15)ˆ4-form (2 b ) − fermionic : R ˆ X = Υ ˆ X + θH ˆ X + i N − θ ¯ θ ( ˙Υ ˆ X − ψ H ˆ X ) , (5.16)(1 ,
4) (2 b ) : Z a = z a + θκ a + i N − θ ¯ θ ( ˙ z a − ψ κ a ) , (5.17)where f i , g P and H ˆ X are bosonic auxiliary fields. These auxiliary fields can, of course, notbe obtained from the reduction (since 11-dimensional supersymmetry is realised on-shell)and have to be filled in “by hand”. Full, off-shell supersymmetry transformations for allthe above components are given in Appendix C. Having identified the relevant supermultiplets and their components our next step is towrite down an N = 2 superspace version of the one-dimensional effective theory. Forthe most part, an appropriate form for the superspace action can be guessed based onthe bosonic action (4.14). Basically, all one has to do is to promote the bosonic fieldsin this action to their associated superfields, replace time derivatives by super-covariantderivatives D or ¯ D and integrate over superspace. In addition, we need to implement theconstraint (4.30) on the 3-form fermions Λ P at the superspace level. The superpartner ofthe constraint (4.30) turns out to be g P = N − ∆ QP ˙ X Q + N − ( ψ Λ P − ¯ ψ ¯Λ P ) . (5.18)Note that since the only object in this equation depending on the complex structure moduliis ∆ QP , it follows that ∆ QP ,a ˙ X Q = 0. Constraints (4.30) and (5.18) form a constraintmultiplet and can hence be obtained from a single complex superspace equation P −P Q ( Z, ¯ Z ) DX P = 0 , (and c.c.) , (5.19)where P −P Q ( Z, ¯ Z ) is the superspace version of the projection operator P −P Q definedin eq. (B.104). The superspace constraint (5.19) follows from a superspace action byintroducing a set of b ( X ) complex fermionic Lagrange multiplier superfields L P L P = L (0) P + θL (1) P + ¯ θL (2) P + 12 θ ¯ θL (3) P . (5.20)The action for the fermionic Lagrange multiplier superfields is then given by − l Z dτ d θ E (cid:0) L Q P −P Q ( Z, ¯ Z ) DX P − ¯ L Q P + P Q ( Z, ¯ Z ) ¯ D ¯ X P (cid:1) . (5.21)– 27 –his takes care of all but the fermionic multiplets in the ˆ4-form sector whose superfieldaction has to be inferred from the fermionic component action (4.43), (4.45). In particu-lar, the ˆ4-form part of the superspace action should be such that the bosons H ˆ X in thefermionic multiplets are non-dynamical. As for the 3-form case, we need to implement theconstraint (4.33) on the ˆ4-form fermions Υ ˆ X at the superspace level. The superpartner ofthe constraint (4.33) is simply P + ˆ Y ˆ X H ˆ Y = H ˆ X , (and c.c.) . (5.22)Eqs. (4.33) and (5.22) are part of a single superspace equation P − ˆ Y ˆ X ( Z, ¯ Z ) R ˆ Y = 0 , (and c.c.) , (5.23)which can be obtained from a superspace action principle − l Z dτ d θ E (cid:16) L ˆ X P − ˆ Y ˆ X ( Z, ¯ Z ) R ˆ Y − ¯ L ˆ X P + ˆ Y ˆ X ( Z, ¯ Z ) ¯ R ˆ X (cid:17) (5.24)by means of a set of 2 h , ( X ) complex fermionic Lagrange multiplier superfields L ˆ X , whichhave the same component expansion as in eq. (5.20). P ± ˆ Y ˆ X ( Z, ¯ Z ) are the superspaceversions of the projection operators P ± ˆ Y ˆ X defined in eq. (B.130).Combining all this, the suggested superspace action is S = − l Z dτ d θ E n G (1 , ij ( T ) D T i ¯ D T j + G (3) PQ ( T , Z, ¯ Z ) DX P ¯ DX Q − G (ˆ4)ˆ X ˆ Y ( T ) R ˆ X ¯ R ˆ Y +4 V ( T ) G (1 , a ¯ b ( Z, ¯ Z ) D Z a ¯ D ¯ Z ¯ b + (cid:16) L Q P −P Q ( Z, ¯ Z ) DX P + L ˆ X P − ˆ Y ˆ X ( Z, ¯ Z ) R ˆ Y + c.c. (cid:17)o . (5.25)This action can be expanded out in components using the formulæ presented earlier andsystematically developed in Appendix C. The result can be split into (1 , ,
4) parts by writing S = l Z dτ n L (1 , + L (3) + L (ˆ4) + L (1 , o . (5.26)For these four parts of the Lagrangian in (5.26) we find, after taking into account theconstraints (4.30) and (5.18) and using the formulæ provided in Appendix B.4.1 L (1 , = 14 N − G (1 , ij ( t ) ˙ t i ˙ t j − i G (1 , ij ( t )( ψ i ˙¯ ψ j − ˙ ψ i ¯ ψ j ) + 14 N G (1 , ij ( t ) f i f j + i N − G (1 , ij ( t )( ψ i ψ + ¯ ψ i ¯ ψ ) ˙ t j + 12 N − G (1 , ij ( t ) ψ ¯ ψ ψ i ¯ ψ j − N G (1 , ij,k ( t )( ψ i ¯ ψ j f k − ψ k ¯ ψ j f i − ψ i ¯ ψ k f j ) + i G (1 , ij,k ( t )( ψ k ¯ ψ i + ¯ ψ k ψ i ) ˙ t j − N G (1 , ij,kl ( t ) ψ i ¯ ψ j ψ k ¯ ψ l , (5.27)– 28 – (3) = 12 N − G (3) PQ ( t, z, ¯ z ) ˙ X P ˙ X Q − i G (3) PQ ( t, z, ¯ z )(Λ P ˙¯Λ Q − ˙Λ P ¯Λ Q )+ iN − G (3) PQ ( t, z, ¯ z )(Λ P ψ + ¯Λ P ¯ ψ ) ˙ X Q + N − G (3) PQ ( t, z, ¯ z ) ψ ¯ ψ Λ P ¯Λ Q − N G (3) PQ ,i ( t, z, ¯ z )Λ P ¯Λ Q f i + iG (3) PQ ,i ( t, z, ¯ z )( ψ i ¯Λ P + ¯ ψ i Λ P ) ˙ X Q − G (3) PQ ,i ( t, z, ¯ z )Λ P ¯Λ Q ( ψ ψ i − ¯ ψ ¯ ψ i ) − N G (3) PQ ,ij ( t, z, ¯ z )Λ P ¯Λ Q ψ i ¯ ψ j − i G (3) PQ ,a ( t, z, ¯ z )Λ P ¯Λ Q ( ˙ z a − ψ κ a ) + i G (3) PQ , ¯ a ( t, z, ¯ z )Λ P ¯Λ Q ( ˙¯ z ¯ a + 2 ¯ ψ ¯ κ ¯ a )+ G (3) PQ ,a ( t, z, ¯ z ) κ a ¯Λ P ˙ X Q − G (3) PQ , ¯ a ( t, z, ¯ z )¯ κ ¯ a Λ P ˙ X Q − N G (3) PQ ,a ¯ b ( t, z, ¯ z )Λ P ¯Λ Q κ a ¯ κ ¯ b − iN G (3) PQ ,ia ( t, z, ¯ z )Λ P ¯Λ Q ¯ ψ i κ a − iN G (3) PQ ,i ¯ a ( t, z, ¯ z )Λ P ¯Λ Q ψ i ¯ κ ¯ a , (5.28) L (ˆ4) = − i G (ˆ4)ˆ X ˆ Y ( t )(Υ ˆ X ˙¯Υ ˆ Y − ˙Υ ˆ X ¯Υ ˆ Y ) + 3 N G (ˆ4)ˆ X ˆ Y ( t ) H ˆ X ¯ H ˆ Y + 3 iN G (ˆ4)ˆ X ˆ Y ,i ( t )( ψ i Υ ˆ X ¯ H ˆ Y + ¯ ψ i ¯Υ ˆ Y H ˆ X ) + 32 N G (ˆ4)ˆ X ˆ Y ,i ( t )Υ ˆ X ¯Υ ˆ Y f i + 3 N G (ˆ4)ˆ X ˆ Y ,ij ( t )Υ ˆ X ¯Υ ˆ Y ψ i ¯ ψ j − G (ˆ4)ˆ X ˆ Y ,i ( t )Υ ˆ X ¯Υ ˆ Y ( ψ ψ i − ¯ ψ ¯ ψ i ) , (5.29) L (1 , = 4 N − V G (1 , a ¯ b ( z, ¯ z ) ˙ z a ˙¯ z ¯ b − iV G (1 , a ¯ b ( z, ¯ z )( κ a ˙¯ κ ¯ b − ˙ κ a ¯ κ ¯ b ) − N − V G (1 , a ¯ b ( z, ¯ z )( ψ κ a ˙¯ z ¯ b − ¯ ψ ¯ κ ¯ b ˙ z a ) + 4 N − V G (1 , a ¯ b ( z, ¯ z ) ψ ¯ ψ κ a ¯ κ ¯ b + 2 iV G (1 , a ¯ b,c ( z, ¯ z ) κ a ¯ κ ¯ b ˙ z c − iV G (1 , a ¯ b, ¯ c ( z, ¯ z ) κ a ¯ κ ¯ b ˙¯ z ¯ c − N K i G (1 , a ¯ b ( z, ¯ z ) κ a ¯ κ ¯ b f i − N K ij G (1 , a ¯ b ( z, ¯ z ) κ a ¯ κ ¯ b ψ i ¯ ψ j − K i G (1 , a ¯ b ( z, ¯ z ) ψ i ¯ κ ¯ b ( ˙ z a − ψ κ a ) + 13! K i G (1 , a ¯ b ( z, ¯ z ) ¯ ψ i κ a ( ˙¯ z ¯ b + 12 ¯ ψ ¯ κ ¯ b ) . (5.30)We should now compare this Lagrangian with our result obtained from dimensional reduc-tion in the previous section. To do this, we first have to integrate out the auxiliary fields f i and H ˆ X . A quick inspection of their equations of motion derived from eqs. (5.27)–(5.30)shows that they are given by fermion bilinears. Hence, integrating them out only leads toadditional four-fermi terms. Since we have not computed four-fermi terms in our reduc-tion from 11 dimensions they are, in fact, irrelevant for our comparison. All other terms,that is purely bosonic terms and terms bilinear in fermions, coincide with our reductionresult (4.14), (4.43) and (4.45). This shows that eq. (5.25) is indeed the correct superspaceaction.Both the lapse function N and the gravitino ψ are non-dynamical and their equationsof motion lead to constraints. For the lapse, this constraint implies the vanishing of theHamiltonian associated with the Lagrangian (5.27)–(5.30) and it reads (after integratingout the (1 ,
1) and ˆ4-form auxiliary fields f i and H ˆ X )14 G (1 , ij ( t )( ˙ t i + 2 iψ i ψ + 2 i ¯ ψ i ¯ ψ ) ˙ t j + 12 G (3) PQ ( t, z, ¯ z )( ˙ X P + 2 i Λ P ψ + 2 i ¯Λ P ¯ ψ )) ˙ X Q + 4 V G (1 , a ¯ b ( z, ¯ z )( ˙ z a ˙¯ z ¯ b − ψ κ a ˙¯ z ¯ b + ¯ ψ ¯ κ ¯ b ˙ z a ) + (fermi) = 0 . (5.31)– 29 –he equation of motion for ψ generates the superpartner of this Hamiltonian constraintand implies the vanishing of the supercurrent.Let us now discuss some of the symmetries of the above one-dimensional action. Theaction (5.25) is manifestly invariant under super-worldline reparametrizations { τ, θ, ¯ θ } →{ τ ′ ( τ, θ, ¯ θ ), θ ′ ( τ, θ, ¯ θ ), ¯ θ ′ ( τ, θ, ¯ θ ) } , which, in particular, includes worldline reparametriza-tions τ → τ ′ ( τ ) (that is, one-dimensional diffeomorphisms) and local N = 2 supersym-metry. Note that the super-determinant of the supervielbein E , which transforms as asuper-density, is precisely what is needed to cancel off the super-jacobian from the changeof dτ d θ , so that dτ d θ E is an invariant measure.In particular, the theory is invariant under worldline reparametrizations, τ → τ ′ ( τ )which can be seen as a remnant of the diffeomorphism invariance of the eleven dimensionalaction (2.1). Here, the lapse function, N , plays the same rˆole as the “vielbein” and it trans-forms as a co-vector under worldline reparametrizations. The transformation properties ofthe different types of component fields under worldline reparametrizations are summarizedin Table 2. The bosonic matter fields t i , X P and z a and the bosonic auxiliary fields f i , Name WR transformation τ → τ ′ ( τ )scalar z a → z a ′ ( τ ′ ) = z a ( τ )co-vector N → N ′ ( τ ′ ) = ∂τ∂τ ′ N ( τ )spin-1 / κ a → κ a ′ ( τ ′ ) = κ a ( τ )spin-3 / ψ → ψ ′ ( τ ′ ) = ∂τ∂τ ′ ψ ( τ ) Table 2:
Worldline reparametrization (WR) covariance. g P and H ˆ X transform as scalars, whereas the fermionic matter fields ψ i , Λ P , Υ ˆ X and κ a transform as spin-1 / ψ transforms as a spin-3 / X P arise as zero-modes of the M-theory three-form A and, hence,they are axion-like scalars with associated shift transformations acting as X P ( τ ) → X P ′ ( τ ) = X P ( τ ) + c P , (5.32)where the c P are a set of complex constants. It is easy to see that the component ac-tion (5.27)–(5.30) only depends on ˙ X P but not on X P and that, hence, the action isinvariant under the above shifts. Also in the 3-form sector, there is a local fermionicsymmetry of the form δ Λ P = P −QP l Q as discussed around eq. (5.12).
6. Flux and the one-dimensional scalar potential
We have seen that, unless one works with a Calabi-Yau five-fold X satisfying c ( X ) = 0,flux and/or membranes are required in order to satisfy the anomaly condition (3.6). Atorder β , both flux and membranes are expected to contribute to a scalar potential in theone-dimensional effective theory. So far, we have worked at zeroth order in β but, in thissection, we will calculate the leading order β contributions to the scalar potential. Giventhe need for flux and/or membranes in many five-fold compactifications this potential isclearly of great physical significance. – 30 – .1 Calculating the scalar potential from 11 dimensions There are three terms in the 11-dimensional theory which can contribute at order β to a scalar potential in the one-dimensional effective theory: The non-topological R terms (2.13) evaluated on the five-fold background, the kinetic terms G ∧ ∗ G for the four-form field strength if flux is non-zero and the volume term in the membrane action (2.14)provided wrapped membranes are present. We will now discuss these terms in turn startingwith the R one.Due to its complicated structure, the reduction of this term on a Calabi-Yau five-fold background is not straightforward. Also, this term depends on the unknown four-curvature of the five-fold and the only hope of arriving at an explicit result is that itbecomes topological when evaluated on a five-fold background. A fairly tedious, althoughin principle straightforward calculation shows that this is indeed the case and that it canbe expressed in terms of the fourth Chern class, c ( X ), of the five-fold. Explicitly, we findthat eq. (2.13) reduces to lβ Z dτ N c i ( X ) t i , (6.1)where β = (2 π ) β/v / and we have expanded the fourth Chern class as c ( X ) = c i ( X )˜ ω i into a basis of harmonic (4 , ω i dual to the harmonic (1 , ω i .Next, we consider the contribution of a membrane wrapping a holomorphic curve C in X with second homology class W . Using the explicit parametrisation X = σ , X µ = X µ ( σ ), X ¯ µ = X ¯ µ (¯ σ ), where σ = ( σ + iσ ) / √ C , the first term in themembrane action (2.14) reduces to − lβ Z dτ N W i t i . (6.2)Here, we have expanded the membrane class as W = W i ˜ ω i into our basis of harmonic(4 , g (see eq. (3.4)) the ansatz for flux can be written as g = 12 n X O X = n e σ e + ( m x ̟ x + c . c . ) , (6.3)where { O X } with X , Y , . . . = 1 , . . . , b ( X ) is a basis of real harmonic 4-forms, { σ e } with e, f, . . . = 1 , . . . , h , ( X ) is a basis of real harmonic (2 , { ̟ x } with x, y, . . . =1 , . . . , h , ( X ) is a basis of harmonic (1 , , ,
1) and (2 ,
2) parts. The factor of 1 / n X bean even (odd) integer depending on whether c ( X ) is even (odd). An essential ingredientin the reduction is the 10-dimensional Hodge dual of g . From the results in eq. (B.57) wesee that this is given by ∗ g = n e (cid:18) J ∧ σ e − i J ∧ ˜ σ e + 112 ˜˜ σ e J (cid:19) − ( m x J ∧ ̟ x + c . c . ) . (6.4)– 31 –e recall from Appendix B.3 that ˜ σ e is a harmonic (1 , σ e by a contraction with the inverse metric g µ ¯ ν . Likewise, ˜˜ σ e is a scalar on X , obtained from σ e by contraction with two inverse metrics. Following the discussion in Appendix B.4 theseobjects can be written as ˜ σ e = ik ie ω i , ˜˜ σ e = − κ k ie κ i , (6.5)where k ie is a set of (moduli-dependent) coefficients. Combining these results the four-formkinetic term κ R M ( − ) G ∧ ∗ G reduces to − lβ Z dτ N (cid:20) n e n f (cid:18) d efi t i + 12 k if d eijk t j t k − κ k ie κ i d ejkl t j t k t l (cid:19) − (cid:0) m x ¯ m ¯ y d x ¯ yi t i + c . c . (cid:1)(cid:21) , (6.6)where we have used some of the intersection numbers introduced in Appendix B.4.We introduce a one-dimensional scalar potential U by S B , pot = − l Z dτ N U . (6.7)This expression should be added to the bosonic action (4.14). Then, by combining thethree contributions above, we find that U = β (cid:20)(cid:18)
12 ( g ∧ g ) (2 , i −
12 ( g ∧ g ) (1 , i + W i − c i ( X ) (cid:19) t i (6.8)+ 14 n e n f k if (cid:18) G (1 , ik − κ i κ k κ (cid:19) G (1 , kj d ejlm t l t m (cid:21) . (6.9)Let us pause to discuss this result. The first line is linear in the K¨ahler moduli t i withcoefficients which are almost identical to the components of the anomaly condition (3.6). Infact, only the sign of ( g ∧ g ) (1 , , the contribution from the (1 ,
3) part of the flux, is oppositeto what it is in the anomaly condition (3.6). The sign difference between the (2 ,
2) and(1 ,
3) flux parts in eq. (6.8) can be traced back to a sign difference in the formulæ (B.57)for the Hodge duals which read ∗ σ = J ∧ σ + . . . for (2 ,
2) forms and ∗ ̟ = − J ∧ ̟ for (1 , ,
3) flux. As will become clear in thefollowing such a term is not consistent with one-dimensional N = 2 supersymmetry. Onthe other hand, the second part (6.9) of the potential which only depends on (2 ,
2) fluxcan be written in a supersymmetric form, as we will see. Hence, (2 ,
2) flux is consistentwith one-dimensional N = 2 supersymmetry while (1 ,
3) flux breaks it explicitly. Thisconclusion is also supported by analysing the eleven-dimensional Killing spinor equationsand the conditions for N = 2 supersymmetry in the presence of fluxes [34]. While theremay not be anything wrong with this explicit breaking, we have set out to study M-theory compactifications which preserve one-dimensional N = 2 supersymmetry. We will,therefore, focus on (2 ,
2) flux and set the (1 ,
3) flux to zero in the subsequent discussion.– 32 –he decomposition in eq. (6.3) of four-form flux into (1 , ,
1) and (2 ,
2) pieces de-pends on the complex structure and therefore the condition for unbroken N = 2 supersym-metry, namely that the (1 ,
3) and (3 ,
1) parts of the four-form flux vanish, g (1 , = g (3 , = 0, a priori leads to a potential for the complex structure moduli. In other words, the complexstructure moduli are only allowed to fluctuate in such a way as to keep the four-form fluxpurely of (2 ,
2) type. With the decomposition (B.113) inserted into eq. (6.3), the condition g (1 , = 0 becomes m x ( z, ¯ z ) = n X D X x ( z, ¯ z ) = 0 , (and c.c.) . (6.10)However, it is not known whether the D X x and hence the resulting potential for the z a can be calculated explicitly. It is important to recall that in our analysis of bosonic andfermionic 4-form fields we are restricting to Calabi-Yau five-folds that satisfy eq. (B.117)and, in this case, the potential vanishes, that is the complex structure moduli are restoredas flat directions in the moduli space, because for such manifolds the split of a four-forminto a (2 , ,
3) + (3 , n ˆ X = 0, with the help of the decomposition (B.120). Moreover,the (2 ,
2) flux itself, g (2 , = n ˜ X O ˜ X = n e σ e , becomes a complex structure independentquantity.This leaves us with the second part (6.9) of the scalar potential and, in order to writethis into a more explicit form, we need to compute the coefficients k ie . This has, in fact,been done in eq. (B.92). Inserting these results and using eqs. (B.80) and (B.81) we finallyfind for the scalar potential U = 12 G (1 , ij W i W j , W i = ∂ W ∂t i , (6.11)where the “superpotential” W is given by W ( t ) = √ β d eijk n e t i t j t k (6.12)and G (1 , ij is the inverse of the physical (1 ,
1) moduli space metric (4.18). The fact thatthe scalar potential can be written in terms of a superpotential in the usual way suggestsit can be obtained as the bosonic part of a superfield expression. This is indeed the caseand the term we have to add to the superspace action (5.25) is simply S pot = − l Z dτ d θ E W ( T ) . (6.13)Indeed, combining this term with the (1 ,
1) kinetic term in the superspace action (5.25)and working out the bosonic component action using eq. (5.27) one finds the terms l Z dτ N (cid:16) G (1 , ij f i f j − f i W i (cid:17) , (6.14)which, after integrating out the (1 ,
1) auxiliary fields f i = G (1 , ij W i , (6.15)– 33 –eproduce the correct scalar potential.It is, perhaps, at first surprising that the formula (6.11) for the scalar potentialin terms of the superpotential looks exactly like the one in global supersymmetry anddoes not seem to have the usual supergravity corrections such as the analogue of the fa-mous − |W| term in four-dimensional N = 1 supergravity. However, we have to keepin mind that the physical moduli space metric G (1 , ij differs from the standard modulispace metric G (1 , ij and it this difference which encodes the supergravity corrections tothe scalar potential. Specifically, let us formally introduce a “K¨ahler covariant derivative” D i W = W i + ∂K (1 , ∂t i W = W i − t i W , where we recall that K (1 , = − ln κ and we haveused eq. (B.78) in the second equality. Moreover, we note that, from eq. (B.81), the inverse G (1 , ij can be written as G (1 , ij = 18 V (cid:18) G (1 , ij − t i t j (cid:19) . (6.16)Combining these results and using κ = 5! V we can also write the scalar potential (6.11) as U = 152 e K (1 , (cid:18) G (1 , ij D i W D j W − W (cid:19) , (6.17)which resembles the expression for the four-dimensional N = 1 supergravity potential quiteclosely.Finally, we should point out that the superpotential (6.12) can be obtained from aGukov-type formula W ( t ) = 13 Z X G flux ∧ J . (6.18)This integral is, in fact, the only topological integral, linear in flux, one can build using thetwo characteristic forms J and Ω of the five-fold and G flux . In this sense, it is the naturalexpression for the superpotential. Here, we have explicitly verified by a reduction form 11dimensions that it gives the correct answer.When (2 ,
2) flux is non-vanishing, another set of bosonic terms arises from the Chern-Simons term A ∧ G ∧ G in the 11-dimensional action (2.2). Writing the complete ansatzfor the four-form field strength G , including flux and zero modes, one has G = G (2 , + ˙ X P dτ ∧ N P = 2 πT n e σ e + ˙ X P dτ ∧ N P . (6.19)Here, we recall that { N P } , where P , Q , . . . = 1 , . . . , b ( X ), are a basis of real harmonic3-forms and X P are the associated 3-form zero modes. Inserting this ansatz into the11-dimensional Chern-Simons term one finds S B , CS = − l Z dτ √ β d PQ e n e ˙ X P X Q , (6.20)where d PQ e = R X N P ∧ N Q ∧ σ e . Note that (6.20) is linear in flux and, hence, appears atorder √ β . It represents a one-dimensional Chern-Simons term.– 34 – .2 A closer look at the bosonic action and the scalar potential We would now like to discuss some features of the bosonic effective action. To begin with,we summarise our result for the complete bosonic action up to and including order β . Thebosonic action depends on three sets of fields, the real (1 ,
1) moduli t i , the real 3-formmoduli X P and the complex (1 ,
4) moduli z a . It can be written as a sum of three parts S B = S B , kin + S B , pot + S B , CS (6.21)which, from eqs. (4.14), (6.7), (6.11) and (6.20), are given by S B , kin = l Z dτ N − (cid:26) G (1 , ij ( t ) ˙ t i ˙ t j + 12 G (3) PQ ( t, z, ¯ z ) ˙ X P ˙ X Q + 4 V ( t ) G (1 , a ¯ b ( z, ¯ z ) ˙ z a ˙¯ z ¯ b (cid:27) , (6.22) S B , pot = − l Z dτ N U , (6.23) S B , CS = − l Z dτ √ β d PQ e n e ˙ X P X Q , (6.24)with the scalar potential U and superpotential WU = 12 G (1 , ij W i W j , W ( t ) = √ β d eijk n e t i t j t k . (6.25)The (1 ,
1) metric G (1 , ij has been defined in eq. (4.18), the 3-form metric G (3) PQ in eq. (4.21)and the (1 ,
4) metric G (1 , a ¯ b in eq. (4.20). The first two parts of this action can be schemat-ically written as S B , kin + S B , pot = l Z dτ (cid:26) N − G IJ ( φ ) ˙ φ I ˙ φ J − N U ( φ ) (cid:27) , (6.26)where we have collectively denoted the various types fields of fields by ( φ I )=( t i , X P , z a , ¯ z ¯ b )and G IJ is a block-diagonal metric which contains the above moduli space metrics in theappropriate way. The associated equations of motion then have the general form1 N ddτ ˙ φ I N ! + Γ IJK ˙ φ J N ˙ φ K N + 14 G IJ ∂ U ∂φ J + C I = 0 , (6.27)where Γ IJK is the Christoffel connection associated to G IJ and C I is the contribution fromthe Chern-Simons term. Since the Chern-Simons term only depends on X P , we have C i = C a = C ¯ b = 0.Are there any static solutions, that is, solutions with all φ I = const in the presence ofa flux potential? Since the potential U only depends on the (1 ,
1) moduli, it is certainlyconsistent with the equations of motion (6.27) to set all other fields to constants. Forvacua without (2 ,
2) flux (but possibly with membranes) this can also be done for the (1 , t i . In this case the scalar potential vanishes identically and the moduli space iscompletely degenerate. – 35 –n the presence of (2 ,
2) flux the situation is more complicated. First, one shouldlook for vacua with constant scalars which preserve the N = 2 supersymmetry of the one-dimensional theory. Finding such vacua amounts to setting the supersymmetry variationsof all fermions to zero and solving the resulting Killing spinor equations, as usual. For thevarious 2 b multiplets the supersymmetry variations of their fermion components vanishesautomatically for constant scalar fields and vanishing fermions, as can be seen directlyfrom the results in Appendix C.2. On the other hand, the supersymmetry variations of thefermions residing in 2 a multiplets require a bit more care. For the 3-form fermions Λ P onehas from eqs. (C.76) and (C.77) δ ǫ Λ P = 0 , δ ǫ ¯Λ P = − ǫg P = 0 , (6.28)after inserting the constraint in eq. (5.18) determining g P . For the (1 ,
1) fermions ψ i thetransformations lead to δ ǫ ψ i = 0 , δ ǫ ¯ ψ i = − ǫf i = − ǫG (1 , ij W j , (6.29)again assuming vanishing fermions and constant scalars. Hence, constant scalar field vacuawhich preserve N = 2 supersymmetry are characterised by the “F-term” equations W i = 0 . (6.30)Eq. (6.25) shows that solutions to these F-term equations are stationary points of thescalar potential, although, unlike in four-dimensional N = 1 supergravity, they need not beminima since the (1 ,
1) metric G (1 , is not positive definite. Another interesting differenceto four-dimensional supergravity is that the scalar potential always vanishes for solutionsof the F-term equations.Let us now consider explicit examples to see whether the F-term equations have inter-esting solutions for our examples. From the general form of W in eq. (6.25), it is clear thatfor a single (1 ,
1) modulus, that is, h , ( X ) = 1, the only solution to the F-term equations is t = 0. This corresponds to vanishing Calabi-Yau volume so we should certainly not trustour one-dimensional effective theory at this point. Moving on to Calabi-Yau manifoldswith h , = 2 we start with the second example in Table 6, a co-dimension one CICY inthe ambient space A = P × P with configuration matrix X ∼ " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (6.31)The discussion in Appendix B.2 shows that the anomaly condition for this CICY can besatisfied for a range of fluxes and an appropriate number of membranes. Since h , ( X ) = 3,we have three flux parameters n , n , n and flux can explicitly be written as g = n J + n J J + n J . Then, one finds for the K¨ahler potential and superpotential κ = 40 t t + 40 t t , W = 43 n t + 4( n + n ) t t + 43 n t + 4( n + n ) t t . (6.32)– 36 –t is easy to see that setting, for example, n = n = 3 and n = − t = t . Moreover, this flat direction consists ofminima of the potential with zero cosmological constant. The existence of such minimais of considerable importance for our M-theory compactifications. A general problem ofcompactifications with flux is the tendency of producing large potential energies above thecompactification scale due to the quantised nature of flux. Such high scales of potentialenergy are of course problematic as they invalidate the low-energy effective theory. Wehave just seen an example where this problem can be avoided due to a flat direction withvanishing vacuum energy in the two-dimensional K¨ahler moduli space. This means, at leastto first order in our β expansion, self-consistent five-fold compactifications of M-theory with(2 ,
2) flux exist.As the next example shows this is by no means automatic. Consider the first examplein Table 6, a co-dimension one CICY in the ambient space A = P × P with configurationmatrix X ∼ " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (6.33)As in the previous example, the anomaly condition can be satisfied for a range of fluxesand with the appropriate number of membranes (see the discussion around eq. (B.34)).This manifold has h , ( X ) = 2 and the flux can be written as g = n J J + n J withtwo flux parameters n and n . The K¨ahler potential and superpotential for the model aregiven by κ = 30 t t + 2 t , W = 6 n t t + (cid:18) n + 23 n (cid:19) t . (6.34)In this case, the F-term equations imply that t = 0 and the above expression for theK¨ahler potential shows that the Calabi-Yau volume vanishes for this value. Hence, thereis no viable supersymmetric minimum in this case.We should now discuss the scalar potential in some of the cases where solutions to theF-term equations cannot be found. In general, we note that under a rescaling t i → λt i the(1 ,
1) metric scales as G (1 , ij ( λt ) = λ G (1 , ij ( t ) and the superpotential as W ( λt ) = λ W ( t ).This means that the scalar potential scales as U ( λt ) = λ U ( t ), so is homogeneous withdegree one. When discussing the implications of this scaling behaviour it has to be keptin mind that the metric G (1 , ij has signature ( − , +1 , . . . , +1) with the negative direction u i given by u i ∼ t i . Whether this negative direction is “probed” by the scalar potentialdepends on the structure of the superpotential and its derivatives. If it is, the potentialwill be of the form U = − cλ , where c is a positive constant. This indicates an instabilitywhich leads to a rapid growth of the Calabi-Yau volume and decompactification. Clearly,this is always the case for examples with h , ( X ) = 1 where the metric is just a negativenumber. For h , ( X ) > | κ = 7 t , W = 352 p β t , (6.35)where t is the single (1 ,
1) modulus. After a short computation, using eqs. (4.18) and (6.25)this leads to the scalar potential U = − β t . (6.36)As expected, the potential is negative and results in a rapid growth of the volume. Ourcompactification can only be trusted for large Calabi-Yau volume, that is t ≫
1. In thiscase the scale of the scalar potential (6.36) is quite large and it is questionable if we cantrust our low-energy theory.For an example with h , ( X ) = 2 we return to the manifold in eq. (6.33) with K¨ahlerand superpotential as in eq. (6.34) which did not exhibit F-flat directions. We find for thescalar potential U = β (15 n + 2 n )6(15 t + t ) (cid:0) (3 n − n ) t − n t t (cid:1) . (6.37)We recall that the K¨ahler cone of this CICY is given by t > t >
0. Now choosethe fluxes to be n = 0 and n = 1 /
2. Then the above potential is strictly negative in theK¨ahler cone of X and such that both t and t will grow. For n = 1 and n = − / t contracts and, as a result, the total volume goes to zero (while t slowlyexpands). As for the septic, for large volume, t ≫ t ≫
1, the scalar potential is largeand it is not clear that the low-energy theory is valid.In summary, a first look at the one-dimensional effective theory at order β indicates anumber of possibilities to obtain self-consistent compactifications with vanishing vacuumenergy. First of all, for some Calabi-Yau five-folds the anomaly condition can be satisfiedwithout the inclusion of flux, either if c ( X ) = 0 or if a non-zero c ( X ) can be compensatedfor by membranes, and, in this case, the scalar potential vanishes identically. An interestinggeneral feature of the scalar potential is that it vanished for supersymmetric vacua, thatis, for solutions to the F-term equations. We have shown that such solutions to the F-termequations do indeed exist for some five-folds and that they correspond to flat directions ofthe potential. The general structure of the scalar potential means that the vacuum energyvanishes along those flat directions. If supersymmetric flat directions do not exist, thescalar potential, which is homogeneous of degree one in the K¨ahler moduli, is generallylarge for large volume and it is questionable whether one can trust the effective theory.Taken at face value, this scalar potential may either lead to a rapid expansion or a rapidcontraction of the Calabi-Yau volume, depending on the case. Calabi-Yau five-folds with h , = 1 such as the septic do not have F-flat directions and always contract. For h , >
7. Conclusion and Outlook
In this paper, we have considered compactifications of M-theory on Calabi-Yau five-fold– 38 –ackgrounds, leading to one-dimensional effective theories with N = 2 supersymmetry. Inthe absence of flux and membranes, such five-fold backgrounds are solutions to M-theory atzeroth order in the β ∼ κ / expansion of the theory but at first order in β one encountersa non-trivial consistency condition (3.6). This condition ensures the absence of a gaugeanomaly of the M-theory three-form A on a five-fold background. It requires a cancellationbetween the fourth Chern class, c ( X ), of the Calabi-Yau five-fold X , the square, G ∧ G ,of the flux G and the charge, W , of a membrane wrapping a holomorphic curve in X .We have studied explicit examples of Calabi-Yau five-folds to check whether and howthis condition can be satisfied. The simplest possibility is to use a five-fold satisfying c ( X ) = 0, without any membranes or flux. We have constructed an explicit example ofsuch a five-fold with vanishing fourth Chern class, based on a quotient of a 10–torus by afreely-acting Z symmetry. Although such a torus quotient has merely Z rather than SU(5)holonomy, it still breaks supersymmetry by a factor of 16 and, hence, all our subsequentresults apply to this example. As another class of examples, we have studied completeintersection Calabi-Yau five-folds (CICY five-folds) which are defined as the common zerolocus of homogeneous polynomials in a projective space or a product of projective spaces.The simplest example of such a CICY five-fold is the septic in P , the analogue of thefamous quintic Calabi-Yau three-fold in P . We have shown for a wide range of CICYfive-folds that c ( X ) = 0 and it is conceivable that this holds for all CICY five-folds. Itremains an open question as to whether Calabi-Yau five-folds with full SU(5) holonomyand c ( X ) = 0 exist, for example among toric five-folds.For CICY five-folds we have shown that the anomaly condition can frequently besolved by a cancellation between c ( X ) = 0 and appropriate flux and/or membranes. Inparticular, this can be achieved for the septic in P when both flux and membranes areincluded. Given the large number of topologically different Calabi-Yau five-folds and thefact that many of the simplest examples can already be made to work we can expect a largeand rich class of consistent M-theory five-fold compactifications. It is for such anomaly-free compactifications that we have set out to compute the associated one-dimensional N = 2 effective theory. To this end, we have developed the general properties of Calabi-Yau five-folds with regards to their topology, differential geometry and moduli spaces. Inparticular, there are six a priori independent Hodge numbers, h , ( X ), h , ( X ), h , ( X ), h , ( X ), h , ( X ) and h , ( X ). However, the Calabi-Yau condition c ( X ) = 0 together withthe index theorem leads to one linear relation between those six Hodge numbers, so thatonly five of them are effectively independent.M-theory zero modes on five-folds can be classified according to the sector of harmonic( p, q ) forms they are related to. For the bosonic zero modes, we have metric K¨ahler moduli,related to the (1 ,
1) sector and the metric complex structure moduli, related to the (1 , ,
1) sector arise from the three-form A . Allthese bosonic zero modes have associated fermionic partners which originate from the same( p, q ) sector of the five-fold. In addition, we also find (1 ,
3) fermionic zero modes that donot have any bosonic partners, a feature which seems at first puzzling from the viewpointof supersymmetry.After identifying these zero modes, we have reduced both the bosonic and fermion– 39 –ilinear terms in 11 dimensions to obtain the one-dimensional effective action, initially atzeroth order in the β expansion. In order to understand the supersymmetry of this effectiveaction, we have systematically developed one-dimensional global and local N = 2 super-space, extending previously known results. Based on these results, it turned out that the(1 ,
4) zero modes reside in 2 b multiplets while the (1 ,
1) multiplets reside in 2 a multiplets.The complex (2 ,
1) zero modes are best described collectively as real 3-form fields forming2 a multiplets and subject to a constraint halving the number of fermions. This was neces-sary in order to keep under control the otherwise intricate intertwining with the complexstructure moduli. It was found that the fermionic (1 ,
3) zero modes are compatible withsupersymmetry. However, the complex structure moduli also intertwine with those modes.For this sector, we restricted our analysis to five-folds whose (2 , , ,
3) and (3 ,
1) modes together, or ˆ4-formmodes for short, could then be described by constrained fermionic 2 b multiplets. For allthose multiplets, we have then written down a non-linear supersymmetric sigma modelin superspace and we have verified that the component version of this sigma model pre-cisely reproduces our reduction result from 11 dimensions. Interesting properties of thissigma model are the “non-standard” form of the (1 ,
1) sigma model metric which differsfrom the standard Calabi-Yau moduli space metric and the mixing between 2 a and 2 b multiplets. We also stress that local one-dimensional N = 2 supersymmetry is required inorder to properly describe the constraints which are the remnants of (super)gravity in onedimension.In a next step we have extended our results to order β effects and we have computedthe one-dimensional scalar potential which arises at this order. After imposing the anomalycondition, it turns out that this potential has two parts, depending on (1 ,
3) and (2 ,
2) flux,respectively. We have not been able to find a supersymmetric description of the (1 , ,
3) flux is not compatible with one-dimensional N = 2 supersymmetry. Since this is a complex structure dependent statement,keeping full N = 2 supersymmetry in the presence of non-zero four-form flux inducesrestrictions on the complex structure moduli. The explicit form of these restrictions andhow they can be implemented, for example in terms of a potential, is not known. In order tononetheless make concrete statements about flux, we therefore focussed on Calabi-Yau five-folds for which (1 ,
3) flux can be set to zero without imposing restrictions on the complexstructure moduli. In particular, we restricted to Calabi-Yau five-folds whose (2 , , , ,
1) 2 a multiplets and is given interms of a superpotential which only depends on (1 ,
1) moduli and is cubic in those fields.We find that this superpotential can also be directly obtained from a Gukov-type formula.A first look at the properties of the effective theory suggests different possibilities forself-consistent compactifications with small, or rather vanishing vacuum energy. For com-pactifications without flux (but possibly with membranes) the potential vanishes identically.In the presence of (2 ,
2) flux and depending on the case, there may be supersymmetric flat– 40 –irections with vanishing vacuum energy. The property of zero vacuum energy for super-symmetric solutions is facilitated by the form of the scalar potential which vanishes forvanishing F-terms. Models with flux but without flat directions have a rather large scalarpotential and it is not clear if the effective theory can be trusted. Na¨ıvely, such scalarpotentials can lead to a rapid expansion or contraction of the Calabi-Yau volume, depend-ing on the Calabi-Yau manifold and the choice of fluxes. We have constructed explicitexamples for all these cases.Our results open up a whole range of applications, particularly in the context of modulispace “cosmology”. For example, the question as to whether the system can evolve towardsa state with three large and seven small spatial dimensions can be studied as a dynamicalproblem in the five-fold moduli space. The effect of a scalar potential, from flux or other,non-perturbative sources not discussed in the present paper, is of course crucial in such adiscussion. Another interesting aspect of such a cosmological analysis might be the study ofvarious types of topological phase transitions for Calabi-Yau five-folds. These and relatedissues are centred around the question of how a one-dimensional theory can evolve toeffectively become four-dimensional and thereby become a viable description of the “late”universe. Such a question might even be studied in a “mini-superspace” quantised versionof our one-dimensional theory.There are also a number of more theoretical issues in relation to our results. It wouldbe interesting to find the “uplift” of certain solutions to our one-dimensional theory bystudying supersymmetric solutions to the 11-dimensional theory based on Calabi-Yau five-folds. In particular, our results for the flux scalar potential indicate that such solutionsshould not exist in the presence of (1 ,
3) flux. Another interesting aspect concerns the pos-sibility of mirror symmetry for Calabi-Yau five-folds. One might speculate that M-theoryon five-folds is mirror symmetric to F-theory on five-folds (times a circle). Both compact-ifications lead to one-dimensional N = 2 supersymmetric theories in one dimension and afirst test for mirror symmetry would be provided by a comparison of the one-dimensionaltheories derived in the present paper with the ones obtained from an F-theory reductionon five-folds. Several of these problems are currently under investigation. Acknowledgments
The authors are very grateful to A. Barrett for collaboration in the early stages of this work.We would also like to thank P. Candelas, X. de la Ossa and G. Papadopoulos for discussionsand we are grateful to the referee for careful reading and constructive comments. A.S.H. ac-knowledges the award of a postgraduate studentship by the Institute for Mathematical Sci-ences, Imperial College London, and thanks the Albert-Einstein-Institute for hospitalityand generous financial support. A.L. is supported by the EC 6th Framework ProgrammeMRTN-CT-2004-503369 and would like to thank the Albert-Einstein-Institute for hospi-tality. The research of K.S.S. was supported in part by the EU under contract MRTN-CT-2004-005104, by the STFC under rolling grant PP/D0744X/1 and by the Alexander vonHumboldt Foundation through the award of a Research Prize. K.S.S. would like to thankthe Albert-Einstein-Institute and CERN for hospitality during the course of the work.– 41 – ymbols range meaning
A, B, C, . . . , θ, ¯ θ one-dimensional N = 2 superspace indices M, N, P, . . . , . . . , D = 11 space-time indices m, n, p, . . . , . . . , D = 10 Euclidean indices µ, ν, . . . , . . . , D = 10 Euclidean holomorphic indices¯ µ, ¯ ν, . . . ¯1 , . . . , ¯5 D = 10 Euclidean anti-holomorphic indices Table 3:
Curved space-time indices and superspace indices. Tangent space indices are denoted bythe same letters but are underlined. symbols range meaning i, j, . . . , . . . , h , (1 , p, q, . . . , . . . , h , (2 , x, y, . . . , . . . , h , (1 , e, f, . . . , . . . , h , (2 , a, b, . . . , . . . , h , (1 , P , Q , . . . , . . . , b X , Y , . . . , . . . , b X , ˆ Y , . . . , . . . , h , ˆ4-form moduli (ˆ4 = (1 ,
3) + (3 , A , B , . . . , . . . , b Table 4:
Indices for Calabi-Yau five-fold cohomology.
AppendixA. Index conventions and spinors
In this section, we summarise notations and conventions used throughout the paper. Indicesfor space-time or superspace in the various relevant dimensions are listed in Table 3. Indicesin this table are curved indices and we refer to their corresponding tangent space indicesby underlining the same set of letters. Multiple indices are always symmetrized or anti-symmetrized with weight one. In addition, we need a range of index types for the variouscohomology groups of Calabi-Yau five-folds. They are listed in Table 4. For the indextypes p, q, . . . , x, y, . . . and a, b, . . . , the barred versions are also present and are used tolabel the complex conjugate of the respective moduli fields.We now turn to our spinor conventions and start in 11 dimensions. We denote the11-dimensional coordinates by x M and choose the 11-dimensional Minkowski metric η MN to have mostly plus signature, so η MN = diag( − , +1 , . . . , +1). The eleven dimensionalgamma matrices Γ M satisfy the Clifford algebra { Γ M , Γ N } = 2 η MN × . (A.1)Dirac spinors Ψ in 11 dimensions have 32 complex components and are anti-commutingobjects. We are working in the Majorana representation in which the charge conjugationmatrix is equal to ∗ = Ψ. The gamma– 42 –atrices in this representation are also real, (Γ M ) ∗ = Γ M , and all spatial gamma matri-ces are symmetric, (Γ m ) T = Γ m , whereas the timelike gamma matrix is anti-symmetric,(Γ ) T = − Γ . These properties combine into the following formulæ:(Γ M ) † = Γ Γ M Γ and (Γ M ) T = Γ Γ M Γ . (A.2)Curved gamma matrices Γ M are constructed by contracting with an inverse vielbein Γ M = e MN Γ N .In 10 Euclidean dimensions with coordinates x m we introduce complex coordinates by z µ = 1 √ (cid:0) x µ + i x µ +5 (cid:1) , ¯ z ¯ µ = 1 √ (cid:0) x ¯ µ − i x ¯ µ +5 (cid:1) . (A.3)Tensors transform from real to complex coordinates accordingly.The 10-dimensional gamma matrices γ m satisfying the Clifford algebra { γ m , γ n } = 2 δ mn × . (A.4)In accordance with our 11-dimensional conventions they are chosen to be real matrices andare, hence, also symmetric. The ten dimensional chirality operator γ (11) is given by γ (11) = iγ · · · γ , (A.5)and it satisfies the relations ( γ (11) ) = × , ( γ (11) ) ∗ = − γ (11) , ( γ (11) ) T = − γ (11) and { γ (11) , γ m } = 0. Ten-dimensional Dirac spinors η are 32-dimensional complex, as in 11dimensions, and are taken to be commuting. Positive (negative) chirality spinors η are thendefined by γ (11) η = η ( γ (11) η = − η ). Written in complex coordinates the anti-commutationrelations for the gamma matrices read { γ µ , γ ¯ ν } = 2 δ µ ¯ ν × , { γ µ , γ ν } = { γ ¯ µ , γ ¯ ν } = 0 . (A.6)As usual, the gamma matrices in complex coordinates can be interpreted as creation andannihilation operators. If one defines a “ground state” η by γ ¯ µ η = 0 (A.7)then η has positive and η ⋆ negative chirality. The other spinor states are obtained by actingwith up to five creation operators γ µ on η .Finally, in one dimension, there is only one gamma matrix, a 1 × − i . One-dimensional Dirac spinors ψ are complex one-component anti-commuting objects and we often denote their complex conjugate by ¯ ψ := ( ψ ) ∗ . Spinorialdifferentiation and Berezin integration are the same operations and satisfy the relations ∂ θ θ = 1 , ∂ θ ¯ θ = 0 , ∂ ¯ θ θ = 0 , ∂ ¯ θ ¯ θ = 1 ,∂ θ = 0 , ∂ θ = 0 , { ∂ θ , ∂ ¯ θ } = 0 , (A.8)where ∂ θ := ∂/∂θ and ∂ ¯ θ := ∂/∂ ¯ θ = − ( ∂ θ ) ∗ . Complex conjugation of a product of twoanti-commuting objects is defined to be ( ψ ψ ) ∗ = ¯ ψ ¯ ψ . Note the change of order on the– 43 –ight hand side. The rules for Berezin integration can be read off by replacing ∂ θ → R dθ and ∂ ¯ θ → R d ¯ θ . We also abbreviate d θ = dθd ¯ θ so that Z d θ θ ¯ θ = − . (A.9)The relation between 11-, 10- and one-dimensional gamma matrices is summarised bythe decomposition Γ = ( − i ) ⊗ γ (11) , Γ m = × ⊗ γ m , (A.10)where the tensor product between a complex number and a 32 ×
32 matrix has beenintroduced solely to make contact with similar formulæ for compactifications to more thanone dimension. As can be checked quickly, the matrices (A.10) indeed satisfy the 11-dimensional anti-commutation relations (A.1) and (A.2), provided the γ m satisfy the 10-dimensional anti-commutation relations (A.4). Dirac spinors Ψ in 11-dimensions can bewritten as (linear combinations of) tensor products of the form ψ ⊗ η , where ψ and η areone- and 10-dimensional spinors, respectively. An 11-dimensional Majorana spinor Ψ canbe decomposed as Ψ = ψ ⊗ η + ¯ ψ ⊗ η ⋆ . (A.11) B. Calabi-Yau five-folds
In this appendix, we develop the necessary tools to deal with Calabi-Yau five-folds andpresent some examples relevant to our discussion in the main text. Of course, much of theformalism will be analogous to Calabi-Yau three-folds and four-folds and we will borrowheavily from the literature, particularly from Refs. [16, 35, 36].
B.1 Basic topological properties
For the purpose of this paper, we define a Calabi-Yau five-fold to be a five complex-dimensional compact K¨ahler manifold X with vanishing first Chern class, c ( X ) = 0, andholonomy Hol( X ) ⊂ SU(5) sufficiently large to allow only one out of 16 supersymmetries.By the last condition we mean that in the decomposition Spin(10) → [ + ¯ + ] SU(5) (B.1)of the spinor representation under SU(5) only the SU(5) singlet is invariant underHol( X ). Hence, for positive chirality, we have precisely one covariantly constant spinor η . In particular, this means that 10-dimensional tori, direct products such as betweenthree-folds and four tori and similar spaces are excluded from our considerations. Thecorrespondence between covariantly constant spinors and harmonic (0 , p ) forms then impliesthat the Hodge numbers of X are constrained by h ,p ( X ) = h p, ( X ) = 0 for p = 1 , , , h , ( X ) = h , ( X ) = h , ( X ) = h , ( X ) = 1. Consequently, the Hodge diamond of a– 44 –alabi-Yau five-fold has the following general form10 00 h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h ,
00 01 (B.2)with six, a priori independent Hodge numbers. For the Betti numbers b i ( X ) this implies b ( X ) = 1 b ( X ) = 0 b ( X ) = h , ( X ) b ( X ) = 2 h , ( X ) b ( X ) = 2 h , ( X ) + h , ( X ) b ( X ) = 2 + 2 h , ( X ) + 2 h , ( X )and b i ( X ) = b − i ( X ) for i >
5. The Euler number η ( X ) of X can, therefore, be writtenas η ( X ) ≡ X i =0 ( − i b i ( X ) = 2 h , ( X ) − h , ( X ) + 4 h , ( X ) + 2 h , ( X ) − h , ( X ) − h , ( X ) . (B.4)For Calabi-Yau four-folds it is known [15] that one additional relation between the Hodgenumbers can be derived by an index theorem calculation using the Calabi-Yau condition c ( X ) = 0. As we will now see, a similar procedure can be carried out for Calabi-Yaufive-folds. First recall the general form of the index theorem χ ( X, V ) ≡ dim( X ) X i =0 ( − i dim H i ( X, V ) = Z X ch( V ) ∧ Td(
T X ) , (B.5)for a vector bundle V on X . We would now like to apply this theorem to the specificbundles V = ∧ q T ⋆ X , where q = 0 , , ,
3. The cohomology groups of these bundles canbe written as H i ( X, V ) = H i ( X, ∧ q T ⋆ X ) ≃ H i,q ( X ) and they are, hence, directly relatedto the Hodge numbers of X . For the subsequent calculation, it is convenient to use thesplitting principle and write the Chern class and character of the tangent bundle as c ( T X ) = 1 + c ( T X ) + c ( T X ) + · · · = Y i (1 + x i ) , ch( T X ) = X i e x i . (B.6)Then we havech( ∧ q T ⋆ X ) ∧ Td(
T X ) = X i >i > ··· >i q e − x i . . . e − x iq Y j x j − e − x j . (B.7)– 45 –xpanding this expression and re-writing it in terms of Chern classes using eq. (B.6) wefind from the index theorem (B.5) χ = h , − h , + h , − h , + h , + h , = 11440 Z X (cid:2) − c c + c c − c c + 3 c c (cid:3) χ = h , − h , + h , − h , + h , − h , = 1480 Z X (cid:2) − c c + c c − c c + 3 c c − c (cid:3) χ = h , − h , + h , − h , + h , − h , = 1720 Z X (cid:2) − c c + c c − c c + 3 c c + 330 c (cid:3) χ = h , − h , + h , − h , + h , − h , = − Z X (cid:2) − c c + c c − c c + 3 c c + 330 c (cid:3) where we have used the short-hand notation χ q = χ ( X, ∧ q T ⋆ X ) and c i = c i ( T X ). Insertingthe non-trivial information about Hodge numbers from the Hodge diamond (B.2) togetherwith c ( X ) = 0 the above equation for χ is trivially satisfied while the one for χ isequivalent to the one for χ . The remaining two relations for χ and χ turn into χ = − h , + h , − h , + h , = − Z X c ,χ = − h , + h , − h , + h , = 1124 Z X c . (B.8)Subtracting these two equations from one another and comparing with eq. (B.9) results inthe standard formula η ( X ) = Z X c ( X ) (B.9)for the Euler number η ( X ) of the five-fold X . Eliminating c , on the other hand, leads tothe relation11 h , ( X ) − h , ( X ) − h , ( X ) + h , ( X ) + 10 h , ( X ) − h , ( X ) = 0 (B.10)which only depends on Hodge numbers. Hence, five-folds are characterised by five ratherthan six independent Hodge numbers.Other relevant topological invariants of Calabi-Yau five-folds, apart from the Hodgenumbers and the Euler number, are the Chern classes c ( X ), c ( X ) and c ( X ), the inter-section numbers d i ...i of five eight-cycles and various other intersection numbers which wewill introduce later.As we have seen in the main part of the paper, compactification of M-theory requiresa Calabi-Yau five-fold X , a fourth cohomology class g ∈ H ( X ) and an effective secondcohomology class W ∈ H ( X, Z ) satisfying the integrability and quantisation conditions c ( X ) − g ∧ g = 24 W , g + 12 c ( X ) ∈ H ( X, Z ) . (B.11)– 46 –hysically, g corresponds to a four-form flux and W is the homology class of a holomorphiccurve in X which is wrapped by membranes. Clearly, there are a number of qualitativelydifferent ways one might try to solve these conditions. Probably the simplest possibility isto find a Calabi-Yau five-fold X with c ( X ) = 0. In this case, one can set the flux g andthe membrane class W to zero. For Calabi-Yau five-folds with c ( X ) = 0 one can ask if theconditions can be satisfied with either flux or membranes individually or by a combinationof both. We should now discuss if and how these possibilities can be realised and to do sowe need to turn to specific examples of Calabi-Yau five-folds. B.2 Examples of Calabi-Yau five-foldsB.2.1 Complete intersection Calabi-Yau five-folds
Perhaps the simplest class of Calabi-Yau manifolds is obtained from complete intersectionsin a projective space or a product of projective spaces (see, for example, Ref. [35] for areview). In the case of Calabi-Yau three-folds, the best known example of such completeintersection Calabi-Yau manifolds (CICY) is the quintic in P , defined as the zero locus of ahomogeneous degree five polynomial in P . For the case of Calabi-Yau five-folds, the directanalogue of the quintic in P is the septic in P , that is the zero locus of a homogeneousdegree seven polynomial in P .In order to define CICY five-folds more generally, we first introduce an ambient space A = N mr =1 P n r , as a product of m projective spaces with dimension n r . Each of theseprojective spaces comes equipped with a K¨ahler form J r which we normalise such that Z P nr J n r r = 1 . (B.12)We are interested in the common zero locus of polynomials p α , where α = 1 , . . . , K , whichare homogenous of degree q rα in the coordinates of the factor P n r in A . In order for thiszero locus to be five-dimensional we need, of course, K = m X r =1 n r − . (B.13)It is useful to summarise the dimensions of the various projective spaces together with the(multi-) degrees of the polynomials in a configuration matrix [ n | q ] = n ... n m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q . . . q K ... ... q m . . . q mK (B.14)We note that every column in the q part of this matrix corresponds to the multi-degreeof one of the defining polynomials. As an example, using this short-hand notation, theseptic in P can be written as [6 | c ([ n | q ]) = Q mr =1 (1 + J r ) n r +1 Q Kα =1 (1 + P ms =1 q sα J s ) . (B.15)– 47 –nd the various individual Chern classes c q ([ n | q ]) can be obtained by expanding the aboveexpression and extracting terms of order q in the K¨ahler forms J r . For the first Chern classthis leads to c ([ n | q ]) = m X r =1 n r + 1 − K X α =1 q rα ! J r . (B.16)Hence, the Calabi-Yau condition c ( X ) = 0 translates into the simple numerical constraints K X α =1 q rα = n r + 1 , ∀ r = 1 , . . . , m (B.17)on the multi-degrees of the defining polynomials. This means the rows in the q part of theconfiguration matrix always have to sum up to the dimension of the associated projectivespace plus 1 in order for the complete intersection to be a Calabi-Yau space. In ourapplication to M-theory compactifications, higher Chern classes and c ( X ) in particular,play a crucial rˆole. By expanding eq. (B.15) we find for CICYs c ([ n | q ]) = c rs J r J s = 12 m X r,s =1 " − ( n r + 1) δ rs + K X α =1 q rα q sα J r J s , (B.18) c ([ n | q ]) = c rst J r J s J t = 13 m X r,s,t =1 " ( n r + 1) δ rst − K X α =1 q rα q sα q tα J r J s J t , (B.19) c ([ n | q ]) = c rstu J r J s J t J u = 14 " − ( n r + 1) δ rstu + K X α =1 q rα q sα q tα q uα + 2 c rs c tu J r J s J t J u , (B.20) c ([ n | q ]) = c r ...r J r · · · J r = 15 " ( n r + 1) δ r ...r − K X α =1 q r α · · · q r α + 5 c ( r r r c r r )2 J r · · · J r , (B.21)where c ([ n | q ]) = 0 has been used to simplify the expressions. The fourth Chern classshould be written in terms of a set of harmonic eight-forms { ˜ J r } as c ( X ) = ˜ c r ˜ J r . If wechoose these forms to be dual to the K¨ahler forms J r , that is, Z X J r ∧ ˜ J s = δ sr , (B.22)it is easy to see that c r can be obtained from the coefficients which appear in the for-mula (B.20) by ˜ c r = d rstuv c stuv , d i ...i = Z X J i ∧ · · · ∧ J i . (B.23)The intersection numbers d i ...i can be explicitly computed from the identity Z X w = Z A w ∧ µ , µ = K ^ α =1 m X r =1 q rα J r ! , (B.24)– 48 –hich converts integration of a 10-form w over X into an integration over the ambientspace. In carrying out the latter the normalisation (B.12) must be taken into account. Thecalculation of Hodge numbers is straightforward for CICYs with q rα > r and α .In this case, repeated application of the Lefshetz hyperplane theorem (see, for example,Ref. [35]) shows that H p,q ( X ) ≃ H p,q ( A ) for p + q = 5 . (B.25)Hence, all cohomology groups except the middle ones are isomorphic to the ambient spacecohomology groups for such CICYs. The only non-vanishing Hodge numbers of P n are h p,p ( P n ) = 1 and, by applying the K¨unneth formula H n ( Y ⊗ Z ) = L i + j = n H i ( Y ) ⊗ H j ( Z ), one can easily compute the Hodge numbers of the ambient space A from thisresult. Combining these facts, one finds for CICYs with all q ra > h , ( X ) = h , ( A ) = m (B.26) h , ( X ) = 0 (B.27) h , ( X ) = 0 (B.28) h , ( X ) = h , ( A ) = m ( m − { r | n r ≥ } . (B.29)The first of these equations means that the restrictions of the ambient space K¨ahler forms J r to X form a basis of the second cohomology. The last equation implies that the four-forms J r ∧ J s span H , ( X ). Let us define the six-cycles C rs = [ n | q e r e s ] ⊂ X , where e r isa vector with one in the r th entry and zero elsewhere. The measure µ rs for these six-cyclesis given by µ rs = µ ∧ J r ∧ J s where µ is the measure of the CICY as in eq. (B.24). It followsthat R C rs w = R A w ∧ µ rs = R X w ∧ J r ∧ J s for all six-forms w . Hence, the forms J r ∧ J s are Poincar´e dual to the six-cycles C rs and are, therefore, integral. Two remaining Hodgenumbers need to be determined, namely h , ( X ) and h , ( X ). This can be accomplishedby calculating the Euler number from eqs. (B.9), (B.21) and then using the two constraints(B.4) and (B.10).For CICYs where some q rα vanish a more refined version of the above reasoning cansometimes be applied [35]. In more complicated cases, the Hodge numbers must be calcu-lated using spectral sequence methods [37]. For CICY three-folds it is known that h , ( X )can be larger than m in such cases so that not all (1 ,
1) classes descend from the ambientspace. A similar phenomenon can be expected for CICY five-folds. In general, one canalso expect h , ( X ) and h , ( X ) to acquire non-zero values. The detailed analysis of theseissues is somewhat outside our main line of investigation and will not be pursued here.Another useful feature of CICYs whose second cohomology is spanned by the ambientspace K¨ahler forms J r is that the Mori cone, the cone of effective cohomology classes in H ( X, Z ) ≃ H ( X, Z ), is given by positive integer linear combinations n r ˜ J r of the eight-forms ˜ J r dual to J r .It is useful to have some explicit examples of CICYs available. The simplest sub-class consists of CICY five-folds which can be defined in a single projective space. In thiscase, a linear polynomial constraint simply amounts to a reduction of the ambient space– 49 – n | q . . . q K ] c ( X ) /J c ( X ) /J η ( X ) h , ( X ) h , ( X )[6 |
7] 21 819 − | − | − | − | − | − | − | − | − | − | − Table 5:
The 11 CICY five-folds which can be defined in a single projective space. The Hodgenumbers h , ( X ) = h , ( X ) = 0 and h , ( X ) = h , ( X ) = 1 for all manifolds. [ n | q ] c ( X ) c ( X ) η ( X ) h , ( X ) h , ( X ) h , ( X ) h , ( X ) " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J J +15 J J +4542 ˜ J − " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J +16 J J +6 J J +3600 ˜ J − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J +12 J J +6 J +6 J J +8 J J
84 ˜ J +114 ˜ J +130 ˜ J − Table 6:
Examples of CICY five-folds defined in a product of projective spaces. The Hodgenumbers h , ( X ) = h , ( X ) = 0 for all manifolds. dimension by one. In other words, a configuration matrix of the form [ n | q . . . q K − P n is equivalent to a configuration matrix [ n − | q . . . q K − ] in P n − . Hence, we canrequire that all q α > c ( X ) = 0, without flux and membranes. Do CICYfive-folds with c ( X ) = 0 exist? From eq. (B.23) we see that the intersection numbersare positive, that is, d i ...i ≥
0. Further, it is clear from eq. (B.20) that all components c rstu ≥
0. This means the fourth Chern class of CICY five-folds is positive in the sensethat ˜ c r ≥ r . If the configuration matrix is such that q rα ≥ r and α , thecoefficients c rstu are strictly positive. From the first eq. (B.23) this shows that c ( X ) = 0for all such CICY five-folds. In particular, it follows that all CICY five-folds defined in asingle projective space ( m = 1) and all co-dimension one five-folds ( K = 1) have c ( X ) = 0.(The former fact is, of course, confirmed by Table 5.) So, we are left with CICY five-foldssatisfying m > K > q rα < m ≤ K ≤ c ( X ) = 0 we have foundare spaces such as X ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.30)which correspond to the direct product of a Calabi-Yau three-fold Y (the quintic in theabove example) with two tori T . Clearly, c ( Y × T × T ) = 0 but such a space onlyhas holonomy SU(3). It breaks a quarter of the supersymmetry and is, therefore, not aCalabi-Yau five-fold in the sense defined at the beginning of this appendix. In summary,for m ≤ K ≤ c ( X ) = 0. We cannot exclude that larger configurations with this propertyexist although we have not been able to find any explicit examples.Given the lack of CICY five-folds with c ( X ) = 0, we can ask if the conditions (B.11)can be satisfied by including flux and membranes. To analyse this question let us startwith the five-folds in a single projective space which are listed in Table 5. We write thefourth Chern class as c ( X ) = C J , where the numbers C can be read off from Table 5and the flux as g = kJ for some number k . In the absence of membranes ( W = 0) theanomaly condition (B.11) is then solved for flux values k = ± r C . (B.31)For the 11 cases in Table 5, it can be checked that the resulting values of k are neverrational. This means, it is impossible to satisfy the flux quantisation condition (B.11) forsuch values of k . We conclude that, in the absence of membranes the 11 CICY five-foldsin a single projective space cannot be used for consistent M-theory compactifications.Does the inclusion of membranes help? We begin with the septic, [6 | c ( X ) − g = (819 − k ) J , g − c ( X ) = (cid:18) k + 212 (cid:19) J . (B.32)Setting the flux to k = 15 /
2, the anomaly condition can then be satisfied for a membranewrapping a holomorphic curve with class W = 6 J . Recalling that J is an integral class,the flux quantisation condition is also satisfied for this value of k . Hence, by including fluxand membranes the M-theory conditions can be satisfied for the septic.While the M-theory conditions for CICY five-folds in a single projective space cannotbe satisfied with flux only, can they be satisfied for membranes only? Let us look at theexample [7 | µ = 12 J . This means Z X J ∧ J = 12 . (B.33)Comparing this with the definition R X J ∧ ˜ J = 1 of the dual eight-form ˜ J we learn that˜ J = J /
12 and that this is an integral class. Given that c ( X ) = 454 J , the anomalycondition can then be satisfied by setting the flux to zero and by including a membranewhich wraps a holomorphic curve with class W = 227 ˜ J .In order to find viable examples with flux only we need to consider CICYs defined inproducts of projective spaces. Let us start with the first example in Table 6, a co-dimensionone CICY five-folds with configuration matrix X ∼ " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.34)defined in the ambient space A = P ⊗ P . Writing the flux as g = k , J J + k , J onefinds for the right-hand-sides of the anomaly and quantisation condition (B.11) c ( X ) − g = (2610 − k , ) ˜ J + (4542 − k , k , − k , ) ˜ J , (B.35) g + 12 c ( X ) = ( k , + 6) J J + (cid:18) k , + 152 (cid:19) J . (B.36)For the anomaly to vanish without membranes we need a non-rational flux parameter k , = ±√ / k , and any half-integer k , the coefficients on the right-hand-sideof eq. (B.35) are divisible by 24 and, for sufficiently small flux integers, positive. Hence,the anomaly condition can be satisfied by inclusion of a membrane.Next, we consider the second example in Table 6, the co-dimension one CICY five-foldsin A = P ⊗ P with configuration matrix X ∼ " (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B.37)– 52 –ith the flux parameterised as g = k , J + k , J J + k , J one finds for the right-hand-sides of the anomaly and quantisation condition (B.11) c ( X ) − g = (3600 − k , − k , k , − k , k , ) ˜ J + (3600 − k , k , − k , − k , k , ) ˜ J , (B.38) g + 12 c ( X ) = ( k , + 3) J + ( k , + 8) J J + ( k , + 3) J . (B.39)Again, without membranes, it can be checked that the anomaly condition cannot be satis-fied for integers k , , k , and k , . However, as the right-hand-side of eq. (B.38) is divisibleby 24, a complete model can always be obtained be inclusion of membranes as long as theflux integers are not too large.For the above examples, we have h , ( X ) = 2 or 3 flux parameters and h , ( X ) =2 equations from the anomaly condition, so it is perhaps not surprising that a rationalsolution without membranes cannot be found. In fact, a similar obstruction can be foundfor other simple CICYs defined in a product of two projective spaces. This suggests lookingat more complicated examples in products of more than two projective spaces, so that h , ( X ) > h , ( X ). To this end, we consider the CICY in the third row of Table 6, definedin a product of three projective spaces and with configuration matrix X ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B.40)Flux can be parameterized as g = k , J J + k , J J + k , J + k , J J + k , J and wefind for the right-hand-sides of the anomaly and quantisation conditions c ( X ) − g = (130 − k , k , − k , k , − k , k , − k , − k , k , − k , k , ) ˜ J + (114 − k , k , − k , k , − k , k , − k , k , ) ˜ J + (84 − k , − k − , k , − k , k , ) ˜ J , (B.41) g − c ( X ) = (3 + k , ) J J + (4 + k , ) J J + (cid:18)
32 + k , (cid:19) J + (6 + k , ) J J + (3 + k , ) J . (B.42)A quick scan reveals that both conditions can be satisfied for the choice ( k , , k , , k , , k , ,k , ) = (1 , , / , , h , ( X ) > h , ( X ),solutions with flux only can be found as well. Unfortunately, we have not managed to findCICY five-folds with holonomy SU(5) and c ( X ) = 0 and such CICY five-folds may wellnot exist. However, an example with c ( X ) = 0 which allows for a “clean” compactificationwithout flux or membranes is still highly desirable and we, therefore, turn to another classof Calabi-Yau five-folds. – 53 – .2.2 Torus quotients The Chern classes of a torus vanish and it is, therefore, a promising starting point for theconstruction of Calabi-Yau five-folds with c ( X ) = 0. Specifically, we start with a product T = T × · · · × T of five two-tori, each with a complex coordinate z µ , where µ = 1 , . . . , z µ ∼ z µ + 1 and z µ ∼ z µ + i . Then we consider the symmetry Z defined bythe four generators γ ( z , . . . , z ) = ( − z + 1 / , − z + i/ , z + 1 / , z , z ) (B.43) γ ( z , . . . , z ) = ( z , − z + 1 / , − z + i/ , z + 1 / , z ) (B.44) γ ( z , . . . , z ) = ( z , z , − z + 1 / , − z + i/ , z + 1 /
2) (B.45) γ ( z . . . , z ) = ( z + 1 / , z , z , − z + 1 / , − z + i/ . (B.46)It is straightforward to check that the 16 elements of this group all act freely on T . Hence,the quotient X = T / Z is a manifold. Clearly, it inherits the property of vanishing Chernclasses from the torus and, in particular, c ( X ) = 0. The holonomy of X is of course just Z but the four Z symmetries are still sufficient to reduce the number of supersymmetriesby a factor of 1 /
16. Therefore, X is a Calabi-Yau five-fold in this sense defined at thebeginning of this appendix.What are the properties of X ? Clearly, c i ( X ) = 0 for i = 1 , . . . , η ( X ), also vanishes. The Hodge numbers are obtained by counting thenumber of Z invariant ( p, q ) forms dz µ ∧ · · · ∧ dz µ p ∧ d ¯ z ν ∧ · · · ∧ d ¯ z ν q . This results in h , ( X ) = 5 , h , ( X ) = 0 , h , ( X ) = 0 ,h , ( X ) = 10 , h , ( X ) = 5 , h , ( X ) = 10 . (B.47)Presumably five-folds from tori divided by other discrete symmetries can be constructedalong similar lines. We will not pursue this explicitly, having shown the existence of Calabi-Yau five-folds with c ( X ) = 0 by the simple example above. It remains an open questionwhether Calabi-Yau five-folds with full SU(5) holonomy and c ( X ) = 0 exist. We are notaware of a general mathematical reason which rules this out and it would be interesting tolook for such manifolds, for example among toric five-folds. B.3 Some differential geometry on five-folds
As discussed earlier, on a Calabi-Yau five-fold X we have a spinor η , unique up to normal-isation, which is invariant under the holonomy group Hol( X ). This means, η is covariantlyconstant with respect to the Levi-Civita connection associated to the Ricci-flat metric g .Its direction can be defined by imposing the five annihilation conditions γ ¯ µ η = 0 . (B.48)Given the definition (A.5) of the 10-dimensional chirality operator, η has positive and η ⋆ negative chirality, that is γ (11) η = η, γ (11) η ⋆ = − η ⋆ . (B.49) For our conventions on 10-dimensional gamma matrices and spinors, see Appendix A. – 54 –s usual, we normalize η such that η † η = 1 . (B.50)It can be shown that η satisfies the Fierz identity (see, for example, Ref. [19], Proposition5, or Ref. [38], eq. (2.3)) η ⋆ η T = − g µ ¯ ν γ µ ¯ ν , (B.51)which will be useful in our reduction of the fermionic terms. Apart from the normalisa-tion (B.50), there exist two other non-zero spinor bilinears, namely the K¨ahler form J andthe holomorphic (5 ,
0) form Ω defined by J µ ¯ ν = iη † γ µ ¯ ν η , Ω µ ...µ = || Ω || η † γ µ ...µ η ⋆ , (B.52)where || Ω || = Ω µ ...µ ¯Ω µ ...µ / J and Ω are covariantly constant as a directconsequence of η being covariantly constant. The complex structure J is defined by theequation J mn = J mp g pn and the metric g is hermitian with respect to J . The projectionoperators P ± = ( ∓ i J ) / X into ( p, q ) “index types”with p holomorphic and q anti-holomorphic indices. As usual, we will work in local com-plex coordinates such that J µν = iδ νµ and J ¯ µ ¯ ν = − iδ ¯ ν ¯ µ . In this basis, the (2 ,
0) and (0 , J µ ¯ ν = ig µ ¯ ν . (B.53)For a ( p, q ) form ω ( p,q ) with p > q > p − , q − ω with the inversemetric g µ ¯ ν . In the following, it will be convenient to introduce the short-hand notation˜ w ( p,q ) for this ( p − , q −
1) form. Note that ˜ ω ( p,q ) is harmonic if ω ( p,q ) is, since the metricis covariantly constant. This short-hand notation for the contraction of forms is useful towrite down explicit formulæfor the Hodge duals of ( p, q ) forms which are required in manyphysics applications. Straightforward but in part somewhat tedious component calculationsshow that(0 ,
1) : ∗ ζ = i J ∧ ζ, (1 ,
1) : ∗ ω = − J ∧ ω − i
4! ˜ ωJ , (2 ,
1) : ∗ ν = i J ∧ ν + 13! J ∧ ˜ ν, (3 ,
1) : ∗ ̟ = − J ∧ ̟ − i J ∧ ˜ ̟, (4 ,
1) : ∗ χ = iχ + J ∧ ˜ χ, (2 ,
2) : ∗ σ = J ∧ σ − i J ∧ ˜ σ + 112 ˜˜ σJ , (3 ,
2) : ∗ φ = − iφ − J ∧ ˜ φ − i J ∧ ˜˜ φ. (B.54)Some simplifications of these equations arise for harmonic ( p, q ) forms. We recall thatCalabi-Yau five-folds have vanishing Hodge numbers h p, ( X ) = h ,p ( X ) for p = 1 , , , p,
0) and (0 , p ) forms do not exist and consequently˜ ω ( p, = ˜ ω (1 ,p ) = 0 for harmonic ( p,
1) and (1 , p ) forms with p >
1. (B.55)– 55 –oreover, a harmonic (0 ,
0) form is a constant and, hence,˜ ω (1 , = const . for harmonic (1 ,
1) forms. (B.56)Combining these facts with the formulæ (B.54), one finds for the Hodge dual of harmonic( p, q ) forms on Calabi-Yau five-folds(1 ,
1) : ∗ ω = − J ∧ ω − i
4! ˜ ωJ , (2 ,
1) : ∗ ν = i J ∧ ν, (3 ,
1) : ∗ ̟ = − J ∧ ̟, (2 ,
2) : ∗ σ = J ∧ σ − i J ∧ ˜ σ + 112 ˜˜ σJ , (4 ,
1) : ∗ χ = iχ, (3 ,
2) : ∗ φ = − iφ − J ∧ ˜ φ , (B.57)where we should keep in mind that ˜ ω and ˜˜ σ are constants and ˜ σ is a harmonic (1 ,
1) form.The volume V of the five-fold can be written as V ≡ Z X d x √ g = 15! Z X J . (B.58)Then, acting with J ∧ on the (1 ,
1) part of eq. (B.57), using that J ∧ ∗ w = − id x √ g ˜ w and integrating over X we learn that˜ w = 5 i R X J ∧ w R X J . (B.59)A further useful relation for a Hodge dual is ∗ ( σ ∧ J ) = ˜˜ σJ − i ˜ σ . (B.60)where σ is a (2 ,
2) form. In the next sub-section, we will use this relation to explicitlycompute ˜ σ and ˜˜ σ . B.4 Five-fold moduli spaces
For Calabi-Yau three-folds the moduli space of Ricci-flat metrics is (locally) a direct productof a K¨ahler and complex structure moduli space which are associated to harmonic (1 , ,
1) forms, respectively. For Calabi-Yau five-folds the situation is analogous and wewill naturally borrow from the literature for three-folds (in particular, see Ref. [16], for anexplicit description). Just as for three-folds, the K¨ahler deformations of a five-fold metricare associated to harmonic (1 ,
1) forms while the complex structure deformations can bedescribed in terms of (1 ,
4) forms. All other harmonic forms on five-folds are unrelated tometric deformations but some of them still do play a rˆole in M-theory compactifications.In particular, the (2 ,
1) forms determine the zero modes of the M-theory three-form field.No bosonic degrees of freedom can be associated with the (1 ,
3) forms but, as we discussin the main part of the paper, they give rise to a set of fermionic zero modes. In summary,all harmonic ( p,
1) (or, equivalently, (1 , p )) forms, where p = 1 , , ,
4, are relevant for thezero-modes expansion of the M-theory fields. In addition, harmonic (2 ,
2) forms play a rˆole– 56 –hen flux is included in the compactification. It is useful to introduce sets of harmonicbasis forms for these cohomologies as follows H (1 , ( X ) : { ω i } i =1 ,...,h , ( X ) , (B.61) H (2 , ( X ) : { ν p } p =1 ,...,h , ( X ) , (B.62) H (1 , ( X ) : { ̟ x } x =1 ,...,h , ( X ) , (B.63) H (2 , ( X ) : { σ e } e =1 ,...,h , ( X ) , (B.64) H (1 , ( X ) : { χ a } a =1 ,...,h , ( X ) , (B.65)with ω i and σ e real and all other forms complex. These forms can be used to constructvarious intersection numbers d i ...i = R X ω i ∧ · · · ∧ ω i , d p ¯ qij = R X ν p ∧ ¯ ν ¯ q ∧ ω i ∧ ω j ,d eijk = R X σ e ∧ ω i ∧ ω j ∧ ω k , d p ¯ qe = R X ν p ∧ ¯ ν ¯ q ∧ σ e ,d efi = R X σ e ∧ σ f ∧ ω i , d x ¯ yi = R X ̟ x ∧ ¯ ̟ ¯ y ∧ ω i . (B.66)which will play a rˆole later on.We begin with the metric moduli. As usual, the basic requirement is that a variation g mn → g mn + δg mn of the metric leaves the Ricci tensor zero at linear order in δg . Workingthis out in detail, reveals that the (1 ,
1) part of δg can be expanded in terms of harmonic(1 ,
1) forms while the (2 ,
0) and (0 ,
2) parts can be expressed in terms of harmonic (1 , δg µ ¯ ν = − iw i,µ ¯ ν δt i , δg µν = − || Ω || Ω µ ¯ ρ ... ¯ ρ χ a,ν ¯ ρ ... ¯ ρ δz a , (B.67)with the variations δt i and δz a in the K¨ahler and complex structure moduli. The standardmoduli space metric on the space of metric deformations is defined by G ( δg, e δg ) = 14 V Z X d x √ g g mn g pq δg mp e δg nq . (B.68)This metric splits into a K¨ahler and a complex structure part which can be worked outseparately. Let us first discuss the K¨ahler deformations. A straightforward calculation,inserting the first eq. (B.67) shows that G (1 , ij ( t ) = 12 V Z X ω i ∧ ∗ ω j . (B.69)Using the expression in eq. (B.57) for the dual of (1 ,
1) forms together with eq. (B.59), thiscan be written in terms of topological integrals which involve J and the forms ω i . Then,defining the K¨ahler moduli by J = t i ω i , (B.70) The term “intersection number” is a slight misnomer in this context, as, in fact, all of these integrals,except d i ...i , in general depend on the complex structure (due to the use of complex ( p, q )-forms) and thusdo not represent topological invariants. – 57 –ne finds G (1 , ij ( t ) = − κ ij κ + 252 κ i κ j κ , (B.71)where we have introduced the notation κ = Z X J = 5! V = d i ...i t i . . . t i , (B.72) κ i = Z X ω i ∧ J = d ii ...i t i . . . t i , (B.73) κ ij = Z X ω i ∧ ω j ∧ J = d iji i i t i t i t i , (B.74)... (B.75)and so on for versions of κ with more than two indices. With this notation, eq. (B.59) canbe re-written as ˜ ω i = 5 i κ i κ . (B.76)It is easy to check that the above moduli space metric (B.71) can be obtained from a“K¨ahler potential” K (1 , as G (1 , ij = ∂ i ∂ j K (1 , , where K (1 , = −
12 ln κ . (B.77)We can use the moduli space metric to define lower index moduli t i via t i = G (1 , ij t j . Fromthe explicit form (B.71) of the metric, it is easy to verify the useful relation t i = 5 κ i κ . (B.78)A further useful observation is related to “metrics” of the form˜ G ij = G (1 , ij + c κ i κ j κ (B.79)for any real number c . A short calculation, using eq. (B.78) and κ i t i = κ repeatedly, showsthat ˜ G ij (cid:18) G (1 , jk + ˜ c κ j κ k κ (cid:19) = δ ki + (cid:18) c + ˜ c + 25 c ˜ c (cid:19) κ i κ k κ , (B.80)where ˜ c is an arbitrary real number. Here, the standard moduli space metric G (1 , ij and itsinverse G (1 , ij have been used to lower and raise indices. The above relation shows thatfor all c = − / G ij = G (1 , jk + ˜ c κ j κ k κ , ˜ c = − c c . (B.81)These relations will be helpful when calculating the flux potential in the one-dimensionaleffective theory.To summarise the main points, the K¨ahler moduli space for five-folds can be treated incomplete analogy with the one for three-folds. The main difference is that the moduli spacemetric is now governed by a quintic pre-potential κ instead of a cubic one for three-folds.– 58 –e now move on to the complex structure moduli. Evaluating the standard modulispace metric (B.68) for the (2 ,
0) variation of the metric in eq. (B.67), one finds G (1 , a ¯ b = 1 V || Ω || Z X χ a ∧ ∗ ¯ χ ¯ b . (B.82)Using the result in eq. (B.57) for the Hodge dual of (4 ,
1) forms together with the relation V || Ω || = i R X Ω ∧ ¯Ω then leads to the standard result G (1 , a ¯ b ( z, ¯ z ) = R X χ a ∧ ¯ χ ¯ b R X Ω ∧ ¯Ω . (B.83)Kodaira’s relation ∂ Ω ∂z a = k a Ω + χ a (B.84)can be shown exactly as in the case of Calabi-Yau three-folds [16]. It implies, via direct dif-ferentiation, that the moduli space metric (B.83) can be obtained from the K¨ahler potential K (1 , as G (1 , a ¯ b = ∂ a ∂ ¯ b K (1 , , where K (1 , = ln (cid:20) i Z X Ω ∧ ¯Ω (cid:21) . (B.85)In order to express K (1 , more explicitly in terms of moduli, we introduce a symplecticbasis ( A A , B B ) of five-cycles and a dual basis ( α A , β B ) of five-forms satisfying Z A B α A = Z X α A ∧ β B = δ BA , Z B A β B = Z X β B ∧ α A = − δ BA . (B.86)Then, the period integrals are defined in the usual way as Z A = Z A A Ω , G A = Z B A Ω . (B.87)and the periods G A can be shown to be functions of Z A , just as in the three-fold case. Inthe dual basis ( α A , β B ) the (5 ,
0) form can then be expanded as Ω = Z A α A − G A β A andinserting this into the expression (B.85) for the K¨ahler potential yields K (1 , = ln (cid:2) i ( G A ¯ Z A − Z A ¯ G A ) (cid:3) . (B.88)By virtue of Kodaira’s relation, R X Ω ∧ ∂ Ω ∂ Z A = 0 which immediately leads to G A = ∂∂ Z A ( G B Z B ). Hence, the periods G A can be obtained as derivatives G A = ∂ G ∂ Z A (B.89)of a pre-potential G which is homogeneous of degree two in the projective coordinates Z A .This is formally very similar to the three-fold case. However, an important difference isthat the five-forms here contain not only (5 , , ,
1) and (1 ,
4) pieces but also (3 , ,
3) parts. That is, A , B , . . . = 0 , , . . . , h , + h , . As a consequence, the periods Z A do not simply serve as projective coordinates on the complex structure moduli space,– 59 –hough they can in principle be computed as functions of the z a . However, their vastredundancy renders them much less useful as compared to the three-fold case.When flux is included, the one-dimensional effective theory depends on yet anotherset of moduli-dependent functions which arises from the contractions, ˜ σ e and ˜˜ σ e , of theharmonic (2 ,
2) forms σ e which appear in the relation (B.57) for the Hodge dual of (2 , σ e must be a harmonic (1 , ω i . Concretely, we write˜ σ e = ik ie ω i (B.90)with some coefficients k ie which, in general, depend on the (1 ,
1) moduli t i . Applying onemore contraction to this relation and using eq. (B.76) we learn that˜˜ σ e = − κ k ie κ i . (B.91)Hence, we can deal with all the contractions of harmonic (2 ,
2) forms if we are able tocompute the coefficients k ie . This can be accomplished by multiplying eq. (B.60) with ω j and integrating over the Calabi-Yau five-fold X . This results in k ie = 14 V (cid:18) G (1 , ij − κ i κ j κ (cid:19) d ejkl t k t l , (B.92)where G (1 , ij is the inverse of G (1 , ij . B.4.1 Real vs. complex forms
For the purpose of disentangling and clarifying the intertwining of (2 , , , , h , ( X ) = 0 for Calabi-Yau five-folds ensures that a real 3-form is exclusivelymade up of a (2 , , , , A and B from real 3-forms to complex (2 , ν p = A p Q N Q (and: ¯ ν ¯ p = ¯ A ¯ p Q N Q ) , (B.93) All differential forms occurring in this subsection are henceforth implicitly assumed to be harmonic. – 60 – P = B P q ν q + ¯ B P ¯ q ¯ ν ¯ q , (B.94)where { N P } P =1 ,...,b ( X ) is a real basis of H ( X ) and { ν p } p =1 ,...,h , ( X ) is a basis of H (2 , ( X ).To avoid confusion with symbols defined elsewhere, we use Fraktur font letters to denotemaps translating between real and complex forms and calligraphic letters for real formindices. Note that A p Q and B P q are complex and have dependence A p Q = A p Q ( z, ¯ z ), B P q = B P q ( z, ¯ z ), where z a and ¯ z ¯ a are the complex structure moduli of the Calabi-Yaufive-fold. The equations above have two faces, for they can either be written in localreal ten dimensional coordinates or in local (complex) Darboux coordinates. For example,eq. (B.94) in real coordinates is N P ,m m m = B P q ν q,m m m + ¯ B P ¯ q ¯ ν ¯ q,m m m , (B.95)whereas in Darboux coordinates it reads N P ,µ µ ¯ ν = B P q ν q,µ µ ¯ ν , (and c.c.) , (B.96)where forms with unnatural index types are to be translated manually using eq. (A.3).Inserting eq. (B.93) into eq. (B.94) and vice versa, we learn relations between the A and B maps: A p Q B Q q = δ pq (and c.c.) , (B.97) A p Q ¯ B Q ¯ q = 0 (and c.c.) , (B.98) B P q A q Q + ¯ B P ¯ q ¯ A ¯ q Q = δ P Q . (B.99)For the complex structure dependence, one finds: ∂ a N P = 0 , ∂ a ν p = A p Q ,a B Q q ν q + A p Q ,a ¯ B Q ¯ q ¯ ν ¯ q , (B.100) ∂ ¯ a N P = 0 , ∂ ¯ a ν p = A p Q , ¯ a B Q q ν q + A p Q , ¯ a ¯ B Q ¯ q ¯ ν ¯ q . (B.101)Using eqs. (B.93)-(B.94) and eq. (B.57), one can compute the Hodge star of the real 3-form N P : ∗ N P = 12 ∆ P Q N Q ∧ J , (B.102)where ∆ P Q := i ( B P q A q Q − ¯ B P ¯ q ¯ A ¯ q Q ). The linear map ∆ provides a complex structureon the moduli space of real 3-forms induced by the complex structure of the Calabi-Yaufive-fold itself. It satisfies∆ P Q ∆ QR = − δ P R , (∆ P Q ) ∗ = ∆ P Q , tr ∆ = 0 . (B.103)Using the complex structure ∆, we define projection operators P ±P Q := 12 ( ∓ i ∆) P Q (B.104)satisfying P ±P Q P ±QR = P ±P R , P + P Q P −QR = P −P Q P + QR = 0 , ( P ±P Q ) ∗ = P ∓P Q . (B.105)– 61 –n terms of the A and B maps, they are explicitly given by P + P Q = B P q A q Q , P −P Q = ¯ B P ¯ q ¯ A ¯ q Q . (B.106)The standard metric on the moduli space of real 3-forms is G (3) PQ = Z X N P ∧ ∗ N Q . (B.107)Using the expression for the Hodge star (B.102), we can rewrite this so as to make thedependence on the moduli more explicit: G (3) PQ ( t, z, ¯ z ) = 12 ∆ ( P R d Q ) R ij t i t j , (B.108)where we have defined a new intersection number d PQ ij := R X N P ∧ N Q ∧ ω i ∧ ω j , whichis purely topological. Note that d PQ ij = − d QP ij . The metric anti-commutes with thecomplex structure: ∆ P Q G (3) QR + G (3) PQ ∆ RQ = 0 , (B.109)which, in fact, becomes a Hermiticity condition on the metric G (3) : G (3) PQ = ∆ P R ∆ QS G (3) RS . (B.110)Thus, the 3-form moduli space is a Hermitian manifold with G (3) being a Hermitian metric.A real 4-form, which is topologically invariant, can be decomposed into the sum of(1 , , , X .In the same spirit as for the 3-forms, we introduce linear maps C , D , E and F to translatebetween real 4-forms and their (1 , , , ̟ x = C x X O X (and: ¯ ̟ ¯ x = ¯ C ¯ x X O X ) , (B.111) σ e = E e X O X , (B.112) O X = D X x ̟ x + ¯ D X ¯ x ¯ ̟ ¯ x + F X e σ e , (B.113)where { ̟ x } is a basis of H (1 , ( X ), whereas { σ e } and { O X } are real bases of H (2 , ( X )and H ( X ), respectively. Unlike C and D , E and F are real. All linear maps C , D , E and F a priori depend on the complex structure moduli z a and ¯ z ¯ a . By consecutively insertingeqs. (B.111)-(B.113) into each other, we learn relations among the linear maps C x X D X y = δ xy , ¯ C ¯ x X ¯ D X ¯ y = δ ¯ x ¯ y , E e X F X f = δ ef , (B.114) C x X ¯ D X ¯ y = C x X F X e = E e X D X x = 0 , (and c.c.) , (B.115) D X x C x Y + ¯ D X ¯ x ¯ C ¯ x Y + F X e E e Y = δ X Y . (B.116)The wedge product of two harmonic (1 , , , , H (2 , ( X ) = H (1 , ( X ) ∧ H (1 , ( X ) . (B.117)All examples of Calabi-Yau five-folds presented in Appendix B.2 satisfy eq. (B.117). Thesignificance of this restriction is that, since the (1 , , , σ e , E e X and F X e are allindependent of the complex structure moduli (or of any moduli fields, in fact). Since theleft hand side and the last term on the right hand side of eq. (B.113) are independent ofthe complex structure, the same must be true for the sum of the first two terms on theright hand side. This observation allows us to treat the (1 ,
3) and (3 ,
1) part together in acomplex structure independent way.Let us now choose the basis { O X } such that the first 2 h , ( X ) indices lie in the(1 ,
3) + (3 ,
1) directions and the remaining indices lie in the (2 ,
2) direction, that is wedivide the index range X = ( ˆ X , ˜ X ), where ˆ X = 1 , . . . , h , ( X ) and ˜ X = 1 , . . . , h , ( X ).This rearrangement is also independent of the complex structure. Eqs. (B.111)-(B.113)then become ̟ x = C x ˆ X O ˆ X (and: ¯ ̟ ¯ x = ¯ C ¯ x ˆ X O ˆ X ) , (B.118) σ e = E e ˜ X O ˜ X , (B.119) O ˆ X = D ˆ X x ̟ x + ¯ D ˆ X ¯ x ¯ ̟ ¯ x , O ˜ X = F ˜ X e σ e , (B.120)where O ˆ X , O ˜ X , F ˜ X e , E e ˜ X and σ e are independent of the complex structure moduli, whereasall other objects are dependent on them. Instead of eqs. (B.114)-(B.116) we have C x ˆ X D ˆ X y = δ xy , ¯ C ¯ x ˆ X ¯ D ˆ X ¯ y = δ ¯ x ¯ y , E e ˜ X F ˜ X f = δ ef , (B.121) C x ˆ X ¯ D ˆ X ¯ y = 0 , (and c.c.) , (B.122) D ˆ X x C x ˆ Y + ¯ D ˆ X ¯ x ¯ C ¯ x ˆ Y = δ ˆ X ˆ Y , F ˜ X e E e ˜ Y = δ ˜ X ˜ Y . (B.123)The relations between C x ˆ X , D ˆ X y , O ˆ X and ̟ x are very similar to the relations between A p P , B P q , N P and ν p for the 3-form case discussed above. The complex structure dependencein the (1 , C x ˆ X and D ˆ X y ∂ a O ˆ X = 0 , ∂ a ̟ x = C x ˆ Y ,a D ˆ Y y ̟ y + C x ˆ Y ,a ¯ D ˆ Y ¯ y ¯ ̟ ¯ y , (B.124) ∂ ¯ a O ˆ X = 0 , ∂ ¯ a ̟ x = C x ˆ Y , ¯ a D ˆ Y y ̟ y + C x ˆ Y , ¯ a ¯ D ˆ Y ¯ y ¯ ̟ ¯ y . (B.125)Using eqs. (B.118), (B.120) and (B.57), one can compute the Hodge star of the real 4-form O ˆ X : ∗ O ˆ X = − O ˆ X ∧ J . (B.126) In the Calabi-Yau four-fold literature, the right hand side of eq. (B.117) is often referred to as thevertical part, denoted H (2 , V , of H (2 , (see, for example, Ref. [6]). The total space H (2 , is given by H (2 , = H (2 , V ⊕ H (2 , H , where H (2 , H comprises all (2 , not be obtained by the productof two (1 , X for which H (2 , ( X ) = H (2 , V ( X ) and H (2 , H ( X ) = 0. – 63 –henever we use the forms O ˆ X to describe (1 , , G (ˆ4)ˆ X ˆ Y = Z X O ˆ X ∧ ∗ O ˆ Y . (B.127)Using the expression for the Hodge star (B.126), we can rewrite this so as to make thedependence on the moduli more explicit: G (ˆ4)ˆ X ˆ Y ( t ) = − d ˆ X ˆ Y i t i , (B.128)where we have defined a new intersection number d ˆ X ˆ Y i := R X O ˆ X ∧ O ˆ Y ∧ ω i , which is purelytopological. Note that d ˆ X ˆ Y i = d ˆ Y ˆ X i .Similarly to the 3-form case, there is a complex structure ∆ ˆ X ˆ Y on the ˆ4-form modulispace inherited from the complex structure of the Calabi-Yau five-fold and given by∆ ˆ X ˆ Y = i ( D ˆ X x C x ˆ Y − ¯ D ˆ X ¯ x ¯ C ¯ x ˆ Y ) . (B.129)It satisfies the relations of eq. (B.103). The projection operators are P ± ˆ X ˆ Y := 12 ( ∓ i ∆) ˆ X ˆ Y , (B.130)which satisfy eq. (B.105) and are explicitly given by P + ˆ X ˆ Y = D ˆ X y C y ˆ Y , P − ˆ X ˆ Y = ¯ D ˆ X ¯ y ¯ C ¯ y ˆ Y . (B.131)Note, however, that unlike in the 3-form case, the standard ˆ4-form metric (B.127) is notHermitian with respect to the complex structure ∆ ˆ X ˆ Y . C. N = 2 supersymmetry in one dimension In this appendix we will review and develop one-dimensional N = 2 supersymmetry to thelevel necessary for the theories which arise from our M-theory reductions. One-dimensionalsupersymmetry has previously been discussed in the literature (see, for example, [10, 39, 40]and references therein), notably in the context of black hole moduli spaces [17]. However, todescribe the effective actions which arise from M-theory reduction on Calabi-Yau five-foldsa number of generalisations and extensions of the one-dimensional N = 2 theories studiedin the literature are required. For example, we find that we require theories in which thetwo main types of multiplets, the 2 a and 2 b multiplets, are coupled. Some of the five-foldzero modes fall into fermionic (2 b ) multiplets so we need to introduce and develop thesemultiplets properly. Even though gravity in one dimension is non-dynamical, it leads toconstraints which cannot be ignored. This means we need to consider one-dimensional local supersymmetry. Finally, when we include M-theory four-form flux we need to incorporatea potential and an associated superpotential into the 2 a sector of the theory. All thosefeatures have not been fully worked out in the literature. We have, therefore, opted for asystematic exposition of one-dimensional N = 2 global and local supersymmetry, in orderto develop a solid base for our application to M-theory.– 64 – .1 Global N = 2 supersymmetry Before turning to one-dimensional N = 2 curved superspace, we will briefly recapitulatethe case of global N = 2 supersymmetry in one dimension [10]. One-dimensional super-space ( supertime ) is most easily obtained by dimensional reduction from d = 2, which hasattracted a lot of attention in view of formulating superstring actions in superspace [41, 42].In d = 2, there are Majorana, Weyl and Majorana-Weyl spinors and hence the same amountof supersymmetry can be realized by different choices of spinorial representation for thesupercharges (see, for example, Ref. [43]). For N = 2, the two options are (1 ,
1) and (2 , N = 2 su-persymmetry lead to two different one-dimensional N = 2 super multiplets, referred to as2 a (descending from two-dimensional (1 ,
1) supersymmetry) and 2 b (descending from two-dimensional (2 ,
0) supersymmetry) multiplets. These two multiplets will play a central rˆolein our discussion. Off-shell, the 2 a multiplet contains a real scalar as its lowest componentplus a complex fermion and a real scalar auxiliary field while the 2 b multiplets containsa complex scalar as its lowest component, accompanied by a complex fermion. The 2 b multiplet does not contain an auxiliary field. Other off-shell multiplets, not obtained froma standard toroidal reduction, are the fermionic 2 a and 2 b multiplets and the non-linearmultiplet [39]. From those we will only need and discuss in detail the fermionic 2 b multi-plet. It has a complex fermion as its lowest component which is balanced by a complexscalar at the next level.Flat N = 2 supertime, R | , is parametrised by coordinates { x = τ ; θ, ¯ θ } , where θ is acomplex one-dimensional spinor. In the following, we use indices A, B, . . . = 0 , θ, ¯ θ to labelsupertime tensors. The supersymmetry algebra is generated by two supercharges Q and ¯ Q defined as Q = ∂ θ − i θ∂ , ¯ Q = − ∂ ¯ θ + i θ∂ (C.1)where ∂ θ = ∂∂θ , ∂ ¯ θ = ∂∂ ¯ θ = − ( ∂ θ ) ∗ , ∂ = ∂∂x = ∂∂τ . Using the conventions for one-dimensional spinors summarised in Appendix A it is easy to verify that they satisfy thealgebra { Q, ¯ Q } = i∂ = H, { Q, Q } = 0 , { ¯ Q, ¯ Q } = 0 . (C.2)Supersymmetry transformations of N = 2 supertime are parameterised by a complex one-dimensional spinor ǫ and act as δ ǫ = iǫQ , δ ¯ ǫ = i ¯ ǫ ¯ Q . (C.3)This choice ensures that the total supersymmetry variation δ ǫ, tot. = δ ǫ + δ ¯ ǫ is real. As usual,we introduce the associated covariant derivatives D and ¯ D which anti-commute with thesupercharges, that is { D, Q } = { D, ¯ Q } = { ¯ D, Q } = { ¯ D, ¯ Q } = 0, and are explicitly givenby D = ∂ θ + i θ∂ , ¯ D = − ∂ ¯ θ − i θ∂ . (C.4)They satisfy the anti-commutation relations { D, ¯ D } = − i∂ = − H, { D, D } = 0 , { ¯ D, ¯ D } = 0 . (C.5)– 65 –lthough not really required for the global case it is useful for comparison with local su-persymmetry later on to develop the geometry of flat supertime. To this end, we introducethe notation ( ∂ A ) = ( ∂ , ∂ θ , ∂ ¯ θ ) for the partial derivatives and similarly for the covariantderivatives, ( D A ) = ( D , D θ , D ¯ θ ). These two types of derivatives are generally related by D A = E AB ∂ B , (C.6)where E AB is the inverse of the supervielbein E BA , that is E AC E C B = δ AB . For flatsupertime we have D = ∂ , D θ = D and D ¯ θ = ¯ D with D and ¯ D given in eq. (C.4). Ashort computation using eq. (C.6) then shows that the supervielbein of flat supertime isgiven by E = 1 , E θ = 0 , E θ = 0 ,E θ = − i θ, E θθ = 1 , E θ ¯ θ = 0 ,E ¯ θ = − i θ, E ¯ θθ = 0 , E ¯ θ ¯ θ = − . (C.7)The torsion tensor T ABC and curvature tensor R ABrs can be obtained from the generalrelation [43, 44] (cid:2) D A , D B (cid:9) = − T ABC D C − R ABrs M rs , (C.8)where M rs are the Lorentz generators. In d = 1, the Lorentz indices only run over onevalue and hence the single Lorentz generator and the curvature tensor vanish. To computethe torsion tensor of flat superspace we use the flat superspace covariant derivatives (C.4)in the above relation (C.8) for the torsion tensor. One finds that the only non-vanishingcomponent is T θ ¯ θ = i . (C.9)Finally, we find for the super-determinant of the flat supervielbein (C.7)sdet E AB = − . (C.10)Now we need to introduce superfields. One-dimensional N = 2 superfields are functionsof the supertime coordinates τ , θ and ¯ θ . As usual, their component field content can beworked out by expanding in θ and ¯ θ . Since θ = ¯ θ = 0, only the terms proportional to θ , ¯ θ and θ ¯ θ arise, in addition to the lowest, θ -independent component. Different types ofirreducible superfields can be obtained by imposing constraints on this general superfield.We now discuss these various types in turn.A 2 a superfield φ = φ ( τ, θ, ¯ θ ) is a real superfield, that is, a superfield satisfying theconstraint φ = φ † . A short calculation shows that the most general component expansionconsistent with this constraint is φ = ϕ + iθψ + i ¯ θ ¯ ψ + 12 θ ¯ θf , (C.11)where ϕ and f are real scalars and ψ is a complex fermion. The highest component f willturn out to be an auxiliary field so that a 2 a superfield contains one real physical scalar– 66 –eld. From eqs. (C.1) and (C.3), the supersymmetry transformations of these componentsare given by δ ǫ ϕ = − ǫψ, δ ǫ ψ = 0 , δ ǫ ¯ ψ = i ǫ ˙ ϕ − ǫf, δ ǫ f = − iǫ ˙ ψ, (C.12) δ ¯ ǫ ϕ = ¯ ǫ ¯ ψ, δ ¯ ǫ ¯ ψ = 0 , δ ¯ ǫ ψ = − i ǫ ˙ ϕ −
12 ¯ ǫf, δ ¯ ǫ f = − i ¯ ǫ ˙¯ ψ . (C.13)For a set, { φ i } , of 2 a superfields the most general non-linear sigma model can be writtenin superspace as [10, 17, 45] S a = 14 Z dτ d θ (cid:8) ( G ( φ ) + B ( φ )) ij Dφ i ¯ Dφ j + L ij ( φ ) Dφ i Dφ j + M ij ( φ ) ¯ Dφ i ¯ Dφ j + W ( φ ) (cid:9) , (C.14)where G ij is symmetric, B ij , L ij , M ij are anti-symmetric and W is an arbitrary functionof φ i . The component version of W ( φ ) is obtained by a Taylor expansion about ϕ i : W ( φ ) = W ( ϕ ) + iθψ i W ,i ( ϕ ) + i ¯ θ ¯ ψ i W ,i ( ϕ ) + 12 θ ¯ θ ( W ,i ( ϕ ) f i + 2 W ,ij ( ϕ ) ψ i ¯ ψ j ) . (C.15)The , i notation denotes the ordinary derivative with respect to ϕ i . From this and the otherformulægiven in this appendix it is straightforward to work out the component action ofthis superspace action. Here, we will not present the most general result but focus on thefirst and last term in eq. (C.14) which are the only ones relevant to our M-theory reduction.One finds S a = 14 Z dτ d θ (cid:8) G ij ( φ ) Dφ i ¯ Dφ j + W ( φ ) (cid:9) (C.16)= 14 Z dτ (cid:26) G ij ( ϕ ) ˙ ϕ i ˙ ϕ j − i G ij ( ϕ )( ψ i ˙¯ ψ j − ˙ ψ i ¯ ψ j ) + 14 G ij ( ϕ ) f i f j − G ij,k ( ϕ )( ψ i ¯ ψ j f k − ψ k ¯ ψ j f i − ψ i ¯ ψ k f j ) + i G ij,k ( ϕ )( ψ k ¯ ψ i + ¯ ψ k ψ i ) ˙ ϕ j (C.17) − G ij,kl ( ϕ ) ψ i ¯ ψ j ψ k ¯ ψ l − W ,i ( ϕ ) f i − W ,ij ( ϕ ) ψ i ¯ ψ j (cid:27) . Apart from the standard kinetic terms we have Pauli terms (coupling two fermions and thetime derivative of a scalar), Yukawa couplings and four-fermi terms. We also see that thehighest components f i are indeed auxiliary field. The f i equation of motion can be solvedexplicitly and leads to f i = G ij W ,j + . . . (C.18)where G ij is the inverse of G ij . The dots indicate fermion bilinear terms which we have notwritten down explicitly. Using this solution to integrating out the f i produces additionalfour-fermi terms and the scalar potential S a, pot = − Z dτ U , U = 12 G ij W ,i W ,j . (C.19) For an introduction to supersymmetric non-linear sigma models in one and two dimensions, see, forexample, Ref. [40]. – 67 –he other major type of multiplet is the 2 b multiplet Z = Z ( τ, θ, ¯ θ ) which is definedby the constraint ¯ DZ = 0. Working out its most general component expansion one finds Z = z + θκ + i θ ¯ θ ˙ z, (C.20)where z is a complex scalar and κ is a complex fermion. We note that, unlike for the 2 a multiplet, the highest component is not an independent field but simply ˙ z . Hence, a 2 b multiplet contains a complex physical scalar field and no auxiliary field. This differencein physical bosonic field content in comparison with the 2 a multiplet will be quite usefulwhen it comes to identifying which supermultiplets arise from our M-theory reduction.Eqs. (C.1) and (C.3) lead to the component supersymmetry transformations δ ǫ z = iǫκ, δ ǫ ¯ z = 0 , δ ǫ κ = 0 , δ ǫ ¯ κ = ǫ ˙¯ z, (C.21) δ ¯ ǫ ¯ z = i ¯ ǫ ¯ κ, δ ¯ ǫ z = 0 , δ ¯ ǫ ¯ κ = 0 , δ ¯ ǫ κ = ¯ ǫ ˙ z . (C.22)A general non-linear sigma model for a set, { Z a } , of 2 b multiplets has the form [10, 17, 45] S b = 14 Z dτ d θ (cid:26) G a ¯ b ( Z, ¯ Z ) DZ a ¯ D ¯ Z ¯ b + (cid:20) B ab ( Z, ¯ Z ) DZ a DZ b + c.c. (cid:21) + F ( Z, ¯ Z ) (cid:27) , (C.23)where G a ¯ b is hermitian, B ab is anti-symmetric and F is an arbitrary real function. Thecomponent version of F ( Z, ¯ Z ) is obtained by a Taylor expansion about z a and ¯ z ¯ a : F ( Z, ¯ Z ) = F ( z, ¯ z ) + θκ a F ,a ( z, ¯ z ) − ¯ θ ¯ κ ¯ a F , ¯ a ( z, ¯ z )+ 12 θ ¯ θ n iF ,a ( z, ¯ z ) ˙ z a − iF , ¯ a ( z, ¯ z ) ˙¯ z ¯ a + 2 F ,a ¯ b ( z, ¯ z ) κ a ¯ κ ¯ b o . (C.24)The component form of the action (C.23) can again be worked out straightforwardly fromthe above formalæ but we will not pursue this here. Instead, we focus on a slightly differentsuperspace action which is better adapted to what we need in the context of our M-theoryreduction. First, we drop the term proportional to B ab which does not arise from M-theory.Secondly, we introduce a slight generalisation in that we allow the sigma model metric G a ¯ b to also depend on 2 a superfields φ i , in addition to the 2 b superfields Z a and their complexconjugates. A multi-variable Taylor expansion of a function G ( φ, Z, ¯ Z ) depending on 2 a as well as 2 b superfields yields the component form: G ( φ, Z, ¯ Z ) = G ( ϕ, z, ¯ z ) + θ [ iψ i G ,i ( ϕ, z, ¯ z ) + κ a G ,a ( ϕ, z, ¯ z )] + ¯ θ [ i ¯ ψ i G ,i ( ϕ, z, ¯ z ) − ¯ κ ¯ a G , ¯ a ( ϕ, z, ¯ z )] + θ ¯ θ (cid:20) G ,i ( ϕ, z, ¯ z ) f i + G ,ij ( ϕ, z, ¯ z ) ψ i ¯ ψ j + iG ,ia ( ϕ, z, ¯ z ) ¯ ψ i κ a + iG ,i ¯ a ( ϕ, z, ¯ z ) ψ i ¯ κ ¯ a + G ,a ¯ b ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b + i G ,a ( ϕ, z, ¯ z ) ˙ z a − i G , ¯ a ( ϕ, z, ¯ z ) ˙¯ z ¯ a (cid:21) . (C.25)The relevant action is S b = 14 Z dτ d θ n G a ¯ b ( φ, Z, ¯ Z ) DZ a ¯ D ¯ Z ¯ b + F ( Z, ¯ Z ) o – 68 – 14 Z dτ (cid:26) G a ¯ b ( ϕ, z, ¯ z ) ˙ z a ˙¯ z ¯ b − i G a ¯ b ( ϕ, z, ¯ z )( κ a ˙¯ κ ¯ b − ˙ κ a ¯ κ ¯ b ) (C.26) − i G a ¯ b,c ( ϕ, z, ¯ z )( κ a ¯ κ ¯ b ˙ z c − κ c ¯ κ ¯ b ˙ z a ) + i G a ¯ b, ¯ c ( ϕ, z, ¯ z )( κ a ¯ κ ¯ b ˙¯ z ¯ c + 2 κ a ¯ κ ¯ c ˙¯ z ¯ b ) (C.27) − G a ¯ b,c ¯ d ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b κ c ¯ κ ¯ d − G a ¯ b,i ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b f i − G a ¯ b,ij ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ψ i ¯ ψ j (C.28) − iG a ¯ b,ic ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ¯ ψ i κ c − iG a ¯ b,i ¯ c ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ψ i ¯ κ ¯ c − G a ¯ b,i ( ϕ, z, ¯ z ) ψ i ¯ κ ¯ b ˙ z a (C.29)+ G a ¯ b,i ( ϕ, z, ¯ z ) ¯ ψ i κ a ˙¯ z ¯ b − i F ,a ˙ z a − F , ¯ b ˙¯ z ¯ b ) − F ,a ¯ b κ a ¯ κ ¯ b (cid:27) . (C.30)Note that the function F gives rise to a Chern-Simons type term (and fermion mass terms)but not to a scalar potential.The 2 a and 2 b superfields introduced above are bosonic superfields in the sense thattheir lowest components are bosons. However, for both types of multiplets there alsoexists a fermionic version, satisfying the same constraint as their bosonic counterparts butstarting off with a fermion as the lowest component. In our context, we will only needfermionic 2 b superfields so we will focus on them. The details for fermionic 2 a superfieldscan be worked out analogously.Fermionic 2 b superfields R = R ( τ, θ, ¯ θ ) have a spinorial lowest component and aredefined by the constraint ¯ DR = 0. Their general component expansion reads R = ρ + θh + i θ ¯ θ ˙ ρ , (C.31)where ρ is a complex fermion and h is a complex scalar. For its component supersymmetrytransformations one finds δ ǫ ρ = iǫh, δ ǫ ¯ ρ = 0 , δ ǫ h = 0 , δ ǫ ¯ h = − ǫ ˙¯ ρ, (C.32) δ ¯ ǫ ¯ ρ = − i ¯ ǫ ¯ h, δ ¯ ǫ ρ = 0 , δ ¯ ǫ ¯ h = 0 . δ ¯ ǫ h = ¯ ǫ ˙ ρ . (C.33)A set, { R x } , of fermionic 2 b superfields can be used to build non-linear sigma models whereonly fermions are propagating. A class of such models is given by S b, F = 14 Z dτ d θ G x ¯ y ( φ ) R x ¯ R ¯ y (C.34)= 14 Z dτ (cid:26) i G x ¯ y ( ϕ )( ρ x ˙¯ ρ ¯ y − ˙ ρ x ¯ ρ ¯ y ) − G x ¯ y ( ϕ ) h x ¯ h ¯ y − G x ¯ y,i ( ϕ ) ρ x ¯ ρ ¯ y f i (C.35) − iG x ¯ y,i ( ϕ )( ψ i ρ x ¯ h ¯ y + ¯ ψ i ¯ ρ ¯ y h x ) − G x ¯ y,ij ( ϕ ) ρ x ¯ ρ ¯ y ψ i ¯ ψ j (cid:9) . (C.36)Here, we have allowed the sigma model metric to depend on a set, { φ i } , of 2 a moduli, asituation which will arise from M-theory reductions. Note that the bosons h x are indeedauxiliary fields and only the fermions ρ x have kinetic terms. C.2 Local N = 2 supersymmetry The goal of this subsection is to develop one-dimensional N = 2 curved superspace to anextent that will allow us to write down actions over this superspace and compare their– 69 –omponent expansion with our result from dimensional reduction of M-theory on Calabi-Yau five-folds. Eventually, we are using the results of this subsection to write our one-dimensional effective action in full one-dimensional N = 2 curved superspace therebymaking the residual supersymmetry manifest.The on-shell one-dimensional N = 2 supergravity multiplet comprises the lapse func-tion (or “einbein”) N , which is a real scalar, and the “lapsino” ψ , which is a one-componentcomplex spinor. In all expressions provided in this sub-section, flat superspace (and thusthe equations of the previous subsection) can be recovered by gauge fixing the supergravityfields to N = 1 and ψ = 0. From a more geometric viewpoint, curved N = 2 supertimelooks locally like flat N = 2 supertime R | .The well-known case of N = 1 in four dimensions [46, 47, 48, 49, 50, 51] and supergrav-ity theories in two-dimensions [52, 53] will guide us in constructing our curved supertimehere. Modulo some subtleties, which are explained below, many textbook formulæ carryover to the case of N = 2 supertime with the index ranges adjusted appropriately.The geometrical description of curved superspace follows ordinary Riemannian geom-etry, however with the range of indices extended to include the spinorial coordinates. Inparticular, certain (super-)tensors, such as the supervielbein E AB , super-spin-connectionΩ ABC , supertorsion T ABC and supercurvature R ABC D , play an important rˆole when work-ing with curved superspace. As in the previous subsection, the indices
A, B, . . . = 0 , θ, ¯ θ are used to label supertime tensors and underlined versions A, B, . . . correspond to localLorentz indices. As local coordinates, we choose { x = τ ; θ, ¯ θ } , where θ is a complex one-dimensional spinor. The supervielbein can be used to convert curved to flat indices andvice versa, so that V A = E AB V B , V A = E AB V B . (C.37)In the second relation the inverse of the supervielbein has been used, which is defined via E AB E BC = δ AC , E AB E BC = δ AC . (C.38)Note that one may use the superdifferential dz A together with the graded wedge-product ∧ to write all the aforementioned supertensors as super-differential forms, for example E A = dz B E BA , T A = 12 dz B ∧ dz C T CBA . (C.39)The supertorsion is defined as covariant derivative of the supervielbein: T A = dE A + E B ∧ Ω BA . (C.40)The rˆole of the local Lorentz indices is rather subtle in N = 2 supertime. In order torecover flat supertime, these indices are taken to be valued in the bosonic Lorentz groupSO(1), which is just the trivial group, and not in the full super-Lorentz group SO(1 | A, B, . . . do not transform under any group action but should We shall closely follow Refs. [43, 44], here. – 70 –erely thought of as labels. They label the two different representations of Spin(1), namelyfor A = 0 the “vector” representation, which is nothing but the real numbers in onedimension, and for A = θ the spinor representation which are real Grassmann numbers. Inaddition, the fact that we want to realize N -extended supersymmetry (with N >
1) meanswe need in principle another index, say i, j = 1 , . . . , N on the A = θ components to labelthe N -extendedness of the spinorial components (cf. the notation used in four dimensional N = 2 superspace [54, 55, 56, 57]). Here, N = 2 and hence A, B, . . . = 0 , θ , θ . For easeof notation, we combine the two θ i into a combination of one complex index θ = θ + iθ (and similarly ¯ θ = θ − iθ ) thereby suppressing the additional N -extension index i . Afterthis step, the local Lorentz indices A, B, . . . range over 0, θ and ¯ θ . Note that this coincidesprecisely with the notation used for curved indices except for the additional underlineadded for distinction. In summary, even though the local Lorentz indices can take on threedifferent values, there is no group acting on them. Objects carrying an anti-symmetrizedcombination of two or more local Lorentz indices vanish identically, since the Lorentzgenerator in each representation of Spin(1) is zero and there are no representation-mixingLorentz transformations. This immediately implies Ω ABC = 0 and R ABC D = 0, whichprofoundly simplifies the further discussion.Since the on-shell supergravity multiplet contains only one real scalar, we take thegeometrical supertime tensors to be 2 a superfields, which means they comprise four com-ponent fields when expanded out in powers of θ and ¯ θ (see eq. (C.11)). The supervielbein E AB , in general, consists of a set of 3 × a superfields, which totals to 9 × E AB = E A (0) B + iθE A (1) B + i ¯ θE A (¯1) B + 12 θ ¯ θE A (2) B . (C.41)This is a large number of apparently independent fields given that on-shell, we just havethree, namely N , ψ and ¯ ψ . In order to not obscure the physical content and to formulatesupertime theories in the most efficient way, it is important to find a formulation withthe minimum number of component fields. This can be achieved by imposing covariantconstraints on the supervielbein and by gauging away some components using the super-general coordinate transformations δE AB = ξ C ( ∂ C E AB ) + ( ∂ A ξ C ) E C B , (C.42)with infinitesimal parameters ξ A , which comprise a set of three four-component 2 a su-perfields (that is, 12 component fields in total). The lowest component of ξ | = ζ is theinfinitesimal parameter of worldline reparametrizations, whereas the lowest components ofthe spinorial parameters ξ θ | = iǫ and ξ ¯ θ | = i ¯ ǫ correspond to the infinitesimal local N = 2supersymmetry parameters. The notation φ | is a shorthand for φ | θ =¯ θ =0 , that is denotingthe lowest component of the superfield φ . An infinitesimal local N = 2 supersymmetrytransformations with parameters ǫ and ¯ ǫ on a general superfield φ can be written by meansof the supercharges Q and ¯ Q as δ ǫ φ = iǫQφ, δ ¯ ǫ φ = i ¯ ǫ ¯ Qφ. (C.43)– 71 –f we use the following general component expansion for φ : φ = φ | + θ ( D φ | ) − ¯ θ ( ¯ D φ | ) + 12 θ ¯ θ ([ D , ¯ D ] φ | ) , (C.44)then the components of φ transform as δ ǫ ( φ | ) = iǫQφ | , δ ǫ ( D φ | ) = iǫQ D φ | , δ ǫ ( ¯ D φ | ) = iǫQ ¯ D φ | , δ ǫ ([ D , ¯ D ] φ | ) = iǫQ [ D , ¯ D ] φ | . (C.45)Both (C.44) and (C.45) are manifestly super-covariant expressions since we used the tan-gentized covariant super-derivative of curved supertime D A = E AB ∂ B for building them.Note that, similarly to D and ¯ D in the flat case, the tangentized, spinorial super-covariantderivatives are abbreviated as D := D θ = E θA ∂ A and ¯ D := D ¯ θ = E ¯ θA ∂ A . From the generalfact that Q | = D| = ∂ θ , it follows that one may replace Q s by D s everywhere in (C.45)and hence knowing the component expansion of D is enough for working out the entirecomponent version of (C.43), namely: δ ǫ ( φ | ) = iǫ D φ | , δ ǫ ( D φ | ) = iǫ D φ | = 0 , δ ǫ ( ¯ D φ | ) = iǫ D ¯ D φ | , δ ǫ ([ D , ¯ D ] φ | ) = − iǫ D ¯ DD φ | , (C.46)and similarly for the ¯ ǫ -transformations. In the second and fourth equation in (C.46), weused the property D = 0.Now continuing our quest for finding the minimal formulation of off-shell N = 2, d = 1supergravity, we have here opted for the analogue of the Wess-Zumino gauge in d = 4and the way to formulate it in the present case will be explained in the following. Sincewe have three physical components in the supergravity multiplet, we shall use 9 = 12 − ξ A to gauge fix 9 out of the 36 components of E AB , namely E θ | = E ¯ θ | = E θ ¯ θ | = E ¯ θθ | = D E ¯ θθ | = D E ¯ θ ¯ θ | = 0 , (C.47)¯ D E θ | = i , E θθ | = 1 , E ¯ θ ¯ θ | = − . (C.48)The three remaining parameters in ξ A act on the three physical fields N , ψ and ¯ ψ , whichwe choose to identify in the following way: E | = N, E θ | = ψ , E θ | = − ¯ ψ . (C.49)We will now discuss our choice of covariant constraints. Usually, they are imposed oncertain components of the tangentized supertorsion T ABC . “Trial and error” and “educatedguesses” eventually lead to a combination of constraints that yield the minimum numberof fields in the θ -expansion of the supervielbein E AB . The main idea is to take the systemof constraints from N = 1, d = 4 and restrict the index ranges appropriately. Doing this,we obtain the following torsion constraints: T θ ¯ θ = i, T θ ¯ θθ = 0 , (conventional constraints) , (C.50) T ¯ θ ¯ θ = 0 , T ¯ θ ¯ θθ = 0 , (representation preserving constraints) , (C.51)– 72 – θθθ = 0 , (“type 3” constraint) , (C.52)and their complex conjugates, of course. We are equating superfields to superfields hereand hence each of the above relations is manifestly (super-)covariant. The first line is theanalogue of the conventional constraints in N = 1, d = 4 and are characterized by beingalgebraically solvable. In the absence of R AB , the torsion is directly related to the gradedcommutator of two super-covariant derivatives via[ D A , D B } = − T ABC D C . (C.53)The conventional constraints now stem from imposing (cf. eq. (C.4)) {D , ¯ D} = − i D , (C.54)which guarantees that the tangentized covariant super-derivatives of curved superspace, D and ¯ D , satisfy the flat algebra. A 2 b superfield Z by definition satisfies ¯ D Z = 0.The representation preserving constraints listed in (C.51) follow from the correspondingintegrability condition, that is from { ¯ D , ¯ D} Z = 0 ∀ b superfields Z. (C.55)For the constraint in (C.52), we do not have a direct motivation from a one-dimensionalviewpoint, so we impose it purely by analogy to the conformal constraint of N = 1 in d = 4.In general superspace theory, the torsion and curvature tensors satisfy the two Bianchiidentities (BIs) ∇ T A = E B ∧ R BA , (C.56) ∇ R AB = 0 , (C.57)where ∇ = d + Ω ∧ . Specializing to N = 2 supertime, the second BI identically vanishesdue to R AB = 0 and the first BI becomes dT A = 0 ⇔ D [ A T BC } D + T [ AB E T | E | C } D = 0 . (C.58)In the presence of constraints, consistency requires that the BIs are sill obeyed and thisneeds to be checked by explicit calculation. In this respect, the BIs become “contentful”(rather than being genuine identities) when constraints are present and then the BIs mustbe imposed . For the case at hand, one learns from the BI (C.58) that all remaining torsioncomponents which are not already fixed by the constraints (C.50)-(C.52) must be zero.From the definition of the supertorsion (C.40), the choice of gauge fixing (C.47)-(C.49)and torsion constraints (C.50)-(C.52) and the imposition of the BI (C.58), all 36 compo-nents in the supervielbein expansion (C.41) are fixed uniquely to E = N + iθ ¯ ψ + i ¯ θψ , (C.59) E θ = ψ , E θ = − ¯ ψ , (C.60)– 73 – θ = − i θ, E ¯ θ = − i θ, (C.61) E θθ = 1 , E θ ¯ θ = 0 , E ¯ θθ = 0 , E ¯ θ ¯ θ = − . (C.62)Note that the minimal set of fields of off-shell pure N = 2, d = 1 supergravity does notcomprise any auxiliary fields. From eq. (C.38) we compute the component expansion ofthe inverse supervielbein E = N − − i θN − ¯ ψ − i θN − ψ − θ ¯ θN − ψ ¯ ψ , (C.63) E θ = − N − ψ − i θN − ψ ¯ ψ , E θ = − N − ¯ ψ + i θN − ψ ¯ ψ , (C.64) E θ = i θN − − θ ¯ θN − ¯ ψ , E ¯ θ = − i θN − − θ ¯ θN − ψ , (C.65) E θθ = 1 − i θN − ψ − θ ¯ θN − ψ ¯ ψ , E θ ¯ θ = − i θN − ¯ ψ , (C.66) E ¯ θ ¯ θ = − i θN − ¯ ψ + 14 θ ¯ θN − ψ ¯ ψ , E ¯ θθ = i θN − ψ . (C.67)Since D A = E AB ∂ B , the above expressions allow us to write down the component expansionof the tangentized, spinorial super-covariant derivative D = (cid:18) − i N − ¯ θψ − N − θ ¯ θψ ¯ ψ (cid:19) ∂ θ + (cid:18) i N − ¯ θ − N − θ ¯ θ ¯ ψ (cid:19) ∂ − i N − ¯ θ ¯ ψ ∂ ¯ θ , (C.68)and similarly for ¯ D . By comparing the component expansion of eq. (C.42) with eqs. (C.59)-(C.62), we learn how the supergravity fields transform under local N = 2 supersymmetry δ ǫ N = − ǫ ¯ ψ , δ ¯ ǫ N = ¯ ǫψ , δ ǫ ψ = i ˙ ǫ, δ ǫ ¯ ψ = 0 , δ ¯ ǫ ψ = 0 , δ ¯ ǫ ¯ ψ = − i ˙¯ ǫ. (C.69)In order to build curved superspace actions that are manifestly invariant under local N = 2supersymmetry, we need the analogue of √− g to construct an invariant volume form. Itturns out that this is given by the super-determinant of the supervielbein, denoted simplyby E , and defined, in general, as E := sdet E AB = (det E ab )(det[ E αβ − E αc ( E dc ) − E dβ ]) − , (C.70)where a, b, . . . and α, β, . . . denote vector and spinor indices, respectively. Specializing to N = 2 supertime and inserting eqs. (C.59)-(C.62), one finds for the super-determinant ofthe supervielbein E = − N − i θ ¯ ψ − i θψ . (C.71)Since there is no θ ¯ θ -component in this expression, it follows that the canonical action ofpure supergravity vanishes as expected, that is S pure sugra = Z dτ d θ E = 0 . (C.72)– 74 –lso, as an additional consistency check, one may verify that E is super-covariantly con-stant, so that Z dτ d θ DE = Z dτ d θ ¯ DE = (total derivative) = 0 . (C.73)This allows us to use the partial-integration rule for superspace.In analogy to the flat superspace case, we will now present the different irreduciblemultiplets. We begin with the 2 a multiplet, defined by the constraint φ = φ † . The generalsolution to this constraint leads to the component expansion φ = ϕ + iθψ + i ¯ θ ¯ ψ + 12 θ ¯ θf , (C.74)where the component fields are labelled as in eq. (C.11). This can also be written in amanifestly super-covariant fashion as φ = φ | + θ ( D φ | ) − ¯ θ ( ¯ D φ | ) + 12 θ ¯ θ ([ D , ¯ D ] φ | ) . (C.75)For the supersymmetry transformations of the 2 a component fields one finds δ ǫ ϕ = − ǫψ, δ ǫ ψ = 0 , δ ǫ ¯ ψ = i N − ǫ ˙ ϕ − ǫf + 12 N − ǫ ( ψ ψ + ¯ ψ ¯ ψ ) ,δ ǫ f = − iN − ǫ ˙ ψ + i N − ǫ ¯ ψ ˙ ϕ + 12 N − ǫ ¯ ψ f − N − ǫψψ ¯ ψ , (C.76) δ ¯ ǫ ϕ = ¯ ǫ ¯ ψ, δ ¯ ǫ ψ = − i N − ¯ ǫ ˙ ϕ −
12 ¯ ǫf − N − ¯ ǫ ( ψ ψ + ¯ ψ ¯ ψ ) , δ ¯ ǫ ¯ ψ = 0 ,δ ¯ ǫ f = − iN − ¯ ǫ ˙¯ ψ + i N − ¯ ǫψ ˙ ϕ − N − ¯ ǫψ f + 12 N − ¯ ǫ ¯ ψψ ¯ ψ . (C.77)This is obtained by plugging in the component expansions (C.68) and (C.74) into thegeneral formula (C.46). A standard kinetic term of a single 2 a superfield φ and its associatedcomponent action are given by S , kin = − Z dτ d θ E D φ ¯ D φ = 14 Z dτ L , kin , L , kin = 14 N − ˙ ϕ − i ψ ˙¯ ψ − ˙ ψ ¯ ψ ) + 14 N f + i N − ( ψψ + ¯ ψ ¯ ψ ) ˙ ϕ + 12 N − ψ ¯ ψ ψ ¯ ψ. (C.78)In the context of M-theory five-fold compactifications we need to consider more generalactions, representing non-linear sigma models for a set of 2 a fields φ i which also include a(super)-potential term. The superspace and component forms for such actions read S = − Z dτ d θ E { G ij ( φ ) D φ i ¯ D φ j + W ( φ ) } = 14 Z dτ L , L = 14 N − G ij ( ϕ ) ˙ ϕ i ˙ ϕ j − i G ij ( ϕ )( ψ i ˙¯ ψ j − ˙ ψ i ¯ ψ j ) + 14 N G ij ( ϕ ) f i f j + i N − G ij ( ϕ )( ψ i ψ + ¯ ψ i ¯ ψ ) ˙ ϕ j + 12 N − G ij ( ϕ ) ψ ¯ ψ ψ i ¯ ψ j − N G ij,k ( ϕ )( ψ i ¯ ψ j f k − ψ k ¯ ψ j f i − ψ i ¯ ψ k f j ) + i G ij,k ( ϕ )( ψ k ¯ ψ i + ¯ ψ k ψ i ) ˙ ϕ j − N G ij,kl ( ϕ ) ψ i ¯ ψ j ψ k ¯ ψ l − N W ,i ( ϕ ) f i − N W ,ij ( ϕ ) ψ i ¯ ψ j − W ,i ( ϕ )( ψ i ψ − ¯ ψ i ¯ ψ ) , (C.79)– 75 –ith a sigma model metric G ij ( φ ) and a superpotential W ( φ ) . Here, G ...,i denotes differ-entiation with respect to the bosonic fields ϕ i . Note that the fields f i are indeed auxiliary.Solving their equations of motion leads to f i = G ij W j + G ij G kl,j ψ k ¯ ψ l − G ij G jk,l ( ψ k ¯ ψ l + ψ l ¯ ψ k ) , (C.80)where G ij is the inverse of G ij and W i = W ,i = ∂ W ∂ϕ i . Inserting this back into the componentaction leads, among other terms, to the scalar potential S a, pot = − Z dτ N U , U = 12 G ij W i W j , (C.81)for the scalars ϕ i in the 2 a multiplets. We will also need a slight generalization of eq. (C.79),namely an action for a set of 2 a superfields X p coupling to a set of other 2 a superfields φ i and to a set of 2 b superfields Z a via the sigma model metric G pq ( φ, Z, ¯ Z ): S , gen . = − Z dτ d θ E { G pq ( φ, Z, ¯ Z ) D X p ¯ D X q } = 14 Z dτ L , gen . , L , gen . = 14 N − G pq ( ϕ, z, ¯ z ) ˙ x p ˙ x q − i G pq ( ϕ, z, ¯ z )( λ p ˙¯ λ q − ˙ λ p ¯ λ q ) + 14 N G pq ( ϕ, z, ¯ z ) g p g q + i N − G pq ( ϕ, z, ¯ z )( λ p ψ + ¯ λ p ¯ ψ ) ˙ x q + 12 N − G pq ( ϕ, z, ¯ z ) ψ ¯ ψ λ p ¯ λ q − N G pq,i ( ϕ, z, ¯ z )( λ p ¯ λ q f i − ψ i ¯ λ p g q + ¯ ψ i λ p g q ) + i G pq,i ( ϕ, z, ¯ z )( ψ i ¯ λ p + ¯ ψ i λ p ) ˙ x q − N G pq,ij ( ϕ, z, ¯ z ) λ p ¯ λ q ψ i ¯ ψ j − i G pq,a ( ϕ, z, ¯ z ) λ p ¯ λ q ( ˙ z a − ψ κ a )+ i G pq, ¯ a ( ϕ, z, ¯ z ) λ p ¯ λ q ( ˙¯ z ¯ a + ¯ ψ ¯ κ ¯ a ) − i N G pq,a ( ϕ, z, ¯ z ) κ a ¯ λ p g q − i N G pq, ¯ a ( ϕ, z, ¯ z )¯ κ ¯ a λ p g q + 12 G pq,a ( ϕ, z, ¯ z ) κ a ¯ λ p ˙ x q − G pq, ¯ a ( ϕ, z, ¯ z )¯ κ ¯ a λ p ˙ x q − N G pq,a ¯ b ( ϕ, z, ¯ z ) λ p ¯ λ q κ a ¯ κ ¯ b − iN G pq,ia ( ϕ, z, ¯ z ) λ p ¯ λ q ¯ ψ i κ a − iN G pq,i ¯ a ( ϕ, z, ¯ z ) λ p ¯ λ q ψ i ¯ κ ¯ a . (C.82)Next we turn to 2 b multiplets. They are defined by the constraint ¯ D Z = 0 which leadsto the component expansion Z = z + θκ + i N − θ ¯ θ ( ˙ z − ψ κ ) . (C.83)Here, N and ψ are the components of the supergravity multiplet and the other fields arelabelled in analogy with the globally supersymmetric case (C.20). Expression (C.83) isequivalent to the manifestly super-covariant version: Z = Z | + θ ( D Z | ) − θ ¯ θ ( ¯ DD Z | ) . (C.84)By plugging in the component expansions (C.68) and (C.83) into the general for-mula (C.46), the component field supersymmetry transformations are derived and read δ ǫ z = iǫκ, δ ǫ ¯ z = 0 , δ ǫ κ = 0 , δ ǫ ¯ κ = N − ǫ ( ˙¯ z + ¯ ψ ¯ κ ) , (C.85)– 76 – ¯ ǫ z = 0 , δ ¯ ǫ ¯ z = i ¯ ǫ ¯ κ, δ ¯ ǫ κ = N − ¯ ǫ ( ˙ z − ψ κ ) , δ ¯ ǫ ¯ κ = 0 . (C.86)A standard kinetic term for a single 2 b multiplet Z can be written and expanded intocomponents as S , kin = − Z dτ d θ E D Z ¯ D ¯ Z = 14 Z dτ L , kin , L , kin = N − ˙ z ˙¯ z − i κ ˙¯ κ − ˙ κ ¯ κ ) − N − ( ψ κ ˙¯ z − ¯ ψ ¯ κ ˙ z ) + N − ψ ¯ ψ κ ¯ κ. (C.87)The generalization to a non-linear sigma model for a set, { Z a } , of 2 b multiplets is given by S = − Z dτ d θ E G a ¯ b ( Z, ¯ Z ) D Z a ¯ D ¯ Z ¯ b = 14 Z dτ L , L = N − G a ¯ b ( z, ¯ z ) ˙ z a ˙¯ z ¯ b − i G a ¯ b ( z, ¯ z )( κ a ˙¯ κ ¯ b − ˙ κ a ¯ κ ¯ b ) − N − G a ¯ b ( z, ¯ z )( ψ κ a ˙¯ z ¯ b − ¯ ψ ¯ κ ¯ b ˙ z a )+ N − G a ¯ b ( z, ¯ z ) ψ ¯ ψ κ a ¯ κ ¯ b − i G a ¯ b,c ( z, ¯ z )( κ a ¯ κ ¯ b ( ˙ z c − ψ κ c ) − κ c ¯ κ ¯ b ˙ z a )+ i G a ¯ b, ¯ c ( z, ¯ z )( κ a ¯ κ ¯ b ( ˙¯ z ¯ c + 2 ¯ ψ ¯ κ ¯ c ) − κ a ¯ κ ¯ c ˙¯ z ¯ b ) − N G a ¯ b,c ¯ d ( z, ¯ z ) κ a ¯ κ ¯ b κ c ¯ κ ¯ d . (C.88)Here, G ...,a means differentiation with respect to the bosonic fields z a . In our applicationto M-theory, we need a variant of this action where the sigma model metric G a ¯ b is alsoallowed to depend on a set of 2 a multiplets φ i in addition to Z a and ¯ Z ¯ b . This leads to acoupling between 2 a and 2 b multiplets. The action for this case reads S = − Z dτ d θ E G a ¯ b ( φ, Z, ¯ Z ) D Z a ¯ D ¯ Z ¯ b = 14 Z dτ L , L = N − G a ¯ b ( ϕ, z, ¯ z ) ˙ z a ˙¯ z ¯ b − i G a ¯ b ( ϕ, z, ¯ z )( κ a ˙¯ κ ¯ b − ˙ κ a ¯ κ ¯ b ) − N − G a ¯ b ( ϕ, z, ¯ z )( ψ κ a ˙¯ z ¯ b − ¯ ψ ¯ κ ¯ b ˙ z a ) + N − G a ¯ b ( ϕ, z, ¯ z ) ψ ¯ ψ κ a ¯ κ ¯ b − i G a ¯ b,c ( ϕ, z, ¯ z )( κ a ¯ κ ¯ b ( ˙ z c − ψ κ c ) − κ c ¯ κ ¯ b ˙ z a )+ i G a ¯ b, ¯ c ( ϕ, z, ¯ z )( κ a ¯ κ ¯ b ( ˙¯ z ¯ c + 2 ¯ ψ ¯ κ ¯ c ) − κ a ¯ κ ¯ c ˙¯ z ¯ b ) − N G a ¯ b,c ¯ d ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b κ c ¯ κ ¯ d − N G a ¯ b,i ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b f i − N G a ¯ b,ij ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ψ i ¯ ψ j − iN G a ¯ b,ic ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ¯ ψ i κ c − iN G a ¯ b,i ¯ c ( ϕ, z, ¯ z ) κ a ¯ κ ¯ b ψ i ¯ κ ¯ c − G a ¯ b,i ( ϕ, z, ¯ z ) ψ i ¯ κ ¯ b ( ˙ z a − ψ κ a )+ G a ¯ b,i ( ϕ, z, ¯ z ) ¯ ψ i κ a ( ˙¯ z ¯ b + 12 ¯ ψ ¯ κ ¯ b ) . (C.89)This result can be readily specialized to G a ¯ b ( φ, Z, ¯ Z ) = f ( φ ) G a ¯ b ( Z, ¯ Z ), for a real function f = f ( φ ), which is the case relevant to M-theory compactifications.Finally, we need to discuss fermionic 2 b multiplets, that is, super-multiplets R with afermionic lowest component and satisfying ¯ D R = 0. Their component expansion is givenby R = ρ + θh + i N − θ ¯ θ ( ˙ ρ − ψ h ) , (C.90)– 77 –here the notation for the component fields is completely analogous to the globally su-persymmetric case (C.31). The component supersymmetry transformations follow fromplugging in the component expansions (C.68) and (C.90) into the general formula (C.46)and are given by δ ǫ ρ = iǫh, δ ǫ ¯ ρ = 0 , δ ǫ h = 0 , δ ǫ ¯ h = − N − ǫ ( ˙¯ ρ − ¯ ψ ¯ h ) , (C.91) δ ¯ ǫ ρ = 0 , δ ¯ ǫ ¯ ρ = − i ¯ ǫ ¯ h, δ ¯ ǫ h = N − ¯ ǫ ( ˙ ρ − ψ h ) , δ ¯ ǫ ¯ h = 0 . (C.92)A simple kinetic term for a single fermionic 2 b superfield R takes the form S − f , kin = − Z dτ d θ E R ¯ R = 14 Z dτ L − f , kin , L − f , kin = i ρ ˙¯ ρ − ˙ ρ ¯ ρ ) − N h ¯ h . (C.93)Note that the only bosonic field, h , in this multiplet is auxiliary and, hence, we are left withonly fermionic physical degrees of freedom. This observation will be crucial for writing downa superspace version of the effective one-dimensional theories obtained from M-theory. Asfor the other types of multiplets, we need to generalise to a sigma model for a set, { R x } ,of fermionic 2 b multiplets. The sigma model metric G x ¯ y = G x ¯ y ( φ ) should be allowed todepend on 2 a multiplets φ i . Such an action takes the form S − f = − Z dτ d θ E G x ¯ y ( φ ) R x ¯ R ¯ y = 14 Z dτ L − f , L − f = i G x ¯ y ( ϕ )( ρ x ˙¯ ρ ¯ y − ˙ ρ x ¯ ρ ¯ y ) − N G x ¯ y ( ϕ ) h x ¯ h ¯ y − iN G x ¯ y,i ( ϕ )( ψ i ρ x ¯ h ¯ y + ¯ ψ i ¯ ρ ¯ y h x ) − N G x ¯ y,i ( ϕ ) ρ x ¯ ρ ¯ y f i − N G x ¯ y,ij ( ϕ ) ρ x ¯ ρ ¯ y ψ i ¯ ψ j + 12 G x ¯ y,i ( ϕ ) ρ x ¯ ρ ¯ y ( ψ ψ i − ¯ ψ ¯ ψ i ) . (C.94) References [1] P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten,
Vacuum configurations forsuperstrings , Nucl. Phys.
B258 (1985) 46–74.[2] I. Brunner and R. Schimmrigk,
F-theory on Calabi-Yau fourfolds , Phys. Lett.
B387 (1996)750–758, [ hep-th/9606148 ].[3] C. M. Hull and P. K. Townsend,
Unity of superstring dualities , Nucl. Phys.
B438 (1995)109–137, [ hep-th/9410167 ].[4] E. Witten,
String theory dynamics in various dimensions , Nucl. Phys.
B443 (1995) 85–126,[ hep-th/9503124 ].[5] K. Becker and M. Becker,
M-theory on eight-manifolds , Nucl. Phys.
B477 (1996) 155–167,[ hep-th/9605053 ].[6] M. Haack,
Calabi-Yau fourfold compactifications in string theory , Fortsch. Phys. (2002)3–99.[7] A. Kumar and C. Vafa, U-manifolds , Phys. Lett.
B396 (1997) 85–90, [ hep-th/9611007 ]. – 78 –
8] G. Curio and D. Lust,
New N = 1 supersymmetric 3-dimensional superstring vacua fromU-manifolds , Phys. Lett.
B428 (1998) 95–104, [ hep-th/9802193 ].[9] H. Lu, C. N. Pope, K. S. Stelle, and P. K. Townsend,
String and M-theory deformations ofmanifolds with special holonomy , JHEP (2005) 075, [ hep-th/0410176 ].[10] R. A. Coles and G. Papadopoulos, The geometry of the one-dimensional supersymmetricnonlinear sigma models , Class. Quant. Grav. (1990) 427–438.[11] B. Julia, Group disintegrations , . Invited paper presented at Nuffield Gravity Workshop,Cambridge, Eng., Jun 22 - Jul 12, 1980.[12] T. Damour, M. Henneaux, and H. Nicolai,
E(10) and a ’small tension expansion’ of Mtheory , Phys. Rev. Lett. (2002) 221601, [ hep-th/0207267 ].[13] H. Ooguri, C. Vafa, and E. P. Verlinde, Hartle-Hawking wave-function for fluxcompactifications , Lett. Math. Phys. (2005) 311–342, [ hep-th/0502211 ].[14] J. B. Hartle and S. W. Hawking, Wave Function of the Universe , Phys. Rev.
D28 (1983)2960–2975.[15] S. Sethi, C. Vafa, and E. Witten,
Constraints on low-dimensional string compactifications , Nucl. Phys.
B480 (1996) 213–224, [ hep-th/9606122 ].[16] P. Candelas and X. de la Ossa,
Moduli space of Calabi-Yau manifolds , Nucl. Phys.
B355 (1991) 455–481.[17] G. W. Gibbons, G. Papadopoulos, and K. S. Stelle,
HKT and OKT geometries on solitonblack hole moduli spaces , Nucl. Phys.
B508 (1997) 623–658, [ hep-th/9706207 ].[18] A. Bilal and S. Metzger,
Anomaly cancellation in M-theory: A critical review , Nucl. Phys.
B675 (2003) 416–446, [ hep-th/0307152 ].[19] A. Miemiec and I. Schnakenburg,
Basics of M-theory , Fortschr. Phys. (2006) 5–72,[ hep-th/0509137 ].[20] E. Cremmer, B. Julia, and J. Scherk, Supergravity theory in 11 dimensions , Phys. Lett.
B76 (1978) 409–412.[21] M. J. Duff, J. T. Liu, and R. Minasian,
Eleven-dimensional origin of string / string duality:A one-loop test , Nucl. Phys.
B452 (1995) 261–282, [ hep-th/9506126 ].[22] M. B. Green and P. Vanhove,
D-instantons, strings and M-theory , Phys. Lett.
B408 (1997)122–134, [ hep-th/9704145 ].[23] M. B. Green, M. Gutperle, and P. Vanhove,
One loop in eleven dimensions , Phys. Lett.
B409 (1997) 177–184, [ hep-th/9706175 ].[24] J. G. Russo and A. A. Tseytlin,
One-loop four-graviton amplitude in eleven-dimensionalsupergravity , Nucl. Phys.
B508 (1997) 245–259, [ hep-th/9707134 ].[25] S. P. de Alwis,
A note on brane tension and M-theory , Phys. Lett.
B388 (1996) 291–295,[ hep-th/9607011 ].[26] S. P. de Alwis,
Anomaly cancellation in M-theory , Phys. Lett.
B392 (1997) 332–334,[ hep-th/9609211 ].[27] J. H. Schwarz,
Superstring Theory , Phys. Rept. (1982) 223–322. – 79 –
28] E. Witten,
On flux quantization in M-theory and the effective action , J. Geom. Phys. (1997) 1–13, [ hep-th/9609122 ].[29] P. Candelas, A. M. Dale, C. A. Lutken, and R. Schimmrigk, Complete IntersectionCalabi-Yau Manifolds , Nucl. Phys.
B298 (1988) 493.[30] A. C. Cadavid, A. Ceresole, R. D’Auria, and S. Ferrara,
Eleven-dimensional supergravitycompactified on calabi-yau threefolds , Phys. Lett.
B357 (1995) 76–80, [ hep-th/9506144 ].[31] B. S. DeWitt,
Quantum Theory of Gravity. 1. The Canonical Theory , Phys. Rev. (1967)1113–1148.[32] C. N. Pope, lecture notes , . http://faculty.physics.tamu.edu/pope/ihplec.ps.[33] D. Roest,
M-theory and gauged supergravities , Fortsch. Phys. (2005) 119–230,[ hep-th/0408175 ].[34] J. Gillard, U. Gran, and G. Papadopoulos, The spinorial geometry of supersymmetricbackgrounds , Class. Quant. Grav. (2005) 1033–1076, [ hep-th/0410155 ].[35] T. Hubsch, Calabi-Yau manifolds: A Bestiary for physicists , . Singapore, Singapore: WorldScientific (1992) 362 p.[36] P. Candelas,
Lectures on complex manifolds , . In *Trieste 1987, Proceedings, Superstrings’87* 1-88.[37] P. S. Green, T. Hubsch, and C. A. Lutken,
All Hodge Numbers of All Complete IntersectionCalabi-Yau Manifolds , Class. Quant. Grav. (1989) 105–124.[38] S. Naito, K. Osada, and T. Fukui, Fierz identities and invariance of 11-dimensionalsupergravity action , Phys. Rev.
D34 (1986) 536–552.[39] J. W. van Holten, d = 1 supergravity and spinning particles , hep-th/9510021 .[40] W. Machin, Supersymmetric sigma models, gauge theories and vortices , hep-th/0311126 .[41] P. S. Howe, Superspace and the Spinning String , Phys. Lett.
B70 (1977) 453.[42] E. J. Martinec,
Superspace geometry of fermionic strings , Phys. Rev.
D28 (1983) 2604.[43] P. C. West,
Introduction to supersymmetry and supergravity , . Singapore, Singapore: WorldScientific (1990) 425 p.[44] J. Wess and J. Bagger,
Supersymmetry and supergravity , . Princeton, USA: Univ. Pr. (1992)259 p.[45] C. M. Hull,
The geometry of supersymmetric quantum mechanics , hep-th/9910028 .[46] J. Wess and B. Zumino, Superspace Formulation of Supergravity , Phys. Lett.
B66 (1977)361–364.[47] R. Grimm, J. Wess, and B. Zumino,
Consistency Checks on the Superspace Formulation ofSupergravity , Phys. Lett.
B73 (1978) 415.[48] J. Wess and B. Zumino,
Superfield Lagrangian for Supergravity , Phys. Lett.
B74 (1978) 51.[49] K. S. Stelle and P. C. West,
Minimal Auxiliary Fields for Supergravity , Phys. Lett.
B74 (1978) 330.[50] S. Ferrara and P. van Nieuwenhuizen,
The Auxiliary Fields of Supergravity , Phys. Lett.
B74 (1978) 333. – 80 –
51] M. F. Sohnius and P. C. West,
An Alternative Minimal Off-Shell Version of N=1Supergravity , Phys. Lett.
B105 (1981) 353.[52] P. S. Howe,
Super Weyl transformations in two-dimensions , J. Phys.
A12 (1979) 393–402.[53] M. F. Ertl,
Supergravity in two spacetime dimensions , hep-th/0102140 .[54] P. Breitenlohner and M. F. Sohnius, Superfields, auxiliary fields, and tensor calculus for N=2extended supergravity , Nucl. Phys.
B165 (1980) 483.[55] L. Castellani, P. van Nieuwenhuizen, and S. J. Gates,
The constraints for N=2 superspacefrom extended supergravity in ordinary space , Phys. Rev.
D22 (1980) 2364.[56] K. S. Stelle and P. C. West,
Algebraic derivation of N=2 supergravity constraints , Phys. Lett.
B90 (1980) 393.[57] S. J. Gates,
Supercovariant derivatives, super weyl groups, and N=2 supergravity , Nucl. Phys.
B176 (1980) 397.(1980) 397.