JHEP01(2020)195 ond c,d Springer , August 19, 2019 January 1, 2020 January 31, 2020 December 4, 2019 : : : : rgravity action on ing Vacua, Super- 1 heterotic structure in the large 2 Revised Received = Accepted G Published nctional of the corresponding N and Eirik E. Svanes s of d moduli problem of instanton b uilding, Published for SISSA by https://doi.org/10.1007/JHEP01(2020)195 . We make some comments on the role ′ is three dimensional maximally symmetric α 1) , (2 is a compact seven dimensional manifold with Matthew Magill M Y b [email protected] , [email protected] , , where Y × 1) , (2 . M 3 = 1 theory. We confirm that extrema of this functional corresp Magdalena Larfors, N a 1904.01027 manifolds. The Authors. 2 c Effective Field Theories, Flux compactifications, Superstr G

We dimensionally reduce the ten dimensional heterotic supe , [email protected] structure. In doing so, we derive the real superpotential fu 2 Strada Costiera 11, Trieste 34151, Italy E-mail: Department of Uppsala Physics SE-751 andL¨agerhyddsv¨agen, 20, Astronomy, Uppsala Sweden University, Department of Physics, KingsLondon College WC2R London, 2LS, U.K. Abdus Salam International Centre for Theoretical Physics, Mathematical Institute, Oxford University,Woodstock Andrew Road, Wiles Oxford B OX2 6GG, U.K. [email protected] b c d a of the superpotential functionalbundles with over respect to the couple Keywords: ArXiv ePrint: strings and Heterotic Strings volume, weak coupling limit to first order in to supersymmetric heterotic compactifications on manifold Anti de Sitter orG Minkowski space, and spacetimes of the form three dimensional Open Access Article funded by SCOAP Abstract: Xenia de la Ossa, Superpotential of three dimensional supergravity JHEP01(2020)195 ) 4 5 6 8 1 4 7 ′ 19 21 11 12 12 16 18 α ( O -flux and gauge H g from a mathematical per- eometry, ometric moduli of heterotic theory is an intrinsic part of for the understanding of the ally sheaves) over Calabi-Yau , such as Donaldson-Thomas uggest that a more complete od to offer advantages to the nfinitesimally, and indicates a e multiplets. On the downside, systems is required. ons where manifold singularities cal and quantum moduli spaces. luding corrections at order rious dimensions it is common to , etc. One then studies instanton bun- 2 G – 1 – system as a differential 2 G systems 2 G kinetic terms and curvature scales 3 structures 2 G The study of such coupled geometries is also very interestin A.1 Clifford algebra A.2 AdS 3.2 Three dimensional3.3 Einstein-Hilbert term Relation to a three dimensional superpotential 2.1 2.2 Heterotic 2.3 The heterotic 3.1 Gravitino mass terms bundle in an intricate way.compactifications This can implies not that be therich meaningfully gauge structure separated, and in even ge i modulieffective space physical that theory is of ontheoretical pivotal understanding space-time. importance of the These moduli subtleties space of s heterotic dles over these geometries,This and their often corresponding allows classi forinvariants in the the case definition of of holomorphic geometric bundles (or invariants more gener due to the Green-Schwarz anomaly cancellation) couple the g phenomenologist. This is largelythe due low-energy to theory, the in contrast fact to thatand Type a brane II gauge intersection compactificati patterns arethe needed supersymmetry to constraints engineer and gaug Bianchi identities (inc 1 Introduction Heterotic string compactifications have long been understo A Conventions 4 The superpotential and the5 supersymmetry conditions Conclusions and outlook 3 Dimensional reduction Contents 1 Introduction 2 Background spective. Traditionally, when investigating bundlesfix in a va background structure, such as Calabi-Yau, JHEP01(2020)195 ] ) It 3 35 1 , ]. 8 34 – , 6 ifications 15 )[ , Q ( 14 ]. The current ¯ 1 on the internal D )[ 15 way tend to have H ]. In light of such , Q 5 ( an SU(3) instanton 14 , ˇ 1 D 4 H l 7-manifold must admit ]. In these constructions, nsional space time to be ]. We will refer to such odaira-Spencer theory for round Calabi-Yau variety. 17 an SU(3) structure, where ic system of base and bun- to be Anti de Sitter (AdS 34 , xample of such an invariant s case, supersymmetry and which in general experiences rotic system is interesting in being nilpotent is equivalent – ppear and it is even debated -flux, and the gauge bundle t operator, ve rise to a certain nilpotent oduli of the system: they are ant is determined by a certain 16 ˇ 32 H ion appears complicated at first D ral. This approach has already structures [ g coupled moduli space to see if structures [ 2 m [ 2 G G system and this is simply referred to 2 ] for earlier work on the infinitesimal moduli of ] and G = 1 supersymmetric compactifications of ]. Again, there is a corresponding nilpotent 26 13 ], see also [ – – N 15 6 23 31 – – 2 – systems are captured by 27 2 . This perspective on the Hull-Strominger system G (which is, however, not an extension bundle), and the Q systems [ Q = 1 heterotic 2 ] and refs. [ = 1 supersymmetric compactifications of the heterotic G ]. 22 -flux, this reduces to the Calabi-Yau compactification of N – 15 H N 19 -flux. The simplest solutions of this type arise from compact = 1 heterotic system in [ instanton condition [ H 2 2 N G G ]. It is known, however, that the “invariants” defined in this on an extension bundle ] for a slightly different approach). Note that 3 on an associated bundle – ¯ 36 ˇ 1 D D ]. It can be shown that part of the supersymmetry constraints manifolds with an instanton bundle. In general, the interna -flux specifies the torsion, and the gauge bundle must satisfy structure, whose torsion is again specified by the 18 2 In this paper, we study In compactifications to four dimensions, H 2 The reader is referred to refs. [ G 1 G is particularly useful whencaptured determining by the classes infinitesimal in the m first cohomologies of the nilpoten as the heterotic the heterotic string are given by the Hull-Strominger syste to a slight generalisation of the (see also [ Hull-Strominger systems. configurations as operator is noteworthy how thisdeformations structure of of a the complex moduli manifold. space mimics K the geometry and bundle, specificallyoperator the F-term conditions, gi paper fits well within this program. supersymmetry and maximal symmetryMinkowski. constrain the Furthermore, four the dime internal 6-manifold must have issues, it appears moredle natural from to the start, studyany thereby the invariants investigating can coupled the be geometr correspondin definedthis for regard: the total although geometry. the Green-Schwarz Thesight, anomaly hete cancellat from a certainproven perspective useful it for is heterotic SU(3) actually structures rather [ natu residual dependence on theis background the structure. dimension of A the simplejumps moduli e space across of the a holomorphic complexIn bundle, structure general, moduli more complicated spaceif wall-crossing of meaningful phenomena the invariants can can backg a be defined in the case of infinitesimal deformations of heterotic varieties [ condition. When there is no a on must satisfy a ref. [ string to threemaximal dimensions symmetry in constrain the the three supergravity dimensional limit. geometry In thi or Minkowski, and the threecomponent dimensional of cosmological the const JHEP01(2020)195 systems 2 ], the finite G and Hessian 36 y a reasonably – W 34 , 15 , , structure. We provide 2 14  = 1 heterotic s, and in particular to go ϕ G rs [ of this paper. Presently, we N ∧ e, rely on holomorphicity of case, this superpotential has ide a good description of the these configurations, and the Hitchin functional in the text. pecifically, we have ϕ is is that the tools of complex ng this perspective leads to an as a tool to better understand ven by study the superpotential of the We intend to come back to this ce of the system, and its critical d rs the usual nonrenormalisation ature of supergravity theories is straints. While the infinitesimal hree dimensional superpotential nal effective action and based on for infinitesimal moduli. We also 2 1 t it reproduces all supersymmetry , superpotential ty there is less control of quantum W 2 s enough to determine a third-order is a constant which determines the / − 3 h ψ , M = 1 supersymmetry in four dimensions, = 1 supersymmetry in three dimensions ∧ Y W, determine the ) 2 and, correspondingly, restrict to the large N N ′ K/ h ϕ α ]. e ϕ, ψ + 11 to capture the low-energy physics of the super- = – 3 – 2 H / ( W 3  M systems at its critical locus. Computing its second φ 2 2 − G e ]. As required by Y Z 39 – 4 1 37 is the dilaton, and = φ W = 1 heterotic -flux on the internal 7-manifold H N , we can deduce the third. The gravitino mass can be computed b K is the H curvature scale, We thus expect the superpotential In future works, we hope to use the superpotential In this paper, we take the first step in a similar analysis of To obtain further information about heterotic moduli space 3 understanding of the structureholomorphicity of of the the superpotential finite provides constraint deformationsMaurer-Cartan of equation for finite moduli [ it is a holomorphic functional onlocus the corresponds off-shell to parameter the spa infinitesimal moduli space. Adopti where deformations are still poorlygeometry, understood. which One proved reason so for usefulthe superpotential, th in a the property four wequestion lose dimensional in in cas future three dimensions. publications. by determining the associated superpotential.a A relation general between fe gravitino mass and the superpotential. S so if we know any two out of three of the mass volume, weak coupling limitphysics. where supergravity should prov gravity regime of the heterotic string. However, corrections. We will comment morecontent ourselves on to this work in at the leading conclusions order in is less constraining than inis four necessarily dimensions. real, Notably, so the we t theorems. lack the Consequently, in holomorphicity three-dimensional that supergravi powe order variations will therefore reproducecomment on the the constraints relation between this superpotential and the additional evidence for this superpotential byconstraints showing tha of AdS moduli space of these systems have been studied in recent yea the finite-order moduli space and the resulting physical con potential the resulting form, we propose that the superpotential is gi straightforward dimensional reduction of the ten dimensio beyond the infinitesimal limit, iteffective is lower useful to dimensional determine theory. and been determined In in the refs. [ four dimensional JHEP01(2020)195 2 2 G G fact = 0, (2.1) (2.3) (2.2) (2.4) 2 . The ts that τ λ y if , which also is integrable determines a A Y ϕ ix 2 with totally antisym- G 2 ace of heterotic systems. G e-form with components: ] for more details on , where the Hodge dual is , to identify the physical ∇ 45 , ϕ erpotential for the effective – structure on ∗ ) ions of further evidence that it plays 2 ϕ 40 = ( . , G , irreducible representation of T 3 3 i ψ τ τ 14 ntial, it was important to correctly ∗ and give some indications of future = 27 − , 5 3 + ψ τ , structure if it admits a non-degenerate ijk ∗ ϕ 7 2 , 2 τ y ϕ ) ∧ 1 + ∗ G τ ψ 1 = ( ϕ τ + ), which give the structure equations for the − is in the T λ ∧ i ψ we compute the mass functional of the three 3 ϕ 1 – 4 – + 3 τ ijk ∧ = τ 0 ϕ, ψ 3 τ is orientable and spin. The form Γ 1 ψ structures in terms of spinors. Indeed, a seven † ψ τ 6 1 0 + 3 2 Y τ ∧ iλ G 1 ψ − = 4 = ]. In the case that the forms, τ ) = 0 τ ϕ ϕ ψ k 46 structure exists when the first and second Stiefel-Whitney ( 7 7 , = 4 = d d 2 structure if it admits a nowhere vanishing spinor T . We refer the reader to [ 34 ϕ are G ψ 2 ϕ , and a coassociative 4-form 7 7 – g k G d d Y τ 32 ]. , on it is shown that the critical locus of this functional does in 45 by briefly reviewing the inherent geometry and the constrain 29 structure admits a totally antisymmetric torsion if and onl , ϕ 4 , 2 metric . Such a 2 g ] the structure equations can be written 28 ϕ 2 28 G , [ 47 G 15 space. These technical calculations are recorded in append 14 are trivial, that is when 3 = 0) [ Y structures 2 τ , in fact defines a globally-defined, nowhere-vanishing thre ) is in fact the torsion of the unique metric connection 2 λ in the ϕ ( G 2 τ T The basic mathematical setting of this paper is the moduli sp Alternatively, we may discuss The exterior derivative of the forms ( structure, can be decomposed into irreducible representat 2 metric torsion [ 2 Background 2.1 A seven dimensional manifold is said to have a the part of a superpotential.applications. In We conclude order to in correctly section normalise identify the our superpote fieldsmass and, in since AdS wesummarises include our external conventions. space flux We begin in section string theory puts ondimensional them. gravitino and In extractphysics. section from this In a section reproduce candidate the known sup supersymmetry constraints, providing classes of and associative 3-form spinor, where the torsion classes We remark that a dimensional manifold has Riemannian metric and for G structures. (that is, taken using the JHEP01(2020)195 2 al G 2.1 (2.6) (2.9) (2.7) (2.8) (2.10) (2.11) and Θ satisfies B λ , A , (see section H Y be a vector bundle V ] is defined to be the satisfies an instanton 15 F ]. We call the resulting A , 34 , – and let  27 ϕ 3 (Θ)) system [ , A -field. The fields = 0 (2.5) ] 2 ed above. Furthermore, which is also an instanton 2 3 structure. B λ G , 2 Y ,H − CS + jk . , G ) ) Γ of A Θ) = 0 A ϕ d structure ( ( is the is identified with the torsion of the = 0 ijk ψ 2 whose curvature ∧ T T TY TY, ψ CS B ∧ G T ( ( A 8 1 A = , ∧ ′ )  4 α − – 5 – H F (Θ) ) and it is the totally antisymmetric torsion of a λ R + i V,A 2.3 ( D B , ) = tr ) (Θ), and A = ( = d CS λ system. A heterotic Y, ϕ 2 CS 2 H [( compatible with the G i G ) which preserve minimal supersymmetry. This requirement 2 ∇ G structure on the seven dimensional manifold ∇ 2 Y,V . We are interested in compactifications of the ten dimension G A systems 2 ) is the Chern-Simons form of the connection G ) is given in equation ( = 1 heterotic A ( ϕ (Θ) is the curvature of Θ. ( N CS R T is the Levi-Civita connection and is a gauge bundle with connection is a three form defined by is an integrable D be a seven dimensional manifold with a with connection where where with a similar definition for (unique) connection for definitions). are constrained such that V condition ϕ where Θ is a connection on the tangent bundle H Y • • • • Y on Let constrains the allowed geometry of the compactifications [ heterotic superstring on ( 2.2 Heterotic structure. where geometry an quadruple the following differential equation which is identified with the associative three-form discuss where JHEP01(2020)195 7 is = ) is and h Q Q ˇ (2.13) (2.12) (2.14) ( D ∗ f torsion: ), where ˇ Ω 2 G TY the torsion, elow that tor ) End( given by is a connection on ⊕ ) , i.e. the connection D 2 TY V Y,TY G ( and those corresponding ∇ ! 2 ) +1 ) p G y α alar part of the ( ( Ω ∇ = End( F F ⊕ , G

) . It acts on forms with values in ) G A Q ( n, and higher degree forms to zero. . connection Y, of ( , +1 F , one can construct the bundle p p F is the Levi-Civita connection. In fact, 8 +1 y ˇ + Ω p ) given by projecting two-forms to the 2 E d , Ω G BPS φ , A Q 0 is and connection on × ( system is equivalent to the existence of a d τ ) we have ∇ mp ∗ + ζ 8 2 1 1 3 n G 2 −→ Ω −→ 7→ ζ E G is the corresponding differential, and – 6 – = = Y, ) ⊆ ) ( = Γ and G 1 ˇ p h ) D τ Q Ω m ( G Y, . Q p n ( ( = d + T ∈ p ∗ ζ ˇ Ω ˇ − Ω D Ω α : is ⊕ ) ˇ ζ ), where D ! ˇ denotes the appropriate projection, and D y α , ), and π ) ) together with a vector bundle

system as a differential Y,TY Q 2 ( ( p ). In this subsection, we will recapitulate the definitions o ∗ Y, ϕ ] for more details and relevant references on this topic. G ˇ Y,TY ] that the heterotic Ω Q ( ( 15 p ∗ 15 , where Ω :Ω spinor. Note that the connection symbols in is constructed using the curvature 2 ∈ ◦ D F are not the same except when G y F π structure and vector bundle data, one can construct an opera +Θ is the connection on ζ are the connection symbols of the 2 = 1 supersymmetry imposes a relation between the dilaton and = A G ˇ mp D will be defined in terms of the external flux below. We will see b . For and consider the complex N = n G ) has structure group contained in h A = d + ), referring to [ ) which we will now define. V Given a manifold ( Explicitly, we have The map or ζ ⊕ G Q Q ( ( ∗ ∗ ˇ It was noticed in [ constant, i.e. coordinate independent. 2.3 The heterotic a sub-complex of Ω and that the external part of the flux is proportional to the sc Ω where differential complex ( Here, Finally, to d preserving the End( the torsion of the connection d Ω TY given by where Γ TY namely representation, three-forms to the singletUsing representatio the JHEP01(2020)195 ˇ D (3.2) (3.3) (3.1) space 3 . system, 2 structure.  G 2 G NPQ H ). We will start to  ds to a minimally , as defined in sec- Q ] particular that this Q 1 ( τ Ψ , ˇ 49 1 D P
, rise from the fermionic = 0 and the operator H Γ κ 2 48 ψ N e three dimensional super- ˇ , perpotential considerations D Ψ or Minkowski space time. µκ . ravitino mass in AdS e ten dimensional action that 37 Γ 3 ) b µ tic and mass terms h three dimensional gravitinos. + 6 l reduction once we have settled y [ x e that the dimensional reduction ψ d R tring on manifolds of m b Ψ , a + c F x . κ . Furthermore, the infinitesimal moduli 5 d ∧ ψ system with exact ν Q bc α W. 2 D F MNP QR K is nilpotent, i.e. tr( G e Γ ∧ ′ ˇ µνκ b 4 D M α = Γ y – 7 – Ψ 2 µ p p /  ψ 3 1) 1) 1 M − − 24 g structure and bundles satisfy the heterotic − ]: we will first determine all contributions to the three − 2 = ( φ √ 37 2 P , and then read off the superpotential from the relation ) = ( a G x − 2 ) y 3 Ψ / ( d α 3 N ( ] be a heterotic g e F M D Z F − ,H 2 3 √ 1 κ ] that the operator x Θ) 2 MNP 10 15 Γ d − = 1) effective field theory on either AdS TY, M 10 ( Ψ , N M )  Z systems were also shown to correspond to classes in is itself an instanton connection on 2 2 10 1 V,A D κ G ( 2 , the rather complicated heterotic Bianchi identity. Note in ) − . Compactification of the heterotic string on this system lea = Y, ϕ 2.2 The contributions to the three dimensional gravitino mass a In this section we use dimensional reduction to determine th ,f 0 S is a differential if and only if the It was then shown in [ tion with care. of heterotic dimensional gravitino mass supersymmetric ( 3 Dimensional reduction Let [( The three dimensional action of the gravitino contains kine have two three dimensional Clifford matricesFinding contracted such wit terms is straightforward, asthe normalisation is of the our dimensiona fields. Inresults particular, in we must a ensur canonical Einstein-Hilbert term and define the g and we can identify contributions to the mass from terms in th part of the ten dimensional action of heterotic supergravit potential arising from compactifications of the heterotic s implies that where including Our analysis follows the logic of [ of the current paper in the outlook part of section connect some threads between this moduli analysis and the su JHEP01(2020)195 ). ). ve 2.5 2.4 (3.5) (3.4) (3.6) ational we have 3 . ce of these ) θ  ) ⊗ λ , proportional to given by ( i structure of the β λ ϕ D 2 i ⊗ G Γ ν al three dimensional † ρ ( . iλ i α − D ( onal gravitino and hence ) +3 α . 1 β In the second line σ α three-form tive along the internal space +3 m the dimensional reduction r and Γ matrices discussed in de scale factors (summarised β . ted to the ⊗ 2 +3 the scale factor 2+2 in the next subsection. ) β i itino mass as G n/ ϕ 2+2 Γ 10 κ , 2+2 ∧ +3 n/ ⊗ φ n/ 2 +3 ijkl dϕ − φ +3 ( 2 e )(Id ), all three terms contribute to the three φ 7 dx − 2 2 ∗ ) e × ijk − 2 3.2 7 ekl e α = ϕ , appendix A], that σ g · ϕ ( +3 e ψ √ 15 ϕ ijk β y ⊗ y ν – 8 – ij T 7 ∧ -invariant vector and must therefore vanish. Ψ T 7 2 d dϕ . i Id 2+2 g 1 4 G A D dϕ n/ = ⊗ Z √ = y +3 − The ten dimensional gravitino kinetic term contributes Z µiν 7 νµ φ ijk  d Γ 2 1 8 dϕ ϕ ) -structure with positive µ − structure. It is therefore related to the exterior derivati 2 ν )(Γ − Z e Ψ 2 ρ 2 ijk G 7 φ 1 8 σ = 2 T g G µν † − 1 − θ satisfies the seven dimensional Killing spinor equation ( Γ √ e µ 3 M = ⊗ λ g ρ corresponds to a 10 (¯ † 1 g − 3 λ λ Some elements of the calculation and, in particular, our not i g M − √ Γ 2 ⊗ y † − √ 7 λ µ √ X ρ determines a xd x = 1 for this calculation, reinstating factors of (¯ 10 3 3 )) when decomposing the ten dimensional fields. The importan λ d d d 10 × κ Z Z Z . In particular, motivated by considerations of the canonic A.16 ; however, A = = is the torsion of the λ )–( i Γ T † λ We now recall that ) : A.12 A priori, we should also include a term from the derivative of We will set β 3 2 ϕ i D compact 7-manifold. which implies, after using an identity from [ of in ( factors will be discussed in more detail below. Consequently, Using this, we can rewrite the mass contribution as used the decomposition ofappendix the ten dimensional metric,Einstein-Hilbert spino term and gravitino kinetic term, we inclu Mass term 1 — threeto Γ-term. the three dimensionaldirections, mass i.e. by taking the covariant deriva dimensional gravitino mass term.and rewrite In the this respective section, contributions we in a perfor languageconventions adap are relegated to appendix In particular, this determines the contribution to the grav where In the last line,interpret we the find expression a in the quadratic square term bracket for as the mass three term. dimensi ( In dimensionally reducing the fermionic action ( 3.1 Gravitino mass terms JHEP01(2020)195 (3.9) (3.7) (3.8) ijk . Recog- . H A . ) θ  λω 3 ) contributes g λω 3 ⊗ ijk g 3.2 κξ 3 λ H g κξ 3 ) ⊗ g λ ν κσλ ρ ijk κσλ H Γ )( . † H 1  ) ever, in three dimensions σ dices are along the three iλ 1 4 θ  ). Consequently, the mass . ree dimensional spacetime, − ⊗ ψ − ssed in appendix α ⊗ ( ave indices along the internal itino, it does not yet have the  α ∧ λ NPQ +3 ijk )  β )), the mass-contribution in the +3 Γ ω H ⊗ H β ρ ( Q , λω (10) ω ⊗ σ 2.4 2+2 7 g 3 ρ Ψ Γ ∗ 2+2 g n/ ξ P )( ρ 2 − n/ κξ (10) Γ ×  ), namely (¯ +3 σ g β φ √ β N +3 Id)(Id 2 ijk φ 3.2 ⊗ Ψ +2 µν − +2 2 H ⊗ α κσλ e α φ Γ − ν Id 2 · e H +2 Id +2 − × Ψ 7 ϕ n ⊗ ω σµν n e α g 2 – 9 – 2 ǫ ⊗ 1 4 σ ijk ∧ Ψ − − 2 1 √ +3 2 σ Γ 2 − y β νµ H Γ − 7 )(Γ n/  n/ νµ 7 ξ 2 d The final contribution to the three dimensional grav- ∗ Γ = 2+2 Ψ +3 )(Γ σ +3 µ 10 † 2 φ φ Z φ σ n/ g 2 Z θ 2 σ 2 The second term in the fermionic action ( Ψ Γ † − − − 1 4 1 − +3 24 e ⊗ θ 1 e e 24 φ √ 7 †  2 7 1 4 = ⊗ g − ) λ g − X 2 † ν − e √  √ ρ ⊗ λ 10 3 7 , which we can rewrite as 6  3 M φ d g ξ g 2 g µν ⊗ ρ 10 ijk − − √ Γ (¯ − ] g Z µ e 3 µ H √ ρ g √  − ρ 50 (¯ 10 y (¯ 1 4 y − ijk g √ 7 7  3 ϕ − √ g − X 1 x d xd 24  y − √ 10 3 3 7 three-form in the spinor bilinear (cf. eq. ( − d d d √ × X xd x 2  10 3 3 Z Z Z G d d d × = = Z Z Z = = itino mass originates from the single Γ-term in ( Mass term 3 — single Γ-term. dimensional reduction gives When both ten dimensional gravitino indices are along the th where the computation again relies on the conventions discu While this is a quadraticright term Γ for matrix the three structurethe dimensional to grav Clifford be duality interpreted [ as a mass term. How nizing the last row contains to the gravitino massdimensional spacetime when and both the remainingspace. ten three That dimensional Γ-matrices is: h gravitino in contribution is Mass term 2 — five Γ-term. JHEP01(2020)195 )).  ) ν (3.10) (3.12) (3.14) (3.11) (3.13) λ . δ A.20 µ κ  , whence δ f λω 3 . Indeed, it − 7 2 g ν κ − κξ 3 δ A.2 g µ λ = can be shown to 3 δ ( g h ˜ h , . − . . λω se since we have to 3 ) ) √ g  f f  β κξ 3 , 7 7 ψ g σλκ ∗ ∗ +2 ordingly (cf. eq. ( orm
he appendix, this can be α ∧ H κ cetime will be proportional , which is shifted by a term (3) . ) poses that the scale factors ρ . arameter µ +2 + 2 + 4 3 terms of the flux parameter H th result β n σµν µκ g ∇ 2 hϕ . Moreover, ǫ supersymmetric solutions have ϕ ϕ 3 Γ ˜ h +2 − ∗ ˜ − h 1 8 µ 2 + α ∧ ∧ ve  ρ 1 8 √ n/ )). Rewriting the gravitino kinetic ) +2 2  H H H / ω n = ) +3 7 7 3 ρ + ω φ space, and we find these conventions ∗ ∗ + ( φ 2 ρ M A.19 µν 2 2 2 3 − µνσ ϕ − Γ for some µν e ǫ + ξ − − ∧ Γ ρ κ space discussed in appendix ξ f e (¯ ϕ ϕ (3) ρ ρ µνσ σλκ 3 7 β ν (¯ dϕ H ∧ ∧ g β 1 2 ˜ hǫ ∇ +2 3 ˜ h ǫ √ – 10 – α ∗ +2 3 − dϕ dϕ y = ( ( α 7 g µνκ 7 +2  d g n n n Γ 3! − n +2 2 µ + + √ σλκ n + √ Z − φ φ ρ 2 y φ 2 2 2 H 7 2 − 4 1 − − = 2 g d n/ − e e − e − n/ 7 7 We now simply collect together the three mass contribu- +3 Z 7 (3) Z Z φ = √ Z +3 2 1 4 H )) 1 8 8 1 3 φ x 3 − 4 1 2 3 . T − ): e ∗ − − − d 2  n 7 = e ν g = = A.21 7 2 Z ρ = 3.10 / g 2 2 √ 3 / / α µν 3 √ 3 3 : g 3 Γ M − ˜ (cf. ( g µ M M − (3) ¯ ρ = − √ (3) 3 H ) and ( √ 3 β g y H ∗ 7 y − 3.7 3 7 ∗ xd √ 3 2 xd x ), ( d 3 3 n/ d d 3 3.6 Z − Z Z e Clearly, the three-form flux along the three dimensional spa This is not, however, the mass contribution we would like to u = = We may use this to write the gravitino mass term in a succinct f Finally, the massf contribution is simplified if rewritten in useful for the analysisaccomplished using performed a in three dimensional this covariant paper. derivati As shown in t be related to Total mass contribution. tions ( where term in terms ofThe this result operator, is we that must the shift gravitino is the governed mass by term the acc action is possible to use conventionsa for the massless gravitino gravitino mass so both that in Minkowski and AdS to the totally antisymmetric form, This expression simplifies somewhat if we introduce the flux p after we have usedare that related consistency by of the Clifford algebra im proportional to the cosmological constant (cf. eq. ( account for the corrected kinetic term in AdS can be inserted in the dimensionally reduced action term, wi JHEP01(2020)195 ). + R s is not (3.16) (3.15) (3.18) (3.17) − A.12 e n → . and internal n R − bert term φ  e 2 3 . Using this fact n/ R R + 2 structure manifold. φ 2 2 n/ dilaton − the seven dimensional G +3 e φ 4 7 , namely the fluctuations 2 g v − l and its variation to zero, = 1 theory has a canonical e √ tional to ion nal fields around the vacuum y 7 7 g . It turns out that if pendence on the non-compact N d √ δV . . 3 ) Z g en dimensional metric as in ( 2 .  . − 2 ln n/ 3 3 3 . √ + R − R , independent of the non-compact co- R 7 φ X 3 3 y 2 g 3 g g g − δφ 10 √ e − − d − y , the Ricci scalar transforms as ( 7 g √ √ 7 + 4 √ d s Z g x x ), can only stay constant if x e n part of the torsion and we do not get agreement 3 3 3 = 1 theory we are striving to determine, the √ – 11 – Z 2 10 d d d 7 1 y must be constant internally as well. = κ → 7 3.17 A = 2 N Z Z d g n Z δn 2 3 V 2 3 2 10 1 κ Z 1 1 κ + κ 2 → − 2 = n R − v φ → 2 = to be a function of may depend on the coordinates of the − n e ). We can then define the constant volume scale n φ , defined by ( y ( 10 v g φ has to be constant in the sense that it does not depend on the − = n √ φ ), the dimensional reduction is: , we then recover the canonically-normalised Einstein-Hil X 10 /v d A.12 2 10 κ Z = 2 10 1 2 3 κ κ 2 ) about the vacuum expectation values of, respectively, the − Under a conformal transformation Indeed, in the three dimensional , where the ellipsis indicates irrelevant terms, not propor δφ, δV with the known BPS equation. Hence constant internally, then this changes the dynamical fields are given bysolution. fluctuations of Let the us ten focus dimensio on the two fluctuations that can change We recall that the dilaton In general, this necessitates a conformal rescaling of the t internal coordinates either. Indeed, as we will show in sect Note that this constrains ( BPS equations can beor derived equivalently the from gravitino setting mass the and superpotentia its variation to zero We here fix this factor. Setting However, three dimensional Lorentz invariancedimensions, forbids and a so de and equation ( Under such fluctuations, ordinates. In fact, As discussed above we require that the three dimensional 3.2 Three dimensional Einstein-Hilbert term volume ... Einstein-Hilbert term JHEP01(2020)195 ) 2 f is G n nical (4.1) A.22 (3.21) (3.20) (3.19) of the W duce that me by ( parameter space s of infinitesimal system discussed 2 2 . G G = 1 theory must be tential is a functional  . , ϕ N string on a heterotic   ∧ have tentatively identified ould be determined by a ϕ ϕ nsional Bianchi identity it terotic ϕ e = 1 supersymmetry in three ∧ ∧ d nicely matches the expected ϕ ϕ 1 2 heterotic s conclusion. N n d d − 2 2 1 1 of ψ − − ∧ ψ ψ ) ∧ ∧ W. ) ) h ϕ K n , + ]. e according to h ϕ h ϕ depends on the parameters. ≃ 15 H = + + ( n , W – 12 – 2 K  / H H 14 3 ( ( φ   M 2 − φ φ e 2 2 Y − − = 1 theories, this result has the following interpretation. ] is given by e e Z Y Y N K ,H Z Z e that, up to an overall constant, the superpotential 1 1 4 4 1 4 Θ) 3 = = = are essential for the three dimensional theory to have a cano systems of refs. [ TY, 2 ( n / 2 W W , and a superpotential 3 ) ). With this match in mind, we will from now on assume that G M K is related to the curvature of the three dimensional spaceti V,A ( is constant on the internal manifold. From this we can also de f , 2 7 δφ, δV . Our presentation uses machinery developed for the analysi ) ˜ h − 2.2 Y, ϕ = h is the dilaton field. In this section we show that this superpo , implying that the superpotential of the three dimensional n φ Using the above discussion of the Einstein-Hilbert term, we Finally, let us remark that the required variation of = 1 effective theory obtained by compactifying the heterotic ≃ in section dimensions, or equivalently, the conditions that define the K The above dimensional reduction fixes the gravitino mass to b must be constant. 3.3 Relation to a three dimensional superpotential spanned by ( In the next section, we will provide further evidence4 for thi The superpotential andWe the have supersymmetry shown in conditions section N system [( where and moduli of heterotic whose critical points give the conditions for preservation gravitational sector. This determines how constant in vacuum, so that In terms of threeWith dimensional our conventions,Hessian the potential gravitino mass of such theories sh form of a Hessian potential for the metric on the part of the he Thus, fluctuations of is a reasonablefollows identification. that Furthermore, by the ten dime JHEP01(2020)195 ∈ a x (4.3) (4.5) (4.2) (4.6) (4.4) (4.8) (4.7) d b a must be M H . . . Note that ) ) = } ϕ ψ b ( ( Θ . Note that this ˚ ˚ . M M M i i , ab odel with values in M TY ) = ) = δϕ , δψ . ith respect to the three ( ( ) ) , ψ 27 ϕ 27 ( ( φ, ϕ, ψ, B, A, { ˚ ˚ Θ)) M M that the variation of i δ i . onents of the three dimensional tric vacua by requiring that its and to first order ψ , π ϕ , π Θ) ϕ ) + ) + (Θ) y δ ) ∧ ψ ϕ R = 0, so in order to check that our ( ( Θ ϕ ) and that the variations corresponding y m m . ϕ as the two form corresponding to the tr( − i i ) y δσ m δW δϕ ( − ab + + m ( m ) − ϕ as a three form into irreducible representa- ψ M A δA = 0 (4.9) ) ) = 1 supersymmetry conditions, thus deserving ) = ( ) = δϕ tr( M M F δA Φ = 2 ′ ψ – 13 – ϕ δ 4 δW ( N ( α (tr (tr ab m (tr( m i 3 7 i ′ 7 4 − δg 2 α is the traceless symmetric part of ) = δB ) = + ) = ) = ˚ ϕ ψ M B = δψ δϕ ( ( ( ( B 7 7 M M i i = d is therefore given by , and = = does not contribute to irreducible representations exist for any differential form W ab δH 2 m ϕ , π δϕ ϕ , π δψ G M can be written in terms of a one-form valued in ) ) 14 metric invariant as ϕ are determined by the variations of π M M 2 ψ G (tr (tr 4 7 3 7 4 ) = ) = : 2 ) δψ δϕ G leave the ( ( 1 1 π π m We can view our three dimensional effective theory as a sigma m The variation of The critical locus of 7 Y,TY Similar partitions into π ( 4 1 , being (the dual of) a 3d field strength ought not be varied. where Here, in the second equality, we have defined Ω The Green-Schwarz anomaly cancellation mechanism implies decomposition is precisely the partition of functional truly reproduces the heterotic tions of dimensional scalars. In the present case, these are given by to be named superpotential, we must compute the variations w The variations of Note moreover that to h antisymmetric part of where, up to a closed form, we have defined some target space, locally parameterisedsupermultiplets. by the scalar The comp superpotentialvariations controls supersymme with respect to these scalars vanish, JHEP01(2020)195 = 0) (4.11) (4.10) (4.12) 2 τ + : ψ ψ W ∧ ∧ structure. Not- of ) 2  ) (Θ))] G hδϕ ϕ δW R instantons. ∧ gives + Θ 2 δ φ G δφ  δH . must be integrable ( tr( ) + d } , Y − + ( δϕ everal conditions, one for each ϕ Θ ) variation ), and integrated by parts the ) d ψ ∧   . ( 4.8 ϕ ϕ − ϕ ˚ . M ψ ∧ ∧ i δA F = 0 , δA, δ + d ϕ ϕ ∧ h ψ B − ψ = 0 ( , d d ) [tr( y ϕ 0 ′ ∧ 1 1 2 2 ) and ( ˚ ϕ ψ τ and Θ must be M, 2 ∧ (Θ) ( α d structure on , φ . 2 1 − − ) R m 4.7 δϕ A d 2 ) should be the supersymmetric locus in the 1 2 ) = 0 i δϕ 1 2 ψ ψ m − ψ G ) + + gives ( − + 4.9 (d φ h 7 ∧ ∧ ψ = 2 – 14 – B ϕ ) ) 5 2 1 φ δψ − 1 + y 2 = 0 = τ e ) and Θ vanish if and only if ∧ − is invariant up to a closed form. M ψ ϕ − h ϕ h ϕ M, π ψ e ) y d( tr hϕ A B tr + + ∧ d( H δψ 3 7 hϕ + φ correspond to the variations of the 1 7 F 2 H H ∧ δφ, ( ( + e = ) H { (   = 0 on the supersymmetric locus. . We demand that this expression should vanish for any H δW δϕ hϕ ∗ δφ δφ δϕ W Φ = + + ( 2 2 − B ∧ δ with respect to the dilaton variations with respect to − − H   and d +( δW δW φ φ with respect to 2 2 B − − e e W ) we find Y Y is gauge invariant, hence Z Z 2.2 B = = δW Φ represents any of the variations δ The vanishing of The vanishing of For consistency, the locus specified by ( The remaining terms in Note that d 5 which are the conditions that the connections The variations of ing that Then, using ( and that terms containing d parameter space of heterotic strings. Consider a first order variations of the field, andindependent so the field equation variation. decomposes into s where Note that this implies that which is equivalent to the statement that the where, for the second equality, we have used ( JHEP01(2020)195 (4.13) and the .  . W ]. We leave ) 3 structures on ψ τ whose critical 13 ( 2 − , ˚ ϕ M G i 10 ∧ ) = , = 1 supersymmetry 9 )] H pectation, however, is ( N erpotential torsion. } H 27 ( 2 = 0. δψ 27 about the gauge structure G π } ∧ W ) , + tential on the moduli space of terature [ δψ ψ , π s form is very analogous to the 3 , y τ c compactifications is related to ∧ h ϕ ) 0 ] is a functional . τ ) φ 0 + 43 3 1 τ M ) + [ h ϕ are equivalent to requiring a heterotic 3 1 H + d ψ = tr ( 1 ψ . + = τ W ψ y + ( m 3 1 i H ∧ τ ) give ψ , − ∧ δϕ  − ∧ ) + ( )] , h 4.12 ϕ ∧ ∗ H 0 H ( ) = ( τ ) = – 15 – δϕ ) ( Y 1 φ , h 7 1 6 H Z H π ∧ d ( π ( h ϕ ) 7 7 1 2 + structures when restricted to closed + π + ) = ) ϕ 2 h ψ = ψ ψ H y 1  G y ( + ) τ h 1 φ ϕ , π 4.11 φ ϕ π d d +  ∧ 0 h − − ]. The Hitchin functional [ τ φ 1 ϕ − ) together with 7 3 τ 51 d 0 − τ + d 2.2 3 7 ϕ − ∗ + [(3 d (  {  − ( φ φ 2 2 { ) = − − φ e e 2 H = 0 gives the supersymmetry conditions necessary for ( − Y Y 1 e Z Z structures relevant to heterotic compactifications. The ex π Y δW 2 = = Z G Finally, let us remark briefly on the relation between the sup Summarising our results, the critical points of holonomy manifolds [ system (cf. section 2 2 Hitchin functional The first of these equations together with ( that this functional, consideredthe in metric the on context the of modulicorresponding heteroti space SU(3) of heterotic structure structures, functionals as defined it in the li points correspond to torsion-free or the a fixed cohomology class. In this context it has nothing to say which in the literature isG sometimes referred to as a superpo That is, and the second, together with ( this for future research. Hence we write the remaining terms as Thus, both the internal and external flux is determined by the G in three dimensions. Moreover, on the supersymmetric locus JHEP01(2020)195 – ept ] or 52 and , 17 n , 37 rections , 16 , and the 33 Q , = 1 heterotic , Theorems 6 superpotential 32 15 N = 1 supergravity vacua. In this paper imensional hetorotic N hese 3 structure manifold (see ns is a very interesting systems may offer more 3d, 2 erotic configurations. For = 1 supergravity, the su- 2 onal Hull-Strominger sys- ry resulting from heterotic G G ns between the order olume, weak coupling limit. N r dimensional systems: both f ative corrections. It follows stem only allows Minkowski e pointed out, however, that been argued [ iyah-like bundles relates to the four dimensional id for the infinitesimal moduli heterotic string [ solutions, for example the rela- es of supersymmetric [ structure and take a particular provides a new tool that can be W 3 2 systems and Strominger-Hull sys- G 2 W . This argument shows that correc- , between heterotic systems and the ′ -valued cohomology classes. An im- ′ for the three dimensional Yang-Mills G α α Q W vacua. spacetime and , whose cohomology is then related to the expansion, which is physically expected but 3 ˇ 3 D ′ α systems. The crux of the matter is that, given – 16 – 2 G = 1 compactifications of the heterotic string on manifolds ]. The superpotential N 35 systems allow both Minkowski and AdS 2 G , to the infinitesimal moduli space of heterotic systems are k ′ α ]. Such a study is also expected to throw light on possible cor 39 – 37 ). It would be interesting to use the leverage provided by the A.2 structure in the large volume, weak string coupling limit. T 2 expansion, the heterotic Bianchi identity imposes relatio G ′ The results of this paper may also be of interest for other het Another important difference between heterotic systems are three dimensional analogues of the four dimensi systems, both around Minkowski and AdS in more detailed studies of the low energy effective field theo α ]. A recent study of such connections between three and four d + 1 terms, enabling a bootstrapping procedure in 2 2 protection than supersymmetry alone. In particular, it has decompactification limit in order to connect to different typ perpotential may receive boththat our perturbative results and are valid nonperturbEven only so, so there long are as some we signs stay that in the the string large theory v origins o lacks nonrenormalisation theorems and so, in contrast to 4d example, we may restrict the torsion of the heterotic tems is the reduced amount of supersymmetry. In particular, space. How thisdirection may for change future when research. we consider finite deformatio not mathematically guaranteed, and in addition, is only val n used to explore this relation, for example by studying how an G under control by thethe heterotic argument Bianchi presumes identity. the existence It of should an b supersymmetry-breaking four dimensional solutions of the systems can be found in ref. [ superpotential [ and 7] that there is an equivalence, at all orders in W tions, perturbative in 56 In this paper, wesupergravity have that proposed results a from superpotential 5 Conclusions and outlook nilpotency of a certain differential operator, with infinitesimal moduli space of heterotic tem. There is acorrespond striking to similarity between certain the nilpotent three operators and fou on associated At G infinitesimal deformations are captured by certain appendix tion between the curvature scales of the AdS we have reported on some of the properties of these AdS portant difference, however,vacua, is whereas that heterotic the Hull-Strominger sy JHEP01(2020)195 2 er G (5.1) (5.2) ). This G , denotes the ) 3 i Φ , ) imposes that, ), and that the δ h ( Q ( 5.1 O 1 tions, it is useful to vanish on a locus in ] is non-supersymmetric + ˇ Ω of the superpotential , and the three-dimensional 2 58 W , viewed as a functional Φ X , with ( δ vanish on a locus in field ) 0 X ∈ W this regard, we recall the 3 Φ and the trace on | itesimal moduli of heterotic W X al ions. These supersymmetry ( h that the higher order terms W . Further, a generalised gauge finitesimal moduli of heterotic potential. We have shown, in 2 O , i.e. TY otential for the Hull-Strominger Q δ system if and only if , has the formal expansion Q + 2 1 0 2 on 0 section 7 of ref. [ G ψ and ). One can wonder what the higher 0 ϕ Φ + ˇ g D δ 5.2 0 , -valued one-form Φ | Q DX i ∧ 6 , ]. = 0 δW ] that a generic deformation of the heterotic 59 hX -exact one-forms, so the infinitesimal variations . – 17 – – 1 0 ˇ ˇ ] + D DX 15 S φ 0 , but it is less clear if the structure of higher order 57 systems. Let us therefore end with a discussion of 2 ′ × guarantee that second order variations of the super- − α 2 [Φ 6 e 4 G ]. In the same paper, it was argued that this structure W M ]. Z 15 = 11 1 2 )[ for the definitions of W (given by the metric = Q ( ˇ 1 Q D 2.3 ) look like. In particular, we would like to understand wheth W 3 H ] + ∆ ∆ 0 X ( [Φ O X ∈ W ]. As a consequence, the second order variations of will reproduce these infinitesimal constraints on the varia 15 [Φ] = W W , that the first order variations of the superpotential 4 . Another interesting avenue is to explore the relation with In more detail, it was shown in ref. [ Although the results of section Some comments are in order concerning eq. ( The main motivation for our study is, however, the relevance systems [ systems. However, note that the three-dimensional theory derived in W 6 2 2 deformation preserves the conditions of the heterotic system can be thought of as a one-form with values in there is some parametrisationin of the the finite superpotential deformations expansion suc truncate at finite order. In space that accurately reproducesconstraints contain the the supersymmetry conditions condit G used to determine the in natural inner-product on where the first two terms vanish by supersymmetry. Agreement where we refer to section higher order, and eventuallysection finite, variations of the super order deformations and arises from a compactification on a CY to second order in the variations, we have up to a constant potential is expected to persist to all orders in parameter space that correspondsG to (a subspace of) the infin Chern-Simons theories studies in refs. [ for finite deformations of heterotic deformations will enjoy similar protection. where we rewrite the variation in terms of the sketch how this comes about. A variation of the superpotenti of transformation turns out to give rise to expression is analogous to thesystem finite derived variation of recently the in superp ref. [ are represented by of the fields Φ, around the supersymmetric locus Φ = Φ JHEP01(2020)195 - ) 7 3 7 X g L ⊕ 3 ˜ g fined for ], though 4 structure. Y, nfigurations 2 × G 3 M uncil (VR) under ]. Donaldson and n would include an 11 manifolds [ 2 1/2, so it is the order nd Marjorie Schillo for he given -structure manifold (see pported by a grant from G 2 ]. This is also the natural ve from quartic terms in RC grant EP/J010790/1. z Institute for Theoretical G k was initiated. We would 11 ase that the corresponding the form ( se, we might expect that a mation of the superpotential tions that the superpotential marginal in four dimensions, otential [ Yau manifolds equipped with at the invariants defined using ra can be described as an ion of the superpotential in al [ d for analog of holomorphic Chern-Simons ential terms give rise to quartic 2 tions within holomorphic Chern- G holonomy, which appear to be well 2 as an action which can be quantised. G W structures. We hope to test this conjecture 2 ) is an integrable 7 G – 18 – algebra. We hope to address these questions in Y, g 4 L ], and it has been suggested that similar invariants may be de ] where the one-loop partition function of the systems provide a string theoretic setting that includes co 3 – 60 2 1 G ] and [ is maximally symmetric and ( 5 3 M do not suffer from an anomalous dependence, as their definitio Another interesting direction is to view ∆ In three dimensional theories the chiral field has dimension See e.g. [ 7 W theory was computed. A Conventions We are interested in heterotic supergravity backgrounds of illuminating discussions. XDThe is research supported of in MLgrant part number and 2016-03873 by MM and the is 2016-03503.the EPS financed Simons The Foundation work by (#488569, of the Bobby ESS Swedish Acharya). is Research su Co Acknowledgments The authors would like toPhysics express (MITP) a for special itsalso thanks hospitality to like and the to support Main when thank this Anthony wor Ashmore, Mateo Galdeano Solans, a in the future. integration over the full parameter space of consisting of instanton bundles on manifolds with suited for the study of∆ invariants. Indeed, one might hope th the Hull-Strominger system usingSegal the have suggested corresponding that superp similarit invariants has could be been define debated to what extent such invariants depend on t expectation from a physical point ofterms view, in as the cubic four superpot dimensionalsince action. there the Such mass terms dimension areand of exactly a its chiral variation field vanish is implies 1. that The condi the deformation algeb Simons theory [ Indeed, the famous Donaldson-Thomasholomorphic invariants sheaves of can Calabi- be interpreted as correlation func deformation algebra now becomes an six terms in thethe action superpotential. that become Inparametrisation of marginal. analogy the with deformations These thetruncates exist will where Hull-Strominger at deri the ca quartic expans order. Similarly, it might also be the c The heterotic could be parametrised as a Chern-Simons like cubic function Hull-Strominger system where it was shown that a finite defor where the near future. algebra. JHEP01(2020)195 × 3 M (A.7) (A.5) (A.1) (A.2) (A.3) (A.4) es. ]: , while seven . It has the 32 [ with the precise (10) MN 8 3 g ˜ g mensional metric is n µ, ν, . . . = − e ons with an arbitrary are real. This choice is ¯ N ¯ ) osition on the space := M 2 (3) pace Clifford matrices: η noted 3 σ ) (A.6) ginary and as a consequence ¯ g 2 N N with the canonical Einstein- e have non-zero cosmological ⊗ σ E . ric, Id ⊗ ¯ M ) M 1 ⊗ , under the group decomposition , E Id , σ ¯ which algebra the Γ matrices are in. They 2 1 ) M i ⊗ i ¯ ¯ µ (3) σ σ ⊗ . Γ ⊗ ⊗ ,E µ (3) ¯ µ µ (7) i ]. ! ¯ µ µ e i Γ ! ˜ e (Γ ( Id i (7) ¯ 32 0 2 2 2 − Γ ⊗ ⊗ , we give our definition of the kinetic operator n/ n/ i n/ 0 1 1 0 0 ; ten dimensional coordinates are labeled by ⊗ e − − ¯ ( µ (3) e e

A.2 – 19 – = = = (Id . = = 3 = Γ = Id −→ 1 2 g i, j, . . . σ σ ¯ ¯ µ (10) (10) i ¯ M M ¯ µ (10) i (10) ¯ Γ Γ E i Γ Γ i ¯ ¯ µ µ E E = for this calculation and discussion). = i (10) 3.2 µ (10) Γ Γ SO(7) we have the Clifford algebra decomposition be the ten dimensional vielbein, × to be purely imaginary matrices, while the Γ ¯ M M 2) are the Pauli matrices , E (7) ). A bar on the indices indicates that they are flat space indic 2 i σ , y SO(1 µ where we define a heterotic structure). The physical three di ֓ x and is the vielbein associated to ( ← 1 ¯ µ µ 2.2 e σ → 9) , ) This decomposition gives the relationship between curved s Throughout this paper, three dimensional indices will be de This algebra decomposition induces a Clifford bundle decomp In the next section, we specify the Clifford algebra conventi The subscripts in parentheses are to explicitly keep track of M 8 . Indeed, let X where decomposition Y rescaling dictated by comparingHilbert the term (see dimensional section reduction SO(1 for the three dimensionalconstant. gravitinos, which is subtle when w a conformal rescaling of the restricted ten dimensional met will not appear in the main text. where A.1 Clifford algebra Starting with the flat space 10d Clifford algebra, Γ section made to be consistent with the conventions of [ We will assume that the tenwe dimensional choose matrices the are Γ pure ima ( dimensional indices are indicated by conformal scaling. After that, in appendix JHEP01(2020)195 0 (A.9) (A.12) (A.13) (A.14) (A.16) (A.10) is a definite , and seven is a ten di- ρ θ , be the spinor 10 λ S , where variable θ ifying the three dimen- , only the three dimen- , ⊗ , where λ 7 ional spinor, are in the same ten dimen- with decomposition: ⊗ 10 µ al Clifford matrices. s distinct from other metric nsider the seven dimensional ρ ρ 2 ⊗ S M ional fermion, uced induce scaling here, too. , , parent. ional chirality. We will only be 3 gravitino to be light, relative to 1 ) S , ,..., 2 . and σ ) = 0, keeping every conformal scale onal gravitino kinetic term obtained from † θ ∼ = ψ nos, Ψ = 0 = 3 θ 2 β µ ⊗ σ ) (A.15) ⊗ 10 + 1 M M λ θ S † ⊗ 9 σ 2 (A.8) s the Einstein-Hilbert term is. λ α 2 (A.11) = = ⊗ ⊗ Id ⊗ n/ 7 µ ⊗ λ n/ . We note that in the final equation, the Γ g ρ − ⊗ µ ( i (7) β ⊗ ρ 2 ⊕ Γ := (¯ − µ µ (3) – 20 – for more details). := θ µ 3 n/ α θ i ρ β Γ g ⊗ e ( + α ⊗ ⊗ n α β β 2.1 e e e e 2 = i ¯ λ λ −→ = = = Id = = n/ ⊗ µ ⊗ ¯ µ µ µ − 10 Ψ ρ ρ ¯ Ψ Ψ g is the dilatino. As a consequence, we find that the correct µ (10) i (10) = ( Γ Γ ψ α ∼ ¯ contribution; although M Ψ α , we write the ten dimensional spinor as λ structure (see section 2 adds the G 0 Γ † a positive-chirality two dimensional spinor. In this paper θ Gathering the above, we have the following conventions: However, we have to be a little bit careful in correctly ident As we did for the Clifford algebra above, we introduce an extra Next, we explicitly realise the isomorphism It can be checked that with this definition, the three dimensi ¯ Ψ = Ψ 9 sional gravitino since theWe can conformal deduce factors the we correct factor have by introd observing that Γ dimensional reduction is canonically normalised, so long a contributions, we introduce To keep the scaling factors associated with Clifford matrice factor explicit makes the dimensional reduction more trans decomposition to the three dimensional gravitino is: sional supermultiplet, where in where we should recall that in order to keep the origin of each factor explicit. chirality two dimensional spinor specifyinginterested in the ten ten dimensional dimens positive-chirality graviti as the factor in the relation between ten and three dimension with sional gravitinos are relevant andgravitino. we Furthermore, will in have order no forthe cause the compactification to three scale, co dimensional we require that the seven dimens defining the mensional Majorana-Weyl spinor space. Given a three dimens dimensional spinor, JHEP01(2020)195 and θ (A.18) (A.17) ⊗ η ⊗ ρ = is not necessarily ǫ 3 M ) = 0 θ µ , and can be determined Γ ree dimensional Γ-duality 2 ⊗ we write a η , chapter 6]. The first thing c space ishing superpotential. The (3) 2 , that annihilates the Killing ]): ticular, with this convention Killing spinor equation, then µ − H ⊗ 61 A.1 62 ing the physical mass terms as . , we will discuss the curvature ∇ 3 ρ cal kinetic operator, in the sense s a mass-like source term in the th this subtlety by absorbing the ( ∗ e,  ρ . = 0 . µ ǫ = 2 3 Γ Id) . Hence the Killing spinor equation g . σ  α ρ , ⊗ Γ 2 − (3) e µ NP √ H = 0 Γ Id σ Γ νκµ (3) 3 2 a ρ Γ ⊗ ∗ H H = 0  = σ µ 3 νκ 3 ǫ νκσ δ MNP – 21 – ∗ ρ ǫ Γ νκσ µνκ  ǫ H 1 4 α (3) µ = 3 H 1 8 2 νκ = 2 g e D = Γ ( νκ 3 νκ − − ρ Γ , appendix B]: Γ µ √ M α νκµ µνκ µνκ 50 2 D D H H H e = 8 8 8 1 1 1  νκσ solutions. µνκ − − − ǫ 3 µ µ µ H D D D νκ    Γ ⇒ ⇒ = = kinetic terms and curvature scales space, the Killing spinor equation is (see e.g. [ 3 3 is related to the cosmological constant, Λ, via Λ = a In AdS The two Γ-matrices are related to a single Γ-matrix via the th becomes: where we have recognised spinors. This operator can bethat identified as its the eigenvalues correct physi correspondsupersymmetric to solutions the physical have mass.point a of massless In view par gravitino pursued here and was van learnt from Cecotti’s book, [ from the ten dimensional Killing spinor equation: It follows that: use this to identifyscales the of correct the heterotic kinetic AdS operator. Secondly we will do is identify the precise cosmological constant and Using the Clifford algebra decomposition given in appendix where We are studying backgrounds in which theMinkowski. maximally symmetri Since a non-trivialKilling cosmological spinor constant equations, give weopposed must to be mass-like careful kinetic contributions. aboutcosmological identify We will constant deal into a wi redefined covariant derivativ A.2 AdS compute: that can be found in Polchinski [ JHEP01(2020)195 ). ), we where A.19 (A.19) (A.21) (A.23) (A.20) (A.22) 4.13 (3) f . H ,( 2 ∗ 0 compactifica- τ n/ 3 3 = e variant derivative f = . f , metric. There exist κ 2 ρ 3 ; | | 2 ast infinitesimally, one 3 2 µκ | G 10 g H 3 Γ g τ − µ 27 ice. This can also be com- | , since the kinetic operator − ten dimensional metric, i.e. parated AdS ρ π √ 2 1 sor. | α calar field, p 2 1 2 − . e = 2 µ − , | Γ 2 2 (3) and the three dimensional cosmo- as follows: . On the other hand, the internal | τ µκ α 2 | 0 ]. The scalar curvature, for instance, 2 φ H µνλ 2 Γ τ . 1 2 7 ǫ e (3) 3 n/ 45 d − . α ∗ | κλ 3 H 2 − n (3) g 3 fe e = 4 4 1 2 e 30 ∗ 1 2 | H 2 ν νσ 3 1 (3) + 3 τ g f Γ | = ∗ H κ 1 2 µρ 3 = 1 heterotic system is: Λ + 3 a ρ is a natural object from the three dimensional g 4 1 µνκ n ∗ ν − ǫ – 22 – − N + 30 2 1 D − e ρσκ (3) 2 = 0 3 µ τ ]. = H ν Λ = fε µνκ  3 D 8 Γ 3 a Γ 21 ∗ 63 2 g µ 21 := ρ 3! − + µνκ  µ 1 Γ . √ = ¯ τ τ − ∇ † y κ , we find that = d φ τ ρ ) gives mass the interpretation na¨ıve of the extra term in ( curvature scale is set by 7 n ν is proportional to the internal torsion component := 3 − ∇ µνλ 2 f | ǫ = 12 3.12 = τ | = 6∆ µνκ R α µνλ Γ R µ H ¯ ρ 3 g 3! − √ now identified we can define the physical three dimensional co includes the extra term proportional to: 10 is the seven dimensional Laplacian defined with the = a κ 7 Ψ (3) ν H ∇ Since the different components of the torsion decouple, at le Note that this shift really does contribute a mass-like term With As a consequence, the kinetic term in the 3d action is: Finally, we note that although Using the fact that 3 Here, we use notation ∗ 10 µνκ could hope to tune the internal torsion and produce a scale se tion. We leave this possibility to explore in future. curvature depends on all components of the torsion, [ where we have used theputed using three the dimensional GAMMA Γ-matrix package duality [ tw comparable expressions for the Riemann tensor and Ricci ten logical constant is Γ In particular, since 2 where ∆ in particular, the scalar curvature for an can observe that the AdS is given by: Comparing to equation ( Therefore, we must have the Hodge star iswithout taken the with conformal respect factor. to This the is restriction related of to the perspective, our results are simplified by introducing the s JHEP01(2020)195 , 2 ty Finite ommons ∞ ]. L ]. P. Tod and , ]. SPIRE nd bundles on ]. IN , [ ]. SPIRE arXiv:1610.09836 holonomy manifolds ]. IN SPIRE , ]. SPIRE (2017) 539 2 IN ][ arXiv:0902.3239 , in proceedings of the IN G SPIRE , ][ IN 369 SPIRE ][ SPIRE IN honor of Sir Roger Penrose’s (1986) 253 redited. ]. IN , p. 31 [ Canonical metrics on ][ ]. nstable and E.E. Svanes, ][ ]. -manifolds r, 2 c Chern-Simons and an G SPIRE B 274 The Universal Geometry of Heterotic SPIRE IN , Ph.D. Thesis, University of Oxford, [ SPIRE Math. Ann. IN IN , ]. ][ arXiv:1704.08717 ][ [ -folds in Infinitesimal moduli for the Strominger Holomorphic string algebroids 3 arXiv:1607.03473 [ math.AG/9806111 [ SPIRE A Metric for Heterotic Moduli The Infinitesimal Moduli Space of Heterotic G Infinitesimal moduli of Nucl. Phys. arXiv:1402.1532 IN [ , – 23 – ][ arXiv:1605.05256 Algebroids, Heterotic Moduli Spaces and the [ (2018) 727 (2016) 016 (2000) 367 arXiv:1803.01873 Gauge theory in higher dimensions 360 Connections, Field Redefinitions and Heterotic Supergravi , 11 arXiv:1806.08367 (2014) 037 ]. [ ), which permits any use, distribution and reproduction in 54 arXiv:1810.00879 [ ]. (2017) 567 Gauge Theory in higher dimensions, II 07 JHEP SPIRE 356 , SPIRE arXiv:1409.3347 IN [ IN JHEP [ ][ , (2018) 179 (2019) 038 CC-BY 4.0 Superstrings with Torsion 10 J. Diff. Geom. This article is distributed under the terms of the Creative C 02 Gauge theory on Calabi-Yau manifolds A Holomorphic Casson invariant for Calabi-Yau three folds a , Oxford, U.K., 25–29 June 1996, S.A. Huggett, L.J. Mason, K. , (2014) 008 Conjectures on counting associative Commun. Math. Phys. ]. ]. JHEP , JHEP 12 , , fibrations SPIRE SPIRE 3 arXiv:1503.07562 IN IN Strominger System Systems arXiv:1807.10329 Vacua Commun. Math. Phys. deformations from a heterotic superpotential: holomorphi JHEP system and Killing spinors[ in generalized geometry [ [ Oxford U.K. (1997). Conference on Geometric Issues in65th Foundations Birthday of Science in S.T. Tsou eds., Oxford University Press, Oxford U.K. (1998) holomorphic Courant algebroids with instanton bundles K algebra [1] R.P. Thomas, [5] D. Joyce, [6] L.B. Anderson, J. Gray and E. Sharpe, [8] M. Garcia-Fernandez, R. Rubio and C. Tipler, [9] P. Candelas, X. de la Ossa and J. McOrist, [7] X. de la Ossa and E.E. Svanes, [3] R.P. Thomas, [4] S. Donaldson and E. Segal, [2] S. Donaldson and R.P. Thomas, [14] X. de la Ossa, M. Larfors and E.E.[15] Svanes, X. de la Ossa, M. Larfors and E.E.[16] Svanes, A. Strominger, [12] M. Garcia-Fernandez, R. Rubio and C. Tipler, [13] P. Candelas, X. De La Ossa, J. McOrist and R. Sisca, [10] M. Garcia-Fernandez, R. Rubio, C. Shahbazi and C. Tiple [11] A. Ashmore, X. De La Ossa, R. Minasian, C. Strickland-Co any medium, provided the original author(s) and source are c References Open Access. Attribution License ( JHEP01(2020)195 ]. s ] ion in , , , SPIRE ] IN , ]. ][ ]. , ]. , -models (1986) 357 structure σ SPIRE SPIRE Structures and IN SPIRE ]. [ IN hep-th/9502009 2 B 178 IN SU(3) [ ][ G ]. ][ and geometry of (2011) 032 7 arXiv:1409.7539 SPIRE non-linear [ IN Deformations of Linear 10 ]. 2) (2018) 067 arXiv:1102.0011 SPIRE ][ , [ ]. 2) structure moduli spaces with ]. IN ]. , (0 Vacuum Configurations for 09 -Heterotic String Solitons ]. (1995) 169 Phys. Lett. , ][ (0 , SPIRE JHEP IN , SPIRE SPIRE SU(3) math/0112201 (2015) 837 SPIRE [ ][ IN SPIRE IN JHEP ]. arXiv:1709.06974 Spin(7) IN B 351 arXiv:1010.0255 ][ , IN [ 19 ][ ]. ]. , ][ 2 Stability Walls in Heterotic Theories Stabilizing All Geometric Moduli in Stabilizing the Complex Structure in The Atiyah Class and Complex Structure ][ G (2011) 106011 Massless spectrum for hybrid CFTs SPIRE Superstrings with intrinsic torsion Nigel Hitchin’s 70th Birthday Conference (2003) 1 SPIRE SPIRE IN math.DG/0102142 Restrictions of Heterotic Exploring [ [ IN IN 48 D 83 – 24 – arXiv:1402.1751 ][ ][ Phys. Lett. [ (2011) 088 math/0312094 , [ hybrid models 02 2) , Heterotic domain wall solutions and (1985) 46 arXiv:1205.6208 hep-th/0302158 [ arXiv:1001.2104 [ (2002) 303 [ -Instantons and arXiv:1110.1886 [ On marginal deformations of ]. 6 Seven-dimensional octonionic Yang-Mills instanton and it Phys. Rev. JHEP ]. (2014) 221 (2005) 79 , , Parallel spinors and connections with skew-symmetric tors Killing spinor equations in dimension B 258 88 Adv. Theor. Math. Phys. J. Geom. Phys. G-structures and Domain Walls in Heterotic Theories SPIRE , , 259 , in proceedings of the SPIRE arXiv:1005.5302 arXiv:0905.1748 IN [ [ (2012) 099 IN (2011) 044 ][ (2011) 529 ][ 08 (2004) 086002 07 Asian J. Math. manifolds structures Nucl. Phys. , , Madrid, Spain, 5–16 September 2016, , 2 2 B 705 Compactifications of the Heterotic Superstring D 69 JHEP (2011) 151 (2009) 026 JHEP G G , , ]. ]. ]. 01 09 SPIRE SPIRE SPIRE -models IN IN arXiv:1712.04976 arXiv:1107.5076 IN string theory Superstrings (Hitchin 70) integrable [ JHEP Phys. Rev. Commun. Math. Phys. extension to an heterotic[ string soliton Phys. Lett. Proc. Symp. Pure Math. [ Stabilization in Heterotic Calabi-Yau Compactifications [ Heterotic Calabi-Yau Vacua Heterotic Calabi-Yau Vacua [ JHEP σ manifolds integrable Instanton Connections [35] X. de la Ossa, M. Larfors and E.E. Svanes, [28] T. Friedrich and S. Ivanov, [18] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten [34] X. de la Ossa, M. Larfors and E.E. Svanes, [32] A. Lukas and C. Matti, [33] J. Gray, M. Larfors and D. L¨ust, [29] T. Friedrich and S. Ivanov, [30] J.P. Gauntlett, D. Martelli and D. Waldram, [31] P. Ivanov and S. Ivanov, SU(3) [27] M. and G¨unaydin H. Nicolai, [25] M. Bertolini, I.V. Melnikov and M.R. Plesser, [26] M. Bertolini and M.R. Plesser, (0 [23] M. Kreuzer, J. McOrist, I.V. Melnikov and M.R.[24] Plesser, I.V. Melnikov and E. Sharpe, [20] L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, [21] L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, [22] L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, [19] L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, [17] C.M. Hull, JHEP01(2020)195 , 55 , Compt. ]. ]. ]. . (2004) 126009 ]. , 2 G J. Diff. Geom. SPIRE Spin(7), , SPIRE , Cambridge University IN D 70 SPIRE IN , ou ][ ]. IN ][ (1987) 525 2 ]. ][ ]. G . structures and the Strominger 126 , SPIRE 2 arXiv:0709.2987 IN SPIRE [ SPIRE -manifolds G Phys. Rev. ][ 2 IN , Cambridge University Press, IN , ]. G ][ ][ Five-branes wrapped on SLAG three , Oxford Mathematical Monographs, -manifolds: Intermediate Jacobians and ]. 2 ]. hep-th/0110034 ]. G [ SPIRE Moduli stabilisation in heterotic string Ann. Math. Moduli Stabilising in Heterotic Nearly arXiv:1210.5933 IN [ , Moduli of (2009) 297 [ arXiv:1107.3573 [ math.DG/0305124 SPIRE SPIRE , IN 99 SPIRE [ IN The Heterotic Superpotential and Moduli IN ][ – 25 – hep-th/0507173 ][ ]. [ Heterotic Calabi-Yau Compactifications with Flux (2001) 018 Superstring Theory . (2013) 015 arXiv:0709.1932 (1989) 439 [ Bundles over Homogeneous Nearly-K¨ahler Spaces in arXiv:1606.05221 Heterotic String Compactifications on Half-flat Heterotic on half-flat 11 (2011) 100 [ -structures SPIRE 01 2 The Quartic Effective Action of the Heterotic String and Hodge Theory for 09 IN G ][ Dolbeault cohomology for . B 328 (2006) 005 JHEP Riemannian manifolds with structure group (1982) 19 ]. , JHEP 03 , JHEP (2007) 081 32 arXiv:1607.01219 , (1966) 127. (2018) 1031 arXiv:1305.0594 arXiv:1509.08724 [ [ 12 SPIRE 7, (1998) 57 IN 262 Proc. London Math. Soc. JHEP 108 ][ , , 70 Nucl. Phys. , JHEP String theory. Volume 2: Superstring theory and beyond The geometry of three-forms in six and seven dimensions Some remarks on Metrics with exceptional holonomy , Compact manifolds with special holonomy On the Effective Field Theory of Heterotic Vacua math.DG/0010054 Sur riemanniennes les `agroupe d’holonomie vari´et´es (2013) 034 (2016) 049 09 01 hep-th/0408121 Abel-Jacobi maps Rendus Acad. Sci. Press, Cambridge U.K. (2007). Oxford University Press, Oxford U.K. (2000). system in dimension Heterotic String Theory JHEP Compactifications K´ahler compactifications Manifolds. II Geom. Dedicata cycles and related geometry Annali Mat. Pura Appl. Lett. Math. Phys. [ JHEP (2000) 547 [ Supersymmetry Cambridge U.K. (1987). [42] M. and Fern´andez A. Gray, [51] S. Karigiannis and N.C. Leung, [45] R.L. Bryant, [55] M. Klaput, A. Lukas and E.E. Svanes, [56] M. Klaput, A. Lukas, C. Matti and E.E. Svanes, [52] B. de Carlos, S. Gurrieri, A. Lukas and A.[53] Micu, S. Gurrieri, A. Lukas and A. Micu, [54] M. Klaput, A. Lukas and C. Matti, [50] J. Polchinski, [47] M. and Fern´andez L. Ugarte, [48] M. Green, J. Schwarz and E. Witten, [46] J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, [43] N.J. Hitchin, [44] D.D. Joyce, [39] J. McOrist, [40] R.L. Bryant, [41] E. Bonan, [37] S. Gurrieri, A. Lukas and A. Micu, [38] X. de la Ossa, E. Hardy and E.E. Svanes, [36] A. Clarke, M. Garcia-Fernandez and C. Tipler, [49] E.A. Bergshoeff and M. de Roo, JHEP01(2020)195 , , , gebra and Fierz , ualities e-dimensions? strings ]. 2 G ]. ]. SPIRE IN [ Open SPIRE SPIRE IN IN [ supersymmetric Chern-Simons ][ solutions in Heterotic Supergravity = 2 ]. 3 N ]. AdS ]. SPIRE (1993) 3371 hep-th/0105086 IN SPIRE – 26 – , ][ IN [ SPIRE A 8 arXiv:0902.0655 [ IN [ Remarks on the Chern-Simons theories with supersymmetries in Supersymmetric (1992) 72 , Cambridge U.K. (2015). (2009) 631 (1992) 1805 hep-th/0611080 [ B 281 122 Int. J. Mod. Phys. A 7 , Does heterotic string generate Chern-Simons action in thre Supersymmetric Field Theories: Geometric Structures and D (2008) 012 GAMMA: A Mathematica package for performing gamma matrix al Phys. Lett. , 02 theories Prog. Theor. Phys. JHEP Cambridge University Press three-dimensions Mod. Phys. Lett. transformations in arbitrary dimensions [63] U. Gran, [60] J. de Boer, P. de Medeiros, S. El-Showk[61] and A. S. Sinkovics, Cecotti, [62] H. Kunitomo and M. Ohta, [58] H. Nishino and S.J. Gates Jr., [59] H. Nishino, [57] S.J. Gates Jr. and H. Nishino,