Compactification of the Heterotic Pure Spinor Superstring I
Osvaldo Chandia, William D. Linch III, Brenno Carlini Vallilo
aa r X i v : . [ h e p - t h ] A ug YITP-SB-09-21
Compactification of the Heterotic Pure SpinorSuperstring I
Osvaldo Chand´ıa ♠ , William D. Linch, iii ♣ and Brenno Carlini Vallilo ♠ ♠ Departamento de Ciencias F´ısicas, Universidad Andres BelloSazie 2315, Santiago, Chile ♣ C.N. Yang Institute for Theoretical Physics and Department of MathematicsSUNY, Stony Brook, NY 11794-3840, USA
Abstract
In this paper we begin the study of compactifications of the pure spinor formalism forsuperstrings. As a first example of such a process we study the case of the heterotic stringin a Calabi-Yau background. We explicitly construct a BRST operator imposing N = 1four-dimensional supersymmetry and show that nilpotence implies K¨ahler and Ricci-flatnessconditions. The massless spectrum is computed using this BRST operator and it agreeswith the expected result. email: [email protected] email: [email protected] email: [email protected] ontents N = 1 Ten dimensional Supersymmetry . . . . . . . . . . . . . . . . . . . . 52.3 Complex and K¨ahler Geometry Using Frames . . . . . . . . . . . . . . . . . 8 d -operator algebra and BRST Charge 104 Physical State Conditions and Spectrum 13 Since the realization that string theory could give rise to anomaly-free chiral theories,compactifications have been studied in many different contexts in attempts to make contactwith the observed four-dimensional world. The process of compactification usually involvesbreaking of the extended supersymmetry present in higher dimensional supersymmetrictheories.The compactification procedure for the RNS superstring is well-known for backgroundswith pure NS fields. If one wants to include RR fields in the case of Type II string the-ories, then worldsheet methods are not available and one is forced to study it using onlysupergravity.In the case of the RNS superstring, compactifications to Calabi-Yau manifolds and theirorbifold limits are standard knowledge in the field and many interesting physical propertiesare derived using worldsheet methods. Alternative descriptions of the RNS superstring incompactified backgrounds, known as hybrid formalisms, were developed for two [1], four[2], and six [3] dimensions by Berkovits and collaborators. Since the roots of the hybridformalism are in the RNS superstring, it was not known until recently [4] how to studycompatifications with RR fields. One of the interesting aspects of the hybrid formalismapproach to RR flux compactifications is that the N = (2 ,
2) superconformal algebra, anessential ingredient in standard CY compactifications, is still preserved. On the otherhand, a drawback of it is that it is not known how the procedure works if the startingpoint of the compactification is not a CY manifold. Furthermore, computations involvingcompactification-dependent states are subtle, and appropriate care should be taken in thiscase. One of the authors (BCV) would like to thank Massimo Bianchi and Pierre Vanhove for pointing out theseproblems and for discussions on these issues. or these reasons we would like to have another formalism in which it is possible tostudy more general flux compactifications. The pure spinor formalism [5] is the appropriateone. However, the pure spinor formalism has superspace coordinates corresponding toall supersymmetries and curved superspaces are not known explicitly except for maximallysymmetric cases and the recent construction of the full Type IIA superspace for AdS × C P [6]. For the eleven-dimensional case a systematic procedure was developed in [7]. Althoughone could use this procedure, four-dimensional supersymmetry arguments are more effectiveto attack the present problem.Compactifications of the pure spinor formalism is the theme of this paper. As a firststep toward more general backgrounds in heterotic and type II theories, we will studycompactifications of the heterotic string on a Calabi-Yau 3-fold. The pure spinor formalismwas studied in cases with reduced supersymmetry previously in [8, 9, 10, 11, 12]. What ismissed by some of these previous works is the input from the geometry of the Calabi-Yauand the full pure spinor constraint from ten dimensions. These two ingredients give extraterms to the BRST charge and these extra terms allow us to derive on-shell equations forthe four-dimensional multiplets. Chiral Superspace and Chiral Coordinates
As an example of the constructionin the next sections let us consider first the simple case of N = 1 four-dimensional su-persymmetry. In this case the superspace coordinates are given by ( x α ˙ α , θ α , θ ˙ α ). Thesupersymmetric derivatives are given by D α = ∂ α + iθ ˙ α ∂ α ˙ α , D ˙ α = − ∂ ˙ α − iθ α ∂ α ˙ α . (1)It is well-known that a consistent non-trivial constraint on superfields is D ˙ α Φ = 0. Theeasiest way to solve this constraint is to realize that the chiral variable y α ˙ α = x α ˙ α + iθ α θ ˙ α is annihilated by D ˙ α , i.e. D ˙ α y β ˙ β = 0. We then construct superfields depending only on( y α ˙ α , θ α ). Furthermore, the supersymmetric derivatives and supercharges in these variablesare given by D α = ∂ α + 2 iθ ˙ α ∂ α ˙ α , D ˙ α = ∂ ˙ α ,Q α = ∂ α , Q ˙ α = − ∂ ˙ α + 2 iθ α ∂ α ˙ α , (2)and we have that Q α y β ˙ β = 0. This means that any background field Φ( y α ˙ α ) is invariantunder chiral supersymmetries. Of course, in Minkowski signature, it is not possible toconsider theories invariant only under the anti-chiral supersymmetry. (In a Euclidean sig-nature the chiral and anti-chiral supersymmetries are not related by complex conjugationand such a symmetry is consistent.) The fact that the y α ˙ α is not real forces us to includeits complex conjugate and one should also consider functions which are not holomorphic in y α ˙ α . This means that chiral coordinates are not useful for reducing supersymmetry sincewe cannot have theories constructed only on subspace parameterized by y α ˙ α . (Of course Note that we could also consider dependence on θ but that is not a physical superfield, that is, not arepresentation of the supersymmetry algebra. he superpotential is a function on this subspace and one can use holomorphicity to provenon-renormalization theorems but there is also the D -term.)It turns out that higher dimensional superspaces also have chiral-like variables. Wewill see that after we break ten-dimensional Lorentz invariance, type I supersymmetryin ten dimensions will have “chiral” variables invariant under four-dimensional N = 1supersymmetry. Organization
In the next section we introduce the pure spinor formalism and discussgeneral concepts that are useful in later sections. In section 3 we construct curved-space d -operators and the BRST charge for a complex six-dimensional internal manifold and showthat nilpotence and four-dimensional supersymmetry require that the internal space is aCalabi-Yau manifold. Section 4 contains a discussion of the spectrum obtained from thecohomology of the BRST operator constructed in section 3. The final section containsfuture directions and open problems. In this section we discuss preliminary material needed for later sections. We begin witha short review of the pure spinor formalism. After that we discuss type I supersymmetrypreserving only four dimensional Lorentz symmetry. We close this section with a review ofcomplex geometry using frames.
The action of the heterotic string in a flat background is given by S = Z d z [ 12 ∂X b m ∂X b m + p b α ∂θ b α + ω b α ∂λ b α ] + S λ + S R , (3)where ( X b m , θ α ) parameterize the D = 10, N = 1 superspace and p b α is the fermionicconjugate momentum. S λ is the action for the pure spinor λ b α which is defined to satisfythe constraint λγ b m λ = 0 for b m = 0 to 9 . (4)Although an explicit form of S λ in terms of λ and its conjugate momentum ω requiresbreaking SO (9 ,
1) (or its Euclidean version SO (10)) to a subgroup, the OPE of λ b α withits Lorentz current N b m b n = ωγ b m b n λ is manifestly SO (9 ,
1) covariant. The condition (4)implies that ω is defined only up the gauge invariance δω b α = Λ b m ( γ b m λ ) b α , (5) or any Λ b m . Finally, S R is the action for the right-moving degrees of freedom which describethe reparametrization ghosts and the heterotic fermions.It is useful to define the supersymmetric operators in terms of the free worldsheet fields d b α = p b α − (Π b m − θγ b m ∂θ )( γ b m θ ) b α , Π b m = ∂X b m + θγ b m ∂θ, (6)which satisfy the OPE’s d b α ( y ) d b β ( z ) → − γ b m b α b β Π b m ( y − z ) − , d b α ( y )Π b m ( z ) → ( γ b m ∂θ ) b α ( y − z ) − . (7)The BRST operator and left moving stress energy tensor are given by Q = I λ b α d b α , T = − ∂X b m ∂X b m − p b α ∂θ b α + T λ (8)where λ b α carries ghost-number 1. Nilpotency is easily checked using the OPE’s (7) andthe pure spinor condition (4). It can be shown that the cohomological conditions give theequations of motion and gauge invariances of linearized N = 1 , D = 10 supergravity.In the right moving sector we have the heterotic fermions, Ψ A , and the reparametrizationghosts, ( b, c ). The action for them is given by S R = Z d z [Ψ A ∂ Ψ A + b∂c ] . (9)The right moving energy momentum tensor is T = − ∂X b m ∂X b m − b∂c − ∂ ( bc ) + T A , (10)where T A is the c = 16 stress energy tensor coming from the heterotic fermions. Finally,the right moving BRST charge is given by Q = I ( cT + c∂cb ) . (11)Physical vertex operator should be in the cohomology of both Q and Q .The action in a general curved background can be constructed by adding the integratedvertex operator to the flat action of (3) and then covariantizing with respect to the back-ground super-reparametrization invariance. The result of doing this is [13] S = Z d z
12 Π b a Π b b η b a b b + 12 Π b A Π b B B b B b A + d b α Π b α + ω b α ∇ λ b α + Ψ A ∇ Ψ A (12) d b α J I W b αI + λ b α ω b β J I U I b α b β + S F T + S bc , where Π b A = ∂Z c M E c M b A and J I = K I AB Ψ A Ψ B with the K s denoting the generators of thegauge group. The covariant derivatives are defined as ∇ λ b α = ∂λ b α + λ b β Ω b β b α , ∇ Ψ A = ∂ Ψ A + A I K I AB Ψ B , where Ω b β b α = Π b A Ω b A b β b α , A I = Π b A A I b A with Ω b A b β b α being the background connection forLorentz and scaling transformations, and A I b A the connection for background gauge trans-formations. The Fradkin-Tseytlin term S F T is given by S F T = 12 π Z d zr Φ , (13)where r is the world-sheet curvature and Φ is the dilaton superfield. Although this term isnot necessary for having a covariant action, it is required to have a quantum conformallyinvariant sigma-model action [14] [15]. N = 1 Ten dimensional Supersymmetry
We are interested in a background preserving N = 1 supersymmetry in four dimen-sions. The corresponding supersymmetric derivative algebra is a sub-algebra of the tendimensional supersymmetric derivative algebra. A 16-component, 10-dimensional spinordecomposes into → ( , ) + ( , ) (14)representations of SL (2 , C ) and SU (4). We will denote the four-dimensional coordinatesas x a or x α ˙ α and the six dimensional coordinates by y i where the index i goes from 1 to6. To relate vector and spinor representations of the Lorentz group we use standard sigmamatrices. The six dimensional sigma matrices are σ IJi where
I, J = 1 , . . . , SU (4)spinor indices and sigma is antisymmetric in I and J . These sigma matrices are related tothe ones with indices down by σ iIJ = 12 ǫ IJKL σ i KL . (15)Other useful identities that the six dimensional sigma matrices satisfy are σ IJi σ iKL = δ IK δ JL − δ IL δ JK , σ IJi σ i KL = ǫ IJKL . (16)The 16 supersymmetries are now parameterized by complex spinors ( η Iα , η ˙ αI ) and theworldsheet spinor variables are now ( θ Iα , θ ˙ αI ). The supersymmetry transformations of thebosonic variables are δx m = iθ I σ m η I − iη I σ m θ I (17) y i = iθ α σ i η α − iη ˙ α σ i θ ˙ α , (18)where we suppressed the index contractions with the sigma matrices. As in four dimensionsit is useful to consider y IJ = y i σ IJi , y IJ = y i σ iIJ , (19)subject to the reality condition( y IJ ) † = 12 ǫ IJKL y KL , (20)inherited from (15).Since we are interested in preserving only N = 1 supersymmetry in four dimensions,we split the SU (4) index to ( i , · ) where i = 1 to 3 and the · denotes a singlet under the SU (3) subgroup of SU (4). Now the odd superspace variables are ( θ α , θ ˙ α , θ i α , θ ˙ α i ) and thesupersymmetry transformations are given by δx m = iθσ m η − iησ m θ + iθ i σ m η i − iη i σ m θ i , (21) δy i = iθ α i η α − iθ α η i α − iǫ i j k η ˙ α j θ ˙ α k , (22) δy i j = iθ α i η j α − iθ α j η i α − iǫ i j k η ˙ α k θ ˙ α + iǫ i j k η ˙ α θ ˙ α k . (23)Note that if we write y i j as y i = ǫ i j k y j k the reality condition is just ( y i ) † = y i whichmeans that ( y i , y i ) are usual complex coordinates. In the standard SU (4) → SU (3) × U (1)decomposition, the spinors θ i have U (1) charge − and the singlets θ have charge (andthe opposite charges for the conjugate spinors). This is reflected in the supersymmetrytransformations above since y i has +1 charge. In this notation, the algebra of supersymmetric derivatives in flat space is given by { d α , d β } = 0 { d α , d ˙ α } = − i∂ α ˙ α { d ˙ α , d ˙ β } = 0 { d α , d β i } = − iε αβ ∂ i { d α , d ˙ α i } = 0 { d ˙ α , d ˙ β i } = − iε ˙ α ˙ β ∂ i { d α i , d β j } = − iε αβ ǫ i j k ∂ k { d α i , d ˙ α j } = − iδ i j ∂ α ˙ α { d ˙ α i , d ˙ β j } = − iε ˙ α ˙ β ǫ i j k ∂ k . (24)A realization of this algebra in terms of the superspace coordinates is given by d α = ∂ α + iθ ˙ α ∂ α ˙ α + iθ i α ∂ i ,d ˙ α = − ∂ ˙ α − iθ α ∂ α ˙ α − iθ i ˙ α ∂ i ,d α i = ∂ α i + iθ ˙ α i ∂ α ˙ α − iθ α ∂ i − iǫ i j k θ j α ∂ k d ˙ α i = − ∂ ˙ α i − iθ α i ∂ α ˙ α + iθ ˙ α ∂ i + 2 iǫ i j k θ j ˙ α ∂ k (25) Taking care to keep track of the U (1) charges, we can raise and lower all SU (3) indices at will with theunderstanding that we only apply Einstein summation convention when the index carriers have opposite U (1)charges. ince we are in flat space, there exist corresponding supercharges which commute withall these supersymmetric derivatives. However, as we will not need their full expressionhere, we will not write them.The interesting property of the realization (25) using the notation described earlier isthat there exist chiral-like coordinates analogous to the four-dimensional case described inthe introduction: z i = y i − iθ α i θ α , z i = y i − iθ i ˙ α θ ˙ α . (26)These are invariant under the SU (3) singlet supersymmetries generated by ( η α , η ˙ α ) butunlike the four-dimensional case, we can consistently consider functions of ( z i , z i ) and stillhave Minkowski signature in spacetime. Furthermore, when written in these variables therealization (25) simplifies to d α = ∂ α + iθ ˙ α ∂ α ˙ α + 2 iθ i α ∂ i ,d ˙ α = − ∂ ˙ α − iθ α ∂ α ˙ α − iθ i ˙ α ∂ i ,d α i = ∂ α i + iθ ˙ α i ∂ α ˙ α − iǫ i j k θ j α ∂ k d ˙ α i = − ∂ ˙ α i − iθ α i ∂ α ˙ α + 2 iǫ i j k θ j ˙ α ∂ k , (27)where now the derivatives ( ∂ i , ∂ i ) are taken with respect to ( z i , z i ). Note that the algebra(24) is preserved. Furthermore, in these new variables, the corresponding supercharges forthe supersymmetries generated by ( η α , η ˙ α ) are given by q α = ∂ α − θ ˙ α ∂ α ˙ α , q ˙ α = − ∂ ˙ α + iθ α ∂ α ˙ α , (28)which means that the variables ( z i , z i ) are invariant under the SU (3) singlet supersymme-tries. Note also that the new “chiral” variables ( z i , z i ) are annihilated by d ˙ α z i = 0 , d α i z j = 0 , (29)and this is consistent with the algebra (24). In other words, the constraints d ˙ α Ψ = d α i Ψ = 0 (30)on a general superfield Ψ are integrable.In what follows, we will assume that our background fields depend only on these vari-ables. Since the supercharges in (28) are independent of ( z i , z i ) any background constructedwith them will be invariant under this N = 1 supersymmetry. This also means that a back-ground preserving this amount of supersymmetry is naturally almost-complex. Of coursewe still have to check that the background is on-shell. This will be the subject of section 3where we will generalize the realization (27) to a curved six-dimensional background. .3 Complex and K¨ahler Geometry Using Frames The appropriate language to construct the pure spinor superstring sigma model in ageneral background uses frames. Since we want to study the heterotic string in a Calabi-Yau background, it is useful to review complex and K¨ahler geometry in this language. Thereader familiar with this material, or willing to accept the interpretations of the relevantformulæ given in the subsequent sections, can skip ahead to section 3. This discussion isbased on the definitions and conventions of [16].A tangent complex index will be denoted by i , as in the previous subsection, and acoordinate (or “curved”) index will be denoted by i . In a complex manifold of dimension n a hermitian metric is given in local coordinates by ds = g i j dz i ⊗ dz j . (31)The Riemannian metric on this manifold is given by Re( ds ) and the imaginary part of ds is given by ω = ig i j dz i ∧ dz j , (32)and is called the associated (1 , coframe is definedby two matrices ( E i i , E i j ) such that ds = g i j dz i ⊗ dz j = E i i E i j dz i ⊗ dz j = E i ⊗ E i , (33)where E i = E i i dz i and E i = E i i dz i . Using the coframe, the associated (1 , ω = iE i ∧ E i . (34)As usual, the exterior derivative is d = ∂ + ∂ = dz i ∂ i + dz i ∂ i . We can compute the exteriorderivative of the coframe giving dE i = ( dE i i ) ∧ dz i = [( ∂E i i ) E i j − E i i ∂E j i ] ∧ E j + T i , (35)where T i is a (2 , T i = ( ∂E i i ) E i j ∧ E j + ( ∂E j i ) E i i ∧ E j (36)with E i i = ( E i i ) − . The complex manifold is K¨ahler if T i = 0. Equation (35) can bewritten in the form dE i = Ω i j ∧ E j + T i (37) A bar over an antiholomorphic index will not be used unless it is necessary. here Ω i j = ( ∂E i i ) E i j − E i i ∂E j i and satisfies Ω + Ω † = 0. Such a connection Ω iscompatible with both the metric and complex structure. To see this more clearly, note that d ( E i ∧ E i ) = T i ∧ E i − E i ∧ T i . (38)From this last equation we can also see the standard definition of a K¨ahler manifold, thatis, dω = 0 if T i = 0. The equations above allow us to define covariant exterior derivatives ∇ and ∇ given by ∇ = ∂ + ( E − ∂E ) i j , ∇ = ∂ − ( ∂EE − ) i j . (39)With this definition we can say that E i is covariantly holomorphic ∇ E i = 0 , (40)while the holomorphic covariant exterior derivative ∇ defines the torsion ∇ E i = T i , (41)In the case of vanishing torsion, these last two equations say ∇ i E i j = 0 , ∇ i E i j = ∇ j E i i , (42)where the second equation translates to the usual ∂ i g j k = ∂ j g i k . One should be careful tonote that the definition of covariant derivatives acts differently on the frames E i and E i , i.e. ( ∇ ) † = ∇ . This is because the connection Ω defined above is skew-hermitian:Ω + Ω † = 0 → ( d + Ω) † = d − Ω , (43)so the analogous expressions for the covariant derivatives (39) for E i have opposite signsand we have, in the case of vanishing T i , ∇ i E i j = 0 , ∇ i E i j = ∇ j E i i . (44) Curvature
We can define new covariant derivatives using the inverse of the coframematrices. ∇ i = E i i ∇ i , ∇ i = E i i ∇ i . (45)Also, note that because of (42) we have ∇ i E j i = 0 , E i j ∇ i E k i = E i i ∇ j E k i . (46) sing the new covariant derivatives above we can rewrite these expressions as ∇ j E j i = 0 , ∇ i E k j = ∇ j E k i . (47)As usual, the curvature is defined from the commutators of covariant derivatives. Sincethe manifold is hermitian, we have [ ∇ i , ∇ j ] = 0. The same will be true for ∇ i precisely ifthe second equation in (47) holds. So, in terms of the new covariant derivatives the K¨ahlercondition is [ ∇ i , ∇ j ] = 0. We will see how this condition arises from nilpotence of theBRST charge in section 3.The non vanishing part of the curvature matrix can be defined as R i j = [ ∇ i , ∇ j ] . (48)Since the first equation of (47) holds we have that R i j = E i i E j j R i j = [ ∇ i , ∇ j ] . (49)Due to all the symmetries the curvature matrix has when the manifold is K¨ahler, there arethree equivalent ways to write the Ricci-flatness condition. The first is the usual Ric = 0,the second is Tr( R i j ) = 0 and the last one is δ i j R i j = δ i j [ ∇ i , ∇ j ] = 0. Again, in section 3we will see how this last equation appears from nilpotence of the BRST charge. Vector bundles
Since we are studying the heterotic string, we know vector bundlesalso appear in the theory and couple to the background in a non-trivial way. Consider thatour manifold comes with additional structure given by a gauge 1-form A = A i E i + A i E i = A Σ i T Σ E i + A Σ i T Σ E i , (50)where T Σ are the gauge algebra generators. We generalize the covariant derivatives aboveto include this gauge 1-form connection D i = ∇ i − A i , D i = ∇ i − A i . (51)Computing again the conditions that give K¨ahler and Ricci-flatness [ D i , D j ] = 0 and δ i j [ D i , D j ] = 0 they factorize into original K¨ahler and Ricci-flatness and holomorphic YMequations, i.e. F i j = 0 and δ i j F i j = 0. d -operator algebra and BRSTCharge The expression for the d -operators in a general curved background was derived in [13].It is given by d b α = E b α c M h P c M + 12 B c M b N ( ∂Z b N − ∂Z b n ) − Ω c M b β b γ λ b γ ω b β − A Σ c M J Σ i , (52) here P c M are the momenta conjugate to the worldsheet variables defined as P c M = δS/δ ( ∂ Z c M ).The nilpotence of the BRST charge is computed using Poison brackets [ P c M , Z b N ] P B = δ b N c M and [ λ b α , ω b β ] P B = δ b α b β . Note that the background field B c M b N does not mix with the otherbackground fields in (52) when we compute the nilpotence condition. This mixing onlyoccurs when computing holomorphicity of the BRST current. In a flat background the d operator reduces to d b α = E b α c M P c M (ignoring the contribution from the flat B c M b N ) and usingthe expression for the flat frame in the 4 + 6 notation we get precisely (25) after replacingthe conjugate momenta by the corresponding derivatives. The flat space BRST charge is Q = I ( λ α d α + λ ˙ α d ˙ α + λ α i d α i + λ ˙ α i d ˙ α i ) , (53)and it will square to zero if the ghosts satisfy the pure spinor constraint, reduced to 4 + 6notation λ α λ ˙ α + λ α i λ ˙ α i = 0 , (54) λ α λ i α − ǫ i j k λ j ˙ α λ ˙ α k = 0 , (55) λ ˙ α λ i ˙ α − ǫ i j k λ j α λ α k = 0 . (56)We want to generalize this to a flat four-dimensional background plus a curved six dimen-sional one. We must find the appropriate generalization of the d operators for this case. Thefirst thing to note is that if they are generalized to covariant derivatives ( ∇ α , ∇ ˙ α , ∇ α i , ∇ ˙ α i )satisfying the following algebra {∇ α , ∇ β } = 0 {∇ α , ∇ ˙ α } = − i ∇ α ˙ α {∇ ˙ α , ∇ ˙ β } = 0 {∇ α , ∇ β i } = − iε αβ ∇ i {∇ α , ∇ ˙ α i } = 0 {∇ ˙ α , ∇ ˙ β i } = − iε ˙ α ˙ β ∇ i {∇ α i , ∇ β j } = − iε αβ ǫ i j k ∇ k {∇ α i , ∇ ˙ α j } = − iδ i j ∇ α ˙ α {∇ ˙ α i , ∇ ˙ β j } = − iε ˙ α ˙ β ǫ i j k ∇ k (57)the BRST charge will be nilpotent. Here, the covariant derivatives ( ∇ α ˙ α , ∇ i , ∇ i ) aredefined by these equations. Using the variables defined in section 2.2 we can write thespinor covariant derivatives as ∇ α = ∂ α + iθ ˙ α ∇ α ˙ α + 2 iθ i α ∇ i , ∇ ˙ α = − ∂ ˙ α − iθ α ∇ α ˙ α − iθ i ˙ α ∇ i , ∇ α i = ∂ α i + iθ ˙ α i ∇ α ˙ α − iǫ i j k θ j α ∇ k ∇ ˙ α i = − ∂ ˙ α i − iθ α i ∇ α ˙ α + 2 iǫ i j k θ j ˙ α ∇ k , (58)The higher order dependence on θ s come from the derivatives ( ∇ α ˙ α , ∇ i , ∇ i ). Note thatthe equations (58) can be put in the form (52) with the spin connection term Ω c M b β b γ λ b γ ω b β and the gauge connection term A Σ c M J Σ inside the bosonic covariant derivatives. Since thebackground does not break four-dimensional Lorentz symmetry, the covariant derivative ∇ α ˙ α is just ∂ α ˙ α + O ( θ ) and nothing will depend on x α ˙ α . Moreover, since we are imposing hat the background is invariant under the N = 1 supersymmetry, the background cannotdepend on ( θ α , θ ˙ α ). We will now derive the restrictions imposed by these conditions.Repeated application of the Jacobi identities( − ) AC [ ∇ A , [ ∇ B , ∇ C }} + ( − ) BA [ ∇ B , [ ∇ C , ∇ A }} + ( − ) CB [ ∇ C , [ ∇ A , ∇ B }} = 0 , for the covariant derivatives will show that the background is on-shell. Here A , B and C corresponds to any tangent space index. At dimension 3 / ∇ α , ∇ β ˙ β ] = ε αβ W ˙ β , [ ∇ α , ∇ i ] = F α i , [ ∇ ˙ α , ∇ i ] = 0 , (59)together with their complex conjugates. Note that the first and last equations are a con-sequence of the algebra (57) plus Jacobi identities, while the second is the definition of F α i . To proceed, we have to solve order-by-order in θ s using the Jacobi identities. Four-dimensional Lorentz invariance implies that the first components of the superfields definedabove vanish and their second components should be four-dimensional scalars, as discussedabove. The field-strengths ( W α , F α i ) have an expansion in powers of θ s. In particular wehave the components W α = θ α D + θ i α h i + ... F α i = θ α F i + θ j α R i j + ... (60)where the ellipses denote components that do not concern us at the moment. The back-ground defined by (57) will be N = 1 supersymmetric if and only if these components vanishsince all field-strengths should be invariant under shifts of ( θ α , θ ˙ α ). This is related to theusual N = 1 field theory requirement that in order to have a supersymmetric vacuum, D and F terms should vanish. The h i and R i j components are, at this stage, not requiredto vanish and are related to the geometry of the compactified space. We will now calculatethe values of these components in terms of higher-dimension field-strengths.Using the Jacobi identities again we can alternative forms of the field-strengths:[ ∇ α i , ∇ β ˙ β ] = ε αβ F ˙ β i , [ ∇ α i , ∇ j ] = − ǫ i j k F α k , [ ∇ α i , ∇ j ] = − δ i j W α . (61)At lowest order in θ the F i component inside F α i is given by {∇ α , F β i } = ε αβ F i . However,using (61) we can write F α i as F α i = 12 ǫ i j k [ ∇ α j , ∇ k ] . (62)Now, the {∇ α , [ ∇ β j , ∇ k ] } Jacobi identity implies thatF i = i ǫ i j k [ ∇ j , ∇ k ] (63)and since [ ∇ j , ∇ k ] is anti-symmetric, it follows that the vanishing of the component F i implies that [ ∇ j , ∇ k ] = 0. As we saw in section 2, these two conditions imply that thecompactification manifold is K¨ahler and that the vector bundle over it is holomorphic. n a similar way, the component D of W α is the lowest component of {∇ α , W β } = ε αβ D.The computation of its value in terms of higher dimension field-strengths has one additionalstep. First we have to use the Jacobi identity with {∇ α , [ ∇ β i , ∇ j ] } to find δ i j {∇ α , W α } = {∇ α i , F α j } + 4 i [ ∇ i , ∇ j ] (64)Next, we use the Jacobi identity with {∇ α i , [ ∇ β j , ∇ k ] } to find {∇ α i , F α j } = − i [ ∇ i , ∇ j ] + 2 iδ i j δ k l [ ∇ k , ∇ l ] . (65)Plugging this result back into (64) we find {∇ α , W α } = 2 iδ k l [ ∇ k , ∇ l ] . (66)This means that the D component of W α vanishes when δ k l [ ∇ k , ∇ l ] = 0. This equation isthe second condition imposed by four-dimensional supersymmetry.In summary, we have found that the vanishing of F -terms in the superfield F α i impliesthe K¨ahler condition on the compactified manifold and part of holomorphic YM equationsfor the gauge background. The vanishing of the D -term in the W α field-strength impliesRicci-flatness and the remaining equation for the set of holomorphic YM equations. Onecan proceed to find the values of the other components of the field-strengths and computethe expression for the curved d -operators in (57) explicitly. For example, one can use theJacobi identity with {∇ α i , [ ∇ β j , ∇ k ] } to find that R i j = − i [ ∇ j , ∇ i ]. The component h i of W α vanishes due to the K¨ahler condition and the Jacobi identity with {∇ α i , [ ∇ β j , ∇ k ] } . Now that we have a BRST operator for the compactified background we want to checkthat Q ( V ) = 0 on a ghost number one vertex operator V gives the correct spectrum for thecompactification. In order to do this we will first show that we get the correct equations ofmotion for a super-Maxwell multiplet plus three chiral fields with N = 1 supersymmetryin four dimensions and then generalize to the full string. notation The propose of this section is to see how standard N = 1 superfield equations of motionappear when we perform a toroidal reduction of the ten dimensional ghost number onevertex operator and BRST charge. The vertex operator takes the form V = λ α A α + λ ˙ α A ˙ α + λ α i A α i + λ ˙ α i A ˙ α i (67) here ( A α , A ˙ α , A α i , A ˙ α i ) are superfields of the full superspace. The solution of QV = 0where Q is given by equation (53) is d α A β + d β A α = 0 d α A β i + d β i A α = ε αβ Φ i d α i A β j + d β j A α i = 2 ε αβ ǫ i j k Φ k d α A ˙ α + d ˙ α A α = A α ˙ α d α i A ˙ α j + d ˙ α j A α i = δ i j A α ˙ α d ˙ α A ˙ β + d ˙ β A ˙ α = 0 d ˙ α A ˙ β i + d ˙ β i A ˙ α = ε ˙ α ˙ β Φ i d ˙ α i A ˙ β j + d ˙ β j A ˙ α i = 2 ε ˙ α ˙ β ǫ i j k Φ k d α A ˙ β i + d ˙ β i A α = 0 d ˙ α A β i + d β i A ˙ α = 0 , (68)where the d s are defined in (27) and ( A α ˙ α , Φ i , Φ i ) are defined by these equations. Thevertex operator V also has the gauge invariance δV = Q Λ with a real superfield Λ. Interms of its components, this translates to δA α = d α Λ , δA α i = d α i Λ (69)together with their complex conjugates. The first equation in (68) implies that A α = d α V for some complex superfield V . The equations of motion imply the following gaugeinvariance δA α ˙ α = ∂ α ˙ α Λ , δ Φ i = ∂ i Λ , δ Φ i = ∂ i Λ (70)We can use the algebra of the supersymmetric derivatives to derive various relations onthe fields defined by (68). It is possible to solve all the Bianchi identities for a general set of( A α ˙ α , Φ i , Φ i ) but since our goal is to generalize this to the case of a CY compactification,we will take another route. First, note that it is possible to fix ( A α , A ˙ α ) to vanish withouttrivializing the system of equations. The gauge transformation that preserves this choicehas to satisfy d α Λ = d ˙ α Λ = 0 , (71)which, by use of the d -operator algebra, means Λ is just a constant in four dimensions.This implies that the degrees of freedom described by (0 , , A α i , A ˙ α j ) do not have gaugeinvariance from the four-dimensional point of view.When A α = A ˙ α = 0 the equations (68) simplify to d α A β i = ε αβ Φ i , d ˙ α A β i = 0 d α i A β j + d β j A α i = 2 ε αβ ǫ i j k Φ k , d α i A ˙ α j + d ˙ α j A α i = 0 d ˙ α i A ˙ β j + d ˙ β j A ˙ α i = 2 ε ˙ α ˙ β ǫ i j k Φ k d ˙ α A ˙ β i = ε ˙ α ˙ β Φ i (72) he first equation can be used to show that d α Φ i = 0, so it describes an anti-chiralfield. The second equation together with the first shows that d Φ i = 0, which is themassless equation for a chiral field. Then, using the commutator [ d , d ˙ α i ] = − id ˙ α ∂ i and acombination of the equations above, we find that d Φ k ǫ k i j = − i ( ∂ i Φ j − ∂ j Φ i ) = 0 , (73)which indicates that the massless field equation of the chiral superfield is related to thecohomology of 1-forms. If we set the higher ( θ i α , θ i ˙ α ) components to zero, we have preciselya triplet of chiral and anti-chiral fields. We also need to determine the higher ( θ i α , θ i ˙ α )components. This is accomplished by computing d α i Φ j and d ˙ α i Φ j . Using the equationsabove and the d algebra, we find that d α i Φ j = − i∂ j A α i , d ˙ α i Φ j = − i∂ j A ˙ α i , (74)so the higher ( θ i α , θ i ˙ α ) components do not describe new degrees of freedom. If the fields donot depend on ( z i , z i ) we have a triplet of four-dimensional chiral fields, as desired.It is easy to check that if we try to impose A α i = A ˙ α i = 0, we get a trivial system. Simi-larly, a solution where Φ i = 0 and Φ i = 0 is trivial because the vector field strength W α is ahigher component in Φ. There is no covariant way to solve the constraints containing onlythe gauge part. However, if the fields do not depend on the internal coordinates, it is possibleto isolate the four-dimensional gauge part. Instead of following this path, it is worthwhile toderive the equations of motion from (68) for a general ( A α ˙ α = i [ d α , d ˙ α ] V, Φ i , Φ i ). Repeatedapplication of the d -algebra gives d ˙ α (Φ i + 2 ∂ i V ) = 0 , d α (Φ i − ∂ i V ) = 0 , (75) d Φ i + 2 iǫ i j k ∂ j Φ k = 2 ∂ i d α A α , d Φ i + 2 iǫ i j k ∂ j Φ k = 2 ∂ i d ˙ α A ˙ α , (76) d α d d α V − δ i j ( ∂ i Φ j − ∂ j Φ i ) = 0 , (77)where higher components of ( θ i , θ j ) (which are consequences of the equations above) areset to zero. These are the linearized equations of motion for ten dimensional superYM in1 + 3 notation obtained long ago in reference [17]. If the fields do not depend on the internalcoordinates, we get three chiral fields and a vector multiplet. The higher components areagain determined by equation (68). The spectrum of the heterotic string is calculated in a similar way by repeated appli-cation of the curved space derivative algebra and the equations of motions coming from A = 0. Additionally, we now have to remember that the covariant derivatives act appro-priately on each section of the various vector bundles over the CY. We will see that whenthe section of vertex operator is not in the cohomology of ∇ i , the state corresponds to aKaluza-Klein mode obeying a massive superspace equation of motion.We begin with the compactification dependent sector. The complete heterotic stringvertex operator must be tensored with the right-moving dimension 1 currents given by( ∂x a , ∂y i , ∂y i , J Σ ), where Σ is a general index for the two E algebras. Although notdiscussed in the present paper, the anomaly cancelation condition of the B -field shouldbe taken into account. The simplest way to solve it is by the standard embedding. Thisembedding breaks one of the E factors into E → E × SU (3). The Kac-Moody currentsare decomposed into J Σ → ( J σ , J ρ , J i A , J j A , J i j ) (78)where σ is the index of the adjoint representation of E , ρ is an index for the adjointrepresentation of E , A is the index for the fundamental representation of E , and ( i j ) areindices for endomorphisms of the holomorphic tangent bundle.The BRST charge is now Q = I ( λ α ∇ α + λ ˙ α ∇ ˙ α + λ α i ∇ α i + λ ˙ α i ∇ α i ) . (79)We proceed exactly as in the previous section. The equations from the BRST physical statecondition are of the form (68) with the operators d replaced by the operators ∇ of (79). Asin the previous section, we will set A Γ α = A Γ˙ α = 0 where Γ denotes any right-moving index.After doing this, we obtain the equations ∇ α A Γ β i = ε αβ Φ Γ i , ∇ ˙ α A Γ β i = 0 ∇ α i A Γ β j + ∇ β j A Γ α i = 2 ε αβ ǫ i j k Φ Γ k , ∇ α i A Γ˙ α j + ∇ ˙ α j A Γ α i = 0 ∇ ˙ α i A Γ˙ β j + ∇ ˙ β j A Γ˙ α i = 2 ε ˙ α ˙ β ǫ i j k Φ Γ k ∇ ˙ α A Γ˙ β i = ε ˙ α ˙ β Φ Γ i (80)Using these equations and the algebra (57) we obtain ∇ α Φ Γ i = 0 and ∇ Φ Γ i = 0 . Notethat the commutator [ ∇ , ∇ ˙ α i ] = − i ∇ ˙ α ∇ i still holds for the covariant derivatives. Thisimplies that the chiral fields Φ Γ i satisfy ∇ i Φ Γ j − ∇ j Φ Γ i = 0 . (81)Thus, for each type of index Γ, the corresponding chiral field is in the cohomology ring H , ( T ), where T is the vector space corresponding to the index Γ. This is the expected In order to get a dimension (0 ,
0) vertex operator we should also multiply by the right-moving ghost c . In the pure spinor formalism this comes from conservation of the BRST current and the anomaly in theconservation of ghost and gauge currents. We define ∇ = ∇ α ∇ α and ∇ = ∇ ˙ α ∇ ˙ α . esult for the matter part of a CY compactification. The analysis of the higher θ -componentsproceeds as in the previous section. In particular, the Φ Γ do not describe additional degreesof freedom at the massless level.To derive the equations of motion for the compactification-independent part, we haveto solve the generalization of equations (68) with covariant derivatives without setting thesuperfields A Γ α and A Γ˙ α to zero. Again, we obtain the generalization of (75): ∇ ˙ α (Φ Γ i + 2 ∇ i V Γ ) = 0 , ∇ α (Φ Γ i − ∇ i V Γ ) = 0 , (82) ∇ Φ Γ i + 2 iǫ i j k ∇ j Φ Γ k = 2 ∇ i ∇ α A Γ α , ∇ Φ Γ i + 2 iǫ i j k ∇ j Φ Γ k = 2 ∇ i ∇ ˙ α A Γ˙ α , (83) ∇ α ∇ ∇ α V Γ − δ i j ( ∇ i Φ Γ j − ∇ j Φ Γ i ) = 0 . (84)If the fields do not depend on the compactification, the three possible right movingindices are the four-dimensional vector index, the adjoint E index, and the adjoint E index. This completes the massless spectrum of the heterotic string in the CY background.As a final remark, since the equations above do not impose that the fields are harmonicforms on the CY (see equation 83), they also describe in superspace the KK spectrum ofthe compactification. In this paper, we began the study of superstring compactifications using the pure spinorformalism. Although only N = 1 supersymmetry in four dimensions is preserved, the de-scription of the BRST operator and spectrum given here uses the full superspace inheritedfrom ten dimensions. We first considered some algebraic aspects of the compactification,mainly the BRST operator and the spectrum. In a second paper we will discuss furtheraspects, such as the construction of the sigma model describing the dynamics of the com-pactification and the anomaly cancelation condition, which comes from the conservation ofthe BRST current. In this discussion the B -field, which played no role in the present work,will be included.One interesting direction for future work could be to see how the well known non-renormalization theorems of Calabi-Yau compactifications arise in the supersymmetric de-scription given here. This will require knowledge of the zero-mode measure for scatteringamplitudes (which will be presented elsewhere). It is possible that the non-renormalizationis just a consequence of the superspace integration arising from this measure.A more important line of research is to generalize these results to Type II strings,especially in the case of flux compactifications (for a review see e.g. [18]). Most of the resultsin the literature use only supergravity methods and little is known about α ′ corrections andthe spectrum. Even though it is unlikely that a sigma model including all powers of θ can bewritten explicitly, partial knowledge will already be enough to address important questionspertaining to the form of the effective action of the light modes in a flux compactification.We plan to address flux compactifications of the pure spinor formalism in the future. cknowledgments We would like to thank P.A. Grassi, L. Mazzucato and D. Sorokin for useful discussions.WDL and BCV also thank KITP at Santa Barbara, where part of this work was done, fortheir kind hospitality. OC would like to thank Galileo Galilei Institute for TheoreticalPhysics at Arcetri for their kind hospitality, where parts of this work were done. WDLthanks UNAB for the warm hospitality, where this work was started. This work is supportedby FONDECYT grants 1061050 and 7080027, UNAB grants DI-03-08/R and AR-02-09/R,NSF grants PHY 0653342, DMS 0502267, PHY 05-51164.
References [1] N. Berkovits, S. Gukov and B. C. Vallilo,
Superstrings in 2D backgrounds with R-R fluxand new extremal black holes,
Nucl. Phys. B , 195 (2001) [arXiv:hep-th/0107140].[2] N. Berkovits,
Covariant quantization of the Green-Schwarz superstring in a Calabi-Yaubackground,
Nucl. Phys. B , 258 (1994) [arXiv:hep-th/9404162].[3] N. Berkovits and C. Vafa,
N=4 topological strings,
Nucl. Phys. B , 123 (1995)[arXiv:hep-th/9407190].N. Berkovits,
Quantization of the type II superstring in a curved six-dimensional back-ground,
Nucl. Phys. B (2000) 333 [arXiv:hep-th/9908041].[4] W. D. Linch III and B. C. Vallilo,
Hybrid formalism, supersymmetry reduction, andRamond-Ramond fluxes,
JHEP , 099 (2007) [arXiv:hep-th/0607122].W. D. Linch III, J. McOrist and B. C. Vallilo, Type IIB Flux Vacua from the StringWorldsheet,
JHEP , 042 (2008) [arXiv:0804.0613 [hep-th]].[5] N. Berkovits,
Super-Poincare covariant quantization of the superstring,
JHEP ,018 (2000) [arXiv:hep-th/0001035].[6] J. Gomis, D. Sorokin and L. Wulff,
The complete AdS(4) x CP(3) superspace for thetype IIA superstring and D-branes,
JHEP , 015 (2009) [arXiv:0811.1566 [hep-th]].P. A. Grassi, D. Sorokin and L. Wulff,
Simplifying superstring and D-brane actions inAdS(4) x CP(3) superbackground, arXiv:0903.5407 [hep-th].[7] D. Tsimpis,
Curved 11D supergeometry,
JHEP , 087 (2004)[arXiv:hep-th/0407244].[8] P. A. Grassi and P. van Nieuwenhuizen,
Harmonic superspaces from superstrings,
Phys.Lett. B , 271 (2004) [arXiv:hep-th/0402189].[9] N. Berkovits,
Pure spinor formalism as an N = 2 topological string,
JHEP , 089(2005) [arXiv:hep-th/0509120].[10] P. A. Grassi and N. Wyllard,
Lower-dimensional pure-spinor superstrings,
JHEP ,007 (2005) [arXiv:hep-th/0509140].[11] O. Chandia,
D = 4 pure spinor superstring and N = 2 strings,
JHEP , 105 (2005)[arXiv:hep-th/0509185].
12] I. Adam, P. A. Grassi, L. Mazzucato, Y. Oz and S. Yankielowicz,
Non-critical purespinor superstrings,
JHEP , 091 (2007) [arXiv:hep-th/0605118].[13] N. Berkovits and P. S. Howe,
Ten-dimensional supergravity constraints fromthe pure spinor formalism for the superstring,
Nucl. Phys. B , 75 (2002)[arXiv:hep-th/0112160].[14] O. Chandia and B. C. Vallilo,
Conformal invariance of the pure spinor superstring ina curved background,
JHEP , 041 (2004) [arXiv:hep-th/0401226].[15] O. A. Bedoya and O. Chandia,
One-loop conformal invariance of the type II pure spinorsuperstring in a curved background,
JHEP , 042 (2007) [arXiv:hep-th/0609161].[16] P. Griffiths and J. Harris,
Principles of Algebraic Geometry , Wiley Classics LibraryEdition, 1994.[17] N. Marcus, A. Sagnotti and W. Siegel,
Ten-Dimensional Supersymmetric Yang-MillsTheory In Terms Of Four-Dimensional Superfields,
Nucl. Phys. B , 159 (1983).[18] M. Grana,
Flux compactifications in string theory: A comprehensive review,
Phys.Rept. , 91 (2006) [arXiv:hep-th/0509003]., 91 (2006) [arXiv:hep-th/0509003].