(Non)triviality of Pure Spinors and Exact Pure Spinor - RNS Map
aa r X i v : . [ h e p - t h ] N ov (Non)triviality of Pure Spinorsand Exact Pure Spinor - RNS Map Dimitri Polyakov † Center for Advanced Mathematical SciencesAmerican University of BeirutBeirut, Lebanon
Abstract
All the BRST-invariant operators in pure spinor formalism in d = 10 can be repre-sented as BRST commutators, such as V = { Q brst , θ + λ + V } where λ + is the U (5) componentof the pure spinor transforming as 1 . Therefore, in order to secure non-triviality of BRSTcohomology in pure spinor string theory, one has to introduce “small Hilbert space” and“small operator algebra” for pure spinors, analogous to those existing in RNS formalism.As any invariant vertex operator in RNS string theory can also represented as a commu-tator V = { Q brst , LV } where L = − c∂ξξe − φ , we show that mapping θ + λ + to L leads toidentification of the pure spinor variable λ α in terms of RNS variables without any ad-ditional non-minimal fields. We construct the RNS operator satisfying all the propertiesof λ α and show that the pure spinor BRST operator H λ α d α is mapped (up to similaritytransformation) to the BRST operator of RNS theory under such a construction..October 2008 † [email protected],[email protected]; the address after January 5,2009: National Insti-tute for Theoretical Physics, Department of Physics and Centre for Theoretical Physics, Universityof the Witwatersrand, Wits 2050, South Africa ntroduction Pure spinor formalism for superstrings has been proposed by Berkovits several yearsago [1] as an alternative method of covariant quantization of Green-Schwarz superstringtheory [2]. It involves the remarkably simple worldsheet action: S = Z d z { ∂X m ¯ ∂X m + p α ¯ ∂θ α + ¯ p α ∂ ¯ θ α + λ α ¯ ∂w α + ¯ λ α ∂ ¯ w α } (1)where p α is conjugate to θ α [3] and the commuting spinors λ α and w α are the bosonicghosts which, roughly speaking, are related to the fermionic gauge κ -symmetry in GSsuperstring theory. The action (1) is related to the standard GS action by substitutingthe constraint d α = p α −
12 ( ∂X m + 14 θγ m ∂θ )( γ m θ ) α = 0 (2)and the corresponding BRST operator Q brst = I dz iπ λ α d α ( z ) (3)is nilpotent provided that λ α satisfies the pure spinor condition: λ α γ mαβ λ β = 0 (4)reducing the number of independent components of λ from 16 to 11. An example ofunintegrated massless vertex operator in such a BRST cohomology is given by U = λ α A α ( X, θ ) (5). This operator is physical provided that the space-time superfield A α is on-shell: γ αβm ...m D α A β = 0 (6)(this particularly implies the standard Maxwell equation for the bosonic vector com-ponent of A ) and thus the vertex operator (5) is identified with the emission of a photon bythe superstring [1], [4], [5]. The integrated version of this operator ∼ H dz iπ V ( z ) satisfying[ Q brst , V ] = ∂U can also be constructed, with V obviously having ghost number zero [6].Physical vertex operators (both massless and massive) considered in pure spinor formalism1hus typically have ghost number 1 in unintegrated form and number zero in the integratedversion.The important question is how the PS approach is related to other descriptions ofsuperstring , such as RNS formalism. While such a relation exists and can be constructed,the construction is not straightforward and the constructions considered so far particularlyrequired the introduction of additional non-minimal fields by hands [7], [6]Another natural question is whether the PS superstring could contain any additionalphysical operators, e.g. with higher ghost numbers. It is far from obvious that suchoperators could exist at all. For example, a straightforward naive attempt to generalizethe unintegrated operator (5) to the ghost number 2 case: U = λ α λ β F αβ ( X, θ ) (7)fails as the on-shell conditions for the field F αβ : γ αβm ...m D γ F αβ ( X, θ ) = 0 (8)imply the triviality of the U operator: U = { Q brst , θ α λ β F αβ } (9)Similarly, naive construction of ghost number n operators ∼ λ n leads to BRST-exactexpressions, provided the on-shell constraint on the corresponding background fields. De-spite that, below we shall demonstrate that vertex operators with non-standard couplingto pure spinors do appear in BRST cohomology. In general, the question of non-trivialityof BRST cohomology in the PS formalism appears more subtle compared to RNS. Thatis, since { Q, θ α } = λ α and [ Q, λ α ] = 0, any invariant operator V in pure spinor stringtheory can be written as an exact BRST commutator. For example, consider the standardU(5)-invariant parametrization of λ α : λ α = ( λ + , λ ab , λ a )( a, b = 1 , ..,
5) with λ ab = − λ ba and λ a = ǫ abcde λ bc λ de . Then any invariant V can be written as V = [ Q brst , θ + λ + V ] (10). This poses a question whether BRST cohomology of PS string theory is empty (similarobservations have also been made in [8]) 2n fact, the identity (10) is reminiscent of the similar relation in the RNS formalismwhere any invariant V can be written as V = { Q RNSbrst , LV } (11)where L = − ce χ − φ = − c∂ξξe − φ (12)with the ghost fields bosonized as [9] b = e − σ , c = e σ β = e χ − φ ∂χ ≡ ∂ξe − φ , γ = e φ − χ (13)It is easy to check { Q RNSbrst , L } = 1 (14)In RNS approach, however, the relation (12) does not lead to the triviality of states sincethe L -operator is not in the small operator algebra, as it explicitly depends on ξ = e χ (rather than its derivatives). So the only way to bail out pure spinors is to introducesimilar classification for the PS formalism as well. Such a classification, however, isn’tas obvious as in the RNS case. In the RNS case we exclude the operators with explicit ξ -dependence because the bosonization relations for the ghost fields β and γ depend on thederivative of ξ , but not on ξ itself ( ξ can only be expressed as a generalized step function of β : ξ = Θ( β )) In the PS formalism, however, the analogue of the L -operator is given by theratio θ + λ + consisting of fields already present in the theory. For this reason, the distinctionbetween “large” and “small” operator algebras appears more obscure in the PS approach.One possible approach is to try to construct a direct map between P S and
RN S variables,which in particular would identify θ + λ + with the L -operator of RNS formalism. Once sucha map is constructed, it would transform the states from the little Hilbert space in RNSformalism to those in the little Hilbert space in the pure spinor description. So we startwith the map c∂ξξe − φ ∼ θ + λ + (15)and will try to deduce the correspondence between PS ans RNS variables by using thisisomorphism. Since the Green-Schwarz variable θ α is known to be related to RNS spinoperator by the field redefinition θ α ∼ e φ Σ α (16)3 we write θ + = e φ Σ + where Σ + is the component of Σ α with five pluses (+ + + + +)in the ( ± ) representation. For our purposes, it is convenient to split 32-component spinoperator into two 16-component spin operators Σ α and ˜Σ α with opposite GSO parities.Then the RNS expression for λ + ≡ ( λ + ) − which OPE with θ + gives L is given by( λ + ) − = ce χ − φ ˜Σ + (17)where ˜Σ + is the ( − − − − − ) component of the 32-component spin operator (so it hasGSO parity opposite to Σ + ). One can easily check that the OPE of ( λ + ) − with θ + isnon-singular, with the zero order term given precisely by L . Next, the λ + operator can beread off the OPE ( λ + ) − ( z ) λ + ( w ) ∼ O ( z − w ) (18)It is easy to see that λ + = be φ − χ Σ + (19)is precisely the operator satisfying this OPE identity. Note that λ + and θ + have the sameGSO parity. It is now straightforward to identify λ α ∼ be φ − χ Σ α (20)however such an identification is not yet complete for the following reason. On one hand,the expression (20) of λ α in terms of RNS variables does have some basic properties ofpure spinors: it is the dimension zero primary field, it is a commuting spinor (since it ismultiplied by the b-ghost which is worldsheet fermion) however its full OPE does not yetsatisfy the pure spinor constraint as λ α ( z ) λ β ( w ) ∼ z − w ) ∂bbe φ − χ γ mαβ ψ m + 14 ∂bbe φ − χ γ mαβ ∂ ψ m + ∂bbe φ − χ γ αβm ...m ψ m ...ψ m + ... (21)so the while the second term of the normally ordered part of this OPE would vanishafter substituting into the left hand side of the pure spinor constraint λγ m λ (since itwould produce the factor proportional to ∼ T r ( γ m γ m ...m ) = 0, the first term would stillcontribute. In addition, the OPE (21) has a double pole singularity while the OPE of two λ ’s in the pure spinor formalism is known to be non-singular [6] The reason is that boththe OPE singularity and the violation of the pure spinor constraint are related to BRSTnon-invariance of the operator (20), while the actual pure spinor must be BRST-invariant.4or this reason, one has to add the correction terms to the r.h.s. of (20) to ensure theBRST-invariance. This can be done by replacing λ α → λ α − Lρ α (22)where ρ α = [ Q brst , be φ − χ Σ α ] is the BRST commutator with the right hand side of(20). Since { Q brst , L } = 1 and [ Q brst , ρ α ] = 0, the modified λ α will be BRST-invariant byconstruction. Evaluating ρ α and its normally ordered product with L we find the completeRNS representation for the pure spinor variable λ α to be given by λ α = be φ − χ Σ α + 2 e φ − χ γ mαβ ∂X m ˜Σ β − ce φ Σ α ∂φ − ce φ ∂ Σ α (23)It is straightforward to check that this expression for λ α does satisfy the pure spinorcondition (4) (see the Appendix). Note that λ α = − { Q brst , θ α } where the factor of − λ α is annihilated by inverse picture-changing operator Γ − = 4 c∂ξe − φ and therefore cannot be transformed to pictures lowerthan , such as − or − . In the next section we will use the RNS expression (23) for λ α in order to map the BRST charge of pure spinor string theory into RNS BRST charge. RNS BRST Operator from Pure Spinor BRST Operator
In this section we will use the RNS representation (23) for the pure spinor variable λ α to construct the exact map relating pure spinor BRST charge and RNS BRST charge.To demonstrate this relation we have to calculate the normally ordered expression of thepure spinor BRST current : λ α d α : in the RNS formalism. The constraint operator d α = p α − θ β γ αβm ∂X m −
18 ( θγ m ∂θ )( γ m θ ) α (24)consists of three terms, so we are to calculate the normally ordered OPE’s of these termswith λ α one by one. A useful formula for our calculation is the OPE between two spinoperators: Σ α ( z )Σ β ( w ) ∼ γ mαβ ψ m ( z + w )( z − w ) + 16 ( z − w ) γ mnpαβ ψ m ψ n ψ p ( z + w ... Σ α ( z ) ˜Σ β ( w ) ∼ δ αβ ( z − w ) + 12 ( z − w ) ∂ψ m ψ m ( z + w ... (25)where we skipped higher order OPE terms as well as those not contributing to the normallyordered expression for : λ α d α :. Note that, as the ordered RNS expressions for three terms524) of d α contain zero, one and three gamma-matrices respectively (see below), only theterms with one or three gamma-matrices in the ΣΣ or ˜Σ ˜Σ operator products and only theterms proportional to δ αβ in the Σ ˜Σ OPE contribute to the BRST current. All other OPEterms (i.e. those with the number of antisymmetrized gamma-matrices other than 0,1 or3) are irrelevant to us since their contributions to : λ α d α : produce terms proportional tovanishing traces of antisymmetrized gamma-matrices.We start with the p α term of d α Since p α is canonically conjugate to θ β : p α ( z ) θ β ( w ) ∼ δ βα z − w (26)the RNS representation for p α is easily deduced to be p α = e − φ ˜Σ α (27)i.e. it is simply the space-time supercurrent at picture − . Then the normally orderedOPE’s of p α with the first two terms of λ α are easily evaluated to give p α ( z ) be φ − χ Σ α ( w ) = ( z − w ) be φ − χ ( z + w O ( z − w ) p α ( z )2 e φ − χ γ mαβ ˜Σ( w ) = ( z − w ) e φ − χ ψ m ∂X m ( z + w O ( z − w ) (28)so the result is given by easily recognizable (up to normalization factors) ghost andmatter supercurrent terms of j brst in the RNS formalism. The OPE of p α with the re-maining part of λ α , namely, ce φ Σ α (2 ∂σ − ∂φ ) − ce φ ∂ Σ α is a bit more tedious butstraightforward to calculate producing terms with the structure ∼ cG (2) ( ψ, σ, φ, χ ) with G being an operator of conformal dimension two, consisting of ψ, φ, χ and σ worldsheetfields, giving a hint on the relevance of this contribution to the cT + b∂cc part of Q brst inthe RNS description. Performing the calculation and collecting all the terms together weobtain the contribution of : p α λ α : to j brst to be given by: p α λ α := δ αα { γ b + γψ m ∂X m + c ( 114 ∂ψ m ψ m − ∂φ ) + ∂ φ + 116 ( ∂σ ) − ∂ σ − ∂φ∂σ ) } (29)The next step is to calculate the contribution stemming from the normally orderedOPE of λ α with the second term of d α , given by − ∂X m ( γ m θ ) α . However, an importantremark should be made first. Since the RNS expressions (16) for θ α and (23) for λ α are bothat the ghost picture , the straightforward evaluation of their OPE would give an operator6t picture 1. This is not quite what we are looking for since all the terms of j brst are atpicture zero. Since we expect that the OPE of λ α and − ∂X m ( γ m θ ) α reproduces only apart of j brst , the resulting operator is generally off-shell, so one cannot picture transformit in a straightforward manner. As for λ α , although it is on-shell, inverse picture-changingstill isn’t applicable to it, as was noted above. For this reason, in order to get a picturezero result for this contribution, instead of taking θ α in the standard form (16) one has totake it in its equivalent form θ α = − ce χ − φ Σ α (30)which is at picture − . Although picture-changing transformation isn’t well-defined foroff-shell variables such as θ α , the expressions (16) and (30) are equivalent since they bothsatisfy the same canonical relation with the conjugate momentum p α . Indeed, since theworldsheet integral of p α is on-shell, one can transform it to picture obtaining p α = − e φ γ mαβ Σ β ∂X m − be φ − χ ˜Σ α (31). Applying p α of (31) to θ β of (30) one easily finds that, while the first term of p α doesn’tcontribute to the simple pole of its OPE with θ β , the second term’s OPE with θ producesprecisely the simple pole leading to the standard canonical relation. Thus − θ β γ mαβ ∂X m = 2 ce χ − φ Σ β γ mαβ ∂X m (32)Evaluating the OPE of this term with λ α of (23) we obtain2 ce χ − φ Σ β γ mαβ ∂X m ( z ) be φ − χ Σ α ( w ) = ( z − w ) δ αα e φ − χ ψ m ∂X m ( z + w O ( z − w )(33)for the product of (32) with the first term of λ α ce χ − φ Σ β γ mαβ ∂X m ( z )2 e φ − χ ˜Σ γ γ mαγ ∂X m ( w )= δ αα c { ∂X m ∂X m − ∂ψ m ψ m − ∂ σ − ∂σ ) − ∂φ ) − ∂χ ) + 24 ∂χ∂φ − ∂χ∂σ + 18 ∂φ∂σ } ( z + w O ( z − w ) (34)for the product of (32) with the second term of λ α ce χ − φ Σ β γ mαβ ∂X m ( z ) − ce φ Σ α ∂φ − ce φ ∂ Σ α )( w )= − δ αα ∂cce χ − φ ψ m ∂X m ( z + w O ( z − w ) (35)7or the product of (32) with the third term of λ α Note the appearance of an extra γψ m ∂X m term on the r.h.s. of the OPE (33) thatensures the correct normalisation of the matter supercurrent term with respect to the ghostsupercurrent term in j brst . The final contribution to j brst comes from the OPE of λ α and − ( θγ m ∂θ )( γ m θ ) α = θ β θ ρ ∂θ λ γ mβλ ( γ m ) αρ . To ensure that the contribution of this OPE to j brst is at picture zero, it is convenient to take θ β and θ ρ at the picture − representation(30) while keeping ∂θ λ at the picture version (16). Using the OPE (25) one easily finds: θ β θ ρ : ( z ) = 83 γ mnpβρ ∂cce χ − φ ψ m ψ n ψ p ( z ) (36)Calculating the operator product of (36) with ∂θ λ = ∂ ( e φ Σ λ ) using (25) gives −
18 : ( θγ m ∂θ )( γ m θ ) α : ( z ) = − ∂cce φ − φ ( ∂ Σ α − ∂φ Σ α )( z ) (37)The calculation of the OPE of (37) with the first term of λ α (23) gives − ∂cce φ − φ ( ∂ Σ α − ∂φ Σ α )( z ) be φ − χ Σ α ( w )= ( z − w ) δ αα c { ∂ψ m ψ m − ∂φ ) − ∂ φ + 10( ∂χ ) + 14 ∂ χ − ∂σ ) + 16716 ∂ σ + 24 ∂χ∂φ + 594 ∂φ∂σ − ∂χ∂σ } ( z + w O ( z − w ) (38)The OPE of (37) with the second term of λ α gives − ∂cce φ − φ ( ∂ Σ α − ∂φ Σ α )( z )2 e φ − χ γ αβm ∂X m ˜Σ β = − ∂cce χ − φ ψ m ∂X m ( z )+ O ( z − w )(39)Finally, the OPE of (37) with the third term of λ α produces − ∂cce φ − φ ( ∂ Σ α − ∂φ Σ α )( z ) ce φ ( − ∂ Σ α − α ∂φ )( w )= 32 ∂ c∂cce χ − φ ( z + w O ( z − w ) (40)Collecting together all the terms in (29) - (40) we find the overall normally ordered productof d α and λ α to be given by:116 : λ α d α := 2 γψ m ∂X m + γ b + c { ∂X m ∂X m + 2 ∂ψ m ψ m − ∂χ ) + 14 ∂ χ − ∂φ ) − ∂ φ −
252 ( ∂σ ) + 152 ∂ σ +48 ∂χ∂φ − ∂χ∂σ + 32 ∂φ∂σ } − ∂cce χ − φ ψ m ∂X m + 32 ∂ c∂cce χ − φ (41)8here the factor of in front of the pure spinor BRST current is to absorb the factorof δ αα = 16 always appearing on the right hand side of the operator products (29) - (40).Although the RNS expression (41) for the pure spinor BRST current looks tedious, it isstraightforward to check that, up to an overall numerical factor and BRST trivial terms,it is equivalent to the BRST current in RNS formalism. Indeed, using the bosonizedexpression for RNS BRST current: j RNSbrst = cT + b∂cc − γψ m ∂X m − bγ = c {− ∂X m ∂X m − ∂ψ m ψ m −
12 ( ∂φ ) − ∂ φ + 12 ( ∂χ ) + 12 ∂ χ + 98 ( ∂σ ) + 18 ∂ σ } − e φ − χ ψ m ∂X m − be φ − χ (42)and the commutator: [ Q RNSbrst , ∂cce χ − φ ∂χ ] = ∂ c∂cce χ − φ − ∂cce χ − φ ψ m ∂X m − c { ∂ φ − ∂ χ − ∂ σ + 4( ∂φ ) + 2( ∂χ ) + ( ∂σ ) − ∂χ∂φ + 3 ∂χ∂σ − ∂φ∂σ } (43)one easly finds 116 j purespinorbrst = − j RNSbrst + 32[ Q RNSbrst , ∂cce χ − φ ∂χ ] (44)This concludes the calculation identifying the BRST charges in RNS and pure spinorapproaches. Note that a shift of a BRST charge by any BRST trivial term (that particularlyoccurs in (44)): Q brst → Q brst + [ Q brst , R ] (45)where R is some operator, is equivalent to the similarity transformation Q brst → e − R Q brst e R (46)considered in [7]. In our case, R = 32 I dz iπ ∂cce χ − φ ∂χ ( z ) (47)Note that the R -operator isn’t generally required to be in the “small operator algebra”and, as a matter of fact, both the R -operator (47) and the R -operator used in the similaritytransformation in [7] are outside the small algebra: the R -operator (47) contains the factor9f e χ ∂χ = ∂ ξξ , while the R-operator used by Berkovits explicitly depends on λ + which,when translated into RNS language, isn’t in the small algebra as well. Discussion. Vertex Operators with Non-trivial Pure Spinor Couplings
In this letter we have proposed an exact map expressing the pure spinor variable λ α in terms of BRST invariant RNS operator of conformal dimension zero, satisfying purespinor constraint. The map is based on identifying the θ + λ + operator in the pure spinorformalism and the L -operator in the RNS description satisfying { Q brst , L } = 1 This mapparticularly leads to the identification (23) of pure spinor and RNS BRST operators, up tosimilarity transformation (or BRST-trivial terms). The non-triviality of vertex operators inpure spinor approach requires the introduction of “small” and “large” operator algebra inpure spinor approach, similarly to the classification existing in RNS approach. However,classifying the full operator algebra in terms of “large” and “small” appears somewhatambiguous in the pure spinor formalism, compared to RNS formalism, where such a classi-fication is clear and is based on the bosonization relations for superconformal ghosts. Thesmall operator algebra of the pure spinor formalism should particularly exclude operatorsinverse to pure spinor components, such as λ + , but such a constraint appears too relaxedand also somewhat artificial since, unlike RNS variable ξ , which can only be expressedas a generalized step function of superconformal β -ghost: ξ = Θ( β ) pure spinor operator λ + is the function of a variable manifestly present in the theory. This particularly leadsto the pure spinor BRST cohomology containing operators which physical meaning is un-clear. In particular any function F ( λ ) is an invariant operator in pure spinor formalism. If F ( λ ) is polynomial, e.g. F ( λ ) ∼ λ α ...λ α n , it can be represented as a BRST commutator F ( λ ) = { Q brst , θ α λ α ...λ α n } , i.e. it is BRST exact. If, however, F ( λ ) isn’t a polynomialfunction (e.g. F ( λ ) ∼ log ( λ )) then the only way to represent it as a BRST commutatorseems to be F ( λ ) = { Q brst , θ + λ + F ( λ ) } , but this doesn’t make an operator unphysical, dueto the small/large algebra classification. Apparently not all these operators, while formallyin the cohomology, are of physical significance. For this reason, one needs to find the wayto eliminate these clearly excessive states, which apparently requires better understandingof how operator formalism works in the pure spinor approach. Acknowledgements
As I’m preparing to join the research group at the University of the Witwatersrand inJohannesburg, I wish to express my heartfelt gratitude to Wafic Sabra and other membersof Center for Advanced Mathematical Sciences (CAMS) in Beirut for many illuminatingdiscussions and for the friendship during the years I have spent in Lebanon.10 ppendix
In this short appendix we demonstrate that the BRST-invariant RNS expression (23)for λ α satisfies the pure spinor condition (4). Since λ α ∼ { Q brst , θ α } , it is sufficient toshow that the operator N m = ( θγ m λ ) is BRST-invariant. Taking θ α = e φ Σ α accordingto (16) and evaluating its OPE with λ α of (23) using (25) it is straightforward to calculate −
14 : θγ m λ := − γ mαβ : e φ Σ α ( z )[ be φ − χ Σ β +2 e φ − χ γ mβλ ∂X m ˜Σ λ − ce φ Σ β ∂φ − ce φ ∂ Σ β ]( z ) := − e φ ( ψ n ∂X n ) ∂X m + 12 ce φ [ 12 ∂ ψ m + ∂ψ m ( ∂φ − ∂χ ) + 12 ( ∂ φ − ∂ χ +( ∂φ − ∂χ ) ) ψ m ] − ∂ ce φ ψ m + 12 c∂ ( e φ ∂χψ m ) − e φ − χ [2 ∂ φ + 2 ∂ χ − ∂ σ + (2 ∂φ − ∂χ − ∂σ ) ] ∂X m − e φ − χ [( ψ n ∂X n ) ∂φ + ∂ ( ψ n ∂X n ) ψ m − ∂χ∂ X m − ∂χ ( ∂φ − ∂χ ) ∂X m + 12 ∂ X m + ( ∂φ − ∂χ ) ∂ X m + 12 ( ∂ φ − ∂ χ + ( ∂φ − ∂χ ) ) ∂X m ] − be φ − χ [(2 ∂φ − ∂χ − ∂σ )(2 ∂φ − ∂χ − σ ) + 2 ∂ φ − ∂ χ − ∂ σ ] ψ m −
14 [ Q brst , e φ ( ψ m ∂φ + 12 ∂ψ m )] (48)Up to the BRST trivial terms, the right hand side of (48) can be recognized as[ Q brst , ξV photon ] where V photon = c∂X m + γψ m is the unintegrated photon vertex op-erator at zero momentum. For this reason, the RNS expression for − θγ m λ is given bythe unintegrated photon vertex operator at superconformal ghost picture 1 at zero mo-mentum (which of course is BRST-invariant) plus BRST trivial terms. Therefore − θγ m λ is BRST-invariant and its BRST commutator, given by λγ m λ , is identically zero. Thisconcludes the proof that λ α satisfies the pure spinor constraint (4).11 eferenceseferences