Nuclear Physics B 848 [FS] (2011) 155–215 www.elsevier.com/locate/nuclphysb Notes on β-deformations of the pure spinor superstring in AdS × S55 Oscar A. Bedoya a, L. Ibiapina Beviláqua a, Andrei Mikhailov b,c,∗, Victor O. Rivelles a a Instituto de Física, Universidade de São Paulo, C.P. 66.318, CEP 05315-970, São Paulo, SP, Brazil b Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan c Institute for Theoretical and Experimental Physics, Bol. Cheremushkinskaya 25, 117259 Moscow, Russia Received 30 August 2010; received in revised form 11 February 2011; accepted 14 February 2011 Available online 18 February 2011 Abstract We study the properties of the vertex operator for the β-deformation of the superstring in AdS × S55 in the pure spinor formalism. We discuss the action of supersymmetry on the infinitesimal β-deformation, the application of the homological perturbation theory, and the relation between the worldsheet description and the spacetime supergravity description. © 2011 Elsevier B.V. All rights reserved. Keywords: AdS/CFT; String worldsheet theory; Pure spinor formalism; Deformation theory 1. Introduction Historically, the development of the pure spinor formalism was mostly concentrated on the special case of flat space. But in fact the flat space case is a degenerate case. In many ways the general background is qualitatively different, the flat space being a special degenerate limit. The general, “typical” background has a non-degenerate Ramond–Ramond bispinor field. Among such non-degenerate examples the most symmetric one is AdS5 × S5. Therefore the study of this background is important for the string theory in general. * Corresponding author at: Instituto de Física Teórica, Universidade Estadual Paulista, R.Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra-Funda CEP 01140-070, São Paulo, Brazil. E-mail address: andrei@theory.caltech.edu (A. Mikhailov).0550-3213/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.02.012 156 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215During the last several years, continuous progress has been made in this direction. One of the observations made recently in [3] is that the pure spinor Lagrangian is invariant under the action of the global symmetry group PSU(2,2|4). This is in contrast with the case of flat space, where the Lagrangian is invariant only up to total derivatives. This observation was generalized in [4] where it was argued that the vertex operators for massless supergravity states can be chosen in a PSU(2,2|4)-covariant way. At this time there are two explicit examples of vertices: the vertex for the zero mode of the dilaton (the descent of the Lagrangian) introduced in [5] and the vertex for the β-deformation introduced in [4]. In this paper we will study the vertex for the β-deformation. We will be mostly concerned with the following subjects: • how the supersymmetry acts on β-deformations; • extension of an infinitesimal β-deformation; to a finite β-deformation; the homological per- turbation theory; • the spacetime picture. First steps towards the pure spinor description of the β-deformed AdS5 × S5 were made in [6], although our approach is somewhat different.1 We will now briefly outline our paper. 1.1. Deformations of the pure spinor action The Type IIB string worldsheet theory, in the pure spinor formulation, has the following struc- ture: 1. An action S which is assumed to be local and conformally invariant; 2. A pair of BRST operators QL and QR with the properties: Q2L =Q2R = {QL,QR} = 0; The “total” BRST operator Q is the sum of QL and QR : Q=QL +QR; 3. Two ghost number operators ghL and ghR , such that ghL(QL) = 1, ghL(QR) = 0, ghR(QL)= 0, and ghR(QR)= 1; 4. The composite b-ghost b++, b−−, which satisfy: {Q,b++} = T++, {Q,b−−} = T−−. Given a worldsheet theory with these axioms satisfied, we ask ourselves: how can such a theory be deformed? It turns out that the infinitesimal deformations are parametrized by integrated vertex operators2 (2)V1 : 1 The authors of [6] followed the method of twisted boundary conditions previously used in [7–9] in the context of Green–Schwarz approach. We are using a more straightforward approach, using the vertex operator and the homological perturbation theory. 2 The subindex 1 in (2)V shows that this is the 1-st infinitesimal deformation, and the superindex (2) indicates a 2-form.1 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 157∫ = + (2)S S0 ε V1 , Q=Q0 + εQ1, (1) where S0 is the undeformed original action, invariant under Q0. The integrated vertex operator should be a total derivative under the original BRST transfor- mation: (2) Q0V1  d(smth), (2) where  means that “equals on-shell”. The condition (2) guarantees that the deformed action is BRST-invariant at the first order; notice that the BRST transformation itself gets deformed, unless (2) is satisfied off-shell (which is usually not the case). Generally speaking, given the first infinitesimal deformation (2)V1 , it should be possible to construct the series: ∫ ∫ = + (2) + 2 (2)Sexact S0 ε V1 ε V2 + · · · , Qexact =Q0 + εQ1 + ε2Q2 + · · · (3) and obtain the full deformed theory. 1.2. Special case of β-deformation In this paper we will consider an example: the so-called β-deformations. These deformations were introduced in field theories by Leigh and Strassler in [10]. 1.2.1. First order in ε Let us first consider the β-deformation at the linearized level. In the pure spinor formalism the corresponding vertex operator has a very simple form [4]: (2) V1 = 1 Babj[a ∧ jb], (4) 2 where ja are the conserved currents corresponding to the global symmetries, and Bab is a con- stant antisymmetric tensor, the parameter of the deformation3: B ∈ (g ∧ g)0/g, (5) where: • g = psu(2,2|4) is the global symmetry algebra; indices a, b enumerate the generators of g; • the subindex 0 means that the “inner commutator” is zero, see Eq. (112); • the subspace g ⊂ (g ∧ g)0 is generated by f bca tb ∧ tc (for B ∈ g ⊂ g ∧ g we find that (4) is a total derivative); in other words we consider B1 and B2 equivalent if: Bab −Bab = f ab c1 2 cG . (6) 3 Here we consider the full supermultiplet of the linearized β-deformations, which is obtained when we act on the β-deformation by the supersymmetry which it breaks. To the best of our knowledge, the full supermultiplet of the β- deformations has not been previously studied. But there is a construction of the deformations of the AdS part of AdS ×S55 in [11,12], which must be related to the deformations of the sphere by the supersymmetry. 158 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215We will summarize the details about the symmetries of the vertex in Section 6. Notice that the representation in which the vertex transformation is finite-dimensional, and in particular it is not unitary. It is not unitary because the perturbations of the SUGRA fields are not square integrable. Notice that Ref. [13] classifies both normalizable and non-normalizable SUGRA solutions. 1.2.2. Second order in ε We want to construct the series of the form (3) so that Qexact is a symmetry of Sexact and Q2exact = 0. It follows from the general principles of string theory, that this should be always possible starting from the first order (2)V1 given by (4). The second order correction (2)V2 depends on B quadratically. It turns out that the dependence of (2)V2 on B is rather subtle. Notice that the space of linearized β-deformations (5) is fibered by the orbits of PSU(2,2|4). The structure of (2)V2 depends on which orbit B belongs to. The formula for (2)V2 is relatively simple when B satisfies a certain quadratic equation. This equa- tion says that the Schouten–Nijenhuis–Gerstenhaber bracket [[B,B]] is equivalent to zero. The standard definition of this bracket is: [[B,B]] ∈ g ∧ g ∧ g, (7) [[B,B]]abc = Be[af b Bc]fef . (8) However this definition does not respect the equivalence relation (6). The construction which does respect this equivalence relation is this one: [[B,B]] mod L, where (9) L is generated by f [ab A|m|c] m . (10) In our terminology, the β-deformation is called real if: [[B,B]] ∈ L. (11) We distinguish β-deformations of the following three types: real, complex, and obstructed. 1.3. Types of β-deformations 1.3.1. Real β-deformations In the special case when B ∈Λ2su(4), the condition (11) for real β-deformations is equivalent to: [[B,B]] = 0. (12) We explicitly constructed (2)V2 for real β-deformations in Section 8.5, Eqs. (219), (229). We find that (2)V2 is a polynomial function of the currents j and the group element g. For such B , we conjecture that the polynomial dependence of (2)Vn on the currents and the group element will persist at higher orders. This agrees with the formula for the obstruction suggested in [2]. In fact, we suspect4 that (2)V2 is always a polynomial function, but we point out that the formula is much 4 Because it was proven in [2] that there the obstruction to the existence of a polynomial solution only appears at the third order in , thus we expect to have problems only with (2)Vn for n > 2. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 159simpler when (11) is satisfied. In fact we do not even know the explicit formula in the case when B does not satisfy (11). Notice that for the real Lunin and Maldacena [1] solutions B satisfies a stronger condition: Beaf bef B cf = 0 (13) (no antisymmetrization of abc). Does (12) imply (13)? The condition (12) can be interpreted as a classical Yang–Baxter equa- tion for the r-matrix r = B [14]. If this condition is satisfied, then the antisymmetric tensor B defines a left-invariant Poisson structure on the supergroup PSU(2,2|4). In this context the so- lutions of (12) have been previously studied in the mathematical literature. For a compact Lie group (such as SU(4)) it was proven in [15] that (12) implies that B lies in the exterior product of an abelian subalgebra a ⊂ g: B ⊂ a ∧ a. (14) This means that in this case (12) implies (13). However, for non-compact groups (such as SU(2,2)) there are more general solutions. Solutions of the rank 8 for SU(2,2) were constructed in [16]. Therefore the results of [16] suggest that there are solutions more general than those consid- ered in [1], corresponding to the deformation of the AdS part of AdS5 × S5. But at this time we have not proven that such solutions would not be obstructed at the cubic and higher orders. 1.3.2. Obstructed β-deformations What happens for a general B? Generally speaking, any solution of the linearized supergravity can be “repaired” to the full exact nonlinear solution, if we dress it appropriately with the corrections to self-interaction. In other words, it is always possible to construct the series of the form (3) order by order in ε. But for a generic B the nonlinear solution will not be a polynomial in the current and the group element. In particular, the solution for a general B will not be periodic in the global time of AdS5. In other words, the nonlinear solution will not be a universal cover of any manifold with a nontrivial π1 (while AdS5 was a universal cover of the hyperboloid). See [17] for a discussion of a similar phenomenon in the classical string dynamics, and explanations therein. 1.3.3. Complex β-deformations It is natural to ask the following question: what is the condition on B necessary and sufficient for the nonlinear solution to be, order by order in ε, of the polynomial type? At this time, we do not have a full answer to this question in our approach. The condition (11) is probably sufficient, although we have only proven this at the order ε2. But it is not necessary. It appears too strong. For some B violating (11) there are still polynomial solutions. This can be seen using the solution-generating technique of [1]. As we will review in Sections 6.1.3 and 10.1 the space of linearized β-deformations has a complex structure, i.e. there is an operator I commuting with the PSU(2,2|4) such that I2 =−1. The results of [1] imply that if B corresponds to a polynomial solution then eIφB also corresponds to a polynomial solution. But the action of I violates the condition (11). In our terminology, the complex β-deformations are those which can be connected to the real β-deformations by eIφ . The analysis of [2] implies that the obstruction first appears at the third order of perturbation theory (i.e. ε3). The authors of [2] suggest the formula for the obstruction, which we review and supersymmetrize in Section 10.2—see Eq. (263) and its supersymmetric generalization (273). It 160 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215is interesting that this obstruction can almost be expressed in terms of the Schouten bracket, but not quite—see Section 10.2.3. Also, it appears that there are cases when B satisfies (273) but is not in the orbit eIφB of the real B . This means, provided that B satisfying (273) are indeed unobstructed, that not all of these solutions can be obtained by the solution-generating trick of [1] from the real solutions. Complex β-deformations receive α′-corrections. The known results from the field theory side [18] combined with the AdS/CFT correspondence imply that the complex solutions receive accu- mulating5 α′-corrections [7]. This suggests that there should be a proof of finiteness to all orders in α′ which works for B satisfying (11) and does not work for the complex β-deformation. The picture presented in the current literature [2,1,7,18] (as we understand it) is the following: • real β-deformations are periodic in global time, including the α′ corrections; • complex β-deformations are classically periodic, but receive accumulating α′-corrections quantum mechanically; • the β-deformations which we call “obstructed” are not periodic even classically. Brief review of the literature. As pointed out in [1], one can generate the Lunin–Maldacena background by performing a TsT chain of transformation on the AdS5 × S5 background: a T- duality transformation along one angular variable ϕ1, a shift in another angular variable ϕ2 and again a T-duality along ϕ1. The parameter of the deformation is introduced by the shift and it is therefore real. On the other hand, the β-deformed field theory is allowed to have complex β [10]. In order to generate a complex parameter in the dual geometry, we have to apply S-duality before and after the TsT chain, so we would have a STsTS chain (for further discussion, see [19]). Note that, since we perform twice the S-duality transformation, the original and the deformed solutions are in the same coupling regime. However, as it was pointed out by Frolov in [8], the S-duality step departs from the worldsheet treatment, as opposed to T-duality. Indeed, as it is discussed in Frolov, Roiban and Tseytlin’s paper [7], the T-duality can be implemented directly at the level of the worldsheet, so the starting point may be the classical Green–Schwarz action on AdS × S55 . They also say that there is no good reason to believe that the S-dual background will not be deformed by α′/R2 corrections, while TsT does not introduce any correction. In- deed, while T-duality and a coordinate shift preserve the 2d conformal invariance of the string theory, with S-duality things are very different and we may need to modify the classical su- perstring action by extra α′/R2 correction terms in order to ensure its quantum 2d conformal invariance. 1.4. Plan of the paper We start by reviewing the Lunin–Maldacena supergravity solution in Section 2 and discussing a remarkable duality relation between the RR and NSNS fields special for this solution. In Sec- tion 3 we list basic formulas for the pure spinor superstring in AdS5 × S5 and briefly discuss the descent procedure. Then in Section 4 we introduce the vertex operator which corresponds to the β-deformation at the linearized level. In Section 5 we discuss an additional constraint 5 These α′ corrections are “accumulating” in the sense that the corrected background is not periodic in the global time of AdS; the deviation from periodicity corresponds to the anomalous dimension on the field theory side. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 161on the parameter B , which we don’t understand as well as we would want to. We then dis- cuss the symmetries of the vertex in Section 6, with a surprising conclusion that our vertex is not strictly speaking covariant. In Section 7 we discuss the general deformation theory of the classical worldsheet action, and in Section 8 apply it to the β-deformation. In particular, in Sec- tions 8.5 and 8.6 we obtain the explicit formula for the second order correction (2)V2 . It turns out that the Schouten bracket [[B,B]] plays an important role, and we further study its prop- erties in Section 9. In Section 10 we discuss the complex β-deformations and the equation for the obstruction proposed by Aharony, Kol and Yankielowicz in [2]; we discuss the supersym- metric generalization of their formula. In Section 11 we show (at the linearized level) that the target space supergravity fields of our worldsheet theory agree with the known supergravity de- scription of the β-deformation. In Section 12 we explain (at the linearized level) the relation between our approach and the approach of [7–9,6] which uses the twisted boundary condi- tions. 1.5. Open questions We will list here several open questions: 1. The constraint on the internal commutator described in Section 5 has to be explained. 2. Even under the condition (11) we have only constructed the action up to the second order in . It should be possible to find an explicit expression for (2)Vn for n > 2. 3. Is it true that the condition for the existence of the classical periodic solution is given by Eq. (263) of Section 10.2? We must understand the relation to the covariant subcomplex of [4] suggested in Section 8.4.2. 4. It would be nice to prove the nonrenormalization theorem for real β-deformations without invoking the TsT-transformations argument of [19]. In the language of twisted boundary conditions, which we review in Section 12, how do we derive (11)? It has to be related to the BRST symmetry of the twisted boundary conditions. 5. Solutions of [16] mentioned in Section 1.3.1 have to be studied explicitly. Are they ob- structed at the higher orders of ε? 2. Lunin–Maldacena background 2.1. The solution of [1] Let us first briefly review the solution presented by Lunin and Maldacena. The Lunin–Maldacena solution was first introduced in [1], where the authors presented a method to generate solutions to supergravity. Their method apply to any theory with a U(1)× U(1) symmetry, meaning that we can choose suitable coordinates for the metric such that a torus is parametrized by two of its coordinates, ϕ1 and ϕ2. The idea of [1] is think about this as a compactifica√tions on this two torus and perform an SL(2,R) transformation on the parameter τ ≡√B12 + i g, where B12 is the NSNS B-field component along the directions on the torus, and g is the volume of the two torus. Actually, to get nonsingular metric from an originally nonsingular metric, the SL(2,R) ele- ment to act on τ must be judiciously chosen, such that the actual transformation one needs to do is 162 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215τ → ττγ = 1 + .γ τ Lunin and Ma[ldacena ap∑plied this method to the AdS5 × S 5 so(lu∑tion, an)d g]ot 6: 2 ds2 2 2 ( 2 2 2) 2 2 2 2 str =R dsAdS + dμi +Gμi dφi + γ̂ Gμ1μ2μ3 dφ5 i , ∑ i i3 with μ2i = 1, i=1 G−1 = 1 + γ̂ 2(μ2 )μ2 +μ2μ2 +μ2μ2 , γ̂ =R2γ, R4 ≡ 4πeφ01 2 2 3 1 3 N, e2φ = e2φ0G, NS = 2 (B γ̂R G μ2μ2 dφ dφ +μ2 2 )1 2 1 2 2μ3 dφ dφ 2 22 3 +μ3μ1 dφ3 dφ1 , C2 =−3γ (16πN)w1dψ, with dw = c s31 α αsθ cθ dα dθ, C4 = (16πN)(w4 +Gw1 dφ1 dφ2 dφ3), F5 = (16πN)(ωAdS5 +GωS5), ωS5 = dw1 dφ1 dφ2 dφ3, ωAdS5 = dw4 (15) where γ̂ is the deformation parameter, and (μi,φi) are the embedding coordinates of S5 into R6: μ1 = sinα cosψ, φ1 = θ + ϕ1 + ϕ2, (16) μ2 = sinα sinψ, φ2 = θ − ϕ1, (17) μ3 = cosα, φ3 = θ − ϕ2. (18) This is not the only possible deformation on AdS × S5, since one can select a different pair of U(1) symmetries, and combine them, constructing a multiparameter deformed solution. The solution (15) is called β-deformed geometry because it is claimed to be dual to the β-deformed Yang–Mills theory. 2.2. A relation between NSNS and RR fields in the Lunin–Maldacena background 2.2.1. A ∗-duality relation In this section we will discuss the relation between the NSNS and the RR 2-form fields of the Lunin–Maldacena solution, the supergravity dual of the Leigh–Strassler’s β-deformed gauge theory. It will turn out that the RR 3-form dC2 is in fact Hodge dual to the BNS: BNS klmij = cij ∂kClm (19) where c is some coefficient. This relation is not gauge-invariant under the gauge transformations of B , and is only valid in the special gauge which we are using. This is the “covariant gauge”, characterized by the following property: the correspondence between the NSNS field strength 3-form HNS and the B-field BNS: HNS → BNS such that HNS = dBNS (20) commutes with the action of SO(6) rotations of S5. 6 To save space, we follow [1] and define sα = sinα, cα = cosα, etc. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 163This relation is interesting, because it has an interesting worldsheet interpretation, which we will describe in Section 11.3 (at the linearized level). We will now check (19), first for the linearized case, and then for the full solution. 2.2.2. At the linearized level We will start by discussing the relation (19) at the first order in the deformation parameter. Notice that Maldacena and Lunin identify ϕ1 and ϕ2 as cyclic coordinates corresponding to their two U(1) symmetries; those two U(1) act as ∂ and ∂ . On the other hand Eq. (3.2) of their ∂ϕ1 ∂ϕ2 paper implies that in terms of φ1 and φ2 they act as: ∂ = ∂ − ∂ ∂ =− ∂ + ∂, . (21) ∂ϕ1 ∂φ2 ∂φ3 ∂ϕ2 ∂φ1 ∂φ2 For any vector field vμ, we can contract it with the metric and get a one form g(v), defined as: g(v)μ =(g v)νμν . Then Eq. (3.11) from Lunin–Maldacena implies: ∂ g( )= μ22 dφ 22 −μ3 dφ3, (22)∂ϕ1 ∂ g =−μ2 dφ +μ21 1 2 dφ2. (23)∂ϕ2 Now we see: ( ) ( ) ∂ ∂ BNS = γ̂ R2g ∧ g . (24) ∂ϕ1 ∂ϕ2 Then w(e can use tha)t for any two vector fields v and u:∗ g(v)∧ g(u) = ιvιu (volume-form). (25) The volume form of S5 in terms of (α, θ,ψ,ϕ1, ϕ2) is: s3αcαcθ sθ dα ∧ dθ ∧ dψ ∧ dφ1 ∧ dφ2. (26) This equation with our Eq. (24) and equation for C2 on p. 9 of Maldacena and Lunin imply our Eq. (19). 2.2.3. Exact relation for the full solution The deformed metric is: ds2 = dα2 + s2(dθ2 + (G 1 + 9γ̂ 2s4c2 2 2) 2 2 2 ( 2 2 2) 2γ α α αsθ cθ dψ ++Gsα dϕ1 +G sαcθ + cα dϕ2+ 2Gs2 2α cθ − 2) ++ (s dψ dϕ 2G s2θ 1 αc2θ − c2) 2 2α dψ dϕ2 + 2Gsαcθ dϕ1 dϕ2 (27) where G= 1 = 1 . (28) 1 + γ̂ 2(μ2 21μ2 +μ2μ2 +μ2 2 2 2 2 2 2 21 3 2μ3) 1 + γ̂ sα(cα + sαsθ cθ ) We may write it a 2 = 2 +( ( s: 2 + )2) 2 + (( + 2) 2 2 2) 2 +( ( 2 +) 2)dsγ dα μ2 μ3 dθ G 1 9γ̂ μ1μ2μ3 dψ G μ2 μ3 dϕ21+G μ21 +μ22 dϕ22 + 2G μ22 −μ23 dψ dϕ1 + 2G μ22 −μ21 dψ dϕ2 + 2Gμ2 dϕ1 dϕ2. (29)2 164 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215The determinant of th(e metric a)bove is g = 9μ2μ2μ2G2γ 1 2 3 μ2 22 +μ3 , (30) and the nonzero components of the inverse matrix are: gααγ = 1, gθθ 1 γ = , μ2 22 +μ3 gψψ = 1 (μ2μ2 +μ2μ2 2 2)γ 9 2 2 2 1 2 1 3 +μ2μ3 ,μ1μ2μ3 gψϕ1γ = 1 ( μ2 ) μ2 +μ2μ2 − 2μ2μ2 , 9μ21μ 2 2μ 2 3 = 1 ( 1 3 2 3 1 2 gψϕ2 μ2μ2 +μ2μ2 − 2μ2 )γ 1 2 1 3 2μ23 ,9 2(μ1μ 2 2 2μ3) μ2 +μ2ϕ ϕ − (μ2 −μ2)2g 1 1 = μ2 +μ2γ 1 2 γ̂ 2 + 1 2 1 2 ,9μ2μ21 2μ23 2 2 2 2 2 gϕ2ϕ2 = (μ2 +μ2)γ̂ 2 + μ2 +μ3 − (μ2 −μ3)γ 2 3 ,9μ21μ2 22μ3 4 2 2 2 2 2 gϕ1ϕ2 =−μ2γ̂ 2 μ2 +μ1μ2 −μ2μ3 −μ2γ 2 + .9μ2μ2μ21 2 3 The RR field strength F3 = dC2 is given by: = 1F3 !Fμνρ dx μ ∧ dxν ∧ dxρ (31) 3 where the only nonzero component is F 2αθψ = −12γπNsαs2αs2θ . Its Hodge dual on the de- formed sphere S5γ is: ∗F3 =√ ( ) g gααgθθ(F( gψψ ϕ1ψ ϕ2ψγ γ γ αθψ γ dϕ1 ∧ dϕ2 − )gγ dψ ∧ dϕ2 − gγ dϕ1 ∧ dψ=−1(6γπNG μ2μ2 +μ2 2 2 21 2 1μ3)+μ2μ3 dϕ1 ∧ dϕ2+ (μ21μ22 − 2μ2μ22 3 +μ23μ21 dψ ∧ dϕ1+ 2μ2μ2 ) )1 2 −μ22μ2 −μ23 3μ21 dψ ∧ dϕ2 . (32) From B2 = c∗F(3(and B = γ̂ R(2G μ2μ2 +μ2μ2 2 2) ( 2 2 2 2 2 2)2 1 2 1 3 +μ2μ3 dϕ1 ∧ dϕ2 + μ1μ2 − 2μ2μ3 +μ3μ1 dψ ∧ dϕ1+ 2μ2 21μ2 − 2 2 − 2 2)μ2μ3 μ3μ1 dψ ∧ )dϕ2 , (33) we get 4 c=− R . (34) 16πN 3. Notations and properties of the action In this section we will use [20–22]. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1653.1. Notations The global symmetry algebra is g = psu(2,2|4). It has the Z4 grading g = g0 + g1 + g2 + g3. Various worldsheet fields take values in g; the lower index will denote the Z4 grade of the field. For example, consider this field: w1+. The index 1 means that it takes values in g1, and the index + means that it has the conformal dimension (1,0); similarly the field w3− has the conformal dimension (0,1) and takes values in g3. The pure spinor ghosts are λ3 and λ̃1. The tilde over λ̃1 is redundant; it is to stress that this field would be right-moving in the free field limit. We will then sometimes write λ and λ̃ for short. The corresponding conjugate momenta are w1+ and w̃3−; the kinetic term in the action is given by Eq. (289) below. Once again, our notations are highly excessive because there is no such things as for example w1−. Therefore, we could have just written w+ and w− instead of w1+ and w̃3−. The pure spinor action is constructed out of the right-invariant current J =−dg g−1, (35) which is invariant under g→ gH , with H ∈ PSU(2,2|4) being a global parameter. We use notations from Section 2 of [22]. As in that paper, the spectral parameter of the Lax operat[or will be denoted by z. T]he Lax equation is ∂+ +L+[z], ∂− +L−[z] = 0, (36) where L+[z] = J −10+ −N+ + z J3+ + z−2J + + z−32 J1+ + z−4N+, (37) L−[z] = J0− −N− + zJ1− + z2J2− + z3J3− + z4N−. (38) We will also introduce l by l = log z. (39) Then the density o∣∣f the global conserved charges can be written as= dLj g−1 ∣∣ g. (40)dl l=0 Following Berkovits and Howe, we also denote by z, z̄ the worldsheet coordinates. We think that the meaning of z will be always clear from the context. For any x ∈ g we denote: xa = Str(xta). (41) In particular: ( ∣ ) j = Str g−1 dL ∣∣a ∣ gta . (42)dl l=0 166 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2153.2. BRST transformation of S0 2 ∫ ( S0 = R 2 1 3 1d zStr J2+J2− + J1+J3− + J3+J1− π 2 4 4 ) +w1+∂−λ3 +w3−∂+λ1 +N0+J0− +N0−J0+ −N0+N0− , (43) the ghost currents are: N0+ =−{w1+, λ3}, N0− =−{w3−, λ1}, (44) The BRST transformations of the currents are: QLJ3 =−D0λ3, QRJ3 =−[J2, λ1], QLJ2 =−[J3, λ3], QRJ2 =−[J1, λ1], QLJ1 =−[J2, λ3], QRJ1 =−D0λ1, QLJ0 =−[J1, λ3], QRJ0 =−[J3, λ1]. (45) We also use the Maurer–Cartan equation: D0+J1− −D0−J1+ + [J3+, J2−] + [J2+, J3−] = 0. (46) We get: QLL=−1d Str(λ3J1), (47) 4 QL=−1d Str(λ3J1 − λ1J3). (48) 4 This equation is the first step of the descent procedure for the Lagrangian itself. The second step is: ′QStr(λ3J1 − λ1J3)= ′ d Str(λ3λ1). (49) Notice that Eq. (48) can( be rewritt)en in the following way: QL=−1d Str Λg−1 dg . (50) 4 Using the notations o(f Section 7).1:(1) IQ =− 1 Str Λg−1 dg . (51) 0 4 3.3. Adding antifields The Q defined so far is only nilpotent on-shell. Indeed, we get: Qw1+ =−J1+, QJ1+ =−D0+λ̃1 − [J2+, λ3]. (52) To make Q nilpotent off-shell we have to introduce, following [23], the fermionic antifields w1+ and w3− inside the ac∫tion as a te(rm −w + 1 R2 2 ) w3−: S  0 → S0 − d z Str w π 1+ w3− . (53) O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 167The{se antifie}lds {satisfy the} constraints: λ̃1,w  1+ = λ3, w̃3− = 0. (54) We must also modify the BRST transformations: Qw1+ =−J1+ −w 1+, Qw̃3− =−J3− − w̃3−, Qw1+ =D0+λ̃1 − [N+, λ̃1], Qw̃3− =D0−λ3 − [N−, λ3]. (55) With this modific[a(tion, we obtain:2 = + { }) ] [ ]Q w1+ J2+ w1+, λ̃1 , λ3 + {λ3, λ̃1},w1+ (56) which is a combination of the Lorentz gauge transformation and the pure-spinor-constraint gauge transformation of w. Now we wou(ld like∣∣to modify th)e currents to include antifields. We propose: ĵ+ = g−1 dL+( ∣∣∣ − 4w  g, (57) dl ∣∣l= 1+ 0 ) dL ĵ− = −g−1 + 4w g. (58) dl ∣ = 3−l 0 We need the off-shell ver(sion of Eq. (35)7), an(d with Q)(modified according to (55):=− [z] 1 + + − 1 )QL+(z) D(+ λ)3 zλ1 z D0+λ − [N ,λ ]z [ ] z3 1 + 1+ 11 (− w1+, λ)3 , ( ) (59)z4 [z] 1 QL−(z)=−D(− )λ[3 + zλ1 ] + 1 − 3 (z D0−λ3 − [ )N−, λ3] (60) z z + 1 − z4 w3−, λ1 . (61) This means tha(t: ) Qĵ = d g−1(λ3 − λ1)g . (62) 4. Vertex corresponding to β-deformation 4.1. The vertex and its descent As discussed in Appendix A, we introduce a set of formal anticommuting constants , ′, ′′, . . . . 4.1.1. Definition of the vertex It was proposed in [4] that the unintegrated vertex corresponding to the β-deformation is given by this expression: V beta ( ) ab ,  ′ = (g−1 ) ( )(λ − λ )g g−13 1 ′(λ3 − λa 1)g . (63)b Here the indices a and b enumerate the adjoint representation of psu(2,2|4). Notice that (63) is antisymmetric under the exchange of a and b. Therefore this vertex is in the antisymmetric 168 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215product of two adjoint representations of psu(2,2|4). We will parametrize the β-deformations by a constant a(ntisym) metric(tensor Bab:[ ] ) ( )V B , ′ = Bab g−1(λ −1 ′3 − λ1)g g  (λ3 − λ1)g . (64)a b 4.1.2. Equivalence relation The antisymmetric product of two adjoint representations is not an irreducible representation. In particular, it has a subspace consisting of Bab of the form: Bab = f abc Ac. It turns out that such B corresponds to B[R(ST-exact vertices: f ab V beta ) ( )] c ab = g−1((λ3 − λ )g , g−1)′1 (λ3 − λ1)g c= Q g−1BRST ′(λ3 + λ1)g . (65)c Therefore the tensors Bab and Bab + f abAcc give the same β-deformation: Bab  Bab + f abAcc . (66) We will explain in Section 4.2 that the gauge transformation (66) should be accompanied by the change of variables (field redefinition). This is because the corresponding integrated vertex is only invariant on-shell. 4.1.3. Descent procedure and integrated vertex The deformation of the action corresponding to (63) follows from the standard descent proce- dure. Let us denote: Λa()= ( g−1 ) (λ3 − λ1)g . (67)a The operator Λa() corresponds to the local conserved currents in the following sense: dΛa()= Q(ja), (68) where ja±(τ+, τ−) is the density of the local conserved charge corresponding to the global symmet(ries. Theref(ore: d Λ[ ()Λ ] ′ )) a b = ( ) 2Qj[ ′aΛb]  (69) and ( ) 1 d j[aΛb]() =− Q(j[a ∧ jb]). (70) 2 We conclude that for any constant antisymmetric matrix Bab we can infinitesimally deform the worldsheet action as fo∫llows: S → S + 1Bab j[a ∧ jb]. (71) 2 4.1.4. Summar(y of the d(esce)nt procedure(0) (1) ( ) (2) ) (d +Q) V1 [B] , ′ + V1 [B] ′ + ′V1 [B] = 0, (72) where O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 169(0)[ ]( ′)= 1 ( ) ( )V1 B (,) Ba(b g−1(λ − λ )g) g−1′3 1 (λa 3 − λ1)g ,2 b(1) V [B] ′ = Babj g−1′1 a (λ3 − λ1)g ,b (2)[ ] = 1V B Bab1 ja ∧ jb. (73)2 In Eq. (72) we assume that d commutes with . 4.1.5. “Bosonic” example Consider for example(B∫ab in the directions of S5. We)get: S → S +B[kl][mn] X[k dXl] ∧X[m dXn] + · · · (74) where dots denote fermionic (θ -dependent) terms, and Xj describes the embedding of S5 into R6: X21 +X22 + · · · +X26 = 1. (75) These θ -dependent terms appear because ja includes θ . The subspace g ⊂ g ∧ g corresponds to B of the following form: B[kl][mn] = δkmAln − δlmAkn + δlnAkm − δknAlm (76) where Amn is antisymmetric matrix; then the corresponding deformation of the Lagrangian is a total derivative d(AmnXm dXn). The complementary space has real dimension 90, it corresponds to the representation 45C of so(6)—see Fig. 2 of [13], the point k = 2, M2 = 0. Indeed, that is the only subspace of classical solutions having the fields constant along AdS, and transforming under the so(6) as 45C. 4.2. What happens to the integrated vertex when Bab is proportional to f ab Acc ? 4.2.1. In this case the integrated vertex becomes a total derivative Indeed, consider the descent procedure. When Bab is proportional to the structure constant, this means that the vertex operator is of the form [g−1 dλg, g−1′ dλg]. Here we use: dl dl l = log z, (77) see Section 3.1. We want to apply the descent procedure and obtain the corresponding integrated vertex op- erator. The first step is to take the derivative of our unintegrated vertex and see that it is BRST exact: [ ] [ ] −1 dλ −1 ′ dλ dL dλd g  g,g  g =−2Q g−1 g,g−1′ g . (78) dl dl dl dl But now a special thing happens; on the right-hand side Q is taken of the expression which is d of someth[ing plus Q of someth]ing: ( 2 ) − dL2 g−1 g,g−1 dλ = −1 d L + ( g Q g g d g−1 )λg . dl dl dl2 This formula can be derived as follows: 170 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215( ) [ ] −1 d2L d2L d2Q g g = g−1 [ , λ g − g−1 (Dλ)gdl2 dl2 ] dl2 =− −1 dL d λ − (2g , g d g−1 )λg . (79) dl dl The second (and the last) step of the descent procedure is to take the d of −2[g−1 dLg,g−1 dλg] dl dl and see that it is Q of some expression, which is then the corresponding integrated vertex oper- ator. But we can see directly from (79) that in fact d of −2[g−1 dLg,g−1 dλg] is equal to Q of dl dl d(g−1 d2L2 g). This means that, indeed, the corresponding integrated operator is a total derivative.dl This can be easily seen explicitly. The integrated vertex operator is [j∧, j ] = −g−1[ dL∧, dl dL ]g. W(e observe:dl 2 )d L d2L d g−1 g = g−1D g dl2 [dl2 ] ( ) =−g− 2 1 dL∧ dL 1, g + g−1 d DL g. (80) dl dl 2 dl2 2 Notice that the second term on the right-hand side 1g−1( d2 2 DL)g is proportional to the equationsdl of motion. Therefore, this term should be canceled by an infinitesimal field redefinition. 4.2.2. Field redefinition 2 More precisely, Str(t g−1( da 2 DL)g) is the result of the variation of the action with respect todl the infinitesimal left shift of g by 8(gt g−1a )1 − 8(gt g−1a )3, plus some variation of λ and w. Let us denote this ve(ctor fiel(d Xa :2 ) ) X daS = Str t g−1a DL g (81) dl2 (where S is the action). We will not need the explicit form of Xa in this paper. We conclude that: • the infinitesimal gauge transformation Bab → Bab + f ab Acc changes the vertex by a to- tal derivative plus terms which can be absorbed into an infinitesimal field transformation corresponding to the vector field X aaA . 5. Constraint on the internal commutator 5.1. Additional constraint on B The vertex (64) cannot as such be the right description of the β-deformation because it gives extra states which are not present in the supergravity description. For example, consider B of the form: { ab c Bab = fc A if both a and b are even (bosonic) indices, (82) 0 otherwise where A ∈ so(6)⊂ psu(2,2|4). The corresponding linearized excitation of AdS ×S55 is constant in the AdS directions, and transforms in the adjoint representation of so(6) (rotations of S5). But there is no such state in the supergravity spectrum [13]. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1715.1.1. Conjecture It is necessary for the consistency of the deformed worldsheet theory that the vertex V (0) is given by a primary operator. This condition was investigated in [24]; it was found that the double pole of the vertex operator with the energy–momentum tensor is proportional to the action of the Laplacian on psu(2,2|4). In ou(r case, when)V (0) = BabΛaΛb, this is proportional to: Babf cf deΛ Λ =Q Babf cab c d e ab Λ̄c . (83) Therefore if Babf cab = 0 then the unintegrated vertex operator is not a conformal primary of the weight zero. Therefore we must impose this condition on B: Babf cab = 0. (84) 5.1.2. The descent of the anomalous dimension The worldsheet anomalous dimension7 of V (0) is Q0-exact: V (0) =Q (0)0U . (85) Let us act on this by d , and then use that dV (0) =Q V (1)0 : (− 1)dV (0) =Q dU(0)0 , (86) (− 1)Q V (1)0 =Q0 dU(0). (87) Then use that there is no Q0-cohomology in the conformal dimension 1 and ghost number 1. Therefore exists U(1) such that: (− 1)V (1) = dU(0) +QU(1), (88) (− 2)dV (1) =QdU(1), (89) (− 2)V (2) = dU(1). (90) Theref[ore:anomalous dimension ] [ anomalous dimension ] of the unintegrated vertex ⇒ of the integrated vertex . (91) is BRST exact is a total derivative 2 In our case Eq. (88) is log  times Eq. (79); i.e. U(1) is proportional to g−1 d L2 g.dl 5.1.3. Renormalization of the integrated vertex We can demonstrate that the integrated vertex has a nonzero anomalous dimension in the case when Babf cab = 0. Let us pick a point in AdS × S55 and consider the near flat space expansion around this fixed point as in [22]. This means that we write g = eϑ/Rex/R and expand around the selected point x = ϑ = 0. Suppose that the only nonzero components of B are in g0 ∧ g0 where g0 corresponds to the rotations around the selected point. In other words B is B[μν][ρσ ], where Greek letters are in the tangent space of S5. Then the integrated vertex V (2) contains the terms: B[μν][ρσ ]xμ dxν ∧ xρ dxσ . (92) The log divergence comes from the contraction of xμ and xρ : 7 Notice that the classical worldsheet dimension of V (0) is zero. 172 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215log εg B[μν][ρσ ]μρ dxν ∧ dxσ . (93) This is indeed proportional to Babf cab . 5.2. Example of an unintegrated vertex violating the constraint Babf cab = 0 Pick a(cons)tant A ∈( g[2(and consider the f)ollo(wing( vertex operators: V , ′ = Str A g−1 − −1 ′ ′ ) ) ])A (λ3 λ1)g 2, g  λ3 −  λ1 g 0 . (94) This operator{corresponds to the following Bab: Bab = f abcAc for a in g2 and b in g0 or a in g2 and b in g0, (95) 0 otherwise. The internal commutator is: Babf cab = Cso(6)Ac = 0 (96) where Cso(6) is the adjoint Casimir of so(6). Therefore the internal commutator constraint is not satisfied for this vertex. There is no such state in Type IIB SUGRA on AdS × S55 . 5.3. What happens with the vertex in the flat space limit In order to better understand this vertex we will consider its flat space limit. We will use the flat space expansion similar to the one used in [22]. We will write: g = ex2+θ3+θ1 (97) and consider x and θ small. To reproduce the flat space BRST operator we consider the “flat space scaling”: x R−2, θ R−1, λR−1. (98) R is the radius of the AdS space entering the action as in (43). Notice that there are two differences with [22]; [22] used a different gauge g = eθ ex ; also that paper used the “uniform” scaling x  θ  λ  R−1 which is different from the “flat space” scaling (98) which we use here. In the flat space limit the “flat space scaling” gives the BRST operator λα( ∂ + Γ m β ∂α αβθ m ), which is the correct BRST operator in flat space. While the∂θ ∂x “uniform” scaling x  θ  λR−1 used in [22] gives λα ∂ . ∂θα With these notations the vertex operator becomes a function of x, θ, λ. The BRST operator in terms of x, θ, λ is calculated in Appendix C. The expansion of the vertex (94) starts with the following( term)s:′ = ([ ([[ [ ] [ ]′] ]] [ [ ]]VA ,  Str A θ3, λ ′3 , θ1,  λ3 [+ [θ1, λ[1], θ3, ]]λ1− [ ′ ))θ3, λ1], θ3,  λ3 − [θ1, λ1], θ1, ′λ3 + · · · (99) where · · · stands for the terms of higher order in 1/R expansion. We used a Mathematica program to recast VA(, ′) in various BRST-equivalent forms. It turns out that VA(, ′) is BRST equivalent to the following expression: O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 173( ( −8 [[ ] [ [ ]]]Str A θ3, [θ , λ ′3 3] , θ3, θ3,  λ3 − 8 [[ 9 [ ] [ [ ]]] ))θ1, θ1, λ1] , θ1, θ ′1,  λ1 + · · · . (100) 9 On the ot(her( h[and[, V ′A(,  )[is also e]q]u]ival[ent to this:[ [ [ ][ [ [ [ ]]]]Str A 4 x, θ1, λ1 , θ3[, ′λ3 ]−]]] [θ1, )λ)1], θ3, θ ′3, θ3,  λ3− [θ3, λ3], θ ′1, θ1, θ1,  λ1 + · · · . (101) In both (100) and (101) · · · stands for terms of the order R−8 and higher in 1/R expansion. We will call (100) “the (2,0)+ (0,2)-gauge” and (101) “the (1,1)-gauge”. 5.3.1. Flat space notations Eqs. (100) and (101) are written in terms of the algebraic structures of psu(2,2|4), the com- mutator and the supertrace. It is possible to rewrite them using the gamma-matrices. The (2,0)+ (0,2)-gauge expression (100) reads: −8 ( ) ( )(θ3Γklmθ3) θ k l3Γ λ3 Ā θ3Γ m′λ3 9 − 8 ( ) ( )(θ1Γklmθ1) θ1Γ kλ Āl m ′1 θ1Γ  λ1 + · · · (102) 9 where we de{noted: l = Al if l ∈ {0, . . . ,4},Ā − l (103)A if l ∈ {5, . . . ,9}. On the other hand, the (1, (1) expressi)o(n (101) rea+ ( )( ′ ) ds: ( )( ) 2(Ā[ x ] A[ x̄ ]) θ Γ mm n m n 1 λ1 θ Γ n3  λ3 − )An(θ Γ λ ) θ Γ̄ [mΓ n]Γ lθ ′1 m 1 3 3 θ3Γl λ3−A (θ Γ λ ) θ Γ̄ [mΓ n]Γ ln 3 m 3 1 θ1 θ1Γl′λ1 . (104) We leave the target-space interpretation of these states (even in flat space) as an open question. 6. Symmetries of the vertex In this section we will collect the necessary fact from the representation theory and discuss the symmetries of our vertex. 6.1. Representation theory We will denote: g = psu(2,2|4), (105) ĝ = su(2,2|4), (106) ĝ′ = u(2,2|4). (107) 174 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2156.1.1. Matrix notations Elements(of ĝ′ are )block matrices:i i ua = uj uαb α α (108)ui uβ satisfying some hermiticity property. The precise form of the hermiticity property will not be important for us. The indices i, j, k, . . . correspond to the fundamental representation of the su(4) and the indices α,β, γ, . . . to the fundamental of the su(2,2). The letters a, b, c, . . . stand for either i, j, k, . . . or α,β, γ, . . . . 6.1.2. Exterior product of two adjoint representations Let us consider the exterior product of two adjoint representations of ĝ′: ĝ′ ∧ ĝ′. (109) In matrix notations, this is the space of matrices bacbd satisfying the antisymmetry property: bac = (−)(ā+b̄)(c̄+d̄)+1bcabd db (110) where Ī is 0 if I is the index of su(2,2) and 1 if I is the index of su(4). The difference between g and ĝ′ is in the central charge c and the differentiation s. The central charge is the unit 8 × 8 matrix, and the differentiation is diag(1,1,1,1,−1,−1,−1,−1). Let Rs denote the 1-dimensional linear space spanned by the differentiation and Rc the 1- dimensional linear space spanned by the central charge. Let us consider ĝ′ ∧ ĝ′ as a representation of g. We observe that ĝ′ = g + Rs + Rc. Therefore: ĝ′ ∧ ĝ′ = g ∧ g + Rs ⊗ g + Rc ⊗ g + Rs ⊗ Rc. (111) We observe the following facts about g ∧ g: 1. The representation g ∧ g is n∑ot irreducible, because∑it contains two invariant subspaces: (g ∧ g)0 consisting of xI ∧ yI such that [xI , yI ] = 0, (112) I I g ⊂ (g ∧ g)0 spanned by f bca tb ∧ tc. (113) 2. We therefore have two exact sequences: 0 → g → (g ∧ g)0 → (g ∧ g)0/g → 0, (114) 0 → (g ∧ g)0 → (g ∧ g)→ g → 0. (115) Both of them do not split. This means that there is no complementary subspace to (113) in (g ∧ g)0 and no complementary subspace to (112) in g ∧ g. We introduce an “inner com(m∑utator” map)F : ∑ F : g ∧ g → g, F xI ∧ yI = [xI , yI ]. (116) I I With this notation (g ∧ g)0 = KerF . Notice that (x, y) = Str(xy) is a non-degenerate symmet- ric scalar product, but this scalar product is not positive definite. Let F ∗ denote the conjugate O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 175to F with respect to this scalar product. Because the Casimir operators vanish in the adjoint representation we have: FF ∗ = 0. (117) On the other hand, F ∗F : g ∧ g → (g ∧ g)0 (118) is nonzero. In fact F ∗F can be identified with the action of the quadratic Casimir of psu(2,2|4) on g ∧ g:  ab ∗2 = C tatb = F F. (119) Note that 2 on g ∧ g is nilpotent: (2)2 = 0. We conclude that: • The space of linearized β-deformations (g ∧ g)0/g can be identified with Ker2Im .2 6.1.3. Complex structure Notice that ĝ′ ∧ ĝ′ has a complex structure, which acts as a multiplication by i and the ex- change of the upper indices: IbIK = ibKIJL JL. (120) Notice that I2 =−1. Let us discuss the action of I on the decomposition (111). We get: I(Rc ⊗ g)= (g ⊂ g ∧ g), (121) I(g ⊂ g ∧ g)= (Rc ⊗ g). (122) Generally speaking I(x ∧ y) has a component in Rs ⊗ g, but when restricted on (g∧ g)0 it lands into (g ∧ g)0 + Rc ⊗ g. We conclude that: • the operation I induces a complex structure on the space of linearized β-deformations (g ∧ g)0/g. 6.2. Our vertex is not covariant The linearized β-deformations transform in the following representation of g = psu(2,2|4): (g ∧ g)0/g. (123) But the B tensor satisfying Babf cab = 0 is in (g ∧ g)0. The object transforming in (g ∧ g)0/g is the equivalence class of Bab  Bab + f abcAc. Let us denote this equivalence class [B]. The short exact sequence (114) does not split. Therefore it is not possible to pick a representative for B in the equivalence class [B] in a way consistent with the supersymmetry. It was argued in [4] that there is always a way to choose the vertex covariantly, i.e. in a way consistent with the supersymmetry. Generally speaking, the action of the supersymmetry on the vertex operator agrees with the action on the corresponding state (i.e. on the BRST cohomology class) only up to Q-exact terms. It was proven in [4] that in AdS5 × S5 it is possible to choose a representative vertex in each cohomology class, so that the action of the supersymmetry on this representing vertex 176 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215agrees with the action on the cohomology class; the agreement is precise, in a sense that no Q-exact terms are needed. This is called the “covariant vertex”. However the proof used the assumption that the representation of g in which the state transforms has a sufficiently large spin when restricted to so(6)⊂ g. The β-deformation is a low-spin case, so there is an obstacle to choosing the vertex in a covariant way. 7. General deformation theory In this section we will discuss the general theory of extending an infinitesimal deformation of a BRST-invariant string worldsheet action to a finite deformation. We have not been able to find such a discussion in the literature. 7.1. Some general notations We will consider the deformation of the Lagrangian of the following form: L(2) = L(2) + (2) (2)deformed εV 21 + ε V2 + · · · . (124) Here the upper index (2) indicates that the object is a 2-form (e.g. the action density L(2)). When we say that some equation is valid “on-shell” we will generally speaking mean on-shell with respect to the undeformed Lagrangian L(2). We will write: F  0 (125) when F is zero up to the equations of motion of L(2). Given some infinitesimal field transforma- tion ξ we will define (1)Iξ by the following formula: ξ.L(2)  (1)dIξ . (126) This equation holds on-shell, but we will consider it in situations where it actually defines (1)Iξ also off-shell. The only ambiguity would be to add to (1)Iξ some local conserved current, but for those ξ which we need there will be no local conserved currents with appropriate symmetries. We conclude that: • for every vector field ξ there is a 1-form (1)Iξ defined by (126); it is defined up to d of something. 7.2. Deforming with integrated vertex operator Let us return to the descent procedure discussed in Section 4.1, Eq. (72). The general relation is: (d + ( )Q) V (0) + V (1) + V (2) = 0. ∫ (127) Given∫the vertex oper∫ator V , w∫e can perturb ∫the action by adding to it V (2): [ (2) (2) (2)dg dλdw]e L → [dg dλdw]e (L +εV1 +···) (128) where ε is an infinitesimally small parameter. (The lower index 1 in (2)V1 is to indicate that (2) V1 is the coefficient of the first power of ε.) O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1777.2.1. BRST invariance at the first order in ε The vertex operator (2)V1 in (128) should be such that on-shell (2) Q0V1 is a total derivative. Generally speaking this means that there exists an odd vector field, which we will call Q1, and a 1-form (1)X1 such that: (2) Q V +Q L(2) = (1)0 1 1 dX1 . (129) Comment 1. Notice that Eq. (129) determines Q1 up to an infinitesimal transformation of ghost number one which leaves the action invariant. We assume that there are no such infinitesimal transformations except for Q0 (in other words the BRST symmetry is the only symmetry with the ghost number one). Under this assumption the infinitesimal transformation Q1 is defined by (129) unambiguously. Comment 2. The combined Q0+εQ1 is a symmetry of the action off-shell. But is this a nilpotent symmetry? In other words, is it true that {Q0,Q1} = 0? In fact this is true, for the following reason. Observe that if {Q0,Q1} is not zero, then it would be a symmetry of the unperturbed action: {Q0,Q1}S0 =−Q2V (2)0 = 0. (130) Under the assumption that there are no symmetries of the ghost number 2, we conclude that {Q0,Q1} should be zero.8 Therefore Q0 + εQ1 is automatically nilpotent. 7.2.2. BRST invariance at the second order in ε Eq. (129) guarantees that the deformation (128) exists at the first order in ε. Similarly, the consistency condition at the second order in ε is: (2) Q1V1 + (2) (1)Q0V (2)2 +Q2L = dX2 . (131) This equation is a definition of and (2); the existence of and (2)Q2 V2 Q2 V2 satisfying (131) is the consistency condition. But it is more convenient to describe the consistency condition in terms of the dimension zero operators (unintegrated vertices). We will now translate the consistency condition (131) from the dimension two language to the dimension zero language. 7.2.3. The descent of (2)Q1V1 Let us act on (129) w(ith Q 91. We get :− (2)  (1) (1))Q0Q1V1 d Q1X1 − I 2 . (132)Q1 Therefore (1)− (1)d(Q0Q1X1 Q0I 2 ) 0. In fact this is also true off-shell because there are no localQ1 conserved( charges of the)ghost number three:(1) − (1) = (0)Q0 Q1X1 I 2 dW2 . (133)Q1 This equation is the definition of (0)W2 . An alternative notation for (0) W2 could be − (2)(Q1V )(0)1 . 8 See footnote on p. 7 of [21]. A.M. would like to thank V. Puletti for a discussion about this. 9 The only reason why this equation would not hold off-shell is the use of Q21L= (1)dI 2 .Q1 178 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215Condition on (0)W2 . Now we want to derive a constraint on W 0 2 following from (131). On the left-hand side of (132), let us replace: (2) (1) (2) Q1V1 → dX2 −Q0V2 −Q2L as follow( s from (131), and use L (1) ) Q2( dIQ . We get:2− (1) (1) (1)d Q0X2 +Q0IQ  d Q1X1 − (1))I 2 . (134)2 Q1 This implies: − (1) (1) (1) (1)Q0X2 +Q0IQ =Q1X1 − I 2 + d(smth). (135)2 Q1 Therefore in this case (0)W2 defined by (133) is Q0-exact: (0) W2 =Q0(smth). (136) 7.2.4. Going back Now suppose that (136) is satisfied: (0) (0) W2 =Q0T2 . Then we (get: (1) Q0 Q1X1 − (1)I 2 − (0) ) dT2 = 0. (137)Q1 Let us assume that the following is true: • the cohomology of Q0 on 1-forms of the ghost number 2 is trivial. Then (137) implies the existence of (1)X2 such that: (1) − (1)Q1X1 I 2 = (0) (1)dT2 −Q0X2 . (138)Q1 Let us compare this to (132): − (2) (1) (1)Q ( ) 0 Q1V1  d(Q1X1 − I 2 ). We get:Q1 (2) (1) Q0 Q1V1 − dX2  0. (139) Let us assume that: • the covariant cohomology of Q0 on 2-forms of the ghost number 1 is trivial. Then (139) implies the existence of (2)V2 and Q2 such that (131). This means that (136) is the necessary and sufficient condition for the deformation to exist at the second order in ε. Comment about Q2. As is shown in Section 7.3, in our particular case (the beta-deformation) (0) W2 = 0 implies Q21 = 0. This implies that Q2 = 0 (because Eq. (139) holds true off-shell; see the footnote before Eq. (132)). O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1797.3. Higher orders of perturbation theory 7.3.1. Going forward (necessary condition) Suppose that we have identified (2)Vp and Qp up to the order n, i.e. for p = 1,2, . . . , n, so that: (2) (2) (2)Q V (2) (1)0 p +Q1Vp−1 + · · · +Qp−1V1 +QpL = dXp , Q0Qp +Q1Qp−1 + · · · +Qp−1Q1 +QpQ0 = 0. (140) Then: Q Q V (2) + (2) (2) (2)1 0 n Q1Q1Vn−1 +Q1Q2Vn−2 + · · · +Q1Q − V +Q Q L(2)n 1 1 1 n + (2) (2) (2) (2)Q (2)2Q0Vn−1 +Q2Q1Vn−2 + · · · +Q2Qn−2V1 +Q2Qn−1L +Q3Q0Vn−2 + · · ·(+ − (2)Q3Qn 3V1 +Q3Q − L(2)n 2 + ·)· · + (2)Q Q V +Q Q L(2)n 0 1 n 1−Q( (2) (2) (2) (1)0 Q1Vn +Q2Vn−1 + · · · +QnV1) + dIQ1Qn+Q2Qn−1+···+QnQ1 (1) (1)d Q1X(1)n +Q2Xn−1 + · · · +QnX1 . (141) This impl(ies the existence of (0) Wn+1 such that: (1) (1) ) (0) Q0 Q X (1) 1 n + · · · +QnX1 − IQ1Qn+···+Q = dWnQ1 n+1. (142) In other words, the validity of (140) for p ∈ {1, . . . , n} allows us to define (0)Wn+1 by Eq. (142). Now, suppose that we can construct (2)Vn+1 and Qn+1, so that (2) Q0Vn+1 +Q V (2)1 n + · · · + (2)Q V +Q + L(2)n 1 n 1 = (1)dXn+1. (143) Then this(implies that (0) Wn+1 satisfi)es some(conditions. Ind)eed, applying Q0 to (143) we get:(2) (1) (1) Q (2)0 Q1Vn + · · · +QnV1  dQ0 Xn+1 − IQ + . (144)n 1 Now, re(turning to (141) we derive: d Q (1)1Xn +( (1) (1) (1) ) Q2Xn−1 + · ·)· +QnX1 − IQ1Qn+Q2Qn−1+···+QnQ1=− (1)dQ0 Xn+1 − (1)IQ + (145)n 1 and therefore there exists a (0)Tn+1 such that: Q X(1) (1) 1 n +(Q2Xn−1 + · · ·)+ (1)QnX1 − (1)IQ1Qn+Q2Qn−1+···+QnQ1=− (1)Q0 Xn+1 − (1) (0)IQn+ + dT1 n+1. (146) This implies: (0) (0) Wn+1 =Q0Tn+1. (147) 180 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2157.3.2. Going back (sufficient condition) Now s(uppose that (147) holds. Then Q Q X(1)0 1 n + · · · + (1) − (1)QnX1 IQ Q +···+Q Q − (0) ) dTn+1 = 0. (148)1 n n 1 Assuming the triviality of the cohomology of Q0 on the 1-forms of the ghost number 2, we conclude the existence of (1)Xn+1 such that: Q X(1)1 n + · · · + (1)QnX1 − (1) (0) (1)IQ1Qn+···+Q Q = dTn+1 −Q0Xn+1. (149)n 1 This and (141) imply, under the assumption that the cohomology of Q0 on the 2-forms of the ghost number 1 is zero, the existence of (2)Vn+1 such that: (2) + (2) + · · · + (2) (1) (2)Q1Vn Q2Vn−1 QnV1  dXn+1 −Q0Vn+1. (150) This means that (0)Wn+1 being Q0-exact is not only a necessary, but also a sufficient condition to be able to extend the deformation to the order n+ 1. The off-shell version of Eq. (150): (2) (2) (2) (2) (1)Q0Vn+1 +Q1Vn +Q2Vn−1 + · · · +QnV1 +Qn+1L= dXn+1. (151) This is( the definition of Qn+1. Notice that so)defined(Qn+)1 satisfies: Q0 + εQ1 + ε2Q2 + · · · + 2εn+1Q + n+2n 1 =O ε . (152) The proof goes by induction. We start with Q20 = 0. The induction hypothesis is (Q0 + εQ1 + · · ·+ εnQ )2n =O(εn+1), and this guarantees that (Q0 + εQ1 +· · ·+ εn+1Q 2n+1) is also at least as sma(ll as O(εn+1): Q0 + εQ1 + · · · + εn+1 )2 ( ) Q n+1n+1 =O ε . (153) By con∫st(ruction: Q (+ εQ) + · · · + εnQ + εn+1 )( Q + L(2) + (2)εV + · · · + εn+1 (2) ) 0 1 n n 1 1 Vn+1 =O εn+2 . (154) Theref∫or(e: )2( (2) (2) ) Q0 (+ εQ1) + · · · + εnQ n+1 (2) n+1n + ε Qn+1 L + εV1 + · · · + ε Vn+1=O εn+2 . (155) This a∫nd((153) imply: Q0 + εQ1 + · · · + εnQn + +1 )2 ( ) εn Q + L(2)n 1 =O εn+2 . (156) Let Pn(+1 denote the coefficient of εn+1: )2 Q0 + εQ1 + · · · + εnQ n+1n + ε Q + = εn+1n 1 Pn+1 + · · · . ∫ (157) Then (156) implies that Pn+1 is a symmetry of the undeformed theory: Pn+1L= 0. Under the assumption that the pure spinor superstring in AdS 55 × S does not have conservation laws of the ghost number two, it follows that Pn+1 = 0. This completes the step of the induction. Conclusion. Provided that the deformation of the action is defined up to the order n in ε, the obstacle to defining the deformation to the order n+ 1 is the Q0 cohomology class of (0)Wn+1. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1818. Applying the general theory to beta-deformation Here we will apply the general theory developed in the previous section to the particular case of beta-deformation. 8.1. Calculation of Q1 The of(f-shell version)of Eq. (70) for (2)QV1 is: 1 Q Babja ∧ j = Babb dΛa()∧ jb. (158) 2 For the deformed action to be BRST-invariant off-shell, we need to modify the BRST transfor- mation: Q=Q0 +Q1 (159) where Q0 is the original (pure AdS × S55 ) BRST transformation, and Q1 is the modification. We will now argue that Q1 is in fact a psu(2,2|4) transformation with the spacetime- dependent parameter. First of all, notice that under the global rotations: g → gg0, g0 = const (160) the Lagrangian is invariant. But what will happen if we allow g0 to depend on τ±? Consider an infinitesimal transformation: ( ) δg = gα where α = α τ+, τ− . (161) Then: δJ =−g dα g−1 (162) and the variation(of the Lagrangia)n is: δL= 1 dLStr g−1 g ∧ dα 4 dl = 1 [( + −1 ) (Str ĵ+ 4g w+g ∂−α − ĵ− − 4g−1 ) ]w1 3−g ∂+α dτ+ ∧ dτ−. (163)4 In order to derive this formula, it is useful to rewrite the Lagrangian (43) in the following inter- esting form: ( L= 1 dLStr( J+ − ∣∣ ∣∣ ∣∣ ) +w1+∂−λ3 +w3−∂+λ1 −N0+N0− −w 4 dl = 1+ w3− l 0 = −1 dLStr + ∣∣ )J− +w1+∂−λ3 +w3−∂+λ1 −N0+N0− −w w4 dl = 1+ 3− (164)l 0 and then use (162). We are now ready to calculate Q1. To start, let us consider the following infinitesimal trans- formation Ξα : ( ) Ξ α: Ξαg = gα(, Ξαw) 3− =P g∂−αg−131 3, Ξ  −1αw =P13 g∂+αg . (165)1+ 1 182 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215This transformation combines the “localized” rotation (161) with the shift of the antifields w. Here the projectors P13 and P31 are defined by the formulas: P13A1 =A1 + [λ3, smth2], [λ1,P13A1] = 0, P31A3 =A3 + [λ1, smth2], [λ3,P31A3] = 0. (166) These projectors are non-polynomial as functions of λ. The purpose of these projectors in (175) is to enforce the constraints (54). 8.1.1. On the definition of P In this paragraph we will prove the existence of the projector P13 satisfying these properties. We will start with the following lemma: Lemma 8.1.1. If {[S2, λ3], λ1} = 0 then [S2, λ3] = 0. Proof. Notice that {[S2, λ3], λ1} ∈ C ⊗ g2 (the complexification of g2). Using the spinor no- tations: {[S2, λ3], λ1}m = (λ1,ΓmF̂Sn2Γnλ3) where F̂ is the Ramond–Ramond 5-form field strength of AdS ×S55 contracted with the gamma-matrices. Then {[S2, λ3], λ1} = 0 would imply that ( ) λ1,X m 2 Γ F̂S n m 2Γnλ3 = 0 (167) for any vector X2. Let us introduce the notation X̄2 for the vector with the components Xm2 for m ∈ {0, . . . ,4} and −Xm2 for m ∈ {5, . . . ,9}. Let X̄2 run over the space of vectors annihilating λ3. Then for such X2 (167) becomes10: Str(λ1λ3)Str(X̄2S2). (168) Therefore our assumption that {[S2, λ3], λ1} = 0 implies that the scalar product of S2 and X̄2 is zero for any X̄2 such that [X̄2, λ3] = 0 (i.e. for X̄2 ∈ Ann(λ3)). Because Ann(λ3) has the maximal dimension possible of a null-plane, this implies that S2 itself belongs to the annihilatorof λ3, i.e. [S2, λ3] = 0. This proves the lemma. If this was not the case, then the action would have a gauge symmetry w → w +[S2+, λ3] with S2+ satisfying {[S2+, λ3], λ1} = 1+ 1+ 0. We use the notation for the annihilator of a pure spinor: Ann(λ3)= the subspace of g2 consisting of vectors X2 such that [X2, λ3] = 0. (169) Notice that Ann(λ3) is a 5C-dimensional null-subspace11 of the complexification of g2. Our Lemma 8.(1.1 imp→ { lies that: }) Ker X2 [X2, λ3], λ1 = Ann(λ3). (170) Therefore the image of the map X2 → {[X2, λ3], λ1} is also a 5C-dimensional space (because C ⊗ g2 is 10C-dimensional). This image is a subspace of Ann(λ1). But notice that Ann(λ1) is 10 Notice that Str(X̄2S2) is the scalar product of X̄2 and S2. 11 A pure spinor defines a null-plane of the maximal possible dimension. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 183itself 5C-(dimensi{onal. This im→ [ ] } p)lies: Im X2 X2, λ3 , λ1 = Ann(λ1). (171) Given that {λ1,A1} is in the annihilator of λ1, we conclude that there exists X2 such that {λ1,A1} + {λ1, [λ3,X2]} = 0. This proves the existence of the projector P13. The existence of P31 can be proven similarly. The Lagrangian depends on the antifields w through the last term in (43); therefore the part of the Lagrangian inv AF =− ( olving w)is: L Str w+w1 3− . (172) When we calculate the variation Ξ LAFα , the projectors P13 and P31 drop out because of the constraint (54). We get: Ξ LAF =− ( ) ( )Str ∂+αgw−g−1 − Str gw+g−1α 3 1 ∂−α . (173) Combining (173) and (163) we get: 1 1 ΞαL= Str[j+∂−α − j−∂+α]dτ+ ∧ dτ− =− Strdα ∧ j. (174) 4 4 Now we obse[rve that Q1 is an examp(le of(Ξα for)a )particular value of α, namely α =ΛaB abtb: Q = 4 Bab + ab P −1 α̇ δ1 Λatb ∂+ΛaB 13 gtbg ( ( ) ) ] 1 δwα̇1++ α δ∂−Λ ab −1aB P31 gtbg 3 α . (175)δw3− Indeed, Eqs. (174) and (158) imply that so defined Q1 satisfies: Q1L+ (2)Q0V1 = 0 (176) which is the defining equation of Q1. In the first term BabΛatb of Eq. (175) tb stands for the generators of the psu(2,2|4) rotations, which act only on the matter fields: δag = gta (they do not touch the ghosts and the antifields). 8.2. Deformation of the BRST current We have seen that the BRST transformation is deformed, Eq. (175). Not surprisingly, the BRST current is also deformed. We will now calculate the deformation of the BRST current. 8.2.1. General procedure for calculating the current density To calculate the deformed charge density we will use a well-known general procedure. Given the action S[φ] invariant under some global symmetry transformation δφa = ξa we consider the position-dependent transformation δuφa = u(τ,σ )ξa and calculate the variation of the action. This should b∫e proportional to the derivatives of u and (as any variation) should vanish on-shell: δuS = (j+∂−u− j−∂+u) 0. (177) Then it follows that the current j± is conserved: ∂+j− − ∂−j+ = 0. 184 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2158.2.2. Particular case of BRST transformation Original (undeformed) BRST current. Let us start by deriving the BRST charge in the case of pure AdS5 × S5. Let us concentrate on QL. In this subsection δ will stand for QL, and δu for QL with the replacements: • δg = λ3g replaced with δug = uλ3g; • δw1+ =−J + − w+ replaced with δ w + =−uJ + − uw1 u 1 1 ;• δw3− =D0−λ3 − [ 1 1+ N0−, λ3] replaced with δuw3− = uD0−λ3 − u[N0−, λ3]. We can ta[utologically]rewrite: δ uS g,λ, = [w,w −1 ]+ ( [ ]− [ −1 ])(δS) g,uλ,u w,w δuS g,λ,w,w (δS) g,uλ,u w,w . (178) We observe that (δS)[g,uλ,u−1w,w] = 0 because the action is BRST-invariant. On the other hand, in the second line of (178), the difference between δuS[g,λ,w,w] and (δS)[g,uλ,u−1w, w] is in two places: • the variation of the term (w1+∂−λ3); • the variation of −w1+w3−. Therefore: [ ] [ ] δuS∫g,λ,w,w − ((δ(S) g,uλ,u−)1w,w= )dτ dσ ∂−uStr  J + +w1 1+ λ3 +w1+λ3 . (179) The terms containing w1+ cancel out, and we get: jL+ =−Str(J1+λ3), (180) jL− = 0. (181) Notice that the deformation of the BRST transformation given by Eq. (175) does not contribute to the deformation of the BRST current. Deformation∫of the BRST(curr(ent. The additio)na)l te(rm in(the action is: ) ) S(2) = dL+ dL−dτ dσ Bab g−1 − 4w1+ g g−1 + 4wdl dl 3− g . (182)a b Remember th[at Qw3− =D] 0−λ(3 − [N2 ∫ −( ( 2 )[ −, λ3] and therefo]re we get: δ S( ) g, λ,w,w  δS( ) g,uλ,u−1u )w,)w (183) =− dτ dσ Bab −1 dL+g − ( )4w1+ g g−1(4∂−uλ3)g . (184)dl ba This means that the deformed BRST curr(ent is:−1 )j abL+ =−Str(J1+λ3)− 4B ja+ g λ3g , (185)b jL− = 0. (186) O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1858.2.3. Conservation of the deformed current Deformed equation(s of m)otio(n. Under th)e variation δξ3g = ξ3g we get: δ L+ =−D+ z−1ξ + z−5 − z−1ξ 3 [N+, ξ3] + ( z−4 − )1 [J1+, ξ3], (187) δξL− =− ( −1 )+ ( )D− z ξ3 z3 − z−1 [ (N−, ξ ] + z−1 − z3)3 D0−ξ3. (188) In particu∣lar: d ∣∣∣ [ ]= dL∣ δξL+ D+ + ξ3 − [ , ξ3]− 4[N+, ξ3] − 4[J1+, ξ3], (189)dl d ∣l=0 dl∣∣ dL−δξL− =D−ξ3 − , ξ3 + 4[N−, ξ3] − 4D0−ξ3. (190)dl l=0 dl The variation∫of the undeformed action gives: δξS0 = ( ( )) Str ξ3 D0−J1+ − [N−, J1+] + [J1−,N+] . (191) The variation of th(e “small case currents” is): ( ) δξ j+ = g−1(−4[N+, ξ3] − 4[J1+, ξ ] g + ∂+ g−13 ξ3g , (192) δξ j− = g−1 4[ ) ( ) N−, ξ3] + 4[J1− + J − + J −, ξ ] g − 3∂− g−12 3 3 ξ3g . (193) Therefor∫e: ( ( ) ) δ dτ dσ Babj +j − = Bab −1ξ a b g −(4[N(+, ξ3] − 4[J1+, ξ3] g ja b−+ ) )Babj(a+(g−1 4[N)−, ξ3] + 4[J1− (+ J2− + J3−, ξ3] g b ∫ + ) ) Bab ∂+ g−1ξ3g jb− − 3j −1a a+∂− g ξ3g . (194)b The last term is equivalent to 4(∂−ja+)Bab(g−1ξ3g)b . We conclude that the deformed equation of motion for J1+ is: D −J + − [N−[ , J +] + [J −,N+] + [4 N+ + J]+, gt g−1] ab0 1 1 1 1 a 1B jb−− 4j aba+B N− + + ( )J1− J2− + J3−, gtbg−1 + 4Bab(∂−j +) gt g−11 a b 1 = 0. (195) We will also need the deformed equations of motion for λ3, which is obtained by varying the action with respect to w1+: [( D −λ − [N−, λ ] − 4Bab gt g−1 ) ] 0 3 3 a 0, λ3 jb− = 0. (196) Holomorphicity of t(he cu(rrent. Consider the d)eriv(ative of the current given by Eq. (185):− = ) )∂−jL+ Str J1(+ D0−λ(3 − [N−, )λ)3] + D0−J1+ − [N−, J1+] λ3+ ∂− 4Babj −1a+ g λ3g . (197)b Substitution of (195) and (196) into this formula gives ∂−jL+ = 0. 186 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2158.3. Relation between (0)W2 and the Schouten bracket on Λ •g Eq. (136) tells us that a necessary condition for the deformed theory to be BRST invariant to the order ε2 is that (0)W2 is BRST exact. Here we will explicitly calculate (0) W2 for the beta- deformation and express it in terms of the Schouten bracket on Λ2g. We start with the observation that actually: (2) Q V +Q L(2)0 1 1 = 0. (198) In other words (1)X1 = 0. Notice that generally speaking we only have Eq. (129), but in our particular case V = 1 2B abja ∧ jb we claim a stronger equation (198). It remains to calculate I 2Q2 . Let us first calculate Q1. Let us split Q1 = QF1 +QAF1 where1 QF = 4Λ Bab1 a tb is the first term on the right-hand side of (175) and QAF1 is the sum of the remain[ing two term(s ()i].e. F stands for field(s a)nd AF for antifields). We get: QF (),QF ′ = ( )1 1 16Λa()BapΛ ′ Bbqf r ap r ′ bqb pq tr(+ )16 × 2Λa()B fpb Λr  B tq= 16 × 3Bp[af bBc]qpq Λa()Λ ′b  tc. (199) This has similar structure to QF1 ; namely it is a psu(2,2|4)-rotation, but not a global symmetry because the parameter of the rotation is spacetime-dependent. We have [QAF1 (),QAF(′1 )] = 0, and [Q[F1 (),QAF1 ((′)])i]s given by: QF (),QAF ′ w1 1 1+ = ( ( )) ( ) 16BcdBebf a∂+ Λ ′ −1de c()Λa ( )P13 (gtbg− )116BcdBebf a ′de Λc()∂+Λb  (P)13 gt(ag−1 1= 3 × 16Bd[ )cf a b]ede B ∂+Λc()Λ ′a  P13 gt g−1b 1. (200) This implies: (1) = × [ ] ( )I 3 16Bp af b c q ′2 pq B Λa()Λb  jc (201)Q1 and therefore: (0) W2 = 3 × 16Bpaf b cqpq B ΛaΛbΛc. (202) This means that (0)W2 is expressed in terms of the Schouten bracket on Λ •g: [[ ]]abc = [a|e| b |f |c]B1,B2 B1 fef B2 . (203) 8.4. Could a nonzero [[B,B]] be harmless? 8.4.1. Operator (0)W2 may be Q-exact We have seen that the obstacle to extending the deformation to the second order in ε is [[B,B]]abcΛaΛbΛc. But a nonzero [[B,B]] does not yet mean that the deformation is obstructed, because [[B,B]]abcΛaΛbΛc can still be QBRST -exact: [[ ?B,B]]abcΛaΛbΛc =Q0T . (204) O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 187Example of a Q-exact expression of the ghost number 3. Let us consider the following oper- ator of the ghost number 2: T =AmaΛ̄mΛa (205) where Ama is some tensor which does not need to have any special symmetry properties under the exchange m↔ a. In this case we get: Q ma bc0T =−A fm ΛbΛcΛa. (206) Therefore, if: [[B,B]]abc =Am[af bc]m (207) then such [[B,B]] is harmless. In particular, such harmless [[B,B]] arise in the following situa- tion. Consider [[B ,B ]]abc1 2 ΛaΛbΛc in the special case when Bab =Glf ab1 l . We get: Glf e[a b c]fl fef B2 =− [ [ ]] c]f 1 c]f Gl ta, tb, tf B = f gGlf [ab B . (208)l 2 f l2 g 2 This expression is proportional to f abg . Therefore in this case [[B1,B ]]abc2 ΛaΛbΛd is BRST- exact: [[ ( )B abc a m nb1,B2]] ΛaΛbΛc =Q f mnG B2 Λ̄aΛb . (209) This implies that the condition that [[B,B]] is exact is correctly defined on the equivalence classes of Bab ∼ Bab+f abcGc in agreement with Sections 4.1 and 6.2. Comparing (209) with (207) we see that in this case Ama = fmpqGpBqa . Notice that this Ama is not antisymmetric in a ↔m; but the antisymmetrization of A is a Schouten bracket [[G,B]]. 8.4.2. But W(0) is of ghost number 3; isn’t it always Q0-exact? There is no nontrivial BRST cohomology in the ghost number 3, therefore strictly speaking (0) W2 is always Q0-exact. Since (0) Q0W2 = 0 we should always be able to find (0)T such that W2 = Q0T . However, this is not true if we also impose some additional constraints on T . There are two possible constraints on T : 1. Covariance, i.e. we demand T to transform covariantly under psu(2,2|4). A priori it only transforms covariantly modulo KerQ0; see [4]. 2. Absence of resonant terms; in other words T is periodic in the global time of AdS5. Notice that this would be automatically satisfied if we impose the covariance. Considerations similar to [4] show that there are the following obstructions to the covariance of T : ( ( [ ])) 1 ∧ ∧ physicalH (g,HomC(g g g,[ ]),states ) H 2( g,HomC g ∧ ∧ conserved g g, , charges H 3 ) g,HomC(g ∧ g ∧ g,C) . (210) 188 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–2158.5. Calculation of (2)V2 We will start with calculating Q1ja+. We get: ( (( ) Q1ja+ = 4Λ bc e bcb(B fca )je+ +(4∂+ΛbB) Str tag−1 gt −1cg 3+ 2 gt g−1 −1 ( −1) ) )c 2 + 3 gtcg 1 − 4P(13 gtc(g ( 1 g , (211) Q1ja− = ) 4Λ bc eb(B fca )je− −( 4∂−Λb)Bbc Str t g−1 3 gt g−1a c 3+ 2 gt g−1 + gt g−1 ( ) ) )c 2 c 1 − 4P31 gt g−1c 3 g . (212) This implies: ( ) Q Babj +j − = 4Bcd ab e e1 a b B Λc fda je+jb− +(fdb ja+(je−+ 4( ( ) ( )BcdBabQ) 0(j+ j −()Str gt)g)−)1 gt g−1 −1c b a d 3 + 2 gtdg 2+ 3 gtdg−1 1 − 4P13 gt g−1d 1 . (213) Now we are going to use the condition (207). Let us first assume that A= 0, and then consider the case when A is nonzero. 8.5.1. The case when [[B,B]] = 0 Taken into acc(ount that [[B,B]] = 0 we ca)n transform in the first line of (213): BcdBabΛc f e e da je+jb− + fdb j +j − =−Bdca e BabΛef eda jc+jb−. (214) Observe that: (([ −1(( −1) ( −1) ( −1) ( −1) ))Q0L Str gtag gtd]g(( 3 + 2) gtdg( 2 + 3) gtdg( 1 − 4)P13 gtdg( 1= ) ))Str λ( −13, gtag ([ g(tdg−1 −1 −1 −13)+]2 gt[dg ( 2 + 3 g)td]g [1 − (4P13 gtd+ −1 )g])) 1Str gtag λ3, gtdg−1 0 + 2 λ , gt g−13 d 3 + 3 λ , gt −13 dg 2 . (215) Using that Str([λ ,gt g−13 a ]P13(gt g−1d ) −1 −11)= Str([λ3, gtag ](gtdg )1) this can be rewritten as follows: ( Q Str gt( g−[ 1(( −1) ( ) ( ) ( ) ))0L a gtdg 3 + ]2)gtdg−1 2 + 3 gtdg−1 1 − 4P13 gt −1dg 1=−Str λ gt g−1, gt g−1 =− (f e g−1 )3 a d ad λ3g . (216)e Similarly, let u(s calcul(a(te the ac)tion of(Q0R on the same expression: Q Str(gt g−1 gt g−)1 + 2(gt g−1)) + ( −1)0R a d 3 d 2 3 gtdg 1 − ( ) ))4P13 gt g−1d 1= Str g−1λ e −11g[ta, td ] = fad g λ1g . (217)e Combining (216) and (217) with (213) and (214) we get: (2) Q1V1 =− (2)Q0V2 (218) where (2) V =− ( (( ) ( ) ( ) ( )4BcdBab2 jc)+(jb− Str )gt g−(1 gt )g−1 (+ 2 gt )g−)1 gt g−1a 1 d 3 a 2 d 2+ 3 gt g−1a gt g−1 − 4 gt g−1d a P13 gt g−1d . (219)3 1 3 1 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 189Notice that (2)V2 is non-polynomial in pure spinors because the projectors P13 defined by (166) are non-polynomial. But we will see that the nonlocality actually cancels out in the classical action if we substitute the classical values for the antifields w (which are nonzero after the deformation). In other words, we remove the terms with P by a shift of w. 8.5.2. The case when [[B,B]] is of the form (207) Now we have to explain what happens when [[B,B]] is nonzero, but is Q-exact in the sense of (207). Then (214) fails and consequently instead of (218) we are getting this: (2) Q1V1 =− (2)Q V + 3Am[af bc]0 2 m Λajb+jc−. (220) The second term on the right-hand side can be transformed as follows: Amaf bcm ((Λajb ∧(jc +Λbjc ∧)ja)+Λcja(∧ jb) ( ) ) − ma d 2L d2L d A Λa g −1 g +Q Ama −1( 0 ja g gdl2 2m dl m − d2 ) −Ama Lj 1 maa ∧Q0 g g + 2A [Λ,j ]m ∧ ja. (221) dl2 m Here we ca(n use: ) −1 d2LQ0 g g =−2[Λ, j ] − d(Λ̄) (222) dl2 and finally obtain: Amaf b(cm (Λajb ∧(jc +Λbjc ∧)ja +Λcja ∧ jb)) ( ( ) ) −d AmaΛ g− d 2 2 1 Lg +Amaa Λ̄mja +Q Ama d L0 j −12 a g g . (223)dl 2m dl m Therefore with A = 0(we get: ( d2 ) ) (2) (2) L Q1V −Q ma −11 0 V2 +A ja g g + d(smth). (224)dl2 m 8.6. Taking into account nonzero classical values of the antifields 8.6.1. Second order correction to the deformed action The term(s in L+ ) (2)V1 c(ontaining )the antifields are the f(ollowing:− )Str w+w1 3− − 4 g−1w1+g Babj + 4j Bab g−1w g . (225)a b− a+ 3− b This mean∣∣s that the classical values of the antifields are:w3−∣∣ =− ( −( 1)4P gt g Bab31 a 3 )jb−, (226)cl w1+ = 4P ab −1cl 13ja+B gtbg 1. (227) When we s∣∣ubstitu∣te these classical values bac(k(into the)action(, we get): )w1+ w−∣ =−16j +j −BcdBab Str gt g−1 P gt g−1cl 3 cl c b d 1 31 a 3 . (228) Combining this with (219) we get: 190 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215(2) + ∣∣ ∣ (( ) ( )V w+ w−∣ =−4B(cdBab2 1 3 jc)+(jb− Str )gtag−cl cl + −1 −1 + ( 1 gt g−11 d 3 2 gt g gt g 3 gt g−1 ) ( −1) ) a 2 d 2 a 3 gtdg 1 . (229) This formula describes the second order deformation of the classical action. 8.6.2. BRST transformation of the shifted antifields Taking into account (2∣26) and (227) we(define th)e shifted antifields: w3− =w 3− −w3−∣∣ =w + 4P gt g−1 Babj , (230)cl 3− 31 a ( 3 b)− w1+ =w+ −w+∣1 1 =w1+ − 4P j +Bab13 a gt g−1b 1. (231)cl In terms of these shifted antifields the BRST transformation Q0 + εQ1 (where Q1 is given by (175)) is: + [( ) ](Q εQ )w+ =D +λ − [N+, λ ] + 4εj +Bab gt g−10 1 1 0 1 1 a b 0, λ1 , (Q0 + [( ) ] εQ )w1 3− =D0−λ3 − [N−, λ ab −13] + 4εja−B gtbg 0, λ3 . (232) 8.7. Final formula for the deformed action at the second order in ε2 The action a∫t the seco(nd order is given by: R2= 1 3 1S d2z Str J2+J2− + J1+J3− + J3+J1− +w1+∂−λ3 +w3−∂+λ1 π 2 4 4 + 1N0+J0− +N ab0−J0+ −(N(0+N0−)+( εB j)[a ∧ jb] (233)2− 4ε2BcdBabj +j − Str gt g−1 g)t g−1 + (2 gt g−1) ( )( ) (c b ) ) a 1 d 3 a 2 gtdg −1 2 + 3 gt −1ag 3 gt g−1 −wd 1 1+w3− . (234) 8.8. Comments 8.8.1. About higher orders We have started with (2)(( V)1 ( = ab (2)B j)a+jb−( and ob)tai(ned V2 )=−4B apMpqB qbjc+jb− where M = Str (gt g−1 ) (gt g−1 ) −1 −1pq a 1 d 3 )+ 2 gtag 2 gtdg 2+ 3 gtag−1 3 gt −1dg 1 . (235) Notice that while (2)V1 is parity-odd (2) V2 is not. This corresponds to the fact that there is a nonzero deformation of the metric at the second order. Also, notice the schematic pattern in going from the first order vertex to the second order vertex: Bab → BapM Bqbpq . (236) We conjecture that higher orders follow the same pattern. O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 1918.8.2. About the gauge transformation Bab → Bab + f abcGc As we explained in Section 4.2 the gauge transformation Bab → Bab + f abcGc (237) should be accompanied by a field redefinition GaXa . Therefore the condition of the gauge in- variance at the second order in  is: ab c δ (2)f cG V2 +GaX (2)aV1 = d(smth). (238)δBab This means that (2)V2 is not invariant under the gauge transformation (237) in the naive sense, but rather in the sense of Eq. (238). 9. Properties of the Schouten bracket on Λ•g 9.1. Projection to g ⊗ g Given a ∧ b ∧ c⊂Λ3g we consider: [a ∧ b ∧ c] = [a, b] ⊗ c− [a, c] ⊗ b+ [b, c] ⊗ a ∈ g ⊗ g. (239) If the i[nternal c]ommutator of B (defined in Section 5) vanishes, then:[[B,B]] ∈ g • g (240) where • means the symmetric product. More precisely, this is f a f b BkmBlnkl mn . 9.2. From the r-matrix point of view Suppose that B satisfies: [[B,B]] = 0. (241) (Which is slightly stronger than Eq. (207).) Eq. (241) is known in the mathematical literature as the classical Yang–Baxter equation [25]. In this context, B should be thought of as a classical r-matrix. It defines the Poisson bracket on g, and therefore the structure of the Lie algebra on g∗, in the following way [25]: [ (1)X,Y ]B = ι(B)d(X ∧ Y)= ad∗ bc Y − (X↔ Y). (242)XbB tc In this formula d(X ∧ Y) is the differential on Λ•g∗, which is the same d as the Lie algebra cohomology defines. This differential is “dual” to the Lie bracket on g, in the following sense. Remember that in our notations the coordinates of an element ξ ∈ g are enumerated with the upper indices: ξ = ξata ∈ g. (243) The commutator is [ξ, η] = ξaηbf cab tc . Therefore the elements of g∗ have lower indices, so the pairing of X ∈ g∗ and ξ ∈ g is 〈X,ξ〉 =Xaξa . The structure of a Lie algebra on g determines the differential on Λ•g∗, which is continued by the polylinearity from: (dX)ab = f cab Xc. (244) 192 O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215When B is decomposable, i.e. B = b1 ∧ b2, we can rewrite (242) as follows: [ (1)X,Y ] = ι ad∗B b[1 b ]X ∧ Y. (245)2 The Ja[cobi identity for [ ] (1) , B gives: [ ](1) ](1) ( )X,Y B ,Z ± (cycl) = 2ι ∗B b[1adb ] (X ∧ Y ∧Z)2 = [[B,B]]abcι ∗ta ιt adt (X ∧ Y ∧Z) (246)b c where the last equality is true also if B is not decomposable. This means that [ , ]B satisfies the Jacobi identity iff [[B,B]] = 0. But what happens if [[B,B]] is not zero, but is Q-exact in the sense of (207)? Then [[B,B〈][]abc =A[a|m|f bc]m and we get:[X,Y ](1) ](1)B ,Z ± 〉(cycl. X,Y,Z), tB q = 6A[a|m|f bc] ′m X[aYbZc′]f c cq = 1Aamf cmq Xa[Y,Z] + 1Acm[ ′c X,Y ]mZ ′f c ± (cycl. X,Y,Z) (247) 3 3 c cq where [X,Y ]a = f bca XbYc (so defined [X,Y ] is a bracket on g∗ which turns g∗ into a Lie algebra isomorphic to g). Suppose that A is antisymmetric: Aam = −Ama . In this case, let us define a new op(eration [ )](2), ∗A : g ∧ g∗ → g∗; in coordinates:[ (2)X,Y ]A = 〈AamX bafmq Yb +〉Acmf aq cq XaYm − (X↔ Y)= (X ∧ Y), [[tq ,A]] . (248) Conclusion. If [[B,B]] is Q-exact in the sense of (207) with antisymmetric A then the following bracket on g∗: [ ] + [ ](1) + 2[ ](2), ε , B ε , A (249) is satisfies the Jacobi identity up to the order 2. 9.3. The space of solutions to [[B,B]] = 0 Unfortunately we do not have an explicit description of the space of solutions to [[B,B]] = 0. Here we will discuss a subspace which corresponds to the Lunin–Maldacena solution. Then we will argue that this subspace does not exhaust all the solutions. In other words, there are beta- deformations other than the Lunin–Maldacena solution. 9.3.1. The solutions of Lunin–Maldacena type Let us introduce the basis in gl(m|n) consisting of the m|n matrices Eij , which have 0 in all positions exc(ept for 1 in)the i-th row and j -th column. For example, for E23 ∈ gl(3) we have:0 0 0 E23 = 0 0 1 . 0 0 0 Notice that jE is a diagonal matrix.j O.A. Bedoya et al. / Nuclear Physics B 848 [FS] (2011) 155–215 193It is straigh∑tforward to see that these matrices satisfy [[B,B]] = 0: j B = hijEii ∧Ej . (250) 1i