Deformations with maximal supersymmetries part 2: off-shell formulation

Journal of High Energy Physics, Apr 2016

Continuing our exploration of maximally supersymmetric gauge theories (MSYM) deformed by higher dimensional operators, in this paper we consider an off-shell approach based on pure spinor superspace and focus on constructing supersymmetric deformations beyond the first order. In particular, we give a construction of the Batalin-Vilkovisky action of an all-order non-Abelian Born-Infeld deformation of MSYM in the non-minimal pure spinor formalism. We also discuss subtleties in the integration over the pure spinor superspace and the relevance of Berkovits-Nekrasov regularization.

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Deformations with maximal supersymmetries part 2: off-shell formulation

HJE Deformations with maximal supersymmetries part 2: Cambridge 0 U.S.A. 0 Chi-Ming Chang 0 2 Ying-Hsuan Lin 0 2 Yifan Wang 0 1 Xi Yin 0 2 0 Cambridge , MA 02139 , U.S.A 1 Center for Theoretical Physics, Massachusetts Institute of Technology , USA 2 Jefferson Physical Laboratory, Harvard University Continuing our exploration of maximally supersymmetric gauge theories (MSYM) deformed by higher dimensional operators, in this paper we consider an offshell approach based on pure spinor superspace and focus on constructing supersymmetric deformations beyond the first order. In particular, we give a construction of the BatalinVilkovisky action of an all-order non-Abelian Born-Infeld deformation of MSYM in the non-minimal pure spinor formalism. We also discuss subtleties in the integration over the pure spinor superspace and the relevance of Berkovits-Nekrasov regularization. Extended Supersymmetry; Superspaces; Supersymmetric Effective Theories - 4 5 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 4.1 4.2 5.1 5.2 5.3 Other examples Noncommutative deformation The 5-form deformation Regularization by smearing A smearing operator Shifted pure spinor variables 6 Discussion A Siegel gauge and the b ghost 1 Introduction 2 Pure spinor superspace 3 Super-Yang-Mills theory and the pure spinor superfield The non-minimal pure spinor superspace Descendant pure spinor superfields Recovering the on-shell Yang-Mills superfields The first order deformation Non-Abelian Born-Infeld deformation at the second order No obstruction to all order Superspace Lagrangian deformations using smeared fields metries manifest have been developed, in the on-shell formulation [1] based on the algebra of super-gauge covariant derivatives and its deformations [2–5], and in the off-shell formulation based on pure spinor superspace by Cederwall and Karlsson [6, 7], after the work of Berkovits [8]. The algebraic, on-shell approach was explored in our previous paper [9] to classify infinitesimal deformations that preserve 16 supersymmetries, while allowing the possibility of breaking either Lorentz or R-symmetry. In this approach, the problem of finding higher order deformations (or identifying the obstructions) can be formulated systematically as a cohomology problem. In practice, however, it was very difficult to compute the relevant obstruction classes and to verify their triviality. – 1 – In this paper, we adopt the off-shell approach based on pure spinor superspace. This formalism was first developed in the context of superstring perturbation theory [10–24]. It was known for some time that the standard two-derivative, undeformed, MSYM can be reformulated as a Chern-Simons-like theory in pure spinor superspace [1, 8], in close analogy with cubic open string field theory [25]. Although, it was not immediately obvious how to write down higher derivative deformations in this language. It was explained in [6] how the Born-Infeld deformation, to first order, can be constructed in the non-minimal pure spinor superspace formalism, and that the first order deformation in the Abelian case already gives a consistent action to all orders. We will develop this construction further, and show that the non-Abelian Born-Infeld deformation can be extended to all orders. (See [26–35] pure spinor variables is used in writing the descendant superfields, and the higher derivative terms in pure spinor superspace. This could potentially lead to divergences in the integration over the tip of the pure spinor cone. Such divergences do not seem to appear in our construction of F-term deformations, but this is not a priori obvious. We find it useful to consider a regularization introduced by Berkovits and Nekrasov [24], which amounts to smearing the superfields in pure spinor superspace in a manner that preserves the BV master equation. This allows us to demonstrate the absence of divergences in simple examples. We suspect that it is relevant for the construction of general D-terms in this formalism as well. In section 2 we will review the pure spinor superspace and the descendant pure spinor superfields of [6], and set up our notations and conventions. In section 3, we apply this formalism to the Born-Infeld deformation, and demonstrate that the construction can be extended to all orders in the deformation parameter, solving the BV master equation [36, 37] order by order. Other examples such as noncommutative deformations, and the 5-form deformation in the IKKT matrix model [38], are discussed in section 4. We introduce the Berkovits-Nekrasov regulator in the context of MSYM theories in section 5, and discuss their role in regularizing potential divergences in the pure spinor integral, and possibly the construction of D-terms. We conclude with some open questions in section 6. 2 Pure spinor superspace In this section we review the construction of the action of maximally supersymmetric YangMills theories based on pure spinor superspace. A first attempt at constructing an action based on the Yang-Mills superfield involves a Chern-Simons type functional defined by an integration over the “minimal” pure spinor superspace. We will see that this action gives – 2 – rise to the correct SYM equation of motion up to pure gauge terms, provided that a truncation on the superfield is implemented. The truncation condition breaks manifest supersymmetry, however. To fix the problem, one extends the superfield to one defined over the non-minimal pure spinor superfields [6]. Instead of a classical gauge invariant action, in this formalism one find a Batalin-Vilkovisky action functional [6, 36, 37]. A conventional BRST invariant action may be obtained by imposing the Siegel gauge condition that effectively eliminates the BV anti-fields in the pure spinor superfield. Working with the BV action has the advantage that deformations of BRST transformations need not be introduced explicitly, but rather is determined via the BV anti-bracket. The problem of finding supersymmetric higher derivative deformations turns into the problem of constructing higher derivative terms that solve the BV master equation [6]. We will also see later that the closure of BV master equation order by order can be reformulated as a cohomology problem. 2.1 Super-Yang-Mills theory and the pure spinor superfield Let us begin by considering the classical action of N = 1 SYM in 10 dimensions. The dimensional reduction to d dimensional (undeformed) MSYM will be straightforward. Let (xm, θα) be superspace coordinates, m = 0, · · · , 9, α = 1, · · · , 16. The ordinary Yang-Mills superfield is written as Aα(x, θ). The super-derivative is written as It obeys the anti-commutator Let λα be a pure spinor variable, namely it obeys the constraint The ordinary SYM equation of motion can be written in the form [1] dα = ∂ ∂θα − (Γmθ)α ∂xm . {dα, dβ} = −2Γαmβ ∂xm . λαΓαmβλβ = 0. ∂ ∂ λαλβ(dαAβ + AαAβ) = 0. QΨ + Ψ2 = 0, Q = λαdα δΨ = QΩ + [Ψ, Ω]. 1Although Q is analogous to the BRST charge in the worldsheet formulation of pure spinor string theory, here in the context of spacetime gauge theory it is merely a differential and should not be confused with the BRST charge. – 3 – If we write a pure spinor superfield Ψ(x, θ, λ) as λαAα(x, θ), then the equation of motion can be put in the simple form where1 is a nilpotent differential, namely Q2 = 0, by virtue of (2.2) and the pure spinor constraint on λ. Super-gauge transformations δAα = dαΩ + [Aα, Ω] can be expressed in terms of Ψ as (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) As an example, in the Abelian case, when Aα(x, θ) obeys the equation of motion, there is a gauge in which Aα can be put in the form 1 2 1 12 Aα(x, θ) = (Γmθ)αam(x) + (θΓmnpθ)(Γmnp)αβχβ(x) + (θΓmnpθ)(Γpθ)α∂man + · · · , 1 16 where · · · involves derivatives of am and χα. am(x) and χα(x) are the component fields for the gauge boson and the gaugino. So far the pure spinor superfield Ψ is by definition linear in λα, or in other words, it has ghost number 1, where the ghost number here simply counts the degree in λ (the notion where · · · stands for terms that involve more than 3 powers of λ (such terms will not play any role in the minimal formalism). Here C(x, θ) is the ordinary ghost superfield, A∗mnpqr and Cα∗βγ are BV anti-fields. Note that the pure spinor constraint implies that λα1 · · · λαn transforms under Lorentz group or Spin(10) in a single irreducible representation of Dynkin label [0000n]. A first attempt of writing a superspace action in connection with the Yang-Mills superfield equation is the following Chern-Simons-like action [8] Here Tr stands for the trace over the gauge index of Ψ. h· · · i amounts to an integration over the minimal pure spinor superspace, with a peculiar choice of measure. It is defined to be nonzero only when evaluated on the spin(10) singlet constructed out of λ3θ5, (λΓmθ)(λΓnθ)(λΓpθ)(θΓmnpθ) = 1, and vanishes on any other monomials of the form λkθℓ. This seemingly ad hoc definition has a natural explanation in the language of non-minimal pure spinor superspace, which will be reviewed in section 2.3. Note that while this measure has the property that hQ(· · · )i is a total derivative, h∂θα (· · · )i generally is not a total derivative. If we restrict Ψ to its ghost number 1 component, of the form λαAα(x, θ), then the expression (2.10) reduces to a gauge invariant functional of Aα(x, θ). The problem is that varying it with respect to Aα(x, θ) does not quite reproduce the equation of motion QΨ + Ψ2 = 0. For instance all terms involving 5 or more θ’s in Ψ drop out of the action functional. Such terms will end up as pure gauge, but still (2.10) is not quite the correct action in the conventional sense. This problem will be resolved in the non-minimal pure spinor formalism, where infinitely many more auxiliary fields are introduced. – 4 – (2.8) (2.10) (2.11) It is instructive nonetheless to inspect explicitly the functional (2.10) restricted to Ψ = λαAα(x, θ). Since the resulting functional is gauge invariant under δAα = dαΩ + [Aα, Ω], let us first restrict the form of Aα(x, θ) using such a gauge transformation. For simplicity, we will illustrate with the example zero-dimensional MSYM (also known in its component field form as the IKKT matrix model), where Aα is a function of θ only, and the gauge transformation takes the form ∂ ∂θα δAα(θ) = Ω + [Aα, Ω]. (2.12) We can remove Aα(0) with a linear gauge parameter in θ. Let A(αn) be the degree n component of Aα in θ. Now the minimal action can be written as S = λαλβλγ Tr A(α1) ∂∂θβ A(γ5) [00030] + A(α2) ∂∂θβ A(γ4) [10020] + 12 A(α3) ∂∂θβ A(γ3) + A(α1)A(β1)A(γ3) + A(α1)A(β2)A(γ2) . Note that here we only retain dependence on the representation component [00030] in A(5) (which contains Λ5[00001] = [00030] ⊕ [11010]), and the component [10020] in A(4) (which contains Λ4[00001] = [02000] ⊕ [10020]). Varying with respect to A(5) and A(4) gives the following equations λαλβ∂αA(β1) = 0 ⇒ A(α1) = (Γmθ)αam, λαλβ∂αA(β2) = 0 ⇒ A(α2) = (θΓmnpθ)(Γmnp)αβχβ. These conditions remove some of the gauge redundancy in A(α1) and A(α2) while retaining the physical degrees of freedom, the gauge boson am and the gaugino χα. Varying with respect to A(3) gives λ λ α β ∂αA(β3) + A(α1)A( 1 ) β = 0, which is the precisely the degree 2 component of the equation QΨ + Ψ2 = 0 with A(0) set to zero. Here we have used the fact that θ3 contains only 1 irreducible representation of spin(10), namely Λ3[00001] = [01010]. We have also used the fact that in the minimal pure spinor superspace integral we can integrate by parts on Q (but not on ∂θα by itself). Next, if we vary A(2) and A( 1 ), we obtain λ λ α β ∂α A(5) β [00030] + {A(α1), A(β3)} λ λ α β ∂α A(4) the [10020] representation component of A(4). The second equation is the restriction of QΨ+Ψ2 = 0 to degree 4 and the representation [10020], keeping only the [00030] component – 5 – (2.13) (2.14) (2.15) (2.16) of A(5). Note that the degree 4, [02000] component of the equation of motion is missing here. However, this component of QΨ + Ψ2 = 0 would have only involved the [11010] component of A(5), which has dropped out the minimal superspace action altogether. As a result, we do get the correct equation of motion for A(5)|[00030]. To summarize, once we have fixed on the gauge condition A(0) = 0, the only components of Aα(θ) that appears in the minimal pure spinor action S are given by give the correct SYM equations for am and χα, in zero dimension. The price to pay, if we make the restriction (2.17), is that the super-gauge invariance is no longer manifest, since the gauge variation δAα generally does not maintain the form (2.17). This is expected, since we cannot implement 16 off-shell supersymmetries with finitely many auxiliary fields [39]. The way to cure this problem is to introduce the nonminimal pure spinor variables, which allows for writing down the superspace action with a conventional measure, and no restriction of the form (2.17) on the pure spinor superfield will be needed. 2.3 The non-minimal pure spinor superspace In order to write down higher order terms in the pure spinor superfield, one needs some way of taking derivative with respect to the pure spinor variable λ, as the naive ∂/∂λα is generally not well defined due to the constraints. This is achieved through the nonminimal pure spinor variables, as was first introduced in the context of pure spinor string theory [23]. We must pay a hefty price however: infinitely many more auxiliary fields are introduced, and generally we will need to work in the BV formalism [40]. One introduces a new “conjugate” pure spinor λ¯α, of the opposite chirality as λα, that obeys λ¯Γm λ¯ = 0. It will be also necessary to introduce a Grassmannian variable rα that obeys λ¯γmr = 0. rα can be identified with the differential dλ¯α, and we will sometimes use this notation when it does not cause confusion. The differential Q will be modified to on λ¯. Note that the combination rα∂λ¯α annihilates λ¯γmλ¯ due to the constraint on r, and thus is well defined. This is also clear if we think of rα∂λ¯α = dλ¯α∂λ¯α as taking exterior derivative Now we will extend the pure spinor Yang-Mills superfield Ψ(x, θ, λ) to one that depends on λ¯, r also, Ψ(x, θ, λ, λ¯, r). This introduces infinitely many more auxiliary fields, but does not change the number of physical degrees of freedom because the cohomology of Q in the (λ¯, r) sector is trivial. The superspace integration will take the form (2.18) (2.19) Q = λαdα + rα ∂λ¯α . ∂ Z d16θ[dλ][dλ¯][dr], – 6 – where the spin(10) invariant measure factors [dλ], [dλ¯], [dr] are defined as [dλ]λαλβλγ = (ǫT¯)αα1β·γ··α11 dλα1 · · · dλα11 , [dλ¯]λ¯α λ¯β λ¯γ = (ǫT )αα1β·γ··α11 dλ¯α1 · · · dλ¯α11 , [dr] = (ǫT¯)αα1β·γ··α11 λ¯α λ¯β λ¯γ ∂rα1 · · · ∂ ∂ ∂rα11 . d16θ[dλ]e−ζλ¯λ (−ζdλ¯θ)11 λαλβλγ = d16θ(ǫT )αα1β·γ··α11 θα1 · · · θα11 = T αβγα1···α5 ∂ ∂θα1 · · · ∂ ∂θα5 θ=0 and ǫT its contraction with the 16-dimensional anti-symmetric tensor. T¯ is the same tensor with chiral and anti-chiral spinors exchanged. In performing the integration of (λ, λ¯) over the pure spinor superspace, λ¯α will be regarded as the complex conjugate variable of λα. Note that we could have also simplified our notation by identifying rα with dλ¯α and write the integration measure as while the d11λ¯ factor will be supplied from the integrand which is now regarded as a differential form in λ¯α rather than a function of rα. The superfield will be regarded as an analytic function in the pure spinor variables λ, λ¯. In order for the integration over the pure spinor space to converge as λ, λ¯ → ∞, one multiplies the integration measure with a regulator of the form It is crucial that such a regulator formally differs from 1 by a Q-exact expression, so as to ensure that Q-exact integrands integrate to zero. A convenient choice is Λ = λ¯αθα, {Q, Λ} = λ¯αλα + rαθα. Note that the BV action constructed by integrating with this regulator, as a functional of Ψ, will generally depend on ζ, since the integrand isn’t Q-closed. Note that the dependence on ζ would drop out if we restrict to the part of integrand of homogeneous degree 3 in λ and r. Now the superspace SYM action is written as S = Z d10xd16θ[dλ][dλ¯][dr]e−ζ(λ¯λ+rθ) Tr ΨQΨ + 1 2 3 If we restrict Ψ to be independent of λ¯, r, then the (θ, λ, λ¯, r) measure factor may be replaced by Z d16θ[dλ]e−ζλ¯λ (−ζdλ¯θ)11 which is nonzero only when evaluated with the integrand (λ3θ5), giving Let us denote collectively Z = (λ, θ, λ¯, r), and the regularized non-minimal superspace integration measure as [dZ] = d10xd16θ[dλ][dλ¯][dr]e−ζ{Q,Λ}. Given two functionals F, G of Ψ, one may define a Batalin-Vilkovisky antibracket [40] by (F, G) = −Tr [dZ] Z δF δG δΨ(Z) δΨ(Z) The sign convention has to do with the fact that our measure factor [dZ] is odd. The extension of the nilpotency of BRST transformation in the BV formalism is the (classical) BV master equation (S, S) = 0. 2.4 Descendant pure spinor superfields A key ingredient introduced by [6] is the construction of descendant superfields from Ψ by acting with certain linear differential operators. The first few descending operators are Aˆα, Aˆm, χˆα, Fˆmn, ηbnα. They obey the descending relations (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) HJEP04(216)7 Explicitly, they are given by [Q, Aˆα] = −dα − 2(Γmλ)αAˆm. {Q, Aˆm} = ∂m − λΓmχˆ, [Q, χˆα] = − (Γmnλ)αFˆmn, 1 2 {Q, Fˆmn} = 2λΓ[mηˆn]. 1 1 32 1 192 1 32 1 4 1 2 1 8 1 4 Aˆα = −(λλ¯)−1 1 4 λ¯αN + (Γmnλ¯)αNmn , 8 Aˆm = − (λλ¯)−1(λ¯Γmd) + (λλ¯)−2(λ¯Γmnpr)N np, χˆα = (λλ¯)−1(Γmλ¯)α∂m − (λλ¯)−2(λ¯Γmnpr)(Γmnpd)α − (λλ¯)−3(Γmλ¯)α(rΓmnpr)Nnp, Fˆmn = (λλ¯)−2(λ¯Γmnpr)∂p + (λλ¯)−3(rΓmnpr)(λ¯Γpd)− (λλ¯)−4(λ¯Γmnpr)(rΓpqrr)Nqr, where N = λα ∂λ∂α and Nmn = λα(Γmn)αβ ∂λ∂β . It will be convenient to introduce an operator Δm, – 8 – 1 32 1 8 1 32 1 64 1 256 1 2 1 16 χˆα = (λλ¯)−1(Γmλ¯)αΔm, Fˆmn = (λλ¯)−2(λ¯Γmnpr)Δp, ηˆmα = − (λλ¯)−3(Γnλ¯)α(rΓmnpr)Δp = − (λλ¯)−2(rΓmnpr)(Γnpχˆ)α. Δm ≡ ∂m + (λλ¯)−1(rΓmd) − (λλ¯)−2(rΓmnpr)N np. Δm is analogous to ∂m but has a nontrivial commutator with Q, 1 2 [Q, Δm] = (λλ¯)−1(rΓmΓnλ)Δn. This property will be useful later in constructing deformations of the BV action. The descending operators χˆα, Fˆmn and ηˆmα are related to Δm by Z S = [dZ]Tr ΨQΨ + 1 2 where S(n)[Ψ] will be an integral over the non-minimal superspace of a function of linear descendant fields χˆΨ, FˆΨ, etc. The BV master equation will be solved order by order in the deformation parameter ǫ. Since Ψ(x, θ, λ, λ¯, r) now contains infinitely many auxiliary fields, here we would like to describe how to recover a deformed equation of motion for an ordinary Yang-Mills superfield Aα(x, θ). We will consider an analogous expansion of a ghost number 1 superfield Ψ in ǫ, ∞ n=1 X ǫnΨn(x, θ, λ, λ¯, r), Ψ = Ψ0(x, θ, λ) + Ψ0 = λαAα(x, θ). (2.37) Suppose Ψ0 solves the equation QΨ0 + Ψ20 = 0 of the undeformed MSYM theory. We would like to construct a nearby solution of the deformed theory. To first order in ǫ, the equation to solve is QΨ1 + {Ψ0, Ψ1} + δS( 1 ) δΨ Ψ0 = 0. The key is to show that δS( 1 )/δΨ evaluated on Ψ0 can be put in the form A useful fact is that all of χˆα and Fˆmn anti-commute or commute with one another.2 Note that λΓmχˆ and λΓnχˆ do not commute,3 though they would commute when their indices are contracted with Fˆmn or Fˆmp · · · Fˆnq · · · . The following relations are also useful: χˆΓmχˆ = 0, (λΓmχˆ)Fˆmn = Fˆmn(λΓmχˆ) = 0, Batalin-Vilkovisky in non-minimal pure spinor superspace, generally of the form where the term E1[Ψ0] involves only the minimal variables, and Λ is a function of Ψ0 and its derivatives that generally involves non-minimal variables. If we can do this, then we would have recovered the first order deformation of the equation on minimal superfield Ψmin(x, θ, λ) as QΨmin + Ψmin + ǫ E1[Ψmin] = O(ǫ2). 2 Here Ψ simply differs from Ψmin by ǫΛ. 2The LiE package [41] and the decomposition of tensor products of r and λ¯ into irreps of SO(10) listed in [6] are useful in verifying the relations among the descendent operators. 3There appears to be an incorrect statement regarding this in [6]. – 9 – (2.35) (2.36) (2.38) (2.39) (2.40) In practice, we can construct E1[Ψ0] from S( 1 ) roughly by replacing the linear descendant fields with the minimal descendant superfields. To illustrate this, let us consider the example of Abelian Born-Infeld theory, with E1[Ψ0] = (λΓmχ)(λΓnχ)Fmn, δS( 1 ) δΨ where χα and Fmn are the minimal descendant superfields, related to Aα(x, θ) via Am = − 116 ΓαmβdαAβ, χα = − 110 Γαmβ(dβAm − ∂mAβ), Fmn = ∂mAn − ∂nAm. (2.43) For λαdα-closed Ψ0(x, θ), both (2.41) and (2.42) are Q-closed. Generally, the existence of E1[Ψ0] is a consequence of the statement that the non-minimal variables λ¯, r do not introduce new Q-cohomology. What we need to see here is that (2.41) and (2.42) differ by a Q-exact term, thus verifying in particular that the off-shell deformation is a nontrivial one. We can write + Q 1 2 1 2 δS( 1 ) δΨ 1 4 1 2 Now we can put (2.41) in the form where (λΓmχ)(λΓnχ)Fmn = (λΓmχ)(λΓnχ) − (λλ¯)−1(rΓmnχ)+ (λλ¯)−2(λr)(λ¯Γmnχ) 1 2 1 2 (2.41) (2.42) = − (λλ¯)−2(rΓmnpλ¯)(λΓmχ′)(λΓnχ′)(λΓpχ′), 1 32 1 8 1 8 1 2 1 32 (λλ¯)−1(λΓmχ)(λΓnχ)(λ¯Γmnχ) ΔmΨ0 = λα∂mAα + (λλ¯)−1(rΓmd)(λA) − (λλ¯)−2(rΓmijr)(λΓijA) = − (λλ¯)−2(rΓmnpλ¯)(λΓmχ)(λΓnχ)(λΓpχ)+Q (λλ¯)−1(λΓmχ)(λΓnχ)(λ¯Γmnχ) . This is now very close to (2.41), but there is still a little difference between ΔmΨ0 and λΓmχ. We have = λΓmχ + Q Am + (λλ¯)−1(λ¯Γmd)(λA) − (λλ¯)−2(λ¯Γmijr)(λΓijA) + (stuff that vanishes upon contraction with rΓmnpλ¯) = λΓmχ + Q (λλ¯)−1(λΓmΓk λ¯)Ak − (λλ¯)−2(λΓmΓkr)(λ¯ΓkA) + (stuff that vanishes upon contraction with rΓmnpλ¯). χ′α = χα + Q (λλ¯)−1(Γkλ¯)αAk − (λλ¯)−2(Γkr)α(λ¯ΓkA) Using the identity Q (λλ¯)−1(λΓm)[α(λΓn)β(λ¯Γmn)γ] = (λλ¯)−2(rΓmnpλ¯)(λΓm)α(λΓn)β(λΓp)γ , (2.48) which in particular implies that the r.h.s. commutes with Q, we see that (2.46) is indeed equal to (2.42) up to Q-exact terms. 3 The Born-Infeld deformation A primary example of interest in this paper is the Born-Infeld deformation of MSYM theory. At the infinitesimal level, this is an F-term deformation of the Lagrangian by a dimension 8 operator. While this deformation is expected to preserve all 16 supersymmetries, in the usual component field formalism the Lagrangian deformation is only invariant under supersymmetries up to terms proportional to the equation of motion, which must be compensated by deformation of the supersymmetry transformations. Such a procedure generally requires adding terms to all orders in the deformation parameter, and there could be potential obstructions in finding higher order terms. The Abelian Born-Infeld theory to all orders in the deformation parameter (a.k.a. α′2 in the context of string theory) was first constructed in [42] by gauge fixing a kappa symmetric D-brane action. It seemed difficult to generalize this approach to the non-Abelian case. In the conventional component field formalism, the second order Born-Infeld deformation was constructed in [34]. Using pure spinor superspace, an all-order Abelian Born-Infeld deformation was constructed in [6]. It was not clear whether the action of [6] upon integrating out auxiliary fields would coincide with the construction from the super D-brane action. A priori they could differ by D-terms. The objective of this section is to extend the construction of [6] in the non-Abelian case to all orders in the deformation parameter. In principle this also gives a solution to the on-shell deformation problem, considered in Part 1 of the paper [9]. 3.1 The first order deformation Let us begin by recalling the construction of the infinitesimal Born-Infeld deformation in BV formalism based on non-minimal pure spinor superspace [6]. This is described by a quartic deformation of the MSYM action,4 [dZ]Tr hΨ ◦ (λΓmχˆΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnΨ)i , where ◦ denotes the symmetric product. Variation with respect to Ψ corrects the equation of motion to QΨ + Ψ2 + ǫ(λΓmχˆΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnΨ) = 0, which is cohomologically equivalent to the on-shell Born-Infeld deformation in terms of minimal pure spinor superfields, in the sense explained in section 2.5. In showing this, one integrates by part with respect to the differential operators (λΓmχˆ) and Fˆmn, making use 4We shall use Sn to denote the part of the BV action S with degree n in Ψ. (3.1) (3.2) of the identities (λΓmχˆ)(λΓnχˆ)(λ¯Γmnpr) = 0 and (λΓmχˆ)Fˆmn = 0. Note that despite the presence of the regulator e−ζ(λλ¯+rθ) in [dZ], χˆα and Fˆmn in fact commute with this regulator. Note also that while χˆα does not commute with λβ, they satisfy λΓmχˆ = χˆΓmλ. This infinitesimal deformation actually does not depend on the value of the parameter ζ in the regulator. If we vary ζ in (3.1), we obtain a term that can be written as (S2 + S3, G), where G = Z [dZ]λ¯αdαTr hΨ ◦ (λΓmχˆΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnΨ)i . (3.3) [dZ](λλ¯)−4(λΓmΓiλ¯)(λΓnΓj λ¯)(rΓmnk λ¯)Tr [Ψ ◦ ΔiΨ ◦ ΔjΨ ◦ ΔkΨ] [dZ](λλ¯)−2(rΓijk λ¯)Tr (ΨΔiΨΔjΨΔkΨ) . The integrand inside (S2 + S3, G) is proportional to the undeformed equation of motion (2.5); hence, it can be absorbed by field redefinition of Ψ. To see that the action of the form S2 + S3 + S4 obeys BV master equation up to order ǫ, one needs to show that (S2, S4) = 0 and (S3, S4) = 0. The manipulations needed to verify these relations will be useful for the extension to higher order deformations later, and so let us recall how this is done. Firstly, we have [dZ]Tr hQΨ ◦ (λΓmχˆΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnΨ) − Ψ ◦ (λΓmχˆQΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnΨ) +Ψ ◦ (λΓmχˆΨ) ◦ (λΓnχˆQΨ) ◦ (FˆmnΨ) − Ψ ◦ (λΓmχˆΨ) ◦ (λΓnχˆΨ) ◦ (FˆmnQΨ) . (3.4) i Using the fact that Q commutes with λΓmχˆ, {Q, Fˆmn} = 2λΓ[mηˆn], and the basic pure spinor identity (λΓm)α(λΓm)β = 0, we see that the integrand is Q-exact and thus To see the vanishing of (S3, S4), it is useful to use the identity Fˆmn = − 14 (λλ¯)−1(rΓmnχˆ), and rewrite S4 as S4 = [dZ](λλ¯)−1(Γmλ)[α(Γnλ)β(Γmnr)γ]Tr Ψ ◦ χˆαΨ ◦ χˆβΨ ◦ χˆγΨ . Further using χˆα = 12 (λλ¯)−1(Γmλ¯)αΔm, we can write (S2, S4) = ǫ Z 4 (S2, S4) = 0. S4 = − = − In going to the second line, we used the pure spinor constraints on λ¯α and rα, which in particular implies (λΓmΓiλ¯)(λΓnΓjλ¯)(rΓmnkλ¯) = 4(λλ¯)2(rΓijk λ¯). In the last trace we can replace the symmetric product by ordinary, due to the symmetry on [ijk]. The BV bracket with S3 is computed as (S3, S4) = − [dZ](λλ¯)−2(rΓijk λ¯)Tr Ψ2ΔiΨΔjΨΔkΨ − ΨΔiΨ2ΔjΨΔkΨ +ΨΔiΨΔjΨ2ΔkΨ − ΨΔiΨΔjΨΔkΨ2 [dZ](λλ¯)−2(rΓijkλ¯)Tr Ψ2 ◦ ΔiΨ ◦ ΔjΨ ◦ ΔkΨ . We’ve chosen to rewrite the last line in terms of a symmetrized product once again, for later convenience. The representation in terms of the Δ’s is particularly useful due to the properties [Δi, Δj] = 0, [Δi, λλ¯] = 0, and R [dZ]Δi(· · · ) = 0 which allows for integration by parts on Δi. Repeatedly applying integration by parts and cyclicity of the trace, as well as the anti-symmetry on [ijk], we can make the following replacement on the integrand 3 2 Tr Ψ2 ◦ ΔiΨ ◦ ΔjΨ ◦ ΔkΨ → Tr ({Ψ, ΔiΨ} ◦ Ψ ◦ ΔjΨ ◦ ΔkΨ) → Tr (ΔiΨ ◦ {Ψ, Ψ ◦ ΔjΨ ◦ ΔkΨ}) → 3Tr ΔiΨ ◦ Ψ2 ◦ ΔjΨ ◦ ΔkΨ + 3Tr ΔiΨ ◦ Ψ ◦ ΔjΨ ◦ ΔkΨ2 → 6Tr Ψ2 ◦ ΔiΨ ◦ ΔjΨ ◦ ΔkΨ . (3.8) This shows that indeed (S3, S4) = 0, thus completing the verification that the Born-Infeld deformation (3.1) solves the BV master equation at order ǫ. Now at order ǫ2, there is a potentially non-vanishing contribution to the BV master equation, (S4, S4) = −3ǫ2 [dZ](λλ¯)−4(rΓijk λ¯)(rΓmnpλ¯)Tr(ΔiΨΔjΨΔkΨΔmΨΔnΨΔpΨ). (3.9) Note that the combination r2λ¯2 appearing in the prefactor of the integrand can only transform in the representation [00120] ⊕ [01011] of spin(10), due to the pure spinor constraints on r and λ¯. In the case of Abelian gauge theory, ΔiΨ · · · ΔpΨ lives in the 6th anti-symmetric tensor representation, or [00011]. It cannot form a singlet by contracting with r2λ¯2, and hence (3.9) vanishes in the Abelian theory. It does not vanish in the non-Abelian case, and a second order deformation of the action must be introduced to cancel this term in the BV master equation. This will be analyzed next. 3.2 Non-Abelian Born-Infeld deformation at the second order Let us now consider (S4, S4) in the non-Abelian theory. Using Baker-Campbell-Hausdorff formula, we can write the integrand in (S4, S4) as (λλ¯)−4(rΓijkλ¯)(rΓmnpλ¯)Tr ΔiΨ ◦ ΔjΨ ◦ ΔkΨ ΔmΨ ◦ ΔnΨ ◦ ΔpΨ = (λλ¯)−4(rΓijkλ¯)(rΓmnpλ¯)Tr ΔiΨ ◦ ΔjΨ ◦ ΔkΨ ◦ ΔmΨ ◦ ΔnΨ ◦ ΔpΨ 3 2 9 1 1 5 − 2 ΔiΨ ◦ ΔmΨ ◦ {ΔjΨ, ΔnΨ} ◦ {ΔkΨ, ΔpΨ} + 3ΔiΨ ◦ ΔmΨ ◦ ΔnΨ ◦ [ΔjΨ, {ΔkΨ, ΔpΨ}] + 34 {ΔiΨ, ΔmΨ} ◦ {ΔjΨ, [ΔnΨ, {ΔpΨ, ΔkΨ}]} − 14 [ΔiΨ, {ΔjΨ, ΔmΨ}] ◦ [ΔnΨ, {ΔpΨ, ΔkΨ}] − 5 ΔiΨ ◦ ΔjΨ, {ΔmΨ, [ΔnΨ, {ΔpΨ, ΔkΨ}]} − 5 ΔiΨ ◦ ΔmΨ, {ΔjΨ, [ΔnΨ, {ΔpΨ, ΔkΨ}]} . 3 = (λλ¯)−4(rΓijkλ¯)(rΓmnpλ¯)Tr ΔiΨ ◦ ΔmΨ ◦ {ΔjΨ, ΔnΨ} ◦ {ΔkΨ, ΔpΨ} − [{ΔiΨ, ΔmΨ}, ΔjΨ] ◦ [ΔnΨ, {ΔpΨ, ΔkΨ}]} . (3.10) In above we used the fact that the term appearing in the second line is zero, for the same reason as in the Abelian case, and simplified the rest using cyclicity of the trace. The resulting expression is nonzero, and we would like to cancel it by an order ǫ2 deformation of the action. A priori, one may try to cancel it with either (S2, S6), by adding to S some sextic term S6, or with (S3, S5), by adding a quintic term S5. It is easy to see that this cannot be done using S6. The reason is that we would have to construct S6 by taking rλ¯3 contracted with the trace of a product of 6 ΔΨ’s. However, rλ¯3 consists of the representations [00040]⊕ [00120] of spin(10), and neither appear in the (unsymmetrized) 6-fold tensor power of the vector representation, and so no such singlet exist as a candidate for S6. On the other hand, it is possible to cancel (S4, S4) by introducing a quintic term S5, such that (S4, S4) + 2(S3, S5) = 0. Let us first consider the term on the r.h.s. of (3.10) that involves a 4-fold symmetric product. Firstly, we have the identity [dZ](λλ¯)−4(rΓijk λ¯)(rΓmnpλ¯)Tr ΔiΨ ◦ ΔmΨ ◦ {ΔjΨ, ΔnΨ} ◦ {ΔkΨ, ΔpΨ} [dZ](λλ¯)−4(rΓijk λ¯)(rΓmnpλ¯)Tr 3Ψ2 ◦ ΔiΨ ◦ ΔmΨ ◦ {ΔnΔkΨ, ΔjΔpΨ} (3.11) − 8Ψ2 ◦ ΔmΨ ◦ ΔjΔnΨ ◦ {ΔkΔpΨ, ΔiΨ} . Z = In this manipulation we used integration by parts on the Δi’s (recall that Δi also commutes with λλ¯), the cyclicity of the trace, and the symmetry on the indices [ijk][mnp]. It is now easy to write down an S5 such that (S3, S5) can be used to cancel the term appearing in (3.11). We must at the same time make sure that S5 has vanishing BV anti-bracket with S2. This can be achieved by rewriting expressions involving Δi’s in terms of λΓmχˆ and (λΓmn)αFˆmn, 1 2 1 8 λΓmχˆ = (λλ¯)−1(λΓmΓnλ¯)Δn, (λΓmn)αFˆmn = (λλ¯)−2(λΓmn)α(λ¯Γmnpr)Δp, both of which commute with Q. One can verify that the quintic term that can be used to cancel the r.h.s. of (3.10) is S5 = − [dZ](λλ¯)−4(rΓijk λ¯)(rΓmnpλ¯)Tr Ψ ◦ ΔiΨ ◦ ΔmΨ ◦ {ΔnΔkΨ, ΔjΔpΨ} − 2Ψ ◦ ΔmΨ ◦ ΔjΔnΨ ◦ {ΔkΔpΨ, ΔiΨ} − Ψ ◦ [{ΔjΨ, ΔiΔmΨ}, {ΔpΨ, ΔnΔkΨ}] = −48 [dZ] Tr Ψ ◦ (λΓiχˆ)Ψ ◦ (λΓmχˆ)Ψ ◦ {(λΓnχˆ)FˆijΨ, (λΓjχˆ)FˆmnΨ} − 2Ψ ◦ (λΓmχˆ)Ψ ◦ (λΓjχˆ)(λΓnχˆ)Ψ ◦ {FˆijFˆmnΨ, (λΓiχˆ)Ψ} − 1 5 Ψ ◦ h{(λΓiχˆ)Ψ, (λΓjχˆ)FˆmnΨ}, {(λΓmχˆ)Ψ, (λΓnχˆ)FˆijΨ}i . (3.13) The way we could solve for an S5 with the property (S3, S5) = − 12 (S4, S4) is no accident. The essential point is that (S4, S4) is closed with respect to (S3, · ), and the operation (S3, · ), which is nilpotent and can be regarded as a coboundary operator on the space of functionals of Ψ, has trivial cohomology in this case. We will demonstrate this more generally in the next subsection. (3.12) i HJEP04(216)7 In this section, we will prove the existence of an all-order formal deformation that solves the BV master equation, whose first order term in the deformation parameter ǫ is S4. Firstly, note that the BV anti-bracket satisfies Jacobi identity (A, (B, C)) = ((A, B), C) + (−1)|A||B|(B, (A, C)), (3.14) where A, B, C are functionals of Ψ. We define an odd differential δbA ≡ (S3, A) on functionals of Ψ, that obeys δb2A = 0 because of the Jacobi identity and (S3, S3) = 0.5 The BV anti-bracket of two δb-closed functionals is δb-closed, and the BV anti-bracket of a δb-closed functional with a δb-exact functional is δb-exact. So in other words, the BV anti-bracket defines a cup product on the cohomology of δb. Note that in fact, the cohomology of δb is defined already on the traces of products of derivatives of Ψ, without the need for integration over pure spinor superspace. Now consider the vector space V spanned by functionals constructed by taking Fˆmn’s and (λΓmχˆ)’s acting on Ψ, with all vector indices on the (λΓmχˆ)’s contracted with those of the Fˆ’s, traced and then integrated over the pure spinor superspace. A typical functional [dZ]tr · · · FˆmnFˆpqΨ(λΓnχˆ)Ψ · · · (λΓqχˆ)Ψ(λΓmχˆ)Ψ · · · (λΓpχˆ)Ψ · · · . (3.15) i of this type looks like Z in V. and λΓmχˆ in terms of Δm, using 1 8 h The virtue of this construction is that, due to (3.12), such a functional has vanishing BV anti-brackets with S2. Furthermore, the BV anti-bracket of two such functionals remains The action of δb on such functionals, on the other hand, is simplified if we express Fˆmn Fˆmn = − (λλ¯)−2(rΓmnΓpλ¯)Δp, and λΓmχˆ = (λλ¯)−1(λΓmΓnλ¯)Δn. (3.16) By construction, here the vector index on λΓmχˆ is always contracted with an index on an Fˆmn, and so the λΓmχˆ will always appears in the combination (rΓmnΓp λ¯)λΓmχˆ, which can be simplified as 1 2 1 2 (rΓmnΓp λ¯)λΓmχˆ = (λλ¯)−1(rΓmnΓp λ¯)(λΓmΓq λ¯)Δq = 1 2 (λλ¯)−1(rΓnpΓmλ¯)(λΓmΓq λ¯)Δq = (rΓnpΓmλ¯)Δm = (rΓmnΓpλ¯)Δm. In other words, on any of the functionals in V, we can replace λΓmχˆ by Δm. Next, because Δm commutes with λλ¯ (and trivially commutes with r, λ¯), after expressing Fˆ in terms of Δ, we can move all explicit factors involving r, λ, λ¯ outside the Δ’s and outside the trace. A functional in V can thus be rewritten as a linear combination of the terms [dZ]T i1···i3n tr h Δiw( 1 ) · · · Δiw(p1) Ψ Δiw(p1+1) · · · Δiw(p2) Ψ · · · Δiw(pm−1)+1 · · · Δiw(pm) Ψ i (3.17) (3.18) 5It is important here that the BV anti-bracket is even and S3 has odd degree by convention of (3.14) (which is shifted by 1 from the usual convention). Of course, it is also easy to verify directly that δb2 = 0. (3.19) HJEP04(216)7 where 0 ≤ p1 ≤ · · · ≤ pm = 3n,6 w is an element of the permutation group S3n on {1, · · · , 3n}, and T i1···i3n = (λλ¯)−2n(rΓi1i2i3 λ¯) · · · (rΓi3n−2i3n−1i3n λ¯). Since T i1···i3n commutes with Δk, we are free to integrate by part on the Δ’s. The tensor T (n) ∼ T i1···i3n transforms in the overlap between the representation content of rnλ¯n (as listed in the table of [6]) and Λn[00100]. We list these representations below:7 The structures in Sn+3 that we will encounter at the n-th order in the deformation parameter are of the schematic form T (n)trΔ3nΨn+3. The cohomology of δb on V is equivalent to a certain invariant cyclic cohomology. Let ti be a set of commutative variables, i = 1, · · · , 10 (they can be thought of as dual variables to the Δi’s that act on a single Ψ inside the trace). Let h = htii be the Abelian Lie algebra generated by commuting variables ti (i.e. the linear vector space spanned by the ti’s), and A = U (h) = C[ti] its universal enveloping algebra. Let Ck = Hom(⊗k+1A, C) be is invariant under the Zk+1 that shifts the k + 1 arguments with sign, namely the Hochschild cochains. The cyclic complex Cλk is obtained by taking the part of Ck that Cλk = {ϕ ∈ Ck : ϕ(ak, a0, · · · , ak−1) = (−1)kϕ(a0, a1, · · · , ak)}. (3.20) The differential δ : Cλk → Ck+1 defined by λ (δϕ)(a0, · · · , ak+1) = X(−)iϕ(a0, · · · , aiai+1, · · · , ak+1) + (−)k+1ϕ(ak+1a0, · · · , ak) k i=0 (3.21) is nilpotent. The cohomology of δ at Cλk defines the cyclic cohomology HCk(A). Next, consider the complex Cℓ,k = Λℓh∗ ⊗ Cλk with the chain map d : Λℓh∗ ⊗ Cλk → Λℓ−1h∗ ⊗ Ck λ 6If pℓ−1 and pℓ coincide then by convention there is no Δ acting on Ψ in the ℓ-th factor. 7Interestingly, the absence of T (11) ensures that we do not have a term with (λλ¯)−11 pole in the integrand, that would come with 11 powers of r. If such a term were present, it would lead to a log divergence in the (λ, λ¯) integral, making the action ill defined. X X(−1)i−1η1 ∧ · · · ∧ ηi−1 ∧ ηi+1 ∧ · · · ∧ ηℓ ⊗ ϕ(a0, · · · , aj−1, ηi(aj ), aj+1, · · · , ak). Now δ induces a map on Hℓ(h∗, Cλk), algebra homology Hℓ(h∗, Cλk). It is easy to see that dδ = δd. Here h∗ is the dual Lie algebra of h, generated by ∂/∂ti, and η(a) is defined as the derivative map for a ∈ A = C[ti], η ∈ h∗. In other words, with Cλk viewed as an h∗-module as above, the cohomology of the complex Λℓh∗ ⊗ Cλk with respect to the differential d defines the Lie The cohomology of δ∗ on H0(h∗, Cλk) defines the invariant cyclic cohomology HCkh∗ (A). The cohomology of δb on the space of functionals V at the n-order (with n + 3 Ψ’s) lies in HCnh+∗2(A); they correspond to the components that transform under spin(10) according to the representations of T (n). The ordinary cyclic cohomology HCk(A) can be computed using a homology version of Grothendieck’s algebraic de Rham complex [43], HCk(A) ≃ Ker d∗|(Ωk)∗ ⊕ Hk−2,dR(A) ⊕ Hk−4,dR(A) ⊕ · · · (3.24) where (Ωk)∗ is the space of de Rham k-currents in the ti’s. d∗ is the transpose of de Rham differential. H∗,dR(A) is the algebraic de Rham homology of A, which coincides with the ordinary de Rham homology on Spec(A) = V ≃ C10, defined in terms of the codifferential on polyvector fields. For odd k, HCk(A) ≃ Ker d∗|(Ωk)∗ is the dual vector space of the cokernal of d : Ωk−1 → Ωk. In this case due to the triviality of de Rham homology we can also identify it as the dual space of dΩk ⊂ Ωk+1. These classes are in correspondence with the (unintegrated) traces of derivatives of Ψ’s of the form8 Δi1 · · · Δim Tr Δj1 Ψ ◦ · · · ◦ Δjk+1 Ψ . (3.26) This statement is familiar in the context of counting BPS operators [44]. But since all of these term will end up giving total derivatives, they will not be relevant in the invariant cyclic cohomology of interest here. For even k, there is an additional part of HCk(A) coming from H0,dR(A) ≃ C. This corresponds to the element TrΨk+1, which is nonzero only for even k. forms in dΩk, 8There is a canonical pairing between integrands of the form (3.26) and algebraic de Rham differential f (Δi)Tr(Δj1 Ψ ◦ · · · ◦ Δjk+1 Ψ)|∂kg(ti)dtk ∧ dtj1 ∧ · · · ∧ dtik = δ[ij11 · · · δjikk f (∂i)∂jk+1]g(ti). (3.25) Note that expressions of the form (3.26) are not all independent: for intance, Δ[kTr(Δj1 Ψ◦· · ·◦Δjk+1]Ψ) ≡ 0. This is precisely consistent with the pairing (3.25). Therefore, we can identify the set of operators (3.26) with (dΩk)∗ ≃ Ker(d∗)|(Ωk)∗ . Now for the invariant cyclic cohomology HCkh∗ (A), there is an analogous relation with the invariant de Rham homology [45] (with respect to the action of h∗, by translation on the affine space in this case), h∗ HCkh∗ (A) ≃ Ker d∗|(Ωkh∗ )∗ ⊕ Hk−2,dR(A) ⊕ Hk−4,dR(A) ⊕ · · · h∗ The invariant de Rham homology on the r.h.s. are simply represented by constant de Rham currents, i.e. Similarly, Ker d∗|(Ωkh∗ )∗ ≃ ΛkV . So we conclude that Hk−2ℓ,dR(A) ≃ Λk−2ℓV. HCkh∗ (A) ≃ M Λk−2ℓV. ℓ≥0 The order ǫM+1 term in the BV master equation takes the form n−1 i=4 2 1 X(Si, Sn+3−i), (S3, Sn) = − n = 5, 6, · · · , M. δbSM+1 = (S3, SM+1) = − 2 n=4 1 XM(Sn, SM+4−n). These class have clear interpretations in terms of the functionals of Ψ, of degree n+3 = k+1. The ΛkV consists of δb-closed integrals of the form Z [dZ]Tr Ψ ◦ Δi1 Ψ ◦ · · · ◦ Δin+2 Ψ . The remaining Λk−2ℓV for ℓ ≥ 1 are represented by integrals of the form Z [dZ]Tr Ψ2ℓ+1 ◦ Δi1 Ψ ◦ · · · ◦ Δin+2−2ℓ Ψ + · · · , where the · · · stands for terms of the same degree in Δ and Ψ but with different orderings in the trace. The key point in conclusion is that the only nontrivial δb-cohomology classes are represented by functionals in V that involve fewer Δ’s than Ψ’s. This is enough to prove the absence of obstruction in solving the BV master equation for the Born-Infeld deformation to all orders. Now we show that there is an all-order formal deformation of the form S = S2 + S3 + ∞ X Sn+3, n=1 where S4 is the first order non-Abelian Born-Infeld deformation (3.1), and S5, S6, · · · are functionals in V. Sn+3 is of order ǫn. We prove this by induction. Suppose S5, S6, · · · , SM are all functionals of the form V, and solve the BV master equation up to order ǫM . Namely, (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) M n=4 S3, X(Sn, SM+4−n) = 2 X (S3, Sn), SM+4−n M n=5 = − X M n−1 X (Si, Sn+3−i), SM+4−n = 0. The r.h.s. of this equation is a functional of the type V, of degree 3M − 3 in Δ and M + 2 in Ψ, and by (3.33) one sees that the r.h.s. of (3.34) is a δb-closed. Namely, (5-form deformation). Noncommutative deformation In above the Jacobi identity on the BV anti-bracket was used repeatedly. We have seen that cohomology of δb on V of such degrees in Δ and Ψ is trivial. This means that the r.h.s. of (3.34) is δb-exact, and a solution for SM+1 of the type V exists. 4 Other examples In this section, we consider two examples of deformations that preserve maximal supersymmetries, but break either Lorentz invariance (noncommutative deformation) or R-symmetry In every spacetime dimension d between 0 and 10, besides the Born-Infeld deformation, there is only one class of maximally supersymmetric single trace F-term deformations that preserve the Spin(10 − d) R-symmetry. This is the noncommutative deformation of MSYM [46]. As was well known, it can be implemented by replacing the product of fields in MSYM action (2.25) with a noncommutative associative ⋆-product, defined by f (x) ⋆ g(x) = f (x) exp ǫ ωij ←∂−i−→∂j g(x), where ωij is a constant 2-form (more precisely, a Poisson structure). We will fix ωij and think of the coefficient ǫ as an expansion parameter. Cyclicity of the trace is maintained up to total derivatives. Consequently the noncommutative-deformed action still solves the BV master equation. Expanding the deformed action in ǫ to the first order, we obtain the undeformed action S2 + S3 plus This differs from S3′ by a term that can be removed by field redefinition at the first order. Namely, their difference is (S2 + S3)-exact: S3′ − (S3′′ + S4′′) = (S2 + S3, G), 9Written this way, the Born-Infeld deformation looks like a noncommutative deformation with the field strength Fij replacing the non-commutativity parameter ωij. Though, of course, such a naive replacement would have resulted in a non-associative star product. (3.35) (4.1) (4.2) (4.3) (4.4) 2 Z 3 An alternative and equivalent way to write the first order deformation in ǫ is9 S3′′ + S4′′ = ǫ [dZ] Tr ωij Ψ(λΓiχˆ)Ψ(λΓj χˆ)Ψ − ǫ [dZ]ωij Tr Ψ3 ◦ Fˆij Ψ 2 Z 3 S3′ = ǫ [dZ] Tr ωij Ψ∂iΨ∂j Ψ . 1 Z 6 Eq. (4.4) follows from the descending relation {Q, Aˆi} = ∂i −λΓiχˆ. Now the r.h.s. of (4.4) is an integral whose integrand is proportional to the undeformed equation of motion. Therefore, the deformations by S3′ and by S3′′ +S4′′ are equivalent up to a field redefinition, modulo O(ǫ2) terms. The 5-form deformation An F-term deformation that is not an R-symmetry singlet exists in zero dimensional MSYM (IKKT matrix model), transforming in the self-dual 5-form representation of the Spin(10) R-symmetry.10 This arises in the world volume theory of multi-D-instantons probing the AdS5 × S5 background of type IIB string theory, when viewed as a deformation of flat spacetime. The first order deformation of the action is given by h S3′ + S4′ = ǫ [dZ]ωαβTr Ψ((Γmλ)αAˆmΨ)((Γnλ)βAˆnΨ) i where Using [Q, Aˆα] = −dα − 2(Γmλ)αAˆm, it is easy to verify that ǫ [dZ]ωαβTrhΨ3([Aˆα, [Q, Aˆβ]]Ψ)i, ωαβ ≡ ωpqrst(Γpqrst)αβ. 2 Z 3 2 Z 1 Z 3 6 Z + 1 Z 16 (4.5) (4.6) HJEP04(216)7 (4.7) (4.8) (4.9) (4.10) where G is the functional To see this, we can compute the BV anti-brackets of G with S2, S3 as ǫ [dZ]ωij Tr Ψ ◦ AˆiΨ ◦ (∂j + (λΓj χˆ))Ψ . (S2, G) = (S3, G) = ǫ [dZ]ωij Tr Ψ ◦ {Q, Aˆi}Ψ ◦ (∂j + (λΓj χˆ))Ψ , ǫ [dZ]ωij Tr Ψ3 ◦ Fˆij Ψ . (S2, S3′) = (S3, S3′) + (S2, S4′) = (S3, S4′) = 0, and so the BV master equation is obeyed at first order in ǫ. One may attempt to extend this deformation to all-order, by representing it as a noncommutative deformation in the superspace, with the Poisson structure given by ωαβ. Namely, we replace the ordinary product in the undeformed action (2.25) by a noncommutative ⋆-product defined as f (θ) ⋆ g(θ) = f (θ) exp ǫ ωαβ←−dα−→d β g(θ). 10There are other R-symmetry breaking F-term deformations in general d dimensions, that transform in the symmetric traceless tensor representation of Spin(10 − d). They may be viewed as a generalization of the Born-Infeld deformation. We will not discussion their off-shell constructions here. This is only well-defined on (0|16) superspace, because in higher spacetime dimensions the superderivatives dα’s do not commute with one another. Expanding the action to first order in ǫ, one has 2 Z 3 S3′′ = ǫ [dZ]Tr ωαβΨ(dαΨ)(dβΨ) . (4.11) This amounts to replacing (λΓm)αAˆm in S3′ by dα. However, such a construction appears problematic because dα does not commute with the regulator exp(−ζ(λλ¯ + rθ)), and so we would not be able to integrate by parts on dα. Perhaps a suitable ζ → 0 limit can be taken, or one may add terms that cancel the ζ-dependence in the BV master equation. 5 Regularization by smearing In the non-minimal pure spinor descendant field construction, the factor (λλ¯)−1 appears in the descending differential operators, which has a pole at the tip of the pure spinor cone. With sufficiently many descendant fields in the integrand, one may worry about a potential divergence in the integration over the pure spinor space. On the other hand, each net factor of (λλ¯)−1 is accompanied by an rα. When there are more than 11 r’s in the numerator, the integrand vanishes due to the pure spinor constraint relating rα and λ¯α. A priori, there could be a logarithmic divergence coming from integrating r11(λλ¯)−11. In the example of Born-Infeld deformation, the coefficients of such terms appear to be zero, but this isn’t immediately obvious. It was suggested by Berkovits and Nekrasov [24] in the context of pure spinor string theory that one can regularize a potential divergence in the pure spinor integral by smearing the vertex operators in pure spinor space, in a way that preserves BRST invariance. In this section, we will adopt the same smearing operator and consider superspace Lagrangian terms built out of smeared descendant pure spinor superfields. In this way, one could eliminate potential divergences in the pure spinor space integral from the start. A related issue is the construction of D-term deformations. The three examples of deformed BV action of MSYM we have constructed so far are all F-term deformations. It is not clear whether D-terms can be expressed as the integral of a local expression of the superfields over the pure spinor superspace. Naively one may try to apply enough descending operators so that r11 appears and turns the fermionic superspace integral into an integration purely over the 16 θ’s. Such attempts seem to fail. In fact if we could write such an expression using the descending operators, we would also encounter a bosonic integration of (λλ¯)−11 which is logarithmically divergent. It would seem that the construction of D-terms must involve non-local terms on the pure spinor superspace,11 where the smearing construction could be useful as well. 11This is not unfamiliar in the context of harmonic superspace. 5.1 First, one introduces a new bosonic pure spinor variable f α and its fermionic counterpart gIα, as well as their conjugate variables f¯α, g¯α, that obey the constraints12 f αΓαmβf β = f αΓαmβgβ = f¯α(Γm)αβf¯β = f¯α(Γm)αβg¯β = 0. We may also identify gα with the odd differential df α, and g¯α with df¯α. The descendant superfields generally contain terms involving some powers of r and (λλ¯)−1. The idea is to consider the exponential of a Q-exact operator that acts on the field, and effectively shifts λ and λ¯ by a small amount, roughly proportional to f and f¯, so as to smear out the pole in (λλ¯). The differential Q will be extended to Q = λαdα + rα ∂λ¯α ∂ + f α ∂ ∂gα + g¯α ∂f¯α . ∂ Z e−f¯f−df¯df exp(ǫ{Q, X})GbΨ. Note that Q is well defined due to the pure spinor constraints on rα, f α, and g¯α. The smearing operator, which may also be viewed as a regulator, acts on a descendant pure spinor field GbΨ as [Q, Wα] = −dα − 2(Γmλ)αUm, 1 4 Um ≡ − (λf¯)−1(f¯Γmd) + (λf¯)−2(f¯Γmnpg¯)N np. 12One can also generalize this construction by introducing several copies of (f, g, f¯, g¯) variables. (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) X = gαWα + f¯αV α, where W α and V α are differential operators in λ and r respectively, 4 Wα = −(λf¯)−1 41 f¯αN + 1 (Γmnf¯)αN mn , 8 V α = −(f λ¯)−1 1 f α(λ¯∂r) + 1 (Γmnf )α(λ¯Γmn∂r) . Note that Wα takes the same form as the descending operator Aˆα, except that λ¯ has been replaced by f¯. It is useful to write down the Q-commutator of Wα and V α, given by Here X is a linear differential operator in the non-minimal pure spinor variables that acts on GbΨ, and so is {Q, X}. ǫ is a smearing parameter. In writing (5.3) we have made the identification g = df, g¯ = df¯, and the integral is understand as that of a differential top form d11f d11f¯ over the pure spinor space of (f, f¯). Note that f¯f + df¯df = {Q, f¯g} is Q-exact. It is somewhat nontrivial to construct the desired X, since various pure spinor constraints must be obeyed and only certain combinations of the derivatives with respect to the pure spinor variables are allowed. The resulting expression is 8 1 32 and {Q, V α} = W α − f r V α, f λ¯ 4 W α ≡ −(λ¯f )−1 1 f αN + 1 (Γmnf )αN mn , N ≡ λ¯∂λ¯ + r∂r, N mn ≡ λ¯Γmn∂λ¯ + rΓmn∂r. in the regulator exponent where we used the notation Be cautious that W α is not the same as Wα simply with λ, f and λ¯, f¯ exchanged, as it has the extra terms involving r-derivatives. We omit lengthy algebra and record the final expression for the differential operator HJEP04(216)7 {Q, X} = fbΠλ∂λ + f¯Πλ¯∂λ¯ + gΠλd + g¯Πλ¯ + (rΓijmλ¯)(λ¯−1Γijf¯)(Γmλ¯−1) ∂r, (5.8) 1 4 4 8 1 16 (5.7) (5.9) (5.10) (5.11) (5.12) (5.13) λα−1 ≡ (λf¯)−1f¯α, (λ¯−1)α ≡ (λ¯f )−1f α. Πλ and Πλ¯ are projectors that ensures the λ and λ¯ derivatives are well defined. Explicitly, they are given by 1 2 2 (Πλ)αβ ≡ δαβ − (Γmλ)α(Γmλ−1)β = − (λ−1)αλβ − (Γmnλ−1)α(λΓmn)β, (Πλ¯)βα ≡ δαβ − 1 (Γmλ¯)β(Γmλ¯−1)α = − 1 (λ¯−1)β λ¯α − 1 (Γmnλ¯−1)β(λ¯Γmn)α. 1 8 8 In (5.8) we have also defined fbα as a shifted version of f α, 1 2 fbα ≡ f α − (λf¯)−1(gΓmλ)(Γmg¯)α. 5.2 Shifted pure spinor variables The operator exp(ǫ{Q, X}) acts on a field by shifting all superspace variables xm, θα, λα, λ¯α, rα. First, consider the terms in {Q, X} that involve only bosonic derivatives, dropping g, g¯ dependence for the moment, The shift of λ and λ¯ by (5.12) was computed by Berkovitz and Nekrasov, eǫ{Q,X} g,g¯=0 = exp hǫ(fbΠλ∂λ + f¯Πλ¯∂λ¯)i . eǫ{Q,X} eǫ{Q,X} g,g¯=0 λα = (λ + ǫf )α − ǫ g,g¯=0 λ¯α = (λ¯ + ǫf¯)α − ǫ (λΓmf )(Γmf¯)α 2(λ + ǫf )f¯ , (λ¯Γmf¯)(Γmf )α As a consistency check, note that the r.h.s. obey the pure spinor constraint for any finite value of ǫ. Thus, it suffices to consider the action of f − (λf¯)−1(df Γmλ)(df¯Γm) Πλ∂λ + df Πλ (∂θ − Γmθ∂m) + f¯Πλ¯∂λ¯ instead of {Q, X}, on a function of (x, θ, λ, λ¯). The last term in (5.16) commutes with the rest. And so we learn that λ¯ǫ = λ¯ + ǫ f¯ − λ¯Γmf¯ We do not know a simple closed formula for xǫ, θǫ, λǫ. They can be computed order by order in the fermionic variables df, df¯. We write below the first two terms in the expansions of xǫ, θǫ, λǫ in df, df¯. Firstly λǫ, which is independent of x, θ, λ¯, takes the form λǫ = λ + ǫ f − λΓmf 2(λ + ǫf )f¯Γmf¯ − ǫ Z 1 2 0 dt df Γmλ (λ+tǫf )f¯−tǫ (df ΓmΓnf¯)(λΓnf ) 2((λ + tǫf )f¯)2 Γmdf¯− df¯(λ+tǫf ) f¯(λ+tǫf ) Γmf¯ +O(df 2df¯2). Up to order df 2df¯2 terms in the expansion in df and df¯, this expression is exact in ǫ. Likewise, θǫ and xǫ can be solved recursively, Now let us include the g, g¯ dependence. The notation is simplified if we now make the identification rα = dλ¯α, gα = df α, and g¯α = df¯α. We can write eǫ{Q,X}F (x, θ, λ, λ¯, dλ¯) = F (xǫ, θǫ, λǫ, λ¯ǫ, dλ¯ǫ) for any superfield F , where (xǫ, θǫ, λǫ, λ¯ǫ) are functions of (x, θ, λ, λ¯) (independent of dλ¯), that also depends on f, f¯, df, df¯. It follows immediately from the structure of {Q, X} that rǫ is recovered from λǫ by differentiation with respect to λ¯ and f¯, namely (rǫ)α = [Q, (λ¯ǫ)α] = ∂¯(λ¯ǫ)α ≡ (r∂λ¯ + g¯∂f¯)(λ¯ǫ)α. (5.14) (5.15) (5.16) (5.17) (5.18) (5.19) (5.20) Z 1 0 Z 1 0 (θǫ)α = θα + ǫ dt (Πλtǫdf )α xǫm = xm − ǫ dt (df ΠλtǫΓmθtǫ). = θα + ǫdf α − (Γmf¯)α ǫ 2 Z 1 0 dt df Γmλ f¯(λ + tǫf ) − tǫ (df ΓmΓnf¯)(λΓnf ) 2(f¯(λ + tǫf ))2 + O(df 2df¯), 5.3 Superspace Lagrangian deformations using smeared fields A simple class of smeared deformations is the following. Suppose S′ is a first order deformation of the BV action, constructed out of a superspace integral of smeared descendant pure spinor superfields, typically of the form S′ = Z [dZ] Tr nΨ · · · [GbiΨ]ǫi · · · [Gbj1ΨGbj2Ψ]ǫj · · · o It is useful to group several descendants together, and act on with a smearing operator, in constructing a deformation that solves the BV master equation. Let us consider a total action of the form S2 + S3 + S′, and the BV master equation at the first order in the deformation parameter, which demands the vanishing of (S2, S′) and (S3, S′).13 Taking the BV anti-bracket (S2, S′) amounts to computing the variation of S′ under δΨ = ηQΨ, where η is an arbitrary odd parameter.14 Consider a smeared descendant superfield that appears in the integrand of S′, Z where X is the first order differential operator defined as in previous subsections, and Gb is a descending operator that involves the non-minimal variables (but only contains derivatives on x, θ and λ). If Gb commutes (when it is even) or anti-commutes (when it is odd) with Q, we would have δ[GbΨ]ǫ = ηQ[GbΨ]ǫ. This is the case with the non-commutative deformation and the 5-form deformation, as discussed before. Basic example of such Gb operators are λΓmχˆ and (λΓmn)αFˆmn. It is also possible that while not all Gbi’s commute with Q, a suitable linear combination of products of such descendant superfields has the desired property (5.22) (5.23) δ[Gb1ΨGb2Ψ]ǫ = ηQ[Gb1ΨGb2Ψ · · · ]ǫ. We have seen this in the example of the Born-Infeld deformation, in the combination (λΓmΨ)(λΓnΨ)(FˆmnΨ). If all smeared factors in (5.20) have this property, then S′ obeys (S2, S′) = 0. On the other hand, it is easy to see by similar arguments that the ǫ-dependence is S2-exact, which means that the deformation by smearing is independent of ǫ, at least In the non-Abelian MSYM theory, we also need to demand the vanishing of (S3, S′), which is equivalent to the invariance of S′ under δΨ = ηΨ2. This is the translation-invariant cyclic cocycle condition as discussed before. It seems difficult to satisfy this cocycle condition with the product of generic smeared superfields. On the other hand, the cocycle condition can be satisfied if we take ǫ → 0 limit on [GbΨ · · · ]ǫ. Note that when the naive product of such field operators vanishes due to more than 11 powers of r’s, the smeared product can potentially be nontrivial in the ǫ → 0 limit (after the pure spinor superspace integral). We have seen that in the descending operators χˆα, Fˆmn, etc., each pole factor (λλ¯)−1 is accompanied by a factor of rα. Whenever there is potentially an n-th order divergence coming from integrating (λλ¯)−11−n over the pure spinor space, we also have a factor formally of 13For the deformation to be nontrivial (not removable by field definition), we also need S′ to be not exact with respect to (S2, · ) and (S3, · ). 14This variation is not to be confused with a BRST or gauge transformation; if one gauge fixes the BV action by fixing the anti-fields, then the vanishing of (S2, S′) implies that the BRST transformation can be deformed in such a way that S2 + S′ is BRST invariant to first order in the deformation parameter of S′. the form r11+n in the numerator that vanishes. After replacing some of the descendant superfields by their smeared versions, some of the r’s will be shifted to rǫ = ∂¯λ¯ǫ, so that the numerator is no longer identically zero, but of order ǫn. In the denominator, some of (λλ¯)’s will be replaced by (λǫλ¯ǫ), and typically the divergent (λ, λ¯)-integral will be of order ǫ−n. After this “regularization”, the resulting functional can stay finite if we take ǫ → 0 in the end.15 It is clear that the new terms in the integrand that arise this way in the ǫ → 0 limit will always contain r11, which then absorbs the Grassmannian r-integral, leaving no room for a θ-dependent factor from the regulator e−ζ(λλ¯+rθ). The result then looks like an integral of descendant superfields over the full θ-superspace. These appear to be D-terms. We don’t yet have a proposal for the construction of the general D-terms, which we leave HJEP04(216)7 for future work. 6 The main result of this paper is a construction of an all-order Born-Infeld deformation of the MSYM theory, in the non-minimal pure spinor superspace formalism. It would be nice to produce the corresponding all-order deformed superfield equation of motion in the ordinary superfield Aα(x, θ), after eliminating the auxiliary fields having to do with the non-minimal variables. In practice, as explained in section 2.5, this amounts to finding the minimal representatives of certain non-minimal pure spinor cohomology classes. An unsatisfying aspect of the story is that we don’t know how to write the general D-terms in the non-minimal superspace formalism (which one might have expected to be the easiest thing). This question is also related to how to write the D-term deformation of the equation of motion in terms of the on-shell superfield Aα(x, θ). The answer to the latter question is nontrivial though in principle known: as explained by [4, 5] and also discussed in [9], a gauge invariant expression tr(G) in component fields is mapped to a deformation of the superfield equation by the composition of the Connes differential with a map δ that amounts to performing a full superspace integral, but is constructed rather inexplicitly through a spectral sequence argument that involves lifting the relevant chain complex to a complex of vector bundles over the projective pure spinor space. We suspect that the D-terms must be written as a non-local expression in pure spinor superspace. This is presumably closely related to the regularization of [24], which is relevant in computing the D-term contributions in higher genus string amplitudes. Though we have constructed an all-order Born-Infeld deformation, in principle it may differ from the BornInfeld theory that arises as the α′-expansion of the low energy effective theory of open strings on D-branes, by some D-term ambiguity. A potential application of our construction of the all-order Born-Infeld action, as well as a test of its relation to the open string effective action, would be to find some nontrivial nonlinear solutions to the equation of motion in the non-minimal pure spinor superfields and compare it with D-brane configurations (along 15One might worry about the terms that involve ((λ + ǫf )f¯)−1 or ((λ¯ + ǫf¯)f )−1 in the formula for the shifted pure spinor variables giving rise to extra poles in ǫ. A more careful inspection of the λǫλ¯ǫ factors in the denominator shows that this doesn’t happen. the lines of [47]). It would also be interesting to directly connect our construction to open string disc amplitudes in the pure spinor formalism. Ultimately, the non-minimal pure spinor formalism for constructing higher derivative terms may be most useful in maximally supersymmetric supergravity theories. In [40, 48] Cederwall wrote down a remarkable manifestly supersymmetric complete BV action for 11dimensional supergravity in pure spinor superspace. It would be interesting to construct the R4 deformation in this formalism. Acknowledgments We are grateful to Shu-Heng Shao for collaboration at the initial stage of the project, and to Clay Cordova, Thomas Dumitrescu, Ken Intriligator, Daniel Jafferis, and Nati Seiberg for helpful discussions. We would like to thank the organizers of the workshop String Geometry and Beyond at Soltis Center, Costa Rica, the KITP program New Methods in Nonperturbative Quantum Field Theory, and especially the support of KITP during the course of this work. This work is supported in part by a KITP Graduate Fellowship, a Sloan Fellowship, a Simons Investigator Award from the Simons Foundation, NSF Award PHY-0847457, and by the Fundamental Laws Initiative Fund at Harvard University. A Siegel gauge and the b ghost In order to go from the BV action functional to a gauge fixed BRST invariant action, a gauge fixing condition must be imposed that determines the anti-fields in terms of the ordinary gauge fields and the ghosts. Note that the gauge fixing procedure in the BV formalism is different from that of an ordinary gauge invariant classical action, in that one should impose the gauge fixing condition before applying the variational principle on the action functional to obtain the equation of motion. In the pure spinor superspace formulation of the BV action of MSYM, it is a priori not clear how to separate Ψ(x, θ, λ, λ¯, r) into ordinary gauge fields and anti-fields. It has been suggested that an appropriate gauge fixing condition is the Siegel gauge [6, 7, 49] HJEP04(216)7 bΨ = 0, {Q, b} = ∂m∂m. 1 1024 where b is a second order differential operator that obeys The b ghost admits the following representation16 1 2 1 64 1 16 b = − (λλ¯)−1(λ¯Γmd)∂m + (λλ¯)−2(λ¯Γmnpr) Nmn∂p − dΓmnpd 1 24 (λλ¯)−3(rΓmnpr)(λ¯Γmd)Nnp − (λλ¯)−4(λ¯Γmnsr)(rΓpqsr)NmnNpq. 16The signs in our formula differ slightly from those of [7]. (A.1) (A.2) (A.3) b = ∂mAˆm − 2 dαχˆα + 1 4 NmnFˆmn. This is reminiscent of the form of the integrated massless vertex operator in pure spinor string theory. Indeed it is easy to verify {Q, b} = ∂m{Q, Aˆm} + ∂m(λΓmχˆα) + 2 dα[Q, χˆα] − 4 (λΓmnd)Fˆmn + Nmn{Q, Fˆmn} 1 4 1 1 2 = ∂m∂m + N mnλΓmηˆn = ∂m∂m. (A.5) HJEP04(216)7 Let us inspect the Siegel gauge condition more explicitly in the simple example of free Abelian theory. Consider a solution to QΨ = 0 that involves only the minimal pure spinor variables of the form Ψ(x, θ, λ) = (λΓmθ)am(x) + (λΓmθ)(θΓmnpθ)∂nap + · · · . 1 4 Such a Ψ does not obey Siegel gauge condition, since 1 2 1 bΨ = − (λλ¯)−1(λ¯ΓmΓnλ)∂man(x) + O(θ) = − 2 ∂mam(x) − (λλ¯)−1(λ¯Γmnλ)∂man(x) + O(θ). 2 While we can set ∂mam to zero by imposing Lorentz gauge condition on am, ∂man is a nontrivial field strength and cannot be removed this way. We would like to add to Ψ some Q-exact terms to go to Siegel gauge. Using the non-minimal variables, we can write (λΓmθ)am + · · · as an exact expression with respect to λαdα (which is not the same as Q in the non-minimal formalism) Ψ(x, θ, λ) = (λαdα) (λλ¯)−1(λ¯Γnpλ)(θΓmnpθ)am + · · · , This expression can be expressed simply in terms of the descending operators QΨ + δΨ δSint + bΛ = 0, Ψ + b δΨ δSint = 0. = − (λλ¯)−1(rΓnpλ)(θΓmnpθ)am + (λλ¯)−2(rλ)(λ¯Γnpλ)(θΓmnpθ)am + · · · (A.4) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) Another property of the b ghost operator is b2 = 0. This is necessary for the Siegel gauge condition to be compatible with BV master equation. After fixing to Siegel gauge, the equation of motion may be obtained from the BV action of the form S2 + Sint as where Λ is an arbitrary Lagrangian multiplier superfield. 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Chi-Ming Chang, Ying-Hsuan Lin, Yifan Wang, Xi Yin. Deformations with maximal supersymmetries part 2: off-shell formulation, Journal of High Energy Physics, 2016, 171, DOI: 10.1007/JHEP04(2016)171