Self-consistency of conformally coupled ABJM theory at the quantum level

We study the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 6 superconformal Chern-Simons matter field theory (the ABJM theory) conformally coupled to a Lorentzian, curved background spacetime. To support rigid supersymmetry, such backgrounds have to admit twistor spinors. At the classical level, the symmetry of the theory can be described by a conformal symmetry superalgebra. We show that the full N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 6 superconformal algebra persists at the quantum level using the BV-BRST method.


Introduction
Supersymmetric field theories in curved space-time have attracted attention in recent years [1][2][3][4][5][6][7][8][9][10][11]. In such theories, gravity is non-trivial but is kept non-dynamical; it is considered as a rigid background on which matter fields propagate, in contrast with supergravity in which the dynamical gravitational field plays the role of the gauge field of local diffeomorphisms. While theories which preserve rigid supersymmetry in curved space at the classical level are well studied in the literature, the issue of realization of supersymmetry at the quantum level is less investigated. The aim of the present work is to study the ABJM theory [12] in Lorentzian curved space-times and to investigate whether this theory can be consistently formulated as a well-defined perturbatively renormalized quantum field theory. In fact, the original theory, formulated in flat space-time, is claimed to remain superconformal at the quantum level. We prove this claim by showing that all the symmetries of this theory are realized at the quantum level, in the sense that we explain in the following.
The ABJM theory [12], usually formulated in three dimensional flat Minkowski space, is an N = 6 superconformal gauge theory which plays a crucial role in the AdS 4 -CFT 3 correspondence. By conformally coupling the ABJM theory to a curved background M , we obtain an action invariant under a conformal symmetry superalgebra S [13]. S contains as the even part conformal Killing vectors of M (if any) and so(6) (the R-symmetry), and as the odd part the set of all twistor spinors on M in the fundamental representation of so (6). 1

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As it turns out, in this theory rigid supersymmetry and local gauge symmetry are intertwined in the sense that supersymmetry transformations close onto field dependent gauge transformations and equations of motion. The correct gauge-fixing for such "open algebras" can be elegantly performed by employing (an extended version of) the BV-BRST formalism [15][16][17][18], which is adapted to curved space-time in [19]. More precisely, all classical symmetries of the theory, including local gauge symmetry and rigid conformal, R and supersymmetry, can be integrated into one nilpotent BRST symmetry, s. However, the realization of s at the quantum level is potentially obstructed by anomalies which are subject to certain consistency conditions of a cohomological nature.
We work out the relevant cohomology class which contains potential anomalies and show that this class is trivial in the case of the ABJM theory. This means that the trivial anomaly can be removed by finite renormalization, i.e., by adding suitable finite counter terms to the action. The local and covariant nature of these counter terms in curved space-time is best illustrated in the framework of locally-covariant QFT [20][21][22][23][24] which yields a perturbative construction of interacting renormalized QFT on an arbitrary curved (Lorentzian) manifold. It is furthermore shown in [25,26] that the absence of the anomaly directly implies that the full symmetry of the underlying classical theory (in our case, rigid conformal supersymmetry S and the local gauge symmetry) are preserved at the quantum level, in the sense that there exists a renormalization scheme in which: (i) the renormalized Noether current of BRST symmetry is conserved, as an operator-valued distribution, (ii) the renormalized BRST charge Q L is nilpotent and can thus be used to define the Hilbert space of physical state as KerQ L /ImQ L , (ii) the scattering matrix commutes with Q L , and (iv) for classical observables O which are annihilated by s, the corresponding renormalized composite quantum field O L (anti-)commutes with Q L .

ABJM theory conformally coupled to a curved space-time
The model written by Aharony, Bergman, Jafferies, and Maldacena (ABJM) [12] is a well-studied example of AdS-CFT correspondence. It is a 2+1 dimensional, N = 6 superconformal gauge theory with gauge group G = U(N ) × U(N ), which consists of two Chern-Simons theories, corresponding to each U(N ) factor, with opposite levels (couplings) k. In this section, we describe the conformally coupled theory.

Conformal symmetry superalgebra S
The algebraic structure underlying rigid supersymmetric theories is that of a Lie superalgebra S. For the case of the conformally coupled theories, S is called a conformal symmetry superalgebra. This notion was defined in [13,19] where a complete classification of S for dimensions 3,4,5,6 and for both Riemannian and Lorentzian case was also given. Here we describe S for the case of the 3 dimensional ABJM theory conformally coupled to a curved space-time background. We begin with the geometrical setting.
Let (M, g) be a Lorentzian 3 dimensional manifold with signature (−, +, +) which admits a spin structure. Given the Levi-Civita connection ∇, the Lie derivative of a vector JHEP11(2017)153 field X(x) ∈ X(M ) along another vector field Y is defined by (2.1) It extends to tensor fields of arbitrary rank via the Leibniz rule. The Lie derivative also defines a representation of X(M ) on the space of all tensor fields, since [L X , A conformal Killing vector field X is defined to satisfy where σ X := − 1 3 ∇ µ X µ . That is, the infinitesimal diffeomorphism generated by a conformal Killing vector (CKV) leaves the metric invariant up to a conformal factor −2σ X . The linear space of all CKVs together with the Lie bracket of vector fields form the Lie algebra X c (M ) of conformal isometries of (M, g). X c (M ) depends only on the conformal class of g. For instance, for conformally flat metrics it is isomorphic to X c (R 1,2 ) = so(2, 3).
The fermionic counterpart of a CKV is a conformal Killing spinor or a twistor spinor. In order to introduce them, we first need to fix some basic notions about the spinors. On a curved space-time, each γ µ is a section of the Clifford bundle C (T M ) over M , associated to a basis element ∂ µ in X(M ). They obey 2 where ∇ µ is the covariant derivative on spinors. We denote the linear space of all twistor spinors by S c (M ). Similar to X c (M ), S c (M ) also depends only on the conformal class of g, since the twistor equation (2.4) is Weyl invariant. An important property of twistor spinors is that they "square to" conformal Killing vectors in the sense that given any two twistor spinors ψ, χ, the bi-linearψγ µ χ is a conformal Killing vector. There is a similar notion of Lie derivative acting on spinor fields given bŷ where L X ψ = ∇ X ψ+ 1 4 (∇ µ X ν )γ µν ψ.L X defines a representation of the conformal isometry algebra on S c (M ), i.e. for X, In the following, we take the supersymmetry parameter, , to be a real bosonic (commuting) twistor spinor carrying the 6 representation of so (6). Let {e I } be a basis on C 6 , with I = 1, 2, 3, 4, 5, 6, and let {e I } be the dual basis. Relative to this basis, = I e I . We denote the components of the so(6)-invariant nondegenerate bilinear form on C 6 by IJ , with inverse IJ (i.e. IK KJ = δ I J ).

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Now, the conformal symmetry superalgebra, for the conformally coupled ABJM theory, is defined to be a Z 2 -graded vector space S = B ⊕ F , where together with the following graded Lie bracket for all X, Y ∈ X c (M ), ρ, σ ∈ so(6), 1 , 2 ∈ F , In the right hand side of (2.7), the brackets are the Lie brackets of vector fields and the Lie algebra so(6), respectively. In the right hand side of (2.8), (ρ · ) I = ρ I J J , with which is shown to be covariantly constant using the twistor spinor equation (2.4). Also in the right hand side of (2.9), and ρ 12 ≡ 1 2 (ρ 1 + 2 − ρ 2 − ρ 1 ) which is constant on M , as required by consistency of (2.9). The graded Lie bracket [−, −] can be shown to satisfy the graded Jacobi identity [13].

Rigid backgrounds admitting twistor spinors
Admitting twistor spinors puts a strong restriction on the underlying background spacetime. A complete classification of such Lorentzian manifolds is given in [27][28][29][30][31]. The maximum number of linearly independent Killing vectors, and twistor spinors in d dimensions are 1 2 (d + 1)(d + 2) and 2 d 2 +1 respectively. The two bounds are saturated for locally conformally flat metrics. In three dimensions, Lorentzian manifolds admitting twistor spinors fall into two distinct classes: (1) Locally conformally flat metrics, 3 such as R 1,2 , dS 3 and AdS 3 . On Minkowski R 1,2 , the general solution to the twistor equation is with 0 begin a constant spinor with 4 linearly independent components, and α, β arbitrary constants. There are also 10 linearly independent conformal Killing vectors which form the algebra so(2, 3). The ABJM superalgebra is S R 1,2 ∼ = osp(6|4).

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(2) Certain types of pp-wave metrics [27,32] which are not conformally flat. The pp-wave metric in Brinkmann coordinates (u, v, x) [33] takes the form where h is an arbitrary smooth function of u and x. All pp-wave metrics whose h are higher than quadratic in x are not conformally flat.
For pp-waves with arbitrary h, any twistor spinor is parallel, i.e. ∇ µ = 0, and for the non-conformally flat class, the maximum number of independent components are 2. The only conformal Killing vector is ξ = ∂ ν which is null.

Supersymmetry transformations and the supersymmetric Lagrangian
Besides rigid conformal supersymmetry, the ABJM theory has a local gauge symmetry with gauge group G = U(N ) × U(N ), with Lie algebra denoted by g = u(N ) ⊕ u(N ). Here we construct an invariant Lagrangian under S ⊕ G, where G = C ∞ (M, g) in component form. It differs from the flat space Lagrangian [12,34] by expected curvature couplings.
Field content. The basic fields are (A,Â, X, Ψ) as explained in the following. 4 • Matter fields, (X A , Ψ A ), with A = 1, 2, 3, 4, are a complex scalar and a Dirac spinor field respectively. They both carry the 4 representation under su(4) ∼ = so(6) Rsymmetry. X A and Ψ A also transform in the bi-fundamental representation (N, N) of u(N ), while X A and Ψ A transform in the anti-bi-fundamental representation (N,N).
Since matter fields X A , Ψ A and the supersymmetry parameters I carry different representations of the R-symmetry, one needs intertwiners between the 6 and 4 representations. These are six 4 × 4 anti-symmetric matrices Γ I AB = −Γ I BA satisfying where ABCD is the so(6)-invariant totally anti-symmetric tensor in the 4 representation. We also denote ( Action of S on fields. For the bosonic part B ⊂ S, the action of a conformal Killing vector X on the space of field configurations is given by where w Φ is the Weyl weight of a generic field Φ given in table 1. In addition, the so(6) R-symmetry acts by Since the ρ I J are covariantly constant (2.11), the variations (2.16) and (2.17) define a representation of B on the space of fields since (2.18) The action of the Fermionic part F ⊂ S is the curved space ganeralization of the usual superconformal transformations of the ABJM theory in flat spacetime [34]: where I is a twistor spinor satisfying (2.4). The superconformal variations (2.19) define a representation of F on fields only up to a field-dependent gauge transformation and equations of motion, by contrast to the case of B. In fact, the commutator of two supersymmetry transformations turns out to be of the form 5 and where δ ρ is defined in (2.17). The field-dependent gauge transformation parameters In verifying (2.20), we have used¯ = 0 and ¯ = 1 2 ξ as well as ∇ 2 = − 3 8 R which follow from the twistor equation for , together with the Lichnerowicz-Weitzenböck identity ∇ 2 ψ = ∇ 2 ψ + 1 4 Rψ for arbitrary spinors ψ.

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where In the particular case of the Minkowski spacetime R 1,2 , the solution to the twistor spinor equation (2.13) is a linear combination of a constant spinor I 0 , and x I 0 ≡ η I . In this case, the above transformations for I = I 0 reduce to the ABJM supersymmetry transformations on R 1,2 , while with I = η I , they reduce to the superconformal transformations, both derived in [34].
Superconformal Lagrangian. To obtain an invariant Lagrangian, we also need to specify how the dynamical fields transform under local gauge symmetry. Under a local gauge transformation with parameters (Λ,Λ) ∈ C ∞ (M ) ⊗ (u(N ) ⊕ u(N )), The gauge covariant derivatives act on fields by The invariant Lagrangian, L N =6 , under the action of S ⊕ G is the conformally coupled version of the flat space Lagrangian given in [12,34]. Explicitly, with k being the Chern-Simons level, where where R is the scalar curvature of (M, g), and wherē

Gauge fixing and BRST symmetry
In this section, we describe the gauge-fixing of the conformally coupled ABJM theory. As can be seen from (2.20), in this theory, supersymmetry transformations not only close onto local gauge transformations but also onto a term proportional to the equations of motion of fields. This defines a so-called "open algebra" whose gauge-fixing requires a suitable extended version of the BV-BRST formalism [15][16][17][18][19]. Here, we work this out for the theory at hand. One associates besides the dynamical ghost for the local gauge symmetry, a set of rigid ghosts for conformal, R, and supersymmetry as well. Moreover, to every field and ghost Φ of the theory, there is an associated anti-field Φ ‡ . Then an extended BRST differential s acting on (Φ, Φ ‡ ) is defined roughly as follows. sΦ is the sum of all (local and rigid) symmetry transformations of Φ with each symmetry parameter replaced by its corresponding ghost, plus certain terms needed to incorporate the open nature of supersymmetry algebra. sΦ ‡ is the equation of motion of Φ coming from an extended action S. Of course, S and the precise form of s has to be made in such a way to satisfy s 2 = 0 and sS = 0 which we intend to explain in the following.

BRST structure of N = 6 superconformal Chern-Simons matter theory
In order to perform the gauge fixing and the BRST quantization, we need to enlarge the field configuration space, containing (A µ ,Â µ , X A , Ψ A ), to include the following elements: • A pair (c,ĉ) of (u(N ) ⊕ u(N ))-valued Grassmann odd scalars, which are dynamical ghosts for the local gauge symmetry parameters. In components, c = c i T i ,ĉ =ĉ iT i (see section 2.3).
• A (B,B) and (c,c) system which are two pairs of (u(N ) ⊕ u(N ))-valued dynamical scalar fields needed for the gauge fixing term.
• Anti-fields Φ ‡ for each field and ghost Φ. For instance A ‡ µ for A µ etc. By definition, each Φ ‡ has the opposite Grassmann parity of the Φ, and if (∆ Φ , g Φ , w Φ ) are dimension, ghost number and Weyl weights of Φ respectively, then Φ ‡ has  We summarize the data for all ghosts in table 2. Let us first define the nilpotent operatorŝ which increases the ghost number by +1, and acts on all fields Φ bŷ We next define the following action functionalŜ is a source term, which couples all BRST transformed fieldsŝΦ i to their corresponding sources Φ ‡ i . In addition, we have added the anti-field action S af which is quadratic The operatorŝ, can now be extended to anti-fields byŝ = Ŝ , − , where is called the anti-bracket and satisfies the graded Jacobi identity. Finally, gauge-fixing is done by performing a "canonical transformation" on the actionŜ, where Ψ is the "gauge fermion" with ghost number −1 given by which implements the Feynman gauge. The gauge-fixed action of the enlarged theory is thus defined to be S = e −,Ψ Ŝ =Ŝ +ŝΨ + 1 2 ŝΨ, Ψ . Note that by contrast to the usual gauge-fixing of the Yang-Mills theory, our gauge-fixed action has an additional term 1 2 ŝΨ, Ψ . This term appears due to the presence of the quadratic action S af in anti-fields Ψ ‡ A and (A ‡ µ ,Â ‡ µ ) which, in turn, is needed for obtaining a nilpotent differential because of the δ e.o.m. -term in (2.20) which occurs for Ψ A and (A µ ,Â µ ). Such a term is characteristic for supersymmetric gauge theories where the algebra of transformations only closes on-shell.
The gauge-fixed action S leads to the BV-BRST differential s, defined by The action of s on all fields coincides with that ofŝ (3.2), except for (A µ ,Â µ ) where it is On anti-fields is given by The s transformation of those anti-fields which are needed later are given in appendix A. The gauge-fixed action S is checked to satisfy S, S = 0, (3.13) and using the graded Jacobi identity for anti-bracket, one shows from this that s 2 = 0, (3.14)

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as an operator. The identity (3.13) expresses the invariance of S under all rigid and local symmetries, i.e. sS = 0. In fact, the Noether current of s is given by which includes the gauge current, supercurrent, R-symmetry current and conformal current: where T µν is the energy-momentum tensor of S N =6 and J af µ is an anti-field dependent part.

Calculation of BRST cohomology
In this part, we present the cohomology analysis of the nilpotent BRST differential s defined in the previous section. We calculate a particular cohomology class which contains the potential anomalies of the classical symmetry at quantum level, and show that this class is trivial in our theory. As mentioned in the introduction, the counter-terms which are used for finite renormalization in curved space-times [22] are local and covariant functionals O of the field configurations (Φ, Φ ‡ ) and the background metric g, which by definition are of the form where each P p q (M ) is defined to be the space of functionals with ghost number q and form degree p. The q-th cohomology ring of s at form degree p is defined by The anomaly A = M a(x)dx is shown [25] to be a formal power series in In our formalism, we are working off-shell (where the equations of motion are not necessarily satisfied) because of the open nature of the supersymmetry algebra (2.20). We must, thus, consider the fields and anti-fields (Φ, Φ ‡ ) and their derivatives as independent and we will refer to them as basic fields.
To analyse the cohomology ring (4.5) which contains potential anomalies, we first collect two well-known results in the homological perturbation theory [15,35].

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We observe that the following fields form s 0 -doublets, . Also, since both N and s 0 (anti-)commute with ∇ µ and d, all derivatives of each doublet form again new doublets, e.g. {∇ ν A µ , ∇ ν ∇ µ c}. Hence, such fields do not belong to the s 0 -cohomology.
A general element O ∈ H 3 1 (s 0 |d, M ) is a 3-form and ghost number +1, and satisfies s 0 O = dθ, for some θ ∈ P 2 2 (M ). Here we are only interested in the anomaly a(x) which is of dimension 3. Let us denote the Hodge dual of a by A which is a scalar. To have ghost number +1, A has to either • be linear in ghosts (c,ĉ), , X, α with ghost number +1, which we denote by A(c,ĉ), A( ), A(X), A(α) respectively, or • depend on X ‡ A , ‡ , α ‡ , X ‡ with negative ghost numbers, and enough number of other ghosts to make a combination with ghost number +1. We denote them by A(X ‡ ), A( ‡ ), A(α ‡ ), A(X ‡ ), respectively.
The most general manifestly gauge and so(6)-invariant form that these terms can take are I AB¯ I γ µ Tr(Ψ A X B X C ∇ µ X C ) +b 3 Γ IAB¯ I γ µ Tr(Ψ A X B X C ∇ µ X C ) + b 4Γ I AB¯ I γ µ γ ν Tr Ψ A ∇ µ ∇ ν X B +b 4 Γ IAB¯ I γ µ γ ν Tr(Ψ A ∇ µ ∇ ν X B ) + b 5Γ I AB Tr ∇ µ (¯ I Ψ A )∇ µ X B +b 5 Γ IAB Tr ∇ µ (¯ I Ψ A )∇ µ X B + b 6Γ I AB ∇¯ I Tr Ψ A X B X C X C +b 6 Γ IAB ∇¯ I Tr Ψ A X B X C X C + b 7 RΓ I AB¯ I Tr Ψ A X B +b 7 RΓ IAB¯ I Tr(Ψ A X B ) A(α) = d 1 α I J Γ IABΓJ CD Tr(X A X C X B X D X E Φ E ) + d 2 α I J Γ IABΓJ CD Tr(Ψ C Ψ A X B X D ) +d 1 α I J Γ IABΓJ CD Tr(X C X B X D X A X E X E ) +d 2 α I J Γ IABΓJ CD Tr(Ψ A Ψ C X D X B ),

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.