Superconformal duality-invariant models and $\mathcal{N} = 4$ SYM effective action

We present $\mathcal{N}=2$ superconformal $\mathsf{U}(1)$ duality-invariant models for an Abelian vector multiplet coupled to conformal supergravity. In a Minkowski background, such a nonlinear theory is expected to describe (the planar part of) the low-energy effective action for the $\mathcal{N}=4$ $\mathsf{SU}(N)$ super-Yang-Mills (SYM) theory on its Coulomb branch where (i) the gauge group $\mathsf{SU}(N)$ is spontaneously broken to $\mathsf{SU}(N-1) \times \mathsf{U}(1)$; and (ii) the dynamics is captured by a single $\mathcal{N}=2$ vector multiplet associated with the $\mathsf{U}(1)$ factor of the unbroken group. Additionally, a local $\mathsf{U}(1)$ duality-invariant action generating the $\mathcal{N}=2$ super-Weyl anomaly is proposed. By providing a new derivation of the recently constructed $\mathsf{U}(1)$ duality-invariant $\mathcal{N}=1$ superconformal electrodynamics, we introduce its $\mathsf{SL}(2,{\mathbb R})$ duality-invariant coupling to the dilaton-axion multiplet.


Introduction
It is believed that the N = 4 super Yang-Mills (SYM) theory is self-dual [1,2] (see also [3]). This conjecture was originally put forward in the late 1970s as a duality between the conventional and soliton sectors of the theory. Twenty years later it was suggested 1 [7] that self-duality might be realised in terms of a low-energy effective action of the theory on its Coulomb branch. Here the gauge group SU(N) is spontaneously broken to SU(N − 1) × U(1) and the dynamics is described by a single N = 2 vector multiplet associated with the U(1) factor of the unbroken group. Two different realisations of selfduality for the N = 4 SYM effective action in N = 2 superspace were proposed: (i) self-duality under Legendre transformation [7]; and (ii) self-duality under U(1) duality rotations [8]. 2 So far, both proposals have not been derived from first principles, although each of them is consistent with the one-loop [10][11][12][13][14] and two-loop [15] calculations. 3 It is worth pointing out that (ii) implies (i), see [8,17] for the technical details.
Eight years ago, the general formalism of supersymmetric duality rotations [8,17,[35][36][37] was combined with the bosonic approach due to Ivanov and Zupnik 5 [24][25][26] to develop new formulations for N = 1 and N = 2 supersymmetric duality-invariant theories coupled to supergravity [44]. The method makes use of an auxiliary unconstrained chiral superfield (a spinor in the N = 1 case and a scalar for N = 2) and is characterised by the fundamental property that U(1) duality invariance is equivalent to the manifest U(1) invariance of the self-interaction. In the N = 1 rigid supersymmetric case, analogous results were independently obtained in [45]. 6 A constructive perturbative scheme to compute N = 2 superconformal U(1) dualityinvariant actions for the N = 2 vector multiplet was described in [8]. In the present paper we will give closed-form expressions for such actions using the formalism of Ref. [44] in conjunction with the results of earlier publications [51,52] devoted to the N = 2 super-Gauss-Bonnet term and super-Weyl anomalies.
While Ref. [8] described the N = 2 superconformal U(1) duality-invariant actions, it did not discuss N = 0 and N = 1 duality-invariant models compatible with (super) conformal symmetry. The consideration in [8,17] was restricted to those N = 0 and N = 1 self-dual systems which are well-defined in a weak-field limit. 7 A year ago, it was shown that there exists a unique model for conformal duality-invariant electrodynamics [53] (see also [54]). Recently its N = 1 superconformal extension has been introduced [55]. Below we will show how the N = 1 superconformal duality-invariant model of [55] naturally occurs within the framework developed in [8,17,44]. This paper is organised as follows. As a warmup exercise, in section 2 we give a new derivation of the recently constructed N = 1 superconformal duality-invariant model [55]. Section 3 describes the new N = 2 superconformal duality-invariant models. The obtained results and some generalisations are discussed in section 4. The main body of the paper is accompanied by three technical appendices. Appendix A reviews the model for conformal duality-invariant electrodynamics [53] (see also [54]). Appendix B collects those results concerning N = 1 supergravity and super-Weyl transformations, which are used in section 2. Appendix C contains similar material but for the N = 2 case. Our two-component spinor notation and conventions correspond to [56,57]. 6 The IZ approach [24,25] has also been generalised to higher dimensions in [46]. It has been used [47] to establish the relation between helicity conservation for the tree-level scattering amplitudes and the electric-magnetic duality. In four dimensions a hybrid formulation has been developed [48] which combines the powerful features of the IZ approach with the Pasti-Sorokin-Tonin formalism [49,50]. 7 This means that the interactions Λ(z,z) in (2.9) and (A.2) were chosen in [8,17] to be real analytic.
As a result, (super)conformal nonlinear systems were automatically excluded.

N = 1 superconformal duality-invariant model
We consider a dynamical system describing an Abelian N = 1 vector multiplet in curved superspace and denote by S[W,W ] the corresponding action functional. The action is assumed to depend on the chiral spinor field strength W α and its conjugateWα which are constructed in terms of a real unconstrained gauge prepotential V [58,59] as The prepotential is defined modulo gauge transformations such that δ λ W α = 0. The gauge-invariant field strengths W α andWα obey the Bianchi identity and thus W α is a reduced chiral superfield. We assume that S[W,W ] does not involve the combination D α W α as an independent variable, and therefore it can unambiguously be defined as a functional of a general chiral superfield W α and its conjugateWα. Then, defining Here the variational derivative δS/δW α is defined by where E denotes the chiral integration measure, and W α is assumed to be an unrestricted covariantly chiral spinor.
Since the Bianchi identity (2.3) and the equation of motion (2.5) have the same functional form, one may consider U(1) duality rotations with λ ∈ R a constant parameter. The condition for duality invariance is the so-called self-duality equation 8) in which W α is chosen to be a general chiral spinor.

Formulation without auxiliary chiral variables
General duality-invariant supersymmetric theories with at most two derivatives at the component level were constructed in [8,17,35,36]. They belong to the family of nonlinear vector multiplet theories of the general form where W 2 = W α W α andW 2 =WαWα, the complex variable u is defined by and Υ is a nowhere vanishing real scalar with the super-Weyl transformation with σ being the super-Weyl parameter, see appendix B for more details. The transformation law (2.11) implies that (2.9) is super-Weyl invariant. We remind the reader that V is inert under the super-Weyl transformations, and therefore the following composite chiral scalar is super-Weyl invariant. In the functional (2.9), the expression in the first line is the free vector multiplet action, while the interaction effects are encoded by Λ(z,z) which is a real function of a complex variable z.
Three different realisations of Υ are possible: 1. One option is that Υ is a composite superfield, which is constructed in terms of the chiral compensator S 0 of old minimal supergravity and matter chiral superfields ϕ i , where K(ϕ,φ) is the Kähler potential of a Kähler manifold. The super-Weyl transformations laws of the chiral compensator and matter chiral superfields are 3. One more option is given by Υ = D α W α =DαWα, which corresponds to the family of superconformal vector multiplet models introduced in [60]: Here H(z,z) is a real function of a complex variable. In general, this action explicitly depends on D α W α , and thus there is no way to unambiguously define it as a functional of an unrestricted chiral spinor W α and its conjugateWα; such models are not compatible with duality invariance. However, all dependence on D α W α disappears provided 8 for real parameters x, y.
In what follows, it will be assumed that Υ is a compensating multiplet independent of the vector multiplet V , which excludes option 3 with the exception of (2.19).
The model (2.9) is U(1) duality-invariant if the interaction Λ(u,ū) satisfies the following differential equation The function Λ(u,ū) was chosen in [8,17] to be real analytic, in order for the model to be well-defined in the weak-field limit. However Λ(u,ū) is not required to be globally analytic. It should be pointed out that setting Υ = g −1 = const in (2.9) and choosing Λ(u,ū) = 1 defines the N = 1 supersymmetric Born-Infeld action [61]. This U(1) duality-invariant theory is a Goldstone multiplet action for partial N = 2 → N = 1 supersymmetry breaking in Minkowski space [62,63], as well as in the following maximally supersymmetric backgrounds [64]: (i) R × S 3 ; (ii) AdS 3 × R; and (iii) a supersymmetric plane wave.

Superconformal duality-invariant model
The model (2.9) is superconformal provided the functional form of Λ(u,ū) is such that (2.9) is independent of Υ. 9 This condition implies that However, only a special choice of f proves to be compatible with duality invariance, eq. (2.20), specifically with x and y real parameters. Now the self-duality equation (2.20) is satisfied iff Due to the identity the superconformal U(1) duality-invariant action takes the form This is the model proposed in [55].

Formulation with auxiliary chiral variables
There exists a different formulation for the models discussed above. Following [44], we consider an action functional of the form The vector multiplet action (2.18) is superconformal. However it is not compatible with duality invariance since the integrand depends on D α W α .
and the auxiliary spinor η α is only constrained to be covariantly chiral,Dβη α = 0. The action is super-Weyl invariant if η α transforms as 10 27) in conjunction with the transformation of Υ, eq. (2.11).
Making use of the equations of motion for η α andηα allows one to integrate out these variable to end up with an action of the form (2.9). For the interaction Λ(u,ū) one obtains see [44] for the technical details. Here the variables u and v are related to each other as follows where the symbol ≈ is used to indicate that the result holds modulo terms proportional to η α andηα or, equivalently, to W α andWα (such terms do not contribute to the action).
Our model (2.26) possesses U(1) duality invariance under the condition see [44] for the technical details.

Superconformal duality-invariant model
The duality-invariant model defined by eqs. (2.26) and (2.30) is superconformal if the action is independent of Υ, which means that the functional form of F is fixed modulo a single real parameter, The auxiliary variables η α andηα can be integrated out using the equation of motion for η α , and its conjugate. Rather lengthy calculations lead to where Λ SC (u,ū) has the form (2.23), with the parameters x and y being given by It is interesting to compare the results obtained in subsections 2.2 and 2.4. Within the approach without auxiliary chiral variables, the most general superconformal coupling is given by eq. (2.22) and involves an arbitrary real function f on S 1 . Requiring the superconformal coupling to obey the self-duality equation (2.20) fixes the functional form of Λ SC (u,ū) to be given by eq. (2.23). The remaining freedom in the choice of Λ SC (u,ū) is the single real parameter κ. On the other hand, when the formulation with auxiliary chiral variables is used, duality invariance is guaranteed by the condition (2.30). Upon imposing this condition, there exists a unique functional form for superconformal coupling, eq. (2.31), and the only remaining freedom is again the coupling constant κ.

Superconformal invariance
To conclude this section, we comment on rigid symmetries of the superconformal models (2.18), including the duality-invariant theory (2.25), in a background curved superspace. We remind the reader that, within the superconformal approach to supergravitymatter systems [65], every theory of Einstein supergravity interacting with supersymmetric matter is realised as a coupling of the same matter multiplets to conformal supergravity and a superconformal compensator, see, e.g., [66]. Truly superconformal theories are independent of any compensator.
The gauge group of conformal supergravity is spanned by general coordinate (ξ B ), local Lorentz (K βγ andKβγ) and super-Weyl (σ andσ) transformations. In the infinitesimal case, they act on the covariant derivatives by the rule where the general coordinate and local Lorentz transformation is given by and the super-Weyl variation δ σ D A is defined in (B.4). The local transformation (2.35) acts on a primary tensor superfield T (with its indices suppressed) as follows Here p and q are constant parameters which are known as the super-Weyl weights of T, with (p + q) being the dimension of T. The action of any supergravity-matter system is invariant under the transformations (2.35) and (2.37), where T is the collective notation for the matter multiplets and, perhaps, the compensator (the latter is present if the theory is not superconformal).
Let us consider a background curved superspace (M 4|4 , D). In accordance with section 6.4 of [57], a supervector field ξ = ξ B E B on (M 4|4 , D) is called conformal Killing if there exists a symmetric spinor K γδ and a covariantly chiral scalar σ such that In other words, the coordinate transformation generated by ξ can be accompanied by certain Lorentz and super-Weyl transformations such that the superspace geometry does not change. It turns out that the parameters ξ α , K αβ and σ are completely determined in terms of ξ a and its covariant derivatives, The real vector ξ a proves to obey the superconformal Killing equation We denote by δ K[ξ] and δ σ[ξ] the transformations associated with the conformal Killing su- The set of all conformal Killing supervector fields forms the superconformal algebra of (M 4|4 , D). If the superspace is conformally flat (for instance, Minkowski M 4|4 or anti-de Sitter AdS 4|4 superspace), its superconformal algebra is isomorphic to su(2, 2|1).
Given a superconformal field theory defined on the background superspace (M 4|4 , D), its action functional is invariant under superconformal transformations However, if the theory under consideration is not superconformal, then the set T also includes a conformal compensator Ξ which is non-dynamical and, instead, it is a part of the background supergravity multiplet (M 4|4 , D, Ξ). In this case the equations (2.39) and (2.40) must be accompanied by the additional condition which singles out the Killing supervector fields of (M 4|4 , D, Ξ).
The above discussion can be naturally modified to be applicable to the N = 2 superconformal theories studied in the next section. Similar symmetry considerations also hold for supergravity-matter theories in d ≤ 6 with up to eight supercharges, where off-shell conformal supergravity always exists, see [67] for the technical details.

N = 2 superconformal duality-invariant models
Let S[W,W ] be the action for an Abelian N = 2 vector multiplet coupled to supergravity. As a curved-superspace extension of the Grimm-Sohnius-Wess formulation [68], the vector multiplet is described by a covariantly chiral scalar superfield W , subject to the Bianchi identity where D ij := D α(i D j) α andD ij :=Dα (iDj)α , while S ij and its conjugateS ij are special components of the superspace torsion. Constraint (3.2) defines a reduced N = 2 chiral scalar superfield. The superspace formulation for N = 2 conformal supergravity developed in [69] is used in this section, see appendix C for the technical details.
The reduced chiral scalar W is a gauge-invariant field strength constructed in terms of Mezincescu's prepotential [70,71], V ij = V ji , which is an unconstrained real SU(2) triplet, The expression for W in terms of V ij was found in [72] to be Here∆ is the covariantly chiral projecting operator (C.5). The prepotential V ij is defined modulo gauge transformations [73] δ with the gauge parameter Λ α kij being arbitrary modulo the algebraic condition given. 11 It is an instructive exercise to show that δ Λ W = 0.
The super-Weyl transformation law of V ij [37] is Making use of the relations collected in appendix C, we then deduce the super-Weyl transformation law of the field strength W [69] δ σ W = σW . (3.6)

Formulation without auxiliary superfields
To describe the equations of motion, we introduce a covariantly chiral scalar superfield M defined as 12 i In terms of M and its conjugateM , the equation of motion for V ij is In the rigid superspace limit, this reduces to the N = 2 self-duality equation derived in [8].

Formulation with auxiliary chiral variables
The duality-invariant models discussed in the previous subsection possess an important reformulation [44]. We consider a locally supersymmetric theory with action in which the scalar superfield η is only constrained to be chiral,Dβη = 0. We require S[W,W , η,η] to be super-Weyl invariant, and therefore the corresponding transformation of η is 13 For S int [η,η] to be super-Weyl invariant, it may depend, in general, on the supergravity compensators. However, in this paper we are mostly interested in superconformal dynamical systems which possess no dependence on the compensators.
We assume that η and its conjugateη are auxiliary superfields in the sense that the equation of motion for η,

Superconformal duality-invariant models
We introduce the super-Weyl invariant functional where W αβ is the super-Weyl tensor, the chiral scalar Ξ is given by eq. (C.9), and a and c are real anomaly coefficients. In accordance with [52], Γ eff is a nonlocal functional of the background supergravity multiplet which generates the super-Weyl anomaly, with its super-Weyl variation being By construction, Γ eff is independent of η andη.
The functional (3.18) satisfies the condition (3.17) due to the following two properties. Firstly, for any covariantly chiral scalar φ, it holds that Secondly, the functionals are topological invariants. In the case of N = 4 SYM theory, the anomaly coefficients a and c coincide, a = c. In what follows, we will assume this relation and make use of c.

Another family of U(1)-invariant couplings is given by
where we have introduced the super-Weyl inert chiral superfield This is a curved superspace generalisation of the conformal primary weight-zero chiral superfield introduced in [75]. The superfield η 2 Ω naturally originates within the approach developed in [51].
With the contribution (−Γ eff ) included, the action defined by eqs. (3.12) and (3.18) is super-Weyl invariant but nonlocal. Instead of working with the nonlocal functional, we can consider the following U(1) duality-invariant action This action generates the N = 2 super-Weyl anomaly, and therefore it can be viewed as the Goldstone multiplet action for spontaneously broken N = 2 superconformal symmetry. 14 It was proposed in [52] to describe the N = 2 dilaton effective action in terms of the reduced chiral scalar W . An alternative realisation was put forward in Ref. [78] where the N = 2 Goldstone multiplet was identified with an unrestricted chiral scalar, such as η. Both superfields are present in (3.24).

Superconformal duality-invariant models in flat space
In the flat-superspace limit, the supergravity covariant derivatives D A , eq. (C.1), turn into the flat ones D A = (∂ a , D i α ,Dα i ), with i = 1, 2, which satisfy the graded commutation relations This action is invariant under standard N = 2 superconformal transformations, see [75] for the technical details. The equation of motion for η is Assuming the validity of (3.27), the action ( 14 The N = 1 dilaton effective action was proposed in [76], see also [77]. −c 2 d 4 x d 4 θ d 4θ 1 η 2 lnη∆ lnη + 1 η 2 ln η∆ ln η , (3.28) where η is assumed to be a functional of W and its conjugate, η = η[W,W ], obtained by solving the equation (3.27). The latter can be solved by iterations as a series in powers of c, Making use of this solution in (3.28) gives This agrees with the perturbative solution derived in [8].
It is interesting to point out that the equation (3.27) is analogous to the N = 1 chiral constraint (in the notation of [17]) which is at the heart of the Bagger-Galperin construction [62] of the vector Goldstone multiplet action for partial N = 2 → N = 1 supersymmetry breaking in Minkowski space.
The kinetic terms in (3.30) is purely classical. As is known, it receives no perturbative [79][80][81] and non-perturbative [82,83] quantum corrections. The c-term in (3.30), originally introduced in [79], contains four-derivative quantum corrections at the component level; these include an F 4 term, with F the U(1) field strength. This term is believed to be generated only at one loop in the N = 4 SYM theory [84]. In particular it is known that non-perturbative F 4 quantum corrections do not occur in this theory [82,83]. The one-loop calculation of c was carried out by several groups [7,[10][11][12][13][14], and the absence of two-loop corrections was shown in [81].
The c 2 -term in (3.30), which was originally introduced in [7] and later proved to be superconformal [75], contains six-derivative quantum corrections at the component level, including an F 6 term. Such a quantum correction is absent at one loop in the N = 4 SYM theory [75]. It is generated at the two-loop order and the numerical coefficient computed in [15] agrees with that predicted by duality invariance. 15 Our superconformal duality-invariant model (3.26) admits a natural generalisation described by the following action where Ω := η −2∆ lnη is the flat-superspace counterpart of (3.23). Here the functional in the second line is invariant under rigid U(1) transformations The action (3.32) includes higher-derivative structures as compared with (3.26).
Modulo higher derivative terms, we identify (3.26) with (the planar part of) the lowenergy effective action of N = 4 SYM on the Coulomb branch. There are several reasons for this identification. Firstly, upon elimination of the auxiliary variables η andη, the resulting action (3.30) agrees with known structure of the N = 4 SYM effective action to the sixth order in derivatives. Secondly, the theory (3.26) possesses U(1) duality invariance. Thirdly, the action (3.26) involves a single coupling constant, the anomaly coefficient c, which is similar to the D3-brane action in AdS 5 × S 5 , eq. (4.2). The latter is expected to be related to he low-energy effective action of N = 4 SYM (see the discussion in the next section), however only upon a nontrivial field redefinition at the component level [7,15]. It remain possible that one has to include a c-dependent higher-derivative term F(ΩΩ) in (3.32) in order to get a complete agreement between the model. This is still an open problem.

Discussion
Similar to its bosonic sector (A.7), the N = 1 superconformal duality-invariant model is uniquely defined by eq. (2.25) or, equivalently, eqs. (2.26) and (2.31). The only free parameter of the theory is the coupling constant κ in (2.31). In the N = 2 case, however, the superconformal duality-invariant model is given by the action (3.32), which involves an arbitrary function of a real variable, F(x). The reason why the functional freedom is larger in the N = 2 case can be explained by looking at the N = 1 components of the N = 2 vector multiplet field strength W . Given an N = 2 superfield U(x, θ i ,θ i ), its N = 1 sub-multiplets can be introduced with the aid of the N = 1 bar-projection defined by U| = U(θ i ,θ i )| θ 2 =θ 2 =0 . The N = 2 chiral field strength W contains two independent chiral N = 1 components √ 2Φ := W | , Modulo certain technical complications related to the need to perform a nonlinear superfield redefinition [7,15] (when switching from the N = 2 to N = 1 superfield descriptions), the N = 1 counterpart of the system (3.32) is given by the relations (2.26) and (2.30), with Υ ∝ ΦΦ, modulo terms involving derivatives of Φ andΦ. The model for N = 1 superconformal duality-invariant electrodynamics (2.25) does not appear to have a N = 2 counterpart.
The AdS/CFT correspondence provides the main evidence to believe in self-duality of the low-energy effective action for the N = 4 SU(N) SYM theory on its Coulomb branch where the gauge group SU(N) is spontaneously broken to SU(N − 1) × U(1). The AdS/CFT correspondence predicts [6,16,85,86] (a more comprehensive list of references is given in [16]) that the N = 4 SYM effective action (in the large-N limit) is related to the D3-brane action in AdS 5 × S 5 where X I , I = 1, · · · , 6, are transverse coordinates, T 3 = (2πg s ) −1 and Q = g s (N − 1)/π. The action S/T 3 is self-dual in the sense that it enjoys invariance under electromagnetic U(1) duality rotations [19][20][21][22]. Self-duality of the D3-brane action is a fundamental property related to the S-duality of type IIB string theory [87,88]. If the N = 4 SYM effective action and the D3-brane action in AdS 5 ×S 5 are indeed related, the former should possess some form of self-duality. The U(1) duality-invariant action (4.2) involves a single coupling constant, Q, the same as the N = 2 superconformal U(1) duality-invariant action (3.26). The D3-brane parameter Q is proportional to the anomaly coefficient c.
Considered as a field theory in Minkowski space M 4 , the D3-brane action (4.2) is manifestly scale-invariant, but it is not invariant under the standard linear special conformal transformations. It is instead invariant under deformed conformal transformations being nonlinear in the fields [6]. At the component level, the complete effective action of the N = 4 SYM also proves to be invariant under quantum-corrected superconformal transformations [89][90][91] which differ from the ordinary linear superconformal transformations that leave the classical action invariant.
In conclusion, we describe simple generalisations of the models for (super)conformal duality-invariant electrodynamics discussed in this paper. Given a U(1) duality-invariant model for nonlinear electrodynamics, L(F ab ) = L(ω,ω), described in appendix A, its compact duality group U(1) can be enhanced to the non-compact SL(2, R) group by coupling the electromagnetic field to the dilaton ϕ and axion a [20][21][22]. Specifically, this is achieved by replacing L(ω,ω), eq. (A.2), with 16 The duality group acts by the rule where G ab (F ) is defined bỹ see [17] for more details. In the case of the conformal model described by eq. (A.7), this leads to In curved space, the model defined by the action S = d 4 x e L conf (ω,ω, S,S) is Weylinvariant provided S is inert under the Weyl transformations. We can add to (4.6) a higher-derivative Lagrangian for the dilaton and axion (see, e.g., [92]) where  is conformal. 17 Within the N = 1 Poincaré supersymmetry, SL(2, R) duality-invariant couplings of the dilaton-axion multiplet to general models for self-dual supersymmetric nonlinear electrodynamics were described in [17], while the case of the supersymmetric Born-Infeld action [61,62] was first considered in [96]. The results of [17] were generalised to supergravity in [35]. These results can be used to couple the N = 1 superconformal U(1) duality-invariant model to the dilaton-axion multiplet described by a chiral scalar Φ and its conjugateΦ. One obtains where Λ SC (u,ū) is given by eq. (2.23). It is assumed that Φ parametrises the lower half-plane, and therefore (Φ −Φ) −1 exists. 18 One can add to (4.10) a higher-derivative super-Weyl and SL(2, R) invariant action for the dilaton-axion multiplet [97] S where κ and α are real parameters, and
Given a model for nonlinear electrodynamics, its Lagrangian L(F ab ) can be expressed in terms of the two independent invariants of the electromagnetic field, Introducing the complex combination ω = α + i β, the Lagrangian can be rewritten in the form Then, the GZGR condition for U(1) duality invariance is given by see [17] for the technical details.
Requiring the action functional to be conformally invariant constrains Λ(ω,ω) in (A.2) to have the following functional form Only a restricted class of such models prove to be compatible with U(1) duality invariance, specifically with x and y being real coefficients. These coefficients turn out to be constrained by requiring the self-duality equation (A.3) to hold, specifically with γ a real parameter. The resulting Lagrangian is which is exactly the model proposed in [53] (see also [54]). It was called "ModMax electrodynamics" in [53].
It is instructive to re-derive the conformal duality-invariant model using the IZ approach [24,25]. Their reformulation of nonlinear electrodynamics is obtained by replacing L(F ab ) → L(F ab , V ab ), where V ab = −V ba is an auxiliary unconstrained bivector, which is equivalent to a pair of symmetric rank-2 spinors, V αβ = V βα and its conjugateVαβ. The new Lagrangian L is at most quadratic with respect to the electromagnetic field strength F ab , while the self-interaction is described by a nonlinear function of the auxiliary variables, L int (V ab ), The original theory L(F ab ) is obtained from L(F ab , V ab ) by integrating out the auxiliary variables. In terms of L(F ab , V ab ), the condition of U(1) duality invariance was shown [24,25] to be equivalent to the requirement that the self-interaction General N = 1 supergravity-matter systems can be naturally described in superspace using the Grimm-Wess-Zumino geometry [98] in conjunction with the super-Weyl transformations discovered in [99]. The structure group in this setting is SL(2, C), and the geometry of curved superspace is described by the covariant derivatives where M βγ = M γβ andMβ˙γ =M˙γβ are the Lorentz generators. The covariant derivatives obey the graded commutation relations (see [57] for the derivation) Here the torsion tensors R, G a =Ḡ a and W αβγ = W (αβγ) satisfy the Bianchi identities: An infinitesimal super-Weyl transformation of the covariant derivatives [99] induces the following variations of the torsion superfields δ σ R = 2σR + 1 4 (D 2 − 4R)σ , (B.5a) δ σ G αα = 1 2 (σ +σ)G αα + iD αα (σ −σ) , (B.5b) Here the super-Weyl parameter σ is a covariantly chiral scalar superfield,Dασ = 0. The super-Weyl transformations belong to the gauge group of conformal supergravity.
Integrals over the full superspace and its chiral subspace are related as follows: where E is the chiral density (see, e.g., [57] for the derivation). The super-Weyl transformation laws of the integration measures are C Elements of N = 2 supergravity In this appendix we give a summary of the so-called SU(2) superspace, the off-shell formulation for N = 2 conformal supergravity developed in [69]. The structure group in this setting is SL(2, C) × SU (2), and the covariant derivatives are Here J kl are the generators of the group SU(2). The spinor covariant derivatives obey the anti-commutation relations Here the real four-vector G αα , the complex symmetric tensors S ij = S ji , W αβ = W βα , Y αβ = Y βα and their complex conjugatesS ij := S ij ,Wαβ := W αβ ,Ȳαβ := Y αβ obey additional differential constraints implied by the Bianchi identities and given in [69].
The covariantly chiral projecting operator [100,101] is The operator∆ relates an integral over the full superspace to that over its chiral subspace: with E the chiral density. 19 The super-Weyl transformation laws of the integration measures are