Auxiliary tensor fields for Sp(2,R) self-duality

The coset Sp(2,R)/U(1) is parametrized by two real scalar fields. We generalize the formalism of auxiliary tensor (bispinor) fields in U(1) self-dual nonlinear models of abelian gauge fields to the case of Sp(2,R) self-duality. In this new formulation, Sp(2,R) duality of the nonlinear scalar-gauge equations of motion is equivalent to an Sp(2,R) invariance of the auxiliary interaction. We derive this result in two different ways, aiming at its further application to supersymmetric theories. We also consider an extension to interactions with higher derivatives.


Introduction
Noncompact Sp(2, R) duality arises in nonlinear electrodynamics interacting with dilaton and axion scalar fields which support a nonlinear realization of Sp(2, R) in the coset Sp(2, R)/U(1) [1]- [5]. The Sp(2, R) self-dual Lagrangian contains a specific interaction of the electromagnetic field F mn and the coset scalar fields S = (S 1 , S 2 ) [3], 2 Axion-dilaton coupling and nonlinear realization of Sp(2, R) The infinitesimal transformation of the group Sp(2, R) ≃ SL(2, R) (i.e. an element of the algebra sp(2, R)) can be parametrized by the 2 × 2 matrix where a, b and c are real numbers. The nonlinear realization of Sp(2, R) is arranged as the transformations of the scalar field S = S 1 + iS 2 [3] δS = b + 2aS − cS 2 , (2.2) where the real scalar fields S 1 and S 2 are connected with the axion A and dilaton, φ The invariant Kähler σ model Lagrangian contains the Kähler metric g SS = − 1 (S−S) 2 , For the electromagnetic field strengths we will use both the bispinor and tensor representations
(2.20), (2.21) mean that the bispinors F αβ and P αβ form a linear Sp(2, R) doublet, so it is natural to use the notation R a := (P, F ) and rewrite (2.20), (2.21) as δR a = B a b R b , where the matrix B a b was defined in (2.1).
For further use we will also introduce the modified NR field strength and dual field strengtĥ in terms of which the self-duality condition (2.14) takes the conventional form [1]- [4]: The modified quantities (2.22) transform nonlinearly under Sp(2, R). The relation between the NR representation (2.22) and the linearly transforming fields P αβ and F αβ can be written in the matrix form asR These relations contain the real coset matrix g and its inverse g −1 (S): Transformations of the coset matrix have the following form: where ρ = cS 2 is the induced parameter of the nonlinear realization, and τ 2 is the Pauli matrix. Thus the NR fields transform covariantly under the nonlinear realization of Sp(2, R) δP αβ = −ρF αβ , δF αβ = ρP αβ . (2.28) The same transformations can be directly derived from the definition (2.22) with taking into account the compatibility constraint (2.14). For what follows, it is useful to explicitly write howP αβ is expressed through P αβ and F αβP (the connection betweenF αβ and F αβ was already given in (2.22)). It is easy to construct, out of the coset fields S 1 , S 2 , the 2 × 2 matrix M ab supporting a linear realization (LR) of Sp(2, R): One can alternatively write the coset Lagrangian (2.5) through this matrix The matrix of Kähler complex structure J a c also admits a simple expression through M J a c (S) = M cb (S)ε ba , J a c J c r = −δ a r . (2.32) The standard U(1) duality relations are reproduced from the Sp(2, R) ones given above in the non-singular limit (2.33) In this limit, g a b → δ a b , M ab → δ ab and Sp(2, R) is reduced to its O(2) ∼ U(1) subgroup with a = 0, b = −c, which preserves (2.33) and is just the standard U(1) duality group.
3 Nonlinear auxiliary fields in Sp(2, R) duality 3.1 Sp(2, R) duality as invariance of the auxiliary interaction We introduce the following complex combinations of the NR bispinor fieldsF andP defined in (2.22): We will treat these fields as independent auxiliary tensor variables of the NR representation of the tensor formulation of the Sp(2, R) self-duality and postulate forF αβ ,P αβ just the NR transformation properties (2.28). The standard expression ofP through the original fields F αβ , S 1 , S 2 as given in (2.22) will arise after eliminatingV αβ ,Vαβ from the appropriate extended action by their equations of motion. In this extended formulation, we will also use the basic transformation of the NR electromagnetic field which is just the result of substituting P αβ = i(F αβ − 2V αβ ) from the definition (3.1) into (2.28). The scalar combinations of the auxiliary fields and their transformation laws are given bŷ By analogy with the auxiliary tensor field formulation of the U(1) duality [7]- [10], in constructing the extended action we start by defining the bilinear in F andV part of the interaction with scalar fields By construction, the F derivative of this Lagrangian, also transforms linearly (3.8) Varying L 2 with respect toV αβ we obtain the Sp(2, R)-covariant auxiliary equation Substituting this solution back into L 2 , we reproduce the bilinear interaction of the electromagnetic field with scalars L sd 2 (S, F ) (eq. (2.15)). The bilinear Lagrangian (3.4) admits the Gaillard-Zumino-type representation where I 2 is the complex Sp(2, R) invariant, and M bc (S) is the LR matrix (2.30). This representation allows us to easily find the Sp(2, R) variation of the bilinear Lagrangian, Now we are prepared to write the total nonlinear Lagrangian in the (S, F,V ) representation. It is a sum of L 2 and the Sp(2, R) invariant terms whereâ is the invariant quartic auxiliary variable defined in (3.3). Since the equations of motion for the electromagnetic field are not modified as compared to the L 2 case, they still exhibit, together with the Bianchi identity, the Sp(2, R) covariance. The equation for the auxiliary field is also manifestly Sp(2, R) covariant. It is analogous to the twisted self-duality constraints considered in [15,16]. Note that the equation of motion for the coset fields (2.7) is modified by a non-zero source depending on the fieldsV , F . It is easy to show that the Sp(2, R) covariance of (2.7) is not affected by this modification, like in the original GR framework.
It is instructive to rewrite (3.14) in the more detailed form The sum of the last four terms in (3.16) precisely coincides with the extended U(1) self-dual Lagrangian of nonlinear electrodynamics of refs. [7] - [10], up to the rescaling F αβ →F αβ = √ S 2 F αβ . Hence, by reasoning of these papers, it should yield the most general self-dual LagrangianL(φ,φ) upon eliminating the auxiliary fieldsV αβ ,Vαβ by their equations of motion. We conclude that (3.14), (3.16) indeed yields the auxiliary bispinor field extension of the general Sp(2, R) self-dual GR Lagrangian (2.12). Let us point out that the whole information about the given Sp(2, R) self-dual system is encoded in the Sp(2, R) invariant function E(â) which is not subject to any constraints. Given some bispinor field representation of the standard U(1) self-dual action, we can promote it to that defining an Sp(2, R) self-dual system just according to the recipe (3.16).
The auxiliary equation (3.15) is solved bŷ By analogy with the U(1) case [10] we can use the perturbative expansion for E(â) where e 1 , e 2 , . . . are some constant coefficients. The corresponding perturbative solution forL readŝ Like in the U(1) duality case, it is useful to define the intermediate (on-shell) representation for the self-dual Lagrangian by expressing (formally) the fieldF in terms ofV from the algebraic equation (3.15): This "on-shell" representation preserves the Sp(2, R) covariance The substitution (3.20) gives The same transform applied to the total Lagrangian (3.14) preserves the GZ form of the latter Substituting here the solutionV (F ) (3.17), one recovers the F representation of the Lagrangian, i.e. (2.12). Being applied to eq. (3.6), the change (3.20) yields This establishes the relation between the LR and NR auxiliary fields (on the shell of the auxiliary equation (3.15)), which involves the scalar coset fields and the invariant interaction E. The corresponding "on-shell" relation between the scalar combinations of the auxiliary fields reads . (3.25)

Legendre transformation for the nonlinear auxiliary fields
The Legendre transformation for the auxiliary field formulation of the U(1) self-dual electrodynamics was discussed in [9,10]. This transformation simplifies solving the auxiliary self-duality equation, which is the central step in deriving the conventional self-dual Lagrangian from the extended one.
To generalize this to the theory with the Sp(2, R) scalars, we introduce some new NR covariant scalar auxiliary fieldsμ andμ with the transformation law and define the following generalized Lagrangian simultaneously involving two types of the auxiliary fields: In this case we obtain the simple expressions for the dual fields These explicit expressions provide the correct transformation laws for the relevant quantities. In fact, the representation (3.27) just defines the Legendre transformation of the Lagrangian (3.16). Indeed, varying (3.27) with respect to the auxiliary fieldμ yieldŝ Using this equation,μ andb can be expressed in terms ofν,ν, assuming that ( dI db ) −1 is not singular atb = 0. After the elimination of µ,μ, the Lagrangian (3.27) takes the form of (3.16) with Then it is easy to show thatμ VaryingL(S,φ,μ) (3.29) with respect toμ , we arrive at the equation for the auxiliary scalar variablesφ It is analogous to the corresponding equation in the U(1) self-dual theory. Solving it forμ =μ(φ,φ), we obtain the equation for determining the self-dual LagrangianL(φ,φ) By analogy with [9,10], as a notorious example, we can obtain the exact Lagrangian for the Born-Infeld (BI) theory extended by the Sp(2, R) scalar fields, proceeding from the invariant interaction In this case, one can find the complete solution to the equation (3.35) Taking L BI (S 2 ϕ, S 2φ ) asL(φ,φ) in eq. (2.12), we recover the standard GR coupling of the scalar fields in the BI theory [3]. Note that the derivation of the BI Lagrangian in theb representation is much easier than in the originalâ formulation.
All other examples of the U(1) self-dual systems in which eq. (3.35) has a closed solution can also be generalized to the Sp(2, R) case. For instance, the invariant auxiliary interaction yields the cubic equation forμ(φ) and gives us the exact expression forL(φ,φ), like in the analogous U(1) model considered in [10]. The scalar coupling in the so called "simplest interaction model" [7,15,10] corresponds to the choice I SI = −2b .

Sp(2, R) duality and higher derivatives
The self-dual theories with higher derivatives in the standard setting were analyzed in [15,17]. The formulation through auxiliary tensor fields for such theories was worked out in [10]. As was shown there, any U(1) self-dual theory with higher derivatives is generated by the appropriate U(1) invariant auxiliary interaction involving space-time derivatives of the auxiliary bispinor fields .
The transformation parameter ρ = cS 2 in the NR representation (3.1) depends on the scalar field, so in the case under consideration we need to properly Sp(2, R) covariantize the space-time derivatives of the auxiliary fields involved. This can be done following the standard routine of the nonlinear realizations [18].
First, we construct the 2 × 2 matrix Cartan 1-forms pertinent to the nonlinear realization of Sp(2, R) in the symmetric coset space Sp(2, R)/U(1) we deal with: Here, g is the coset matrix (2.25) and g −1 , g T are the corresponding inverse and transposed matrices. The 1-form Γ = −iτ 2 dx m ζ m contains the induced connection ζ m (S) which defines the covariant derivatives of fields having the standard transformation properties, i.e. transforming with the induced U(1) parameter ρ, while D specifies the NR-covariant derivative D m (S) of the coset fields 1 . These objects have the following transformation rules The connection 1-form reads The explicit expressions for the component 1-forms collected in the matrix (3.43) are as follows Now we are ready to define the covariant derivatives of the NR auxiliary fieldsV andV (we suppress their Lorentz indices) The corresponding Sp(2, R) invariant auxiliary interaction can be constructed by analogy with the U(1) self-dual theory [10]. We should add to the standard bilinear interaction L 2 (S, F,V ), eq. (3.4), the general Sp(2, R) invariant interaction involving the covariant derivatives of the scalar coset fields and the NR auxiliary fields: The Sp(2, R)-covariant local equations of motion for the auxiliary fields in this case contain the Euler-Lagrange derivative Solving this equation (e.g., by recursions), we finally obtain the Sp(2, R) self-dual Lagrangian in the initial (F, S) representation. The auxiliary interaction E K der specifying the Sp(2, R) self-dual models with higher derivatives involves new dimensionful constants, starting from the coupling constant c of dimension −2, as well as additional dimensionless coupling constants. Examples of interaction with two derivatives are provided by the terms ∼ ∇β βV αβ ∇ξ αVβξ , ∼ ∇ mν ∇ mν , . . . . Non-standard terms with higher derivatives can be generated by the invariant combinations of the scalar and auxiliary fields, e.g., Note that terms with higher derivatives now also appear in the scalar S 1 , S 2 equations.

Alternative auxiliary-field formulation of Sp(2, R) theory
In the previous section we started from the renowned GR action (1.1), (2.12) and picked up the nonlinearly transforming bispinor auxiliary fieldsV αβ so as to construct the natural generalization of the extended formulation of the U(1) self-dual electrodynamics to the case of the Sp(2, R) self-dual systems with the coset scalar fields. In this section we present an alternative construction which starts just from the extended U(1) formulation and produces the GR action as an output. Its basic distinguishing feature is that it starts with the linear realization of Sp(2, R) on the set (V αβ , F αβ ).

λ parametrization of the Sp(2, R) coset
In the alternative construction it will be convenient to use another parametrization of the coset of Sp(2, R), this time by the complex scalar field λ(x): and use the alternative set of the group parameters The new coset field has a simple transformation law which resembles the CP 1 realization of the group SU(2), the only difference being the sign of the last term in (4.5). The scalar Lagrangian (2.5) can be rewritten in terms of λ as The relevant coset element can be represented by the Hermitian matrix with the transformation law whereγ(λ) is the corresponding induced parameter, τ 3 is the Pauli matrix and The previously considered Sp(2, R) spinors R a = (P, F ) linearly transforming with the matrix B are related to the spinors T a transforming with the new matrix Λ as where

11)
The Cartan form in this representation, dGG −1 , contains the corresponding connection and covariant derivatives by analogy with (3.41): The off-diagonal elements in (4.12) are just the covariant differentials Dλ = dx m D m λ and Dλ = dx m D mλ , δDλ = 2iγDλ (the Lagrangian (4.6) is bilinear in the covariant derivatives D m λ , D mλ ), while the diagonal element is the U(1) connection It defines the covariant derivative of some field with the standard transformation law under the nonlinear realization considered where q is the U(1) charge of ψ . The real symmetric LR matrix defined in (2.30) is related to G 2 It is easy to check that the equations of motion and Bianchi identities for F αβ , are covariant under the Sp(2, R) rotations 20) or δF αβ = iγ(F αβ − 2V αβ ) + αV αβ +ᾱ(F αβ − V αβ ) .
is evidently not covariant. The question is how to modify eq. (4.22) in order to make it also Sp(2, R) covariant.
In the free case, with E(ν,ν) = 0, this modification is rather simple: Using the transformation law of λ (4.5), it is easy to show that which implies the covariance of (4.23).
As the next step, the following generalization of (4.23) naturally occurs: The function E(ν,ν, λ,λ) is assumed to become the previous E(ν,ν) = E(a) in the limit λ = 0 yielding the Lagrangian (4.17) with the residual U(1) duality group with the parameter γ . Eq.
. (4.26) For E = 0, using the algebraic equation (4.23), we obtain the following λ-modified free (F,F ) Lagrangian: . In what follows, it will be convenient to deal with the modified interaction function E = E + λν +λν , (4.28) that corresponds to transferring the λ-dependent terms in the l.h.s. of (4.25) to its r.h.s. We will fix the λ-dependence of the interaction E (orÊ) from the requirement of compatibility of the Sp(2, R) variations of the left-and right-hand sides of eq. (4.25). Using the transformation laws (4.20), we find that, on the shell of the auxiliary equation (4.25), Then, taking into account this transformation law together with (4.20), (4.5) and, once again, (4.25), we find the variation of the r.h.s. of (4.25) and compare it with that of the l.h.s., i.e., with (4.24). We find the following conditions 2 on the functionÊ Eq. (4.30) is just the condition of the U(1) invariance of the generalized functionÊ(ν,ν, λ,λ). The mutually conjugated eqs. (4.31) and (4.32) are new. One can check that the same system of equations arises from the requirement that the transformations (4.29) have the correct sp(2, R) closure. Repeatedly using the constraints (4.31) and (4.32), one finds that and An interesting peculiarity is that the transformation laws ofÊ ν andÊν exactly mimic those of λ and λ. We also observe thatÊ is not invariant under the coset Sp(2, R)/U(1) transformations, while it is still invariant under their U(1) closure. Surprisingly, we can construct such Sp(2, R) invariant from the two independent U(1) invariants A similar object already appeared in [9], when performing the Legendre transformation from the variables ν,ν to µ,μ (recall also Subsection 3.2). At this step, we deal with the bispinor field extended Lagrangian (4.26), which is reduced to the extended Lagrangian (4.17) of the U(1) duality systems in the limit λ = 0 and exhibits Sp(2, R) duality under the constraints (4.30) -(4.32) on the interaction functionÊ(ν,ν, λ,λ) . The question is how to solve these constraints via some unconstrained "prepotential" which would be analogous to E(â) of Subsection 3.1. While for the time being we do not know how to achieve this, the problem is radically simplified in the µ representation obtained by Legendre transformation of (4.26), with the Sp(2, R) invariant interaction H, eq. (4.36), instead ofÊ . In this representation, the constraints are linearized .

From the µ representation to the (F,F ) Lagrangian
In the µ representation, the basic algebraic equation (4.25) implies (4.52) It will be convenient to deal with the variableμ defined in (4.43). Keeping in mind that for the Sp(2, R) invariant case we can rewrite (4.52) as whereμ andμ can be checked to transform just as in (3.26), and introducinĝ which, in the S 1 , S 2 parametrization, coincide with the quantities defined in (2.13), we rewrite (4.53) asφ = −(μ + 2b +bμ) I ′ ,φ = −(μ + 2b +bμ) I ′ , and c.c. , Eqs. (4.56) are recognized as the basic equation (3.35) (and its conjugate) of theb representation of the NR formulation presented in the first part of the paper. So, already at this step we conclude that, on the shell of the auxiliary equation (4.25), the b representations of both formulations of the Sp(2, R) self-duality are the same, which implies that both these formulations yield the same eventual answer for the general Sp(2, R) self-dual LagrangianL sd (λ, F ) . It seems instructive to consider here the consequences of eqs. (4.57) in more detail. These equations have the same form as the equations in the b representation for the U(1) case without scalars (eqs. (2.37) in [10]), with the only change ϕ →φ ,φ →φ andμ,μ instead of µ,μ . Hence, as a corollary they imply the same algebraic equation (eq. (2.38) in [10]) which expressesb in terms of the variablesφ,φ defined in (4.55). Hence, the solution forμ,μ is obtained through the substitution ϕ →φ ,φ →φ in the solution for µ,μ of the case without scalar fields. It remains to find the general expression for the Lagrangian in the original (ϕ,φ) representation. The formulas one starts with mimic the U(1) casẽ where the λ dependence of the r.h.s. is hidden in the λ dependence ofL sd ϕ ,L sd ϕ andb. The holomorphic derivativesL sd ϕ ,L sd ϕ are related to the variables µ,μ also by the same relations as in the U(1) casẽ , and c.c. , (4.60) which, with taking into account eqs. (4.52), yields The deviations from the pure U(1) case are revealed, when making use of the basic equations (4.57), (4.58) to eliminate µ andμ in (4.61). After passing to the variablesμ,μ, we obtaiñ Using eqs. (4.57), we obtain the final expression for the Lagrangiañ The LagrangianL accompanied by the algebraic equation (4.58) precisely yields the standard U(1) self-dual Lagrangian with the replacements ϕ →φ,φ →φ, whereφ,φ are defined through ϕ,φ by eqs. (4.55) or (2.13). It satisfies the standard GZ self-duality constraint with respect toφ,φ: The first term in (4.63) is the appropriate modification of the bilinear part of the action. This final answer for the nonlinear self-dual action is in the precise correspondence with the general action of Gibbons and Rasheed [2]. In the S 1 , S 2 parametrization, using the relations (4.1), the Lagrangians (4.63) and (4.64) can be rewritten in the familiar form as Obviously, the Lagrangian (4.63) (or (4.66)) should be accompanied by the coset field Lagrangian L ′ (λ) = L(S) given in (4.6). It is straightforward to check that the λ equations of motion following from the total Lagrangian L sd = L ′ (λ) +L sd enjoy Sp(2, R) covariance.

Yet another derivation of the Sp(2, R) self-dual Lagrangian
Though the extended Lagrangian (4.26) contains a constrained auxiliary interaction E(ν,ν, λ,λ) and we cannot immediately solve the relevant constraints (4.30) -(4.32), we know that it yields the correct description of the general Sp(2, R) self-dual systems, as was shown above by passing to its µ representation. It turns out that the original E formulation is still capable to yield the general Lagrangian L sd (λ, ϕ,φ) of the Sp(2, R) systems without explicitly solving (4.30) -(4.32). These constraints amount to the triplet of alternative Sp(2, R) self-duality constraints onL sd as they given, e.g., in [4].
Starting from the Lagrangian (4.26) and using the auxiliary equation (4.25) together with its corollaries as well as the transformation properties (4.29), (4.34), (4.35) and we find the following simple Sp(2, R) transformation of the on-shell (i.e., with the algebraic equation (4.25) taken into account) Lagrangian: Defining as usual the general dual field strength and employing the auxiliary equation (4.25) together with its corollaries (4.68), it is easy to show that On the other hand, the Lagrangian (4.26) can be rewritten on the shell of the auxiliary equation as Then, using (4.72), we can cast it in the standard form It is straightforward to check that the variation (4.70) is entirely generated by the first term in (4.75), whence it follows that δH = 0 in agreement with (4.36). Substituting the expression (4.73) into the variation (4.70), we can rewrite it in a more standard way in terms of ϕ,φ, P 2 andP 2 . This variation can be also used to find the Sp(2, R) analog of the standard GZ self-duality constraint in the (F,F ) representation of the Lagrangian. To this end we can use the relations of the U(1) self-duality [9,10]  Substituting the explicit expressions for the variations of δλ, δϕ and their conjugates into δL sd (4.77), and comparing the latter with (4.70), we obtain three conditions on the LagrangianL sd which are just the Sp(2, R) extension of the standard GZ constraint: (4.82) Using these relations and going over to the tensorial notation: we can bring (4.79) -(4.81) to the simple equivalent form given in [4] 2(SL sd S +SL sd The unique solution of these constraints is the general GR Lagrangian (1.1), (2.12) (with L(S) subtracted) 3 . So the linear realization version of the bispinor auxiliary field formulation of the Sp(2, R) self-dual systems yields as the output the same GR Lagrangian as the formulation based on the nonlinearly transforming auxiliary fields, even without passing to the µ representation.

More on the interplay between the linear and nonlinear formulations
Here we give more details on the relationship between the auxiliary fields in the LR and NR formulations.
In Subsections 4.2 and 4.3 we observed that the Legendre transformation from the variables ν,ν to µ,μ performed in (4.26) yields in fact the same µ representation as the Legendre transformation from the variablesν,ν toμ,μ in the Lagrangian (3.16). It is interesting to reproduce the NR formulation, starting from the µ representation obtained in the framework of the LR formulation and applying just another type of the inverse Legendre transformation to this µ representation.
In this way the basic formulas of the NR formulation are recovered. As for the Lagrangian (3.16), it can be restored from the requirement that it reproduces the basic relations of the NR formulation, including eq. (4.92). We introduce the bispinor fieldV αβ ,Vαβ, such thatν =V αβV αβ ,ν =VαβVαβ , Varying it with respect toV αβ gives which implies just (4.92). We also need the correct dynamical equations for F αβ on the shell of the algebraic constraint. In other words, we require withV αβ being subjected to (4.95).
As the first step, we rewrite F αβ − 2V αβ in terms of the variablesν Next we expressV αβ in (4.96) through F αβ by (4.95) and compare two expressions for F αβ − 2V αβ , which yields Finally, the Lagrangian (4.94) takes the form of (3.16) with the subtracted L(S). Note that the auxiliary equation (4.95) together with (4.97) yield the same on-shell relations (3.24) ad (3.25) between the LR and NR tensorial fields.
The off-shell Lagrangians (4.26) and (3.16) are essentially different and seem not to be related to each other by any obvious field redefinition, though they yield the same system on the shell of the auxiliary equation.

Conclusions
We investigated Sp(2, R) duality-invariant interactions of scalar and electromagnetic fields, employing two different formulations involving auxiliary bispinors and/or auxiliary scalars ("µ representation"). The main emphasis was on the transformation properties of the relevant Lagrangians and their equations of motion. The formalism of Section 3 started from the nonlinear realization of Sp(2, R) on the basic auxiliary bispinor fields, while in Section 4 those auxiliary fields were taken to transform linearly. Both formalisms admit a Legendre-type transformation to Lagrangians with auxiliary scalar fields. This allowed us to prove that both auxiliary-field formulations yield equivalent self-dual Lagrangians in the standard (S, F ) representation. Like in the U(1) duality case, any choice of a Lagrangian exhibiting Sp(2, R) duality amounts to a particular choice of an Sp(2, R) invariant unconstrained interaction of the auxiliary bispinor fields. The (S, F ) Lagrangian emerges from the extended Lagrangian upon elimination of the auxiliary fields through their equations of motion in terms of the coset scalars S and the electromagnetic field strengths F .
It is rather straightforward to generalize our auxiliary-field formulations to the case of Sp(2N, R) duality as proper extensions of the analogous formulations for U(N) duality [11]. We hope to address this task elsewhere. The formalism of auxiliary superfields in N = 1 supersymmetric self-dual theories [12,13] can also be generalized, with additional chiral coset multiplets, to the case of noncompact dualities. We are curious to learn which of the two (if not both) approaches presented here admit an unambiguous extension to supersymmetric theories. One might also try to construct an Sp(2, R) (and Sp (2N, R)) version of the "hybrid" formulation of U(1) duality [19], which joins the auxiliary-tensor approach with the manifestly Lorentz-and duality-invariant PST formalism (see [20] and references therein).