Manifestly duality-invariant interactions in diverse dimensions

As an extension of the Ivanov-Zupnik approach to self-dual nonlinear electrodynamics in four dimensions [1,2], we reformulate U(1) duality-invariant nonlinear models for a gauge $(2p-1)$-form in $d=4p$ dimensions as field theories with manifestly U(1) invariant self-interactions. This reformulation is suitable to generate arbitrary duality-invariant nonlinear systems including those with higher derivatives.

Nonlinear electrodynamics with U(1) duality symmetry is described by a Lorentz invariant Lagrangian L(F ab ) which is a solution to the self-duality equation In the case of theories with higher derivatives, this scheme is generalised in accordance with the two rules given in [8]. Firstly, the definition of G is replaced with Secondly, the self-duality equation (1.1) is replaced with Duality-invariant theories with higher derivative theories naturally occur in N = 2 supersymmetry [9]. Further aspects of duality-invariant theories with higher derivatives were studied in, e.g., [16][17][18][19].
Self-duality equation (1.1) is nonlinear, and therefore its general solutions are difficult to find. In the early 2000s, Ivanov and Zupnik [1,2] proposed a reformulation of duality-invariant electrodynamics involving an auxiliary antisymmetric tensor V ab , which is equivalent to a symmetric spinor V αβ = V βα and its conjugateVαβ. 1 The new Lagrangian L(F, V ) is at most quadratic in 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 [1,2] to be equivalent to the requirement that the self-interaction is invariant under linear U(1) transformations ν → e iϕ ν, with ϕ ∈ R, and thus where f is a real function of one real variable. The Ivanov-Zupnik (IZ) approach [1,2] has been used by Novotný [21] to establish the relation between helicity conservation for the tree-level scattering amplitudes and the electric-magnetic duality.
The above discussion shows that the IZ approach is a universal formalism to generate U(1) duality-invariant models for nonlinear electrodynamics. Some time ago, there was a revival of interest in duality-invariant dynamical systems [17,22,23] inspired by the desire to achieve a better understanding of the UV properties of extended supergravity theories. The authors of [22] have put forward the so-called "twisted self-duality constraint," which was further advocated in [17,23], as a systematic procedure to generate manifestly dualityinvariant theories. However, these approaches have been demonstrated [24] to be variants of the IZ scheme [1,2] developed a decade earlier.
The IZ approach has been generalised to off-shell N = 1 and N = 2 globally and locally supersymmetric theories [25,26]. In this note we provide a generalisation of the approach to higher dimensions, d = 4p. In even dimensions, d = 2n, the maximal duality group for a system of k gauge (n − 1)-forms depends on the dimension of spacetime. The duality group is U(k) if n is even, and O(k) × O(k) if n is odd [14] (see, e.g, section 8 of [9] for a review). This is why we choose d = 4p. The fact that the maximal duality group depends on the dimension of space-time was discussed in the mid-1980s [27,28] and also in the late 1990s [29,30].

New formulation
In Minkowski space of even dimension d = 4p ≡ 2n, with p a positive integer, we consider a self-interacting theory of a gauge (n − 1)-form A a 1 ...a n−1 with the property that the Lagrangian, L = L(F ), is a function of the field strengths F a 1 ...an = n∂ [a 1 A a 2 ...an] . 2 We assume that the theory possesses U(1) duality invariance. This means that the Lagrangian is a solution to the self-duality equation [14] G a 1 ...an G a 1 ...an + F a 1 ...an F a 1 ...an = 0 , As usual, the notation F is used for the Hodge dual of F , We now introduce a reformulation of the above theory. Along with the field strength F a 1 ...an , our new Lagrangian L(F, V ) is defined to depend on an auxiliary rank-n antisymmetric tensor V a 1 ...an which is unconstrained. We choose L(F, V ) to have the form allows one to integrate out the auxiliary field V to result with L(F ).
It may be shown that the self-duality equation (2.1) is equivalent to the following condition on the self-interaction in (2.4) Introducing (anti) self-dual components of V , (2.10) In four dimensions, the most general solution to this condition is given by eq. (1.7). Similar solutions exist in higher dimensions, a real function of one variable. However more general self-interactions become possible beyond four dimensions.
It is worth pointing out that an infinitesimal U(1) duality transformation leads to the following transformation of V δV = λ V .
There are several interesting generalisations of the construction described. They include (i) coupling to gravity; (ii) coupling to a dilaton with enhanced SL(2, R) duality; (iii) duality-invariant systems with higher derivatives; and (iv) U(k) duality-invariant systems of k gauge (2p − 1)-forms in d = 4p dimensions.
Recently, U(1) duality-invariant theories of a gauge (2p − 1)-form in d = 4p dimensions have been described [31] within the Pasti-Sorokin-Tonin approach [32,33]. It was argued in [31] that the approach of [32,33] is the most efficient method to determine all possible manifestly U(1) duality invariant self-interactions provided Lorentz invariance is kept manifest. Our analysis has provided an alternative formalism. 3