Manifestly duality-invariant interactions in diverse dimensions

TL;DR

This paper extends the Ivanov-Zupnik approach to formulate U(1) duality-invariant nonlinear gauge theories for higher-dimensional forms, enabling the construction of models with manifest duality symmetry and higher derivatives.

Contribution

It introduces a reformulation of duality-invariant models for gauge (2p-1)-forms in 4p dimensions with manifest U(1) invariance, accommodating higher derivatives.

Findings

Reformulation of duality-invariant models with manifest symmetry

Framework for generating arbitrary duality-invariant systems

Extension to models with higher derivatives

Abstract

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.

Equations

$${\tilde{G}}^{ab}{G}_{ab}+{\tilde{F}}^{ab}{F}_{ab}=0,$$

$${\tilde{G}}^{ab}(F):=\frac{1}{2}{\epsilon }^{abcd}{G}_{cd}(F)=2\frac{\partial L(F)}{\partial F_{ab}}.$$

$${\tilde{G}}^{ab}[F]=2\frac{\delta S[F]}{\delta F_{ab}}.$$

$$\displaystyle\int{\rm d}^{4}x\Big{(}\widetilde{G}^{ab}G_{ab}+\widetilde{F}^{ab}F_{ab}\Big{)}=0~{}.$$

$$L(F_{ab},V_{ab})=\frac{1}{4}{F}^{ab}{F}_{ab}+\frac{1}{2}{V}^{ab}{V}_{ab}-V^{ab}{F}_{ab}+L_{\mathrm{int}}(V_{ab}).$$

$$L_{\mathrm{int}}(V_{ab})=L_{\mathrm{int}}(\nu ,\bar{\nu }),\nu :=V^{\alpha \beta }{V}_{\alpha \beta }$$

$$L_{\mathrm{int}}(\nu ,\bar{\nu })=f(\nu \bar{\nu }),$$

$${\tilde{G}}^{a_1\dots a_n}{G}_{a_1\dots a_n}+{\tilde{F}}^{a_1\dots a_n}{F}_{a_1\dots a_n}=0,$$

$${\tilde{G}}^{a_1\dots a_n}(F)=n!\frac{\partial L(F)}{\partial F_{a_1\dots a_n}}.$$

$${\tilde{F}}^{a_1\dots a_n}=\frac{1}{n!}{\epsilon }^{a_1\dots a_n{b}_1\dots b_n}{F}_{b_1\dots b_n}.$$

$$\displaystyle L(F,V)=\frac{1}{n!}\Big{\{}\frac{1}{2}F\cdot F+V\cdot V-2V\cdot F\Big{\}}+L_{\rm int}(V)~{},$$

$$V\cdot F:=V^{a_1\dots a_n}{F}_{a_1\dots a_n}.$$

$$\frac{\partial }{\partial V^{a_1\dots a_n}}L(F,V)=0,$$

$${\tilde{V}}^{a_1\dots a_n}\frac{\partial }{\partial V^{a_1\dots a_n}}{L}_{\mathrm{int}}(V)=0.$$

$$\displaystyle V_{\pm}^{a_{1}\dots a_{n}}=\frac{1}{2}\Big{(}V^{a_{1}\dots a_{n}}\pm{\rm i}\widetilde{V}^{a_{1}\dots a_{n}}\Big{)}~{},\qquad\widetilde{V}_{\pm}=\mp{\rm i}V_{\pm}~{},\qquad V=V_{+}+V_{-}~{},$$

$$\displaystyle\Big{(}V_{+}^{a_{1}\dots a_{n}}\frac{\partial}{\partial V_{+}^{a_{1}\dots a_{n}}}-V_{-}^{a_{1}\dots a_{n}}\frac{\partial}{\partial V_{-}^{a_{1}\dots a_{n}}}\Big{)}L_{\rm int}(V_{+},V_{-})=0~{}.$$

$$L_{\mathrm{int}}({\mathrm{e}}^{\mathrm{i}\phi }{V}_{+},{\mathrm{e}}^{-\mathrm{i}\phi }{V}_{-})=L_{\mathrm{int}}(V_{+},V_{-}),\phi \in \mathbb{R}.$$

$$\displaystyle\delta\left(\begin{array}[]{c}G\\ F\end{array}\right)=\left(\begin{array}[]{cr}0&-\lambda\\ \lambda&0\end{array}\right)\left(\begin{array}[]{c}G\\ F\end{array}\right)$$

$$\delta V=\lambda \tilde{V}.$$

Full text

August, 2019

Manifestly duality-invariant interactions in diverse dimensions

Sergei M. Kuzenko

*Department of Physics M013, The University of Western Australia

35 Stirling Highway, Perth W.A. 6009, Australia  

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.

1 Introduction

As an extension of the seminal work by Gaillard and Zumino [3], the general formalism of duality-invariant models for nonlinear electrodynamics in four dimensions was developed in the mid-1990s [4, 5, 6, 7]. The Gaillard-Zumino-Gibbons-Rasheed (GZGR) approach was generalised to off-shell ${\cal N}=1$ and ${\cal N}=2$ globally [8, 9] and locally [10, 11] supersymmetric theories. In particular, the first consistent perturbative scheme to construct the ${\cal N}=2$ supersymmetric Born-Infeld action was given in [9] (this approach was further pursued in [12]). The GZGR formalism was also extended to higher dimensions [13, 14, 15].

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

[TABLE]

where

[TABLE]

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 $\widetilde{G}$ is replaced with

[TABLE]

Secondly, the self-duality equation (1.1) is replaced with

[TABLE]

Duality-invariant theories with higher derivative theories naturally occur in ${\cal N}=2$ 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 $V_{\alpha\beta}=V_{\beta\alpha}$ and its conjugate $\bar{V}_{{\dot{\alpha}}{\dot{\beta}}}$.111This approach was inspired by the structure on the ${\cal N}=3$ supersymmetric Born-Infeld action proposed in [20]. 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, $L_{\rm int}(V_{ab})$,

[TABLE]

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

[TABLE]

is invariant under linear U(1) transformations $\nu\to{\rm e}^{{\rm i}\varphi}\nu$, with $\varphi\in\mathbb{R}$, and thus

[TABLE]

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 [22, 23, 17] 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 [23, 17], as a systematic procedure to generate manifestly duality-invariant 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 ${\cal N}=1$ and ${\cal N}=2$ 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 ${\rm O}(k)\times{\rm O}(k)$ 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].

2 New formulation

In Minkowski space of even dimension $d=4p\equiv 2n$, with $p$ a positive integer, we consider a self-interacting theory of a gauge $(n-1)$-form $A_{a_{1}\dots a_{n-1}}$ with the property that the Lagrangian, $L=L(F)$, is a function of the field strengths $F_{a_{1}\dots a_{n}}=n\partial_{[a_{1}}A_{a_{2}\dots a_{n}]}$.222We follow the notation and conventions of [9]. We assume that the theory possesses U(1) duality invariance. This means that the Lagrangian is a solution to the self-duality equation [14]

[TABLE]

where we have introduced

[TABLE]

As usual, the notation $\widetilde{F}$ is used for the Hodge dual of $F$,

[TABLE]

We now introduce a reformulation of the above theory. Along with the field strength $F_{a_{1}\dots a_{n}}$, our new Lagrangian $L(F,V)$ is defined to depend on an auxiliary rank-$n$ antisymmetric tensor $V_{a_{1}\dots a_{n}}$ which is unconstrained. We choose $L(F,V)$ to have the form

[TABLE]

where we have denoted

[TABLE]

The last term in (2.4), $L_{\rm int}(V)$, is at least quartic in $V_{a_{1}\dots a_{n}}$. It is assumed that the equation of motion for $V$,

[TABLE]

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)

[TABLE]

Introducing (anti) self-dual components of $V$,

[TABLE]

the above condition turns into

[TABLE]

This means that $L_{\rm int}(V_{+},V_{-})$ is invariant under U(1) phase transformations,

[TABLE]

In four dimensions, the most general solution to this condition is given by eq. (1.7). Similar solutions exist in higher dimensions, $L_{\rm int}(V_{+},V_{-})=f(V_{+}\cdot V_{+}V_{-}\cdot V_{-})$, with $f(x)$ 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

[TABLE]

Equation (2.7) tells us that $L_{\rm int}(V)$ duality invariant.

There are several interesting generalisations of the construction described. They include (i) coupling to gravity; (ii) coupling to a dilaton with enhanced SL$(2,{\mathbb{R}})$ 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.333It is worth pointing out that in four dimensions a hybrid formulation has been developed [34] which combines the powerful features of the IZ approach with thoes advocated in [32, 33].

**Acknowledgements:

**I am grateful to Stefan Theisen for useful comments. This work is supported in part by the Australian Research Council, project No. DP160103633.