Electric-magnetic Duality Rotations Research Articles

On the Electric-Magnetic Duality Symmetry: Quantum Anomaly, Optical Helicity, and Particle Creation

It is well known that not every symmetry of a classical field theory is also a symmetry of its quantum version. When this occurs, we speak of quantum anomalies. The existence of anomalies imply that some classical Noether charges are no longer conserved in the quantum theory. In this paper, we discuss a new example for quantum electromagnetic fields propagating in the presence of gravity. We argue that the symmetry under electric-magnetic duality rotations of the source-free Maxwell action is anomalous in curved spacetimes. The classical Noether charge associated with these transformations accounts for the net circular polarization or the optical helicity of the electromagnetic field. Therefore, our results describe the way the spacetime curvature changes the helicity of photons and opens the possibility of extracting information from strong gravitational fields through the observation of the polarization of photons. We also argue that the physical consequences of this anomaly can be understood in terms of the asymmetric quantum creation of photons by the gravitational field.

Classical and quantum aspects of electric-magnetic duality rotations in curved spacetimes

It is well known that the source-free Maxwell equations are invariant under electric-magnetic duality rotations, F --> F cos {\theta} + *F sin {\theta}. These transformations are indeed a symmetry of the theory in Noether sense. The associated constant of motion is the difference in the intensity between self- and anti-self dual components of the electromagnetic field or, equivalently, the difference between the right and left circularly polarized components. This conservation law holds even if the electromagnetic field interacts with an arbitrary classical gravitational background. After re-examining these results, we discuss whether this symmetry is maintained when the electromagnetic field is quantized. The answer is in the affirmative in the absence of gravity, but not necessarily otherwise. As a consequence, the net polarization of the quantum electromagnetic field fails to be conserved in curved spacetimes. This is a quantum effect, and it can be understood as the generalization of the fermion chiral anomaly to fields of spin one.

Gravity and handedness of photons

Vacuum fluctuations of quantum fields are altered in the presence of a strong gravitational background, with important physical consequences. We argue that a nontrivial spacetime geometry can act as an optically active medium for quantum electromagnetic radiation, in such a way that the state of polarization of radiation changes in time, even in the absence of electromagnetic sources. This is a quantum effect, and is a consequence of an anomaly related to the classical invariance under electric-magnetic duality rotations in Maxwell theory.

Electromagnetic Duality Anomaly in Curved Spacetimes.

The source-free Maxwell action is invariant under electric-magnetic duality rotations in arbitrary spacetimes. This leads to a conserved classical Noether charge. We show that this conservation law is broken at the quantum level in the presence of a background classical gravitational field with a nontrivial Chern-Pontryagin invariant, in parallel with the chiral anomaly for massless Dirac fermions. Among the physical consequences, the net polarization of the quantum electromagnetic field is not conserved.

Dualities near the horizon

Abstract In 4-dimensional supergravity theories, covariant under symplectic electricmagnetic duality rotations, a significant role is played by the symplectic matrix $ \mathcal{M} $ (φ), related to the coupling of scalars φ to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the $ \mathcal{N} $ = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector $ \mathcal{Q} $ and the invariants of the corresponding symplectic representation $ {R_{\mathcal{Q}}} $ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, $ \mathcal{M} $ still depends on flat directions, but not $ \mathcal{M}\mathcal{Q} $ , defining the so-called Freudenthal dual of $ \mathcal{Q} $ itself. This allows for a universal expression of the symplectic vector field strengths in terms of $ \mathcal{Q} $ , in the near-horizon Bertotti-Robinson black hole geometry.

Testing vacuum electrodynamics using “slow light” experiments

A recent proposal to explore vacuum electrodynamics using the speed of propagation of an electromagnetic wave through an ambient constant magnetic field is examined. It is argued that the proposal should be modified so that the background magnetic field, the direction of propagation and the transverse projection of the electric field (with respect to the direction of propagation) are not coplanar. The implications of invariance under Gibbons' electric-magnetic duality rotations are determined in this context.

The general pattern of Kač–Moody extensions in supergravity and the issue of cosmic billiards

In this paper we study the systematics of the affine extension of supergravity duality algebras when one steps down from D = 4 to D = 2 which is instrumental for the study of cosmic billiards. For all D = 4 supergravities (with N ⩾ 3 ) there is a universal field theoretical mechanism promoting the extension, which relies on the coexistence of two non-locally related Lagrangian descriptions of the corresponding D = 2 degrees of freedom: the Ehlers Lagrangian and the Matzner–Misner one. This and the existence of a generalized Kramer–Neugebauer non-local transformation relating the two models, provide a Chevalley–Serre presentation of the affine Kač–Moody algebra which follows a universal pattern for all supergravities. This is an extension of the mechanism considered by Nicolai for pure N = 1 supergravity, but has general distinctive features in extended theories ( N ⩾ 3 ) related to the presence of vector fields and to their symplectic description. Moreover the novelty is that in the general case the Matzner–Misner Lagrangian is structurally different from the Ehlers one, since half of the scalars are replaced by gauge 0-forms subject to SO ( 2 n , 2 n ) electric–magnetic duality rotations representing in D = 2 the Sp ( 2 n , R ) rotations of D = 4 . The role played by the symplectic bundle of vectors in this context suggests that the mechanism of the affine extension can be studied also for N = 2 supergravity, where one deals with geometries rather than with algebras, the scalar manifold being not necessarily a homogeneous manifold U / H . We also show that the mechanism of the affine extension commutes with the Tits–Satake projection of the relevant duality algebras.

ON DUALITY ROTATIONS IN LIGHT-LIKE NONCOMMUTATIVE ELECTROMAGNETISM

We study electric-magnetic duality rotations for noncommutative electromagnetism (NCEM). We express NCEM as a nonlinear commutative U(1) gauge theory and show that it is self-dual when the noncommutativity parameter θ is light-like (e.g. θ0i=θ1i). This implies, in the slowly varying field approximation, self-duality of NCEM to all orders in θ.

Electric-magnetic duality rotations and invariance of actions

For D = 4 theories of a single U(1) gauge field strength coupled to gravity and matters, we show that the electric-magnetic duality can be formulated as an invariance of the actions. The symmetry is associated with duality rotation acting directly on the gauge field. The rotation is constructed in flat space, and an extension to curved spaces is also given. It is non-local and non-covariant, yet generates off-shell extended transformation of the field strength. The algebraic condition of Gaillard and Zumino turns out to be a necessary and sufficient condition for the invariance of actions. It may be used as a guiding principle in constructing self-dual actions in string and field theories.

R Symmetry and the Topological Twist of N = 2 Effective Supergravities of Heterotic Strings

We discuss R symmetries in locally supersymmetric N = 2 gauge theories coupled to hypermultiplets which can be thought of as effective theories of heterotic superstring models. In this type of supergravities a suitable R symmetry exists and can be used to topologically twist the theory: the vector multiplet containing the dilaton–axion field has different R charge assignments with respect to the other vector multiplets. Correspondingly a system of coupled instanton equations emerges, mixing gravitational and Yang–Mills instantons with triholomorphic hyperinstantons and axion instantons. For the tree level classical special manifolds ST(n) = SU(1,1)/U(1) × SO(2,n)/[SO(2) × SO(n)], R symmetry with the specified properties is a continuous symmetry, but for the quantum-corrected manifolds [Formula: see text] a discrete R group of electric–magnetic duality rotations is sufficient and we argue that it exists.

Lectures on Special Kähler geometry and electric—magnetic duality rotations

In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it relates to the typical geometric structures involved by N=2 supersymmetry, namely Special K\"ahler geometry for the vector multiplets and either HyperK\"ahler or Quaternionic geometry for the hypermultiplets. I discuss classical continuous dualities versus non--perturbative discrete dualities. How the moduli space geometry of an auxiliary dynamical Riemann surface (or Calabi--Yau threefold) relates to exact space--time dualities is exemplified in detail for the Seiberg Witten model of an $SU(2)$ gauge theory.

SL(2, [formula omitted]) invariance of non-linear electrodynamics coupled to an axion and a dilaton

The most general Lagrangian for non-linear electrodynamics coupled to an axion a and a dilaton φ with SL(2, R ) invariant equations of motion is − 1 2 (∇φ) 2 − 1 2 e 2φ (∇a) 2 + 1 4 aF μν ★ F μν + L inv (g μν, e − 1 2 φ F ϱσ) where L inv( g μν , F ϱσ ) is a Lagrangian whose equations of motion are invariant under electric-magnetic duality rotations. In particular there is a unique generalization of Born-Infeld theory admitting SL(2, R ) invariant equations of motion.

Electric-magnetic duality rotations in non-linear electrodynamics

We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Langrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagnetic field terms. We discuss some implications for black holes in string theory and an extension to 2 k-forms in 4 k spacetime dimensions.

Charge quantization of axion-dilaton black holes.

We present axion-dilaton black-hole and multi-black-hole solutions of the low-energy string effective action. Under $\mathrm{SL}(2, R)$ electric-magnetic duality rotations only the "hair" (charges and asymptotic values of the fields) of our solutions is transformed. The functional form of the solutions is duality invariant. Axion-dilaton black holes with zero entropy and zero area of the horizon form a family of stable particlelike objects, which we call holons. We study the quantization of the charges of these objects and its compatibility with duality symmetry. In general the spectrum of black-hole solutions with quantized charges is not invariant under $\mathrm{SL}(2, R)$ but only under $\mathrm{SL}(2, Z)$ or one of its subgroups ${\ensuremath{\Gamma}}_{l}$. Because of their transformation properties, the asymptotic value of the axion-dilaton field of a black hole may be associated with the modular parameter $\ensuremath{\tau}$ of some complex torus and the integer numbers ($n, m$) that label its quantized electric and magnetic charges may be associated with winding numbers.

Electric-magnetic duality and supersymmetry in stringy black holes.

We present a generalization of the $U(1)^{2}$ charged dilaton black holes family whose main feature is that both $U(1)$ fields have electric and magnetic charges, the axion field still being trivial. We show the supersymmetry of these solutions in the extreme case, in which the corresponding generalization of the Bogomolnyi bound is saturated and a naked singularity is on the verge of being visible to external observers. Then we study the action of a subset of the $SL(2,R)$ group of electric-magnetic duality rotations that generates a non-trivial axion field on those solutions. This group of transformations is an exact symmetry of the $N=4$ $d=4$ ungauged supergravity equations of motion. It has been argued recently that it could be an exact symmetry of the full effective string theory. The generalization of the Bogomolnyi bound is invariant under the full $SL(2,R)$ and the solutions explicitly rotated are shown to be supersymmetric if the originals are. We conjecture that any $SL(2,R)$ transformation will preserve supersymmetry.