## About this Research Topic

The electric-magnetic duality principle is the best-known symmetry of theoretical physics that originated in the Maxwell equations. This duality principle had a remarkable contribution to developments of Yang-Mill theory. Today, it continues to permeate into string theory, supergravity and M-theory, implying that it should also have a key role in gravitational theories. In the case of gravity, the duality principle has introduced dual formulations of gravity in arbitrary dimensions. The request for dual gravity is also apparent from solutions of 11-dimensional supergravity, the exceptional group E11, and M-theory.

The concept of dual gravitation goes back to the magnetic part of the Weyl curvature tensor (gravitomagnetism), and analogies between the dynamic equations for the Weyl curvature (Bianchi identities) and the Maxwell equations for electromagnetism. The dual formulations of gravity facilitate the interpretation of the metric of an object with magnetic-type gravitational mass, while the ordinary matter is interpreted as electric-type gravitational mass. The Kaluza-Klein monopole in M-theory or magnetic-type gravitational mass is expected to support the dual gravitational field.

There are several ways to formulate dual formulations of gravitation. The most conventional way of describing the curvature of the dual graviton is a Young symmetry type tensor dualizing on one index that holds Lorentz invariance. In this way, the dual graviton in the linearized theory is expressed by rank-2 symmetric tensors similar to the Pauli-Fierz theory in D=4 spacetime dimensions, while it turns out to be the Curtright field in D=5. The electric-magnetic duality of gravity also exhibits hidden symmetries of supergravity and M-theory in D=11, where a (8, 1) Young symmetry type tensor describes the dual graviton. It is well-known that the Pauli-Fierz field equations are algebraically consistent in D=4. However, the Curtright field equations of the linearized dual gravity do not have any consistent interactions under local Poincaré invariance in D=5. Nonlinear representations of dual gravity with non-vanishing cosmological constant, which contain both the original and dual dynamical metrics, yield the interacting theory of gravity in higher than 4 dimensions in the de Sitter or anti-de Sitter space.

The electric-magnetic duality also appears among higher-spin gauge fields that allows to generalize the equations of motion for higher spin fields in arbitrary dimensions, and provides prepotentials for higher spin gauge fields and their dual fields. The linearized equations of motion of higher spin theories are described by the Fronsdal field in Minkowski flat spacetime. Nevertheless, the Fronsdal action cannot have any consistent interaction in flat space. Fully nonlinear Vasiliev theory permits the interacting theory of higher-spin field perturbations around the anti de Sitter (AdS) spacetime. It is understood that supergravity on D-dimensional AdS space is dual to conformal field theory (CFT) in D-1 dimensions, which forms the AdS/CFT correspondence or Maldacena duality. Holographic dualities imply that the Vasiliev nonlinear higher-spin fields are dual to other vector and scalar fields. The AdS/CFT correspondence and holographic dualities correspond to hidden symmetries of M-theory, as well as underlay electric-magnetic symmetry in higher-spin fields.

The aim of this Research Topic is to review the contemporary developments in linear and nonlinear representations of dual gravity in arbitrary dimensions, their correlations with magnetic-type gravitational sources, and their implications for maximal supergravity, the Lorentzian Kac-Moody algebra E11, and M-theory, which will ultimately reveal hidden symmetries of gravity, and the origin of the AdS/CFT correspondence in M-theory.

**Keywords**:
Linearized Gravity, Field Theories in Higher Dimensions, Gauge Fields, Dual Graviton, Nonlinear Gravitational Theories, Pauli-Fierz Field, Curtright Field, Higher-spin Fields, Fronsdal Fields, Vasiliev's Nonlinear Theory, Electric-Magnetic Duality

**Important Note**:
All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

The electric-magnetic duality principle is the best-known symmetry of theoretical physics that originated in the Maxwell equations. This duality principle had a remarkable contribution to developments of Yang-Mill theory. Today, it continues to permeate into string theory, supergravity and M-theory, implying that it should also have a key role in gravitational theories. In the case of gravity, the duality principle has introduced dual formulations of gravity in arbitrary dimensions. The request for dual gravity is also apparent from solutions of 11-dimensional supergravity, the exceptional group E11, and M-theory.

The concept of dual gravitation goes back to the magnetic part of the Weyl curvature tensor (gravitomagnetism), and analogies between the dynamic equations for the Weyl curvature (Bianchi identities) and the Maxwell equations for electromagnetism. The dual formulations of gravity facilitate the interpretation of the metric of an object with magnetic-type gravitational mass, while the ordinary matter is interpreted as electric-type gravitational mass. The Kaluza-Klein monopole in M-theory or magnetic-type gravitational mass is expected to support the dual gravitational field.

There are several ways to formulate dual formulations of gravitation. The most conventional way of describing the curvature of the dual graviton is a Young symmetry type tensor dualizing on one index that holds Lorentz invariance. In this way, the dual graviton in the linearized theory is expressed by rank-2 symmetric tensors similar to the Pauli-Fierz theory in D=4 spacetime dimensions, while it turns out to be the Curtright field in D=5. The electric-magnetic duality of gravity also exhibits hidden symmetries of supergravity and M-theory in D=11, where a (8, 1) Young symmetry type tensor describes the dual graviton. It is well-known that the Pauli-Fierz field equations are algebraically consistent in D=4. However, the Curtright field equations of the linearized dual gravity do not have any consistent interactions under local Poincaré invariance in D=5. Nonlinear representations of dual gravity with non-vanishing cosmological constant, which contain both the original and dual dynamical metrics, yield the interacting theory of gravity in higher than 4 dimensions in the de Sitter or anti-de Sitter space.

The electric-magnetic duality also appears among higher-spin gauge fields that allows to generalize the equations of motion for higher spin fields in arbitrary dimensions, and provides prepotentials for higher spin gauge fields and their dual fields. The linearized equations of motion of higher spin theories are described by the Fronsdal field in Minkowski flat spacetime. Nevertheless, the Fronsdal action cannot have any consistent interaction in flat space. Fully nonlinear Vasiliev theory permits the interacting theory of higher-spin field perturbations around the anti de Sitter (AdS) spacetime. It is understood that supergravity on D-dimensional AdS space is dual to conformal field theory (CFT) in D-1 dimensions, which forms the AdS/CFT correspondence or Maldacena duality. Holographic dualities imply that the Vasiliev nonlinear higher-spin fields are dual to other vector and scalar fields. The AdS/CFT correspondence and holographic dualities correspond to hidden symmetries of M-theory, as well as underlay electric-magnetic symmetry in higher-spin fields.

The aim of this Research Topic is to review the contemporary developments in linear and nonlinear representations of dual gravity in arbitrary dimensions, their correlations with magnetic-type gravitational sources, and their implications for maximal supergravity, the Lorentzian Kac-Moody algebra E11, and M-theory, which will ultimately reveal hidden symmetries of gravity, and the origin of the AdS/CFT correspondence in M-theory.

**Keywords**:
Linearized Gravity, Field Theories in Higher Dimensions, Gauge Fields, Dual Graviton, Nonlinear Gravitational Theories, Pauli-Fierz Field, Curtright Field, Higher-spin Fields, Fronsdal Fields, Vasiliev's Nonlinear Theory, Electric-Magnetic Duality

**Important Note**:
All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

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