Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

Abstract

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">K</a:mi> </a:math> is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi mathvariant="script">K</d:mi> <d:mo stretchy="false">⊗</d:mo> <d:mover accent="true"> <d:mi mathvariant="script">K</d:mi> <d:mo stretchy="false">¯</d:mo> </d:mover> </d:math> and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi mathvariant="double-struck">C</k:mi> <k:mn>3</k:mn> </k:msup> </k:math> this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mi mathvariant="double-struck">C</n:mi> <n:mn>3</n:mn> </n:msup> </n:math> . The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

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