Gravitational core of double field theory

Abstract Double Field Theory (DFT) has emerged as a comprehensive framework for gravity, presenting a testable and robust alternative to General Relativity (GR), rooted in the $$\textbf{O}(D,D)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> symmetry principle of string theory. These lecture notes aim to provide an accessible introduction to DFT, structured in a manner similar to traditional GR courses. Key topics include doubled-yet-gauged coordinates, Riemannian versus non-Riemannian parametrisations of fundamental fields, covariant derivatives, curvatures, and the $$\textbf{O}(D,D)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -symmetric augmentation of the Einstein field equation, identified as the unified field equation for the closed string massless sector. By offering a novel perspective, DFT addresses unresolved questions in GR and enables the exploration of diverse physical phenomena, paving the way for significant future research.