Jeong-Hyuck Park Lecture Notes
Gravitational Core of Double Field Theory
Lecture Notes
Jeong-Hyuck Park
Double Field Theory as the gravitational framework naturally implied by closed string theory.
These lecture notes explore the possibility that Double Field Theory is the gravitational framework naturally implied by closed string theory. They are structured in analogy with a General Relativity course, covering doubled-yet-gauged spacetime, generalized metrics, non-Riemannian geometry, covariant derivatives, curvatures, gravitational waves, and the Einstein Double Field Equation.
Chapters
Preface: A Physicist's Apology
This preface recounts the origin of Double Field Theory, the motivation for writing another review, and the background of these lecture notes.
Introduction
This chapter motivates Double Field Theory by showing how O(D,D) symmetry reorganizes the closed-string massless sector, in analogy with the role of Lorentz symmetry in electrodynamics.
Double Field Theory Minimum
This chapter sets up the minimal language of DFT: doubled indices, O(D,D) symmetry, twofold spin groups, and the conventions needed for the geometric construction.
Doubled-yet-Gauged Spacetime
This chapter introduces the kinematical heart of DFT: coordinates are doubled, while a gauge principle and section condition prevent the doubled coordinates from becoming unphysical degrees of freedom.
Non-Riemannian Geometry
This chapter explains why the fundamental field variables of DFT need not always be parametrized by an invertible spacetime metric and two-form.
Covariant Derivatives
This chapter develops the connection data of DFT and explains how semi-covariant objects combine into fully covariant derivatives.
Curvatures and Gravitational Waves
This chapter moves from connections to curvature, the doubled Einstein-Hilbert action, and the propagation structures that lead to gravitational wave equations.
Einstein Double Field Equation
This chapter formulates the Einstein Double Field Equation as a unified field equation for DFT matter couplings and the closed-string gravitational sector.
Solutions
This chapter collects representative solutions and tests: spherical geometries, post-Newtonian limits, wormholes, and cosmological solutions of the EDFE.
Outlook
This concluding chapter summarizes the role of DFT as a geometric framework for closed string gravity and points toward open directions.
Editorial Note
The HTML presentation of these lecture notes was prepared with the assistance of AI tools under the direction of the author. The scientific ideas, research vision, interpretations, and conclusions are solely those of the author.