Preface: A Physicist's Apology

Preface: A Physicist's Apology

(Footnote: c.f. A Mathematician's Apology by G. H. Hardy, Cambridge University Press, Cambridge: 1940.)

Double Field Theory (DFT) was initially conceived by Siegel in 1993 [Siegel:1993xq, Siegel:1993th] and independently developed, along with its name, by Hull and Zwiebach in 2009 [Hull:2009mi, Hull:2009zb], with further refinements introduced in 2010 [Hohm:2010jy, Hohm:2010pp]. The term “double” reflects the use of doubled coordinates, \(x^{A} = (\tx_{\mu}, x^{\nu})\), a concept originally introduced by Duff in 1990 [Duff:1989tf], to describe \(D\)-dimensional physics, in contrast to the conventional use of \(x^{\mu}\) alone.

Three influential review papers on DFT [Aldazabal:2013sca, Berman:2013eva, Hohm:2013bwa], all written in 2013, provided timely insights into the field during its early development within the framework of string theory. Over the past decade, significant progress has been continuously made, including advancements, in particular, in cosmological, black hole-related, and phenomenological aspects of DFT e.g., [Brandenberger:2018xwl, Brandenberger:2018bdc, Kang:2019owv, Lescano:2021nju, Liu:2021xfs, Liu:2021gjm, Lescano:2022nrb, Brandenberger:2023ver, Lescano:2023gge, Li:2023mhz, Arapoglu:2024umz, Ying:2024jjr, Angus:2024owg], with more developments to be discussed later.

Furthermore, a unified field equation for DFT was proposed in [Angus:2018mep], co-authored by the present author, which generalises the Einstein field equation of General Relativity by incorporating \(\ODD\) symmetry. Through subsequent developments, DFT has, at least in the author's view, matured into a self-sustaining framework for gravity, characterised by its predictive and falsifiable nature.

These lecture notes are intended to provide an accessible introduction to the foundations of DFT, structured in a manner analogous to introductory General Relativity courses. The aim is to engage not only the hep-th community but also the gr-qc community in exploring and testing this remarkable and precise modified theory of gravity, as predicted by string theory.

These notes were developed in response to repeated requests from colleagues and participants at recent workshops held in 2025 [YITP2025, Bangkok2025]. They are based on the author's lectures from the HANGANG Gravity Workshop in 2024 and a graduate-level course at Sogang University in 2025.

Yet, this modest contribution traces its origins to the author's first encounter with DFT during Zwiebach's lecture at a summer school in München in 2010 [Munchen2010]. That lecture sparked the author's journey into the subject, which began in November 2010 with an exploration of the DFT analogue of the Christoffel symbols in collaboration with Imtak Jeon and Kanghoon Lee [Jeon:2010rw].

Although most \(\ODD\)-manifest computations in DFT can be performed with pen and paper, deriving explicit expressions for \(\ODD\)-symmetric covariant derivatives and curvatures in terms of their \(\ODD\)-breaking component fields, such as \(\{g_{\mu\nu}, B_{\mu\nu}, \phi\}\), often requires computational tools like Cadabra [Peeters:2007wn].