Gravitational Core of Double Field Theory

Doubled-yet-Gauged Spacetime

Lecture Notes
Jeong-Hyuck Park

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.

arXiv

Doubled-yet-Gauged Coordinates & Diffeomorphisms

Aim. To introduce doubled coordinates, the section condition, and relevant diffeomorphisms.

The \(\ODD\)-invariant metric \(\cJ_{AB}\) presented in Table 1 plays a crucial role in splitting the doubled coordinates into two parts, \(x^{A} = (\tx_{\mu}, x^{\nu})\), where \(\mu, \nu = 0, 1, 2, \cdots, D{-1}\) denote \(D\)-dimensional spacetime indices. A fundamental requirement in DFT is that all functions in the theory, collectively denoted as \(\{\Phi, \hat{\Phi}, \tilde{\Phi}, \cdots\}\) - including physical fields and local gauge parameters - must satisfy the section condition [Berman:2010is]. This condition restricts their dependence on the doubled coordinates via the constraint:

\[ \partial_{A}\partial^{A}=\partial_{\mu}\tpartial^{\mu}+\tpartial^{\mu}\partial_{\mu}=0\,, \](2.1)

which ensures that any pairwise contraction of (doubled) partial derivatives vanishes:

\[ \begin{array}{lll} \partial_{A}\partial^{A}\Phi=0\,,\qquad&\quad \partial_{A}\Phi\partial^{A}\hat{\Phi}=0\,,\qquad&\quad \partial_{A}\partial^{A}(\Phi\hat{\Phi})=0\,. \end{array} \](2.2)

In practice, the section condition is often solved by setting \(\tpartial^{\mu} \equiv 0\), thereby eliminating dependence on the dual coordinates \(\tx^{\mu}\). The general solution, or “section”, is obtained by applying a global \(\ODD\) rotation to this specific choice. The \(\ODD\) symmetry is spontaneously broken by selecting the \(D\)-dimensional section, such as \(\tpartial^{\mu} \equiv 0\). However, in cases where a configuration admits an isometric direction, e.g. \(\partial_{1}\equiv0\), the \(\mathbf{O}(1,1)\) rotation of the \((\tx_{1},x^{1})\) plane generates new configurations, a procedure known as Buscher's rule [Buscher:1987qj, Buscher:1987sk].

The section condition is equivalent to a certain translational invariance generated by derivative-index-valued shift parameters \(\Delta^{A}\) [Park:2013mpa, Lee:2013hma]:

\[ \begin{array}{llll} \Phi(x)=\Phi(x+\Delta)\,&\quad\mbox{where}\quad& \Delta^{A}=\hat{\Phi}\partial^{A}\tilde{\Phi}\,,\quad&\quad\Delta^{A}\partial_{A}=0\,. \end{array} \](2.3)

The geometric meaning of the section condition, as suggested in [Park:2013mpa], is that the doubled coordinates are gauged by the derivative-index-valued vectors:

\[ \begin{array}{ll} x^{A}~\sim~x^{A}+\Delta^{A}\,,\qquad&\qquad\Delta^{A}\partial_{A}=0\,. \end{array} \](2.4)

In this framework, each gauge orbit corresponds to a single spacetime point. The concept of "derivative-index-valued" vectors is made possible by the \(\ODD\)-invariant metric \(\cJ_{AB}\), which raises the vector index of the partial derivative: \(\partial^{A}=\cJ^{AB}\partial_{B}\).

In DFT, diffeomorphisms are governed by a generalised Lie derivative, denoted as \(\hcL_{\xi}\) [Siegel:1993th, Hull:2009zb]:

\[ \hcL_{\xi}T_{A_{1}\cdots A_{n}}:=\xi^{B}\partial_{B}T_{A_{1}\cdots A_{n}}+\omega\partial_{B}\xi^{B}T_{A_{1}\cdots A_{n}}+\sum_{i=1}^{n}(\partial_{A_{i}}\xi_{B}-\partial_{B}\xi_{A_{i}})T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{n}}\,, \](2.5)

where \(\omega\) is the weight of the tensor or tensor density, and each tensor index \(A_{i}\) is rotated by an infinitesimal \(\mathbf{so}(D,D)\) element, \(\partial_{A_{i}}\xi_{B}-\partial_{B}\xi_{A_{i}}\). For consistency, generalised Lie derivatives are closed under commutation, provided the section condition holds:

\[ \left[\hcL_{\zeta},\hcL_{\xi}\right]=\hcL_{\left[\zeta,\xi\right]_{\rm{C}}}\,, \](2.6)

where the commutator is given by the C-bracket:

\[ \left[\zeta,\xi\right]^{A}_{\rm{C}}=\half\!\left(\hcL_{\zeta}\xi^{A}-\hcL_{\xi}\zeta^{A}\right)= \zeta^{B}\partial_{B}\xi^{A}-\xi^{B}\partial_{B}\zeta^{A}+\half \xi^{B}\partial^{A}\zeta_{B}-\half \zeta^{B}\partial^{A}\xi_{B}\,. \](2.7)

It is noteworthy that the \(\ODD\)-metric is invariant under generalised diffeomorphisms, \({\hcL_{\xi}\cJ_{AB}=0}\), and the generalised Lie derivative itself is covariant [Angus:2018mep]:

\[ \delta_{\xi}(\hcL_{\zeta}T_{A_{1}\cdots A_{n}})= \hcL_{\zeta}(\delta_{\xi}T_{A_{1}\cdots A_{n}})+\hcL_{\delta_{\xi}\zeta}T_{A_{1}\cdots A_{n}}=\hcL_{\xi}(\hcL_{\zeta}T_{A_{1}\cdots A_{n}})\,. \](2.8)

Explicit Gauging of the Doubled Coordinates

The section condition encapsulates the concept of doubled-yet-gauged coordinates [Park:2013mpa] in DFT, where coordinates serve only as labels for the dynamical fields. This contrasts with the particle worldline or string worldsheet actions, where the target spacetime coordinates themselves become dynamical fields and must be explicitly gauged [Lee:2013hma, Park:2016sbw, Ko:2016dxa, Basile:2019pic].

The usual coordinate basis of differential one-forms, \(\rd x^{A}\), is not \(\ODD\)-symmetric under infinitesimal passive coordinate transformations, (Footnote: Active transformations are set by the generalised Lie derivative (2.5).) \(x^{A} \rightarrow x^{A} + \xi^{A}\), as shown below:

\[ \delta(\rd x^{A})=\rd\xi^{A}=\rd x^{B}\partial_{B}\xi^{A}\neq\rd x^{B}(\partial_{B}\xi^{A}-\partial^{A}\xi_{B})\,. \](2.9)

To restore \(\ODD\) symmetry, a derivative-index-valued gauge potential is introduced to explicitly gauge the doubled coordinates. This defines a new gauged differential:

\[ \begin{array}{ll} \rD x^{A}:=\rd x^{A}-\fa^{A}\,,\qquad&\quad \fa^{A}\partial_{A}=0\,. \end{array} \](2.10)

The passive transformations for the coordinates \(x^{A}\) and the gauge potential \(\fa^{A}\) are given by:

\[ \begin{array}{lll} \delta x^{A}=\xi^{A}\,,\quad&\quad\delta \fa^{A}=\rD x^{B}\partial^{A}\xi_{B}\,,\quad&\quad\delta\fa^{A}\partial_{A}=0\,. \end{array} \](2.11)

This leads to an \(\ODD\)-symmetric transformation for the gauged differential:

\[ \delta (\rD x^{A})=\rD x^{B}(\partial_{B}\xi^{A}-\partial^{A}\xi_{B})\,. \](2.12)

To measure the distance between gauge orbits, a proper length can be defined through a path integral [Morand:2017fnv]:

\[ L=-\ln\left[\int\mathfrak{D}\fa~\exp\left(-\int\sqrt{\rD x^{A}\rD x^{B}\cH_{AB}}~\right)\right]\,, \](2.13)

where \(\rD x^{A}\) is the gauged coordinate differential defined in (2.10), \(\fa\) is the auxiliary gauge potential to be integrated out, and \(\cH_{AB}\) is the DFT-metric, or the generalised metric, which will be detailed later.

This definition of proper length naturally leads to a doubled worldline action for a particle [Ko:2016dxa]:

\[ \dis{ S=\frac{1}{\,\ls} \int\rd\tau~\frac{1}{2} e^{-1\,}\rD_{\tau}x^{A}\rD_{\tau}x^{B}\cH_{AB}(x)-\frac{1}{2} (m\,\ls)^{2}e\,,} \](2.14)

which further generalises to a completely covariant, doubled worldsheet action for a string [Hull:2006va, Lee:2013hma],

\[ S=\frac{1}{4\pi\alphap}\displaystyle{\int}\rmd^{2}\sigma~-\half\sqrt{-h}h^{\alpha\beta}\rD_{\alpha}x^{M}\rD_{\beta}x^{N}\cH_{MN} -\epsilon^{\alpha\beta}\rD_{\alpha}x^{M}\fa_{\beta M}\,. \](2.15)

This framework can be extended to a \(\kappa\)-symmetric Green-Schwarz superstring, as discussed in [Park:2016sbw].