Gravitational Core of Double Field Theory

Curvatures and Gravitational Waves

Lecture Notes
Jeong-Hyuck Park

This chapter moves from connections to curvature, the doubled Einstein-Hilbert action, and the propagation structures that lead to gravitational wave equations.

arXiv

Curvatures

Now we turn to curvatures. The semi-covariant four-index Riemann curvature is defined as [Jeon:2011cn]:

\[ S_{ABCD}:=\half\left(\fR_{ABCD}+\fR_{CDAB}-\Gamma^{E}{}_{AB}\Gamma_{ECD}\right)\,. \](2.89)

Here \(\Gamma_{ABC}\) is the DFT Christoffel connection (2.41) and \(\fR_{ABCD}\) represents its `field strength': (Footnote: In accordance with the decomposition of an \(\ODD\) index into \(D\)-dimensional upper and (dual) lower indices, the field strength \(\fR_{ABCD}\) includes \(\fR^{\kappa}{}_{\lambda\mu\nu}\) which should not be confused with the ordinary (undoubled) Riemann curvature \(R^{\kappa}{}_{\lambda\mu\nu}\). Notably, \(S\) follows \(R\) alphabetically, signifying its intended connection within the semi-covariant formalism. )

\[ \fR_{CDAB}=\partial_{A}\Gamma_{BCD}-\partial_{B}\Gamma_{ACD}+\Gamma_{AC}{}^{E}\Gamma_{BED}-\Gamma_{BC}{}^{E}\Gamma_{AED}\,, \](2.90)

which arises in the commutator of the semi-covariant derivatives,

\[ \left[\na_{A\,},\na_{B}\right]T_{C_1C_2\cdots C_n}=-\Gamma^{D}{}_{AB}\na_{D}T_{C_1C_2\cdots C_{n}}+\sum_{i=1}^{n}~\fR_{C_{i}}{}^{D}{}_{AB}T_{C_1\cdots C_{i-1}DC_{i+1}\cdots C_{n}}\,, \](2.91)

satisfies symmetric and projective properties,

\[ \fR_{ABCD}=\fR_{[AB][CD]}=(P_{A}{}^{E}P_{B}{}^{F}+\brP_{A}{}^{E}\brP_{B}{}^{F})\fR_{EFCD}\,, \](2.92)

and, under arbitrary variation of the Christoffel symbols (2.56), transforms infinitesimally as

\[ \delta \fR_{ABCD}=\na_{C}\delta\Gamma_{DAB}-\na_{D}\delta\Gamma_{CAB}+\Gamma^{E}{}_{CD}\delta\Gamma_{EAB}\,. \](2.93)

In particular, under doubled-yet-gauged diffeomorphisms, it varies as

\[ \begin{array}{rll} \delta_{\xi}\fR_{ABCD}&=&\hcL_{\xi}\fR_{ABCD} -2\Gamma^{E}{}_{AB}\partial_{E}\partial_{[C}\xi_{D]} +2\Gamma^{E}{}_{CD}(\cP{+\brcP})_{EAB}{}^{FGH}\partial_{F}\partial_{[G}\xi_{H]}\\ {}&{}&+4\na_{[C}\Big((\cP{+\brcP})_{D]AB}{}^{FGH}\partial_{F}\partial_{[G}\xi_{H]}\Big)\,. \end{array} \](2.94)

Further, the Jacobi identity of the commutators (2.91) implies the following two sets of identities [Jeon:2010rw]:

\[ \begin{array}{ll} \fR_{[A}{}^{D}{}_{BC]}+\na_{[A}\Gamma^{D}{}_{BC]}+\Gamma^{D}{}_{E[A}\Gamma^{E}{}_{BC]}=0\,,\qquad&\quad \na_{[A}{}\fR^{DE}{}_{BC]}+\fR^{DE}{}_{F[A}\Gamma^{F}{}_{BC]}=0\,. \end{array} \](2.95)

Consequently, the semi-covariant Riemann curvature (2.89) appears through a projected commutator,

\[ \left[\cD_{p\,},\cD_{\brq\,}\right]T_{C_1C_2\cdots C_n}=\sum_{i=1}^{n}~2S_{p\brq C_{i}}{}^{D}T_{C_1\cdots C_{i-1}DC_{i+1}\cdots C_{n}}\,, \](2.96)

satisfies symmetric properties, including an algebraic Bianchi identity,

\[ \begin{array}{ll} S_{ABCD}=S_{CDAB}=S_{[AB][CD]}\,,\qquad&\quad S_{A[BCD]}=0\,, \end{array} \](2.97)

and transforms as total (covariant) derivatives under the arbitrary variation of the Christoffel symbols,

\[ \delta S_{ABCD}=\na_{[A}\delta\Gamma_{B]CD}+\na_{[C}\delta\Gamma_{D]AB}\,. \](2.98)

In particular, it is semi-covariant under doubled-yet-gauged diffeomorphisms:

\[ \delta_{\xi}S_{ABCD}=\hcL_{\xi}S_{ABCD}+ 2\na_{[A}\Big(\!(\cP{+\brcP})_{B][CD]}{}^{EFG}\partial_{E}\partial_{F}\xi_{G}\Big) +2\na_{[C}\Big(\!(\cP{+\brcP})_{D][AB]}{}^{EFG}\partial_{E}\partial_{F}\xi_{G}\Big)\,. \](2.99)

Obvious methods of eliminating the anomalous terms via projection end up producing only identically vanishing trivial quantities,

\[ \begin{array}{lll} S_{pq\brp\brq}=0\,,\qquad&\qquad S_{\brp\brq pq}=0\,,\qquad&\qquad S_{p\brp q\brq}=0\,, \end{array} \](2.100)

and there appears to be no fully covariant four-index curvature in DFT [Jeon:2011cn, Hohm:2011si]. Fully covariant curvature tensors are then obtained by contracting the indices: with \(S_{AB}=S_{BA}=S^{C}{}_{ACB}\), we have the fully covariant two-index or Ricci curvature in DFT,

\[ S_{p\brq}=V^{A}{}_{p}\brV^{B}{}_{\brq}S_{AB}\,, \](2.101)

and further the scalar curvature, (Footnote: The subscript \(\scriptstyle{(0)}\) indicates the scalar nature of \(\So\) and serves to distinguish \(\So\) from the notation used for an action, e.g. (2.110). )

\[ \So:=\left(P^{AC}P^{BD}-\brP^{AC}\brP^{BD}\right)S_{ABCD}=S_{pq}{}^{pq}-S_{\brp\brq}{}^{\brp\brq}\,. \](2.102)

These contain both \(\cH_{AB}\) and \(d\) through the Christoffel symbols (2.41): the generalised metric alone cannot generate the covariant curvature [Jeon:2010rw]. Explicitly, we recover the original expression in [Hohm:2010pp],

\[ \!\So=\cH^{AB}\!\left(\textstyle{\frac{1}{8}}\partial_{A}\cH_{CD}\partial_{B}\cH^{CD}+\textstyle{\frac{1}{2}}\partial_{C}\cH_{A}{}^{D}\partial_{D}\cH_{B}{}^{C}-4\partial_{A}d\partial_{B}d+4\partial_{A}\partial_{B}d\right)-\partial_{A}\partial_{B}\cH^{AB}+4\partial_{A}\cH^{AB}\partial_{B}d\,. \](2.103)

Like (2.100), the Ricci curvature satisfies the identities [Jeon:2011cn, Cho:2015lha]:

\[ S_{pr\brq}{}^{r}=S_{p\brr\brq}{}^{\brr}=\half S_{p\brq}\,, \](2.104)

and it arises through the commutators of the covariant differential operators [Coimbra:2011nw, Cho:2015lha]: using (2.96) and (2.104),

\[ \begin{array}{llll} \!\!\!\big[\cD_{p},\cD_{\brq}\big]T^{p}=S_{p\brq}T^{p}\,,~&\, \big[\cD_{\brq},\cD_{p}\big]T^{\brq}=S_{p\brq}T^{\brq}\,,~&\, {}\big[\gamma^{p}\cD_{p},\cD_{\brq}\big]\varepsilon=\half S_{p\brq}\gamma^{p}\varepsilon\,,~&\, {}\big[\brgamma^{\brq}\cD_{\brq},\cD_{p}\big]\varepsilon^{\prime}=\half S_{p\brq}\gamma^{\brq}\varepsilonp\,. \end{array} \](2.105)

Lastly, the scalar curvature satisfies

\[ \begin{array}{ll} S_{pq}{}^{pq}+S_{\brp\brq}{}^{\brp\brq}=0\,,\qquad&\qquad \So=2S_{pq}{}^{pq}=-2S_{\brp\brq}{}^{\brp\brq}\,, \end{array} \](2.106)

and manifests itself through successive application of the Dirac operators (2.77),

\[ \begin{array}{ll} (\gamma^{p}\cD_{p})^{2}\varepsilon +\cD_{\brp}\cD^{\brp}\varepsilon=-\quarter S_{pq}{}^{pq}\varepsilon=-\textstyle{\frac{1}{8}}\So\varepsilon\,,\quad&\quad (\brgamma^{\brp}\cD_{\brp})^{2}\varepsilonp +\cD_{p}\cD^{p}\varepsilonp=-\quarter S_{\brp\brq}{}^{\brp\brq}\varepsilonp=\textstyle{\frac{1}{8}}\So\varepsilonp\,. \end{array} \](2.107)

For the curvatures of the twofold spin connections and their relation to \(\fR_{ABCD}\) and \(S_{ABCD}\), we refer to [Cho:2015lha] (section 2 therein).

Restricting to the Riemannian parametrisation (2.37), (2.39), we have explicitly,

\[ S_{p\brq}=\half e_{p}{}^{\mu}\bre_{\brq}{}^{\nu}\Big[ R_{\mu\nu}+2\trd_{\mu}(\partial_{\nu}\phi)-\quarter H_{\mu\rho\sigma}H_{\nu}{}^{\rho\sigma} +\half e^{2\phi}\trd^{\rho}\big(e^{-2\phi}H_{\rho\mu\nu}\big) \Big]\,, \](2.108)

and

\[ \So=R+4\Box\phi-4\partial_{\mu}\phi\partial^{\mu}\phi-\textstyle{\frac{1}{12}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}\,. \](2.109)

For non-Riemannian parametrisations, we refer to [Cho:2019ofr] (section 4.3 therein).

Doubled Einstein-Hilbert Action & Gravitational Wave

Aim. To formulate the DFT action and examine its variation, conserved currents, and wave equations.

The `pure' DFT action, or the doubled Einstein-Hilbert action, is naturally given by the scalar curvature \(\So\) multiplied by the \(\ODD\)-symmetric integral measure \(e^{-2d}\), integrated over a section \(\Sigma_{D}\):

\[ S_{\DFT}=\int_{\Sigma_{D}}~e^{-2d}\So\,. \](2.110)

From the compatibility condition of the semi-covariant derivative (2.46) and the nice properties of the semi-covariant four-index curvature (2.97), (2.98), it is straightforward to vary the pure DFT Lagrangian:

\[ \delta \big(e^{-2d}\So\big)=4e^{-2d}\big(\brV_{A}{}^{\brq}\delta V^{Ap}S_{p\brq}-\half\delta d\,\So\big)+\partial_{A}\big(e^{-2d}\Theta^{A}\big)\,, \](2.111)

where in the total derivative we have set [Park:2015bza, Blair:2015eba]

\[ \Theta^{A}=2\big(P^{AC}P^{BD}-\brP^{AC}\brP^{BD}\big)\delta \Gamma_{BCD} =4\cH^{AB}\partial_{B}\delta d-\na_{B}\delta\cH^{AB}\,. \](2.112)

It is worthwhile to note different ways of rewriting the variation of the vielbeins in (2.111):

\[ \brV_{A}{}^{\brq}\delta V^{Ap}=- V^{Ap}\delta\brV_{A}{}^{\brq} =\half\big(\brV_{A}{}^{\brq}\delta V^{Ap}- V^{Ap}\delta\brV_{A}{}^{\brq}\big)=\half V^{Ap}\brV^{B\brq}\delta \cH_{AB}\,. \](2.113)

In particular, when the variation is generated by the Kosmann derivative (2.80), i.e. \(\delta_{\xi}=\fcL_{\xi}\), we have

\[ \begin{array}{ll} \brV_{A}{}^{\brq}\delta_{\xi} V^{Ap}=\brV_{A}{}^{\brq}\fcL_{\xi} V^{Ap} =2\cD^{[\brq}\xi^{p]}\,,\qquad&\quad \delta_{\xi}d=\fcL_{\xi}d=\hcL_{\xi}d=-\frac{1}{2}\na_{A}\xi^{A}\,,\\ \delta_{\xi}\cH_{AB}=\fcL_{\xi}\cH_{AB}=\hcL_{\xi}\cH_{AB}= 8\brP_{(A}{}^{C}P_{B)}{}^{D}\na_{[C}\xi_{D]}=8\brV_{(A}{}^{\brq}V_{B)}{}^{p}\cD_{[\brq}\xi_{p]}\,. \end{array} \](2.114)

Applying these to (2.111), i.e. the action principle, we can identify the off-shell conserved Einstein curvature in DFT, which satisfies a differential Bianchi identity [Park:2015bza]:

\[ \begin{array}{ll} G_{AB}=4(PS\brP)_{[AB]}-\half\cJ_{AB}\So=4V_{[A}{}^{p}\brV_{B]}{}^{\brq}S_{p\brq}-\half\cJ_{AB}\So\,,\quad&\quad\na_{A}G^{AB}=0\,. \end{array} \](2.115)

The complete equations of motion of the pure DFT action-derived from the action principle (2.111)-are, a priori, expressed by the independent vanishing of the Ricci curvature and the scalar curvature [Siegel:1993xq, Hohm:2010pp]. However, by utilising the relations, \(G_{A}{}^{A}=-D\So\) and \((PG\brP)_{AB}=2(PS\brP)_{AB}\), the equations of motion can be seen, in a unified manner, as equivalent to the vanishing of the Einstein curvature [Park:2015bza],

\[ \begin{array}{lll} {S_{p\brq}=0}\quad\&\quad{\So=0}\quad&\Longleftrightarrow&\quad G_{AB}=0\,. \end{array} \](2.116)

Although the Einstein curvature is not symmetric, \(G_{AB}\neq G_{BA}\), and does not satisfy \(\na_{B}G^{AB}\neq 0 \), it is possible to symmetrise the curvature by multiplying the generalised metric from the right. Specifically,

\[ \begin{array}{ll} (G\cH)_{AB}=(G\cH)_{BA}=G_{AC}\cH^{C}{}_{B}=-4 V_{(A}{}^{p}\brV_{B)}{}^{\brq}S_{p\brq}-\half\cH_{AB}\So\,, \quad&\quad \na_{A}(G\cH)^{AB}=0\,. \end{array} \](2.117)

Under the \((0,0)\) Riemannian parametrisation, from (2.109), the pure DFT action (2.110) coincides with the NSNS gravity action up to a total derivative (denoted by \(\simeq\)):

\[ S_{\DFT}=\int_{\Sigma_{D}}~e^{-2d}\So~\simeq\,\int\rd^D x~\sqrt{-g}e^{-2\phi}\Big(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\textstyle{\frac{1}{12}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}\Big)\,, \](2.118)

and the upper left \(D{\times D}\) block of \((G\cH)_{AB}\) contains the undoubled Einstein curvature:

\[ (G\cH)^{\mu\nu}=R^{\mu\nu}-\half g^{\mu\nu}R+2\trd^{\mu}(\partial^{\nu}\phi) -2g^{\mu\nu}(\Box\phi-\partial_{\sigma}\phi\partial^{\sigma}\phi)-\quarter H^{\mu\rho\sigma}H^{\nu}{}_{\rho\sigma} +\textstyle{\frac{1}{24}} g^{\mu\nu}H_{\rho\sigma\tau}H^{\rho\sigma\tau}\,. \](2.119)

Moreover, the differential Bianchi identity (2.115) decomposes into two separate identities:

\[ \begin{array}{ll} \trd_{\mu}\big(R^{\mu\nu}-\half g^{\mu\nu}R\big)=0\,,\qquad&\quad \trd_{\mu}\trd_{\nu}\big(e^{-2\phi}H^{\lambda\mu\nu}\big)=0\,. \end{array} \](2.120)

Gamma Squared Action & Noether Currents

Aim. To rewrite the action in \(\Gamma^2\) form and identify associated symmetries.

The doubled Einstein-Hilbert action (2.110) contains terms with two derivatives of the fundamental fields which can be eliminated by a partial integral. Subtracting a specific total derivative [Park:2015bza],

\[ \begin{array}{ll} \cL_{\Gamma^{2}}=e^{-2d}\So-\partial_{A}\big(e^{-2d}B^{A}\big)\,,\quad&~~~ B^{A}=2(P^{AC}P^{BD}-\brP^{AC}\brP^{BD}) \Gamma_{BCD} =4\cH^{AB}\partial_{B}d -\partial_{B}\cH^{AB}\,, \end{array} \](2.121)

we acquire a doubled \(\Gamma^{2}\)-action, free of any two-derivative terms (c.f. [Dyer:2008hb], [Berman:2011kg]),

\[ S_{\Gamma^{2}}=\int_{\Sigma_{D}}~\cL_{\Gamma^{2}}=\int_{\Sigma_{D}}~e^{-2d} \big(P^{AC}P^{BD}-\brP^{AC}\brP^{BD}\big) \big(\Gamma_{AC}{}^{E}\Gamma_{BDE}-\Gamma_{AB}{}^{E}\Gamma_{DCE}+\half\Gamma^{E}{}_{AB}\Gamma_{ECD}\big)\,, \](2.122)

which still admits diffeomorphisms as a Noether symmetry. We note generically from (2.111) and (2.121),

\[ \delta\cH_{BC}\frac{\partial\cL_{\Gamma^{2}}}{\partial(\partial_{A}\cH_{BC})}+ \delta d\frac{\partial\cL_{\Gamma^{2}}}{\partial(\partial_{A}d)}=e^{-2d}\Theta^{A}-\delta\big(e^{-2d}B^{A}\big)\,, \](2.123)

and especially under diffeomorphisms,

\[ \delta_{\xi}\cL_{\Gamma^{2}}=\partial_{A}\left[\xi^{A}e^{-2d}\So-\delta_{\xi}\big(e^{-2d}B^{A}\big)\right]\,. \](2.124)

Thus, with the explicit expression of \(\Theta^{A}\) (2.112) and the commutator relations (2.105), the on-shell conserved Noether current for the doubled-yet-gauged diffeomorphisms is [Park:2015bza]

\[ J_{{\rm{on-shell}}}^{A}=e^{-2d}\big(4\cH^{AB}\partial_{B}\delta_{\xi}d-\na_{B}\delta_{\xi}\cH^{AB}-\xi^{A}\So\big) =\partial_{B}\big(e^{-2d}{K}^{[AB]}\big)+2e^{2d}G^{A}{}_{B}\xi^{B}+\mathfrak{j}^{A}\,, \](2.125)

where \({K}^{AB}\) is a skew-symmetric Noether potential, constituting an off-shell conserved Noether current,

\[ \begin{array}{ll} K^{AB}=4\brV^{[A}{}_{\brp}V^{B]}{}_{q}\big(\cD^{\brp}\xi^{q}+\cD^{q}\xi^{\brp}\big)\,,\quad&\qquad J_{{\rm{off-shell}}}^{A}=\partial_{B}\big(e^{-2d}{K}^{[AB]}\big)\,, \end{array} \](2.126)

and \(\mathfrak{j}^{A}\) denotes a harmless derivative-index-valued vector,

\[ \begin{array}{ll} \mathfrak{j}^{A}= 2e^{-2d}\Big[V_{B}{}^{p}\brV_{C}{}^{\brq}\cD_{(p}\xi_{\brq)}\partial^{A}\cH^{BC} -\partial^{A}\big(\cH^{BC} \na_{B}\xi_{C}\big)\Big]\,,\quad&\quad \mathfrak{j}^{A}\partial_{A}=0\,, \end{array} \](2.127)

which does not contribute to any Noether charge.

Box Operator That Unveils The Riemann Curvature Tensor: Gravitational Wave Equations

Aim. To build a covariant box operator that encodes curvature and governs wave equations.

By combining the results from equations such as (2.71), (2.94), and (2.99), it is possible to construct a pair of fully covariant d'Alembertians that act on an arbitrary tensor density \(T_{A_{1}A_{2}\cdots A_{n}}\) (2.5) and incorporate \(S_{ABCD}\) (2.89) and \(\fR_{ABCD}\) (2.90) [Lee:2025fme]:

\[ \begin{array}{rll} \Deltab T_{A_{1}A_{2}\cdots A_{n}}&:=& P^{BC}\na_{B}\na_{C}T_{A_{1}A_{2}\cdots A_{n}}\\ {}&{}& +\dis{\sum_{i=1}^{n}~2P_{A_{i}}{}^{C}P_{B}{}^{D}\Big(\fR_{[CD]}-\half\Gamma^{EF}{}_{C}\Gamma_{EFD}-\Gamma^{E}{}_{CD}\na_{E}\Big)\,T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{n}}}\\ {}&{}&+\dis{\sum_{i<j}~2\left(\begin{array}{l} P_{A_{i}}{}^{D}P_{B}{}^{E}\fR_{A_{j}CDE}\\ +P_{A_{j}}{}^{D}P_{C}{}^{E}\fR_{A_{i}BDE}\\ -2P_{A_{i}}{}^{D}P_{B}{}^{E}P_{A_{j}}{}^{F}P_{C}{}^{G}S_{DEFG}\end{array}\right)T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{j-1}}{}^{C}{}_{A_{j+1}\cdots A_{n}}}\,, \end{array} \](2.128)

and with the replacement of every explicit occurrence of the projector \(P_{A}{}^{B}\) with the opposite projector \(\brP_{A}{}^{B}\),

\[ \begin{array}{rll} \brDeltab T_{A_{1}A_{2}\cdots A_{n}}&:=& \brP^{BC}\na_{B}\na_{C}T_{A_{1}A_{2}\cdots A_{n}}\\ {}&{}& +\dis{\sum_{i=1}^{n}~2\brP_{A_{i}}{}^{C}\brP_{B}{}^{D}\Big(\fR_{[CD]}-\half\Gamma^{EF}{}_{C}\Gamma_{EFD}-\Gamma^{E}{}_{CD}\na_{E}\Big)\,T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{n}}}\\ {}&{}&+\dis{\sum_{i<j}~2\left(\begin{array}{l} \brP_{A_{i}}{}^{D}\brP_{B}{}^{E}\fR_{A_{j}CDE}\\ +\brP_{A_{j}}{}^{D}\brP_{C}{}^{E}\fR_{A_{i}BDE}\\ -2\brP_{A_{i}}{}^{D}\brP_{B}{}^{E}\brP_{A_{j}}{}^{F}\brP_{C}{}^{G}S_{DEFG}\end{array}\right)T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{j-1}}{}^{C}{}_{A_{j+1}\cdots A_{n}}}\,. \end{array} \](2.129)

While these two mirroring operators annihilate each other identically due to the section condition:

\[ \left(\Deltab+\brDeltab\right)T_{A_{1}A_{2}\cdots A_{n}}=0\,, \](2.130)

their difference defines the fully covariant box operator:

\[ \bBox T_{A_{1}A_{2}\cdots A_{n}}:=\left(\Deltab-\brDeltab\right)T_{A_{1}A_{2}\cdots A_{n}}=\cH^{BC}\na_{B}\na_{C}T_{A_{1}A_{2}\cdots A_{n}}+\sum_{i}~\cdots~+\sum_{i<j}~\cdots\,. \](2.131)

In particular, acting on projected tensors such as \({\brP}_{C_{1}}{}^{D_{1}}\cdots{\brP}_{C_{n}}{}^{D_{n}}T_{D_{1}\cdots D_{n}}\), \(P_{C_{1}}{}^{D_{1}}\cdots P_{C_{n}}{}^{D_{n}}T_{D_{1}\cdots D_{n}}\), and \((PT\brP)_{AB}=P_{A}{}^{C}\brP_{B}{}^{D}T_{CD}\), the pair of d'Alembertians and the box operator produce the two expressions in (2.72) and the following result:

\[ \begin{array}{rrl} \bBox (PT\brP)_{AB}&=& \cH^{CD}\na_{C}\na_{D}(PT\brP)_{AB}-2P_{A}{}^{C}\brP_{B}{}^{D}(\fR_{CEDF}-\fR_{DFCE})(PT\brP)^{EF}\\ {}&{}&+2P_{A}{}^{C}(\fR_{[CD]}-\frac{1}{2}\Gamma^{EF}{}_{C}\Gamma_{EFD}-\Gamma^{E}{}_{CD}\na_{E})(PT\brP)^{D}{}_{B}\\ {}&{}& -2\brP_{B}{}^{C}(\fR_{[CD]}-\frac{1}{2}\Gamma^{EF}{}_{C}\Gamma_{EFD}-\Gamma^{E}{}_{CD}\na_{E})(PT\brP)_{A}{}{}^{D}\,. \end{array} \](2.132)

Equivalent yet more compact expressions are available through contraction with the vielbeins, \(V_{Ap},\brV_{B\brq}\):

\[ \begin{array}{l} \Deltab T_{p\brq}= \cD_{r}\cD^{r}T_{p\brq}+2\fR_{\brq \brs pr}T^{r\brs} +2\big(\fR_{[pr]}-\half\Gamma^{AB}{}_{p}\Gamma_{ABr}-\Gamma^{C}{}_{pr}\cD_{C}\big)T^{r}{}_{\brq}\,,\\ \brDeltab T_{p\brq}= \cD_{\brr}\cD^{\brr}T_{p\brq}+2\fR_{pr\brq\brs}T^{r\brs} +2\big(\fR_{[\brq\brs]}-\half\Gamma^{AB}{}_{\brq}\Gamma_{AB\brs}-\Gamma^{C}{}_{\brq\brs}\cD_{C}\big)T_{p}{}^{\brs}\,. \end{array} \](2.133)

Under the \((0,0)\) Riemannian parametrisation (2.37), (2.39), we set for the two-index tensor,

\[ \begin{array}{lll} T_{p\brq}= V^{A}{}_{p}\brV^{B}{}_{\brq}T_{AB}=\fT^{\mu\nu}e_{\mu p\,}\bre_{\nu\brq}\quad&~\Longleftrightarrow~&\quad \fT_{\mu\nu}=-e_{\mu}{}^{p}\bre_{\nu}{}^{\brq} T_{p\brq}\,, \end{array} \](2.134)

and the box operator reduces to

\[ \hBox\fT_{\mu\nu}=-e_{\mu}{}^{p}\bre_{\nu}{}^{\brq}\bBox T_{p\brq}= e^{2\phi}\trd^{\rho}\big(e^{-2\phi}\trd_{\rho}\fT_{\mu\nu}\big)- H_{\rho\sigma\mu} \trd^{\rho}\fT^{\sigma}{}_{\nu}+ H_{\rho\sigma\nu}\trd^{\rho}\fT_{\mu}{}^{\sigma}+2\hR_{\mu}{}^{\rho}{}_{\nu}{}^{\sigma}\fT_{\rho\sigma}\,, \](2.135)

where \(\hR_{\mu}{}^{\rho}{}_{\nu}{}^{\sigma}\) contains the Riemann curvature tensor and the \(H\)-flux,

\[ \begin{array}{rll} \hR_{\mu}{}^{\rho}{}_{\nu}{}^{\sigma}&=&R_{\mu}{}^{\rho}{}_{\nu}{}^{\sigma} -\half H_{(\mu}{}^{\rho\kappa}H_{\nu)}{}^{\sigma}{}_{\kappa}-\quarter H_{\mu\nu\kappa}H^{\rho\sigma\kappa} +\half\trd_{(\mu}H_{\nu)}{}^{\rho\sigma}+\half\trd^{(\rho}H^{\sigma)}{}_{\mu\nu}\\ {}&{}& +\frac{1}{4}\delta_{\mu}{}^{\rho}\Big[e^{2\phi}\trd_{\kappa}\big(e^{-2\phi}H^{\kappa\sigma}{}_{\nu}\big)-\frac{1}{2}H_{\nu\kappa\lambda}H^{\sigma\kappa\lambda}\Big] -\frac{1}{4}\Big[e^{2\phi}\trd_{\kappa}\big(e^{-2\phi}H^{\kappa\rho}{}_{\mu}\big)+\frac{1}{2}H_{\mu\kappa\lambda}H^{\rho\kappa\lambda}\Big]\delta_{\nu}{}^{\sigma} \,. \end{array} \](2.136)

In the context of applications, the imposition of an \(\ODD\)-symmetric harmonic gauge,

\[ e^{2d}\na_{A}\delta\big(e^{-2d}\cH^{AB}\big)=\na_{A}\delta\cH^{AB}-2\cH^{AB}\na_{A}\delta d=0\,, \](2.137)

leads to a simplification of the linearised equations of motion in pure DFT [Ko:2015rha] (see also [Hohm:2015ugy, Cho:2019npq]). The resulting \(\ODD\)-symmetric gravitational wave equations are, with the box operator (2.132) [Lee:2025fme]:

\[ \begin{array}{ll} \bBox(P\delta\cH\brP)_{AB}=0\,,~\qquad&\quad\cH^{AB}\na_{A}\partial_{B}\delta d=0\,. \end{array} \](2.138)

When applied to a Riemannian background, the \(\ODD\)-symmetric harmonic gauge condition (2.137) decomposes into the following components:

\[ \begin{array}{ll} e^{2\phi}\trd^{\rho}\!\left(e^{-2\phi}\delta g_{\rho\mu}\right) -\frac{1}{2}H_{\mu}{}^{\rho\sigma}\delta B_{\rho\sigma}+2\partial_{\mu}\delta d=0\,,\quad&\qquad e^{2\phi}\trd^{\rho}\!\left(e^{-2\phi}\delta B_{\rho\mu}\right)=0\,, \end{array} \](2.139)

where \(\delta d=\delta\phi-\frac{1}{4}g^{\rho\sigma}\delta g_{\rho\sigma }\), and the wave equations (2.138) reduce further, with (2.135) and (2.136), to

\[ \begin{array}{ll} \hBox(\delta g_{\mu\nu}-\delta B_{\mu\nu})=0\,,\qquad&\qquad e^{2\phi}\trd^{\rho}\big(e^{-2\phi}\partial_{\rho}\delta d\big)=0\,. \end{array} \](2.140)