Gravitational Core of Double Field Theory

Covariant Derivatives

Lecture Notes
Jeong-Hyuck Park

This chapter develops the connection data of DFT and explains how semi-covariant objects combine into fully covariant derivatives.

arXiv

Christoffel Symbols & Spin Connections

Aim. To build a master derivative that unifies diffeomorphisms and local Lorentz symmetries.

The master derivative, denoted as \(\cD_{A}\), is introduced to unify and covariantly describe all the local symmetries: the doubled-yet-gauged diffeomorphisms and the twofold local Lorentz symmetries, \(\Spin(1,D{-1}) \times \Spin(D{-1},1)\). More explicitly, the master derivative is postulated as:

\[ \cD_{A}=\partial_{A}+\Gamma_{A}+\Phi_{A}+\brPhi_{A}\,. \](2.40)

Here, \(\Gamma_{A}\) refers to the Christoffel connection for the doubled-yet-gauged diffeomorphisms, introduced in [Jeon:2011cn] and based on earlier work [Jeon:2010rw]. (Footnote: One key lesson from [Jeon:2010rw] is that the generalised metric alone is insufficient for fully constructing covariant derivatives and curvatures; the inclusion of the dilaton \(d\) is essential.) (Footnote: An alternative formulation was proposed in [Siegel:1993th] and developed in [Hohm:2010xe], where curved \(\ODD\) indices are mapped to flat Lorentz indices via the vielbeins (2.21). Although this appears to bypass the Christoffel connection, the spin connection necessarily reintroduces its content, as in (2.42), so that the framework ultimately preserves parallels with conventional differential geometry.) The connection is defined as:

\[ \begin{array}{ll} \Gamma_{CAB}=&2\left(P\partial_{C}P\brP\right)_{[AB]} +2\left({{\brP}_{[A}{}^{D}{\brP}_{B]}{}^{E}}-{P_{[A}{}^{D}P_{B]}{}^{E}}\right)\partial_{D}P_{EC}\\ {}&-4\left(\textstyle{\frac{1}{P_{M}{}^{M}-1}}P_{C[A}P_{B]}{}^{D}+\textstyle{\frac{1}{\brP_{M}{}^{M}-1}}\brP_{C[A}\brP_{B]}{}^{D}\right)\!\left(\partial_{D}d+(P\partial^{E}P\brP)_{[ED]}\right)\,. \end{array} \](2.41)

Additionally, \(\Phi_{A}\) and \(\brPhi_{A}\) are the spin connections corresponding to the twofold local Lorentz symmetries [Jeon:2011vx], given as:

\[ \begin{array}{ll} \Phi_{Apq}=\Phi_{A[pq]}=V^{B}{}_{p}\na_{A}V_{Bq}\,,\quad&\quad \brPhi_{A\brp\brq}=\brPhi_{A[\brp\brq]}=\brV^{B}{}_{\brp}\na_{A}\brV_{B\brq}\,. \end{array} \](2.42)

In our notation, \(\na_{A}\) is the diffeomorphism-covariant derivative that involves the Christoffel symbols only,

\[ \na_{A}:=\partial_{A}+\Gamma_{A}\,, \](2.43)

and, ignoring any local Lorentz indices, acts on a tensor density with weight \(\omega\) as:

\[ \na_{C}T_{A_{1}A_{2}\cdots A_{n}} :=\partial_{C}T_{A_{1}A_{2}\cdots A_{n}}-\omega_{{\scriptscriptstyle{T\,}}}\Gamma^{B}{}_{BC}T_{A_{1}A_{2}\cdots A_{n}}+ \sum_{i=1}^{n}\,\Gamma_{CA_{i}}{}^{B}T_{A_{1}\cdots A_{i-1}BA_{i+1}\cdots A_{n}}\,. \](2.44)

In particular, in (2.42),

\[ \begin{array}{ll} \na_{A}V_{Bq}=\partial_{A}V_{Bq}+\Gamma_{AB}{}^{C}V_{Cq}\,,\quad&\quad \na_{A}\brV_{B\brq}=\partial_{A}\brV_{B\brq}+\Gamma_{AB}{}^{C}\brV_{C\brq}\,. \end{array} \](2.45)

The Christoffel connection (2.41) is uniquely fixed by requiring the following three properties:

i) Full compatibility with all fundamental fields,

\[ \begin{array}{ll} \cD_{A}P_{BC}=\na_{A}P_{BC}=0\,,\quad&\quad\cD_{A}\brP_{BC}=\na_{A}\brP_{BC}=0\,,\\ \cD_{A}d\,=\na_{A}d\,=-\half e^{2d}\na_{A}(e^{-2d})=\partial_{A}d+\half\Gamma^{B}{}_{BA}=0\,, \end{array} \](2.46)

which implies

\[ \begin{array}{ll} \cD_{A}\cJ_{BC}=\na_{A}\cJ_{BC}=0\,,\quad&\qquad\cD_{A}\cH_{BC}=\na_{A}\cH_{BC}=0\,, \end{array} \](2.47)

such that \(\Gamma_{A}\) is, as expected, \(\mathbf{so}(D,D)\)-valued:

\[ \Gamma_{ABC}=-\Gamma_{ACB}\,. \](2.48)

ii) Torsionless cyclic property, (Footnote: In full-order supersymmetric DFTs [Jeon:2011sq, Jeon:2012hp], the torsions are quadratic in fermionic fields and naturally implement the 1.5 formalism characteristic of supergravity theories.)

\[ \Gamma_{ABC}+\Gamma_{BCA}+\Gamma_{CAB}=0\,, \](2.49)

which makes \(\na_{A}\) compatible with the generalised Lie derivative (2.5) and the C-bracket (2.7) so that ordinary derivatives in these expressions can be freely replaced by \(\na_{A}\):

\[ \hcL_{\xi}T_{A_{1}\cdots A_{n}}=\xi^{B}\na_{B}T_{A_{1}\cdots A_{n}}+\omega\na_{B}\xi^{B}T_{A_{1}\cdots A_{n}}+\sum_{i=1}^{n}~(\na_{A_{i}}\xi_{B}-\na_{B}\xi_{A_{i}})T_{A_{1}\cdots A_{i-1}}{}^{B}{}_{A_{i+1}\cdots A_{n}}\,. \](2.50)

iii) Projection constraints,

\[ \begin{array}{ll} \cP_{ABC}{}^{DEF}\Gamma_{DEF}=0\,,\quad&\quad\bar{\cP}_{ABC}{}^{DEF}\Gamma_{DEF}=0\,. \end{array} \](2.51)

These constraints involve six-index projectors,

\[ \begin{array}{l} \cP_{ABC}{}^{DEF}:=P_{A}{}^{D}P_{[B}{}^{[E}P_{C]}{}^{F]}+\textstyle{\frac{2}{P_{M}{}^{M}-1}}P_{A[B}P_{C]}{}^{[E}P^{F]D}\,,\\ \bar{\cP}_{ABC}{}^{DEF}:=\brP_{A}{}^{D}\brP_{[B}{}^{[E}\brP_{C]}{}^{F]}+\textstyle{\frac{2}{\brP_{M}{}^{M}-1}}\brP_{A[B}\brP_{C]}{}^{[E}\brP^{F]D}\,, \end{array} \](2.52)

which satisfy the following projection properties,

\[ \begin{array}{ll} \cP_{ABC}{}^{DEF}\cP_{DEF}{}^{GHI}=\cP_{ABC}{}^{GHI}\,,\qquad&\qquad \brcP_{ABC}{}^{DEF}\brcP_{DEF}{}^{GHI}=\brcP_{ABC}{}^{GHI}\,, \end{array} \](2.53)

various symmetric relations [Cho:2015lha],

\[ \begin{array}{lll} \cP_{ABCDEF}=\cP_{DEFABC}\,,\quad&\quad\cP_{ABCDEF}=\cP_{A[BC]D[EF]}\,, \quad&\quad \cP_{[AB]CDEF}=\cP_{CAB[EF]D}\,,\\ \brcP_{ABCDEF}=\brcP_{DEFABC}\,,\quad&\quad\brcP_{ABCDEF}=\brcP_{A[BC]D[EF]}\,, \quad&\quad \brcP_{[AB]CDEF}=\brcP_{CAB[EF]D}\,, \end{array} \](2.54)

and traceless conditions,

\[ \begin{array}{ll} \quad P^{AB}\cP_{ABCDEF}=0\,,\qquad&\quad \quad \brP^{AB}\brcP_{ABCDEF}=0\,. \end{array} \](2.55)

Unlike the GR Christoffel symbols, there exist no normal coordinates where the DFT Christoffel symbols would vanish pointwise. The Equivalence Principle holds for point particles but not for extended objects such as strings [Choi:2015bga]. Parallel to the variation of the Christoffel symbols in GR:

\[ \delta\gamma^{\lambda}_{\mu\nu}=\frac{1}{2}(\trd_{\mu}\delta g^{\lambda}{}_{\nu}+\trd_{\nu}\delta g_{\mu}{}^{\lambda}-\trd^{\lambda}\delta g_{\mu\nu})\,, \]

the DFT Christoffel symbols vary infinitesimally by the DFT metric and dilaton: with \(\delta P_{AB}=\frac{1}{2}\delta\cH_{AB}\),

\[ \begin{array}{rll} {\delta\Gamma}_{CAB}&=&2P_{[A}^{~D}\brP_{B]}^{~E}\DO_{C}\delta P_{DE}+2(\brP_{[A}^{~D}\brP_{B]}^{~E}-P_{[A}^{~D}P_{B]}^{~E})\DO_{D}\delta P_{EC}\\ {}&{}&-\textstyle{\frac{4}{D-1}}(\brP_{C[A}\brP_{B]}^{~D}+P_{C[A}P_{B]}^{~D})(\partial_{D}\delta d+P_{E[G}\DO^{G}\delta P^{E}_{~D]})-\Gamma_{FDE\,}\delta(\cP+\brcP)_{CAB}{}^{FDE}\,. \end{array} \](2.56)

Once the DFT Christoffel symbols are fixed, the spin connections (2.42) follow naturally from the compatibility of the master derivative with the DFT vielbeins: from (2.45),

\[ \begin{array}{ll} \cD_{A}V_{Bp}=\na_{A}V_{Bp}+\Phi_{Ap}{}^{q}V_{Bq}=0\,,\qquad&\quad \cD_{A}\brV_{B\brp}=\na_{A}\brV_{B\brp}+\brPhi_{A\brp}{}^{\brq}\brV_{B\brq}=0\,. \end{array} \](2.57)

The master derivative is also compatible with the metrics and gamma matrices of the twofold spin groups,

\[ \begin{array}{llll} \cD_{A}\eta_{pq}=0\,,\quad&\quad \cD_{A}\breta_{\brp\brq}=0\,,\quad&\quad\cD_{A}(\gamma^{p})^{\alpha}{}_{\beta}=0\,,\quad&\quad \cD_{A}(\brgamma^{\brp})^{\bralpha}{}_{\brbeta}=0\,, \end{array} \](2.58)

and thus, analogous to GR,

\[ \begin{array}{llll} \Phi_{Apq}=-\Phi_{Aqp}\,,\quad&\quad \brPhi_{A\brp\brq}=-\brPhi_{A\brq\brp}\,,\quad&\quad \Phi_{A}{}^{\alpha}{}_{\beta}=\quarter\Phi_{Apq}(\gamma^{pq})^{\alpha}{}_{\beta}\,,\quad&\quad \brPhi_{A}{}^{\bralpha}{}_{\brbeta}=\quarter\brPhi_{A\brp\brq}(\brgamma^{\brp\brq})^{\bralpha}{}_{\brbeta}\,. \end{array} \](2.59)

It is worthwhile to note a formula that relates the three connections [Cho:2015lha]:

\[ \Gamma_{CAB}= V_{A}{}^{p}\partial_{C}V_{Bp}+\brV_{A}{}^{\brp}\partial_{C}\brV_{B\brp}+ V_{A}{}^{p}V_{B}{}^{q}\Phi_{Cpq}+ \brV_{A}{}^{\brp}\brV_{B}{}^{\brq}\brPhi_{C\brp\brq}\,, \](2.60)

ensuring that Cartan's structure equations in GR hold analogously in DFT [Cho:2015lha].

Due to the section condition, the Christoffel symbols satisfy

\[ P_{A}{}^{C}\brP_{B}{}^{D}\Gamma^{E}{}_{CD}\partial_{E}=0\,. \](2.61)

From Semi-Covariance to Full Covariance

Aim. To refine the semi-covariant derivative into a fully covariant derivative through projections.

Explicitly, the master derivative (2.40) acts as

\[ \begin{array}{l} \cD_{A}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\bralpha}= \na_{A}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\bralpha}+ \Phi_{Ap}{}^{q} T_{Bq}{}^{\alpha}{}_{\brp}{}^{\bralpha}+\Phi_{A}{}^{\alpha}{}_{\beta}T_{Bp}{}^{\beta}{}_{\brp}{}^{\bralpha}+\brPhi_{A\brp}{}^{\brq} T_{Bp}{}^{\alpha}{}_{\brq}{}^{\bralpha}+\brPhi_{A}{}^{\bralpha}{}_{\brbeta}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\brbeta}\\ =\partial_{A}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\bralpha}-\omega \Gamma^{C}{}_{CA}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\bralpha} {+\Gamma_{AB}{}^{C}}T_{Cp}{}^{\alpha}{}_{\brp}{}^{\bralpha} {+\Phi_{Ap}{}^{q}} T_{Bq}{}^{\alpha}{}_{\brp}{}^{\bralpha}{+\Phi_{A}{}^{\alpha}{}_{\beta}}T_{Bp}{}^{\beta}{}_{\brp}{}^{\bralpha}{+\brPhi_{A\brp}{}^{\brq}} T_{Bp}{}^{\alpha}{}_{\brq}{}^{\bralpha}{+\brPhi_{A}{}^{\bralpha}{}_{\brbeta}}T_{Bp}{}^{\alpha}{}_{\brp}{}^{\brbeta}\,. \end{array} \](2.62)

The master derivative is fully covariant with respect to the twofold local Lorentz symmetries but is, a priori, only semi-covariant under the doubled-yet-gauged diffeomorphisms. To achieve complete covariance under these diffeomorphisms, an additional step of projection is required, which we explain.

Tensorial Covariant Derivatives

Aim. To obtain a fully covariant derivative for tensors.

Under diffeomorphisms, the DFT Christoffel symbols transform as

\[ \delta_{\xi}\Gamma_{CAB}=\hcL_{\xi}\Gamma_{CAB} -2\partial_{C}\partial_{[A}\xi_{B]}+2(\cP+\brcP)_{CAB}{}^{DEF}\partial_{D}\partial_{[E}\xi_{F]}\,, \](2.63)

such that \(\na_{A}\), and consequently \(\cD_{A}\), are not inherently covariant under the diffeomorphisms:

\[ \delta_{\xi}\big(\na_{C}T_{A_{1}\cdots A_{n}}\big)=\hcL_{\xi}\big(\na_{C}T_{A_{1}\cdots A_{n}}\big)+ \dis{\sum_{i=1}^{n}2(\cP{+\brcP})_{CA_{i}}{}^{BDEF} \partial_{D}\partial_{E}\xi_{F}\,T_{A_{1}\cdots A_{i-1} BA_{i+1}\cdots A_{n}}\,.} \](2.64)

However, the anomalous terms consistently appear through the six-index projectors (2.52), making them straightforward to project out. For instance, in (2.64), projecting the derivative index and the tensor indices in opposite ways allows the construction of fully covariant derivatives [Jeon:2011cn]:

\[ \begin{array}{ll} P_{C}{}^{D}{\brP}_{A_{1}}{}^{B_{1}}\cdots{\brP}_{A_{n}}{}^{B_{n}} \DO_{D}T_{B_{1}\cdots B_{n}}\,,\quad&\qquad {\brP}_{C}{}^{D}P_{A_{1}}{}^{B_{1}}\cdots P_{A_{n}}{}^{B_{n}} \DO_{D}T_{B_{1}\cdots B_{n}}\,. \end{array} \](2.65)

Similarly, using (2.54) and (2.55), we obtain fully covariant divergences,

\[ \begin{array}{ll} P^{AB}{\brP}_{C_{1}}{}^{D_{1}}\cdots{\brP}_{C_{n}}{}^{D_{n}}\DO_{A}T_{BD_{1}\cdots D_{n}}\,,\quad&\qquad \brP^{AB}{P}_{C_{1}}{}^{D_{1}}\cdots{P}_{C_{n}}{}^{D_{n}}\DO_{A}T_{BD_{1}\cdots D_{n}}\,. \end{array} \](2.66)

For instance, the divergence of a weightless vector, \(J^{A}\), reads

\[ \na_{A}\!\left(e^{-2d}J^{A}\right)=\partial_{A}\!\left(e^{-2d}J^{A}\right)=e^{-2d}\na_{A}J^{A}=e^{-2d}\left(P^{AB}+\brP^{AB}\right)\na_{A}J_{B}\,. \](2.67)

Nevertheless, when acting on the two-index projectors, \(P_{AB}\) and \(\brP_{AB}\), the anomalous terms in (2.64) are automatically eliminated due to the properties of the six-index projectors, (2.54) and (2.55). This guarantees the full covariance of the compatibility condition, \(\na_{C}P_{AB}=0=\na_{C}\brP_{AB}\), as postulated in (2.46).

In accordance with (2.24), the partial derivative of the generalised metric satisfies a projection property,

\[ \partial_C\cH_{AB}=(P\partial_C\cH\brP)_{AB}+(\brP \partial_C\cH P)_{AB}\,, \](2.68)

where the two free indices are projected in opposite manners. This property essentially precludes the possibility of canceling the anomalous terms in (2.63) by adding any derivatives of the generalised metric or dilaton to \(\Gamma_{CAB}\) in (2.41). While introducing additional complementary fields could, in principle, eliminate the anomalous terms, such fields remain "undetermined" or are not naturally identifiable within string theory.

Notably, the additional projection step (2.65) highlights the more refined and rigid structure of DFT compared to GR. For instance, while DFT provides two available "metrics," namely \(\cJ_{AB}\) and \(\cH_{AB}\), the pairwise contraction of \(\ODD\) indices is significantly restricted. Specifically, when constructing a scalar from the squared derivative \( \na_{A}T_{B_{1}B_{2}\cdots B_{n}}\) for a kinetic term, from (2.65) there are only two viable ways to "square" it, rather than the \(2^{n+1}\) possibilities that might otherwise exist.

As a further illustration, consider a doubled Yang-Mills potential \(\bfA_{A}\). The fully covariant skew-symmetric field strength is expressed as [Jeon:2011kp, Choi:2015bga]

\[ P_{A}{}^{C}\brP_{B}{}^{D}\bfF_{CD}= P_{A}{}^{C}\brP_{B}{}^{D}\big(\na_{C}\bfA_{D}-\na_{D}\bfA_{C}-i\left[\bfA_{C},\bfA_{D}\right]\big)\,, \](2.69)

and there is only one way to construct the doubled Yang-Mills action:

\[ \dis{\int_{\Sigma_D}e^{-2d}P^{AC}\brP^{BD}\Tr(\bfF_{AB}\bfF_{CD})\,,} \](2.70)

where \(\Sigma_{D}\) is a \(D\)-dimensional section over which the integral is taken with the measure, \(e^{-2d}\).

The doubled Yang-Mills potential decomposes into a displacement vector and an ordinary one-form: \(\bfA_{A} = (\varphi^{\mu}, A_{\nu})\). This structure enables the doubled Yang-Mills action to provide a unified description of gluons/photons and (non-Abelian) phonons [Angus:2021jvm]. However, it is always possible to eliminate the displacement phonon vector by imposing an \(\ODD\)-symmetric constraint, \(\bfA^{A} \partial_{A} = 0\), analogous to the section condition (2.1) [Choi:2015bga].

We proceed to construct fully covariant second-order differential operators, for which we need to identify the relevant anomalous terms. Making use of (2.64) repeatedly with care, we get

\[ \begin{array}{rll} \big(\delta_{\xi}-\hcL_{\xi}\big)\big(\na_{B}\na_{C}T_{A_{1}\cdots A_{n}}\big)&=&2 (\cP{+\brcP})_{BC}{}^{DEFG} \partial_{E}\partial_{F}\xi_{G}\,\na_{D}T_{A_{1}\cdots A_{n}}\\ {}&{}&+ \dis{\sum_{i=1}^{n}}\left[\begin{array}{l} 2(\cP{+\brcP})_{CA_{i}}{}^{DEFG}T_{A_{1}\cdots A_{i-1} DA_{i+1}\cdots A_{n}}\na_{B}\left( \partial_{E}\partial_{F}\xi_{G}\right)\\ + 2 (\cP{+\brcP})_{CA_{i}}{}^{DEFG} \partial_{E}\partial_{F}\xi_{G}\,\na_{B}T_{A_{1}\cdots A_{i-1} DA_{i+1}\cdots A_{n}}\\ + 2 (\cP{+\brcP})_{BA_{i}}{}^{DEFG} \partial_{E}\partial_{F}\xi_{G}\,\na_{C}T_{A_{1}\cdots A_{i-1} DA_{i+1}\cdots A_{n}}\end{array} \right]\,. \end{array} \](2.71)

Again from the symmetric and traceless relations of the six-index projectors (2.54), (2.55), it is straightforward to obtain fully covariant d'Alembertians:

\[ \begin{array}{ll} P^{AB}{\brP}_{C_{1}}{}^{D_{1}}\cdots{\brP}_{C_{n}}{}^{D_{n}} \DO_{A}\DO_{B}T_{D_{1}\cdots D_{n}}\,,\quad&\qquad {\brP}^{AB}P_{C_{1}}{}^{D_{1}}\cdots P_{C_{n}}{}^{D_{n}} \DO_{A}\DO_{B}T_{D_{1}\cdots D_{n}}\,. \end{array} \](2.72)

Later, through equations (2.128), (2.129), (2.130), and (2.131), we will encounter a pair of general forms of the d'Alembertian, which yield a box operator. These operators are crafted to act on arbitrary multi-index tensor densities while a priori incorporating four-index (Riemann) curvature.

Spinorial Covariant Derivatives

Aim. To extend covariance to spinors and the RR sector by constructing Dirac operators.

It is possible to freely replace \(\na_A\) in the fully covariant derivatives (2.65), (2.66), and (2.72) with the master derivative \(\cD_A\), while simultaneously contracting their projected, otherwise unconstrained \(\ODD\) vector indices using the DFT vielbeins, \(V_{Ap}\) and \(\brV_{A\brq}\). Due to the compatibility of the DFT vielbeins with the master derivative (2.57), this replacement reduces the fully covariant derivatives to more streamlined forms:

\[ \begin{array}{llllll} \cD_{p}T_{\brq_{1}\cdots \brq_{n}}\,,\quad& \cD_{\brp}T_{q_{1}\cdots q_{n}}\,,\quad& \cD_{p}T^{p}{}_{\brq_{1}\cdots\brq_{n}}\,,\quad& \cD_{\brp}T^{\brp}{}_{q_{1}\cdots q_{n}}\,,\quad& \cD_{p}\cD^{p}T_{\brq_{1}\cdots \brq_{n}}\,,\quad& \cD_{\brp}\cD^{\brp}T_{q_{1}\cdots q_{n}}\,, \end{array} \](2.73)

where \(\cD_{p}=V^{A}{}_{p}\cD_{A}\) and \(\cD_{\brp}=\brV^{A}{}_{\brp}\cD_{A }\).

The fully covariant doubled Yang-Mills field strength (2.69) also reads

\[ \bfF_{p\brq}:=V^{A}{}_{p}\brV^{B}{}_{\brq}\big(\na_{A}\bfA_{B}-\na_{B}\bfA_{A}-i\left[\bfA_{A},\bfA_{B}\right]\big)=\cD_{p}\bfA_{\brq}-\cD_{\brq}\bfA_{p}-i\left[\bfA_{p},\bfA_{\brq}\right]\,. \](2.74)

The full covariance of (2.73) and (2.74) can also be directly confirmed by observing that, from

\[ \begin{array}{ll} \delta_{\xi}\Phi_{Apq}=\hcL_{\xi}\Phi_{Apq}+2\cP_{Apq}{}^{DEF}\partial_{D}\partial_{[E}\xi_{F]}\,,\quad&\qquad \delta_{\xi}\brPhi_{A\brp\brq}=\hcL_{\xi}\brPhi_{A\brp\brq}+2\brcP_{A\brp\brq}{}^{DEF}\partial_{D}\partial_{[E}\xi_{F]}\,, \end{array} \](2.75)

the following projected components of the spin connections are fully covariant under doubled-yet-gauged diffeomorphisms: (Footnote: The quantities in (2.76) are essentially the “generalised fluxes” considered in [Aldazabal:2011nj, Grana:2012rr, Geissbuhler:2013uka] as the building blocks of DFT.)

\[ \begin{array}{llllll} \Phi_{\brr pq}\,,\quad&\quad\brPhi_{r\brp\brq}\,,\quad&\quad\Phi_{[pqr]}\,,\quad&~ \brPhi_{[\brp\brq\brr]}\,,\quad&\quad\Phi^{p}{}_{pq}\,,\quad&\quad \brPhi^{\brp}{}_{\brp\brq}\,, \end{array} \](2.76)

where we have set \(\Phi_{\brr pq}=\brV^{A}{}_{\brr}\Phi_{A pq} \) and \( \Phi_{rpq}=V^{A}{}_{r}\Phi_{A pq}\), etc.

Consequently, when acting on \(\SpinD\) spinors, such as \(\rho^{\alpha}\) or \(\psi_{\brp}^{\alpha}\), or on \(\oSpinD\) spinors, such as \(\rho^{\prime\bralpha}\) or \(\psi^{\prime\bralpha}_{p}\), the fully covariant Dirac operators are [Jeon:2011vx, Jeon:2011sq]

\[ \begin{array}{llllllll} \gamma^{p}\cD_{p}\rho\,,\quad&~ \gamma^{p}\cD_{p}\psi_{\brp}\,,\quad&~ \cD_{\brp}\rho\,,\quad&~ \cD_{\brp}\psi^{\brp}\,,\quad&~ \brgamma^{\brp}\cD_{\brp}\rhop\,,\quad&~ \brgamma^{\brp}\cD_{\brp}\psi^{\prime}_{p}\,,\quad&~ \cD_{p}\rhop\,,\quad&\quad \cD_{p}\psip{}^{p}\,. \end{array} \](2.77)

Further, in the maximally supersymmetric type II DFT [Jeon:2012hp], as well as in the pure spinor formalism [Berkovits:2001ue], the Ramond-Ramond (RR) sector is characterized by a \(\SpinD \times \oSpinD\) bi-fundamental spinorial potential: \(\cC^{\alpha}{}_{\bralpha}\) (c.f. [Rocen:2010bk, Hohm:2011zr, Hatsuda:2014aza, Cederwall:2016ukd, Butter:2022gbc, Butter:2022sfh]). Subsequently, a pair of fully covariant and nilpotent derivatives, \(\cD_{+}\) and \(\cD_{-}\), are introduced [Jeon:2012kd]:

\[ \begin{array}{ll} \cD_{\pm}\cC:=\gamma^{p}\cD_{p}\cC\pm\gamma^{(D+1)}\cD_{\brp}\cC\brgamma^{\brp}\,,\quad&\qquad \cD_{\pm}^{2}\cC=0\,, \end{array} \](2.78)

where, with (2.59), \(\cD_{A}\cC=\partial_{A}\cC+\Phi_{A}\cC-\cC\brPhi_{A}\). In particular, the RR field strength, \(\cF^{\alpha}{}_{\bralpha}\), is given by one of these operators and remains invariant under RR gauge transformations,

\[ \begin{array}{ll} \cF:=\cD_{+}\cC\,,\qquad&\qquad \delta \cC=\cD_{+}\lambda\quad\longrightarrow\quad \delta\cF=0\,. \end{array} \](2.79)

Generalised Kosmann Derivative, aka the Further-Generalised Lie Derivative

Aim. To adapt the Lie derivative so that it respects local Lorentz symmetry.

The generalised Lie derivative is compatible with the semi-covariant derivative, \(\hcL_{\xi}(\partial_{A}) = \hcL_{\xi}(\na_{A})\), as demonstrated in (2.50), and is in fact fully covariant under the doubled-yet-gauged diffeomorphisms (2.8). However, when acting on spinorial tensors, it fails to preserve the local Lorentz symmetries. To achieve full covariance under both diffeomorphisms and local Lorentz rotations, the generalised Lie derivative must be further extended to incorporate spin connections. This approach was originally introduced in the context of GR by Kosmann in 1971 [Kosmann]. (Footnote: See [Kim:2024ewt] for a recent application of the Kosmann derivative.) The DFT counterpart to the Kosmann derivative, also referred to as the further-generalised Lie derivative, is defined in [Angus:2018mep] as

\[ \begin{array}{lll} \fcL_{\xi}T_{Ap\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}&:=&\xi^{B}\cD_{B}T_{Ap\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}+\omega\cD_{B}\xi^{B}T_{Ap\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}+2\cD_{[A}\xi_{B]}T^{B}{}_{p\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}\\ {}&{}&+2\cD_{[p}\xi_{q]}T_{A}{}^{q}{}_{\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}+ \half\cD_{[r}\xi_{s]}(\gamma^{rs})^{\alpha}{}_{\delta}T_{Ap\brp}{}^{\delta}{}_{\beta}{}^{\bralpha}{}_{\brbeta}- \half\cD_{[r}\xi_{s]}(\gamma^{rs})^{\delta}{}_{\beta}T_{Ap\brp}{}^{\alpha}{}_{\delta}{}^{\bralpha}{}_{\brbeta}\\ {}&{}&+2\cD_{[\brp}\xi_{\brq]}T_{Ap}{}^{\brq\alpha}{}_{\beta}{}^{\bralpha}{}_{\brbeta}+ \half\cD_{[\brr}\xi_{\brs]}(\brgamma^{\brr\brs})^{\bralpha}{}_{\brdelta}T_{Ap\brp}{}^{\alpha}{}_{\beta}{}^{\brdelta}{}_{\brbeta}- \half\cD_{[\brr}\xi_{\brs]}(\brgamma^{\brr\brs})^{\brdelta}{}_{\brbeta}T_{Ap\brp}{}^{\alpha}{}_{\beta}{}^{\bralpha}{}_{\brdelta}\,, \end{array} \](2.80)

which consists of the generalised Lie derivative combined with infinitesimal local Lorentz rotations characterized by

\[ \begin{array}{ll} \xi^{A}\Phi_{Apq}+2\cD_{[p}\xi_{q]}={2\partial_{[p}\xi_{q]}+\Phi_{\brr pq}\xi^{\brr}+3\Phi_{[pqr]}\xi^{r}}\,,~~&~~ \xi^{A}\brPhi_{A\brp\brq}+2\cD_{[\brp}\xi_{\brq]}={2\partial_{[\brp}\xi_{\brq]}+\brPhi_{r\brp\brq}\xi^{r} +3\brPhi_{[\brp\brq\brr]}\xi^{\brr}}\,. \end{array} \](2.81)

As listed in (2.76), all of these are fully covariant under the doubled-yet-gauged diffeomorphisms.

Riemannian Parametrisation of the Covariant Derivatives

Aim. To show how covariant derivatives reduce under the \((0,0)\) Riemannian parametrisation.

Under the \((0,0)\) Riemannian parametrisation, (2.37) and (2.39), the projected components of the spin connections (2.76) reduce explicitly to

\[ \begin{array}{ll} \Phi_{\brp pq}=\frac{1}{\sqrt{2}}\bre_{\brp}{}^{\mu}\left(\omega_{\mu pq}+\half H_{\mu pq}\right)\,,\quad&\quad \brPhi_{p\brp\brq}=\frac{1}{\sqrt{2}}e_{p}{}^{\mu}\left(\bromega_{\mu\brp\brq}+\half H_{\mu \brp\brq}\right)\,,\\ \Phi_{[pqr]}=\frac{1}{\sqrt{2}}\left(\omega_{[pqr]}+\textstyle{\frac{1}{6}} H_{pqr}\right)\,,\quad&\quad \brPhi_{[\brp\brq\brr]}=\frac{1}{\sqrt{2}}\left(\bromega_{[\brp\brq\brr]}+\textstyle{\frac{1}{6}} H_{\brp\brq\brr}\right)\,,\\ \Phi^{p}{}_{pq}=\frac{1}{\sqrt{2}}\left(e^{p\mu}\omega_{\mu pq}-2e_{q}{}^{\mu}\partial_{\mu}\phi\right)\,,\quad&\quad \brPhi^{\brp}{}_{\brp\brq}=\frac{1}{\sqrt{2}}\left(\bre^{\brp\mu}\bromega_{\mu\brp\brq}-2\bre_{\brq}{}^{\mu}\partial_{\mu}\phi\right)\,, \end{array} \](2.82)

where we have a pair of undoubled spin connections,

\[ \begin{array}{ll} \omega_{\mu pq}=e_{p}{}^{\nu}(\partial_{\mu}e_{\nu q}-\gamma_{\mu}^{\lambda}{}_{\nu}e_{\lambda q})\,,\qquad&\quad\bromega_{\mu \brp\brq}=\bre_{\brp}{}^{\nu}(\partial_{\mu}\bre_{\nu\brq}-\gamma_{\mu}^{\lambda}{}_{\nu}\bre_{\lambda\brq})\,, \end{array} \](2.83)

which along with the Christoffel connection \(\gamma^{\lambda}_{\mu\nu}\) imply an undoubled master derivative:

\[ \begin{array}{llllll} \trd_{\mu}:=\partial_{\mu}+\gamma_{\mu}+\omega_{\mu}+\bromega_{\mu}\,,&~ \trd_{\mu} e_{\nu}{}^{p}=0\,,&~ \trd_{\mu}\eta_{pq}=0\,,&~ \trd_{\mu}\bre_{\nu}{}^{\brq}=0,&~ \trd_{\mu}\breta_{\brp\brq}=0\,,&~ \trd_{\lambda} g_{\mu\nu}=0\,. \end{array} \](2.84)

For example, from [Jeon:2011vx] we obtain for a \(\SpinD\) tensor,

\[ \cD_{\brp}T_{q_1q_2\cdots q_{n}}= \frac{1}{\sqrt{2}}\bre_{\brp}{}^{\mu}\left[\partial_{\mu}T_{q_1q_2\cdots q_{n}}+\sum_{i=1}^{n}~(\omega_{\mu q_{i}}{}^{r}+\half H_{\mu q_{i}}{}^{r}) \,T_{q_{1}\cdots q_{i-1}rq_{i+1}\cdots q_{n}}\right]\,, \](2.85)

and for a weightless spinor, we produce known expressions from generalised geometry [Coimbra:2011nw],

\[ \begin{array}{ll} \cD_{\brp}\rho=\frac{1}{\sqrt{2}}\!\left(\partial_{\brp} \rho{\, + \frac{1}{4}} \omega_{\brp q r} \gamma^{q r} \rho {\,+ \frac{1}{8}}H_{\brp q r} \gamma^{q r} \rho \right), ~& \gamma^{p}\cD_{p}\rho=\frac{1}{\sqrt{2}} \gamma^{\mu}\!\left( \partial_{\mu} \rho{\, + \frac{1}{4}} \omega_{\mu qr} \gamma^{qr} \rho {\,+ \frac{1}{24}} H_{\mu qr} \gamma^{qr} \rho - \partial_{\mu} \phi \rho \right). \end{array} \](2.86)

Furthermore, choosing the gauge \(e_{\mu}{}^{p} \equiv \bre_{\mu}{}^{\brp}\) reduces the twofold spin groups to a diagonal subgroup, which can be identified with the single spin group of GR. This gauge condition forces the spinors and the RR fields to transform under \(\ODD\) rotations [Jeon:2012hp], and it allows the single RR potential field to be expanded using gamma matrices:

\[ \cC^{\alpha}{}_{\beta}=\sum_{p}~\frac{1}{p!}C_{\mu_{1}\mu_2\cdots\mu_{p}}(\gamma^{\mu_1\mu_2\cdots\mu_p})^{\alpha}{}_{\beta}\,, \](2.87)

such that the conventional (even or odd) RR form fields appear, and the pair of nilpotent differential operators (2.78) reduce to a twisted exterior derivative and its Hodge dual [Jeon:2012kd]:

\[ \begin{array}{ll} \cD_{+}\quad\!\longrightarrow\!\quad\rd+(H-\rd\phi)\,\wedge~\,,\qquad&\qquad \cD_{-}\quad\!\longrightarrow\!\quad\star\,\big[\,\rd+(H-\rd\phi)\,\wedge~\big]\star~\,. \end{array} \](2.88)

For the non-Riemannian parametrisations of the fully covariant derivatives (2.65) and doubled Yang-Mills theory (2.70), we refer to [Cho:2019ofr] (section 4.3 therein) and [Angus:2021jvm], respectively.