Gravitational Core of Double Field Theory

Non-Riemannian Geometry

Lecture Notes
Jeong-Hyuck Park

This chapter explains why the fundamental field variables of DFT need not always be parametrized by an invertible spacetime metric and two-form.

arXiv

Fundamental Field Variables: Non-Riemannian Geometry

Aim. To define the generalised metric and dilaton, introduce projection operators and vielbeins, and set up the non-Riemannian sector.

The fundamental fields of DFT are the generalised metric, \(\cH_{AB}\), and the dilaton, \(d\), which together define the geometric and gravitational structure of the theory. These fields encapsulate the stringy nature of spacetime geometry and play a central role in the formulation of DFT.

By definition, the generalised metric \(\cH_{AB}\) is a symmetric \(\ODD\) element:

\[ \begin{array}{ll} \cH_{AB}=\cH_{BA}\,,\qquad&\qquad\cH_{A}{}^{C}\cH_{B}{}^{D}\cJ_{CD}=\cJ_{AB}\,, \end{array} \](2.16)

and the exponentiated form of the dilaton, \(e^{-2d}\), serves as the integral measure in DFT: it is a scalar density of unit weight. In words, the generalised metric squares to the identity: \(\cH_{A}{}^{B}\cH_{B}{}^{C}=\delta_{A}{}^{C}\). Their generalised Lie derivatives (2.5) are explicitly given by

\[ \begin{array}{ll} \hcL_{\xi}\cH_{AB}=\xi^{C}\partial_{C}\cH_{AB}+2\partial_{[A}\xi_{C]}\cH^{C}{}_{B}+2\partial_{[B}\xi_{C]}\cH_{A}{}^{C}\,,~~~\,&~~~\hcL_{\xi}d=-\frac{1}{2}e^{2d}\hcL_{\xi}\left(e^{-2d}\right)=\xi^{A}\partial_{A}d-\frac{1}{2}\partial_{A}\xi^{A}\,. \end{array} \](2.17)

Combining the invariant metric \(\cJ_{AB}\) and the generalised metric \(\cH_{AB}\), we define symmetric projection matrices [Jeon:2010rw]:

\[ \begin{array}{ll} P_{AB}=P_{BA}=\half(\cJ_{AB}+\cH_{AB})\,,\quad&\quad P_{A}{}^{B}P_{B}{}^{C}=P_{A}{}^{C}\,,\\ \brP_{AB}=\brP_{BA}=\half(\cJ_{AB}-\cH_{AB})\,,\quad&\quad \brP_{A}{}^{B}\brP_{B}{}^{C}=\brP_{A}{}^{C}\,, \end{array} \](2.18)

which are orthogonal and complete:

\[ \begin{array}{ll} P_{A}{}^{B}\brP_{B}{}^{C}=0\,,\qquad&\qquad P_{A}{}^{B}+\brP_{A}{}^{B}=\delta_{A}{}^{B}\,. \end{array} \](2.19)

Taking the "square roots" of the projection matrices, we introduce a pair of vielbeins, \(V_{Ap}\) and \(\brV_{B\brq }\):

\[ \begin{array}{ll} P_{AB}=V_{A}{}^{p}V_{B}{}^{q}\eta_{pq}\,,\qquad&\qquad \brP_{AB}=\brV_{A}{}^{\brp}\brV_{B}{}^{\brq}\breta_{\brp\brq}\,. \end{array} \](2.20)

The vielbeins satisfy the following defining property:

\[ V_{A}{}^{p}V_{B}{}^{q}\eta_{pq}+\brV_{A}{}^{\brp}\brV_{B}{}^{\brq}\breta_{\brp\brq}=\cJ_{AB}\,. \](2.21)

Essentially, viewed as a \(({D+D}) \times ({D+D})\) matrix, the pair \(\big(V_{A}{}^{p}, \brV_{A}{}^{\brp}\big)\) simultaneously diagonalises \(\cJ_{AB}\) and \(\cH_{AB}\) into \(\diag(\eta, +\breta)\) and \(\diag(\eta, -\breta)\), respectively. Furthermore, since the left inverse coincides with the right inverse, this property (2.21) is equivalent to:

\[ \begin{array}{lll} V_{Ap}V^{A}{}_{q}=\eta_{pq}\,,\qquad&\qquad \brV_{A\brp}\brV^{A}{}_{\brq}=\breta_{\brp\brq}\,,\qquad&\qquad V_{Ap}\brV^{A}{}_{\brq}=0\,. \end{array} \](2.22)

It follows that

\[ \begin{array}{llll} P_{A}{}^{B}V_{Bp}=V_{Ap}\,,\qquad&\quad \brP_{A}{}^{B}\brV_{B\brp}=\brV_{A\brp}\,,\qquad&\quad P_{A}{}^{B}\brV_{B\brp}=0=\brP_{A}{}^{B}V_{Bp}\,. \end{array} \](2.23)

The presence of twofold vielbeins, \(V_{A}{}^{p}\) and \(\brV_{A}{}^{\brp}\), and spin groups, \(\SpinD \times \oSpinD\), is a distinctly string-theoretic feature. This reflects the existence of two separate locally inertial frames: one associated with the left-moving sector, and the other associated with the right-moving sector of closed strings [Duff:1986ne]. Naturally, there are also two types of spinorial fermions, as seen in supersymmetric DFTs [Jeon:2011sq, Jeon:2012hp] and as conjectured in the Standard Model of particle physics coupled to DFT [Choi:2015bga]. This dual structure could offer testable predictions for DFT and string theory. Moreover, it facilitates the unification of type IIA and IIB supergravities in the \({D=10}\) maximally supersymmetric DFT [Jeon:2012hp] that has been explicitly constructed to the complete, quartic order in fermions.

The generalised metric and the vielbeins are constrained by the properties (2.16), (2.21), and (2.22). Consequently, their infinitesimal variations satisfy the following relations: (Footnote: Observe the implicit index contractions in \((P\delta\cH\brP)_{AB}\), where the expression expands as \(P_{A}{}^{C}\delta\cH_{CD}\brP^{D}{}_{B}\).)

\[ \delta\cH_{AB}=(P\delta\cH\brP)_{AB}+(\brP\delta\cH P)_{AB}\,, \](2.24)

and

\[ \begin{array}{ll} \delta V_{Ap}=\brV_{A\brq}\brV^{B\brq}\delta V_{Bp}+V_{A}{}^{q}V_{B[q}\delta V^{B}{}_{p]}\,,\quad&\quad \delta \brV_{A\brp}=V_{Aq}V^{Bq}\delta \brV_{B\brp}+\brV_{A}{}^{\brq}\brV_{B[\brq}\delta \brV^{B}{}_{\brp]}\,, \end{array} \](2.25)

where the second terms on the right-hand sides correspond to local Lorentz rotations.

Parametrisations: Riemannian vs. non-Riemannian

Aim. To present explicit parametrisations of the generalised metric in both Riemannian and non-Riemannian forms.

DFT and its supersymmetric extensions offer a unified framework to describe stringy spacetime dynamics via the fundamental fields, \(\{\cH_{AB},d\}\) and \(\{V_{Ap},\brV_{B\brq},d\}\). These fields satisfy key defining relations, such as (2.16) and (2.21), and allow for the classification of DFT geometries through the introduction of two non-negative integers, \((n,\brn)\), whose sum is bounded: \(0\leq n+\brn\leq D\).

With \(1\leq i,j\leq n\) and \(1\leq \bri,\brj\leq\brn\), the DFT metric satisfying (2.16) takes the most general form [Morand:2017fnv],

\[ \cH_{AB}=\left(\begin{array}{cc}\cH^{\mu\nu}& -\cH^{\mu\sigma}B_{\sigma\lambda}+Y_{i}^{\mu}X^{i}_{\lambda}- \brY_{\bri}^{\mu}\brX^{\bri}_{\lambda}\\ B_{\kappa\rho}\cH^{\rho\nu}+X^{i}_{\kappa}Y_{i}^{\nu} -\brX^{\bri}_{\kappa}\brY_{\bri}^{\nu}\quad&~~ ~~\cK_{\kappa\lambda}-B_{\kappa\rho}\cH^{\rho\sigma}B_{\sigma\lambda} +2X^{i}_{(\kappa}B_{\lambda)\rho}Y_{i}^{\rho} -2\brX^{\bri}_{(\kappa}B_{\lambda)\rho}\brY_{\bri}^{\rho} \end{array}\right)\,. \](2.26)

Here, \(\cH\) and \(\cK\) are symmetric, \(B\) is skew-symmetric,

\[ \begin{array}{lll} \cH^{\mu\nu}=\cH^{\nu\mu}\,,\quad&\qquad \cK_{\mu\nu}=\cK_{\nu\mu}\,,\quad&\qquad B_{\mu\nu}=-B_{\nu\mu}\,; \end{array} \](2.27)

\(\cH\) and \(\cK\) admit kernels,

\[ \begin{array}{ll} \cH^{\mu\nu}X^{i}_{\nu}=0=\cH^{\mu\nu}\brX^{\bri}_{\nu}\,,\quad&\qquad \cK_{\mu\nu}Y_{j}^{\nu}=0=\cK_{\mu\nu}\brY_{\brj}^{\nu}\,; \end{array} \](2.28)

and a completeness relation must hold,

\[ \cH^{\mu\rho}\cK_{\rho\nu} +Y_{i}^{\mu}X^{i}_{\nu}+\brY_{\bri}^{\mu}\brX^{\bri}_{\nu} =\delta^{\mu}{}_{\nu}\,. \](2.29)

The linear independence of the kernel’s zero-eigenvectors implies that

\[ \begin{array}{lllll} Y^{\mu}_{i}X_{\mu}^{j}=\delta_{i}{}^{j}\,,&\,\, \brY^{\mu}_{\bri}\brX_{\mu}^{\brj}=\delta_{\bri}{}^{\brj}\,,&\,\, Y^{\mu}_{i}\brX_{\mu}^{\brj}=0= \brY^{\mu}_{\bri}X_{\mu}^{j}\,,&\,\, \cH^{\rho\mu}\cK_{\mu\nu}\cH^{\nu\sigma}=\cH^{\rho\sigma}\,,&\,\, \cK_{\rho\mu}\cH^{\mu\nu}\cK_{\nu\sigma}=\cK_{\rho\sigma}\,, \end{array} \](2.30)

and hence the \(\ODD\)-invariant trace of the DFT metric amounts to \(\cH_{A}{}^{A}=2(n-\brn)\).

The precise expression of the \((n,\brn)\) DFT metric (2.26) and the fundamental algebraic relations (2.28), (2.29) are all invariant under \(\mathbf{GL}(n)\times\mathbf{GL}(\brn)\) local rotations,

\[ \begin{array}{ccc} \left(X^{i}_{\mu}\,,\,Y_{i}^{\mu}\,,\,\brX^{\bri}_{\mu}\,,\,\brY_{\bri}^{\nu}\right)\quad&\longmapsto&\quad \left(X^{j}_{\mu}\,R_{j}{}^{i}\,,\,R^{-1}{}_{i}{}^{j}\,Y_{j}^{\nu}\,,\,\brX^{\brj}_{\mu}\,\brR_{\brj}{}^{\bri}\,,\,\brR^{-1}{}_{\bri}{}^{\brj}\,\brY_{\brj}^{\nu}\right)\,, \end{array} \](2.31)

and, with arbitrary local parameters, \(V_{\mu i},\brV_{\mu\bri}\), under the following transformations,

\[ \begin{array}{cll} Y_{i}^{\mu}~~&\longmapsto&~~ Y_{i}^{\mu}+\cH^{\mu\nu}V_{\nu i}\,,\\ \brY_{\bri}^{\mu}~~&\longmapsto&~~\brY_{\bri}^{\mu}+\cH^{\mu\nu}\brV_{\nu\bri}\,,\\ \cK_{\mu\nu}~~&\longmapsto&~~ \cK_{\mu\nu}-2X^{i}_{(\mu}\cK_{\nu)\rho}\cH^{\rho\sigma}V_{\sigma i}-2\brX^{\bri}_{(\mu}\cK_{\nu)\rho}\cH^{\rho\sigma}\brV_{\sigma\bri}+(X_{\mu}^{i}V_{\rho i}+\brX_{\mu}^{\bri}\brV_{\rho\bri})\cH^{\rho\sigma}(X_{\nu}^{j}V_{\sigma j}+\brX_{\nu}^{\brj}\brV_{\sigma\brj})\,,\\ B_{\mu\nu}~~&\longmapsto&~~ B_{\mu\nu} -2X^{i}_{[\mu}V_{\nu]i}+2\brX^{\bri}_{[\mu}\brV_{\nu]\bri} +2X^{i}_{[\mu}\brX^{\bri}_{\nu]}\left(Y_{i}^{\rho}\brV_{\rho\bri} +\brY_{\bri}^{\rho}V_{\rho i}+V_{\rho i}\cH^{\rho\sigma}\brV_{\sigma\bri}\right)\,. \end{array} \](2.32)

In fact, these two symmetries (2.31), (2.32) are identifiable as the twofold local Lorentz symmetries.

On the one hand, in the case \((n,\brn)=(0,0)\), \(\cK_{\mu\nu}\) and \(\cH^{\mu\nu}\) coincide with the usual (invertible) Riemannian metric and its inverse. In this Riemannian limit, the \((0,0)\) DFT metric takes the familiar form [Giveon:1988tt],

\[ \cH_{AB}=\left(\begin{array}{cc} \quad g^{\mu\nu}\quad&\quad -g^{\mu\lambda}B_{\lambda\tau}\quad\\ \quad B_{\sigma\kappa}g^{\kappa\nu}\quad&\quad g_{\sigma\tau}-B_{\sigma\kappa}g^{\kappa\lambda}B_{\lambda\tau}\quad \end{array}\right)\,. \](2.33)

The \(\ODD\)-symmetric proper length (2.13), as well as the doubled particle and string actions (2.14), (2.15), all reduce to their conventional (undoubled) forms after integrating out the auxiliary variables \(\fa^{A}\).

On the other hand, cases with \((n,\brn) \neq (0,0)\) are inherently non-Riemannian, as they lack an invertible Riemannian metric. Specifically, in the cases \((n,\brn) = (D,0)\) or \((0,D)\), \(\cK_{\mu\nu}\) and \(\cH^{\mu\nu}\) vanish, the relations \(Y_{i}^{\mu}X^{i}_{\nu}=\delta^{\mu}{}_{\nu}\) or \(\brY_{\bri}^{\mu}\brX^{\bri}_{\nu}=\delta^{\mu}{}_{\nu}\) hold respectively, and the \(B\)-field becomes irrelevant. In these extremely non-Riemannian limits, the DFT metric coincides with the \(\ODD\)-invariant metric up to sign,

\[ \cH_{AB}=\pm\cJ_{AB}=\left(\begin{array}{cc} ~ 0\quad&\quad\pm 1~\\ ~ \pm 1\quad&\quad0~ \end{array}\right)\,. \](2.34)

These cases represent two perfectly \(\ODD\)-symmetric vacua in bosonic DFT that are maximally non-Riemannian, characterized by the saturation, \({n+\brn=D}\). Remarkably, these vacua exhibit infinitely many isometries [Blair:2020gng], as any local parameter \(\xi^{A}\) automatically satisfies the Killing equation, defined by the condition \({\hcL_{\xi}\cH_{AB}=0}\).

In this scenario, the ordinary Riemannian spacetime (2.33) emerges after spontaneous symmetry breaking of the fully \(\ODD\)-symmetric vacua (2.34), while the Riemannian metric and the \(B\)-field are identified as Nambu-Goldstone bosons [Berman:2019izh, Park:2020ixf].

On a generic \((n, \brn)\) non-Riemannian background, a particle described by the doubled action (2.14) freezes in \(n+\brn\) untilde directions,

\[ \begin{array}{ll} \dis{X^{i}_{\mu}\frac{\rd x^{\mu}}{\rd\tau}=0}\,,\qquad&\qquad \dis{\brX^{\bri}_{\mu}\frac{\rd x^{\mu}}{\rd\tau}=0}\,, \end{array} \](2.35)

while the string in (2.15) becomes chiral and anti-chiral in the \(n\) and \(\brn\) untilde directions, respectively:

\[ \begin{array}{ll} X^{i}_{\mu}\left(\partial_{\alpha}x^{\mu}+\textstyle{\frac{1}{\sqrt{-h}}}\epsilon_{\alpha}{}^{\beta}\partial_{\beta}x^{\mu}\right)=0\,, \qquad&~~\quad \brX^{\bri}_{\mu}\left(\partial_{\alpha}x^{\mu}-\textstyle{\frac{1}{\sqrt{-h}}}\epsilon_{\alpha}{}^{\beta}\partial_{\beta}x^{\mu}\right)=0\,. \end{array} \](2.36)

This behaviour arises as the components of the auxiliary gauge potential \(\fa^{A}\) act as Lagrange multipliers [Morand:2017fnv].

Nonetheless, the \((n,\brn)\)-classification of the generalised metric (2.26) described above remains valid for bosonic DFT geometries. In the presence of fermions, particularly in supersymmetric DFTs [Jeon:2011sq, Jeon:2012hp], the twofold spin groups are a priori fixed, and the permissible values of \((n,\brn)\) are constrained by their dimensions and signatures. Specifically, the Euclidean spin group, \(\Spin(D)\times\Spin(D)\), does not permit any non-Riemannian geometries, whereas the Minkowskian case, \(\Spin(1,{D-1})\times\Spin({D-1},1)\), as assumed in Table 1, allows \((1,1)\) non-Riemannian geometry [Lee:2013hma, Ko:2015rha], which corresponds to non-relativistic [Gomis:2000bd, Danielsson:2000gi, Gomis:2005pg] or Newton-Cartan string theories [Christensen:2013lma, Hartong:2015zia, Harmark:2017rpg, Harmark:2018cdl, Bergshoeff:2018yvt, Bergshoeff:2019pij, Harmark:2019upf, Bergshoeff:2021bmc, Oling:2022fft, Hartong:2022lsy]. Furthermore, certain known singularities in supergravity theories can be appropriately identified as regular \((1,1)\) non-Riemannian geometries [Morand:2021xeq]; see section Spherically Symmetric Solution: Generalisation of the Schwarzschild Geometry for an example.

While we refer to [Morand:2017fnv] for the general \((n,\brn)\) cases, here we merely spell out the \((0,0)\) Riemannian parametrisation of the DFT vielbeins,

\[ \begin{array}{ll} V_{Ap}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}e_{p}{}^{\mu}\\ e_{\nu}{}^{q}\eta_{qp}+B_{\nu\sigma}e_{p}{}^{\sigma}\end{array}\right)\,,\quad&\quad \brV_{A\brp}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}\bre_{\brp}{}^{\mu}\\ \bre_{\nu}{}^{\brq}\breta_{\brq\brp}+B_{\nu\sigma}\bre_{\brp}{}^{\sigma}\end{array}\right)\,. \end{array} \](2.37)

Here \(e_{\mu}{}^{p}\) and \(\bre_{\mu}{}^{\brp}\) are a pair of (Riemannian) vielbeins which square to the same metric,

\[ e_{\mu}{}^{p}e_{\nu p}=-\bre_{\mu}{}^{\brp}\bre_{\nu\brp}=g_{\mu\nu}\,. \](2.38)

Their dual presence allows \(\ODD\) to safely rotate the capital-letter indices of the vielbeins exclusively [Jeon:2011cn].

Lastly, the DFT dilaton is parametrised by the Riemannian metric and the (weightless) string dilaton:

\[ e^{-2d}=\sqrt{-g}\,e^{-2\phi}\,. \](2.39)