Gravitational Core of Double Field Theory

Einstein Double Field Equation

Lecture Notes
Jeong-Hyuck Park

This chapter formulates the Einstein Double Field Equation as a unified field equation for DFT matter couplings and the closed-string gravitational sector.

arXiv

Einstein Double Field Equation: Unified Field Equation

Aim. To couple matter fields to DFT consistently and derive the unified field equation.

By employing the fully covariant derivatives (2.65), (2.66), (2.73), (2.77), and (2.79), it is possible to couple the pure DFT action (2.110) to various forms of matter,

\[ S_{\DFT-\Matter}=\int_{\Sigma_{D}}~e^{-2d}\So\,+\,\cL_{\Matter}(\Psi,\cD\Psi;d,V,\brV)\,, \](2.141)

where \(\cL_{\Matter}(\Psi,\cD\Psi;d,V,\brV)\) represents a Lagrangian density of unit weight for generic matter fields \(\Psi\) which are minimally coupled to the gravitational sector \(\{d,V_{Ap},\brV_{A\brp}\}\).

Generalising the case of pure DFT (2.111), while ignoring any surface integral and assuming that \(\cL_{\Matter}\) is local Lorentz invariant (2.25), the infinitesimal variation of the full action (2.141) is given by [Angus:2018mep]:

\[ \delta S_{\DFT-\Matter}=\int_{\Sigma_{D}}\, e^{-2d}\Big[4\brV_{A}{}^{\brq}\delta V^{Ap}\big(S_{p\brq}-K_{p\brq}\big)-2\delta d\big(\So-\To\big)\Big]+\delta\Psi\frac{\delta \cL_{\Matter}}{\delta\Psi}\,. \](2.142)

Here \(\frac{\delta \cL_{\Matter}}{\delta\Psi}\) denotes the Euler-Lagrange equations for the matter field. Naturally, we are led to define

\[ \begin{array}{ll} \dis{K_{p\brq}=\frac{1}{4}e^{2d}\left(V_{Ap}\frac{\delta \cL_{\Matter}}{\delta\brV_{A}{}^{\brq}}-\brV_{A\brq}\frac{\delta \cL_{\Matter}}{\delta V_{A}{}^{p}}\right)\,,}\qquad&\quad \dis{\To=\frac{1}{2}e^{2d\,}\frac{\delta \cL_{\Matter}}{\delta d}\,,} \end{array} \](2.143)

which constitute the on-shell conserved, \(\ODD\)-symmetric energy-momentum tensor in DFT:

\[ \begin{array}{ll} T_{AB}=4V_{[A}{}^{p}\brV_{B]}{}^{\brq}K_{p\brq}-\half\cJ_{AB}\To\,,\qquad&\quad\na_{A}T^{AB}=0\,, \end{array} \](2.144)

where the conservation condition requires the Euler-Lagrange equations for the matter field. Notably, when matter couples directly to the generalised metric rather than the vielbeins, we simply have

\[ K_{p\brq}=-e^{2d}\frac{\delta \cL_{\Matter}}{\delta\cH_{AB}}V_{Ap}\brV_{B\brq}\,. \](2.145)

The unified field equation of DFT is obtained by equating the Einstein curvature (2.115) with the energy-momentum tensor (2.144):

\[ G_{AB} = T_{AB}\,, \](2.146)

which was dubbed the Einstein Double Field Equation (EDFE) [Angus:2018mep].

On a \((0,0)\) Riemannian background, the energy-momentum tensor can be parametrised as

\[ T_{AB}=\left(\begin{array}{cc} -K^{[\mu\nu]}& K^{(\mu\lambda)}g_{\lambda\sigma} +K^{[\mu\lambda]}B_{\lambda\sigma}-\half\delta^{\mu}{}_{\sigma}\To\\ \!\!-g_{\rho\kappa}K^{(\kappa\nu)}-B_{\rho\kappa}K^{[\kappa\nu]}-\half\delta_{\rho}{}^{\nu}\To&\quad K_{[\rho\sigma]}+B_{\rho\kappa}K^{[\kappa\lambda]}B_{\lambda\sigma}+B_{\rho}{}^{\kappa}K_{(\kappa\sigma)}+K_{(\rho\lambda)}B^{\lambda}{}_{\sigma} \end{array}\right)\,, \](2.147)

and the EDFE yields the following three sets of equations:

\[ \begin{array}{rll} R_{\mu\nu}+2\trd_{\mu}(\partial_{\nu}\phi)-\quarter H_{\mu\rho\sigma}H_{\nu}{}^{\rho\sigma} &\!=&\! K_{(\mu\nu)}\,,\\ \half e^{2\phi}\trd^{\rho}\!\left(e^{-2\phi}H_{\rho\mu\nu}\right)&\!=&\! K_{[\mu\nu]}\,,\\ R+4\Box\phi-4\partial_{\mu}\phi\partial^{\mu}\phi-\textstyle{\frac{1}{12}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}&\!=&\!\To\,. \end{array} \](2.148)

Each equation can be understood as the equation of motion of the full action \(S_{\DFT-\Matter}\) for \(g_{\mu\nu}\), \(B_{\mu\nu}\), and \(d\), respectively (rather than \(g,B,\phi\)). In view of \(d=\phi-\frac{1}{2}\ln\sqrt{-g}\) (2.39), the traditional energy-momentum tensor in GR is given by \(K_{(\mu\nu)}\) and \(\To\) [Lee:2023boi]:

\[ T_{\mu\nu}^{\scriptscriptstyle{\mathbf{GR}}}=e^{-2\phi}\left[K_{(\mu\nu)}-\frac{1}{2}g_{\mu\nu}\To\right]\,. \](2.149)

The on-shell conservation (2.144) reduces to

\[ \begin{array}{ll} \trd^{\mu}K_{(\mu\nu)}-2\partial^{\mu}\phi\,K_{(\mu\nu)}+\half H_{\nu}{}^{\lambda\mu}K_{[\lambda\mu]}-\half\partial_{\nu}\To=0\,,\quad&\qquad \trd_{\mu}\!\left(e^{-2\phi}K^{[\mu\nu]}\right)=0\,, \end{array} \](2.150)

which are consistent with the geometric counterparts (2.120). In particular, the middle equation in (2.148) and the latter in (2.150) represent the stringy two-form generalisations of Maxwell’s equations (1.2) and the corresponding conservation laws: \(\trd_{\lambda}F^{\lambda\mu}=J^{\mu}\) and \(\trd_{\mu}J^{\mu}=0\). Strings, unlike point particles, couple to the two-form \(B\)-field rather than the one-form Maxwell vector potential.

It is worth noting that, after subtracting the third equation from the trace of the first in (2.148), we can derive a Klein-Gordon-type equation [Choi:2022srv]:

\[ \Box\big(e^{-2\phi}\big)=\big(K_{\mu}{}^{\mu}-\To+\textstyle{\frac{1}{6}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}\big)e^{-2\phi}\,, \](2.151)

where the quantity inside the parentheses on the right-hand side of the equality can be interpreted as the effective or “Chameleon mass” [Khoury:2003aq, Brax:2004qh] of the dilaton scalar field, \(e^{-2\phi}\). This interpretation links the dilaton dynamics to a scalar field with an effective mass that varies based on other fields and their interactions, similar to scalar-tensor theories where the scalar field's mass adjusts to its environment.

\(\ODD\) Tells Matter How to Couple to DFT: Equivalence Principle Holds in String Frame

The EDFE (2.146) provides an \(\ODD\)-symmetric extension of Wheeler's famous insight into gravity: “Matter tells spacetime how to curve.” The complementary notion, “Spacetime tells matter how to move,” is realised through the \(\ODD\)-symmetric minimal coupling of the gravitational fields \(\{V_{Ap}, \brV_{B\brq}, d\}\) to matter, as demonstrated in (2.14), (2.15), (2.70), (2.85), (2.86), and (2.141). Consequently, the coupling of the Riemannian trio \(\{g_{\mu\nu}, B_{\mu\nu}, \phi\}\) to the Standard Model of particle physics is fully dictated by the \(\ODD\) symmetry principle, ensuring the emergence of a fixed structure that would not otherwise arise [Choi:2015bga]. Schematically, from (2.70) and (2.86), the action for a scalar field \(\Phi\), electromagnetic fields \(A_{\lambda}, F_{\mu\nu}\), and a spinorial fermion \(\psi\) with diffeomorphic weight \(\omega = \frac{1}{2}\) is given by:

\[ \dis{\int_{\Sigma_D}\sqrt{-g}e^{-2\phi}\left(-g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-{\frac{1}{4\alpha}}F_{\mu\nu}F^{\mu\nu}\right)+\bar{\psi}\gamma^{\lambda}\left( \trd_{\lambda}\psi-iA_{\lambda}\psi + {\frac{1}{24}} H_{\lambda\mu\nu} \gamma^{\mu\nu}\psi \right)}\,. \](2.152)

This indicates that bosons couple to the string dilaton, while fermions couple to the \(H\)-flux. As a result, they contribute distinct components to the energy-momentum tensor: bosons generate \(\To\), and fermions generate \(K_{[\mu\nu]}\), alongside the shared components \(K_{(\mu\nu)}\). With additional components, the gravitational physics in DFT is inherently richer than in General Relativity: \(D^2+1\) vs. \(\frac{1}{2}D(D+1)\) (off-shell) degrees of freedom.

However, after integrating out the auxiliary potential \(\fa^{A}\), the \(\ODD\)-symmetric doubled particle action (2.14) reduces, on a Riemannian background (2.33), to the standard (undoubled) relativistic point-particle action minimally coupled solely to the string frame metric \(g_{\mu\nu}\). Since the \(\ODD\) singlet dilaton \(d\), or \(e^{-2d}\), carries a nontrivial diffeomorphism weight, it cannot couple to the diffeomorphism-invariant doubled particle action (2.14). Consequently, the free-falling motion of particles along geodesics - free from any fifth force - preserves the equivalence principle, not in Einstein frame but in string frame [Ko:2016dxa]. This result aligns with string theory's foundational premise that matter consists of vibrating tiny strings, which naturally couple to the string frame metric.

Importantly, in the string frame, the kinetic term of the string dilaton has an opposite sign, with the square of its time derivative carrying a negative coefficient (2.118). This property enables the formation of wormholes and drives the accelerated expansion of the Universe, as discussed below.