Gravitational Core of Double Field Theory
Double Field Theory Minimum
Lecture Notes
Jeong-Hyuck Park
This chapter sets up the minimal language of DFT: doubled indices, O(D,D) symmetry, twofold spin groups, and the conventions needed for the geometric construction.
Double Field Theory Minimum
In this main section, we delve into the geometric foundations of Double Field Theory (DFT). After establishing the necessary notation, we introduce the concept of double-yet-gauged coordinates and the fundamental field variables, followed by the formulation of semi-covariant derivatives and curvatures. The notion of 'semi-covariance' acts as an intermediate step in the formalism, leading to fully covariant derivatives and curvatures. The overarching goal is to present the action principle and derive the unified field equation, the EDFE (1.5), along with the conservation law (1.6).
Symmetry & Notation
Aim. To establish index conventions and identify the symmetry group \(\ODD\) as well as the twofold Lorentz symmetries.
DFT can be viewed as a top-down extension of GR, guided by the \(\ODD\) symmetry principle, necessitating the introduction of notation for its fundamental symmetry groups: \(\ODD\) and \(\Spin(1, D-1) \times \Spin(D-1, 1)\). As summarised in Table 1, capital letters are used for \(\ODD\) vector indices, while small letters are reserved for the twofold local Lorentz symmetries. To distinguish between the indices of the two distinct spin groups, barred and unbarred notations are employed: unbarred indices correspond to \(\Spin(1, D-1)\), while barred indices correspond to \(\Spin(D-1, 1)\). These conventions prepare the ground for consistently formulating doubled geometry.
| Symmetry | Index | Metric for raising/lowering indices |
|---|---|---|
| \(\ODD\) | \(A,B,\cdots,M,N,\cdots\) | \(\cJ_{AB}=\begin{pmatrix}0&1\\1&0\end{pmatrix}\) |
| \(\SpinD\) | \(p,q,\cdots\) | \(\eta_{pq}=\mathrm{diag}(-++\cdots+)\) |
| \(\oSpinD\) | \(\brp,\brq,\cdots\) | \(\breta_{\brp\brq}=\mathrm{diag}(+--\cdots-)\) |
Vectorial indices and “metrics” for \(\ODD\), \(\Spin(1,D-1)\), and \(\Spin(D-1,1)\):
i) \(\cJ_{AB}\) denotes the invariant metric for the \(\ODD\) group.
ii) \(\eta_{pq}\) and \(\breta_{\brp\brq}\) are flat \(D\)-dimensional Minkowskian metrics associated with \(\Spin(1,D-1)\) and \(\Spin(D-1,1)\), respectively. These metrics exhibit opposite signatures.