Gravitational Core of Double Field Theory

Introduction

Lecture Notes
Jeong-Hyuck Park

This chapter motivates Double Field Theory by showing how O(D,D) symmetry reorganizes the closed-string massless sector, in analogy with the role of Lorentz symmetry in electrodynamics.

arXiv

Introduction

In electrodynamics, the electric field is conventionally represented by the capital letter \(\mathbf{E}\) for obvious reasons. However, the magnetic field is represented by either \(\mathbf{B}\) or \(\mathbf{H}\) rather than \(\mathbf{M}\). This convention arises because Maxwell did not use vector notations when formulating his equations in 1861. Instead, Maxwell used the letters \(\{E, F, G\}\) to denote components of the electric field and neighboring letters, such as \(\{B, C, D\}\) or \(\{H, I, J\}\), for the magnetic field. It was Heaviside-or perhaps the rotational \(\SO(3)\) symmetry-that reformulated Maxwell's equations into their modern vectorial form:

\[ \begin{array}{llll} \dis{\na\cdot\mathbf{E}=\rho\,,}\quad&\quad\dis{\na\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\,,}\quad&\quad\dis{ \na\cdot\mathbf{B}=0\,,}\quad&\quad\dis{ \na\times\mathbf{B}=\mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}\,.} \end{array} \](1.1)

Minkowski, or the \(\SO(1,3)\) Lorentz symmetry, then introduced further simplifications in 1908, unifying space and time into a four-dimensional spacetime framework and recasting Maxwell's equations in a more compact and elegant form:

\[ \begin{array}{ll} \partial_{\lambda}F^{\lambda\mu}=J^{\mu}\,,\qquad&\qquad \partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}=0\,. \end{array} \](1.2)

In string theory, a similar unification has been accomplished through the \(\ODD\) symmetry principle. The vanishing of the three beta-functions on a closed string worldsheet,

\[ \begin{array}{r} R_{\mu\nu}+2\trd_{\mu}(\partial_{\nu}\phi)-\quarter H_{\mu\rho\sigma}H_{\nu}{}^{\rho\sigma} =0\,,\\ \half e^{2\phi}\trd^{\rho}\!\left(e^{-2\phi}H_{\rho\mu\nu}\right)=0\,,\\ R+4\Box\phi-4\partial_{\mu}\phi\partial^{\mu}\phi-\textstyle{\frac{1}{12}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}=0\,, \end{array} \](1.3)

is unified into a single formula characterized by the vanishing of the \(\ODD\)-symmetric augmentation of the Einstein curvature tensor [Park:2015bza],

\[ {G}_{AB}=0\,, \](1.4)

which corresponds to the vacuum case of the more general unified field equation in Double Field Theory (DFT), dubbed the Einstein Double Field Equation (EDFE) [Angus:2018mep]:

\[ G_{AB}=T_{AB}\,. \](1.5)

Hereafter, the capital letters \(A, B, \cdots\) denote \(\ODD\) vector indices, which span the doubled spacetime dimensions, \(D+D\). As reviewed below, the DFT Einstein curvature, \(G_{AB}\), on the left-hand side of the equation, can be - but not necessarily - constructed from the trio \(\{g_{\mu\nu},B_{\mu\nu},\phi\}\). This trio represents the traditional closed string massless sector, known as the Neveu-Schwarz Neveu-Schwarz (NSNS) sector, and constitutes the stringy gravitational degrees of freedom. On the right-hand side, the DFT energy-momentum tensor, \(T_{AB}\), accounts for other sectors (or “matter”), including the Ramond-Ramond (RR) sector and NSR fermions. Like General Relativity, these quantities are governed by conservation laws derived from the action principle:

\[ \begin{array}{ll} \nabla_{A}G^{A}{}_{B}=0~~~:~~~\mbox{off-shell}\,,\qquad&\qquad \nabla_{A}T^{A}{}_{B}=0~~~:~~~\mbox{on-shell}\,. \end{array} \](1.6)

This conservation principle highlights the fundamental symmetries and consistency of the theory, mirroring General Relativity (GR). With a greater number of components in the energy-momentum tensor, the gravitational physics in DFT becomes inherently richer compared to GR.