Gauging spacetime inversions in quantum gravity

TL;DR

The paper argues that spacetime inversion symmetries, notably CRT, must be gauge symmetries in quantum gravity and develops a framework for their gauging. It shows that gauging CRT is naturally realized by summing over Euclidean spacetime topologies, implies a real Hilbert space for quantum gravity in closed universes, and necessitates including Lorentzian time-nonorientable manifolds to reproduce boundary duals in AdS/CFT. Through background-field constructions, explicit AdS/CFT examples, and discussions of chronology and spin structure, the authors derive concrete consequences for topology, causality, and fermions, while connecting to JT gravity and matrix ensembles. The work advances a coherent picture in which topology and symmetry gauging shape the nonperturbative structure of quantum gravity and holography, with implications for completeness, fractionalization, and the emergence of conventional quantum mechanics from a real-structure foundation.

Abstract

Spacetime inversion symmetries such as parity and time reversal play a central role in physics, but they are usually treated as global symmetries. In quantum gravity there are no global symmetries, so any spacetime inversion symmetries must be gauge symmetries. In particular this includes $\mathcal{CRT}$ symmetry (in even dimensions usually combined with a rotation to become $\mathcal{CPT}$), which in quantum field theory is always a symmetry and seems likely to be a symmetry of quantum gravity as well. In this article we discuss what it means to gauge a spacetime inversion symmetry, and we explain some of the more unusual consequences of doing this. In particular we argue that the gauging of $\mathcal{CRT}$ is automatically implemented by the sum over topologies in the Euclidean gravity path integral, that in a closed universe the Hilbert space of quantum gravity must be a real vector space, and that in Lorentzian signature manifolds which are not time-orientable must be included as valid configurations of the theory. In particular we give an example of an asymptotically-AdS time-unorientable geometry which must be included to reproduce computable results in the dual CFT.

Figures (22)

• Figure 1: Moving a charged operator $O$ around a closed path in the presence of a background gauge field for a discrete global symmetry. As $O$ crosses the symmetry insertion $U(g)$ it picks up a transformation matrix $D(g)$.
• Figure 2: Gauging an internal $\mathbb{Z}_2$ symmetry in $1+1$ dimensions. We need to sum over trivial holonomy, a $\mathbb{Z}_2$ holonomy around the temporal direction, a $\mathbb{Z}_2$ holonomy around the spatial direction, and $\mathbb{Z}_2$ holonomies around around both directions. The locations of the $U$ insertions implementing these holonomies are shown as red dashed lines, and for simplicity we have drawn a square torus with $\theta=0$. The first two terms give a projection onto singlets in the untwisted sector, while the last two terms give a projection onto singlets in the twisted sector.
• Figure 3: Deriving the Rindler representation of the ground state of a relativistic field theory.
• Figure 4: Background fields for a spatial rotation.
• Figure 5: Background fields for $\mathcal{R}$ symmetry in $1+1$ dimensions. Inserting $U_\mathcal{R}$ wrapping a spatial circle results in the partition function on the Klein bottle, while inserting $U_\mathcal{R}$ on a temporal circle results in the partition function on a spatial interval of length $L/2$ with Neumann boundary conditions at both ends. Inserting $U_\mathcal{R}$ on the temporal circle and $U_{\mathcal{R}'}$ on the spatial circle gives a Möbius strip.