Connection I — Diffeomorphism Gauging: DeTurck ↔ BMS
Infinite-Dimensional Gauge Symmetries
Both Perelman's Ricci flow programme and the BMS description of asymptotically flat spacetimes confront the same core difficulty: the dynamical variables carry redundancy generated by infinite-dimensional diffeomorphism groups. Extracting physics means quotienting or gauge-fixing these symmetries in a controlled way.
DeTurck Side: Riemannian Moduli Space
On a closed manifold $M$, the space of Riemannian metrics modulo diffeomorphisms defines the Riemannian moduli space:
$$\mathcal{M} = \mathrm{Met}(M)\,/\,\mathrm{Diff}(M)$$
The Ricci flow $\partial_t g_{ij} = -2R_{ij}$ is only weakly parabolic because of diffeomorphism invariance. DeTurck's trick introduces a background metric $\hat{g}$ and adds a Lie derivative term to break the gauge degeneracy. The resulting Ricci–DeTurck flow:
$$\partial_t g_{ij} = -2R_{ij} + \mathcal{L}_{W}g_{ij}, \qquad W^k = g^{pq}\bigl(\Gamma^k_{pq} - \hat{\Gamma}^k_{pq}\bigr)$$
is strictly parabolic, yielding short-time existence and uniqueness by standard PDE theory.
BMS Side: Asymptotic Symmetry Group
At null infinity $\mathscr{I}^+$, the diffeomorphism group of the unphysical boundary decomposes as:
$$\mathrm{Diff}(\mathscr{I}^+) = \mathrm{Diff}_0 \rtimes \mathrm{BMS}$$
where $\mathrm{Diff}_0$ denotes the trivial (pure gauge) diffeomorphisms and the BMS group encodes the physical asymptotic symmetries. The physical gravitational phase space is the quotient:
$$\mathcal{P}_{\rm grav} = \{\text{radiative data on } \mathscr{I}^+\}\,/\,\mathrm{Diff}_0$$
BRST Complexes
The gauge redundancy in each framework is most cleanly encoded through BRST cohomology.
DeTurck BRST operator: The ghost field $c^i$ generates infinitesimal diffeomorphisms, and the BRST transformation of the metric reads:
$$s_{\rm DT}\,g_{ij} = \mathcal{L}_c\,g_{ij} = \nabla_i c_j + \nabla_j c_i$$
BMS BRST operator: On $\mathscr{I}^+$, the shear tensor $C_{AB}$ transforms under a BMS vector $(Y^A, f)$ (superrotation + supertranslation):
$$s_{\rm BMS}\,C_{AB} = \mathcal{L}_Y C_{AB} + 2D_{\langle A}D_{B\rangle}f$$
In both cases, physical observables are elements of the BRST cohomology: they are $s$-closed (gauge-invariant) but not $s$-exact (not pure gauge artifacts).
Structural Theorem
Both the DeTurck gauge-fixing of Ricci flow and the BMS reduction of the gravitational phase space solve instances of the same abstract problem:
$$\textit{Given an infinite-dimensional group } G \text{ acting on a space of fields } \mathcal{F},$$
$$\textit{construct a well-posed dynamical system on } \mathcal{F}/G$$
$$\textit{whose solutions are in bijection with } G\textit{-orbits of solutions on } \mathcal{F}.$$
DeTurck achieves this via a gauge-breaking vector field $W^k$ that renders the PDE strictly parabolic; the BMS programme achieves it by identifying the radiative phase space $\mathcal{P}_{\rm grav}$ as the symplectic reduction $\mathcal{F}\,/\!/\,\mathrm{Diff}_0$.