Sheaves for spacetimes

Joint work with Benoît Jubin [JS16] (see also [Sc26]).  
A causal manifold $(M,\lambda)$ is a manifold $M$ endowed with a closed convex proper cone $\lambda$ in its cotangent bundle $T^*M$. In this situation, one defines the $\lambda$-topology on $M$ as well as the past $I^-_\lambda(A)$ of any subset $A$, the closure of $A$ for this topology, or its future.

It has been proved for long (see [Ge70]) that under suitable conditions, there exists a so-called time function, a smooth surjective causal map $q\colon M\to\mathbb{R}$. Using a time function, we show that if the micro-support of a sheaf $F$ does not intersect $\lambda\cup -\lambda$ outside of the zero-section, then for any Cauchy hypersurface $N_t=q^{-1}(t)$, the restriction morphism $
\mathrm{R}\Gamma(M;F)\to \mathrm{R}\Gamma\!\left(N_t;\,F|_{N_t}\right)
$ is an isomorphism. As an application, we get that the Cauchy problem is globally well-posed for hyperfunction solutions of hyperbolic systems, such as the wave operator.

Finally we have a glance to “before the Big Bang” (see [MM14, Pe20]). We consider the case where $N_t$ is a closed ball or a sphere of radius $t>0$ and show that a natural extension of $M$ for $t<0$ is a “shifted space”, the family of open balls or spheres, shifted by the dimension.

References

- [Ge70] Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437–449.
- [JS16] Benoît Jubin and Pierre Schapira, Sheaves and D-modules on causal manifolds, Letters in Mathematical Physics 16 (2016), 607–648.
- [MM14] Yuri Manin and Mathilde Marcolli, Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology, SIGMA 10 (2014), arXiv:1402.2158.
- [Pe20] Roger Penrose, Cycles of Time: an extraordinary view of the universe, Bodley Head, 2012.
- [Sc26] Pierre Schapira, Topologies and sheaves on causal manifolds, International Mathematics Research Notices 2026 (1) (2026), arXiv:2505.10364.