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In quantum field theory a vacuum state is supposed to be a quantum state that expresses the absence of any particle excitations of the fields.
On Minkowski spacetime the vacuum state for a free field theory is the standard Hadamard state. On general globally hyperbolic spacetimes there are always Hadamard states, and they do play the role of the vacuum state in the construction of AQFT on curved spacetimes, see at locally covariant perturbative AQFT. Notably the choice of such a Hadamard state fixes the Feynman propagator, hence the time-ordered product of quantum observables and thus the perturbative S-matrix away from coinciding interaction points (the extension of these distributions to coinciding interaction points is the process of renormalization).
However, since on a general globally hyperbolic spacetime there is no globally well-defined concept of particles, there is in general no concept of vacuum state. But under good conditions (such as existence of suitable timelike Killing vectors) one may identify Hadamard states which deserve to be thought of as vacuum states (Brum-Fredenhagen 13).
Stephen J. Summers, Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State (arXiv:0802.1854)
Marcos Brum, Klaus Fredenhagen, “Vacuum-like” Hadamard states for quantum fields on curved spacetimes (arXiv:1307.0482)
Michael Dütsch, def. 2.12 in From classical field theory to perturbative quantum field theory, 2018