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I was attending lectures om holography where the lecturer kept on mentioning that a QFT lives on a Cauchy slice. What does that mean?
Is it such that each point of the slice is associated to a unique vector in the Hilbert space?
Can this be generalized to QM? Does that mean that the harmonic oscillator lives on an integer lattice?
Is this geometric correspondence the motivation behind the operator state correspondence?
There are various layers to how a quantum field theory lives on a manifold. The simplest one is to note that a (classical) field is a mapping between a spacetime manifold and some target space like the reals. When you quantize the theory, the operators corresponding to the fields will again be functions of the spacetime. Moreover, at least for Lagrangian field theories, the integration domain is spacetime and the Lagrangian depends on the choice of background metric.
When one says that the QFT lives on a Cauchy slice, one is working in the Heisenberg picture where the field operators are functions of the spatial coordinates at a fixed time. Then, one can use the Hamiltonian to generate evolution in time and move the operators outside the Cauchy slice.
Another layer is that in algebraic quantum field theory, a QFT is just an assignment of spacetime regions to algebras of field operators localized to the region of choice. In this picture, it is clear that the QFT "lives" on the spacetime manifold.
Quantum mechanics is basically quantum field theory in 0+1 dimensions. Thus, we usually do not model the background for quantum mechanical systems.
Finally, the operator-state correspondence is a special property of conformal field theories. In any QFT, one can map from local operators to states by simply acting by the operator on some state. The other way around says that one can extract a unique local operator from any state. The reason this is true for CFT is because one can radially quantize a CFT so that evolution in the radial direction from the origin is unitary. This means that one could always retrace this evolution to the origin and define the corresponding operator.
In a general quantum field theory, one would have to extract a single local operator in the far past ($t \to -\infty$) from any given state. This clearly fails as $t \to -\infty$ defines a whole three-dimensional spatial manifold in Minkowski space. Thus, there is not only a single operator associated to any state, rather there is a lot of them that are not localized to a single point. The problem gets even worse for QFT in curved space.
