  We show that correlators of local operators in four dimensional free scalar
field theory can be expressed in terms of amplitudes in a two dimensional
topological field theory (TFT2). We describe the state space of the TFT2, which
has $SO(4,2)$ as a global symmetry, and includes both positive and negative
energy representations. Invariant amplitudes in the TFT2 correspond to surfaces
interpolating from multiple circles to the vacuum. They are constructed from
SO(4,2) invariant linear maps from the tensor product of the state spaces to
complex numbers. When appropriate states labeled by 4D-spacetime coordinates
are inserted at the circles, the TFT2 amplitudes become correlators of the
four-dimensional CFT4. The TFT2 structure includes an associative algebra,
related to crossing in the 4D-CFT, with a non-degenerate pairing related to the
CFT inner product in the CFT4. In the free-field case, the TFT2/CFT4
correspondence can largely be understood as realization of free quantum field
theory as a categorified form of classical invariant theory for appropriate
SO(4,2) representations. We discuss the prospects of going beyond free fields
in this framework.
