I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:

Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \times S^1$, with this spin structure, is the boundary of a “1/2-K3 surface,” that is, a four-manifold $M^4$ that maps to a disc $D$ with generic fiber an elliptic curve. In particular, the map $$ M^4 \to D$$ has a section $$s : D \to M^4.$$ (possibly missing contexts) ...

Are these mathematically clear? Or do we require more clarification?

What are some other Mathematical way"s" to think about or to define "(1/2) K3 surface"?

elliptic, meaning that they are fibered over $\mathbb P^1$ with generic fiber an elliptic curve. Since $\mathbb P^1$ is glued from two disks it seems natural to consider the inverse image of (say) the unit disk to be "1/2" of the K3. This would somehow motivate their definition of a 1/2-K3 surface to be just any old elliptic fibration over a disk, except for the fact that an elliptic surface fibered over $\mathbb P^1$ is not in general a K3; the part of the quoted section after "that is" seems like it could just as well deserve to be called a "1/2-Enriques surface"... $\endgroup$ – Dan Petersen Feb 14 at 5:43