We continue our study of the moduli space of a Calabi-Yau (CY) and the topology changes that can occur. We find that as one traverses the moduli space, three things can happen:
Case 1: A curve collapses but the volume of divisors and the CY stays finite (see first picture). This indicates a "flop wall". One can go through this singularity and emerge on the other side in another geometry where the collapsed curve is replaced by another curve. Case 2: A divisor collapses, but the CY volume stays finite (see second picture). This indicates a somewhat mysterious boundary which we call a Zariski wall. The CY volume stays finite, but one nevertheless does not have a geometric description beyond this wall. Case 3: The volume of the whole CY collapses (see third picture). This indicates the end of the effective cone. One would need a GLSM or some other description to make sense of string theory beyond this wall.
For CYs with 2 Kahler parameters in M-Theory, we can solve the geodesics equation of motion analytically (case 3 features every trigonometric function in the book...) in the vector moduli multiplet space of the theory, which essentially means solving the equations along curves of constant overall volume. We find: Case 1: These are always at finite geodesic distance. Case 3: These are always at infinite geodesic distance (this is obvious, since the volume of the CY shrinks in case c, but we are moving along curves of constant volume). Case 2: This is the strangest case. If we wanted to probe what’s beyond the Zariski wall and happily follow our geodesic, we find that the geodesic “bounces off” the Zariski wall and is reflected back into the geometric regime we came from (see fourth picture). And that bounce happens at finite distance. Moreover, this bounce is not benign: the closer we get to the wall, the more the geodesic motion accelerates, reaching infinite speed at the wall. So it crashes into the wall with infinite speed and is re-emitted into the opposite direction, also with infinite speed. Of course our theory is invalidated before reaching infinite speed, and it is suggested that an additional SU(2) theory appears at such walls. Presumably, if we could correctly include the SU(2) vector multiplet, the singularity would be cured, but we don’t attempt this in this paper.