  It has been conjectured that in theories consistent with quantum gravity
infinite distances in field space coincide with an infinite tower of states
becoming massless exponentially fast in the proper field distance. The
complex-structure moduli space of Calabi-Yau manifolds is a good testing ground
for this conjecture since it is known to encode quantum gravity physics. We
study infinite distances in this setting and present new evidence for the above
conjecture. Points in moduli space which are at infinite proper distance along
any path are characterised by an infinite order monodromy matrix. We utilise
the nilpotent orbit theorem to show that for a large class of such points the
monodromy matrix generates an infinite orbit within the spectrum of BPS states.
We identify an infinite tower of states with this orbit. Further, the theorem
gives the local metric on the moduli space which can be used to show that the
mass of the states decreases exponentially fast upon approaching the point. We
also propose a reason for why infinite distances are related to infinite towers
of states. Specifically, we present evidence that the infinite distance itself
is an emergent quantum phenomenon induced by integrating out at one-loop the
states that become massless. Concretely, we show that the behaviour of the
field space metric upon approaching infinite distance can be recovered from
integrating out the BPS states. Similarly, at infinite distance the gauge
couplings of closed-string Abelian gauge symmetries vanish in a way which can
be matched onto integrating out the infinite tower of charged BPS states. This
presents evidence towards the idea that also the gauge theory weak-coupling
limit can be thought of as emergent.
