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It has been conjectured that in theories consistent with quantum gravityinfinite distances in field space coincide with an infinite tower of statesbecoming massless exponentially fast in the proper field distance. Thecomplex-structure moduli space of Calabi-Yau manifolds is a good testing groundfor this conjecture since it is known to encode quantum gravity physics. Westudy infinite distances in this setting and present new evidence for the aboveconjecture. Points in moduli space which are at infinite proper distance alongany path are characterised by an infinite order monodromy matrix. We utilisethe nilpotent orbit theorem to show that for a large class of such points themonodromy matrix generates an infinite orbit within the spectrum of BPS states.We identify an infinite tower of states with this orbit. Further, the theoremgives the local metric on the moduli space which can be used to show that themass of the states decreases exponentially fast upon approaching the point. Wealso propose a reason for why infinite distances are related to infinite towersof states. Specifically, we present evidence that the infinite distance itselfis an emergent quantum phenomenon induced by integrating out at one-loop thestates that become massless. Concretely, we show that the behaviour of thefield space metric upon approaching infinite distance can be recovered fromintegrating out the BPS states. Similarly, at infinite distance the gaugecouplings of closed-string Abelian gauge symmetries vanish in a way which canbe matched onto integrating out the infinite tower of charged BPS states. Thispresents evidence towards the idea that also the gauge theory weak-couplinglimit can be thought of as emergent.

Infinite Distances in Field Space and Massless Towers of States