Let me start with a couple of examples of rationality.

  1. Let $X$ be a nonsingular, projective Calabi-Yau threefold. Let $\beta\in H_2(X)$ be a homology class. The rationality of the reduced Donaldson-Thomas partition function $Z_\beta(X,q)$ in the class $\beta$ was one of the MNOP conjectures, and was proved by Bridgeland and by Toda some years ago. Equivalently, we now know that the Pandharipande-Thomas stable pair partition function of $X$ is the Laurent expansion of a rational function.
  2. In a completely different (but not totally unrelated) setting, one can consider the motivic zeta function $\mathsf Z_f(T)$, attached (by Denef-Loeser) to any regular function $f:V\to \mathbb A^1$ on a smooth scheme $V$. This is a series with coefficients in the relative (monodromic) Grothendieck group $K^{\hat{\mu}}(\textrm{Var}_{V_0})$, where $V_0=f^{-1}(0)$. Denef and Loeser proved the rationality of this series over the ring $K^{\hat{\mu}}(\textrm{Var}_{V_0})[\mathbb L^{-1}]$ obtained by inverting the Lefschetz motive $\mathbb L$.

Even though I can understand the interest of finding that some meaningful series is the expansion of a rational function, I usually do not understand what it means, and what it implies. So here are my questions:

What is the meaning of the rationality described in the above examples? Are there any evident geometric consequences of it? Is there, maybe, a reason from Physics to expect rationality in example 1? Finally, are these two rationality examples related in some way?



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