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arXiv · 2608.19720

Tropical and Stringy Integrals for In-In Correlators

Song He·Xiang Li·Yong-Xiang Su·Fan Zhu
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Abstract

We introduce tropical and stringy integrals for fixed-graph contributions to cosmological in-in correlators of conformally coupled scalars. The full-time representation factorizes into a graph-dependent vertex-space Laplace integral, with one real variable for each graph vertex, and an elementary edge-space Laplace integral, with one real variable for each internal edge. The vertex-space exponent is a sum of absolute values associated with sites and relative edge times; as a piecewise-linear function, it is the support function of the in-in zonotope. Each absolute value is also the tropical limit of a positive Laurent binomial. Retaining these binomials before tropicalization defines a finite-$\alpha'$ vertex-space stringy integral, so both the polytope and its stringy integral are read directly from the physical time integral. In the $\alpha' \to 0$ limit this integral becomes the normalized dual volume of the in-in zonotope, while the edge-space factor deforms independently into a product of beta integrals and restores the elementary propagator normalization. We derive the field-theory rational form from augmented-graph chambers, as well as exact finite-$\alpha'$ parallel-edge reduction, factorization formulas for edge-energy and partial-energy poles, and even descendant towers. As an alternative geometric realization of fixed-graph correlators, we find an ambient Minkowski-sum and stringy-integral realization of the graph correlahedron for a tree graph as the so-called graph cubeahedron of its line graph. For completeness, we also record the logarithmic critical equations and generic reference degrees of the associated affine divisor arrangement.

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Song He, Xiang Li, Yong-Xiang Su, Fan Zhu. 2026-08-20. Tropical and Stringy Integrals for In-In Correlators. https://arxiv.org/abs/2608.19720

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