Tropical and Stringy Integrals for In-In Correlators

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-$α'$ vertex-space stringy integral, so both the polytope and its stringy integral are read directly from the physical time integral. In the $α' \to 0$ 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-$α'$ 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.

• The paper constructs tropical and finite-$\alpha'$ stringy integrals for in-in correlators of conformally coupled scalars.
• Each correlator factorizes into a zonotopal factor and an edge-space factor, with discussions on melonic reduction, edge-deletion, and partial-energy factorization.
• The authors introduce an augmented graph representation of well-defined rational integrals and introduce a complementary Born-rule computational geometry.

Overview and main construction

This paper, by He, Li, Su, and Zhu (2608.19720), constructs tropical and finite-$\alpha'$ "stringy" integral representations of fixed-graph contributions to equal-time in-in (Schwinger–Keldysh) correlators of conformally coupled scalars. The central observation is that the full-time representation of the fixed-graph correlator is already a Laplace integral whose exponent is a piecewise-linear function of the vertex time variables; this function is precisely the support function of the in-in zonotope $Zono(G)$ introduced by Glew (Glew, 26 Jan 2026). Because each absolute value in the action arises as the max-plus tropical limit of a positive Laurent binomial, retaining those binomials before tropicalization yields a canonical finite-$\alpha'$ stringy integral. In this way both the polytope and its positive Laurent representation are read directly from the physical time integral, rather than guessed from the Newton polytope alone — addressing the known non-uniqueness whereby different positive polynomials share a Newton polytope.

The paper's structural result parallels the ABHY associahedron story for $\operatorname{Tr}(\phi^3)$ amplitudes: there, $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$ is the $\alpha'\to 0^+$ limit of an ordered Koba–Nielsen integral (Arkani-Hamed et al., 2019). Here the analogous statement is

$$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$

where $H_\square(G)$ is a centered hypercube recording only propagator normalizations.

Factorization of the full-time representation

For a connected loopless multigraph $G=(V,E)$ with site energies $x_v$ (sums of external momentum magnitudes at vertex $Zono(G)$0) and edge energies $Zono(G)$1, the full-time representation assigns $Zono(G)$2 to each site and the kernel $Zono(G)$3 to each internal edge. The correlator factorizes as $Zono(G)$4, with

• Vertex-space factor: $Zono(G)$5, where $Zono(G)$6;
• Edge-space factor: $Zono(G)$7, itself a product Laplace integral over one variable per internal edge.

The two Laurent binomials $Zono(G)$8 and $Zono(G)$9 tropicalize to $\alpha'$0 and $\alpha'$1 respectively. Their weighted Minkowski sum of Newton segments gives exactly the in-in zonotope,

$\alpha'$2

and radial integration over its polar body yields the identity $\alpha'$3. The edge-space deformation is elementary: it deforms independently into a product of beta integrals $\alpha'$4, restoring the propagator normalization in the field-theory limit. All dependence on graph adjacency therefore resides in the zonotopal factor.

Rational form from augmented-graph chambers

To obtain a uniform rational formula valid for graphs with cycles, the authors introduce the augmented graph $\alpha'$5, adjoining a universal vertex $\alpha'$6 connected to every site, with weights $\alpha'$7. The action becomes a sum of $\alpha'$8 terms over edges of $\alpha'$9, and the ordering cones of the braid fan on $\operatorname{Tr}(\phi^3)$0 (projected along the diagonal axis onto $\operatorname{Tr}(\phi^3)$1) refine the normal fan of $\operatorname{Tr}(\phi^3)$2. On each cone, indexed by a permutation $\operatorname{Tr}(\phi^3)$3, the action is linear with coefficients given by cut energies $\operatorname{Tr}(\phi^3)$4 of prefix sets, so the chamber contribution factorizes into a product of simple poles:

$\operatorname{Tr}(\phi^3)$5

A key structural point follows from the facet classification of graphical zonotopes: opposite facets of $\operatorname{Tr}(\phi^3)$6 are indexed by nonempty connected vertex subsets $\operatorname{Tr}(\phi^3)$7, with facet variable the partial energy $\operatorname{Tr}(\phi^3)$8. Individual chamber terms may contain denominators $\operatorname{Tr}(\phi^3)$9 for disconnected $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$0 (internal rays of the refinement); these are auxiliary and cancel upon grouping centrally symmetric chambers, leaving only physical partial-energy poles. This cancellation is demonstrated explicitly for $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$1, where the spurious denominator $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$2 disappears via the partial-fraction identity, and for stars and cycles generally. The resulting rational form is algebraically equivalent to the pole organization of Glew (Glew, 9 Jul 2025).

Concrete results include $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$3, the full ten-term expression for $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$4, and the twelve-term expression for the triangle $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$5 — the latter notable because every subset of $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$6 is connected, so no auxiliary denominator appears at all. Setting an edge energy to zero distinguishes cycles from trees already at the graph level: deleting an edge of $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$7 produces $$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$8 rather than a disconnected product.

Finite-$$m_n^{\operatorname{Tr}(\phi^3)} = \Omega(\mathcal A_{n-3})$$9 structure

Three exact identities hold at finite $\alpha'\to 0^+$0, not merely in the field-theory limit:

Melonic reduction. Parallel-edge bundles combine through their total energy because all edges in a bundle contribute the same hyperbolic factor:

$\alpha'\to 0^+$1

Consequently, after stripping propagator poles, melonic reduction leaves only connected vertex-induced poles — a strictly smaller set than the wavefunction pole set for graphs with cycles. The count of parallel edges survives only in the elementary hypercube factor via a ratio of beta functions.

Edge-deletion factorization. At simultaneous edge-energy poles $\alpha'\to 0^+$2 ($\alpha'\to 0^+$3), the zonotopal factor evaluates to the product over connected components of $\alpha'\to 0^+$4, while the residue $\alpha'\to 0^+$5 is carried entirely by the elementary beta functions.

Partial-energy factorization. At a pole $\alpha'\to 0^+$6 with $\alpha'\to 0^+$7 connected, translating all vertices of $\alpha'\to 0^+$8 collectively to $\alpha'\to 0^+$9 isolates the singular region, and the residue factorizes as

$$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$0

where $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$1 is an internal block for $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$2 and each $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$3 is a shifted zonotopal integral for a complementary component, tilted by exponential factors $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$4 with $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$5 the total cut-edge energy incident to $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$6. When $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$7 is a tree, the internal block collapses to a product of beta functions, $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$8; when $$\langle G\rangle_V = \Omega(Zono(G)) \times \Omega(H_\square(G)) \xleftarrow{\;\alpha'\to 0^+\;} I_G^{\rm corr}(\alpha'),$$9 contains cycles, one constraint per independent cycle leaves a coupled integral. The worked example of the $H_\square(G)$0 cut of $H_\square(G)$1 reproduces the sign-shift structure of tree-level correlator factorization found in (Arkani-Hamed et al., 29 Dec 2025).

Descendant towers. Expanding the hyperbolic factors near either endpoint of the collective translation produces only even powers of $H_\square(G)$2, so meromorphic continuation of $H_\square(G)$3 can have poles only in the evenly spaced tower

$H_\square(G)$4

with odd levels absent. The $H_\square(G)$5 members are finite-$H_\square(G)$6 descendants of the physical partial-energy poles; their residues are not computed here.

The cubeahedral alternative and critical points

An appendix develops a complementary Born-rule geometry for trees: the graph correlahedron of a tree $H_\square(G)$7 is identified with the graph cubeahedron of its line graph $H_\square(G)$8, denoted $H_\square(G)$9, with an explicit Minkowski-sum realization and a corresponding stringy integral built from truncated-cube polynomials $G=(V,E)$0. Matching the generic Minkowski weights to physical poles requires generally signed parameters obtained by Möbius inversion, and the physical specialization can degenerate the polytope (for $G=(V,E)$1, the maximal summand weight vanishes). The authors state plainly that a naive extension to graphs with cycles or parallel edges fails to reproduce the cut-dependent powers of two in the Born-rule expansion — already for two parallel edges plus a third edge no tube-variable redefinition makes the weights consistent — though they make no general no-go claim.

A second appendix records logarithmic critical equations for the master function $G=(V,E)$2, which in sign-invariant variables $G=(V,E)$3 define an affine hyperplane arrangement with hyperplanes $G=(V,E)$4, $G=(V,E)$5, $G=(V,E)$6. Finite-field transfer-matrix computations give closed-form generic critical-point degrees: Pell-number growth for paths, $G=(V,E)$7 for stars, and $G=(V,E)$8 for cycles. These are explicitly reference quantities for generic independent logarithmic weights; whether they survive the constrained physical exponent specialization, and whether a CHY-type pushforward to $G=(V,E)$9 exists, are left open.

Limitations and open questions

Several caveats bear directly on the results. The stringy interpretation is algebraic only — no microscopic string origin is claimed. The cubeahedral realization is restricted to trees and requires signed, possibly degenerate Minkowski weights at the physical point. The generic arrangement degrees do not address the physically relevant critical-point count. Open questions stated by the authors include: whether the sum over all graphs in a theory such as conformally coupled $x_v$0 admits a single tropical integral with a positive Laurent lift, and how its Newton geometry relates to cosmohedra and correlator polytopes (Arkani-Hamed et al., 2024, Figueiredo et al., 24 Jun 2025); whether the zonotopal (full-time) and cubeahedral (Born-rule) geometries are related by subdivision, projection, or pushforward of canonical forms; the residues and possible recursion structure of the descendant poles at $x_v$1; which hidden zeros, kinematic splits, and differential relations persist in the stringy deformation; and whether poles, residues, zeros, and asymptotic data characterize $x_v$2 recursively.

Conclusion

The paper establishes that fixed-graph in-in correlators of conformally coupled scalars admit a direct tropical description as normalized dual volumes of the in-in zonotope, together with a canonical finite-$x_v$3 lift whose positive Laurent data are dictated by the physical time integral rather than chosen ad hoc. The framework yields a uniform chamber-sum rational form with only physical partial-energy poles, exact finite-$x_v$4 melonic reduction and edge-deletion identities, factorized residues at partial-energy poles with beta-function internal blocks for tree subgraphs, and even-spaced descendant towers. The remaining gaps — the graph-summed uplift, the relation between the two geometric realizations, and descendant residues — are well-posed questions within the framework the paper sets up.