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arxiv: 2512.10518 · v2 · pith:KWZHEZR2new · submitted 2025-12-11 · ✦ hep-th · math-ph· math.CO· math.MP

Fano and Reflexive Polytopes from Feynman Integrals

Leonardo de la Cruz , Pavel P. Novichkov , Pierre Vanhove This is my paper

Pith reviewed 2026-05-21 17:17 UTC · model grok-4.3

classification ✦ hep-th math-phmath.COmath.MP
keywords fano polytopesreflexive polytopesfeynman integralssymanzik polynomialscalabi-yau varietiesnewton polytopesperiod integralseuler integrals
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The pith

Quasi-finite Feynman integrals correspond to sparse Fano and reflexive polytopes that link directly to Calabi-Yau period integrals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies the Fano and reflexive polytopes that arise when quasi-finite Feynman integrals are analyzed through their Symanzik graph polynomials. These polytopes are formed as scaled Minkowski sums of the Newton polytopes attached to the polynomials, and the authors determine which cases are Fano or reflexive by counting interior points via bivariate Ehrhart polynomials. After examining one-loop graphs, multiloop sunset graphs, and a systematic search over all graphs with up to ten edges and nine loops in generic kinematics, only a handful of such polytopes appear in two and three dimensions. The classification shows that the geometric features of these polytopes, including their correspondence to Calabi-Yau varieties, govern the period integrals of the Feynman integrals. Specific one-loop N-gon cases further reveal that the polytopes encode degenerate Calabi-Yau (N-2)-folds, with explicit ties to del Pezzo surfaces, K3 surfaces, and Calabi-Yau threefolds.

Core claim

The scaled Minkowski sums of the Newton polytopes of the Symanzik polynomials for quasi-finite Feynman graphs produce Fano and reflexive polytopes in only a small number of cases; these polytopes determine the period integrals through their interior-point geometry and Calabi-Yau correspondence. Direct enumeration finds exactly two two-dimensional reflexive polytopes, three three-dimensional reflexive polytopes, and four three-dimensional Fano polytopes among graphs with at most ten edges and nine loops. One-loop N-gon integrals in higher dimensions yield reflexive polytopes that encode degenerate Calabi-Yau (N-2)-folds, while the overall structure connects to del Pezzo surfaces, K3 surfaces,

What carries the argument

Scaled Minkowski sums of the Newton polytopes associated with the Symanzik graph polynomials, whose interior-point counts from bivariate Ehrhart polynomials identify the Fano and reflexive cases and thereby control the period integrals.

If this is right

  • One-loop N-gon integrals encode degenerate Calabi-Yau (N-2)-folds in their reflexive polytopes.
  • The polytopes connect Feynman integrals to del Pezzo surfaces in lower dimensions, K3 surfaces, and Calabi-Yau threefolds.
  • Only specific graph topologies produce reflexive polytopes, making quasi-finite integrals rare.
  • Reflexive polytopes imply that the integrals satisfy differential equations shared with Calabi-Yau periods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Algebraic-geometry algorithms for Calabi-Yau periods could be repurposed to evaluate the corresponding Feynman integrals numerically or symbolically.
  • Extending the search to graphs with more edges or loops might uncover reflexive polytopes for higher-dimensional Calabi-Yau varieties.
  • The link suggests that quasi-finite integrals obey Picard-Fuchs equations identical to those of the associated Calabi-Yau hypersurfaces.

Load-bearing premise

The geometric properties of the polytopes, such as interior points and Calabi-Yau correspondence, are assumed to directly govern the period integrals of the Feynman integrals.

What would settle it

An explicit computation of the period integral for a one-loop N-gon graph whose associated polytope is reflexive, checking whether the result equals the known period of the corresponding degenerate Calabi-Yau (N-2)-fold.

Figures

Figures reproduced from arXiv: 2512.10518 by Leonardo de la Cruz, Pavel P. Novichkov, Pierre Vanhove.

Figure 1
Figure 1. Figure 1: Comparison of the polytopes for the massive triangle in D = 4 dimensions. Polytope (a) is a slice of the Lee–Pomeransky polytope (b) by the plane a0 = 1, parallel to the triangular faces, corresponding to a0 = 0 and a0 = 2. Feynman parameter polytope ∆Γ(L, D; ν) happens to be Fano for some value of nU , this does not guarantee that the corresponding Lee–Pomeransky polytope ∆LP Γ (L, D) is Fano, as the latt… view at source ↗
Figure 2
Figure 2. Figure 2: A two-loop graph with edge label (1, 1, 3). We illustrate the importance of the interior point condition for finiteness with an example of the two-loop graph with edge label (1, 1, 3) (using the notation of ref. [54], see also section A) with four external legs attached to one of the lines of the skeleton graph. The momentum-space representation of this integral is I(1,1,3) = Z R2D 1 (ℓ 2 1 − m2 1 ) ν1 ((ℓ… view at source ↗
Figure 3
Figure 3. Figure 3: Multiloop sunset graph. 4.1 Example: sunset integrals The basic example of reflexive polytopes arising from Feynman integrals is the family of multiloop sunset graphs depicted in fig. 3, with N ≥ 2 edges (N = 2 corresponds to the one-loop bubble graph). They have the following Symanzik polynomials: U N ⊖ = x1x2 · · · xN  1 x1 + · · · + 1 xN  , FN ⊖ = U N ⊖ [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-dimensional reflexive polytopes for the triangle graph. – 22 – [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Three-dimensional reflexive polytopes for the box graph. 5.3 D = 4, L = 2: two-loop integrals in four dimensions The Symanzik graph polynomials for two-loop graphs take the generic expression derived in [54] and are recalled in section A. Thanks to these generic expressions, it is not difficult to search for two-loop graph polytopes that are Fano or reflexive using polymake [56] or sagemath [64]. Solving t… view at source ↗
Figure 6
Figure 6. Figure 6: The kite graph, edge label (1, 2, 2). The kite graph. The graph polynomials of the kite graph, shown in in fig. 6, are given by section A: U(1,2,2) = (y1 + y2 + z1 + z2)x1 + (z1 + z2)(y1 + y2), L(1,2,2) = m2 1x1 + m2 2 y1 + m2 3 y2 + m2 4 z1 + m2 5 z2 , V(1,2,2) = p 2 1 (x1y1y2 + x1y2z1 + y1y2z1 + x1y2z2 + y1y2z2) + p 2 3 (x1y1z1 + x1y2z1 + x1y1z2 + x1y2z2) + p 2 2 (x1y1z2 + x1y2z2 + x1z1z2 + y1z1z2 + y2z1… view at source ↗
Figure 7
Figure 7. Figure 7: The house graph, edge label (1, 2, 3). The house graph. The graph polynomials of the house graph in fig. 7 read U(1,2,3) = x1(y1 + y2 + z1 + z2 + z3) + (y1 + y2)(z1 + z2 + z3), L(1,2,3) = m2 1x1 + m2 2 y1 + m2 3 y2 + m2 4 z1 + m2 5 z2 + m2 6 z3, V(1,2,3) = −2p1 · p2 x1y2(z2 + z3) − 2p1 · p3x1y2z3 + 2p1 · p4x1y2(z1 + z2 + z3) + p1 · p1y2(x1(y1 + z1 + z2 + z3) + y1(z1 + z2 + z3)) + 2p2 · p3z3(x1(y1 + y2 + z1… view at source ↗
Figure 8
Figure 8. Figure 8: The tardigrade graph, edge label (2, 2, 2). – 25 – [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A factorizable graph. In this section, we will present our results for an exhaustive direct search of Fano and reflexive polytopes associated to Feynman integrals with up to ten edges, N ≤ 10, assuming generic external kinematics and generic propagator masses. In generic kinematics the search is facilitated by the fact that the polytope for the second Symanzik polynomial is the Minkowski sum of the polytop… view at source ↗
Figure 10
Figure 10. Figure 10: Three graphs that share the same polytope in generic kinematics 6.3 Results The search proceeds as follows. In a first step, we generate the set of all graphs Γ N all, for N = 1, . . . , 10 with QGRAF. For N = 10, this gives 16744 graphs. We then compute UΓ and FΓ and group our graphs based on the normal ordering of the polynomial pair Normal(PΓ), which we compute using the FeynCalc command FCLoopPakOrder… view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of Fano and reflexive polytopes (N, L, D). Blue points correspond to graphs that are only Fano. Red points are both Fano and reflexive. Each point represents a value for (N, L, D) where at least one Fano polytope exists. 2 4 6 8 10 2 4 6 8 Edges Loops (a) edges vs loops 2 4 6 8 4 8 12 16 Loops Dimension (b) loops vs dimension 2 4 6 8 10 4 8 12 16 Edges Dimension (c) edges vs dimension [PITH_… view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of reflexive graph polytopes from Feynman integrals with massive internal lines up to 10 edges. Each point represents a value for (N, L), (L, D), (N, D) where at least one reflexive polytope exists. 7.1 One-loop N-gon and multiloop sunset Let us begin by analyzing the reflexive polytopes associated with the N-gon and the (N − 1)-loop sunset. The former case has the polytope in D = 2N dimensio… view at source ↗
Figure 13
Figure 13. Figure 13: Distribution of non-reflexive Fano graph polytopes from Feynman integrals with massive internal lines up to 10 edges. Each point represents a value for (N, L), (L, D), (N, D) where at least one non-reflexive Fano polytope exists. (a) (b) (c) (d) (e) [PITH_FULL_IMAGE:figures/full_fig_p034_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Representative graphs with eight propagators that have a reflexive polytope. given by IN−gon(1, N; (1, . . . , 1)) = Z R N−1 + 1 PN i=1 xi N [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Family of graphs with edge label (1, . . . , 1, n). 7.2 Family of graphs of type (1, . . . , 1, n) We consider now the family of graphs in fig. 15 composed by a n-loop sunset with n − 1 legs attached to one edge, which we evaluate in dimension D = 2(n + 1). These graphs have N = 2n edges. For n = 2 this is the ice-cream two-loop graph, for n = 3 this is the last graph in fig. 10, and for n = 4 this is the… view at source ↗
Figure 16
Figure 16. Figure 16: The multiloop graphs with edge label (1, . . . , 1, n, n). The list of reflexive polytopes contains graphs built by attaching the same number of external legs to two lines of the multiloop sunset graph as shown in fig. 16. We label the graph by the number of edges attached to the multiloop sunset graph. The graphs with less than ten edges in this family are the kite graph, shown in fig. 16a, with label (1… view at source ↗
Figure 17
Figure 17. Figure 17: Two graphs that evaluate to zeta values. 7.4 Feynman graphs which evaluate to zeta values The polytope associated with the graph in fig. 17a is ∆W3 (3, 4; (1, . . . , 1)) = 4∆(UW3 ) with UW3 = x1x2 + x3x2 + x1x4x2 + x3x4x2 + x1x5x2 + x3x5x2 + x4x5x2 + x5x2 + x1x3 + x1x4 + x1x3x4 + x3x4 + x1x5 + x1x3x5 + x3x4x5 + x4x5 , (7.19) and the polytope associated with the five-loop diamond circle graph in fig. 17b … view at source ↗
Figure 18
Figure 18. Figure 18: Reflexive polytopes for the two- and three-loop sunset graphs. N 4 5 6 7 8 9 10 h1,N−3 20 101 426 1667 6371 24229 92278 [PITH_FULL_IMAGE:figures/full_fig_p039_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The polytope for the massive box in D = 4. This polytope has the number #4311 in the sagemath [64] database. The one-loop massive box graph has four variables x1, . . . , x4, and the graph polynomials are Ubox = x1 + x2 + x3 + x4, Vbox = X 1≤i<j≤4 (pi + · · · + pj−1) 2xixj , Fbox = Ubox(m2 1x1 + · · · + m2 4x4) − Vbox . (8.16) The Newton polytopes for the massive box integral in D = 4, D = 6, and D = 8 ar… view at source ↗
Figure 20
Figure 20. Figure 20: The polytope for the massless box in D = 4. This polytope has the number #3349 in the sagemath [64] database. (P 1 ) 3 . From the computation of section 8.2 we have that the middle Hodge numbers are h1,1 = 20. The corresponding toric Fano variety is a smooth toric threefold of Picard rank six, and a generic anticanonical hypersurface in this variety is a K3 surface. Using Batyrev’s mirror symmetry constru… view at source ↗
Figure 21
Figure 21. Figure 21: A two-loop graph of type (a, b, c) with a = 4, b = 1 and c = 3. The Symanzik polynomials for two-loop graphs are given by U(a,b,c) = Xa i=1 xi ! X b i=1 yi ! + Xa i=1 xi ! Xc i=1 zi ! + X b i=1 yi ! Xc i=1 zi ! , L(a,b,c) = Xa i=1 m2 i xi + X b i=1 m2 i+a yi + Xc i=1 m2 a+c+i zi , V(a,b,c) = X b i=1 yi + Xc i=1 zi ! X 1≤i<j≤a c x ijxixj + Xa i=1 xi + Xc i=1 zi ! X 1≤i<j≤b c y ijyiyj + Xa i=1 xi + X b i=1 … view at source ↗
read the original abstract

We classify the Fano and reflexive polytopes that arise from quasi-finite Feynman integrals. These polytopes appear as scaled Minkowski sums of the Newton polytopes associated with the Symanzik graph polynomials. For one-loop graphs and multiloop sunset graphs, we identify the Fano and reflexive cases by computing the number of interior points from the associated bivariate Ehrhart polynomials. More generally, we utilize the properties of Symanzik polynomials and their symmetries to conduct a direct search over all Feynman graphs in generic kinematics with up to ten edges and nine loops. We find that such cases are remarkably sparse: for example, we find only two two-dimensional reflexive polytopes, three three-dimensional reflexive polytopes, and four three-dimensional Fano polytopes. We also reveal a surprising feature of one-loop $N$-gon integrals in higher dimensions: their associated reflexive polytopes encode degenerate Calabi--Yau $(N-2)$-folds. We further analyze the geometric structures encoded by these polytopes and exhibit explicit connections with del Pezzo surfaces, $K3$ surfaces, and Calabi--Yau threefolds. Since reflexive polytopes naturally correspond to Calabi--Yau varieties, our classification demonstrates that quasi-finite Feynman integrals, with reflexive polytopes, are intrinsically linked to Calabi--Yau period integrals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper classifies Fano and reflexive polytopes arising from quasi-finite Feynman integrals as scaled Minkowski sums of the Newton polytopes of Symanzik graph polynomials. For one-loop and multiloop sunset graphs, bivariate Ehrhart polynomials determine the number of interior points. A direct search over all graphs with up to ten edges and nine loops in generic kinematics yields only two 2D reflexive polytopes, three 3D reflexive polytopes, and four 3D Fano polytopes. One-loop N-gon integrals in higher dimensions are shown to encode degenerate Calabi-Yau (N-2)-folds, with further connections drawn to del Pezzo surfaces, K3 surfaces, and Calabi-Yau threefolds. The classification is invoked to conclude that quasi-finite Feynman integrals with reflexive polytopes are intrinsically linked to Calabi-Yau period integrals via the standard reflexive-polytope correspondence.

Significance. If accurate, the explicit enumeration of these polytopes supplies concrete examples bridging the combinatorial geometry of Feynman integrals to toric Calabi-Yau varieties. The computational use of Ehrhart polynomials for low-loop cases and the exhaustive search up to moderate size constitute a verifiable foundation that could seed further work on periods and mirror symmetry in QFT. The observed sparsity of reflexive cases is itself a potentially useful result.

major comments (1)
  1. [Conclusion] The central claim that the classification demonstrates an intrinsic link to Calabi-Yau period integrals rests on the reflexivity of the constructed polytopes and the standard toric correspondence, yet the manuscript contains no explicit computation or comparison showing that the actual period integrals (or Picard-Fuchs equations) of the quasi-finite Feynman integrals coincide with or are governed by the periods of the associated toric Calabi-Yau varieties. This verification is load-bearing for the conclusion stated in the abstract.
minor comments (2)
  1. A summary table listing all identified polytopes together with dimension, interior-point count, associated graph, and any degeneracy information would improve readability and allow quick cross-reference.
  2. The description of the direct-search algorithm would benefit from additional detail on how graphs are enumerated, how symmetries of the Symanzik polynomials are used to reduce the search space, and how edge cases (e.g., graphs with vanishing Symanzik polynomials) are handled.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the strength of our central claim. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim that the classification demonstrates an intrinsic link to Calabi-Yau period integrals rests on the reflexivity of the constructed polytopes and the standard toric correspondence, yet the manuscript contains no explicit computation or comparison showing that the actual period integrals (or Picard-Fuchs equations) of the quasi-finite Feynman integrals coincide with or are governed by the periods of the associated toric Calabi-Yau varieties. This verification is load-bearing for the conclusion stated in the abstract.

    Authors: We agree that the manuscript does not contain explicit computations of the period integrals or Picard-Fuchs equations of the quasi-finite Feynman integrals, nor a direct numerical or functional comparison with the periods of the associated toric Calabi-Yau varieties. The argument in the paper rests on the standard toric-geometry correspondence: reflexive polytopes determine Calabi-Yau hypersurfaces (or complete intersections) whose periods are encoded in the polytope data. Our contribution is the classification of which Symanzik polytopes arising from quasi-finite graphs are reflexive or Fano, together with their geometric interpretations (degenerate Calabi-Yau folds, del Pezzo surfaces, K3 surfaces, etc.). We will revise the abstract and the final section to state more precisely that the link to Calabi-Yau period integrals is combinatorial and geometric, via the established reflexive-polytope correspondence, rather than a claim of explicit period matching performed in this work. Explicit verification of the periods themselves is left for future investigation. revision: yes

Circularity Check

0 steps flagged

No circularity: direct polytope classification from Symanzik polynomials with external CY correspondence

full rationale

The paper constructs Newton polytopes explicitly as scaled Minkowski sums of Symanzik graph polynomials, identifies reflexive/Fano cases by direct interior-point counting via bivariate Ehrhart polynomials (for 1-loop/sunset) and exhaustive search (up to 10 edges/9 loops), and notes geometric features such as degenerate CY (N-2)-folds or links to del Pezzo/K3/CY3. The final claim that these yield an intrinsic link to Calabi-Yau period integrals rests on the standard external theorem that reflexive polytopes define toric Calabi-Yau varieties; this theorem is not derived inside the paper, not obtained via self-citation chain, and not a fitted input renamed as prediction. All steps are computational and self-contained against the input graph polynomials; no reduction by construction occurs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on the assumption that Symanzik polynomials generate Newton polytopes whose Minkowski sums yield the relevant geometric objects, plus the standard properties of Ehrhart polynomials for counting lattice points.

axioms (2)
  • domain assumption Symanzik graph polynomials define Newton polytopes whose scaled Minkowski sums produce the Fano and reflexive polytopes under study.
    Invoked throughout the classification procedure described in the abstract.
  • standard math Bivariate Ehrhart polynomials correctly count interior lattice points of the resulting polytopes.
    Used to identify Fano and reflexive cases for one-loop and sunset graphs.

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Forward citations

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