Graph integrals, Feynman periods, and single-valued multiple zeta values

Technical Summary: Graph Integrals, Feynman Periods, and Single-Valued Multiple Zeta Values

Problem Statement
This work addresses the evaluation of "canonical integrals" associated with graphs, which arise in the study of the stable cohomology of the general linear group $GL_n$ and graph complexes. Specifically, for a graph $G$ with $n+1$ vertices and $2n$ edges (satisfying $E=2V-2$), the canonical integral $I_{can}(G)$ is defined by integrating a primitive invariant differential form over a positive coordinate simplex. While these integrals are known to be finite and related to Feynman periods, the specific nature of the numbers they evaluate to remained an open question. The paper seeks to determine the arithmetic nature of these values and their relationship to other known classes of periods, particularly in the context of deformation quantization and the Grothendieck-Teichmüller group.

Methodology
The core methodological contribution is the establishment of an equality between two distinct families of integrals associated with a graph $G$:

1. Canonical Integrals ($I_{can}$): Defined via integration over the space of edge parameters (a simplex), utilizing the Laplacian matrix of the graph.
2. RW-Integrals ($I_{RW}$): Defined via integration over the configuration space of vertex positions in the complex plane $\mathbb{C}$, modulo translation, rotation, and scaling. These integrals involve logarithms and differential forms of the type $d\log(|z|^2)$.

To prove the equality $I_{RW}(G) = I_{can}(G)$, the author constructs an auxiliary integral $I_{aux}$ defined on the product space of the configuration space and the parameter simplex. The proof proceeds by evaluating this auxiliary integral in two different orders:

• Integrating over the simplex first: By utilizing a regularized Feynman parametrization (reversing the standard Schwinger trick) and handling divergent terms via counterterms, the integral reduces to the RW-integral.
• Integrating over the configuration space first: By performing complex Gaussian integration, completing the square, and applying differential operators, the integral reduces to an expression involving the graph matrix and its minors.

A crucial step involves a combinatorial identity (Theorem 7.1) that identifies the resulting determinant-permanent expression from the Gaussian integration with the trace formula defining the canonical form $\beta_{2n-1}$. The proof rigorously handles convergence issues by introducing truncation parameters ($\epsilon$) and utilizing dominated convergence theorems.

Key Contributions and Results

1. Equality of Integrals (Theorem 1.1): The paper proves that for any graph $G$ with $n+1$ vertices and $2n$ edges (where $n$ is odd), the canonical integral equals the RW-integral:
   $$I_{can}(G) = I_{RW}(G)$$
   This establishes a bridge between the parameter-space representation (canonical) and the position-space representation (RW) of these graph integrals.

2. Evaluation to Single-Valued Multiple Zeta Values (Proposition 1.3): By analyzing the structure of the RW-integrals, the author demonstrates that these integrals evaluate to single-valued multiple zeta values (MZVs) of weight $n$. Specifically, $I_{can}(G) \in \mathbb{Z}^{sv}_n$. This provides an analytic explanation for the observed "weight drop" in canonical integrals, where the Hodge weight is lower than the generic bound of $4n-6$ expected for Feynman periods.

3. Generation of Single-Valued MZVs (Proposition 1.5 & Theorem 1.8): The paper shows that the algebra of single-valued MZVs, $\mathbb{Z}^{sv}$, is generated as a $\mathbb{Q}$-algebra by the RW-integrals of all such graphs. Consequently, every single-valued MZV can be realized as a rational linear combination of Feynman periods of graphs with massless propagators. This implies the inclusion $\mathbb{Z}^{sv} \subseteq \mathcal{P}_{Feyn}$.

4. Cohomological Implications (Corollary 1.11): In the context of the even graph complex $\mathcal{GC}_2$, the RW-integrals and canonical integrals define degree-zero cocycles. The equality of the integrals implies that these two cocycles coincide. Furthermore, these cocycles descend to the quotient complex $\mathcal{GC}^{tri}_2$ (graphs modulo those with two-vertex cuts), defining non-trivial cohomology classes.

5. Explicit Formulas (Theorem 7.1 & Proposition 1.13): The proof yields new, more efficient explicit formulas for the canonical forms and RW-integrals.

   • The canonical form $\beta_{2n-1}$ is expressed with a denominator of $\Psi^n$ (where $\Psi$ is the first Symanzik polynomial), accounting for approximately two-thirds of the expected cancellations, improving upon previous trace formulas.
   • The RW-integral formula is simplified by removing the summation over distinguished edges, reducing the computational complexity.

Significance
The paper claims that its primary significance lies in unifying two previously distinct approaches to graph integrals (parameter space vs. position space) and resolving the arithmetic nature of canonical integrals. By proving that these integrals are single-valued multiple zeta values, the work connects the stable cohomology of $GL_n$ and graph homology directly to the arithmetic of MZVs.

Furthermore, the result that $\mathbb{Z}^{sv} \subseteq \mathcal{P}_{Feyn}$ provides a lower bound on the space of Feynman periods, confirming that all single-valued MZVs appear in the context of massless Feynman integrals. The identification of the RW and canonical cocycles in the graph complex resolves a question regarding the consistency of these cohomology classes, which are central to deformation quantization and the study of the Grothendieck-Teichmüller group. The work also offers practical computational advantages by providing simplified formulas for these integrals, which are notoriously difficult to compute explicitly.