Sebastian Mizera
We give a pedagogical review of the recently-introduced notion of a "scalar product" between Feynman integrals and how it helps us understand the analytic structure of the perturbative S-matrix. (This article is a contribution to the proceedings of the workshop "MathemAmplitudes 2019: Intersection Theory and Feynman Integrals" held in Padova, Italy on 18-20 December 2019.)
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Intersection theory, relative cohomology and the Feynman parametrization
该研究提出利用相交理论和相对上同调在费曼参数化中进行圈积分约化的新方法，通过计算相交数实现积分约化，经实例验证正确性并讨论了退化极限的微妙之处。
We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the …
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the derivation of contiguity relations for special functions admitting multi-fold integral representations, and to the decomposition of a …
Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $d\log$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales…
Feynman Integral Reductions by Intersection Theory with Orthogonal Bases and Closed Formulae
提出利用交点理论构造正交基的新方法，并给出闭式公式，实现单圈费曼积分的系统约化。
The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct $d$-dimensional $d\log$-form integrands, and discuss how to convert them to Feynman integral…
Feynman integral reduction with intersection theory made simple
本文提出利用分支表示法简化费曼积分，通过计算最多(3L-3)变量的交数，有效降低多腿积分的计算复杂度，提升计算效率。
We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrar…
This document is a contribution to the proceedings of the MathemAmplitudes 2019 conference held in December 2019 in Padova, Italy. A key step in modern high energy physics scattering amplitudes computation is to express the latter in terms of a minimal set of Feynman integrals using linear relations. In this work we present an innovative approach based on intersection theory, in order to achieve this decomposition. This allows for the direct computation of the reduction, projecting integrals appearing in the scattering amplitudes onto an integral basis in the same fashion as vectors may be pro…
High precision calculations in perturbative QFT often require evaluation of big collection of Feynman integrals. Complexity of this task can be greatly reduced via the usage of linear identities among Feynman integrals. Based on mathematical theory of intersection numbers, recently a new method for derivation of such identities and decomposition of Feynman integrals was introduced and applied to many non-trivial examples. In this note we discuss the latest developments in algorithms for the evaluation of intersection numbers, and their application to the reduction of Feynman integrals.
In this manuscript, which is to appear in the proceedings of the conference "MathemAmplitude 2019" in Padova, Italy, we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, wo…
Feynman integrals that have been evaluated in dimensional regularization can be written in terms of generalized hypergeometric functions. It is well known that properties of these functions are revealed in the framework of intersection theory. We propose a new application of intersection theory to construct a coaction on generalized hypergeometric functions. When applied to dimensionally regularized Feynman integrals, this coaction reproduces the coaction on multiple polylogarithms order by order in the parameter of dimensional regularization.
Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
本文阐述了GKZ系统、扭曲上同调群的de Rham理论与Feynman积分Pfaffian方程的联系，提出了一种更高效的Macaulay矩阵计算算法，用于推导Pfaffian系统并获取超几何积分和Feynman积分的线性关系。
Reduction to master integrals via intersection numbers and polynomial expansions
本文提出一种仅用有理运算、无需积分变换即可计算相交数的新方法，通过多项式级数展开实现费曼积分到主积分基的投影分解，并引入新对偶积分方案消除额外正则化依赖，在有限域上完成单双圈验证。
We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermor…
We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities (LS). Cutting propagators in momentum twistor space amounts to intersecting lines associated with loop and external dual momenta, including the special line associated with the point at infinity, which breaks dual conformal symmetry. We show that cross-ratios of intersection points on these lines, especially those on the infinity line, naturally produce symbol letters for Feynman integrals in $D=4-2ε$, wh…
Intersection matrices associated to geometric-ordered bases of Feynman integrals
本文针对费曼积分分部积分约化中几何序基的主积分被积函数相交矩阵展开研究，明确了不同基下相交矩阵的元素形式，提出可系统消除转换时引入的辅助超越函数的算法，最小化了计算量。
In this review I discuss intersection numbers of twisted cocycles and their relation to physics. After defining what these intersection number are, I will first discuss a method for computing them. This is followed by three examples where intersection numbers appear in physics. These examples are: tree-level scattering amplitudes within the the CHY-formalism, reduction of Feynman integrals to master integrals and correlation functions on the lattice.
We review some relations occurring between the combinatorial intersection theory on the moduli spaces of stable curves and the asymptotic behavior of the 't Hooft-Kontsevich matrix integrals. In particular, we give an alternative proof of the Witten-Di Francesco-Itzykson-Zuber theorem --which expresses derivatives of the partition function of intersection numbers as matrix integrals-- using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers. These techniques extend to a more general interaction potential.
We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces - an algebraic invariant called the intersection number - extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the re…
This course on Feynman integrals starts from the basics, requiring only knowledge from special relativity and undergraduate mathematics. Topics from quantum field theory and advanced mathematics are introduced as they are needed. The course covers modern developments in the field of Feynman integrals. Topics included in this course are: Representations of Feynman integrals, integration-by-parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, coactions and symbols, cluster algebras, elliptic Feynman integrals, motives associated to Fey…
Fourier Calculus from Intersection Theory
本研究利用相交理论计算 Fourier 积分，通过确定主积分基并采用微分方程方法，应用于引力轫致辐射和饱和区深度非弹性散射，得到精确依赖时空维数 D 的新结果。
We study a surprising phenomenon in which Feynman integrals in $D=4-2\varepsilon$ space-time dimensions as $\varepsilon \to 0$ can be fully characterized by their behavior in the opposite limit, $\varepsilon \to \infty$. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on $\varepsilon$ and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either $\varepsilon$ or $1/\varepsilon$. We use the latte…
The first paper of this series introduced objects (elements of twisted relative cohomology) that are Poincaré dual to Feynman integrals. We show how to use the pairing between these spaces -- an algebraic invariant called the intersection number -- to express a scattering amplitude over a minimal basis of integrals, bypassing the generation of integration-by-parts identities. The initial information is the integrand on cuts of various topologies, computable as products of on-shell trees, providing a systematic approach to generalized unitarity. We give two algorithms for computing the multi-va…
In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in the light of the recent developments. Feynman integrals enter in several perturbative methods for solving non linear PDE, starting from Quantum Field Theories and including General Relativity and Condensed Matter Physics. Precision calculations involve several loop integrals, one strategy to address which is to bring them back in terms of linear combinations of a complete set of integrals (the Master Integrals). In this sense Feynman integrals can be thought as defi…
NeatIBP 1.0, A package generating small-size integration-by-parts relations for Feynman integrals
本文介绍 NeatIBP 1.0 软件包，利用合系与模交技术生成传播子次数受控、规模小于 Laporta 算法的 Feynman 积分 IBP 恒等式，并支持扇区级并行及有限域或解析约化。
Critical Points and Syzygies for Feynman Integrals
本文提出临界合冲概念，利用 Baikov 多项式对数的临界轨迹在大 $ε$ 极限下生成全导数，为费曼积分的分部积分约化提供了新的几何方法，并在一圈解析与两圈数值构造中验证了其有效性。
The purpose of this paper is to show that, under certain combinatorial conditions on the graph, parametric Feynman integrals can be realized as periods on the complement of the determinant hypersurface in an affine space depending on the number of loops of the Feynman graph. The question of whether the Feynman integrals are periods of mixed Tate motives can then be reformulated (modulo divergences) as a question on a relative cohomology being a realization of a mixed Tate motive. This is the cohomology of the pair of the determinant hypersurface complement and a normal crossings divisor depend…
We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of $73$ master integrals.
Intersection theory and canonical differential equations
本文探讨了交集理论在规范微分方程中的应用，通过分析交集矩阵揭示规范基的性质，并推导多项式关系以简化计算过程。
R-matrices from Feynman Diagrams in 5d Chern-Simons Theory and Twisted M-theory
本文研究描述扭曲M理论的5维非对易拓扑全纯陈-西蒙斯理论中的R-矩阵类似物，通过费曼图计算关联函数与余积，揭示其与Miura算子、矩阵扩展$W_{\text{∞}}$-代数等的关联。
Feynman integrals in two dimensions and single-valued hypergeometric functions
本文证明二维无质量传播子费曼积分可用单值 Aomoto-Gelfand 超几何函数表示，并将其应用于单圈积分与梯形积分，建立与 Lauricella 函数及广义超几何函数的联系。
By elaborating on the recent progress made in the area of Feynman integrals, we apply the intersection theory for twisted de Rham cohomologies to simple integrals involving orthogonal polynomials, matrix elements of operators in Quantum Mechanics and Green's functions in Field Theory, showing that the algebraic identities they obey are related to the decomposition of twisted cocycles within cohomology groups, and which, therefore, can be derived by means of intersection numbers. Our investigation suggests an algebraic approach generically applicable to the study of higher-order moments of prob…
We formulate and prove Cutkosky's Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in \cite{app-iso}. We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly …
GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
本文基于Mellin-Barnes表示和Miller变换，为任意质量三圈真空图费曼积分构造GKZ超几何系统，并利用其级数解在参数空间子集上构建基本解系，从而得到真空积分的解析表达式。
General Relativity from Intersection Theory
本文将后闵可夫斯基展开与相交理论结合，利用扭曲上链相交数将费曼积分投影到主积分基，成功计算了两个致密天体散射的第二后闵可夫斯基贡献，结果与已有结论一致。
Discrete symmetries of Feynman integrals
本文研究了费曼积分家族的离散对称性，通过变量变换将不同部分联系起来，揭示了这些变换与费曼参数排列的关系，并探讨了其在扭余弦学中的应用及对费曼积分的影响。
Intersection Numbers, Polynomial Division and Relative Cohomology
本文提出一种简化计算微分 $n$-形式相交数递归算法的方法，结合 delta-forms 与多项式除法技术，避免使用消逝解析调节器，并成功应用于推导二圈平面与非平面 Feynman 积分的完整分解及扭曲周期积分关系。
We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of arrangements of subvarieties associated to the subgraphs of a Feynman graph and a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of the variety and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful c…
General Relativity from Intersection Theory and Loop Integrals
该研究聚焦两个Schwarzschild黑洞的二对二引力散射，聚焦2PM修正，通过引入twisted (co)-homology groups与主积分分解公式，首次将相交理论应用于引力量子场论计算主积分基系数，结果与现有文献一致。
Intersection Numbers from Companion Tensor Algebra
本研究揭示相交数的张量结构并结合纤维化方法开发新评估方案，利用伴随矩阵将相交数计算转化为矩阵算子微积分，成功应用于双圈积分样本的数值分解，为费曼积分直接投影到主积分提供重大进展。
This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and L-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-cr…
We comment on the status of "Steinmann-like" constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar ${\cal N}=4$ super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for $n=6,7$ to all multiplicities. As our main result, we find that extended-…
Twisted Riemann bilinear relations and Feynman integrals
本文利用扭曲上同调研究多圈费曼积分及其割线的扭曲黎曼双线性关系，发现其产生推广的二次关系，但无法用于非最大割线或未切割积分。
It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of …
Canonical differential equations and intersection matrices
本文提出利用规范基中的交矩阵推导迭代积分的多项式关系，以解决非dlog形式微分方程的ε因子化问题。
Multiple Mellin-Barnes integrals and triangulations of point configurations
提出基于点构型三角剖分的新方法，快速解析计算多重Mellin-Barnes积分，应用于多圈费曼积分，显著提速并处理104重积分。
$c=1$ strings as a matrix integral
通过谱曲线将 $c=1$ 弦的微扰 $S$-矩阵描述为双标度 (0+0) 维矩阵积分，建立世界 sheet 描述、矩阵量子力学和矩阵积分之间的三重性。
Exponential Periods for Integrals in Physics
本文提出通过同调和上同调理论系统识别并构造指数周期，以分析费曼积分的参数表示，结合维数正则化分析墙交叉结构和Stokes现象，拓展了量子场论和弦论中的积分应用。
Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
研究共形四点Feynman积分的递归结构，发现其在更高维度的推广遵循Toda方程并具有双副本形式，建立了与经典可积性的联系。
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