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Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$

A. V. Shanin, A. I. Korolkov, N. M. Artemov, R. C. Assier

TL;DR

This work develops a matrix-based framework for Picard--Lefschetz--Pham ramification in two complex variables near the real plane. By introducing a stratified universal Riemann domain \tilde{U} and an inflation theorem, surfaces of integration are represented as module-valued vectors, and parameter-space monodromies are captured by matrices over the group ring of the fundamental group. The authors derive explicit elementary transformation matrices for triangle, biangle, circle, and jump degenerations, then combine them to handle more complex configurations, including two-lines-and-a-parabola, establishing additive-crossing identities that elucidate integral decompositions. The approach both simplifies computation of intersection indices and reveals detailed topological information about the parameter space, with promising applications to multidimensional Wiener–Hopf problems and Feynman-type integrals. Overall, the paper provides a concrete, algorithmic, and verifiable framework that connects relative homology, monodromy, and topological identities through a scalable matrix formalism.

Abstract

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. We introduce the universal Riemann domain $\tilde U$, i.e. a sort of ``compactification'' of the universal covering space $\tilde U_2$ over a small tubular neighborhood $N\mathbb{R}^2$ of $\mathbb{R}^2\backslashσ$ in $\mathbb{B}\subset\mathbb{C}^2$, where $\mathbb{B}\subset\mathbb{C}^2$ is a big ball, and $σ$ is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space $\tilde U$ modulo the singularities and the boundary for the standard local degenerations of type $P_1 ,P_2,P_3$ in Pham's [1] notations and for more complicated configurations in $\mathbb{C}^2$. We consider this homology group as a module over the group ring of the $π_1((N\mathbb{R}^2 \cap \mathbb{B})\backslashσ)$ over $\mathbb{Z}$. The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group $H_2(\tilde U_2,\tilde{\partial \mathbb{B}})$) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group $H_2(\tilde U,\tilde U'\cup\tilde{\partial \mathbb{B}})$ (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables.

Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$

TL;DR

This work develops a matrix-based framework for Picard--Lefschetz--Pham ramification in two complex variables near the real plane. By introducing a stratified universal Riemann domain \tilde{U} and an inflation theorem, surfaces of integration are represented as module-valued vectors, and parameter-space monodromies are captured by matrices over the group ring of the fundamental group. The authors derive explicit elementary transformation matrices for triangle, biangle, circle, and jump degenerations, then combine them to handle more complex configurations, including two-lines-and-a-parabola, establishing additive-crossing identities that elucidate integral decompositions. The approach both simplifies computation of intersection indices and reveals detailed topological information about the parameter space, with promising applications to multidimensional Wiener–Hopf problems and Feynman-type integrals. Overall, the paper provides a concrete, algorithmic, and verifiable framework that connects relative homology, monodromy, and topological identities through a scalable matrix formalism.

Abstract

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. We introduce the universal Riemann domain , i.e. a sort of ``compactification'' of the universal covering space over a small tubular neighborhood of in , where is a big ball, and is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space modulo the singularities and the boundary for the standard local degenerations of type in Pham's [1] notations and for more complicated configurations in . We consider this homology group as a module over the group ring of the over . The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group ) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables.

Paper Structure

This paper contains 73 sections, 31 theorems, 358 equations, 73 figures.

Table of Contents

  1. Introduction
  2. Problem under consideration
  3. Basic assumptions
  4. Prior work
  5. The structure of the paper
  6. Riemann domains
  7. Universal Riemann domain
  8. Definition of the universal Riemann domain
  9. Structure of $\tilde{U}_2$
  10. Structure of $\tilde{U}_1$. Locality lemma for $\sigma^{(1)}$
  11. Structure of $\tilde{U}_0$. Locality lemma for $\sigma^{(0)}$
  12. Structure of $\tilde{\partial\mathbb{B}}$
  13. Commutativity of $\pi_1 (N\mathbb{R}^2 \setminus \sigma)$ under some assumptions
  14. An example of $\tilde{U}$
  15. Riemann domain of a function
  16. ...and 58 more sections

Key Result

Proposition 2.1

Let $\gamma'$ be an arbitrarily chosen element of $\Pi(z^*, z)$, $z \in \mathcal{X} \setminus \sigma$. Then any element of $\gamma \in \Pi(z^*, z)$ can be uniquely represented as where $\gamma_1 \in \pi_1 (\mathcal{X} \setminus \sigma)$.

Figures (73)

  • Figure 1: Small ball $D$ and $z \in \sigma$
  • Figure 2: Topology of $W$ for $z\in\sigma_j\backslash\cup_{n\neq j}\sigma_n$.
  • Figure 3: Computation of $\pi_1$ of a torus
  • Figure 4: Topology of $W$ for $z\in\sigma_j\cap\sigma_k$.
  • Figure 5: Topology of $W$ for $z\in\partial\mathbb{B}\cap\sigma_j$
  • ...and 68 more figures

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.1
  • Lemma 2.1: Locality for $\sigma^{(1)}$
  • Definition 2.2
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.2
  • Lemma 2.4
  • Proposition 3.1
  • ...and 46 more