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Regularized Integrals on Configuration Spaces of Riemann Surfaces and Cohomological Pairings

Jie Zhou

TL;DR

This work extends the Li–Zhou regularization of divergent configuration-space integrals on Riemann surfaces and frames it cohomologically. By treating regularized integrals as trace pairings on current cohomology and, for factorizable forms, via Deligne's mixed Hodge structure, the authors provide conceptual explanations for modularity and pole-reduction phenomena and deliver practical smooth-form representatives through conjugate Dolbeault–Cech tools. The main contributions are (i) extending the regularized integral to a broader cohomological setting with a trace map on currents, (ii) establishing two cohomological formulations for regularization (∂- and d-cohomology), (iii) a mixed Hodge-theoretic treatment for factorizable forms including explicit splittings on spectral-sequence pages, and (iv) concrete constructions and examples illustrating the approach. The results offer a principled framework to compute and compare regularized integrals on configuration spaces, with potential applications to 2D CFTs and related geometric-analytic structures.

Abstract

We extend the notion of regularized integrals introduced by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. We then give cohomological formulations for the extended notion using the tools of current cohomology and mixed Hodge structures. We also provide practical ways of constructing representatives of the corresponding cohomology classes in terms of smooth differential forms.

Regularized Integrals on Configuration Spaces of Riemann Surfaces and Cohomological Pairings

TL;DR

This work extends the Li–Zhou regularization of divergent configuration-space integrals on Riemann surfaces and frames it cohomologically. By treating regularized integrals as trace pairings on current cohomology and, for factorizable forms, via Deligne's mixed Hodge structure, the authors provide conceptual explanations for modularity and pole-reduction phenomena and deliver practical smooth-form representatives through conjugate Dolbeault–Cech tools. The main contributions are (i) extending the regularized integral to a broader cohomological setting with a trace map on currents, (ii) establishing two cohomological formulations for regularization (∂- and d-cohomology), (iii) a mixed Hodge-theoretic treatment for factorizable forms including explicit splittings on spectral-sequence pages, and (iv) concrete constructions and examples illustrating the approach. The results offer a principled framework to compute and compare regularized integrals on configuration spaces, with potential applications to 2D CFTs and related geometric-analytic structures.

Abstract

We extend the notion of regularized integrals introduced by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. We then give cohomological formulations for the extended notion using the tools of current cohomology and mixed Hodge structures. We also provide practical ways of constructing representatives of the corresponding cohomology classes in terms of smooth differential forms.
Paper Structure (30 sections, 18 theorems, 164 equations, 3 figures)

This paper contains 30 sections, 18 theorems, 164 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review on regularized integrals on configuration spaces of Riemann surfaces
  3. Preliminaries
  4. Poincaré, Leray, and holomorphic residue maps
  5. Currents
  6. Regularized integrals on configuration spaces of Riemann surfaces
  7. $\partial$-cohomology, regularized integrations as currents
  8. $d$-cohomology, regularized integrations as iterated residues
  9. Regularized integration as trace map on current cohomology
  10. Comparison theorems
  11. Extension of regularized integrals
  12. Extension
  13. Relation to principal value currents
  14. Trace map on current cohomology
  15. Representatives of current cohomology classes in terms of smooth forms
  16. ...and 15 more sections

Key Result

Theorem A

Let the setting be as in eqnsettingofdivergentintegrals. Let $\mathfrak{r}: A^{n,n}_{X}(\log D)+\partial A^{n-1,n}_{X}(\star D)\rightarrow \mathbb{C}$ be an admissible regularized integration operator. Then there exists a linear map such that for any $\omega\in A^{n,n}_{X}(\log D)+\partial A^{n-1,n}_{X}(\star D)$ one has Here $[X]$ is the fundamental class of $X$, and $[\omega_{\mathrm{Dol}}]$ i

Figures (3)

  • Figure 1: Double complex computing $\check{H}^{1}(X,\overline{\Omega}^{1}_{X})$ when $X=C$.
  • Figure 2: Construction of liftings of the splitting to the $E_{0}$ and $E_{1}$ pages, with $n=q-j$.
  • Figure 3: Double complex computing $H^{1}(C,\Omega^{1}_{C})$.

Theorems & Definitions (69)

  • Definition 1.1
  • Definition 1.2: Li:2020regularized
  • Definition 1.3
  • Theorem A
  • Theorem B
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • ...and 59 more