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Aspects of Scattering Amplitudes and Moduli Space Localization

Sebastian Mizera

TL;DR

The paper presents a geometric framework in which tree-level scattering amplitudes are computed as intersection numbers of twisted differential forms on the moduli space ${\cal M}_{0,n}$, with the twist encoded by a local system determined by external momenta. It develops a recursion scheme based on the fibred structure of ${\cal M}_{0,n}$ and braid-matrix data to integrate punctures and reduce intersection numbers to linear-algebra problems, enabling explicit analytic calculations of amplitudes. In the logarithmic (massless) sector, the framework yields simple trivalent-diagram expansions and recovers CHY/Scattering Equations representations, while non-logarithmic twists encode massive states; string-theory amplitudes (open/closed) and KLT relations arise naturally from the duality structure of twisted (co)homologies. The work thereby unifies the genus-zero string and field-theory S-matrices within a single geometric- topological model and provides concrete recursion tools for broad classes of amplitudes, including gauge, gravity, and Kac–Moody correlator inputs. It also clarifies how dualities among homology and cohomology groups generate identities among amplitudes and lays groundwork for future extensions to supermoduli and higher genera.

Abstract

We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant cohomology classes are twisted by representations of the fundamental group that describes how punctures braid around each other on the Riemann surface, which can be used to link it to the space of kinematic invariants. Intersection numbers of said cohomology classes, whose representatives we call twisted forms, can be shown to fully localize on the boundaries of the moduli space, which are in one-to-one map with Feynman diagrams. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams allowing only for massless propagators on the internal and external lines. For physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of the moduli space, which allows for a direct access to the boundaries, to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of the moduli space and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with the previous reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space.

Aspects of Scattering Amplitudes and Moduli Space Localization

TL;DR

The paper presents a geometric framework in which tree-level scattering amplitudes are computed as intersection numbers of twisted differential forms on the moduli space , with the twist encoded by a local system determined by external momenta. It develops a recursion scheme based on the fibred structure of and braid-matrix data to integrate punctures and reduce intersection numbers to linear-algebra problems, enabling explicit analytic calculations of amplitudes. In the logarithmic (massless) sector, the framework yields simple trivalent-diagram expansions and recovers CHY/Scattering Equations representations, while non-logarithmic twists encode massive states; string-theory amplitudes (open/closed) and KLT relations arise naturally from the duality structure of twisted (co)homologies. The work thereby unifies the genus-zero string and field-theory S-matrices within a single geometric- topological model and provides concrete recursion tools for broad classes of amplitudes, including gauge, gravity, and Kac–Moody correlator inputs. It also clarifies how dualities among homology and cohomology groups generate identities among amplitudes and lays groundwork for future extensions to supermoduli and higher genera.

Abstract

We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant cohomology classes are twisted by representations of the fundamental group that describes how punctures braid around each other on the Riemann surface, which can be used to link it to the space of kinematic invariants. Intersection numbers of said cohomology classes, whose representatives we call twisted forms, can be shown to fully localize on the boundaries of the moduli space, which are in one-to-one map with Feynman diagrams. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams allowing only for massless propagators on the internal and external lines. For physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of the moduli space, which allows for a direct access to the boundaries, to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of the moduli space and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with the previous reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space.

Paper Structure

This paper contains 29 sections, 10 theorems, 402 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Intersection Numbers of Twisted Differential Forms
  3. Basics of Moduli Spaces of Punctured Spheres
  4. From Local Systems to Twisted Forms
  5. Definition and Properties of Intersection Numbers
  6. Relations to Other Objects
  7. Open-String Amplitudes
  8. Scattering Equations
  9. Logarithmic Forms, Boundaries, and Dihedral Coordinates
  10. Comment on Shift Relations for Local Systems
  11. Recursion Relations from Braid Matrices
  12. Fibred Moduli Spaces and Twisted Cohomologies
  13. Recursion Relations
  14. Solving for Twisted Cohomologies on All Fibres
  15. Braid Matrices
  16. ...and 14 more sections

Key Result

Proposition 2.1

Let $U\cong V \cong W \cong X$ be four isomorphic complex vector spaces with non-degenerate bilinear pairings denoted by $\langle u | v \rangle$, $\langle u | x \rangle$, $\langle w | x \rangle$, $\langle w | v \rangle$ for $u\in U, v\in V, w\in W, x\in X$, which are normalized such that for every $ for orthonormal bases $\{ v_b^\vee \}_{b=1}^{\dim V} {\in }V$, $\{ w_c^\vee \}_{c=1}^{\dim W} {\in

Figures (1)

  • Figure 2.1: Example Morse function for $n{=}4$. Here we fixed $(z_1,z_3,z_4)=(0,1,2)$ and used massless kinematics with $p_2 {\cdot} p_1 <0$ and $p_2 {\cdot} p_3, p_2 {\cdot} p_4 > 0$. The paths of steepest descent ${\cal J}_1$ and ascent ${\cal K}_1$ (indicated in white) extend from the only critical point located at $z_2^\ast = 2 p_2 {\cdot} p_1 / p_2 {\cdot} (2p_1 {+} p_3)$.

Theorems & Definitions (31)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.1
  • Example 2.2
  • Lemma 3.1
  • ...and 21 more