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A Trace-Path Integral Formula over Function Fields

Yan Yau Cheng

TL;DR

The paper establishes a function-field analogue of trace--path integral relations by proving that the Frobenius trace on a geometric quantisation space $\mathcal{H}$ attached to the $\ell$-torsion $J[\ell]$ of a Jacobian equals a discrete arithmetic path integral over $J[\ell](\mathbb F_q)$ weighted by a pairing $A$ from class field theory, up to an explicitly determined sign. The approach blends physical intuition from topological quantum field theory with arithmetic geometry: $\mathcal{H}$ is realized as global sections of a theta line bundle, the Frobenius action is analyzed via the Heisenberg group $H(J[\ell])$, and the arithmetic action $A$ is shown to coincide with the Abelian Chern–Simons pairing. Under the assumption that the Frobenius acts semisimply on $J[\ell]$, the authors give an explicit formula for the trace in terms of the Legendre symbol and a determinant attached to the action, and compute the arithmetic path integral via finite-field Gauss sums, yielding a real-valued result. This work sharpens the arithmetic–quantum analogy and provides a concrete, sign-determined trace formula linking geometry over finite fields, representation theory of Heisenberg groups, and a function-field analogue of Chern–Simons theory, with potential avenues for generalisation beyond the current $\,\mu_{\ell}\subset \mathbb F_q$ constraint.

Abstract

We show that an arithmetic path integral over the $\ell$-torsion of a Jacobian $J[\ell]$ is equal to the trace of the Frobenius action on a representation of the Heisenberg group $H(J[\ell])$, up to an explicitly determined sign. This is an arithmetic analogue of trace--path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space.

A Trace-Path Integral Formula over Function Fields

TL;DR

The paper establishes a function-field analogue of trace--path integral relations by proving that the Frobenius trace on a geometric quantisation space attached to the -torsion of a Jacobian equals a discrete arithmetic path integral over weighted by a pairing from class field theory, up to an explicitly determined sign. The approach blends physical intuition from topological quantum field theory with arithmetic geometry: is realized as global sections of a theta line bundle, the Frobenius action is analyzed via the Heisenberg group , and the arithmetic action is shown to coincide with the Abelian Chern–Simons pairing. Under the assumption that the Frobenius acts semisimply on , the authors give an explicit formula for the trace in terms of the Legendre symbol and a determinant attached to the action, and compute the arithmetic path integral via finite-field Gauss sums, yielding a real-valued result. This work sharpens the arithmetic–quantum analogy and provides a concrete, sign-determined trace formula linking geometry over finite fields, representation theory of Heisenberg groups, and a function-field analogue of Chern–Simons theory, with potential avenues for generalisation beyond the current constraint.

Abstract

We show that an arithmetic path integral over the -torsion of a Jacobian is equal to the trace of the Frobenius action on a representation of the Heisenberg group , up to an explicitly determined sign. This is an arithmetic analogue of trace--path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space.

Paper Structure

This paper contains 16 sections, 26 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Physical Background
  3. The Path Integral formalism
  4. Twists and Trace of Monodromy
  5. Geometric Quantisation
  6. Trace of Frobenius
  7. Representations of the Heisenberg group
  8. Canonical intertwining morphisms
  9. Symplectic actions on the Heisenberg representations
  10. Invariant Spaces of a Symplectic Vector Space
  11. Determining the sum
  12. An Arithmetic Path Integral
  13. Defining an Arithmetic Action
  14. Relation to the Abelian Chern-Simons pairing
  15. Computation of the Arithmetic Path Integral
  16. ...and 1 more sections

Key Result

Theorem A

Let $J$ be the Jacobian of a genus $g$ curve $X$ over a finite field $\mathbb F_q$. For primes $\ell$ satisfying $q\equiv 1\pmod \ell$, supposing that $\mathop{\mathrm{Fr}}\nolimits_q$ acts semisimply on the $\mathbb F_\ell$ vector space $J[\ell]$. Then there is an equality Where $\overline{\chi_{\mathop{\mathrm{Fr}}\nolimits_q}}$ is the largest monic polynomial dividing the characteristic polyno

Theorems & Definitions (55)

  • Theorem A: \ref{['thm:MainTheoremFull']}
  • Definition 1
  • Theorem 2
  • Definition 3
  • Lemma 5
  • proof
  • Lemma 7
  • proof
  • Corollary 8
  • proof
  • ...and 45 more