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Vanishing of Witten zeta function at negative integers

Kam Cheong Au

TL;DR

This work establishes a robust analytic framework for the Witten zeta function $\zeta_\Phi(s)$ of a root system by deriving a Hurwitz-zeta–based integral representation linked to the Poincaré polynomial $P_\Phi(s)$. Using a key analytic Lemma on integrals of products of linear forms, the authors prove that $\zeta_\Phi(s)$ vanishes to order at least the rank at negative even integers, with refined odd-integer vanishing for certain types, and they describe the leading term in terms of products of zeta-values controlled by the highest root data. A two-part program then yields both analytic and combinatorial descriptions of the leading coefficient: (i) an analytical part showing the coefficient lies in a rational span determined by Hurwitz zeta values, and (ii) a combinatorial part determining the denominators $\mathcal{D}(\Phi)$ as $\mathcal{H}(\Phi^{\vee})\cup\{1\}$ (equivalently $\mathcal{H}(\Phi)\cup\{1\}$). The results connect Weyl-group invariants, highest-root coefficients, and multiple zeta values, providing a conceptual route to the conjectured optimal vanishing and to explicit relations among zeta-values for classical root systems.

Abstract

We prove Witten zeta function of a root system $Φ$ has high-order vanishing at negative even integers, using an integral representation involving the Hurwitz zeta function. This settles a conjecture of Kurokawa and Ochiai for a large class of compact Lie groups. We also provide a qualitative description of the corresponding leading coefficient in terms of Riemann zeta values, in which the highest root of $Φ$ makes a natural appearance.

Vanishing of Witten zeta function at negative integers

TL;DR

This work establishes a robust analytic framework for the Witten zeta function of a root system by deriving a Hurwitz-zeta–based integral representation linked to the Poincaré polynomial . Using a key analytic Lemma on integrals of products of linear forms, the authors prove that vanishes to order at least the rank at negative even integers, with refined odd-integer vanishing for certain types, and they describe the leading term in terms of products of zeta-values controlled by the highest root data. A two-part program then yields both analytic and combinatorial descriptions of the leading coefficient: (i) an analytical part showing the coefficient lies in a rational span determined by Hurwitz zeta values, and (ii) a combinatorial part determining the denominators as (equivalently ). The results connect Weyl-group invariants, highest-root coefficients, and multiple zeta values, providing a conceptual route to the conjectured optimal vanishing and to explicit relations among zeta-values for classical root systems.

Abstract

We prove Witten zeta function of a root system has high-order vanishing at negative even integers, using an integral representation involving the Hurwitz zeta function. This settles a conjecture of Kurokawa and Ochiai for a large class of compact Lie groups. We also provide a qualitative description of the corresponding leading coefficient in terms of Riemann zeta values, in which the highest root of makes a natural appearance.

Paper Structure

This paper contains 9 sections, 27 theorems, 151 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An integral representation of $\zeta_\Phi(s)$
  3. A lemma and deduction of Theorem \ref{['vanishing_theorem']}
  4. Proof of Lemma \ref{['KeyLemma']}
  5. Leading term of $I_\Phi(s)$ at negative integer
  6. Proof of Theorem \ref{['rationality_theorem']} when $\Phi = A_2$
  7. Proof of Theorem \ref{['rationality_theorem']} (analytical part)
  8. Proof of Theorem \ref{['rationality_theorem']} (combinatorial part)
  9. Determining $\mathcal{D}(\Phi)$

Key Result

Theorem 1.2

(a) The order of vanishing of $\zeta_\Phi(s)$ at negative even integer is at least the rank of $\Phi$, that is, (b) If $\Phi$ is one of then $\zeta_\Phi(s)$ vanishes at negative odd integers. More precisely,

Figures (6)

  • Figure 1: The contour $C$.
  • Figure 2: Barycentric subdivision of a $2$-simplex, with $V_i$ and $p_i$ labeled.
  • Figure 3: The contour $C(\theta)$.
  • Figure 4: Triangulation of square $[0,1]^2$ for $I_{B_2}(s)$.
  • Figure 5: Illustrative case when $n=3$.
  • ...and 1 more figures

Theorems & Definitions (58)

  • Conjecture 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 1.7
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • ...and 48 more