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Multiple q-zeta values and traces

Zhenbo Qin

TL;DR

This work extends Bloch–Okounkov’s trace result by showing that a broad class of traces formed from products of $q$-Pochhammer terms, under the constraint $\prod x_j=\prod y_j$, yields generating series whose coefficients are multiple $q$-zeta values. The author develops and exploits the algebra of bi-brackets, Brackets/Brackets, and the spaces ${\rm qMZV}$, ${\bf QM}$, linking them to quasi-modular forms and proving that the traces live in ${\bf qMZV}$-valued formal power series with precise weight bounds. The main theorems establish the simple two-variable case and then generalize to traces $\mathfrak P_N^{a,b}$ for arbitrary $N$, with rigorous weight and degree controls, using Bloch–Okounkov derivative expansions and intricate combinatorial reductions. These results illuminate the structure of traces in representation theory and geometry, align with Okounkov’s conjectures tying $q$-zeta values to geometric invariants, and open pathways for applications to Hilbert schemes and related moduli problems.

Abstract

Let $(a)_\infty = (a; q)_\infty = \prod_{n=0}^\infty (1-aq^n)$. An elegant result of Bloch and Okounkov [BO] states that if $x = e^z$, then $$ \frac{(xq)_\infty (x^{-1}q)_\infty}{(q)_\infty^2}, $$ which appears in various traces in representation theory and algebraic geometry, is a formal power series in $z^2$ whose coefficient for $z^{2k}$ is a quasi-modular form of weight $2k$. Quasi-modular forms are special types of multiple $q$-zeta values. In this paper, we generalize this result of Bloch and Okounkov and prove that certain other traces are related to multiple $q$-zeta values. A simple case of our main results asserts that if $x = e^z$ and $y = e^w$, then $$ \frac{(xq)_\infty (yq)_\infty}{(q)_\infty (xyq)_\infty}, $$ which appears in [CW, Theorem 5] as a trace (the deformed Bloch-Okounkov $1$-point function), is a formal power series in $z$ and $w$ whose coefficient for $z^mw^n$ is a multiple $q$-zeta value (in the sense of [BK3, Oko]) of weight $(m+n)$.

Multiple q-zeta values and traces

TL;DR

This work extends Bloch–Okounkov’s trace result by showing that a broad class of traces formed from products of -Pochhammer terms, under the constraint , yields generating series whose coefficients are multiple -zeta values. The author develops and exploits the algebra of bi-brackets, Brackets/Brackets, and the spaces , , linking them to quasi-modular forms and proving that the traces live in -valued formal power series with precise weight bounds. The main theorems establish the simple two-variable case and then generalize to traces for arbitrary , with rigorous weight and degree controls, using Bloch–Okounkov derivative expansions and intricate combinatorial reductions. These results illuminate the structure of traces in representation theory and geometry, align with Okounkov’s conjectures tying -zeta values to geometric invariants, and open pathways for applications to Hilbert schemes and related moduli problems.

Abstract

Let . An elegant result of Bloch and Okounkov [BO] states that if , then which appears in various traces in representation theory and algebraic geometry, is a formal power series in whose coefficient for is a quasi-modular form of weight . Quasi-modular forms are special types of multiple -zeta values. In this paper, we generalize this result of Bloch and Okounkov and prove that certain other traces are related to multiple -zeta values. A simple case of our main results asserts that if and , then which appears in [CW, Theorem 5] as a trace (the deformed Bloch-Okounkov -point function), is a formal power series in and whose coefficient for is a multiple -zeta value (in the sense of [BK3, Oko]) of weight .

Paper Structure

This paper contains 5 sections, 13 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and notations of multiple $q$-zeta values
  3. Proof of Theorem \ref{['Intro-AAA06261252pm']}
  4. The basic case $\mathfrak P_{2}^{0,0}$
  5. Proof of Theorem \ref{['Intro-yij1toNm>0']}

Key Result

Theorem 1.1

Let $(a)_\infty = (a; q)_\infty = (1-a)(1-aq)(1-aq^2) \cdots$. Fix $r \ge 1$. For $1 \le j \le r$, let $x_j = e^{z_j}$ and $y_j = e^{w_j}$. If $x_1 \cdots x_r = y_1 \cdots y_r$, then the constant term is $1$, and the coefficient of $z_1^{m_1} \cdots z_r^{m_r} w_1^{n_1} \cdots w_{r-1}^{n_{r-1}}$ has weight

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • ...and 20 more