[Paper Review] Witten multiple zeta values attached to sl(4)

Jianqiang Zhao, Zhou Xia|arXiv (Cornell University)|Mar 13, 2009

Advanced Mathematical Identities15 references5 citations

TL;DR

This paper proves that all Witten multiple zeta values (WMZVs) of weight $ w > 3 $ attached to $ rak{sl}(4) $ at nonnegative integers are $ bQ $-linear combinations of multiple zeta values (MZVs) of weight $ w $ and depth $ \leq 3 $, except for nine irregular cases requiring additional Riemann zeta values $ \zeta(w-2) $ and double zeta values of weight $ w-1 $ and depth $ \leq 2 $. The result confirms a structural finiteness property for WMZVs in this Lie algebra setting.

ABSTRACT

In this paper we shall prove that every Witten multiple zeta value of weight w>3 attached to sl(4) at nonnegative integer arguments is a finite rational linear combinations of MZVs of the weight w and the depths three or less, except for the nine irregular cases where the Riemann zeta value zeta(w-2) and the double zeta values of weight w-1 and depth <3 are also needed.

Motivation & Objective

• To determine whether Witten multiple zeta values (WMZVs) attached to $\frak{sl}(4)$ at nonnegative integers can be expressed as $\bbQ$-linear combinations of multiple zeta values (MZVs).
• To investigate the algebraic structure of WMZVs of weight $ w > 3 $, particularly their dependence on MZVs of bounded depth.
• To identify and characterize the exceptional 'irregular' cases where additional zeta values beyond depth-3 MZVs are required.
• To verify the conjecture that the space of WMZV special values of weight $ w > 3 $ is contained in $ \mathcal{MZV}(w,\leq 3) \oplus \mathcal{MZV}(w-1,\leq 2) \oplus \mathcal{MZV}(w-2,1) $, with numerical verification up to weight 12.
• To provide explicit evaluations of specific WMZV values, demonstrating their expression in terms of MZVs and classical zeta values.

Proposed method

• The authors analyze the generalized multiple zeta function $ \zeta_3(\mathbf{s}) $ associated with $ \frak{sl}(4) $, defined via sums over positive integers $ m_1, m_2, m_3 $, with denominators corresponding to sums of consecutive $ m_i $'s raised to powers $ s_{i,j} $.
• They use convergence criteria for generalized MZVs, ensuring convergence when the sum of exponents over all subsets containing a given initial segment exceeds the length of that segment.
• The proof relies on algebraic relations among MZVs and the structure of the Witten zeta function as a special case of the generalized multiple zeta function with non-zero exponents only for consecutive index intervals.
• Explicit evaluations are derived using known identities and numerical verification via Maple and EZface, with results expressed as $ \bbQ $-linear combinations of MZVs and classical zeta values.
• The authors classify the nine irregular cases where $ \zeta(w-2) $ and depth-$ \leq 2 $ double zeta values of weight $ w-1 $ are required, distinguishing them from the regular case.
• A conjecture is formulated and verified numerically for weights up to 12, stating that the space of WMZV values of weight $ w > 3 $ is the direct sum of $ \mathcal{MZV}(w,\leq 3) $, $ \mathcal{MZV}(w-1,\leq 2) $, and $ \mathcal{MZV}(w-2,1) $.

Experimental results

Research questions

• RQ1 Are all Witten multiple zeta values of weight $ w > 3 $ attached to $ \frak{sl}(4) $ at nonnegative integers expressible as $ \bbQ $-linear combinations of MZVs of weight $ w $ and depth $ \leq 3 $?
• RQ2 Which specific cases require additional zeta values beyond depth-3 MZVs, and what are the precise forms of these exceptional contributions?
• RQ3 Can the space of special values of $ \zeta_{\frak{sl}(4)} $ of weight $ w > 3 $ be decomposed as a direct sum of $ \mathcal{MZV}(w,\leq 3) $, $ \mathcal{MZV}(w-1,\leq 2) $, and $ \mathcal{MZV}(w-2,1) $?
• RQ4 How do explicit evaluations of WMZVs (e.g., $ \zeta_{\frak{sl}(4)}(0,0,0,0,0,4) $) confirm the theoretical structure and reveal dependence on $ \zeta(2)^2 $, $ \zeta(3) $, and $ \zeta(2) $?
• RQ5 To what extent do numerical computations using symbolic tools like Maple and EZface validate the theoretical expressions and conjectures for weights up to 12?

Key findings

• Every Witten multiple zeta value of weight $ w > 3 $ attached to $ \frak{sl}(4) $ at nonnegative integers is a $ \bbQ $-linear combination of MZVs of weight $ w $ and depth $ \leq 3 $, except for nine irregular cases.
• The nine irregular cases require the inclusion of the Riemann zeta value $ \zeta(w-2) $ and double zeta values of weight $ w-1 $ and depth $ \leq 2 $, which are not expressible solely via depth-3 MZVs.
• Explicit evaluations show that $ \zeta_{\frak{sl}(4)}(0,0,0,0,0,4) = \frac{1}{2}\zeta(2) - \frac{3}{2}\zeta(3) + \frac{2}{5}\zeta(2)^2 $, confirming dependence on $ \zeta(2)^2 $ and $ \zeta(3) $.
• For higher weights, values such as $ \zeta_{\frak{sl}(4)}(1,1,1,1,1,1) = -\frac{62}{105}\zeta(2)^3 + 2\zeta(3)^2 $ demonstrate that higher-weight MZVs can be expressed purely in terms of $ \zeta(2)^3 $ and $ \zeta(3)^2 $.
• The value $ \zeta_{\frak{sl}(4)}(2,2,2,2,2,2) = \frac{368}{875875}\zeta(2)^6 $ matches known results from previous literature, validating the consistency of the framework.
• The conjecture that the space of WMZV values of weight $ w > 3 $ is $ \mathcal{MZV}(w,\leq 3) \oplus \mathcal{MZV}(w-1,\leq 2) \oplus \mathcal{MZV}(w-2,1) $ has been numerically verified for all weights up to 12 using the MZV table and symbolic computation tools.