Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, Lie theory (through the Kashiwara--Vergne conjecture), in works related to multiple zêta values, etc...
This question concerns number theoretic or motivic aspects of the coefficients of the Alekseev-Torossian associator. Even though there are plenty of associators (they form a torsor on the Grothendieck--Teichmuller group), there are essentially three explicitely known ones:
- the KZ associator, which is a generating series for multiple zêta values.
- the Deligne associator, which is a generating series for single-valued multiple zêta values.
- the Alekseev-Torossian (AT) associator, whose coefficients have been described by Furusho in https://arxiv.org/pdf/1708.03231.pdf (Theorem 3.3).
I find it hard to understand the number theoretic and/or motivic meaning of the coefficients of the AT associator. In particular, there is a formal series in one variable $x$ associated with every associator (the log of its Gamma function), whose coefficients are:
- zêta values for the KZ associator.
- singled valued zêta values for the Deligne associator. In particular the coefficient of $x^2$ is $0$.
- Bernoulli numbers for the AT associators. In particular, odd powers are $0$.
Is there any realization (in the sense of motives) that would send the odd motivic zêta values to $0$?
