Problem 3.25 — (T.

Definitions and mathematical statement

The source writes $\mathcal M_G(\Sigma)$ for its moduli space and $C(\mathcal M_G(\Sigma))$ for an associated function algebra. A common setting is the character/flat-connection moduli space $\operatorname{Hom}(\pi_1\Sigma,G)/G$, with an appropriate Poisson structure, but the group, quotient and function category are not specified in the record.

A formal deformation quantization of a Poisson algebra $A$ has a product \[ f\star g=fg+\sum_{j\geq1}\hbar^jB_j(f,g), \] with associative $\star$ and first-order commutator recovering the Poisson bracket in the chosen normalization. The problem attributed to T. Kohno asks to clarify the relation between a deformation quantization of $C(\mathcal M_G(\Sigma))$ at a special parameter and the Wess--Zumino--Witten conformal-block space.

A WZW conformal-block space requires group/Lie-algebra, level, surface/complex-curve and any marked-point representation data. It is a vector space, whereas a deformation quantization is an algebra. A module action, an endomorphism-algebra quotient, and a vector-space identification are different proposed relations; none is specified here.

Moreover, a purely formal $\hbar$-adic series cannot automatically be evaluated at a chosen nonzero parameter. The quantization model must provide the required specialization, and the source does not name either that model, the special value or its relation to WZW level. Singular-moduli, compact-versus-complex group and continuous/smooth/algebraic function conventions also affect the comparison. No level shift, root-of-unity formula or canonical isomorphism is inferred from the broad request.

Short English statement

Specify how a particular specialized quantized moduli-space algebra relates to WZW conformal blocks, including parameter/level matching and whether the relation is an action, quotient or identification.

Sources

• Local UnsolvedMath registry, permanent ID 10400075, exact source code catalogue entry, Problem 3.25; source attribution T. Kohno, no proposer/year in metadata. Registry status partially solved, classification PARTIAL-PROGRESS; the title is truncated.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 3.25, PDF page 71, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. The source explicitly says section 3.10 was written by T. Kohno. Supplied pdf-font-structure extraction accessed July 29, 2026; provenance distinguishes 231 bold numbered items. No PDF was accessed here.
• The summary calls the Chern--Simons/WZW connection well studied but provides no exact quantization, parameter, theorem or later reference resolving this particular request.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW.