Problem 3.26 — (T.

Definitions and mathematical statement

The primary request is to determine the image and kernel of an "above map" $\tau$, whose definition is omitted. The summary describes it schematically as a map from $\mathcal A_\Sigma$ to functions on $\mathcal M_G(\Sigma)$. The diagram algebra, group, moduli space, function category and construction are absent, so they remain placeholders. Once these are fixed, the target is a complete description of \[ \operatorname{im}\tau=\{\tau(a):a\in\mathcal A_\Sigma\},\qquad \ker\tau=\{a\in\mathcal A_\Sigma:\tau(a)=0\} \] in the intended linear/algebraic setting, not merely examples of values or relations.

The excerpt supplies conformal-block background. Let $p_1,\ldots,p_n$ be marked points on the intended Riemann surface $\Sigma$, let $\mathfrak g$ be the Lie algebra and $H_i$ representations of its affine extension $\widehat{\mathfrak g}$. Let $\mathfrak g_{\mathrm{out}}$ consist of meromorphic $\mathfrak g$-valued functions with poles only at the marked points, acting on $V=H_1\otimes\cdots\otimes H_n$ by Laurent expansions. The given conformal blocks are \[ \{\phi:V\to\mathbb C\text{ linear}:\phi(a\cdot v)=0 \text{ for all }a\in\mathfrak g_{\mathrm{out}},v\in V\}. \] Thus in this formulation they are dual to the coinvariant quotient $V/\mathfrak g_{\mathrm{out}}V$, not literally the quotient itself. The twisted version replaces meromorphic functions by meromorphic sections of a $\mathfrak g$-local system.

The levels, module restrictions, coordinates/local-system data and relationship of these blocks to $\tau$ are not supplied. No kernel formula is inferred merely from this background. The summary says image/kernel are partially understood but gives no exact theorem; that partial-progress label remains an unverified source assessment pending primary-literature review.

Short English statement

Determine the full image and kernel of the source's map $\tau$. The attached WZW conformal-block definition is retained, but the map and its connection to that background are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400076$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 3.26, attributed to T. Kohno, PDF page 72: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No equation for $\tau$, precise partial result or separate resolution reference is supplied.