Problem 9.7 — (Y.

Definitions and mathematical statement

Let $\mathcal C_1$ be a semisimple ribbon category with degenerate $S$-matrix, satisfying the additional hypotheses of the source's Problem 9.6. In the usual finite setting the entries of $S$ are categorical traces of double braidings between simple objects; degeneracy means that $S$ is not invertible. Let $\mathcal C_2$ be the modular tensor category obtained from $\mathcal C_1$ by the particular method in [288]. Write $(V_1,Z_1)$ and $(V_2,Z_2)$ for the two TQFTs referred to in the source as arising from these categories.

Kawahigashi asks to determine their relationship. A precise comparison must first identify the two constructions and their bordism/extra-structure domains, and then state the relation between their state spaces, cobordism maps and closed-manifold values. The source specifies no equality, natural transformation, quotient, extension or numerical formula, so none is asserted as its intended answer. In particular, one cannot use the ordinary modular-category construction to define the first theory merely by treating $S_1$ as invertible: its degeneracy is an explicit hypothesis, and the missing Problem 9.6 must explain the first TQFT. Likewise the method [288] is part of the input, not an arbitrary choice among modular categories associated with $\mathcal C_1$.

Short English statement

Compare the TQFT from a specified degenerate semisimple ribbon category with the TQFT from its modular category constructed by [288]. The missing hypotheses and construction references are essential to identify both theories.

Sources

• Local UnsolvedMath registry, permanent ID $10400171$, exact source code catalogue entry.
• Y. Kawahigashi, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 9.7, PDF page 140: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access 2026-07-29, pdf-font-structure extraction.
• Problem 9.6 and internal reference [288] are invoked by the source; their content and full bibliography are not supplied.