We prove that two subfactors $N \subset M$ and $P \subset Q$
arising from a non-degenerate commuting square have 3-dimensional
topological quantum field theories (based on triangulations)
complex conjugate to each other. Applying the asymptotic
inclusion construction to each subfactor, we have new subfactors
$M \vee (M' \cap M_{\infty}) \subset M_{\infty}$ and $Q \vee (Q'
\cap Q_{\infty}) \subset Q_{\infty}$, then we also prove that the
tensor categories of the $M_{\infty}$-$M_{\infty}$ bimodules and
the $Q_{\infty}$-$Q_{\infty}$ bimodules are isomorphic to each
other, if the two fusion graphs are connected. These results are
based on our previous work and give a finer answer to a question
raised by V. F. R. Jones in June, 1995.
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(C)2000 Nobuya Sato [e-mail:nobuya@mi.cias.osakafu-u.ac.jp]