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202, 65 – 87 (1999) Communications in Mathematical Physics © Springer-Verlag 1999 Topological Approach to Quantum Surfaces Toshikazu Natsume1 , Ryszard Nest2 1 2 School of Mathematics, Nagoya Institute of Technology, Showa-ku, Nagoya 466, Japan Mathematics Institute, University of Copenhagen, Universitetsparken 5, Copenhagen, DK-2100 Ø, Denmark Received: 11 June 1998 / Accepted: 28 July 1998 Abstract: We discuss a topological method to quantize closed surfaces. 1. Introduction One of the more exciting developments of the last decade is the introduction of noncommutative geometry, which subsumes under common structure both the classical Riemannian geometry and various noncommutative situations like discrete groups or their duals, C*-algebras associated to various questions of number theory, quantum mechanical systems and many more.

Let α be the unique connection on the four graphs consisting of the pair of the graphs appearing in Fig. 2, and σ be the connection defined above. Then, the following hold: 1) The eight connections 1, σ, σ 2 , (αα˜ − 1), σ(αα˜ − 1), (αα˜ − 1)σ, σ(αα˜ − 1)σ, ˜ 2 − 3αα˜ + 1) (αα) ˜ 2 − 3αα˜ + 1, σ((αα) are indecomposable and mutually inequivalent. 2) The six connections ˜ + 3α, (αα˜ − 1)σα α, σα, ααα ˜ − 2α, σ(ααα ˜ − 2α), (αα) ˜ 2 α − 4ααα are irreducible and mutually inequivalent. 3) σ(αα˜ − 1)σα ∼ = (αα˜ − 1)σα.

Columns) in the previous table, and g∗∗ ’s denote the vectors corresponding to g∗∗ ’s after being multiplied by suitable gauge numbers respectively. By seeing this table, we easily see that entries other than g∗∗ ’s are invariant to the transformation of (αα˜ − 1) −→ σ(αα˜ − 1)σ, which acts on the table as the relabeling xy → σ(x)σ(y). The remaining problem is whether we have a gauge unitary matrix u cc 2 corresponding to double edges c-c such that u(cc) gσ(∗)σ(∗) g∗∗ −→ or not. We can check by a simple computation that   3 (λ2 −2) 2 −1 λ2√ −2 − i c 6 λ 3  = 2 u 3 (λ2 −2) 2 λ2√ −2 −1 c 2 i i 2 + 6 λ 3 Exotic Subfactors of Finite Depth with Jones Indices 35 Table 5.

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Communications in Mathematical Physics - Volume 202 by A. Jaffe (Chief Editor)

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