Communications In Mathematical Physics - Volume 283 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Rm + , b0m + , B (m + ) ) ) −n − ( (z −n − , a0 , A(−n − ) ), . . , (z −1 , a0−1 , A(−1) ) | (z 1 , a01 , A(1) ), . . 4) ˆ b(i) ∈ R× , B (i) ∈ R and z j ∈ H, a ( j) ∈ C× , A( j) ∈ C for all where ri ∈ R, 0 0 k l i = −m − , . . , −1, 1, . . , m + , j = −n − , . . , −1, 1, . . 6) d (a0 )x d x x|x=w−z j . 4) as TS (m − , m + |n − , n + ). 4) to a different but conformal equivalent disk. It is clear that we have S(m − , m + |n − , n + ) = TS (m − , m + |n − , n + )/S L(2, R). Let S = {S(m − , m + |n − , n + )}m − ,m + ,n − ,n + ∈N .

4) to a different but conformal equivalent disk. It is clear that we have S(m − , m + |n − , n + ) = TS (m − , m + |n − , n + )/S L(2, R). Let S = {S(m − , m + |n − , n + )}m − ,m + ,n − ,n + ∈N . 8) 42 L. Kong act naturally on S(m − , m + |n − , n + ). There are so-called boundary sewing operations [HKo1,Ko2] on S, denoted as i ∞−B j , which sews the i th positively oriented boundary puncture of the first disk with the j th negatively oriented boundary puncture of the second disk. ,i m (m + ) (m + ) (m + ) |l− , l+ ) , k+ +) −−−−−−→ S(m − − m + + k− , k+ |n − + l− , n + + l+ ), 1 (1) (m ) (1) (m ) (r ) where k± = k± + · · · + k± + , l± = l± + · · · + l± + for 1 ≤ ir ≤ k− , r = 1, .

N + ; 2. If ∃ k ∈ {−n − , . . , −1, 1, . . 32) by exchanging the 1op with wα(k) . 38 L. 15. νop is S L(2, R) invariant. Proof. 10. ˜ c → ER+ , which is still denoted as νop . Some interesting Hence νop induces a map D Vop special cases are listed explicitly below: ˆ = (1op , 1op )op idC , νop (ψD (H)) νop (ψD (πD (( (∞, 1, 0)| )))) = 1op , νop (ψD (πD (( |(∞, 1, 0), (0, 1, 0) ))))(u ⊗ v) = (u, v)op , νop (ψD (πD (( (∞, 1, 0), (0, 1, 0)| )))) = op , νop (ψD (πD (( (∞; 1, 0)|(r ; 1, 0), (0; 1, 0) ))))(u ⊗ v) = Yop (u, r )v, ˆ is the single element in D(0, 0) and r > 0.

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