Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0 Jones Jones 3 Chern-Simons Witten [2] 3 S 3 G = SU(2) Chern-Simons S CS [A; S 3 ] = k 4π S 3 tr (A da + 23 A A A ). (3) Chern-Simons 3 K Wilson W R [A; K] Z CS (S 3 ; K R ) = [da]w R [A; K]e is CS[A;S 3]. (4)
R j Witten j = /2 R = Wilson Jones J(K; q) = Z CS (S 3 ; K )/Z SU(2) CS (S 3 2πi ; ), q := e k+2. (5) j- R = (2j) Wilson n Jones J n (K; q) (n = 2j + ) J n (K; q) = Z CS (S 3 ; K (2j) )/Z SU(2) CS (S 3 ; (2j)). (6) Jones n [3, 4] 2π lim n n log J n(k; q = e 2πi n ) = Vol(S 3 \K). (7) Chern-Simons k k + 2 = n Jones 2 2. (7) S 3 \K 3 Thurston 3 S Figure 2: 3 Figure 2 8 4 2
M 3 g ij R ij = 2g ij. (8) Vol(M 3 ) π (S 3 \) =: π (K) 3 3 (8) α, β, γ (α + β + γ = π) 3 AdS 3 2 4 (α, β, γ) 2 0, 3 (shape parameter) Conformal ball model Upper half space model Geodesic Line Conformal Transformation Figure 3: T αβγ AdS 3 [5] Vol(T αβγ ) = Λ(α) + Λ(β) + Λ(γ), (9) Λ(θ) := θ 0 log 2 sin t dt. (0) Λ(θ) Lobachevsky Mosto M 3 π (M 3 ) Vol(M 3 ) 3
2.2 Figure 2 8 4 8 [6, 7] Figure Francis, "A Topological Pictur 4 4 S 3 R 3 8 2 Figure 4: 8 2 4
2. 2 i i = 8 2 =, =. () C A B D A' D' B' C' 2. solid torus 8 2 Meridian µ : = ( ) =. (2) 3 Longitude ν : ( 2 3 ) 2 ( ) 2 ( 2 3 ) 2 = (/) 2 =. (3) C A B D A' B' D' C' 5
8 (), (2), (3) 0 < arg, arg < π = = e πi/3. (4) (9) 2.3 Jones Vol(S 3 \4 ) = 6Λ(π/3)2, 0298832, (5) (7) 8 Jones [8] J n (4, q) = n k=0 j= k (q (n+j)/2 q (n+j)/2 )(q (n j)/2 q (n j)/2 ). (6) n, q = e (n := 2πi) k =: log k d (x; q) k = k i= ( xqi ) e (Li 2(x) Li 2 (x)) [ ] J n (4 ; e 2π /n ) d exp W 4 (), W4 () = Li 2 () + Li 2 ( ), (7) 0 0 = W 4 () = log( )+log( ) = e πi/3 log J n (4 ; q = e 2π /n ) 2π lim = 2Im[Li 2 ( 0 )] = 2, 0298832 = Vol(S 3 \4 ). (8) n n 3 A- 3. Dehn (, 0) (p, q) (p, q) Dehn W.Thurston (p, q) Dehn 3 6
Figure 5: Dehn Dehn x Bulk : ( )( ) = (9) Meridian : ( ) = x (20) Longitude : (/) 2 = (y/x) 2. (2) () Dehn (2) (3) 2 (x, y) 2 (x, y) A 4 (x, y) = y + y (x 2 x 2 x + x 2 ) = 0. (22) A K (x, y) A- SL(2; C) Hom(π (S 3 \K); SL(2; C)) A-x S 3 \K x Vol Chern-Simons CS Vol(S 3 \K x ) + ics(s 3 \K x ) = Vol(S 3 \K) + ics(s 3 \K) + ϕ K (x), (23) x dx ϕ K (x) = log y(x). (24) x ϕ K (x) Neumann-Zagier [9] Chern-Simons 3.2 [0, ] 2π lim n,k log J n (K; q = e 2πi/k ) k = S 0 (K; x) (25) S 0 (K; x) = Vol(S 3 \K) + ics(s 3 \K) + ϕ K (x). (26) 7
x := e 2πin/k K 8 Jones (6) J n (4 ; q) d e W 4 (x,)+o( 0), (27) W 4 (x, ) = Li 2 (x) Li 2 (x) + Li 2 (x ) Li 2 (x ) 2 (log )2 + πi log. (28) W K = 0 = e W 4 (x, 0 )/ = x + x 0 0, (29) = 0 S 0 (4 ; x) W 4 ( 0, x) = S 0 (4 ; x) = Vol(S 3 \4 ) + ics(s 3 \4 ) + ϕ 4 (x), (30) x Neumann-Zagier x S 0(4 ; x) x = log y, y = x 0 x 0, (3) (29) (3) 0 (x, y) (22) A- A 4 (x, y) 8 3.3 WKB Chern-Simons Wilson Chern-Simons [0] AJ [2] Jones q  K (ˆx, ŷ; q)j n (K; q) = 0, ŷˆx = qˆxŷ, (32) ŷf(n) = f(n + ), ˆxf(n) = q n f(n). (33) ÂK(ˆx, ŷ; q) A- q A-  K (x, y; q ) = A K (x, y). (34) 8
8 A- Â 4 (ˆx, ŷ; q) = 3 a j (ˆx; q)ŷ j, (35) j=0 a 0 = t3 ( ˆx)( qˆx)( q 2ˆx 2 )( q 3ˆx 2 ) q 3 ( ˆx)( ˆx 2 )( qˆx)( q ˆx 2 ), a = ( qˆx)( q 3ˆx 2 ) q 3ˆx 2 ( ˆx)( qˆx)( q ˆx 2 ) ( 2qˆx q ( + q 3 q q 2 )ˆx 2 (q + q 2 q 3 )ˆx 3 + 2qˆx 4 q 2ˆx 5), ( q 2ˆx 2 ) a 2 = q 2ˆx 2 ( ˆx 2 )( qˆx) ( 2ˆx + t 2 (q + q 2 q 3 )ˆx 2 + ( + q 3 q q 2 )ˆx 3 + 2q 3ˆx 4 q 3ˆx 5), a 3 =. A- q- ( J n (K; q) exp S 0(K; x) ) 2 δ log + S n+ (K; x) n, (36) WKB [3] Jones n=0 4 Jones [4] [5, 6, 7] A- Eynard-Orantin WKB (36) Jones x x x h x x x xh x x k x i k h i j j q q g = + Σ q g l J g x q l l Figure 6: Eynard-Orantin 9
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