Turaev-Viro TQFTs via Planar Algebras

Transcription

1 Turaev-Viro TQFTs via Planar lgebras hris Schommer-Pries epartment of Mathematics, MIT July 12 th, 2011 hris Schommer-Pries (MIT) July 12 th, / 30

2 How To onstruct a TQFT: 1 lgebraic Input 2 ombinatorial escription of Manifolds 3 Shake them together... hris Schommer-Pries (MIT) July 12 th, / 30

3 How To onstruct a TQFT: 1 lgebraic Input 2 ombinatorial escription of Manifolds 3 Shake them together... 1 Speherical Tensor ategory/planar lgebra 2 Triangulations of 3-Manifolds 3 The Turaev-Viro onstruction... hris Schommer-Pries (MIT) July 12 th, / 30

4 Triangulations of Manifolds Theorem (Pachner) ny two triangulations of d-manifolds can be related by a finite sequence of Pachner moves (k d + 2 k) 1 k d/2 + 1 hris Schommer-Pries (MIT) July 12 th, / 30

5 Triangulations of Manifolds Theorem (Pachner) ny two triangulations of d-manifolds can be related by a finite sequence of Pachner moves (k d + 2 k) 1 k d/ hris Schommer-Pries (MIT) July 12 th, / 30

6 3 Pachner Moves: The 2-3 Move: The 1-4 Move: hris Schommer-Pries (MIT) July 12 th, / 30

7 The Turaev-Viro onstruction Basic Idea: M = 3-manifold, = Spherical ategory. triangulate M 3, order the vertices. hris Schommer-Pries (MIT) July 12 th, / 30

8 The Turaev-Viro onstruction Basic Idea: M = 3-manifold, = Spherical ategory. triangulate M 3, order the vertices. Label triangulation with ata from Get a number for each tetrahedron. hris Schommer-Pries (MIT) July 12 th, / 30

9 The Turaev-Viro onstruction Basic Idea: M = 3-manifold, = Spherical ategory. triangulate M 3, order the vertices. Label triangulation with ata from Get a number for each tetrahedron. Multiply the numbers for each tetrahedron. Sum over labelings. (Prove invariance under 2-3 moves, 1-4 moves, and reordering) Get a Manifold Invaraint! hris Schommer-Pries (MIT) July 12 th, / 30

10 The Turaev-Viro onstruction Basic Idea: M = 3-manifold, = Spherical ategory. triangulate M 3, order the vertices. Label triangulation with ata from Get a number for each tetrahedron. Multiply the numbers for each tetrahedron. Sum over labelings. (Prove invariance under 2-3 moves, 1-4 moves, and reordering) Get a Manifold Invaraint! Miracle: This invariant is not trivial! hris Schommer-Pries (MIT) July 12 th, / 30

11 Properties of Planar lgebras Z σ Simple Objects (,, Z, ) Morphisms hris Schommer-Pries (MIT) July 12 th, / 30

12 Properties of Planar lgebras Simple Objects (,, Z, ) Morphisms ual Morphisms µ Z hris Schommer-Pries (MIT) July 12 th, / 30

13 Properties of Planar lgebras Z Simple Objects (,, Z, ) σ Morphisms ual Morphisms µ Hermitian Inner Product (positive definite!) µ, σ Z Basis {σ i } for Hom(, Z) also gives a daul basis {σi }! for Hom(Z, ). hris Schommer-Pries (MIT) July 12 th, / 30

14 Basis for Hom(, Z) also gives basis for Hom( Z, ) (rotate in planar algebra!) hris Schommer-Pries (MIT) July 12 th, / 30

15 Basis for Hom(, Z) also gives basis for Hom( Z, ) (rotate in planar algebra!) and Hom(Z, ) σ Z hris Schommer-Pries (MIT) July 12 th, / 30

16 Is it orthonormal? σ Z σ hris Schommer-Pries (MIT) July 12 th, / 30

17 Is it orthonormal? σ Z σ Z d(z)d() 1 σ Z σ hris Schommer-Pries (MIT) July 12 th, / 30

18 Is it orthonormal? Z σ d(z)d() 1 Z σ hris Schommer-Pries (MIT) July 12 th, / 30

19 Is it orthonormal? Z σ σ d(z)d() 1 Z hris Schommer-Pries (MIT) July 12 th, / 30

20 Is it orthonormal? Z σ d(z)d() 1 Need to rescale by a factor of d(z) 1 d(). σ Z hris Schommer-Pries (MIT) July 12 th, / 30

21 How to Label a Tetrahedron and get a Number hoose representative simples, B,,,, Z,... hoose basis {σ i } for each Hom( B, ). Label edges with simple objects. Label faces with elements σ i. Then... hris Schommer-Pries (MIT) July 12 th, / 30

22 B σ 2 σ 3 σ2 σ 1 σ 1 B σ 3 σ 4 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

23 Get a formula... Z(M, }{{} τ = triangulation) edge lables face labels If orientation of T disagrees with M, use Z(T ). tetrahedra T Z(T ) How to prove invariance? Is it even invariant? hris Schommer-Pries (MIT) July 12 th, / 30

24 Reordering Vertices What happens when we reorder the vertices? B σ 2 σ 1 σ 3 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

25 Reordering Vertices What happens when we reorder the vertices? Original (13)-edge label: B, (02)-edge label: Z(T 0 1 )/Z(T ) = d(b) 1 d() 1 d()d() (13)-edge:, (02)-edge: B σ 1 σ 2 σ 3 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

26 Reordering Vertices What happens when we reorder the vertices? Original (13)-edge label: B, (02)-edge label: Z(T 0 1 )/Z(T ) = d(b) 1 d() 1 d()d() (13)-edge:, (02)-edge: Z(T 1 2 )/Z(T ) = d(b) 1 d() 1 d( )d( ) (13)-edge:, (02)-edge: B σ 1 σ 2 σ 3 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

27 Reordering Vertices What happens when we reorder the vertices? Original (13)-edge label: B, (02)-edge label: Z(T 0 1 )/Z(T ) = d(b) 1 d() 1 d()d() B σ 2 (13)-edge:, (02)-edge: Z(T 1 2 )/Z(T ) = σ 1 d(b) 1 d() 1 d( )d( ) (13)-edge:, (02)-edge: Z(T 1 3 )/Z(T ) = d(b) 1 d() 1 d()d(b) σ4 (13)-edge: B, (02)-edge: σ 3 hris Schommer-Pries (MIT) July 12 th, / 30

28 Reordering Vertices What happens when we reorder the vertices? Original (13)-edge label: B, (02)-edge label: Z(T 0 1 )/Z(T ) = d(b) 1 d() 1 d()d() (13)-edge:, (02)-edge: Z(T 1 2 )/Z(T ) = d(b) 1 d() 1 d( )d( ) (13)-edge:, (02)-edge: Z(T 1 3 )/Z(T ) = d(b) 1 d() 1 d()d(b) (13)-edge: B, (02)-edge: B σ 1 σ 2 σ 4 σ 3 orrection factor: d(b) 1 d() 1 hris Schommer-Pries (MIT) July 12 th, / 30

29 Invariance: Z(M, }{{} τ = triangulation) edge lables Vertex order 2-3 Pachner Move 1-4 Pachner Move face labels tetrahedra T Z corrected (T ) hris Schommer-Pries (MIT) July 12 th, / 30

30 First Identity: σ σ σ λ λ hris Schommer-Pries (MIT) July 12 th, / 30

31 Second Identity: B B µ µ σ σ,g G σ λ λ hris Schommer-Pries (MIT) July 12 th, / 30

32 2-3 Move hris Schommer-Pries (MIT) July 12 th, / 30

33 B H E F B E B F H hris Schommer-Pries (MIT) July 12 th, / 30

34 σ 3 σ 2 B σ 2 σ 1 B σ 1 σ 3 σ 5 σ 4 E σ 2 B F H σ 7 σ6 H σ 5 B σ 6 F E σ 2 σ 7 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

35 H σ 6 H σ 6 σ 5 F σ 5 F E E B σ 7 σ 7 σ2 σ 2 B B σ 2 σ 1 σ 1 σ 3 σ 3 σ 4 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

36 Second Half of 2-3 Move hris Schommer-Pries (MIT) July 12 th, / 30

37 σ 8 σ 7 E σ 5 E F σ 10 σ 1 G B H σ 8 G σ 9 σ 10 σ 4 G F σ 6 σ 9 H σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

38 H B σ 5 G σ 10 σ 1 E H σ 6 σ 8 E F G σ 8 σ 9 F σ 7 σ 9 G σ 3 H σ 10 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

39 H σ6 B σ5 σ1 E σ8 G F H σ 9 σ9 G σ 10 G σ10 σ8 F E σ7 σ 4 σ 3 hris Schommer-Pries (MIT) July 12 th, / 30

40 H σ6 B σ5 σ1 E σ8 G F H σ 9 σ9 G σ 10 G σ10 σ8 F E σ7 σ 4 σ 3 hris Schommer-Pries (MIT) July 12 th, / 30

41 H σ6 B σ1 σ5 E σ8 G σ9 F σ1 B σ5 H E σ6 H F σ9 G σ 10 σ8 G σ 8 G σ10 E σ8 F σ 7 E σ 7 σ 3 σ3 σ4 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

42 H σ6 H σ6 B σ1 σ5 E σ8 G σ9 F σ1 B σ5 H E σ6 B σ5 E F σ 7 H F σ9 G σ 10 σ8 G σ 8 σ1 G σ8 σ10 F E σ7 E σ3 σ 7 σ4 σ 3 σ3 σ4 σ 4 hris Schommer-Pries (MIT) July 12 th, / 30

43 What about the correction factors? orrection factor of first half (2) d(b) 1 d() 1 d(h) 1 d( ) 1 orrection factor of second half (3) d(b) 1 d(g) 1 d(h) 1 d() 1 d(g) 1 d( ) 1 hris Schommer-Pries (MIT) July 12 th, / 30

44 What about the correction factors? orrection factor of first half (2) d(b) 1 d() 1 d(h) 1 d( ) 1 orrection factor of second half (3) d(b) 1 d(g) 1 d(h) 1 d() 1 d(g) 1 d( ) 1 They differ by d(g) 1 hris Schommer-Pries (MIT) July 12 th, / 30

45 What about the correction factors? orrection factor of first half (2) d(b) 1 d() 1 d(h) 1 d( ) 1 orrection factor of second half (3) d(b) 1 d(g) 1 d(h) 1 d() 1 d(g) 1 d( ) 1 They differ by d(g) 1 Edge orrection Factor: edges d(edge label) hris Schommer-Pries (MIT) July 12 th, / 30

46 Invariance: Z(M, }{{} τ = triangulation) edge lables Vertex order 2-3 Pachner Move 1-4 Pachner Move face labels tetrahedra T Z cor (T ) ( edge factor ) edges hris Schommer-Pries (MIT) July 12 th, / 30

47 What about the 1-4 Move? hris Schommer-Pries (MIT) July 12 th, / 30

48 What about the 1-4 Move? Exercise Prove that the two sides of the 1-4 Move differ by a factor of w = E d(e) 2 where E ranges over a complete set of simples. hris Schommer-Pries (MIT) July 12 th, / 30

49 The Turaev-Viro Invariant Z(M, }{{} τ ) = triangulation Invariance: Vertex order 2-3 Pachner Move 1-4 Pachner Move 1 w #vertices edge lables face labels tetrahedra T Z cor (T ) ( edge factor ) edges hris Schommer-Pries (MIT) July 12 th, / 30

50 Exercise For the 2 planar algebra, for M 3 closed, Z(M) = #(H 1 (M; Z/2)). hris Schommer-Pries (MIT) July 12 th, / 30