Modular categories Higher Gauss sums Arithmetic properties Witt invariance Higher Gauss sums of modular categories Yilong Wang Louisiana State University [email protected] Joint work with Andrew Schopieray and Siu-Hung Ng April 13, 2019 Southern Regional Number Theory Conference 1/18 I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): 2/18 =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces 2/18 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 2/18 I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) 2/18 I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). 2/18 Modular categories Higher Gauss sums Arithmetic properties Witt invariance Fusion categories Fusion category over C (generalization of Rep(G), Rep(H)): I C-linear abelian: Hom spaces are finite-dimensional C-vector spaces =∼ I Monoidal: ⊗, αU;V;W :(U ⊗ V) ⊗ W −! U ⊗ (V ⊗ W) (Pentagon axiom), 1 I Duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1) I Semisimple, Irr(C) finite. C(X; Y) = δX;YC for X; Y 2 Irr(C). 1 2 Irr(C). I Fusion rule M Z X ⊗ Y = NXYZ: Z2Irr(C) K0(C) := spanCfIrr(C)g is the Grothendieck ring with multiplication given by ⊗. 2/18 Rigidity of the duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1). (idV ⊗ evV) ◦ αV;V_;V ◦ (coevV ⊗ idV) = idV Modular categories Higher Gauss sums Arithmetic properties Witt invariance Graphical calculus on morphisms: 3/18 Modular categories Higher Gauss sums Arithmetic properties Witt invariance Graphical calculus on morphisms: Rigidity of the duality: (V_; 1 ! V ⊗ V_; V_ ⊗ V ! 1). (idV ⊗ evV) ◦ αV;V_;V ◦ (coevV ⊗ idV) = idV 3/18 Two ways to define pivotal dimensions (taking values in C(1; 1) = C) A pivotal structure j of C is called spherical if dL(V) = dR(V) for all V 2 C. C spherical fusion ) categorical dimension dim(V) := dL(V). P 2 Global dimension: dim(C) = V2Irr(C) dim(V) . TrC(f ) 2 C(1; 1) for f 2 C(V; V): insert f in suitable position. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Spherical structure A pivotal structure on C is an isomorphism of monoidal functors __ j : idC ! (−) . 4/18 A pivotal structure j of C is called spherical if dL(V) = dR(V) for all V 2 C. C spherical fusion ) categorical dimension dim(V) := dL(V). P 2 Global dimension: dim(C) = V2Irr(C) dim(V) . TrC(f ) 2 C(1; 1) for f 2 C(V; V): insert f in suitable position. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Spherical structure A pivotal structure on C is an isomorphism of monoidal functors __ j : idC ! (−) . Two ways to define pivotal dimensions (taking values in C(1; 1) = C) 4/18 TrC(f ) 2 C(1; 1) for f 2 C(V; V): insert f in suitable position. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Spherical structure A pivotal structure on C is an isomorphism of monoidal functors __ j : idC ! (−) . Two ways to define pivotal dimensions (taking values in C(1; 1) = C) A pivotal structure j of C is called spherical if dL(V) = dR(V) for all V 2 C. C spherical fusion ) categorical dimension dim(V) := dL(V). P 2 Global dimension: dim(C) = V2Irr(C) dim(V) . 4/18 Modular categories Higher Gauss sums Arithmetic properties Witt invariance Spherical structure A pivotal structure on C is an isomorphism of monoidal functors __ j : idC ! (−) . Two ways to define pivotal dimensions (taking values in C(1; 1) = C) A pivotal structure j of C is called spherical if dL(V) = dR(V) for all V 2 C. C spherical fusion ) categorical dimension dim(V) := dL(V). P 2 Global dimension: dim(C) = V2Irr(C) dim(V) . TrC(f ) 2 C(1; 1) for f 2 C(V; V): insert f in suitable position. 4/18 Twist: =∼ θ(V): V −! V . θ(V) = θV idV; 8V 2 Irr(C). (Vafa) For all V 2 Irr(C), θV is a root of unity. Modular categories Higher Gauss sums Arithmetic properties Witt invariance Braided spherical category Braiding: =∼ cVW : V ⊗ W −! W ⊗ V Hexagon axioms. 5/18 Modular categories Higher Gauss sums Arithmetic properties Witt invariance Braided spherical category Braiding: =∼ cVW : V ⊗ W −! W ⊗ V Hexagon axioms. Twist: =∼ θ(V): V −! V . θ(V) = θV idV; 8V 2 Irr(C). (Vafa) For all V 2 Irr(C), θV is a root of unity. 5/18 Pictorially (omitting j): In particular, S1;V = dim(V), and SV;W = SW;V. Source of examples: finite quadratic modules C(G; q), Z(Rep(G)), quantum groups C(g; k) at roots of unity, VOA, subfactor, ... Modular categories Higher Gauss sums Arithmetic properties Witt invariance Modular category Definition A braided spherical fusion category is called modular if its S-matrix SV;W = TrC(cW;V_ ◦ cV_;W); V; W 2 Irr(C) is invertible. 6/18 Source of examples: finite quadratic modules C(G; q), Z(Rep(G)), quantum groups C(g; k) at roots of unity, VOA, subfactor, ... Modular categories Higher Gauss sums Arithmetic properties Witt invariance Modular category Definition A braided spherical fusion category is called modular if its S-matrix SV;W = TrC(cW;V_ ◦ cV_;W); V; W 2 Irr(C) is invertible. Pictorially (omitting j): In particular, S1;V = dim(V), and SV;W = SW;V. 6/18 Modular categories Higher Gauss sums Arithmetic properties Witt invariance Modular category Definition A braided spherical fusion category is called modular if its S-matrix SV;W = TrC(cW;V_ ◦ cV_;W); V; W 2 Irr(C) is invertible. Pictorially (omitting j): In particular, S1;V = dim(V), and SV;W = SW;V. Source of examples: finite quadratic modules C(G; q), Z(Rep(G)), quantum groups C(g; k) at roots of unity, VOA, subfactor, ... 6/18 Modular category ) ρ : SL(2; Z) ! PGL(K0) 0 −1 1 1 7! S; 7! T: 1 0 0 1 I Congruence subgroup property. ∼ I SL(2; Z) = MCG(Σ1;0). ∼ L ∼ L 1 _ I K0 = V2Irr(C) C(V; V) = V2Irr(C) C( ; V ⊗ V ). I Representations of MCG(Σg;n) can be defined on M _ _ C(a1 ⊗ · · · an; b1 ⊗ b1 ⊗ · · · ⊗ bg ⊗ bg ) a1;:::;an;b1;:::;bg2Irr(C) . I They are provided by a (2+1)-TQFT associated to C. Modular categories Higher Gauss sums Arithmetic properties Witt invariance TV;W := θVδV;W, V; W 2 Irr(C). In particular, ord(T) < 1. S- and T-matrices are called the modular data of C. 7/18 ∼ I SL(2; Z) = MCG(Σ1;0). ∼ L ∼ L 1 _ I K0 = V2Irr(C) C(V; V) = V2Irr(C) C( ; V ⊗ V ). I Representations of MCG(Σg;n) can be defined on M _ _ C(a1 ⊗ · · · an; b1 ⊗ b1 ⊗ · · · ⊗ bg ⊗ bg ) a1;:::;an;b1;:::;bg2Irr(C) . I They are provided by a (2+1)-TQFT associated to C.