Classification of Modular Categories Csar Galindo Universidad de - PowerPoint PPT Presentation

Main results Definition of fusion category in coordinates Fusion rules Let L = { X 1 = 1 , X 2 , . . . , X n } be a set of representatives of isomorphism classes of simple objects. There is an involution ∗ : L → L such that 1 ∗ = 1 . k N k X i ⊗ X j = � ij X k , so we have a colection of non-negative integres N k ij , for every i , j , k ∈ { 1 , . . . , n } and satisfy César Galindo

Main results Definition of fusion category in coordinates Fusion rules Let L = { X 1 = 1 , X 2 , . . . , X n } be a set of representatives of isomorphism classes of simple objects. There is an involution ∗ : L → L such that 1 ∗ = 1 . k N k X i ⊗ X j = � ij X k , so we have a colection of non-negative integres N k ij , for every i , j , k ∈ { 1 , . . . , n } and satisfy N b 1 a = δ ab = N b a 1 N 1 ab = δ a ∗ b ae ′ N e ′ N u � N e ab N u � N u abc := ec = bc e e ′

Main results Definition of fusion category in coordinates Fusion rules Let L = { X 1 = 1 , X 2 , . . . , X n } be a set of representatives of isomorphism classes of simple objects. There is an involution ∗ : L → L such that 1 ∗ = 1 . k N k X i ⊗ X j = � ij X k , so we have a colection of non-negative integres N k ij , for every i , j , k ∈ { 1 , . . . , n } and satisfy N b 1 a = δ ab = N b a 1 ae ′ N e ′ N u � N e ab N u � N u abc := ec = bc e e ′

Main results Definition of fusion category in coordinates Fusion rules Let L = { X 1 = 1 , X 2 , . . . , X n } be a set of representatives of isomorphism classes of simple objects. There is an involution ∗ : L → L such that 1 ∗ = 1 . k N k X i ⊗ X j = � ij X k , so we have a colection of non-negative integres N k ij , for every i , j , k ∈ { 1 , . . . , n } and satisfy N b 1 a = δ ab = N b a 1 N 1 ab = δ a ∗ b ae ′ N e ′ N u � N e ab N u � N u abc := ec = bc e e ′ César Galindo

Main results F-matrices (6j-symbols) Without loss of generality we can suppose that ( a ⊗ b ) ⊗ c = a ⊗ ( b ⊗ c ) for all a , b , c , d ∈ L . César Galindo

Main results F-matrices (6j-symbols) Without loss of generality we can suppose that ( a ⊗ b ) ⊗ c = a ⊗ ( b ⊗ c ) for all a , b , c , d ∈ L . F-matrices Define F d abc : Hom C ( a ⊗ b ⊗ c , d ) → Hom C ( a ⊗ b ⊗ c , d ) f �→ f ◦ a a , b , c César Galindo

Main results F-matrices (6j-symbols) Without loss of generality we can suppose that ( a ⊗ b ) ⊗ c = a ⊗ ( b ⊗ c ) for all a , b , c , d ∈ L . F-matrices Define F d abc : Hom C ( a ⊗ b ⊗ c , d ) → Hom C ( a ⊗ b ⊗ c , d ) f �→ f ◦ a a , b , c The set of matrices { F d abc ∈ U ( N d abc ) | a , b , c , d ∈ L } is called the F-matrices and they satisfy the pentagonal identity (pentagon axiom) . César Galindo

Main results Examples Pointed fusion categories, C ( G , ω ) César Galindo

Main results Examples Pointed fusion categories, C ( G , ω ) L = G (a finite group) César Galindo

Main results Examples Pointed fusion categories, C ( G , ω ) L = G (a finite group) fusion rules are the product in G César Galindo

Main results Examples Pointed fusion categories, C ( G , ω ) L = G (a finite group) fusion rules are the product in G a , b , c = ω ( a , b , c ) δ abc , d , so is a function ω : G × 3 → U ( 1 ) F d César Galindo

Main results Examples Pointed fusion categories, C ( G , ω ) L = G (a finite group) fusion rules are the product in G a , b , c = ω ( a , b , c ) δ abc , d , so is a function ω : G × 3 → U ( 1 ) F d Pentagon equation is exactly 3-cocycle condition of group cohomology: ω ( a , b , c ) ω ( b , c , d ) ω ( a , bc , d ) = ω ( ab , c , d ) ω ( a , b , cd ) César Galindo

Main results Examples Fibonnaci theory César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } fusion rules x 2 = 1 + x ( N 1 xx = N x xx = 1) César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } fusion rules x 2 = 1 + x ( N 1 xx = N x xx = 1) � F x xxx = César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } fusion rules x 2 = 1 + x ( N 1 xx = N x xx = 1) φ − 1 φ − 1 / 2 � � F x xxx = φ − 1 / 2 φ − 1 César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } fusion rules x 2 = 1 + x ( N 1 xx = N x xx = 1) φ − 1 φ − 1 / 2 � � F x xxx = φ − 1 / 2 φ − 1 Not every fusion rules admit a set of F -matrices César Galindo

Main results Examples Fibonnaci theory L = { 1 , x } fusion rules x 2 = 1 + x ( N 1 xx = N x xx = 1) φ − 1 φ − 1 / 2 � � F x xxx = φ − 1 / 2 φ − 1 Not every fusion rules admit a set of F -matrices As an example the fusion rules: L k = { 1 , x } x 2 = 1 + kx ( N 1 xx = N x xx = k ), k ∈ Z > 0 define a fusion category if and only if k = 1 (Victor Ostrik). César Galindo

Main results Examples Ising theory L = { 1 , σ, ψ } fusion rules: σ 2 = 1 + ψ, ψ 2 = 1 , ψσ = σψ = σ . � 1 � 1 ψσψ = F ψ F σ 1 , F σ σσσ = σψσ = − 1. √ − 1 2 1 Remarks The ising fusion rules has two possible realization (Isinig or � 1 � 1 σσσ = − 1 Mayorama fermion) F σ . √ 1 − 1 2 Ising categories are particular cases of a more general familily called Tambara-Yamagami categories. César Galindo

Main results Braided fusion category in coordinates If ( C , c ) is a braided fusion, without loss of generality we can suppose that a ⊗ b = a ⊗ b for all a , b ∈ L . César Galindo

Main results Braided fusion category in coordinates If ( C , c ) is a braided fusion, without loss of generality we can suppose that a ⊗ b = a ⊗ b for all a , b ∈ L . R-matrices Define R c a , b : Hom C ( a ⊗ b , c ) → Hom C ( b ⊗ a , c ) f �→ f ◦ c a , b César Galindo

Main results Braided fusion category in coordinates If ( C , c ) is a braided fusion, without loss of generality we can suppose that a ⊗ b = a ⊗ b for all a , b ∈ L . R-matrices Define R c a , b : Hom C ( a ⊗ b , c ) → Hom C ( b ⊗ a , c ) f �→ f ◦ c a , b The set of matrices { R c a , b ∈ U ( N c a , b ) | a , b , c ∈ L } is called the R-matrices and they satisfy the hexagonal identities (hexagon axioms) . César Galindo

Main results Example Pointed braided fusion category César Galindo

Main results Example Pointed braided fusion category If C ( G , ω ) has a braid structure then G is abelian R z xy = c ( x , y ) δ xy , z , so is a function c : G × G → U ( 1 ) Hexagonal equation is exacly the abelian 3-cocycle condition ω ( y , z , x ) c ( x , yz ) ω ( x , y , z ) = c ( x , z ) ω ( y , x , z ) c ( x , y ) ω ( z , x , y ) − 1 c ( xy , z ) ω ( x , y , z ) − 1 = c ( x , z ) ω ( x , z , y ) − 1 c ( y , z ) . César Galindo

Main results Examples R-matrices for Fibonacci theory R 1 ττ = e − 4 π i / 5 , R τ ττ = e 3 π i / 5 . César Galindo

Main results Examples R-matrices for Fibonacci theory R 1 ττ = e − 4 π i / 5 , R τ ττ = e 3 π i / 5 . R-matrices for Ising theory R 1 ψψ = − 1 , R σ σψ = i , R 1 σσ = e − π i / 8 , R ψ σσ = e 3 π i / 8 The Ising category admist tree (non-equivalent) R-matrices. César Galindo

Main results More examples: the Drinfeld center Let C be a (strict) tensor category and let X ∈ C . Definition A half braiding c − , X : � ⊗ X → X ⊗ � for X is a natural isomorphism such that c Y ⊗ Z , X = ( c Y , X ⊗ id Z )( id Y ⊗ c Z , X ) , for all Y , Z ∈ C . César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category objects: pairs ( X , c − , X ) , where X ∈ C and c − , X is a half braiding for X , César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category objects: pairs ( X , c − , X ) , where X ∈ C and c − , X is a half braiding for X , morphisms: Hom Z ( C ) (( X , c − , X ) , ( Y , c − , Y )) = { f ∈ Hom C ( X , Y ) : ( id W ⊗ f ) c W , X = c W , Y ( id W ⊗ f ) , ∀ W ∈ C} , César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category objects: pairs ( X , c − , X ) , where X ∈ C and c − , X is a half braiding for X , morphisms: Hom Z ( C ) (( X , c − , X ) , ( Y , c − , Y )) = { f ∈ Hom C ( X , Y ) : ( id W ⊗ f ) c W , X = c W , Y ( id W ⊗ f ) , ∀ W ∈ C} , tensor product: ( X , c − , X ) ⊗ ( Y , c − , Y ) = ( X ⊗ Y , c − , X ⊗ Y ) , where c − , X ⊗ Y = ( id X ⊗ c − , Y )( c − , X ⊗ id Y ) , César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category objects: pairs ( X , c − , X ) , where X ∈ C and c − , X is a half braiding for X , morphisms: Hom Z ( C ) (( X , c − , X ) , ( Y , c − , Y )) = { f ∈ Hom C ( X , Y ) : ( id W ⊗ f ) c W , X = c W , Y ( id W ⊗ f ) , ∀ W ∈ C} , tensor product: ( X , c − , X ) ⊗ ( Y , c − , Y ) = ( X ⊗ Y , c − , X ⊗ Y ) , where c − , X ⊗ Y = ( id X ⊗ c − , Y )( c − , X ⊗ id Y ) , braiding: σ ( X , c − , X ) , ( Y , c − , Y ) = c X , Y . César Galindo

Main results More examples: the Drinfeld center The Drinfeld center Z ( C ) of C is the following braided fusion category objects: pairs ( X , c − , X ) , where X ∈ C and c − , X is a half braiding for X , morphisms: Hom Z ( C ) (( X , c − , X ) , ( Y , c − , Y )) = { f ∈ Hom C ( X , Y ) : ( id W ⊗ f ) c W , X = c W , Y ( id W ⊗ f ) , ∀ W ∈ C} , tensor product: ( X , c − , X ) ⊗ ( Y , c − , Y ) = ( X ⊗ Y , c − , X ⊗ Y ) , where c − , X ⊗ Y = ( id X ⊗ c − , Y )( c − , X ⊗ id Y ) , braiding: σ ( X , c − , X ) , ( Y , c − , Y ) = c X , Y . Theorem (Muger) The Drinfeld center Z ( C ) is modular if C is a spherical fusion category over C . César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. Let Irr ( C )= { X 0 = 1 , X 1 , . . . , X n } denote the set of isomorphism classes of simple objects in C . César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. Let Irr ( C )= { X 0 = 1 , X 1 , . . . , X n } denote the set of isomorphism classes of simple objects in C . The rank of C is the cardinality of the set Irr ( C ) . César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. Let Irr ( C )= { X 0 = 1 , X 1 , . . . , X n } denote the set of isomorphism classes of simple objects in C . The rank of C is the cardinality of the set Irr ( C ) . Z ∈ Irr ( C ) N Z Fusion rules: X ⊗ Y ≃ � X , Y Z ( X , Y ∈ Irr ( C ) ). César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. Let Irr ( C )= { X 0 = 1 , X 1 , . . . , X n } denote the set of isomorphism classes of simple objects in C . The rank of C is the cardinality of the set Irr ( C ) . Z ∈ Irr ( C ) N Z Fusion rules: X ⊗ Y ≃ � X , Y Z ( X , Y ∈ Irr ( C ) ). The Frobenius-Perron dimension FPdim X ∈ R + of X ∈ C is the largest nonnegative eigenvalue of the matrix ( N Z X , Y ) Y , Z ∈ Irr ( C ) (matrix of left multiplication by X w.r.t ⊗ ). César Galindo

Main results Frobenius-Perron dimensions Set C be a fusion category. Let Irr ( C )= { X 0 = 1 , X 1 , . . . , X n } denote the set of isomorphism classes of simple objects in C . The rank of C is the cardinality of the set Irr ( C ) . Z ∈ Irr ( C ) N Z Fusion rules: X ⊗ Y ≃ � X , Y Z ( X , Y ∈ Irr ( C ) ). The Frobenius-Perron dimension FPdim X ∈ R + of X ∈ C is the largest nonnegative eigenvalue of the matrix ( N Z X , Y ) Y , Z ∈ Irr ( C ) (matrix of left multiplication by X w.r.t ⊗ ). The Frobenius-Perron dimension of C is X ∈ Irr ( C ) ( FPdim X ) 2 . FPdim C = � César Galindo

Main results More definitions A fusion category C is pointed if all the simple objects are invertible ⇔ FPdim X = 1. César Galindo

Main results More definitions A fusion category C is pointed if all the simple objects are invertible ⇔ FPdim X = 1. A fusion category C is called integral if FPdim X ∈ Z + , ∀ X ∈ C ( ⇔ C ≃ Rep H , H semisimple quasi-Hopf [ENO]). César Galindo

Main results More definitions A fusion category C is pointed if all the simple objects are invertible ⇔ FPdim X = 1. A fusion category C is called integral if FPdim X ∈ Z + , ∀ X ∈ C ( ⇔ C ≃ Rep H , H semisimple quasi-Hopf [ENO]). A fusion category C is weakly integral if FPdim C ∈ Z . César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , an element τ ∈ C s.t. | G | τ 2 = 1. César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , an element τ ∈ C s.t. | G | τ 2 = 1. Tambara-Yamagami category T Y ( G , χ, τ ) : is the semisimple category with César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , an element τ ∈ C s.t. | G | τ 2 = 1. Tambara-Yamagami category T Y ( G , χ, τ ) : is the semisimple category with Irr ( T Y ( G , χ, τ )) = G � { X } , X / ∈ G . César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , an element τ ∈ C s.t. | G | τ 2 = 1. Tambara-Yamagami category T Y ( G , χ, τ ) : is the semisimple category with Irr ( T Y ( G , χ, τ )) = G � { X } , X / ∈ G . Fusion rules a ⊗ b = ab , X ⊗ X = � a ∈ G a , a ⊗ X = X . César Galindo

Main results Example: Tambara-Yamagami categories Data: an abelian finite group G , a non-degenerate symmetric bicharacter χ : G × G → k × , an element τ ∈ C s.t. | G | τ 2 = 1. Tambara-Yamagami category T Y ( G , χ, τ ) : is the semisimple category with Irr ( T Y ( G , χ, τ )) = G � { X } , X / ∈ G . Fusion rules a ⊗ b = ab , X ⊗ X = � a ∈ G a , a ⊗ X = X . Duality a ∗ = a − 1 and X ∗ = X . César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. T Y ( G , χ, τ ) admits a braiding ⇔ G is an elementary abelian 2 -group. César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. T Y ( G , χ, τ ) admits a braiding ⇔ G is an elementary abelian 2 -group. Example Ising categories I are Tambara-Yamagami categories with G = < a > ≃ Z 2 . César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. T Y ( G , χ, τ ) admits a braiding ⇔ G is an elementary abelian 2 -group. Example Ising categories I are Tambara-Yamagami categories with G = < a > ≃ Z 2 . In this case, X ⊗ 2 = 1 ⊕ a . César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. T Y ( G , χ, τ ) admits a braiding ⇔ G is an elementary abelian 2 -group. Example Ising categories I are Tambara-Yamagami categories with G = < a > ≃ Z 2 . In this case, X ⊗ 2 = 1 ⊕ a . Then, √ FPdim X = 2 and FPdim T Y = 4. César Galindo

Main results Example: Tambara-Yamagami categories Remark � FPdim X = | G | and FPdim T Y ( G , χ, τ ) = 2 | G | � weakly integral but not necessarily integral. T Y ( G , χ, τ ) admits a braiding ⇔ G is an elementary abelian 2 -group. Example Ising categories I are Tambara-Yamagami categories with G = < a > ≃ Z 2 . In this case, X ⊗ 2 = 1 ⊕ a . Then, √ FPdim X = 2 and FPdim T Y = 4. Moreover, I is modular . César Galindo

Main results Frame problem Recall that the frame problem is: César Galindo

Main results Frame problem Recall that the frame problem is: Problem Classify modular categories. César Galindo

Main results Frame problem Recall that the frame problem is: Problem Classify modular categories. Hard problem! Different approaches, for example: César Galindo

Main results Frame problem Recall that the frame problem is: Problem Classify modular categories. Hard problem! Different approaches, for example: low rank MC, César Galindo

Main results Frame problem Recall that the frame problem is: Problem Classify modular categories. Hard problem! Different approaches, for example: low rank MC, weakly integral MC, César Galindo

Main results Frame problem Recall that the frame problem is: Problem Classify modular categories. Hard problem! Different approaches, for example: low rank MC, weakly integral MC, MC of a given FPdim. César Galindo

Main results Rank finiteness for braided fusion categories Theorem (Bruillard, Ng, Rowell, Wang) 2013 There are finitely many modular categories of a given rank r. Theorem (Bruillard, G., Ng, Plavnik, Rowell, Wang) 2015 There are finitely many braided fusion categories of a given rank r . César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. Theorem If C is a modular category with 2 ≤ Rank C ≤ 5 it is Grothendieck equivalent to one of the following: César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. Theorem If C is a modular category with 2 ≤ Rank C ≤ 5 it is Grothendieck equivalent to one of the following: PSU ( 2 ) 3 (Fibonacci), SU ( 2 ) 1 (pointed), César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. Theorem If C is a modular category with 2 ≤ Rank C ≤ 5 it is Grothendieck equivalent to one of the following: PSU ( 2 ) 3 (Fibonacci), SU ( 2 ) 1 (pointed), PSU ( 2 ) 5 , SU ( 2 ) 2 (Ising), SU ( 3 ) 1 (pointed), César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. Theorem If C is a modular category with 2 ≤ Rank C ≤ 5 it is Grothendieck equivalent to one of the following: PSU ( 2 ) 3 (Fibonacci), SU ( 2 ) 1 (pointed), PSU ( 2 ) 5 , SU ( 2 ) 2 (Ising), SU ( 3 ) 1 (pointed), PSU ( 2 ) 7 , SU ( 2 ) 3 , SU ( 4 ) 1 , products, César Galindo

Main results Known results: Rank C ≤ 5 Results of Bruillard, Hong, Ng, Ostrik, Rowell, Stong, Wang gave the classification of MC of rank at most 5. Theorem If C is a modular category with 2 ≤ Rank C ≤ 5 it is Grothendieck equivalent to one of the following: PSU ( 2 ) 3 (Fibonacci), SU ( 2 ) 1 (pointed), PSU ( 2 ) 5 , SU ( 2 ) 2 (Ising), SU ( 3 ) 1 (pointed), PSU ( 2 ) 7 , SU ( 2 ) 3 , SU ( 4 ) 1 , products, PSU ( 2 ) 9 , SU ( 2 ) 4 , SU ( 5 ) 1 , PSU ( 3 ) 4 . César Galindo

Main results Known results: FPdim C fixed Results of Bruillard, Drinfeld, Etingof, G., Gelaki, Kashina, Hong, Ostrik, Naidu, Natale, Nikshych, P , Rowell help to advance in the classification program. César Galindo

Main results Known results: FPdim C fixed Results of Bruillard, Drinfeld, Etingof, G., Gelaki, Kashina, Hong, Ostrik, Naidu, Natale, Nikshych, P , Rowell help to advance in the classification program. C MC, FPdim C ∈ { p n , pq , pqr , pq 2 , pq 3 , pq 4 , pq 5 } César Galindo

Main results Known results: FPdim C fixed Results of Bruillard, Drinfeld, Etingof, G., Gelaki, Kashina, Hong, Ostrik, Naidu, Natale, Nikshych, P , Rowell help to advance in the classification program. C MC, FPdim C ∈ { p n , pq , pqr , pq 2 , pq 3 , pq 4 , pq 5 } � group-theoretical. César Galindo