Vertex operator algebras with central charges 1 / 2 and 68 / 7 (v.5) - PowerPoint PPT Presentation

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Vertex operator algebras with central charges 1 / 2 and − 68 / 7 (v.5) Kiyokazu Nagatomo Lie algebras, Vertex Operator Algebras, and Related Topics August 14–18, University of Notre Dame, Department of Mathematics Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 1 Introduction 2 The 3rd oder modular linear differential equations – a short course The 3rd oder modular linear differential equations Frobenius method 3 Vertex operator algebras with central charge 1 / 2 MLDE for c = 1 / 2 Theorem ( c = 1 / 2) Proof ( c = 1 / 2) The characters for c = 1 / 2 Remarks 4 Vertex operator algebras with central charge − 68 / 7 Frobenius method for c = − 68 / 7 Thereom ( c = − 68 / 7) MLDE and characters ( c = − 68 / 7) 5 Lattice vertex operator algebras Vertex operator algebras of central charge c = 8 Vertex operator algebras with central charge c = 16 Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Introduction 1 Vertex operator algebras (VOA for short) with central charges 1 / 2 or − 68 / 7 whose sets of characters form fundamental systems of 3rd order modular linear differential equations (MLDE for short) are discussed. Such VOAs are either isomorphic to the minimal series of Virasoro VOAs with central charges c = c 3 , 4 = 1 / 2 or c 2 , 7 = − 68 / 7. 2 We also study VOAs with central charges 8 or 16. √ The lattice vertex operator algebras V L , where L is the 2 E 8 ( c = 8) or the Barnes–Wall lattice ( c = 16) appear. 3 The characters of each VOA are obtained by solving the associated with MLDE (by Frobenius method). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Introduction 1 Vertex operator algebras (VOA for short) with central charges 1 / 2 or − 68 / 7 whose sets of characters form fundamental systems of 3rd order modular linear differential equations (MLDE for short) are discussed. Such VOAs are either isomorphic to the minimal series of Virasoro VOAs with central charges c = c 3 , 4 = 1 / 2 or c 2 , 7 = − 68 / 7. 2 We also study VOAs with central charges 8 or 16. √ The lattice vertex operator algebras V L , where L is the 2 E 8 ( c = 8) or the Barnes–Wall lattice ( c = 16) appear. 3 The characters of each VOA are obtained by solving the associated with MLDE (by Frobenius method). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Introduction 1 Vertex operator algebras (VOA for short) with central charges 1 / 2 or − 68 / 7 whose sets of characters form fundamental systems of 3rd order modular linear differential equations (MLDE for short) are discussed. Such VOAs are either isomorphic to the minimal series of Virasoro VOAs with central charges c = c 3 , 4 = 1 / 2 or c 2 , 7 = − 68 / 7. 2 We also study VOAs with central charges 8 or 16. √ The lattice vertex operator algebras V L , where L is the 2 E 8 ( c = 8) or the Barnes–Wall lattice ( c = 16) appear. 3 The characters of each VOA are obtained by solving the associated with MLDE (by Frobenius method). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Introduction 1 Vertex operator algebras (VOA for short) with central charges 1 / 2 or − 68 / 7 whose sets of characters form fundamental systems of 3rd order modular linear differential equations (MLDE for short) are discussed. Such VOAs are either isomorphic to the minimal series of Virasoro VOAs with central charges c = c 3 , 4 = 1 / 2 or c 2 , 7 = − 68 / 7. 2 We also study VOAs with central charges 8 or 16. √ The lattice vertex operator algebras V L , where L is the 2 E 8 ( c = 8) or the Barnes–Wall lattice ( c = 16) appear. 3 The characters of each VOA are obtained by solving the associated with MLDE (by Frobenius method). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course Vertex operator algebras with central charge 1 Introduction 1 Vertex operator algebras (VOA for short) with central charges 1 / 2 or − 68 / 7 whose sets of characters form fundamental systems of 3rd order modular linear differential equations (MLDE for short) are discussed. Such VOAs are either isomorphic to the minimal series of Virasoro VOAs with central charges c = c 3 , 4 = 1 / 2 or c 2 , 7 = − 68 / 7. 2 We also study VOAs with central charges 8 or 16. √ The lattice vertex operator algebras V L , where L is the 2 E 8 ( c = 8) or the Barnes–Wall lattice ( c = 16) appear. 3 The characters of each VOA are obtained by solving the associated with MLDE (by Frobenius method). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course The 3rd oder modular linear differential equations Vertex operator algebras with central charge 1 Frobenius m 3rd oder modular linear differential equations 1 The 3rd oder modular linear differential equation (MLDE) 8 + c 2 D 3 ( f ) − 1 � 1 h 2 − h 2 − ch 192+ c � � � 2 E ′ 2 E 2 D ( f )+ 2 − D ( f ) E 4 24 � c 12 + 1 + c h − c D = ′ = q d �� � 2 − h E 6 f = 0 , dq . 24 24 2 The c is a central charge and h is the minimal conformal weight and E k ( q ) is the Eisenstein series with weight k . 3 The space of solutions is invariant under the slash 0 action of the full modular group SL 2 ( Z ). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course The 3rd oder modular linear differential equations Vertex operator algebras with central charge 1 Frobenius m 3rd oder modular linear differential equations 1 The 3rd oder modular linear differential equation (MLDE) 8 + c 2 D 3 ( f ) − 1 � 1 h 2 − h 2 − ch 192+ c � � � 2 E ′ 2 E 2 D ( f )+ 2 − D ( f ) E 4 24 � c 12 + 1 + c h − c D = ′ = q d �� � 2 − h E 6 f = 0 , dq . 24 24 2 The c is a central charge and h is the minimal conformal weight and E k ( q ) is the Eisenstein series with weight k . 3 The space of solutions is invariant under the slash 0 action of the full modular group SL 2 ( Z ). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course The 3rd oder modular linear differential equations Vertex operator algebras with central charge 1 Frobenius m 3rd oder modular linear differential equations 1 The 3rd oder modular linear differential equation (MLDE) 8 + c 2 D 3 ( f ) − 1 � 1 h 2 − h 2 − ch 192+ c � � � 2 E ′ 2 E 2 D ( f )+ 2 − D ( f ) E 4 24 � c 12 + 1 + c h − c D = ′ = q d �� � 2 − h E 6 f = 0 , dq . 24 24 2 The c is a central charge and h is the minimal conformal weight and E k ( q ) is the Eisenstein series with weight k . 3 The space of solutions is invariant under the slash 0 action of the full modular group SL 2 ( Z ). Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7

Introduction The 3rd oder modular linear differential equations – a short course The 3rd oder modular linear differential equations Vertex operator algebras with central charge 1 Frobenius m Frobenius method n =0 a n q ε + n with a 0 = 1 and 1 A solution of the form f = � ∞ ε ∈ Q . We suppose that an index is a rational number. 2 The index ε ∈ {− c / 24 , h − c / 24 , c / 12 − h + 1 / 2 } . 3 The Frobenius method determines a n ( n ∈ N ) uniquely by the recursion relation for a given a 0 � = 0: n + ε + c n + ε + c n + ε − c 12 − 1 � − 1 � � − 1 � � − 1 � a n = 24 − h 2 + h 24 n � � � × ( ε − i + n ) 12(2 i − ε − n ) σ 1 ( i ) i =1 + 5 c 2 + 8 c − 24 hc − 96 h + 192 h 2 � � � σ 3 ( i ) 4 ∞ − 7 � � σ k − 1 ( n ) q n ) 96 c ( c − 24 h )( c − 12 h + 6) σ 5 ( i ) a n − i , ( E k = 1 − A k n =1 as far as the denominators do not vanish. Kiyokazu Nagatomo Vertex operator algebras with central charges 1 / 2 and − 68 / 7