Automorphism groups of some orbifold models of lattice VOAs Ching - PowerPoint PPT Presentation

Automorphism groups of some orbifold models of lattice VOAs Ching Hung Lam Academia Sinica Based on joint works with Hiroki Shimakura and Koichi Betsumiya June 25, 2019 C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 1 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. The main idea is to view V as a simple current extension of V L g ⊗ V ˆ g Λ g (proposed by H¨ ohn); C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. The main idea is to view V as a simple current extension of V L g ⊗ V ˆ g Λ g (proposed by H¨ ohn); V L g is a lattice VOA and Λ g is a coinvariant lattice of the Leech lattice Λ . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. The main idea is to view V as a simple current extension of V L g ⊗ V ˆ g Λ g (proposed by H¨ ohn); V L g is a lattice VOA and Λ g is a coinvariant lattice of the Leech lattice Λ . The key step is to compute the stabilizer Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) using the theory of simple current extensions [Shimakura 2007]. C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. The main idea is to view V as a simple current extension of V L g ⊗ V ˆ g Λ g (proposed by H¨ ohn); V L g is a lattice VOA and Λ g is a coinvariant lattice of the Leech lattice Λ . The key step is to compute the stabilizer Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) using the theory of simple current extensions [Shimakura 2007]. It turns out Aut ( V ) = Inn ( V ) Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ), where Inn ( V ) = � exp( a (0) ) | a ∈ V 1 } . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Main question Try to compute the full automorphism group of a holomorphic vertex operator algebra V of central charge 24. Let V 1 ∼ = g as a Lie algebra. The main idea is to view V as a simple current extension of V L g ⊗ V ˆ g Λ g (proposed by H¨ ohn); V L g is a lattice VOA and Λ g is a coinvariant lattice of the Leech lattice Λ . The key step is to compute the stabilizer Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) using the theory of simple current extensions [Shimakura 2007]. It turns out Aut ( V ) = Inn ( V ) Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ), where Inn ( V ) = � exp( a (0) ) | a ∈ V 1 } . We need to know the groups Aut ( V L g ) and Aut ( V ˆ g Λ g ). C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 2 / 26

Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 3 / 26

Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) Set V 1 = V L g and V 2 = V ˆ g Λ g . Let f : ( Irr ( V 1 ) , q 1 ) → ( Irr ( V 2 ) , − q 2 ) be an isometry such that � V = M ⊗ f ( M ) M ∈ Irr ( V 1 ) C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 3 / 26

Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) Set V 1 = V L g and V 2 = V ˆ g Λ g . Let f : ( Irr ( V 1 ) , q 1 ) → ( Irr ( V 2 ) , − q 2 ) be an isometry such that � V = M ⊗ f ( M ) M ∈ Irr ( V 1 ) Then S = { ( M , f ( M )) | M ∈ Irr ( V 1 ) } is a maximal totally singular subspace of ( Irr ( V 1 ) ⊕ Irr ( V 2 ) , q 1 + q 2 ). C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 3 / 26

Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) Set V 1 = V L g and V 2 = V ˆ g Λ g . Let f : ( Irr ( V 1 ) , q 1 ) → ( Irr ( V 2 ) , − q 2 ) be an isometry such that � V = M ⊗ f ( M ) M ∈ Irr ( V 1 ) Then S = { ( M , f ( M )) | M ∈ Irr ( V 1 ) } is a maximal totally singular subspace of ( Irr ( V 1 ) ⊕ Irr ( V 2 ) , q 1 + q 2 ). By [Shimakura 2007], there is an exact sequence 1 → S ∗ → N Aut ( V ) ( S ∗ ) → Stab Aut ( V 1 ⊗ V 2 ) ( S ) → 1 , where Stab Aut ( V 1 ⊗ V 2 ) ( S ) = { g ∈ Aut ( V 1 ⊗ V 2 ) | S ◦ g = S } and S ∗ = dual group of S . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 3 / 26

Stab Aut ( V ) ( V L g ⊗ V ˆ g Λ g ) Set V 1 = V L g and V 2 = V ˆ g Λ g . Let f : ( Irr ( V 1 ) , q 1 ) → ( Irr ( V 2 ) , − q 2 ) be an isometry such that � V = M ⊗ f ( M ) M ∈ Irr ( V 1 ) Then S = { ( M , f ( M )) | M ∈ Irr ( V 1 ) } is a maximal totally singular subspace of ( Irr ( V 1 ) ⊕ Irr ( V 2 ) , q 1 + q 2 ). By [Shimakura 2007], there is an exact sequence 1 → S ∗ → N Aut ( V ) ( S ∗ ) → Stab Aut ( V 1 ⊗ V 2 ) ( S ) → 1 , where Stab Aut ( V 1 ⊗ V 2 ) ( S ) = { g ∈ Aut ( V 1 ⊗ V 2 ) | S ◦ g = S } and S ∗ = dual group of S . Note: Aut ( V 1 ⊗ V 2 ) = Aut ( V 1 ) × Aut ( V 2 ) since V 1 ≇ V 2 . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 3 / 26

Let µ i : Aut ( V i ) → O ( Irr ( V i ) , q i ) , i = 1 , 2 , be the group homomorphism induced from the g -conjugate action of Aut ( V i ) on Irr ( V i ), C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 4 / 26

Let µ i : Aut ( V i ) → O ( Irr ( V i ) , q i ) , i = 1 , 2 , be the group homomorphism induced from the g -conjugate action of Aut ( V i ) on Irr ( V i ), where O ( Irr ( V i ) , q i ) denotes the isometry group of ( Irr ( V i ) , q i ). C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 4 / 26

Let µ i : Aut ( V i ) → O ( Irr ( V i ) , q i ) , i = 1 , 2 , be the group homomorphism induced from the g -conjugate action of Aut ( V i ) on Irr ( V i ), where O ( Irr ( V i ) , q i ) denotes the isometry group of ( Irr ( V i ) , q i ). Lemma Stab Aut ( V 1 ⊗ V 2 ) ( S ) / (ker µ 1 × ker µ 2 ) ∼ = ( Im µ 1 ) ∩ f − 1 ( Im µ 2 ) f . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 4 / 26

Let K ( V ) = { g ∈ Aut ( V ) | g | V 1 = id V 1 } and define Out ( V ) = Aut ( V ) / K ( V ) Inn ( V ) . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 5 / 26

Let K ( V ) = { g ∈ Aut ( V ) | g | V 1 = id V 1 } and define Out ( V ) = Aut ( V ) / K ( V ) Inn ( V ) . Proposition Assume ker µ 2 = id. Then we have Out ( V ) ∼ = µ − 1 L (( Im µ 1 ) ∩ f − 1 ( Im µ 2 ) f ) / W ( V 1 ) , C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 5 / 26

Let K ( V ) = { g ∈ Aut ( V ) | g | V 1 = id V 1 } and define Out ( V ) = Aut ( V ) / K ( V ) Inn ( V ) . Proposition Assume ker µ 2 = id. Then we have Out ( V ) ∼ = µ − 1 L (( Im µ 1 ) ∩ f − 1 ( Im µ 2 ) f ) / W ( V 1 ) , where µ L : O ( L g ) → O ( D ( L g ) , q L g ) is the canonical group homomorphism and W ( V 1 ) the Weyl group of V 1 . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 5 / 26

Let K ( V ) = { g ∈ Aut ( V ) | g | V 1 = id V 1 } and define Out ( V ) = Aut ( V ) / K ( V ) Inn ( V ) . Proposition Assume ker µ 2 = id. Then we have Out ( V ) ∼ = µ − 1 L (( Im µ 1 ) ∩ f − 1 ( Im µ 2 ) f ) / W ( V 1 ) , where µ L : O ( L g ) → O ( D ( L g ) , q L g ) is the canonical group homomorphism and W ( V 1 ) the Weyl group of V 1 . Lemma We have K ( V ) < Inn ( V ) and √ − 1 x (0) ) | x ∈ ˜ Q ∗ / L g } , K ( V ) = { exp( − 2 π √ k i Q i , Q i is the root lattice of g i and where ˜ Q = � s 1 i =1 V 1 ∼ = g ∼ = g 1 ⊕ · · · ⊕ g s . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 5 / 26

For ker µ 1 , let X ( L ) = { h ∈ O ( L ) | h = id on D ( L ) = L ∗ / L } and X (ˆ L ) = { g ∈ O (ˆ L ) | ¯ g ∈ X ( L ) } . Then we have Lemma Im µ 1 ∼ ker µ 1 = Inn ( V L g ) X (ˆ L g ) and = O ( L g ) / X ( L g ) . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 6 / 26

Aut ( V ˆ g Λ g ) C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 7 / 26

Aut ( V ˆ g Λ g ) Recall that Aut ( V L ) = N ( V L ) O (ˆ L ) , � � where N ( V L ) = exp( a (0) ) | a ∈ ( V L ) 1 = Inn ( V L ). C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 7 / 26

Aut ( V ˆ g Λ g ) Recall that Aut ( V L ) = N ( V L ) O (ˆ L ) , � � where N ( V L ) = exp( a (0) ) | a ∈ ( V L ) 1 = Inn ( V L ). Moreover, there is an exact sequence of [FLM88, Proposition 5.4.1] 1 → Hom ( L , Z / 2 Z ) → O (ˆ ϕ L ) → O ( L ) → 1 . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 7 / 26

Aut ( V ˆ g Λ g ) Recall that Aut ( V L ) = N ( V L ) O (ˆ L ) , � � where N ( V L ) = exp( a (0) ) | a ∈ ( V L ) 1 = Inn ( V L ). Moreover, there is an exact sequence of [FLM88, Proposition 5.4.1] 1 → Hom ( L , Z / 2 Z ) → O (ˆ ϕ L ) → O ( L ) → 1 . When L (2) = { x ∈ L | � x , x � = 2 } = ∅ , the normal subgroup N ( V L ) = { exp( λα (0)) | α ∈ L , λ ∈ C } is abelian and we have N ( V L ) ∩ O (ˆ Aut ( V L ) / N ( V L ) ∼ L ) = Hom ( L , Z / 2 Z ) and = O ( L ) . C.H. Lam (A.S.) Orbifold VOAs June 25, 2019 7 / 26