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Cosets of the large N = 4 superconformal algebra and the diagonal - PowerPoint PPT Presentation

Nov 11, 2022 •223 likes •1.42k views

Cosets of the large N = 4 superconformal algebra and the diagonal coset of sl 2 Andrew Linshaw University of Denver Joint work with Thomas Creutzig and Boris Feigin 1. Two-parameter families of vertex algebras Ex : Affine vertex algebra V k ( D

  1. 3. The case g = sl 2 Thm : As a two-parameter VOA, C k 1 , k 2 = C k 1 , k 2 ( sl 2 ) is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Equivalently this holds for generic values of k 1 , k 2 . First stated without proof by Blumenhagen, Eholzer, Honecker, Hornfeck, H¨ ubel, 1995. 1 Step 1 : For k 1 fixed, rescaling generators of V k 2 ( sl 2 ) by √ k 2 , k 2 →∞ C k 1 , k 2 ∼ = V k 1 ( sl 2 ) SL 2 . lim A strong generating set for V k 1 ( sl 2 ) SL 2 will give rise to a strong generating set for C k 1 , k 2 for generic k 2 (Creutzig, L., 2014). Step 2 : Rescaling the generators of V k 1 ( sl 2 ) by 1 √ k 1 , we have k 1 →∞ V k 1 ( sl 2 ) SL 2 ∼ = H (3) SL 2 , lim where H (3) is the rank 3 Heisenberg algebra.

  2. 3. The case g = sl 2 Thm : As a two-parameter VOA, C k 1 , k 2 = C k 1 , k 2 ( sl 2 ) is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Equivalently this holds for generic values of k 1 , k 2 . First stated without proof by Blumenhagen, Eholzer, Honecker, Hornfeck, H¨ ubel, 1995. 1 Step 1 : For k 1 fixed, rescaling generators of V k 2 ( sl 2 ) by √ k 2 , k 2 →∞ C k 1 , k 2 ∼ = V k 1 ( sl 2 ) SL 2 . lim A strong generating set for V k 1 ( sl 2 ) SL 2 will give rise to a strong generating set for C k 1 , k 2 for generic k 2 (Creutzig, L., 2014). Step 2 : Rescaling the generators of V k 1 ( sl 2 ) by 1 √ k 1 , we have k 1 →∞ V k 1 ( sl 2 ) SL 2 ∼ = H (3) SL 2 , lim where H (3) is the rank 3 Heisenberg algebra.

  3. 4. The case g = sl 2 , cont’d Note : Adjoint representation of SL 2 is the same as standard representation of SO 3 . So we can replace H (3) SL 2 with H (3) SO 3 . Strong generating set for H (3) SO 3 give rise to strong generators for V k 1 ( sl 2 ) SL 2 for generic values of k 1 (Creutzig, L., 2014). Need to show that H (3) SO 3 is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory for SO 3 .

  4. 4. The case g = sl 2 , cont’d Note : Adjoint representation of SL 2 is the same as standard representation of SO 3 . So we can replace H (3) SL 2 with H (3) SO 3 . Strong generating set for H (3) SO 3 give rise to strong generators for V k 1 ( sl 2 ) SL 2 for generic values of k 1 (Creutzig, L., 2014). Need to show that H (3) SO 3 is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory for SO 3 .

  5. 4. The case g = sl 2 , cont’d Note : Adjoint representation of SL 2 is the same as standard representation of SO 3 . So we can replace H (3) SL 2 with H (3) SO 3 . Strong generating set for H (3) SO 3 give rise to strong generators for V k 1 ( sl 2 ) SL 2 for generic values of k 1 (Creutzig, L., 2014). Need to show that H (3) SO 3 is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory for SO 3 .

  6. 4. The case g = sl 2 , cont’d Note : Adjoint representation of SL 2 is the same as standard representation of SO 3 . So we can replace H (3) SL 2 with H (3) SO 3 . Strong generating set for H (3) SO 3 give rise to strong generators for V k 1 ( sl 2 ) SL 2 for generic values of k 1 (Creutzig, L., 2014). Need to show that H (3) SO 3 is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory for SO 3 .

  7. 4. The case g = sl 2 , cont’d Note : Adjoint representation of SL 2 is the same as standard representation of SO 3 . So we can replace H (3) SL 2 with H (3) SO 3 . Strong generating set for H (3) SO 3 give rise to strong generators for V k 1 ( sl 2 ) SL 2 for generic values of k 1 (Creutzig, L., 2014). Need to show that H (3) SO 3 is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory for SO 3 .

  8. 5. The case g = sl 2 , cont’d Thm : (Weyl) For n ≥ 0, let V n be a copy of the standard representation C 3 of SO 3 , with orthonormal basis { a 1 n , a 2 n , a 3 n } . Then (Sym � ∞ n =0 V n ) SO 3 is generated by q ij = a 1 i a 1 j + a 2 i a 2 j + a 3 i a 3 k , i , j ≥ 0 , (1) a 1 a 2 a 2 � � � � k k k � a 1 a 2 a 3 � c klm = , 0 ≤ k < l < m . (2) � � l l l � a 1 a 2 a 3 � � m m m � The ideal of relations among the variables q ij and c klm is generated by polynomials of the following two types: q ij c klm − q kj c ilm + q lj c kim − q mj c kli , (3) � � q il q im q in � � � � c ijk c lmn − q jl q jm q jn . (4) � � � � q kl q km q kn � �

  9. 5. The case g = sl 2 , cont’d Thm : (Weyl) For n ≥ 0, let V n be a copy of the standard representation C 3 of SO 3 , with orthonormal basis { a 1 n , a 2 n , a 3 n } . Then (Sym � ∞ n =0 V n ) SO 3 is generated by q ij = a 1 i a 1 j + a 2 i a 2 j + a 3 i a 3 k , i , j ≥ 0 , (1) a 1 a 2 a 2 � � � � k k k � a 1 a 2 a 3 � c klm = , 0 ≤ k < l < m . (2) � � l l l � a 1 a 2 a 3 � � m m m � The ideal of relations among the variables q ij and c klm is generated by polynomials of the following two types: q ij c klm − q kj c ilm + q lj c kim − q mj c kli , (3) � � q il q im q in � � � � c ijk c lmn − q jl q jm q jn . (4) � � � � q kl q km q kn � �

  10. 6. The case g = sl 2 , cont’d Step 3 : We have linear isomorphisms H (3) SO 3 ∼ = gr( H (3) SO 3 ) ∼ = gr( H (3)) SO 3 ∼ � V j ) SO 3 , = (Sym j ≥ 0 and isomorphisms of differential graded rings gr( H (3) SO 3 ) ∼ � V j ) SO 3 . = (Sym j ≥ 0 j ≥ 0 V j ) SO 3 corresponds to a Generating set { q ij , c klm } for (Sym � strong generating set { Q ij , C klm } for H (3) SO 3 , where Q i , j = : ∂ i α 1 ∂ j α 1 + : ∂ i α 2 ∂ j α 2 : + : ∂ i α 3 ∂ j α 3 : , C klm = : ∂ k α 1 ∂ l α 2 ∂ m α 3 : − : ∂ k α 1 ∂ m α 2 ∂ l α 3 : − : ∂ l α 1 ∂ k α 2 ∂ m α 3 : + : ∂ l α 1 ∂ m α 2 ∂ k α 3 : + : ∂ m α 1 ∂ k α 2 ∂ l α 3 : − : ∂ m α 1 ∂ l α 2 ∂ k α 3 : .

  11. 6. The case g = sl 2 , cont’d Step 3 : We have linear isomorphisms H (3) SO 3 ∼ = gr( H (3) SO 3 ) ∼ = gr( H (3)) SO 3 ∼ � V j ) SO 3 , = (Sym j ≥ 0 and isomorphisms of differential graded rings gr( H (3) SO 3 ) ∼ � V j ) SO 3 . = (Sym j ≥ 0 j ≥ 0 V j ) SO 3 corresponds to a Generating set { q ij , c klm } for (Sym � strong generating set { Q ij , C klm } for H (3) SO 3 , where Q i , j = : ∂ i α 1 ∂ j α 1 + : ∂ i α 2 ∂ j α 2 : + : ∂ i α 3 ∂ j α 3 : , C klm = : ∂ k α 1 ∂ l α 2 ∂ m α 3 : − : ∂ k α 1 ∂ m α 2 ∂ l α 3 : − : ∂ l α 1 ∂ k α 2 ∂ m α 3 : + : ∂ l α 1 ∂ m α 2 ∂ k α 3 : + : ∂ m α 1 ∂ k α 2 ∂ l α 3 : − : ∂ m α 1 ∂ l α 2 ∂ k α 3 : .

  12. 6. The case g = sl 2 , cont’d Step 3 : We have linear isomorphisms H (3) SO 3 ∼ = gr( H (3) SO 3 ) ∼ = gr( H (3)) SO 3 ∼ � V j ) SO 3 , = (Sym j ≥ 0 and isomorphisms of differential graded rings gr( H (3) SO 3 ) ∼ � V j ) SO 3 . = (Sym j ≥ 0 j ≥ 0 V j ) SO 3 corresponds to a Generating set { q ij , c klm } for (Sym � strong generating set { Q ij , C klm } for H (3) SO 3 , where Q i , j = : ∂ i α 1 ∂ j α 1 + : ∂ i α 2 ∂ j α 2 : + : ∂ i α 3 ∂ j α 3 : , C klm = : ∂ k α 1 ∂ l α 2 ∂ m α 3 : − : ∂ k α 1 ∂ m α 2 ∂ l α 3 : − : ∂ l α 1 ∂ k α 2 ∂ m α 3 : + : ∂ l α 1 ∂ m α 2 ∂ k α 3 : + : ∂ m α 1 ∂ k α 2 ∂ l α 3 : − : ∂ m α 1 ∂ l α 2 ∂ k α 3 : .

  13. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  14. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  15. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  16. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  17. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  18. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  19. 7. The case g = sl 2 , cont’d Note : Q ij has weight i + j + 2 and C klm has weight k + l + m + 3. As a H (3) O 3 -module, H (3) SO 3 ∼ = M 0 ⊕ M 1 , where M 0 , M 1 are irreducible H (3) O 3 -modules (Dong, Li, Mason, 1998) M 0 ∼ = H (3) O 3 , which has lowest-weight vector 1. M 1 has lowest-weight vector C 012 and contains all cubics C klm . H (3) O 3 generated by Q 0 , 2 , so H (3) SO 3 generated by { Q 0 , 2 , C 012 } . One checks that the following set closes under OPE: { C 01 j | j = 2 , 4 , 5 , 6 , 8 } ∪ { Q 0 , 2 k | k = 0 , 1 , 2 , 3 , 4 } . It follows that this set strongly generates H (3) SO 3 . Minimality follows from Weyl’s second fundamental theorem.

  20. 8. Large N = 4 superconformal algebra V k ,α N =4 Weight 1 : { e , f , h , e ′ , f ′ , h ′ } generate V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) where α k − 1 and ℓ ′ = − ( α + 1) k − 1, where α � = 0 , − 1. ℓ = − α +1 Weight 2 : Virasoro field L of central charge c = − 6 k − 3. 2 : Odd fields G ±± which transform as C 2 ⊗ C 2 under Weight 3 sl 2 ⊕ sl 2 , and satisfy complicated OPE relations. For example, � α � G ++ ( z ) G −− ( w ) ∼ − 2 ( z − w ) − 3 k ( k + 1) + ( α + 1) 2 � α + k + α k h ′ + α (1 + k + α k ) � ( w )( z − w ) − 2 + h (1 + a ) 2 (1 + α ) 2 � α α α 4(1 + α ) 2 : h ′ h ′ : + 2(1 + α ) 2 : hh ′ : + kL + 4(1 + α ) 2 : hh : − α α α k (1 + α ) 2 : e ′ f ′ : + + (1 + α ) 2 : ef : + 2(1 + α ) ∂ h k � 2(1 + α ) ∂ h ′ ( w )( z − w ) − 1 . +

  21. 8. Large N = 4 superconformal algebra V k ,α N =4 Weight 1 : { e , f , h , e ′ , f ′ , h ′ } generate V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) where α k − 1 and ℓ ′ = − ( α + 1) k − 1, where α � = 0 , − 1. ℓ = − α +1 Weight 2 : Virasoro field L of central charge c = − 6 k − 3. 2 : Odd fields G ±± which transform as C 2 ⊗ C 2 under Weight 3 sl 2 ⊕ sl 2 , and satisfy complicated OPE relations. For example, � α � G ++ ( z ) G −− ( w ) ∼ − 2 ( z − w ) − 3 k ( k + 1) + ( α + 1) 2 � α + k + α k h ′ + α (1 + k + α k ) � ( w )( z − w ) − 2 + h (1 + a ) 2 (1 + α ) 2 � α α α 4(1 + α ) 2 : h ′ h ′ : + 2(1 + α ) 2 : hh ′ : + kL + 4(1 + α ) 2 : hh : − α α α k (1 + α ) 2 : e ′ f ′ : + + (1 + α ) 2 : ef : + 2(1 + α ) ∂ h k � 2(1 + α ) ∂ h ′ ( w )( z − w ) − 1 . +

  22. 8. Large N = 4 superconformal algebra V k ,α N =4 Weight 1 : { e , f , h , e ′ , f ′ , h ′ } generate V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) where α k − 1 and ℓ ′ = − ( α + 1) k − 1, where α � = 0 , − 1. ℓ = − α +1 Weight 2 : Virasoro field L of central charge c = − 6 k − 3. 2 : Odd fields G ±± which transform as C 2 ⊗ C 2 under Weight 3 sl 2 ⊕ sl 2 , and satisfy complicated OPE relations. For example, � α � G ++ ( z ) G −− ( w ) ∼ − 2 ( z − w ) − 3 k ( k + 1) + ( α + 1) 2 � α + k + α k h ′ + α (1 + k + α k ) � ( w )( z − w ) − 2 + h (1 + a ) 2 (1 + α ) 2 � α α α 4(1 + α ) 2 : h ′ h ′ : + 2(1 + α ) 2 : hh ′ : + kL + 4(1 + α ) 2 : hh : − α α α k (1 + α ) 2 : e ′ f ′ : + + (1 + α ) 2 : ef : + 2(1 + α ) ∂ h k � 2(1 + α ) ∂ h ′ ( w )( z − w ) − 1 . +

  23. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  24. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  25. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  26. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  27. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  28. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  29. 9. Affine coset of V k ,α N =4 Let D k ,α = Com( V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) , V k ,α N =4 ). Thm : For generic values of k and α , D k ,α is of type W (2 , 4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). k and rescale G ±± by 1 Step 1 : Rescale x , y , h , x ′ , y ′ , h ′ , L by 1 k . √ Then V k ,α admits a well defined limit k → ∞ limit k →∞ V k ,α ∼ lim = H (6) ⊗ T ⊗ G odd (4) . H (6) = lim k →∞ V ℓ ( sl 2 ) ⊗ V ℓ ′ ( sl 2 ) a rank 6 Heisenberg algebra. T has even generator L satisfying L ( z ) L ( w ) ∼ ( z − w ) − 4 . G odd (4) has odd generators φ i , i = 1 , 2 , 3 , 4, satisfying φ i ( z ) φ j ( w ) ∼ δ i , j ( z − w ) − 3 .

  30. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  31. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  32. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  33. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  34. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  35. 10. Affine coset of V k ,α N =4 Step 2 : By a general result of Arakawa, Creutzig, L., Kawasetsu (2017), k →∞ D k ,α ∼ � SL 2 × SL 2 . � lim = T ⊗ G odd (4) Action of SL 2 × SL 2 on C 2 ⊗ C 2 is the same as the action of SO 4 on its standard module C 4 . � SL 2 × SL 2 with � SO 4 . � � We can replace G odd (4) G odd (4) Generator of T has weight 2 and corresponds to the Virasoro field. Suffices to prove that ( G odd (4)) SO 4 is of type W (4 , 6 , 6 , 8 , 8 , 9 , 10 , 10 , 12). Step 3 : This is a formal consequence of Weyl’s first and second fundamental theorems of invariant theory of SO 4 .

  36. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  37. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  38. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  39. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  40. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  41. 11. Isomorphism C k 1 , k 2 ∼ = D k ,α Thm : We have an isomorphism of two-parameter vertex algebras C k 1 , k 2 ∼ = D k ,α . Parameters are related by k 1 = − 1 + k + α k k 2 = − α + k + α k (1 + α ) k , (1 + α ) k . Note : symmetry k 1 ↔ k 2 corresponds to symmetry α ↔ 1 α . Idea of proof : Both algebras are generated by the weight 4 primary field, which is unique up to scaling. It follows from VOA axioms that the full OPE algebra is determined by a small set of structure constants. These can be found directly by computer.

  42. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  43. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  44. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  45. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  46. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  47. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  48. 12. Simple one-parameter quotients of C k 1 , k 2 C k 1 , k 2 is simple as a VOA over C [ k 1 , k 2 ]: for every proper graded ideal I ⊆ C k 1 , k 2 , I [0] � = { 0 } . Equivalently, C k 1 , k 2 is simple for generic k 1 , k 2 . There exist curves in the parameter space C 2 given by polynomials p ( k 1 , k 2 ) = 0, where C k 1 , k 2 degenerates. Ex : p ( k 1 , k 2 ) = k 2 − 1. Then C k 1 , 1 has singular vector in weight 4. Simple quotient C k 1 coincides with 1 Com( V k 1 +1 ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L 1 ( sl 2 )) . This is well-known to be just the Virasoro algebra. Ex : p ( k 1 , k 2 ) = k 2 − n , where n ≥ 1 is a positive integer. Again, C k 1 , n is not simple. Simple quotient C k 1 coincides with n Com( V k 1 + n ( sl 2 ) , V k 1 ( sl 2 ) ⊗ L n ( sl 2 )) .

  49. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  50. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  51. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  52. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  53. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  54. 13. Simple one-parameter quotients, cont’d Thm : In the case n = 2, C k 1 is of type W (2 , 4 , 6). 2 C k 1 is isomorphic as a simple, one-parameter vertex algebra to the 2 Z 2 -orbifold of the N = 1 superconformal vertex algebras. Previously stated without proof in Blumenhagen et al (1995). Thm : In the case n = − 1 2 , C k 1 − 1 / 2 is of type W (2 , 4 , 6), but not generically isomorphic to C k 1 2 . Thm : In the case n = − 4 3 , C k 1 − 4 / 3 is of type W (2 , 6 , 8 , 10 , 12). Rem : ( W 3 ) Z 2 is another one-parameter VOA of type W (2 , 6 , 8 , 10 , 12), but not generically isomorphic to C k 1 − 4 / 3 .

  55. 14. Simple zero-parameter quotients Let k be admissible : k = − 2 + p q where ( p , q ) = 1 and p ≥ 2. Thm : 1. The diagonal homomorphism V k +2 ( sl 2 ) → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) descends to a map L k +2 ( sl 2 ) ֒ → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) . 2. The simple quotient C k , 2 of C k 2 coincides with the coset Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ L 2 ( sl 2 )) . 3. C k , 2 is lisse and rational. Statements (1) and (2) hold if 2 is replaced with an arbitrary positive integer n . We expect (3) to hold as well, but we are unable to prove it.

  56. 14. Simple zero-parameter quotients Let k be admissible : k = − 2 + p q where ( p , q ) = 1 and p ≥ 2. Thm : 1. The diagonal homomorphism V k +2 ( sl 2 ) → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) descends to a map L k +2 ( sl 2 ) ֒ → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) . 2. The simple quotient C k , 2 of C k 2 coincides with the coset Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ L 2 ( sl 2 )) . 3. C k , 2 is lisse and rational. Statements (1) and (2) hold if 2 is replaced with an arbitrary positive integer n . We expect (3) to hold as well, but we are unable to prove it.

  57. 14. Simple zero-parameter quotients Let k be admissible : k = − 2 + p q where ( p , q ) = 1 and p ≥ 2. Thm : 1. The diagonal homomorphism V k +2 ( sl 2 ) → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) descends to a map L k +2 ( sl 2 ) ֒ → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) . 2. The simple quotient C k , 2 of C k 2 coincides with the coset Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ L 2 ( sl 2 )) . 3. C k , 2 is lisse and rational. Statements (1) and (2) hold if 2 is replaced with an arbitrary positive integer n . We expect (3) to hold as well, but we are unable to prove it.

  58. 14. Simple zero-parameter quotients Let k be admissible : k = − 2 + p q where ( p , q ) = 1 and p ≥ 2. Thm : 1. The diagonal homomorphism V k +2 ( sl 2 ) → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) descends to a map L k +2 ( sl 2 ) ֒ → L k ( sl 2 ) ⊗ L 2 ( sl 2 ) . 2. The simple quotient C k , 2 of C k 2 coincides with the coset Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ L 2 ( sl 2 )) . 3. C k , 2 is lisse and rational. Statements (1) and (2) hold if 2 is replaced with an arbitrary positive integer n . We expect (3) to hold as well, but we are unable to prove it.

  59. 15. Simple zero-parameter quotients, cont’d Proof of (3) : Let F (4) be the algebra of 4 free fermions. Regarding F (4) as F (2) ⊗ F (2), it is a simple current extension of L 1 ( sl 2 ) ⊗ L 1 ( sl 2 ). Regarding F (4) as F (3) ⊗ F (1), it is a simple current extension of L 2 ( sl 2 ) ⊗ F (1). Then Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ F (4)) is both a simple current extension of C k , 1 ⊗ C k +1 , 1 , and a simple current extension of C k , 2 ⊗ F (1). Rationality of C k , 2 follows from rationality of C k , 1 ⊗ C k +1 , 1 .

  60. 15. Simple zero-parameter quotients, cont’d Proof of (3) : Let F (4) be the algebra of 4 free fermions. Regarding F (4) as F (2) ⊗ F (2), it is a simple current extension of L 1 ( sl 2 ) ⊗ L 1 ( sl 2 ). Regarding F (4) as F (3) ⊗ F (1), it is a simple current extension of L 2 ( sl 2 ) ⊗ F (1). Then Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ F (4)) is both a simple current extension of C k , 1 ⊗ C k +1 , 1 , and a simple current extension of C k , 2 ⊗ F (1). Rationality of C k , 2 follows from rationality of C k , 1 ⊗ C k +1 , 1 .

  61. 15. Simple zero-parameter quotients, cont’d Proof of (3) : Let F (4) be the algebra of 4 free fermions. Regarding F (4) as F (2) ⊗ F (2), it is a simple current extension of L 1 ( sl 2 ) ⊗ L 1 ( sl 2 ). Regarding F (4) as F (3) ⊗ F (1), it is a simple current extension of L 2 ( sl 2 ) ⊗ F (1). Then Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ F (4)) is both a simple current extension of C k , 1 ⊗ C k +1 , 1 , and a simple current extension of C k , 2 ⊗ F (1). Rationality of C k , 2 follows from rationality of C k , 1 ⊗ C k +1 , 1 .

  62. 15. Simple zero-parameter quotients, cont’d Proof of (3) : Let F (4) be the algebra of 4 free fermions. Regarding F (4) as F (2) ⊗ F (2), it is a simple current extension of L 1 ( sl 2 ) ⊗ L 1 ( sl 2 ). Regarding F (4) as F (3) ⊗ F (1), it is a simple current extension of L 2 ( sl 2 ) ⊗ F (1). Then Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ F (4)) is both a simple current extension of C k , 1 ⊗ C k +1 , 1 , and a simple current extension of C k , 2 ⊗ F (1). Rationality of C k , 2 follows from rationality of C k , 1 ⊗ C k +1 , 1 .

  63. 15. Simple zero-parameter quotients, cont’d Proof of (3) : Let F (4) be the algebra of 4 free fermions. Regarding F (4) as F (2) ⊗ F (2), it is a simple current extension of L 1 ( sl 2 ) ⊗ L 1 ( sl 2 ). Regarding F (4) as F (3) ⊗ F (1), it is a simple current extension of L 2 ( sl 2 ) ⊗ F (1). Then Com( L k +2 ( sl 2 ) , L k ( sl 2 ) ⊗ F (4)) is both a simple current extension of C k , 1 ⊗ C k +1 , 1 , and a simple current extension of C k , 2 ⊗ F (1). Rationality of C k , 2 follows from rationality of C k , 1 ⊗ C k +1 , 1 .

  64. 16. C k , 2 and principle W -algebras of type C Thm : We have the following isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) for n ≥ 2. 4 n ℓ = − ( n + 1) + 1 + 2 n 1. k = − 1 + 2 n , 4(1 + n ), 2. k = 3 − 2 n ℓ = − ( n + 1) + 3 + 2 n , , 4 n n ℓ = − ( n + 1) + 2 n − 1 3. k = 4 n − 6 , 4( n − 1). Rem : In cases (1) and (2), the levels ℓ are nondegenerate admissible, so the rationality of W ℓ ( sp 2 n , f prin ) is already known (Arakawa, Annals of Math. 2015). In case (3), the level ℓ is degenerate admissible . Since C k , 2 is rational and lisse, we obtain new examples of rational and lisse principal W -algebras.

  65. 16. C k , 2 and principle W -algebras of type C Thm : We have the following isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) for n ≥ 2. 4 n ℓ = − ( n + 1) + 1 + 2 n 1. k = − 1 + 2 n , 4(1 + n ), 2. k = 3 − 2 n ℓ = − ( n + 1) + 3 + 2 n , , 4 n n ℓ = − ( n + 1) + 2 n − 1 3. k = 4 n − 6 , 4( n − 1). Rem : In cases (1) and (2), the levels ℓ are nondegenerate admissible, so the rationality of W ℓ ( sp 2 n , f prin ) is already known (Arakawa, Annals of Math. 2015). In case (3), the level ℓ is degenerate admissible . Since C k , 2 is rational and lisse, we obtain new examples of rational and lisse principal W -algebras.

  66. 16. C k , 2 and principle W -algebras of type C Thm : We have the following isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) for n ≥ 2. 4 n ℓ = − ( n + 1) + 1 + 2 n 1. k = − 1 + 2 n , 4(1 + n ), 2. k = 3 − 2 n ℓ = − ( n + 1) + 3 + 2 n , , 4 n n ℓ = − ( n + 1) + 2 n − 1 3. k = 4 n − 6 , 4( n − 1). Rem : In cases (1) and (2), the levels ℓ are nondegenerate admissible, so the rationality of W ℓ ( sp 2 n , f prin ) is already known (Arakawa, Annals of Math. 2015). In case (3), the level ℓ is degenerate admissible . Since C k , 2 is rational and lisse, we obtain new examples of rational and lisse principal W -algebras.

  67. 16. C k , 2 and principle W -algebras of type C Thm : We have the following isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) for n ≥ 2. 4 n ℓ = − ( n + 1) + 1 + 2 n 1. k = − 1 + 2 n , 4(1 + n ), 2. k = 3 − 2 n ℓ = − ( n + 1) + 3 + 2 n , , 4 n n ℓ = − ( n + 1) + 2 n − 1 3. k = 4 n − 6 , 4( n − 1). Rem : In cases (1) and (2), the levels ℓ are nondegenerate admissible, so the rationality of W ℓ ( sp 2 n , f prin ) is already known (Arakawa, Annals of Math. 2015). In case (3), the level ℓ is degenerate admissible . Since C k , 2 is rational and lisse, we obtain new examples of rational and lisse principal W -algebras.

  68. 17. Universal even spin W ∞ -algebra The following was conjectured by physicists Candu, Gaberdiel, Kelm, Vollenweider (2013). There exists a universal 2-parameter VOA W ev ( c , λ ) of type W (2 , 4 , . . . ) with following properties: ◮ Generated by Virasoro field L and weight 4 primary field W 4 . ◮ Freely generated of type W (2 , 4 , 6 , . . . ). ◮ All VOAs of type W (2 , 4 , . . . , 2 N ) for some N satisfying some mild hypotheses, arise as quotients. ◮ This includes principal W -algebras of types B and C , as well as Z 2 -orbifold of type D principal W -algebras. This was recently established in my joint paper with S. Kanade.

  69. 17. Universal even spin W ∞ -algebra The following was conjectured by physicists Candu, Gaberdiel, Kelm, Vollenweider (2013). There exists a universal 2-parameter VOA W ev ( c , λ ) of type W (2 , 4 , . . . ) with following properties: ◮ Generated by Virasoro field L and weight 4 primary field W 4 . ◮ Freely generated of type W (2 , 4 , 6 , . . . ). ◮ All VOAs of type W (2 , 4 , . . . , 2 N ) for some N satisfying some mild hypotheses, arise as quotients. ◮ This includes principal W -algebras of types B and C , as well as Z 2 -orbifold of type D principal W -algebras. This was recently established in my joint paper with S. Kanade.

  70. 17. Universal even spin W ∞ -algebra The following was conjectured by physicists Candu, Gaberdiel, Kelm, Vollenweider (2013). There exists a universal 2-parameter VOA W ev ( c , λ ) of type W (2 , 4 , . . . ) with following properties: ◮ Generated by Virasoro field L and weight 4 primary field W 4 . ◮ Freely generated of type W (2 , 4 , 6 , . . . ). ◮ All VOAs of type W (2 , 4 , . . . , 2 N ) for some N satisfying some mild hypotheses, arise as quotients. ◮ This includes principal W -algebras of types B and C , as well as Z 2 -orbifold of type D principal W -algebras. This was recently established in my joint paper with S. Kanade.

  71. 18. Idea of proof For fields a , b , c in any VOA, and r , s ≥ 0, we have identity r � r � a ( r ) ( b ( s ) c ) = ( − 1) | a || b | b ( s ) ( a ( r ) c ) + � ( a ( i ) b ) ( r + s − i ) c . i i =0 These are called Jacobi relations of type ( a , b , c ). Imposing relations of type ( W 2 i , W 2 j , W 2 k ) for 2 i + 2 j + 2 k ≤ 2 n + 2 uniquely determines OPEs W 2 a ( z ) W 2 b ( w ) for a + b ≤ 2 n . We obtain a nonlinear Lie conformal algebra over ring C [ c , λ ]. W ev ( c , λ ) is the universal enveloping VOA (de Sole, Kac, 2005).

  72. 18. Idea of proof For fields a , b , c in any VOA, and r , s ≥ 0, we have identity r � r � a ( r ) ( b ( s ) c ) = ( − 1) | a || b | b ( s ) ( a ( r ) c ) + � ( a ( i ) b ) ( r + s − i ) c . i i =0 These are called Jacobi relations of type ( a , b , c ). Imposing relations of type ( W 2 i , W 2 j , W 2 k ) for 2 i + 2 j + 2 k ≤ 2 n + 2 uniquely determines OPEs W 2 a ( z ) W 2 b ( w ) for a + b ≤ 2 n . We obtain a nonlinear Lie conformal algebra over ring C [ c , λ ]. W ev ( c , λ ) is the universal enveloping VOA (de Sole, Kac, 2005).

  73. 18. Idea of proof For fields a , b , c in any VOA, and r , s ≥ 0, we have identity r � r � a ( r ) ( b ( s ) c ) = ( − 1) | a || b | b ( s ) ( a ( r ) c ) + � ( a ( i ) b ) ( r + s − i ) c . i i =0 These are called Jacobi relations of type ( a , b , c ). Imposing relations of type ( W 2 i , W 2 j , W 2 k ) for 2 i + 2 j + 2 k ≤ 2 n + 2 uniquely determines OPEs W 2 a ( z ) W 2 b ( w ) for a + b ≤ 2 n . We obtain a nonlinear Lie conformal algebra over ring C [ c , λ ]. W ev ( c , λ ) is the universal enveloping VOA (de Sole, Kac, 2005).

  74. 18. Idea of proof For fields a , b , c in any VOA, and r , s ≥ 0, we have identity r � r � a ( r ) ( b ( s ) c ) = ( − 1) | a || b | b ( s ) ( a ( r ) c ) + � ( a ( i ) b ) ( r + s − i ) c . i i =0 These are called Jacobi relations of type ( a , b , c ). Imposing relations of type ( W 2 i , W 2 j , W 2 k ) for 2 i + 2 j + 2 k ≤ 2 n + 2 uniquely determines OPEs W 2 a ( z ) W 2 b ( w ) for a + b ≤ 2 n . We obtain a nonlinear Lie conformal algebra over ring C [ c , λ ]. W ev ( c , λ ) is the universal enveloping VOA (de Sole, Kac, 2005).

  75. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  76. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  77. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  78. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  79. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  80. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  81. 19. 1 -parameter quotients of W ev ( c , λ ) Each weight space of W ev ( c , λ ) is a free module over C [ c , λ ]. Let I ⊆ C [ c , λ ] be a prime ideal and let I · W ev ( c , λ ) be the VOA ideal generated by I . The quotient W ev , I ( c , λ ) = W ev ( c , λ ) / ( I · W ev ( c , λ )) is a VOA over R = C [ c , λ ] / I . Weight spaces are free R -modules, same rank as before. W ev , I ( c , λ ) is simple for a generic ideal I . But for certain discrete families of ideals I , W ev , I ( c , λ ) is not simple. Let W ev I ( c , λ ) be simple graded quotient of W ev , I ( c , λ ). It is a one-parameter VOA, and V ( I ) is called its truncation curve .

  82. 20. Truncation curve V ( I 2 n ) for W k ( sp 2 n , f prin ) Let I 2 n = ( p 2 n ( c , λ )), where p 2 n ( c , λ ) = f ( c , n ) + λ g ( c , n ) + λ 2 h ( c , n ), and f ( c , n ) = − 204 c 2 − 192 c 3 + 171 c 4 + 952 cn − 4612 c 2 n + 2348 c 3 n − 38 c 4 n + 1568 n 2 − 7708 cn 2 + 1788 c 2 n 2 + 2401 c 3 n 2 − 74 c 4 n 2 + 560 n 3 − 18936 cn 3 + 22280 c 2 n 3 − 2112 c 3 n 3 + 8 c 4 n 3 − 16304 n 4 + 18640 cn 4 + 3420 c 2 n 4 − 364 c 3 n 4 + 8 c 4 n 4 − 17408 n 5 + 27680 cn 5 − 10576 c 2 n 5 + 304 c 3 n 5 − 3264 n 6 − 3072 cn 6 + 2736 c 2 n 6 , g ( c , n ) = − 14( − 1 + c )( − 1 + 2 c )(22 + 5 c )( − 2 + n )( − 1 + n ) (3 c + 10 n + 2 cn + 12 n 2 )(5 c + 28 n + 2 cn + 40 n 2 ) , h ( c , n ) = 49( − 1 + c ) 2 (22 + 5 c ) 2 (21 c 2 + 70 cn − 14 c 2 n + 200 n 2 − 135 cn 2 − 26 c 2 n 2 + 380 n 3 − 176 cn 3 + 8 c 2 n 3 + 436 n 4 + 132 cn 4 + 8 c 2 n 4 + 448 n 5 + 112 cn 5 + 336 n 6 ) .

  83. 21. One-parameter VOAs of type W (2 , 4 , 6) Thm : There are exactly three distinct one-parameter VOAs of type W (2 , 4 , 6) that arise as quotients of W ev ( c , λ ). 1. W k ( sp 6 , f prin ) corresponds to the ideal I 6 . 2. C k 2 corresponds to the ideal J 2 = ( q 2 ( c , λ )) where q 2 ( c , λ ) = 7 λ ( − 1 + c )( − 17 + 2 c )(22 + 5 c ) + 82 − 47 c − 10 c 2 . 3. C k − 1 / 2 corresponds to the ideal J − 1 / 2 = ( q − 1 / 2 ( c , λ )) where q − 1 / 2 ( c , λ ) = 7 λ ( − 41+ c )( − 1+ c )(22+5 c ) − 14+309 c +5 c 2 . The proof of our isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) involves finding intersection points on the curves V ( J 2 ) and V ( I 2 n ).

  84. 21. One-parameter VOAs of type W (2 , 4 , 6) Thm : There are exactly three distinct one-parameter VOAs of type W (2 , 4 , 6) that arise as quotients of W ev ( c , λ ). 1. W k ( sp 6 , f prin ) corresponds to the ideal I 6 . 2. C k 2 corresponds to the ideal J 2 = ( q 2 ( c , λ )) where q 2 ( c , λ ) = 7 λ ( − 1 + c )( − 17 + 2 c )(22 + 5 c ) + 82 − 47 c − 10 c 2 . 3. C k − 1 / 2 corresponds to the ideal J − 1 / 2 = ( q − 1 / 2 ( c , λ )) where q − 1 / 2 ( c , λ ) = 7 λ ( − 41+ c )( − 1+ c )(22+5 c ) − 14+309 c +5 c 2 . The proof of our isomorphisms C k , 2 ∼ = W ℓ ( sp 2 n , f prin ) involves finding intersection points on the curves V ( J 2 ) and V ( I 2 n ).

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