Some Zhu reduction formula and applications Matthew Krauel - PowerPoint PPT Presentation

Some Zhu reduction formula and applications Matthew Krauel California State University, Sacramento June 27th, 2019 Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Zhu’s work: The study of n -point functions. Definition ( n -point functions) Let V be a VOA with Virasoro vector ω of central charge c . For v 1 , . . . , v n ∈ V and a weak V -module M , the n -point function is Z M (( v 1 , x 1 ) , . . . , ( v n , x n ); ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 24 , := tr M Y · · · Y where q := e 2 π i τ with τ ∈ H = { x + iy ∈ C | y > 0 } . Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Zhu’s work: The study of n -point functions. Definition ( n -point functions) Let V be a VOA with Virasoro vector ω of central charge c . For v 1 , . . . , v n ∈ V and a weak V -module M , the n -point function is Z M (( v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 24 , := tr M Y · · · Y where q := e 2 π i τ with τ ∈ H = { x + iy ∈ C | y > 0 } . Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Zhu’s work: The study of n -point functions. Definition ( n -point functions) Let V be a VOA with Virasoro vector ω of central charge c . For v 1 , . . . , v n ∈ V and a weak V -module M , the n -point function is Z M (( v 1 , x 1 ) , . . . , ( v n , x n ); ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 24 , := tr M Y · · · Y where q := e 2 π i τ with τ ∈ H = { x + iy ∈ C | y > 0 } . Zhu’s core results concerning n -point functions: Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Zhu’s work: The study of n -point functions. Definition ( n -point functions) Let V be a VOA with Virasoro vector ω of central charge c . For v 1 , . . . , v n ∈ V and a weak V -module M , the n -point function is Z M (( v 1 , x 1 ) , . . . , ( v n , x n ); ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 24 , := tr M Y · · · Y where q := e 2 π i τ with τ ∈ H = { x + iy ∈ C | y > 0 } . Zhu’s core results concerning n -point functions: Established their modularity. Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Zhu’s work: The study of n -point functions. Definition ( n -point functions) Let V be a VOA with Virasoro vector ω of central charge c . For v 1 , . . . , v n ∈ V and a weak V -module M , the n -point function is Z M (( v 1 , x 1 ) , . . . , ( v n , x n ); ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 24 , := tr M Y · · · Y where q := e 2 π i τ with τ ∈ H = { x + iy ∈ C | y > 0 } . Zhu’s core results concerning n -point functions: Established their modularity. Established their convergence. Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Theorem Suppose V is a rational and C 2 -cofinite and V = M 0 , M 1 , . . . , M k be its inequivalent irreducible modules. Moreover let v s ∈ V [wt v s ] for 1 ≤ s ≤ n. Then 1 each Z M (( v 1 , x 1 ); τ ) converges on H , and � a b � 2 for any ∈ SL 2 ( Z ) we have there exists scalars α ij ∈ C such that c d � � ( v 1 , x 1 ) , . . . , ( v n , x n ); a τ + b Z M i c τ + d k � � wt v j = ( c τ + d ) α ij Z M j (( v 1 , x 1 ) , . . . , ( v n , x n ); τ ) . j =1 Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: The change of coordinate VOA Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: The change of coordinate VOA The ‘Zhu algebra’ Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: The change of coordinate VOA The ‘Zhu algebra’ Introducing the theory of ODEs to study 1-point functions Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: The change of coordinate VOA The ‘Zhu algebra’ Introducing the theory of ODEs to study 1-point functions Reduction formulas. Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story To establish this result Zhu introduced/enhanced a number of tools: The change of coordinate VOA The ‘Zhu algebra’ Introducing the theory of ODEs to study 1-point functions Reduction formulas. Core idea of proof Zhu expressed n -point functions as linear combinations of ( n − 1)-point functions. Reduced the study of n -point functions to the study of 1-point functions. Allowed the creation of ODEs whose solution space consisted of 1-point functions. Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: Original Zhu reduction formula, Part I Original Zhu reduction formula, Part I We have Z M (( a , y ) , ( v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 = tr M v (wt a − 1) Y · · · Y 24 � y − x j � � n � + P m +1 2 π i , τ Z M (( v 1 , x 1 ) , . . . , ( a [ m ] v j , x j ) , . . . , ( v n , x n ); τ ) . j =1 m ≥ 0 Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: Original Zhu reduction formula, Part I Original Zhu reduction formula, Part I We have Z M (( a , y ) , ( v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 = tr M v (wt a − 1) Y · · · Y 24 � y − x j � � n � + P m +1 2 π i , τ Z M (( v 1 , x 1 ) , . . . , ( a [ m ] v j , x j ) , . . . , ( v n , x n ); τ ) . j =1 m ≥ 0 Where ( q w = e 2 π iw ) � P m +1 ( w , τ ) : = ( − 1) m +1 n m q n 1 w 1 − q n − δ m , 0 m ! 2 n ∈ Z \{ 0 } � 1 � n = ( − 1) m d ( P 1 ( w , τ )) . m ! 2 π i dw Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: Original Zhu reduction formula, Part II Original Zhu reduction formula, Part II Let a , v 1 , . . . v n ∈ V . For N ≥ 1 we have Z M (( a [ − N ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 = δ N , 1 tr M v (wt a − 1) Y · · · Y 24 � m + N − 1 � � ( − 1) m +1 + G m + N ( τ ) Z M (( a [ m ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) m m ≥ 0 � m + N − 1 � � x 1 − x j � � n � ( − 1) N +1 + P m + N , τ m 2 π i j =2 m ≥ 0 × Z M (( v 1 , x 1 ) , . . . , ( a [ m ] v j , x j ) , . . . , ( v n , x n ); τ ) . Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: Original Zhu reduction formula, Part II Original Zhu reduction formula, Part II Let a , v 1 , . . . v n ∈ V . For N ≥ 1 we have Z M (( a [ − N ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 = δ N , 1 tr M v (wt a − 1) Y · · · Y 24 � m + N − 1 � � ( − 1) m +1 + G m + N ( τ ) Z M (( a [ m ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) m m ≥ 0 � m + N − 1 � � x 1 − x j � � n � ( − 1) N +1 + P m + N , τ m 2 π i j =2 m ≥ 0 × Z M (( v 1 , x 1 ) , . . . , ( a [ m ] v j , x j ) , . . . , ( v n , x n ); τ ) . Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: Original Zhu reduction formula, Part II Original Zhu reduction formula, Part II Let a , v 1 , . . . v n ∈ V . For N ≥ 1 we have Z M (( a [ − N ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) � � � e x n L (0) v n , e x n � q L (0) − c e x 1 L (0) v 1 , e x 1 = δ N , 1 tr M v (wt a − 1) Y · · · Y 24 � m + N − 1 � � ( − 1) m +1 + G m + N ( τ ) Z M (( a [ m ] v 1 , x 1 ) , . . . , ( v n , x n ); τ ) m m ≥ 0 � m + N − 1 � � x 1 − x j � � n � ( − 1) N +1 + P m + N , τ m 2 π i j =2 m ≥ 0 × Z M (( v 1 , x 1 ) , . . . , ( a [ m ] v j , x j ) , . . . , ( v n , x n ); τ ) . Here, � 1 G k ( τ )(2 π iw ) k − 1 . P 1 ( w , τ ) = 2 π iw − k ≥ 1 Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Rested heavily on the coefficient functions: For k ≥ 1, � 1 G 2 k ( τ ) = ( m τ + n ) 2 k . ( m , n ) ∈ Z 2 \ (0 , 0) Modular forms for k ≥ 2. Quasi-modular form when k = 1. G 2 k ( τ ) holomorphic. Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019

Recap: The original Zhu story Rested heavily on the coefficient functions: For k ≥ 1, � 1 G 2 k ( τ ) = ( m τ + n ) 2 k . ( m , n ) ∈ Z 2 \ (0 , 0) Modular forms for k ≥ 2. Quasi-modular form when k = 1. G 2 k ( τ ) holomorphic. Notes: � � = ( c τ + d ) 2 G 2 ( τ ) − c ( c τ + d ) a τ + b 1 G 2 . c τ + d 2 π i Matthew Krauel (CSUS) Representation Theory XVI June 27th, 2019