Vector Valued Modular Forms in Vertex Operator Algebras

Transcription

1 Vector Valued Modular Forms in Vertex Operator Algebras University of Alberta Alberta Number Theory Days VIII, BIRS Banff, April 2016

2 Overview Vertex Operator Algebra = VOA Origins in deep physics theories that aim beyond QM + GR Philosophy : The relevance of a VOA is found in its rep theory.

3 Overview : modular objects In the VOA theory... C 2 -cofiniteness vs finite # of simple modules vs modularity of characters Following work by Y.Zhu, M.Miyamoto proved that the linear span of trace & pseudo-trace functions of such VOAs is a representation of the modular group.

4 Overview : broad aim An obstacle to non-ss settings : the lack of examples... To this date, a single family of VOAs with C 2 -cofiniteness non-semisimple rep theory has been known... : the W p)-triplet VOAs.

5 Overview : broad aim An obstacle to non-ss settings : the lack of examples... To this date, a single family of VOAs with C 2 -cofiniteness non-semisimple rep theory has been known... : the W p)-triplet VOAs. Broad aim To find new examples of VOAs that are as such.

6 Overview : local aim Several people have been looking for candidate VOAs including D.Adamović, T.Creutzig, A.Milas, D.Ridout, S.Wood. Some of the more accessible candidates with C 2 -cofiniteness non-ss rep theory are constructed out of affine VOAs.

7 Overview : local aim Several people have been looking for candidate VOAs including D.Adamović, T.Creutzig, A.Milas, D.Ridout, S.Wood. Some of the more accessible candidates with C 2 -cofiniteness non-ss rep theory are constructed out of affine VOAs. Local aim To expose the character modular invariance property for the most accessible candidate!!

8 Overview : a candidate The VOA D k from the following diagram : where Coset Extension L k sl 2 ) C k = Com H, L k sl 2 )) D k k < 0 & k + 2 = u v Q >0\ { 1, 1 2, 1 3,...} H = the Heisenberg subalgebra of L k sl 2 )

9 Overview : a candidate The VOA D k from the following diagram : where Coset Extension L k sl 2 ) C k = Com H, L k sl 2 )) D k k < 0 & k + 2 = u v Q >0\ { 1, 1 2, 1 3,...} H = the Heisenberg subalgebra of L k sl 2 ) Then under a suitable assumption on C k... Schur-Weyl + Extension process D k is promising

10 Overview : Schur-Weyl duality Assuming that the vertex tensor theory of HLZ applies for C k... Theorem [T.Creutzig, S.Kanade, A.R.Linshaw, D.Ridout] Then for any a simple L k sl 2 )-module M on which H acts semisimply, we have a decomposition : M = F y Cy M y v M +lattice as a H C k )-module where the F y s are Fock spaces and the Cy M simple C k -modules. are + a few technical properties. Note : H = Com C k, L k sl 2 )).

11 Modular invariance k + 2 = u/v One defines characters as : tr M y k z h0 q L0 c 24 ). We should think : q = e 2πiτ. By some classification work, it is sufficient to consider characters of two types of L k sl 2 )-modules... where... σ l E λ, r,s σ l L r,0

12 Modular invariance k + 2 = u/v One defines characters as : tr M y k z h0 q L0 c 24 ). We should think : q = e 2πiτ. By some classification work, it is sufficient to consider characters of two types of L k sl 2 )-modules... σ l E λ, r,s σ l L r,0 where... l Z & σ is an automorphism of L k sl 2 ) r {1,..., u 1} & s {0,..., v 1} λ 1 v Z

13 Modular invariance L k sl 2 ) C k D k MODULAR DATA AND VERLINDE FORMULAE FOR FRACTIONAL LEVEL WZW MODELS II 9 σ σ σ σ D u r,v 1 L r,0 L r 1 D + u r,v 1 σ σ σ D u r,v s 1 D + r,s σ σ σ σ E λ; r,s Source FIGURE 2. : T.Creutzig, Depictions of the D.Ridout, three typesmodular of familiesdata of admissible and Verlinde irreducibleformulae ŝl2)- for modules when Fractional v > 1. Conformal Level WZW dimensions Models increase II, from Nucl. top Phys. to bottomb and 875 sl2)- 2013) weights increase from right to left I thank the authors for allowing me to use this picture. l )

14 Modular invariance L k sl 2 ) C k D k Decomposing the relevant characters accordingly to the Schur-Weyl result, we get : ch σ l E λ, r,s = n Z ch Fλ+2n+kl ) ch C E r,s,λ+2n q) ) ch σ l L r,0 = n Z ch Fr 1+2n+kl ) ch C L r,r 1+2n q) ) where...

15 Modular invariance L k sl 2 ) C k D k Decomposing the relevant characters accordingly to the Schur-Weyl result, we get : ch σ l E λ, r,s = n Z ch Fλ+2n+kl ) ch C E r,s,λ+2n q) ) ch σ l L r,0 = n Z ch Fr 1+2n+kl ) ch C L r,r 1+2n q) ) where... ch C E r,s,xq) = χvir r,s q) ηq) q 1 4k x 2 v 1 ch Cr,xq) L = 1) d=1 χvir d 1 r,d q) ηq) a=0 q 1 4k x k2av+d))2 )) 2 q 1 4k x k 2a+1)v d

16 Modular invariance L k sl 2 ) C k D k Set p = kv 2 and Γ = 2p Z. Lifting the C k -modules C E r,s,x and C L r,x results in the apparition of lattice Θ-functions and derivatives : Dr,s,ωq) E,0 } {{ } Θ + 0 }{{} Θ L,0 D r,t q) } {{ } Θ + D L,1 r,t q) }{{} Θ where...

17 Modular invariance L k sl 2 ) C k D k Set p = kv 2 and Γ = 2p Z. Lifting the C k -modules C E r,s,x and C L r,x results in the apparition of lattice Θ-functions and derivatives : Dr,s,ωq) E,0 } {{ } Θ + 0 }{{} Θ L,0 D r,t q) } {{ } Θ + D L,1 r,t q) }{{} Θ where... Dr,s,ωq) E,0 = χvir r,s q) ηq) Θ ω 2p +Γ1, q) Dr,t L,0 q) = a linear combination of expressions of the form D E,0 v 1 Dr,t L,1 q) = d=1 1) χvir d 1 r,d q) ηq) r,s,ωq) ) Θ 1, q) r 1+2t)v+kvd 2p +Γ Θ 1, q) r 1+2t)v kvd) 2p +Γ

18 Modular invariance : D E,0 r,s,ωq) 1/2 χ Vir r,s q) Θ 2p ω +Γ1, q) ηq) Consider the generating modular transformations S : τ 1 τ T : τ τ + 1 Span C { D E,0 r,s,ωq) } is then automatically a representation of PSL2, Z)!

19 Modular invariance : D L,1 r,t q) 2/2 v 1 1) d=1 χvir d 1 r,d q) ηq) ) Θ 1, q) r 1+2t)v+kvd 2p +Γ Θ 1, q) r 1+2t)v kvd) 2p +Γ Fix parameters r, t and write Dr,t L,1 ) 1 τ = Coeffr,s ), ω ) χ Vir r,s τ) ητ) Θ 2p ω +Γ τ)

20 Modular invariance : D L,1 r,t q) 2/2 v 1 1) d=1 χvir d 1 r,d q) ηq) ) Θ 1, q) r 1+2t)v+kvd 2p +Γ Θ 1, q) r 1+2t)v kvd) 2p +Γ Fix parameters r, t and write Dr,t L,1 ) 1 τ = Coeffr,s ), ω Fix d. Then for any r, t, one can find that ) χ Vir r,s τ) ητ) Θ 2p ω +Γ τ) 1) d 1 Coeff r,d), r 1+2t )v±kvd = ± [ #r, t, r, t ) ]... and that the irrelevant Coeffs vanish!

21 Modular invariance Result The vector space V = Span C { D E,0 r,s,ωq) + 0, D L,0 r,t q) + D L,1 r,t q) } is a representation of PSL2, Z)!

22 Modular invariance Result The vector space V = Span C { D E,0 r,s,ωq) + 0, D L,0 r,t q) + D L,1 r,t q) } is a representation of PSL2, Z)! More interestingly Span C { D L,1 r,t q) } also is ; where D L,1 r,t S L,1 r,t),r,t ) = X r t ) }{{} 1 or 1/2 ) 1 τ = S L,1 r,t),r,t ) DL,1 r,t τ) 4iτ u 2v u sin π v u rr ) cos π r 1+2t)r 1+2t )v 2v u )