M ∩ Φ 2018 Hecke Relations in Rational Conformal Field Theory (with Y. Wu, arXiv:1804.06860) Jeff Harvey University of Chicago
INTRODUCTION This talk concerns a new relation between characters of two-dimensional rational conformal field theory (RCFT) given by Hecke operators. The relation generalizes previously known Galois symmetry relations between the representations of the modular group provided by RCFT characters. I will discuss these new relations, explain how they explain a number of scattered results in the literature and present some possible applications.
Examples: The Yang-Lee model is a non-unitary minimal model M (5 , 2) (RCFT) with two independent characters χ Y L = q − 1 / 60 X c Y L ( n ) q n 0 0 n =0 1 / 5 = q 11 / 60 X χ Y L c Y L 1 / 5 ( n ) q n n =0 Affine is a unitary rational CFT with two G 2 independent characters χ G 2 = q − 7 / 60 X c G 2 0 ( n ) q n 0 n =0 χ G 2 c G 2 17 / 60 = q 17 / 60 X 17 / 60 ( n ) q n n =0 Although apparently unrelated, there is a subtle relation between the character coefficients:
DATA coefficients of q expansion of coefficients of q expansion of h=1/5 character of vacuum character of affine G2 at Yang-Lee model c=-22/5 level one (c=14/5) divided by 7 Discrepancy 2 6 20 50 120 261 3858/7 1/7 1091 2108 3917 7118 12587 21854 1/7 260072/7 62202 102428 166450 266850 422966 662780 1/7 7198178/7 1579853 2406046 3633082 5443362 8094202 11952388 122716344/7 2/7
RCFT M H = N i, ¯ Hilbert space: i V i ⊗ V ¯ i Representation of chiral i, ¯ i algebra, e.g. Virasoro for minimal models χ i ( τ ) = Tr V i q L 0 − c/ 24 , q = e 2 π i τ Characters: X Partition N i, ¯ Z ( τ ) = i χ i ( τ ) χ ¯ i ( τ ) function: i ∈ ¯ i ∈ I , ¯ I Examples: Ising model, Yang-Lee model, affine Lie algebras, Monster VOA, BM VOA
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cVrRU7pWTStA3NJEOSUcrQH3Hjr7hxoeLGheDfmL7A54ELh3Pu5d57vIBRpW37w4rNzS8sLsWXEyura+sbyc2tSyVCiUkNCybklYcUYZSTmqa katAEuR7jNS9fn k12+IVFTwCz0ISN HXU47FCNtpFay4MqeOILnlXQ+6/pI9zwvuh5moCtpt6eRlOIWnlTSPAtnbnmYaSVTdq5oO8V9B/4mTs4eIwWmqLaSb25b4NAnXGOGlGo4dqCbEZKaYkaGCTdUJEC4j7qkYShHPlHNaPzdEO4ZpQ07QpriGo7VrxMR8pUa+J7pHJ2ofnoj8S+vEerOYTOiPAg14XiyqBMyqAUcRQXbVBKs2cAQhCU1t0LcQxJhbQJNmB mn8L/S 2fK+acs0KqdDxNIw52wC5IAwc gBI4BV QAxjcgQfwBJ6te+vRerFeJ60xazqzDb7Bev8E1B+gxg= </latexit><latexit sha1_base64="BhjtHVsdlC237hC L4NqwqandwM=">A Modular Properties Characters are weakly holomorphic weight zero vector- valued modular functions transforming according to ρ : SL (2 , Z ) → GL ( n, C ) ✓ 0 ◆ ✓ 1 ◆ − 1 1 ( τ → − 1 / τ ) S = T = ( τ → τ + 1) 1 0 0 1 ( ρ ( S )) 2 = ( ρ ( S ) ρ ( T )) 3 = C C 2 = 1 (Bantay) N = order( ρ ( T )) Γ ( N ) ⊂ ker( ρ ) ⇢✓ ◆ � a b Γ ( N ) = ∈ SL (2 , Z ) | a = d = 1 mod N, b = c = 0 mod N c d
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