M 2018 Hecke Relations in Rational Conformal Field Theory (with - PowerPoint PPT Presentation

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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