Abstract
We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.
Highlights
To construct infinite classes of modular functions which have all the properties required of characters of consistent rational conformal field theories (RCFTs)
We have established that the known Galois symmetry connecting modular representations of certain RCFTs has an extension to a relation via Hecke operators between their characters
We showed that Hecke operators connect solutions to Modular Linear Differential Equations (MLDE) which have different numbers of zeroes in the modular Wronskian and used this to show that all the l = 2 characters found in table 1 of [18] are Hecke images of l = 0 solutions
Summary
The modular properties of the characters χi(τ ) will play a central role in what follows. Is a unitary, finite-dimensional representation of the modular group with V a complex vector space and ρ(γ)ab are the matrix elements of ρ with respect to a basis of V. In a RCFT the central charge c and conformal weights hi of primary operators are rational [6] and as a result ρ(T ) will have finite order which we denote by N. It will be useful later to note that upon reduction mod N of each element of a matrix γ ∈ SL(2, Z) we obtain an element of SL(2, Z/N Z) The kernel of this mod N map is Γ(N ) which is a normal subgroup of SL(2, Z). While all characters of RCFT χi are individually modular forms for Γ(N ) and transform according to a representation ρ of SL(2, Z), the converse is not true. In an RCFT the fusion coefficients must be non-negative integers
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
