Abstract
Quasi-characters are vector-valued modular functions having an integral, but not necessarily positive, q-expansion. Using modular differential equations, a complete classification has been provided in arXiv:1810.09472 for the case of two characters. These in turn generate all possible admissible characters, of arbitrary Wronskian index, in order two. Here we initiate a study of the three-character case. We conjecture several infinite families of quasi-characters and show in examples that their linear combinations can generate admissible characters with arbitrarily large Wronskian index. The structure is completely different from the order two case, and the novel coset construction of arXiv:1602.01022 plays a key role in discovering the appropriate families. Using even unimodular lattices, we construct some explicit three-character CFT corresponding to the new admissible characters.
Highlights
Fusion rules, as well as correlation functions, are outputs
Candidate characters for CFT must be holomorphic in moduli space except at infinity, and have three more essential properties: (i) they transform under modular transformations as vector-valued modular functions, so that the partition function is modular invariant, (ii) the ground state in the identity character is non-degenerate, so that the ground state of the full CFT is non-degenerate, as should be the case in any legitimate quantum field theory, (iii) each character has non-negative integer degeneracies of states after suitably normalising the ground state, except for the identity which is normalised to unity
We have proposed a large collection of families of quasi-characters with vanishing Wronskian index ( = 0) and provided evidence for their existence
Summary
For a fixed order p, MLDE’s are classified by an additional piece of data, namely the Wronskian index of the characters This is defined as the integer such that 6 is the number of zeroes of the Wronskian determinant of the independent solutions: χ0 · · · χp−1. Higher orders in the series solution determine the ratios a(ni) a(0i) recursively In general these are rational functions of the input parameters with growing denominators. For some values of the parameters the coefficients are rational with denominators that stabilise for large orders of q (except for the identity where they must be integer from the outset), as well as non-negative. No solutions for ≥ 6 have been generated in the MLDE approach, since the number of parameters in the equation grows too fast to be tractable.
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