Modular forms of rational weights and the minimal models

Kiyokazu Nagatomo
Osaka University
Luigi Stasi Seminar Room, ICTP
Friday, December 8, 2017 - 16:00 to 17:00

After the modular forms of rational weights on Γ(5) (and Γ(7)) were dis- covered, T. Ibukiyama formulated modular forms of weights (N − 3)/2N (N > 3 and odd) on Γ(N) in the millennium, which have remained mysterious until now. In this talk I will gives a new point of view, which has advantages of understanding the factional weights and congruence groups that appear in the theory of Ibukiyama.
I (we) have been working on the minimal models and the associated differential equations which are a higher order generalization of Kaneko-Zagier equation. Recently, we found that the special case of the minimal models “essentially” gives these modular forms of fractional weights, where “essentially” means “after multiplying a power of eta function.” The characters (one-point functions) of (rational) conformal field theories may have negative powers of q when they are expanded as Fourier series. Of course, we can have only non-negative powers by multiplying a power of q. However, the results lose almost all good properties which characters have (including modular invariance property). Now, since the eta function commutes with the Serre derivation, we multiply a power of the eta function to the characters. Moreover, the power must be the so-called effective central charge in the Physics literature. Then the result we will prove is that modular forms of rational weights are obtained by multiplying ηceff to characters.
In a point of view of differential equations such as the Kaneko-Zagier equation, special functions would be defined as solutions of differential equations with regular singularities. Therefore, we may think that modular forms of rational weights would be “special functions.”
This talk requires elementary knowledge of (modular forms), vertex operator algebras, minimal models and modular linear differential equations, which have been (will be) given in series of lectures of Prof. Zagier.
Finally, this is a joint work with Y. Sakai (who is a number theorist) at Kyushu University. 

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