Maths Seminar - Additional events this Friday at ICTP - by Professors Zagier (MPI,Bonn and ICTP) and Nagatomo (Osaka University)

[ICTP Math Section]

TWO M A T H E M A T I C S   S E M I N A R S

Will take place on _Friday 8 December _


in the *_Luigi Stasi Seminar Room_* (ICTP Leonardo da Vinci Building, 
first floor)



1)  From 15:00 to 16:00

*Course on Vertex algebras and modular forms*

By Don B. Zagier (MPI, Bonn/ICTP)

The course normally takes place at SISSA. It has been rescheduled at 
ICTP as SISSA will be closed on Friday




2) From 16:00 to 17:00

Seminar on *
Modular forms of rational weights and the minimal models

*By Kiyokazu Nagatomo  (Osaka University)

Abstract:
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.