Modular forms in quantum field theory

Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms

This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few....

Modular Structure and Duality in Conformal Quantum Field Theory

Making use of a recent result of Borchers, an algebraic version of the BisognanoWichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description i...

Analytic Theory of Modular Forms

Modular Forms and Applications in Number Theory

Modular forms are complex analytic objects, but they also have many intimate connections with number theory. This paper introduces some of the basic results on modular forms, and explores some of their uses in number theory.

Observables in Modular Field Theory

The observables of modular quantisation are studied from the point of view of locality. Such a study allows identification of possible Hamiltonians and also enables us to generalize the fundamental bilinear commutation relations of parafield theory. A comparison of modular field theory with a normal U(m) gauge theory, begun in an earlier publication, is completed with the conclusion that the tw...