A Glimpse of 2D and 3D Quantum Field Theories Through Number Theory

Résumé du livre

The work presented here lies at the intersection between quantum field theory, string theory, and number theory. The latter is the fil rouge connecting extremal conformal field theories, supersymmetric black hole entropy, and 3-manifold invariants. After a general introduction, Chapter 2 reviews relevant number-theoretical objects such as Siegel and Jacobi modular forms, mock modular forms, and Rademacher sums. Chapter 3 analyzes the properties of extremal conformal field theories at small central charges (c d"24). The twining functions of these theories - partition functions with the insertion of an element of the symmetry group - are examined through Rademacher sums. Chapter 4, in turn, is dedicated to the construction of a "mixed" Rademacher sum, which encodes the microscopic degeneracy of single centered 1/4-BPS black holes in Type IIB string theory compactified on K3 × T2. This black hole system provides a rare opportunity to compare the sub-leading terms in the entropy of the brane system with the macroscopic entropy provided by the supergravity theory. Chapter 5 investigates the half-indices (homological invariants of 3-manifolds) of certain 3d N = 2 theories. These supersymmetric theories are 3d-2d coupled systems which can be viewed as three-dimensional Chern-Simons theories on Seifert manifolds via the 3d-3d correspondence. Here an important role is played by false theta functions, mock theta functions, and quantum modular forms.