Theta functions and knots

1. Prologue. 1.1. The history of theta functions. 1.2. The linking number. 1.3. Witten's Chern-Simons theory -- 2. A quantum mechanical prototype. 2.1. The quantization of a system of finitely many free onedimensional particles. 2.2. The quantization of finitely many free one-dimensional particles via holomorphic functions. 2.3. Geometric quantization. 2.4. The Schrodinger representation as an induced representation. 2.5. The Fourier transform and the representation of the symplectic group Sp(2n,[symbol]) -- 3. Surfaces and curves. 3.1. The topology of surfaces. 3.2. Curves on surfaces. 3.3. The mapping class group of a surface -- 4. The theta functions associated to a Riemann surface. 4.1. The Jacobian variety. 4.2. The quantization of the Jacobian variety of a Riemann surface in a real polarization. 4.3. Theta functions via quantum mechanics. 4.4. Theta functions on the Jacobian variety of the torus -- 5. From theta functions to knots. 5.1. Theta functions in the representation theoretical setting. 5.2. A heuristical explanation. 5.3. The skein modules of the linking number. 5.4. A topological model for theta functions -- 6. Some results about 3- and 4-dimensional manifolds. 6.1. 3-dimensional manifolds obtained from Heegaard decompositions and surgery. 6.2. The interplay between 3-dimensional and 4-dimensional topology. 6.3. Changing the surgery link. 6.4. Surgery for 3-dimensional manifolds with boundary. 6.5. Wall's formula for the nonadditivity of the signature of 4-dimensional manifolds. 6.6. The structure of the linking number skein module of a 3-dimensional manifold -- 7. The discrete Fourier transform and topological quantum field theory. 7.1. The discrete Fourier transform and handle slides. 7.2. The Murakami-Ohtsuki-Okada invariant of a closed 3-dimensional manifold. 7.3. The reduced linking number skein module of a 3-dimensional manifold. 7.4. The 4-dimensional manifolds associated to discrete Fourier transforms. 7.5. Theta functions and topological quantum field theory -- 8. Theta functions in the quantum group perspective. 8.1. Quantum groups. 8.2. The quantum group associated to classical theta functions. 8.3. Modeling theta functions using the quantum group -- 9. An epilogue - Abelian Chern-Simons theory. 9.1. The Jacobian variety as a moduli space of connections. 9.2. Weyl quantization versus quantum group quantization of the Jacobian variety.

This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Razvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern-Simons theory will find here an introduction using the simplest case, that of abelian Chern-Simons theory. Moreover, the construction of abelian Chern-Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done. Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study.

Includes bibliographical references (p. 445-449) and index.