Theta Functions and Knots

The following sections are included:

• Dedication
• Preface
• Contents

The main actors of this book are classical theta functions and the Gaussian linking number. They were brought together by Edward Witten using a quantum field theory whose Lagrangian is the Chern-Simons functional. In this chapter we offer the reader a first encounter with theta functions, the linking number, and Witten's Chern-Simons theory. The discussion is less formal, less detailed, and less rigorous then the rest of the book, and should be read like a historical summary. We will revisit these notions later, with more rigor and detail.

The development of physics in the second half of the 20th century gave rise to quantum physical interpretations of several mathematical theories. The theory of theta functions is such an example. To help understand how theta functions arise from quantum mechanics, we first present a fundamental example, that of the quantization of a finite number of free one-dimensional particles. We do this also because this example is prototypical for more general cases of Witten's Chern-Simons theory (cf. [Gelca and Uribe (2010)])

The Stone-von Neumann theorem states that up to a unitary equivalence the quantization model is unique, so our task is apparently simple. But the model can be realized in many ways, each of which offering a different perspective on the subject. The purpose of this chapter is to guide the reader through a variety of models for the quantization of finitely many one-dimensional particles, in order to build the necessary background for subsequent chapters.

In this chapter we review a few facts about surfaces, and about curves embedded in surfaces. These are needed for both the construction of the Jacobian variety, and hence for the definition of Riemann's theta functions, and for establishing the relationship between theta functions and knots, which is the topic of this book. Besides the case of surfaces and curves, we address briefly the case of 3-dimensional manifolds that are cylinders over surfaces. We will return to general 3-dimensional manifolds in Chapter 6. The results that are not part of the buildup of a mathematician without a particular interest in topology are proved…

In short, this chapter is about the quantization of a system of finitely many one-dimensional particles with periodic positions and momenta. But there is more to the story. Such a system is less likely to occur in physics, but is a natural occurrence in algebraic geometry. The torus that constitutes the phase space is the Jacobian variety of a Riemann surface. Hence, by contrast with Chapter 2, there are three players in this game:

• the Riemann surface,
• the Jacobian variety of the Riemann surface, and
• the quantum system.

This chapter connects the theory of theta functions to low dimensional topology. The bridge is representation theory. The main idea belongs to the author and Alejandro Uribe, and was first explained in [Gelca and Uribe (2014)]. It is different from Witten's quantum field theoretical approach.

The next step in our study of theta functions is a more detailed analysis of the discrete Fourier transform defined by an element of the mapping class group. We will be able to relate this discrete Fourier transform to 3- and 4-dimensional topology. To enpower the reader with the necessary background, we devote this chapter to reviewing the necessary facts from low-dimensional topology. For more details the reader can consult [Rolfsen (2003)]

All manifolds are orientable, and for an oriented manifold M, -M denotes the manifold with orientation reversed. To distinguish easily between 3- and 4-dimensional manifolds, we use for the latter the boldface script. Sⁿ and Bⁿ denote the n-dimensional sphere and ball respectively.

In this chapter we will reveal a profound relationship between the discrete Fourier transform that acts on theta functions and 3- and 4-dimensional topology. As an application, we will use 4-dimensional manifolds to resolve the projectivity of the representation of the mapping class group on theta functions. We will construct a topological quantum field theory that brings together, for the Riemann surfaces of all genera, their Hilbert spaces of theta functions, the actions of the finite Heisenberg group and of the mapping class group, as well as the discrete Fourier transforms defined by pairs of canonical bases. Moreover, we will prove that such a topological quantum field theory is unique.

Chapter 5 turned theta functions from analytical into combinatorial objects, rephrasing constructs from the theory of theta functions in the language of skein modules. By their own nature, skeins are endowed with symmetries that arise from ambient isotopies.¹ Of these symmetries, we focus on the third Reidemeister move, an instance of which is depicted in Figure 8.1…

In this concluding chapter we explain how the material from this book relates to Chern-Simons theory the way it is done from Witten's point of view. The discussion of concepts and results is sketchy, the reader can find more details about the algebraic geometry in [Griffiths and Harris (1994)] and [Farkas and Kra (1991)] and about the quantum field theory in [Tyurin (2003)] and [Manoliu (1998a)].

The following sections are included:

• Bibliography
• Index