Theta Functions and Knots 1st Edition by Razvan Gelca ISBN 9814520578 9789814520577

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ISBN 10: 9814520578 
ISBN 13: 9789814520577
Author: Razvan Gelca

Theta Functions and Knots 1st Table of contents:

1. Prologue
1.1 The history of theta functions
1.1.1 Elliptic integrals and theta functions
1.1.2 The work of Riemann
1.2 The linking number
1.2.1 The definition of the linking number
1.2.2 The Jones polynomial
1.2.3 Computing the linking number from skein relations
1.3 Witten’s Chern-Simons theory
2. A quantum mechanical prototype
2.1 The quantization of a system of finitely many free one-dimensional particles
2.1.1 The classical mechanics of finitely many free particles in a one-dimensional space
2.1.2 The Schrödinger representation
2.1.3 Weyl quantization
2.2 The quantization of finitely many free one-dimensional particles via holomorphic functions
2.2.1 The Segal-Bargmann quantization model
2.2.2 The Schrödinger representation and the Weyl quantization in the holomorphic setting
2.2.3 Holomorphic quantization in the momentum representation
2.3 Geometric quantization
2.3.1 Polarizations
2.3.2 The construction of the Hilbert space using geometric quantization
2.3.3 The Schrödinger representation from geometric considerations
2.3.4 Passing from real to Kähler polarizations
2.4 The Schrödinger representation as an induced representation
2.5 The Fourier transform and the representation of the symplectic group Sp(2n, ℝ)
2.5.1 The Fourier transform defined by a pair of Lagrangian subspaces
2.5.2 The Maslov index
2.5.3 The resolution of the projective ambiguity of the representation of Sp(2n, ℝ)
3. Surfaces and curves
3.1 The topology of surfaces
3.1.1 The classification of surfaces
3.1.2 The fundamental group
3.1.3 The homology and cohomology groups
3.1.4 The homology groups of a surface and the intersection form
3.2 Curves on surfaces
3.2.1 Isotopy versus homotopy
3.2.2 Multicurves on a torus
3.2.3 The first homology group of a surface as a group of curves
3.2.4 Links in the cylinder over a surface
3.3 The mapping class group of a surface
3.3.1 The definition of the mapping class group
3.3.2 Particular cases of mapping class groups
3.3.3 Elements of Morse and Cerf theory
3.3.4 The mapping class group of a closed surface is generated by Dehn twists
4. The theta functions associated to a Riemann surface
4.1 The Jacobian variety
4.1.1 De Rham cohomology
4.1.2 Hodge theory on a Riemann surface
4.1.3 The construction of the Jacobian variety
4.2 The quantization of the Jacobian variety of a Riemann surface in a real polarization
4.2.1 Classical mechanics on the Jacobian variety
4.2.2 The Hilbert space of the quantization of the Jacobian variety in a real polarization
4.2.3 The Schrödinger representation of the finite Heisenberg group
4.3 Theta functions via quantum mechanics
4.3.1 Theta functions from the geometric quantization of the Jacobian variety in a Kähler polarization
4.3.2 The action of the finite Heisenberg group on theta functions
4.3.3 The Segal-Bargmann transform on the Jacobian variety
4.3.4 The algebra of linear operators on the space of theta functions and the quantum torus
4.3.5 The action of the mapping class group on theta functions
4.4 Theta functions on the Jacobian variety of the torus
4.4.1 The theta functions and the action of the Heisenberg group
4.4.2 The action of the S map
4.4.3 The action of the T map
5. From theta functions to knots
5.1 Theta functions in the representation theoretical setting
5.1.1 Induced representations for finite groups
5.1.2 The Schrödinger representation of the finite Heisenberg group as an induced representation
5.1.3 The action of the mapping class group on theta functions in the representation theoretical setting
5.2 A heuristical explanation
5.2.1 From theta functions to curves
5.2.2 The idea of a skein module
5.3 The skein modules of the linking number
5.3.1 The definition of skein modules
5.3.2 The group algebra of the Heisenberg group as a skein algebra
5.3.3 The skein module of a handlebody
5.4 A topological model for theta functions
5.4.1 Reduced linking number skein modules
5.4.2 The Schrödinger representation in the topological perspective
5.4.3 The action of the mapping class group on theta functions in the topological perspective
6. Some results about 3- and 4-dimensional manifolds
6.1 3-dimensional manifolds obtained from Heegaard decompositions and surgery
6.1.1 The Heegaard decompositions of a 3-dimensional manifold
6.1.2 3-dimensional manifolds obtained from surgery
6.2 The interplay between 3-dimensional and 4-dimensional topology
6.2.1 3-dimensional manifolds are boundaries of 4-dimensional handlebodies
6.2.2 The signature of a 4-dimensional manifold
6.3 Changing the surgery link
6.3.1 Handle slides
6.3.2 Kirby’s theorem
6.4 Surgery for 3-dimensional manifolds with boundary
6.4.1 A relative version of Kirby’s theorem
6.4.2 Cobordisms via surgery
6.5 Wall’s formula for the nonadditivity of the signature of 4-dimensional manifolds
6.5.1 Lagrangian subspaces in the boundary of a 3- dimensional manifold
6.5.2 Wall’s theorem
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
7.1.1 The discrete Fourier transform as a skein
7.1.2 The exact Egorov identity and handle slides
7.2 The Murakami-Ohtsuki-Okada invariant of a closed 3- dimensional manifold
7.2.1 The construction of the invariant
7.2.2 The computation of the invariant
7.3 The reduced linking number skein module of a 3- dimensional manifold
7.3.1 The Sikora isomorphism
7.3.2 The computation of the reduced linking number skein module of a 3-dimensional manifold
7.4 The 4-dimensional manifolds associated to discrete Fourier transforms
7.4.1 Fourier transforms from general surgery diagrams
7.4.2 A topological solution to the projectivity problem of the representation of the mapping class group on theta functions
7.5 Theta functions and topological quantum field theory
7.5.1 Empty skeins and the emergence of topological quantum field theory
7.5.2 Atiyah’s axioms for a topological quantum field theory
7.5.3 The functor from the category of extended surfaces to the category of finite-dimensional vector spaces
7.5.4 The topological quantum field theory underlying the theory of theta functions
8. Theta functions in the quantum group perspective
8.1 Quantum groups
8.1.1 The origins of quantum groups
8.1.2 Quantum groups as Hopf algebras
8.1.3 The Yang-Baxter equation and the universal R-matrix
8.1.4 Link invariants and ribbon Hopf algebras
8.2 The quantum group associated to classical theta functions
8.2.1 The quantum group and its representation theory
8.2.2 The quantum group of theta functions is a quasitriangular Hopf algebra
8.2.3 The quantum group of theta functions is a ribbon Hopf algebra
8.3 Modeling theta functions using the quantum group
8.3.1 The relationship between the linking number and the quantum group
8.3.2 Theta functions as colored oriented framed links in a handlebody
8.3.3 The Schrödinger representation and the action of the mapping class group via quantum group representations
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

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