This monograph studies the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It introduces many original ideas, methods, and tools developed by the author and provides a list of all known exact results in explicit form – and derives them – for the heat kernel on spheres and hyperbolic spaces. Part I considers the geometry of simple Lie groups and maximally symmetric spaces in detail, and Part II discusses the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. This text will be a valuable resource for researchers and graduate students working in various areas of mathematics – such as global analysis, spectral geometry, stochastic processes, and financial mathematics – as well in areas of mathematical and theoretical physics – including quantum field theory, quantum gravity, string theory, and statistical physics.
Author(s): Ivan G. Avramidi
Series: Frontiers in Mathematics
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 196
City: Cham
Preface
Acknowledgments
Contents
Notation
Part I Manifolds
1 Introduction
1.1 Heat Equation
1.2 Geometric Framework
1.3 Spherical Coordinates
1.4 Hypergeometric Operators
2 Geometry of Simple Groups
2.1 Right-invariant and Left-invariant Basis
2.2 Metric and Connection
2.3 Heat Kernel on Simple Groups
2.4 Spin-tensor Bundles over Simple Groups
2.5 Geometry of SO(n+1) and SO(1,n)
3 Geometry of SU(2)
3.1 Representations of SU(2)
3.2 Right-Invariant and Left-Invariant Basis
3.3 Metric and Laplacian
3.4 Heat Kernel on SU(2)
4 Maximally Symmetric Spaces
4.1 Normal Geodesic Coordinates
4.2 Maximally Symmetric Spaces
4.3 Local Coordinates
4.4 Geodesic Spherical Coordinates
4.5 Isometries
4.6 Lie Derivatives
4.7 Laplacian
5 Three-dimensional Maximally Symmetric Spaces
5.1 Geometry of S3
5.2 Geometry of H3
Part II Heat Kernel
6 Scalar Heat Kernel
6.1 Reduction Formulas
6.2 Scalar Heat Kernel on S1 and R
6.3 Scalar Heat Kernel on S2
6.4 Scalar Heat Kernel on H2
6.5 Scalar Heat Kernel on Sn, n≥3
6.6 Scalar Heat Kernel on Hn, n≥3
7 Spinor Heat Kernel
7.1 Spinor Heat Trace
7.2 Spinor Heat Kernel
8 Heat Kernel in Two Dimensions
8.1 Heat Kernel on S2
8.2 Heat Kernel on H2
9 Heat Kernel on S3 and H3
9.1 Scalar Heat Kernel on S3 and H3
9.2 Spinor Heat Kernel on S3 and H3
9.3 Heat Trace
9.4 Heat Kernel on S3
9.5 Heat Kernel on H3
10 Algebraic Method for the Heat Kernel
10.1 Algebraic Method for Heat Kernel
10.2 Algebraic Method for S2 and H2
10.3 Heat Kernel Diagonal on Sn and Hn
A Integrals, Series, and Special Functions
A.1 Integrals
A.2 Poisson Summation Formula
A.3 Bernoulli Polynomials
A.4 Gamma Function
A.5 Hypergeometric Function
A.6 Legendre Functions
A.7 Polynomials
A.8 Dirac Matrices
References
Index
