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Geometry of physics by Urs Schreiber

Schreiber, U. (2016). Higher prequantum geometry. Retrieved from http://arxiv.org/abs/1601.05956

Geometry of physics on nlab

0. Preliminaries on categories and toposes

I. Geometry

1. Smooth sets

  • Overview

    • Generalized sets equipped with smooth structure called smooth sets
    • Smooth sets are generalized to smooth homotopy types, smooth h-sets, or smooth 0-types
    • The definition of smooth sets is valuable because it is both simpler and more powerful than those it subsumes, which includes smooth manifolds, Frechet manifolds, and diffeological spaces
    • The category of smooth sets is the topos of sheaves on the gros site of Cartesian spaces that contains as full subcategories smooth manifolds and diffeological spaces

  • 1.1 Abstract coordinate systems

    • Overview

      • Every kind of geometry is modeled on a collection of archetypical basic spaces and geometric homorphisms between them

      • In differential geometry the

          1. archetypical spaces are abstract standard Cartesian coordinate systems, \[\mathbb{R}^n\]
          1. archetypical geometric homomorphisms between them are smooth functions \[\mathbb{R}^{n_1} \rightarrow \mathbb{R}^{n_2}\] that are sometimes referred to as coordinate transformations

    • 1.1.1 The continuum real line

      • The fundamental premise for using differential geometry as a model of geometry for physics is that the abstract worldline of any particle can be modeled by the continuum real line \[\mathbb{R}\]
    • 1.1.2 Cartesian spaces and smooth functions
    • 1.1.3 The magic properties of smooth functions
    • 1.1.4 The site of abstract coordinate systems

  • 1.2 Smooth sets

    • 1.2.1 Plots of smooth sets and their gluing
    • 1.2.2 Homomorphisms of smooth sets
    • 1.2.3 Products and fiber products of smooth sets
    • 1.2.4 Smooth mapping spaces and smooth moduli spaces
    • 1.2.5 The cohesive topos of smooth sets
    • 1.2.6 Concrete smooth sets: diffeological spaces
    • 1.2.7 Differential forms
    • 1.2.8 Integration and transgression

2. Smooth homotopy types

3. Stable homotopy types

4. Groups

5. principal bundles

6. Manifolds and orbifolds

7. G-structure and Cartan geometry

8. Representations and associated bundles

9. Modules

10. Flat connections

11. de Rham coefficients

12. Principal connections

13. Integration

14. Super-geometry

I-5. Transition

1. Prequantum geometry

2. WZW terms

3. BPS charges

II. Physics

1. Perturbative quantum field theory

    1. Geometry
    1. Spacetime
    1. Fields
    1. Field variations
    1. Lagrangians
    1. Symmetries
    1. Observables
    1. Phase space
    1. Propagators
    1. Gauge symmetries
    1. Reduced phase space
    1. Gauge fixing
    1. Quantization
    1. Free quantum fields
    1. Interacting quantum fields
    1. Renormalization

2. Physics in higher geometry: motivation and survey

3. Hamilton-Jacobi-Lagrange mechanics via prequantized Lagrangian correspondences

4. Hamilton-de Donder-Wyel field theory via higher correspondences

5. Local topological prequantum field theory

6. Prequantum gauge theory and gravity

7. Quantum mechanics

8. Geometric quantization

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