**Resources for aspiring mathematical physicists**

**Basic online resources**

MIT OpenCourseWare: Physics | Mathematics

The Feynman lectures in physics

t'Hooft's How to become a good theoretical physicist

David Tong: Lectures on Theoretical Physics

Leonard Susskind's Theoretical minimum

Perimeter Institute Recorded Seminar Archive: PIRSA

**Note:** For mathematical subjects mentioned below, please refer to the MSC classification.

**General references on "geometry and physics"**

**Introductory to intermediate**

Jost, Geometry and physics

Fecko, Differential geometry and Lie groups for physicists

Isham, Modern differential geometry for physicists

Burke, Applied differential geometry

Bleecker, Gauge theory and variational principles

Baez and Muniain, Gauge fields, knots and gravity

Frankel, The geometry of physics

Hamilton, Mathematical gauge theory

**Advanced **

Boss and Bleecker, Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics

Nash, Differential topology and quantum field theory

Dunajski, Solitons, instantons and twistors

Deigne et al, Quantum fields and strings: a course for mathematicians

**Recommended books and review articles by subject**

**Classical mechanics**

**Introductory**

Goldstein et al, Classical Mechanics

Arnold, Mathematical Methods of Classical Mechanics

**Intermediate**

Jose and Saletan, Classical dynamics: A contemporary approach

Landau and Lifschitz, Mechanics

**Advanced**

Marsden and Ratiu, Introduction to mechanics and symmetry

Abraham and Marsden, Foundations of Mechanics

Spivak, Physics for mathematicians: Mechanics

Liebermann and Marle, Symplectic geometry and analytical mechanics

Souriau et al, Structure of dynamical systems: a symplectic view of physics

Guillemin & Sternberg, Symplectic techniques in physics

**Relevant mathematics:**

- Lagrangian formulation: basic differential geometry, variational calculus
- Hamiltonian formulation and symplectic reduction: symplectic and Poisson geometry
- Noether theorems: Jet bundles, Lie algebras and Lie groups, Lie group actions on manifolds
- ODEs and basic theory of geometric dynamical systems

**Special relativity**

Gourgoulhon, Special relativity in general frames

Naber, The geometry of Minkowski spacetime

**Relevant mathematics:**

- Linear and multi-linear algebra and tensor calculus
- Quadratic forms and quadratic spaces
- Affine spaces
- Lie groups and Lie algebras
- Basic theory of linear PDEs

**Classical electrodynamics**

**Introductory**

Purcell and Morrin, Electricity and magnetism

Zangwill, Modern Electrodynamics

Brau, Modern problems in classical electrodynamics

Greiner, Classical electrodynamics

Schwartz, Principles of electrodynamics

Landau & Lifschitz, The classical theory of fields

**Intermediate**

Jackson, Classical electrodynamics

Franklin, Classical electromagnetism

**Advanced**

Jentschura, Advanced classical electrodynamics

Hehl and Obukhov, Foundations of classical electrodynamics

Garrity, Electricity and magnetism for mathematicians

Scheck: Classical field theory: On Electrodynamics, Non-Abelian Gauge Theories and Gravitation

**Relevant mathematics:**

- ODEs and some PDEs
- some theory of distributions
- some basic differential geometry, including the geometry of Abelian principal bundles

**Classical field theory**

Soper, Classical field theory

Rubakov, Classical theory of gauge fields

Bleecker, Gauge theory and variational principles

Gockeler & Schucker, Differential geometry, gauge theory and gravity

Choquet-Bruhat and DeWitt-Morette, Analysis, manifolds and physics

Atiyah, Geometry of Yang-Mills fields

Atiyah, Gauge theories

**Relevant mathematics:**

- fiber bundles
- connections on principal and associated bundles
- characteristic classes and Chern-Weyl theory
- geometric PDEs
- spin geometry
- variational calculus

**General relativity**

Wald, General relativity

Kriele, Spacetime

Weinberg, Gravitation and cosmology

Straumann, General relativity

Hawking and Ellis, The large scale structure of space-time

Joshi, Gravitational collapse and spacetime singularities

Poisson, A relativist's toolkit

Choquet-Bruhat, General Relativity and the Einstein Equations

**Relevant mathematics: **

- Riemannian and semi-riemannian geometry
- basic differential topology
- geometric PDEs
- fiber bundles
- spin geometry
- Dirac operators and index theory

**Quantum mechanics**

**Introductory to intermediate**

Shankar, Principles of Quantum mechanics

Sakurai, Modern quantum mechanics

Messiah, Quantum mechanics, vols 1 & 2

Schwabl, Advanced quantum mechanics

Weinberg, Lectures on quantum mechanics

Jackson, Mathematics for quantum mechanics

Gustafson and Sigal, Mathematical concepts of quantum mechanics

**Advanced and special topics**

Tannoudji et al, Quantum mechanics, vols 1--3

Marino, Advanced topics in quantum mechanics

Sternberg, A mathematical companion to quantum mechanics

Prugovecki, Quantum mechanics in Hilbert space

Moretti, Spectral theory and quantum mechanics

Mackey, Mathematical foundations of quantum mechanics

Strocchi, An introduction to the mathematical structure of quantum mechanics

Faddeev and Yakubovskii, Lectures on quantum mechanics for mathematics students

Takhtajan, Quantum mechanics for mathematicians

Hall, Quantum theory for mathematicians

Glimm and Jaffe: Quantum physics: a functional integral point of view

**Relevant mathematics: **

- Basic functional analysis
- operator theory
- linear PDEs
- some measure and probability theory
- basic distribution theory
- C
^{*}algebras - von Neumann algebras
- representation theory of Lie groups (including finite and countable groups).

**Statistical physics**

Cardy, Scaling and renormalization in statistical physics

Dorlas, Statistical mechanics

Sethna, Statistical mechanics: entropy, order parameters and complexity

Gallavotti, Statistical mechanics: a short treatise

Huang, Statistical mechanics

Georgii, Gibbs measures and phase transitions

Ellis, Entropy, large deviations and statistical mechanics

Lavis and Bell, Statistical mechanics of lattice systems, vols 1 & 2

Simon, The statistical mechanics of lattice gases

Friedli and Venenik, Statistical mechanics of lattice systems

Khinchin and Gamow, Mathematical foundations of statistical mechanics

Minlos, Introduction to mathematical statistical physics

Ruelle, Statistical mechanics: rigorous results

Thompson, Mathematical statistical mechanics

Goldenfeld, Lectures on phase transitions and the renormalization group

Yeomans, Statistical mechanics of phase transitions

McCoy and Wu, The two-dimensional Ising model

Bovier, Statistical mechanics of disordered systems

Lanford, Entropy and equilibrium states in classical statistical mechanics

Petersen, Ergodic theory

Bratelli and Robinson, Operator algebra and quantum statistical mechanics, vols 1 & 2

**Manybody theory**

Fetter and Walecka, Quantum theory of many-particle systems

Altland and Simons, Condensed matter quantum field theory

Rammer, Quantum field theory of non-equilibirum states

Negele and Orland, Quantum many-particle systems

Mahan, Many-particle physics

Coleman, Introduction to many-body physics

Abrikosov et al, Methods of quantum field theory in statistical physics

Fradkin, Field theories of condensed matter physics

Danielewicz, Quantum theory of nonequilibrium processes (review paper, parts I and II)

Lehmann, Mathematical methods of many-body quantum field theory

**Quantum Field Theory**

Srednicki, Quantum field theory

Schwartz, Quantum field theory and the standard model

Weinberg, The quantum theory of fields, vols 1 & 2

Itzikson and Zuber, Quantum field theory

Fradkin, Quantum field theory: an integrated approach

Folland, Quantum field theory: a tourist guide for mathematicians

Talagrand, What is a quantum field theory ?

**Relevant mathematics: **

- functional analysis, operator theory and PDEs
- distribution theory
- measure and probability theory
- representations of Lie groups and Lie algebras
- some combinatorics
- some representation theory of associative algebras.

**Quantum field theory on curved spacetimes**

Birrell and Davies, Quantum fields in curved space

Parker and Toms, Quantum field theory in curved spacetime

Fulling, Aspects of quantum field theory in curved spacetime

Wald, Quantum field theory in curved spacetime and black hole thermodynamics

Bar and Fredenhagen, Quantum field theory on curved spacetimes

Hack, Cosmological applications of algebraic quantum field theory in curved spacetimes

**Relevant mathematics:** Everything needed for QFT and GR.

**Algebraic and axiomatic QFT**

Wightman and Streater, PCT, spin and statistics and all that

Haag, Local quantum physics

Araki, Mathematical theory of quantum fields

Brunetti et al, Advances in algebraic quantum field theory

**Relevant mathematics:** Everything needed for QFT.

**Conformal field theory**

Di Francesco et al, Conformal field theory

Schottenloher, A mathematical introduction to conformal field theory

Frenkel and Ben-Zvi, Vertex algebras and algebraic curves

Fuchs, Affine Lie algebras and quantum groups

Kohno, Conformal field theory and topology

Lepowski and Li, Introduction to vertex operator algebras and their representations

Kac, Infinite-dimensional Lie algebras

Kac and Raina, Bombay lectures on infinite-dimensional Lie algebras and highest weight representations

Carter, Lie algebras of finite and affine type

Wakimoto, Lectures on infinite-dimensional Lie Algebra

**Relevant mathematics:**

- representations of infinite-dimensional Lie algebras
- representations of associative algebras
- Riemann surfaces
- some operator theory and theory of distributions
- quantum groups
- some category theory

**Supersymmetry**

Wess and Bagger, Supersymmetry and supergravity

Ceccotti, Supersymmetric field theories

Frappat et al, Dictionary on Lie superalgebras

Scheunert, The theory of Lie superalgebras: an introduction

Gorelik and Papi, Advances in Lie superalgebras

Tuynman, Supermanifolds and supergroups

Rogers, Supermanifolds

De Witt, Supermanifolds

Varadarajan, Supersymmetry for mathematicians

Carmeli et al, Mathematical foundations of supersymmetry

Cattaneo and Schaetz, Introduction to supergeometry

Kapranov, Supergeometry in mathematics and physics

Kessler, Supergeometry, super Riemann surfaces and and the superconformal action functional

Westra, Superrings and supergroups

**Relevant mathematics: **

- Lie superalgebras
- supermanifolds
- basic commutative and supercommutative algebra
- basic notions of scheme theory
- basic theory of quantum groups

**Supergravity**

Ortin, Gravity and strings

Freedman and van Proeyen, Supergravity

Castellani, D'Auria and Fre, Supergravity and superstrings: a geometric perspective

Sezgin, Survey of supergravities

**Basic string theory**

Polchiski, String theory, vols 1 & 2

Hatfield, Quantum field theory of point particles and strings

West, Introduction to strings and branes

Becker, Becker and Schwarz, String theory and M-theory: a modern introduction

Ibanez et al, String theory and particle physics: an introduction to string phenomenology

Deligne et al, Quantum fields and strings: a course for mathematicians

Jost, Bosonic strings: a mathematical treatment

Albeveiro et al, A mathematical introduction to string theory

**Relevant mathematics:** Everything from the above plus:

- theory of (pseudo-)harmonic maps
- some Teichmuller theory
- moduli spaces of Riemann surfaces
- index theory
- some topological K-theory
- some complex and Kahler geometry
- some algebraic geometry
- basic theory of G-structures

**Advanced string theory**

**A. (Super)string perturbation theory**

Witten, Superstring perrturbation theory via super Riemann surfaces: an overview

**B. Calabi-Yau compactifications**

Yau and Nadis, The shape of inner space

Greene, String theory on Calabi-Yau manifolds

Yau, A survey of Calabi-Yau manifolds

Joyce, Compact manifolds with special holonomy

Gross, Joyce & Huybrechts, Calabi-Yau manifolds and related geometries

**C. Topological string theory**

Marino, Chern Simons theory, matrix models and topological strings

Alim, Lectures on mirror symmetry and topological string theory

Nietzke and Vafa, Topological strings and their physcal applications

Marino, An introduction to topological string theory

**D. Mirror Symmetry (ordinary and homological)**

Abramovich et al, Enumerative invariants in algebraic geometry and string theory

Cox and Katz, Mirror symmetry and algebraic geometry

Hori et al, Mirror Symmetry

Bocklandt, A gentle introduction to homological mirror symmetry

Wang Yao, A glimpse into homological mirror symmetry

Aspinwall et al, Dirichlet branes and mirror symmetry

Huybrechts, Fourier-Mukai transforms in algebraic geometry

Auroux, A beginner's introduction to Fukaya categories

Smith, A symplectic prolegomenon

Seidel, Lectures on categorical dynamics and symplectic topology

Seidel, Fukaya categories and Picard-Lefschetz theory

Fukaya et al: Lagrangian intersection Floer theory: anomaly and obstruction, vols 1 & 2

Jarvis and Priddis (eds), Singularities, Mirror Symmetry and the Gauged Linear Sigma Model

**E. Mathematical foundations of string and supergravity theories**

**Relevant mathematics:**

- Complex and Kahler geometry
- Algebraic geometry (including toric geometry)
- noncommutative algebraic geometry
- Symplectic geometry and topology
- Gromow-Witten theory
- G-manifolds
- G-structures
- theory of foliations
- noncommutative geometry in the sense of Connes
- singularity theory and Thom-Mather theory (a.k.a. "catastrophe theory")
- Triangulated and derived categories
- Floer homology and Fukaya categories
- algebraic K-theory
- differential topology
- moduli spaces (differential and algebraic)
- basic theory of differential and algebraic stacks
- algebraic homotopy theory
- theory of gerbes
- simplicial methods and higher category theory
- geometric analysis