Mathematical Physics Courses

List of graduate courses offered at the Department of Mathematics, by members of our research group.
For more details or any questions, just send an email to Gabriel Lopes Cardoso.

Renormalization Group

Level: Master (Fall).

Goal: Introduce the basic notions of quantum field theory and the Feynman integral, including the renormalization group and gauge theory.

Syllabus:

Finite Dimensional Integrals: Partition function and free energy; Saddle points, formulae for asymptotic approximations and Borel summability; Feynman diagrams and Wick's theorem.

Feynman Integral: Classical and quantum action functional; Definition of Feynman integral; Forced harmonic oscillator; Green functions and propagators; Correlation functions and operator formalism.

Scalar Field Theory: λφ⁴ theory, perturbative expansion, divergences and dimensional regularization; Renormalization and β-functions; Cross-sections, unitarity, causality and Lehmann representation.

Renormalization Group: Real and momentum space; Fixed points, anomalous dimensions and critical exponents; Renormalization group, effective action, effective potential; Composite operators and OPE's.

Gauge Theory: Fermions, QED and QCD; Ward-Takahashi identities; Perturbative quantization, Faddeev-Popov, ghosts and unitarity; Renormalization, β-functions and asymptotic freedom.

Mathematical Relativity

Goal: Introduce several mathematical topics in general relativity, with emphasis on singularity theorems and black hole solutions in several dimensions.

Syllabus:

Examples: de Sitter, Anti-de Sitter, FLRW, Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman solutions with/without cosmological constant; Carter-Penrose diagrams; d-dimensional generalizations.

Singularity Theorems: Causal structure, properties of global hyperbolicity and complete geodesics; Definition, description and the character of singularities; Hawking and Penrose theorems.

The Cauchy Problem: Einstein equations, initial data and second order hyperbolic equations; Existence and uniqueness in empty space and with matter; Positive mass theorem, Penrose inequality, cosmic censorship.

The 4 Laws of Black Holes: The wave equation in curved spacetime; Classical and quantum fields in curved spacetime; Area theorem and Hawking effect; Thermodynamical laws; Wald's formula.

Horizon Topology in Several Dimensions: Unicity theorems in d=4; Solutions in dimension d=5 and dimension d≥6; Gregory-Laflamme instability; Black rings, black saturn and blackfolds.

Geometry and Gauge Theory

Goal: Introduce the basic notions of gauge theory and its interaction with both geometry and topology, with a particular emphasis towards the non-perturbative structure of the theory.

Syllabus:

Gauge Theory: Fibre-bundle geometry; Yang-Mills theory and Yang-Mills-Higgs theory; Chern-Simons theory; Self-dual Yang-Mills equations, BPS equations and gauge theory.

BRST and BV: First and second class constraints, quantization of constrained systems; BRST symmetry, ghosts and Koszul-Tate differential; Feynman integral, antifield formalism and BV.

Anomalies: Fermions, classical symmetries, quantums symmetries and the axial current; Fujikawa method; Index theorem; Non-perturbative considerations, anomalies and BRST; Global anomalies.

Monopoles: Solitons, semi-classical methods and collective coordinates; Topological conservation laws; The 't Hooft-Polyakov monopole; Moduli spaces, scattering, Nahm transform and the spectral curve.

Instantons: Tunnel effect, gauge theory and θ-vacua; Topology and boundary conditions; Moduli spaces, ADHM construction and holomorphic vector bundles; Divergent series and Borel summability.

Conformal Field Theory

Goal: Introduce the basics of conformal field theory and construct the corresponding formalism for minimal models (including the relation to matrix models) and WZW models.

Syllabus:

Conformal Symmetry: Phase transitions and scale invariance; Conformal group, Virasoro algebra, OPE's and Ward identities; Superconformal symmetry, RNS algebras and spin fields.

Operator Formalism: Free fields, conformal families and bootstrap; Bosons and fermions on the torus, modular invariance, CFT and Riemann surfaces; Boundary conditions and Verlinde formula.

Minimal Models: Verma modules, representations and Kac determinant; Minimal models, unitarity and fusion rules; Coulomb gas formalism; Modular invariance.

Matrix Models: The 1/N expansion, semi-classical approximation, resolvent and spectral curve; Orthogonal polynomials and correlation functions; Critical points and DSL; Non-perturbative phenomena.

Current Algebras and WZW Models: Simple Lie algebras and affine Lie algebras; WZW models, Sugawara construction and Knizhnik-Zamolodchikov equation; Ishibashi states.

String Theory

Goal: Introduce the basic notions of string theory, together with an introduction to selected advanced research topics.

Syllabus:

Bosonic Strings: Polyakov action, covariant quantization, open strings and closed strings; S-matrix, tree-level and one-loop amplitudes; Riemann surfaces and CFT.

D-Branes and Dualities: Toroidal compactification, closed strings and T-duality; Orbifolds; D-branes, T-duality and Wilson lines; Gauge theory and Born-Infeld electrodynamics.

Superstrings: Superstrings of type I and II, Ramond and Neveu-Schwarz sectors, modular invariance and GSO projection; Superstring interactions; Calabi-Yau compactifications.

More on D-Branes and Dualities: T-duality; D-brane interactions: kinematics, dynamics and bound states; S-duality, U-duality, M-theory and other dualities; Black holes and AdS/CFT.

Topological String Theory: Chern-Simons theory; Kähler and Calabi-Yau geometry; Topological σ-models, A and B models; Mirror symmetry; Large N dualities and matrix models; OSV conjecture.