Combinatorial Quantum Field Theory (CO 739)

Description

Quantum field theory is the mathematical language in which the physical laws of elementary particles are formulated, making it one of the most important theories in physics. One of its most intriguing features is that it combines two very different worlds: analysis and combinatorics. At its core, QFT is formulated analytically, with fields on smooth manifolds described by functional integrals. These integrals are often ill-defined and almost never have exact solutions. To make concrete predictions, we have to use perturbation theory. In this approach, the continuous analytic picture turns into a discrete combinatorial world of sums over graphs and algebraic integrals with rich combinatorial structure. This interplay between the continuous and the discrete is a defining feature of quantum field theory.

Over the last thirty years, this connection has created a powerful feedback loop between mathematics and physics. On one side, QFT has served as a mathematical engine, leading to deep discoveries in geometry, topology, and combinatorics. On the other side, new developments in mathematics have significantly advanced the predictive power of QFT. In this course, we will explore key aspects of this feedback loop from a discrete mathematical perspective.

The course is intended for both mathematicians who wish to broaden their horizons in quantum field theory and for physicists who want to strengthen their mathematical toolkit with powerful discrete methods.

Participants should be comfortable with the basics of real analysis (sequences and series) and ideally will have previous exposure to basic algebra and complex analysis (analytic functions, Cauchy residue theorem). Interested people unsure of their background should email me for more information. I hope that we will have people with a variety of backgrounds who can each bring their different perspectives to the course.

Logistics

Time: Sep 4 to Dec 2, Tuesday and Thursday 10-11:30 (no classes on Sep 25, Oct 14 & 16)

Location: Alice Room, 3rd floor at Perimeter Institute

For non-PI locals: Sign-in at the reception is required, unfortunately. Please be a couple of minutes early to sign in.

Office hours: Room 330, Perimeter Institute - by appointment or email with any questions (for now).

Assignments/Exercises: Will be posted on this website.

For credit: There will be two graded assignments and a final project (making up (25+25+50)% of the final grade, respectively).

Books & Other Resources

Tentative Class Plan

  • Part 1: Analytic graph combinatorics

    What is a graph?, classical graph enumeration, asymptotic analysis, Wick's theorem, Laplace's method, ring of factorially divergent power series, basic resurgence, computational aspects, statistics for graphical enumeration, general 0-dimensional quantum field theory

  • Part 2: Advanced combinatorial quantum field theory

    Matrix models, combinatorial Feynman integrals, generalized permutahedra, graph complexes, Ising model, ribbon graphs, Euler characteristics

Course Material