Lecture series

The Many Faces of the Partition Function

A mini-series for graduate students crossing between mathematics and theoretical physics.

by Sean Downes, The Pasayten Institute

\[ \underbrace{Z=\sum_i e^{-\beta E_i}}_{\text{a sum}} \;\to\; \underbrace{Z=\operatorname{Tr}\, e^{-\beta H}}_{\text{a trace}} \;\to\; \underbrace{Z=\int \mathcal{D}\phi\, e^{-S_E}}_{\text{a path integral}} \;\to\; \underbrace{Z(\tau)=\operatorname{Tr}\, q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{c}{24}}}_{\text{a torus amplitude}} \;\to\; \underbrace{\operatorname{ch} V=\sum_n \dim V_n\, q^n}_{\text{a character}} \]

These notes follow a single object - the partition function - through five re-tellings: the statistical-mechanical sum over states, the quantum-statistical trace, the field-theoretic path integral, the conformal torus amplitude, and the representation-theoretic character. A short interlude abstracts the Boltzmann factor that underlies all of them, and a closing chapter reframes the whole story categorically.

Two concrete examples - the free boson and the free fermion - run through every lecture and converge, at the end, on superconformal field theory. The aim is translation: to let a reader fluent in one of these dialects hear the others as the same sentence.

Download the whole series (PDF)

The lectures

  1. Orientation Orientation: One Idea in Five Disguises

    The plan of the series: the partition function as a weighted trace and graded character, and the three threads that run through every lecture.

  2. Lecture 1 Statistical Mechanics: the Partition Function as a Sum Over States

    The canonical partition function from maximum entropy, why the Boltzmann factor is a homomorphism, and the boson and fermion as single modes.

  3. Lecture 2 Quantum Statistics: the Partition Function as a Trace

    Lifting Z to a basis-independent trace over Fock space, the q-product formulas that count integer partitions, and the chemical potential.

  4. Interlude Interlude: the Boltzmann Factor as a Character

    The conceptual spine: the Boltzmann factor as max-entropy weight, character homomorphism, and Euclidean evolution - one conjugate potential per charge.

  5. Lecture 3 Quantum Field Theory: the Partition Function as a Path Integral

    The trace as a Euclidean path integral on a thermal circle, periodic bosons versus antiperiodic fermions, and the Casimir energy and the -1/12.

  6. Lecture 4 Two-Dimensional CFT: the Partition Function on the Torus

    The CFT partition function as a graded trace on the torus, modular invariance, the Dedekind eta, and the fermion's four spin structures.

  7. Lecture 5 Representation Theory: Partition Functions as Characters

    Partition functions as graded characters, Kac–Peterson and Zhu modularity, the Heisenberg and Clifford VOAs, and the Witten index and elliptic genus.

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