This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind $\eta$-function makes a cameo appearance and found that in physics it's related to the
- statistical parameters of solvable Ising models
(See "The Reasonable and Unreasonable Effectiveness of Number Theory in Statistical Mechanics" by G. Andrews and "Introduction to Exactly Solvable models in Statistical Mechanics" by C. Stacy.)
partition functions (statistical mechanics variety) for colored bosons moving on a line (1/24 is the associated Casimir energy) and one-color fermions
operator traces (characters) for the infinite dimensional Lie algebras $\widehat{su}_n$, equivalent to 2-Dim current algebras
partition function of a microscopic black-hole in a 5-Dim D-brane
string theory threshold corrections to the reciprocal gauge-couplings
(For 2-5, see "Nucleon Structure, Duality and Elliptic Theta Functions" by W. Scott. For item 2, see also "Vertex Operators and Modular Forms" by G. Mason and M. Tuite.)
- partition function in 2+1 dimensions and vanishing chemical potential of non-relativistic fermions in a constant magnetic field
("Nonrelativistic Fermions in Magnetic Fields: a Quantum Field Theory Approach" by O. Espinosa, J. Gamboa, S. Lepe, and F. Mendez)
- physics of gauge theories and the Dirac operator
(See "The Logarithm of the Dedekind $\eta$ Function" by M. Atiyah.)
Michael Atiyah even goes so far as to say, "It seems therefore timely to attempt to survey the whole development of the theory of $\log(\eta)$, putting results in their natural order and in the appropriate general context. This is the aim of the present paper, in which the emphasis will be strongly geometrical. In a sense we shall show that the latest ideas from physics [circa 1987] provide the key to a proper understanding of Dedekind's original results."
- knots and dynamics
(See "Knots and Dynamics" by E. Ghys, and Chapter 2 A New Twist in Knot Theory in Dana MacKenzie's book What's Happening in the Mathematical Sciences Vol. 7.)
In what other contexts in physics does the Dedekind $\eta$ function take a bow?