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


1) statistical parameters of solvable Ising models

(See "[The Reasonable and Unreasonable Effectiveness of Number Theory in Statistical Mechanics][1]" by G. Andrews and "[Introduction to Exactly Solvable models in Statistical Mechanics][2]" by C. Stacy.)  

2) partition functions (statistical mechanics variety) for colored bosons moving on a 
   line  (1/24 is the associated Casimir energy) and one-color fermions

3) operator traces (characters) for the infinite dimensional Lie algebras $\widehat{su}_n$,
   equivalent to 2-Dim current algebras

4) partition function of a microscopic black-hole in a 5-Dim D-brane

5) string theory threshold corrections to the reciprocal gauge-couplings

(For 2-5, see "[Nucleon Structure, Duality and Elliptic Theta Functions][3]" by W. Scott. For item 2, see also "[Vertex Operators and Modular Forms][4]" by G. Mason and M. Tuite.) 

6) 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][5]" by O. Espinosa, J. Gamboa, S. Lepe, and F. Mendez)

7) physics of gauge theories and the Dirac operator

(See "[The Logarithm of the Dedekind $\eta$ Function][6]" 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."

8) knots and dynamics

(See "[Knots and Dynamics][7]" 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][8].)   

**In what other contexts in physics does the Dedekind $\eta$ function take a bow?**

(Edit) Moreover, since this is a community wiki and not a test question with one best answer but an attempt to come to a better understanding of the $\eta$-function and associated math and physics, I invite people to expand on any of the items with specifics (e.g., exact formulas), more references, and/or insightful commentaries (e.g., what you believe are important aspects of the references). 
 


  [1]: http://books.google.co.jp/books?hl=en&lr=&id=j-pLi01XbFUC&oi=fnd&pg=PA21&dq=GE%20Andrews%20reasonable%2band%2bunreasonable%2beffectiveness%2bof%2bnumber%2btheory&ots=tUl15FN6zA&sig=n1cTHjUpWjY42TLrl-OQxkz2xaY#v=onepage&q=GE%20Andrews%20reasonable%2band%2bunreasonable%2beffectiveness%2bof%2bnumber%2btheory&f=false
  [2]: http://www.math.ucdavis.edu/~tracy/selectedPapers/1980s/CV31.pdf
  [3]: http://arxiv.org/abs/hep-ph/9912502
  [4]: http://arxiv.org/abs/0909.4460
  [5]: http://arxiv.org/abs/hep-th/0108022
  [6]: http://www.maths.ed.ac.uk/~aar/papers/atiyahlg.pdf
  [7]: http://www.umpa.ens-lyon.fr/~ghys/articles/ghys-icm.pdf
  [8]: http://books.google.co.jp/books?id=yBL54nHAwXsC&pg=PA14&lpg=PA14&dq=The%20Logarithm%20of%20the%20Dedekind%20eta%20%20Function%20%20Atiyah&source=bl&ots=NsCYEW6gaQ&sig=7Tc3R-nSY4USVutxuCy1pf9zeQ4&hl=en&sa=X&ei=jfa7UMOSCabNmgW43ICYBg&sqi=2&ved=0CDYQ6AEwAw#v=onepage&q=The%20Logarithm%20of%20the%20Dedekind%20eta%20%20Function%20%20Atiyah&f=false