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."

In "Quantum Gauge Theories in Two Dimensions," Edward Witten derives

$$\mathrm{vol}(\mathcal M)=\frac{2(1-2^{1-(2g-2+2k+r)})}{(\sqrt{2}\:\pi)^{2g-2+2k+r}} \zeta(2g-2+2k+r)$$ 

from a volume form for the moduli space $\mathcal M$ of flat connections on a gauge group ($G=SU(2)$) bundle over a compact two-dimensional manifold, a Riemann surface of genus $g$, for a connected sum of an orientable surface of genus $g$ with $k$ Klein bottles and $r$ copies of the projective plane $RP^2$, but this can also be expressed as

$$\mathrm{vol}(\mathcal M)=\frac{2\:\eta(2g-2+2k+r)}{(\sqrt{2}\:\pi)^{2g-2+2k+r}}=\int_{0}^{\infty }\frac{1}{\exp(x)+1}\frac{x^{s-1}}{(s-1)!}dx|_{s=2g-2+2k+r}$$ and so is related to $\eta$ and the moments for Fermi-Dirac statistics.

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). 

Can someone familiar with string theory offer in an answer a description of the corrective term ϵ(g) involving η on pg. 39 of "[Fivebrane instantons][9] ..." and on pg. 11 of "[D3 instantons][10] ..."? (The expression is related to this [MO-Q][11].) 
 


  [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
  [9]: http://arxiv.org/abs/1010.5792
  [10]: http://arxiv.org/abs/1207.1109
  [11]: http://mathoverflow.net/questions/115431/knots-flows-and-the-geometrical-and-combinatorial-significance-of-eigenvalue-eq