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. Tracy.) 

The difference between the average local occupation densities of two sub-lattices of a hard hexagon model of a lattice gas given on pp. 368-371 of Tracy is $R(\tau)=\frac{n(\tau)\eta(5\tau)}{\eta^2(3\tau)}$. 

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 guage corrections

(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.)

From pg. 39 of "[Fivebrane instantons][5] ..." and on pg. 11 of "[D3 instantons][6] ...,"
a correction to the field basis (of the RR axion dual to D3-branes) in type IIB string perturbation theory related to the action of S-duality in ten dimensions: 

$\tilde{c_a} \mapsto \tilde{c_a}-\tilde{c}_{2,a}\:\epsilon(g)$ where, with $g=\binom{a\:\:b}{c\:\:d}$, 

$$\exp(2\pi i \epsilon(g))=\frac{\eta\left [ \frac{a\tau+b}{c\tau+d} \right ]}{\left ( c\tau+d \right )^{\frac{1}{2}}\eta(\tau)}.$$

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][7]" 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][8]" 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][9]" 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][10].)

Ghys presents the equation  $$24\log\eta\left(\frac{a\tau+b}{c\tau+d}\right)=24\: \log(\eta(\tau))+6\: \log(-(c\tau+d)^{2})+2\pi i\:\mathfrak{R}\left(\binom{a\: b}{c\: d}\right)
 $$

where $\mathfrak{R}$ is the Rademacher function, which he relates to the linking number

between two knots related to modular/Lorenz flow: “For every hyperbolic element $A=\binom{a\: b}{c\: d}$ in $PSL(2,Z)$, the linking number between the [modular/Lorenz] knot $k_A$ and the trefoil knot $l$ is equal to $\mathfrak{R}(A)$ ....”    

9) string/brane partition functions, propagators, and metrics


In "[String Theory][11]" by S. Nibbelink, $\eta$ occurs in the denominator of string partition functions for fermionic and bosonic zero modes (pp. 163-7).

A coefficient in the 10-dim metric for a 7-brane is given as $e^{\phi}=\tau_2 \eta^2\bar{\eta}^2|\prod_{i=1}^{k}(z-z_i)^{-\frac{1}{12}}|^2$ on pg. 493 of "[Supergravity vacua and solitons][12]" by G. Gibbons.

**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). 

Other appearances: In Gliozzi's "The Infrared Limit of QCD Effective String" on pg. 14; Panero's "A numerical study of confinement in compact QED" on pg. 4; Zahed's "Holographic Pomeron and Primordial Viscosity" on pg. 1; Caselle and Pinn's "On the Universality of Certain Non-Renormalizable Contributions in Two-Dimensional Quantum Field Theory" on pg. 3; Billo, Casselle, and Pellegrini's "New numerical results and novel effective string predictions for Wilson loops" on pg. 6 and 15; and Basar, Kharzeev, Yee, and Zahed's "Holographic Pomeron and the Schwinger Mechanism" on pg. 7.


  [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/1010.5792
  [6]: http://arxiv.org/abs/1207.1109
  [7]: http://arxiv.org/abs/hep-th/0108022
  [8]: http://www.maths.ed.ac.uk/~aar/papers/atiyahlg.pdf
  [9]: http://www.icm2006.org/proceedings/Vol_I/15.pdf
  [10]: 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
  [11]: http://homepages.physik.uni-muenchen.de/~s.halter/st1/
  [12]: http://arxiv.org/abs/1110.0918