The Dedekind Eta Function in Physics

Question

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" by G. Andrews and "Introduction to Exactly Solvable models in Statistical Mechanics" 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" by W. Scott. For item 2, see also "Vertex Operators and Modular Forms" by G. Mason and M. Tuite.)

From pg. 39 of "Fivebrane instantons ..." and on pg. 11 of "D3 instantons ...," 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" 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" 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" 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.)

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" 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" 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.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Tom Copeland

Comments

Do you consider conformal field theory covered in your question? ...because this is not entirely unrelated to the Ising model.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Steve Huntsman

---

Well, you'll find it discussed in Nash. Gannon also discusses a boson on a circle.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Steve Huntsman

---

If your goal in asking this question goes beyond simply accumulating a list of interesting physics topics, I think it would be important to note how precisely $\eta$ appears, rather than simply pointing out that it is mentioned in a paper. For example, in many statistical partition functions and Fock space characters, it appears in the denominator, since the reciprocal of $\eta$ is a generating function that counts partitions of integers.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user S. Carnahan

Answer 1

The Dedekind eta function shows up in three-dimensional quantum gravity: http://arxiv.org/abs/0712.0155 (Alexander Maloney, Edward Witten, Quantum Gravity Partition Functions in Three Dimensions). On page 17 a basic partition function $Z_{0,1}$ of the theory is calculated as $$Z_{0,1}(\tau)=\frac{1}{|\eta(\tau)|^2}|\bar q q|^{-(k-1/24)}|1-q|^2.$$ It also appears in the calculation of supergravity partition functions in sec.7.

The Dedekind eta function also enters in (supersymmetric) physics through mock modular forms: http://arxiv.org/abs/1208.4074 (Atish Dabholkar, Sameer Murthy, Don Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms).

A good review of mock modular forms is http://mathcs.emory.edu/~ono/publications-cv/pdfs/114.pdf (Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory).

I first heared about Dedekind Eta Function in the physics context via Freeman Dyson's lovely essay "Missed opportunities": http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183533964

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Zurab Silagadze

Comments

Nice examples. Thank you.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Tom Copeland

Answer 2

Apart from adding to the list of examples, I'll say something about why lots of functions in physics and in particular in string theory end up being expressible in terms of the Dedekind eta-function.

The basic reason is that the Dedekind eta-function is one of the main examples of a modular form. These in turn are really sections of a certain canonical line bundle on the  stack of elliptic curves. But since an elliptic curve over the complex numbers is just a complex torus, that means that a modular form in general (and hence the Dedekind eta in particular) is precisely something that 

1. assigns something to each genus-1 closed string worldsheet;

2. subject potentially a conformal anomaly.

So in particular the partition function of a superstring yields a modular form (see at Witten genus for the case of the heterotic superstring) and that accounts for the bulk of the appearances of modular functions and Dedekind etas mentioned above. In particular since the superstring partition function counts BPS states in the effective target space field theory of the string, which in turn means that lots of structures found in supersymmetric field theory in higher dimensions also comes out as being modular forms (since there is secretly the superstring underlying them).

Comments

Nice overview. I took note of D-Eta for the first time as an eigenfunction of some operators related to a Witt Lie algebra pointed out by Givental, so I'm glad to hear of any connections to conformal field theory and strings.

Answer 3

​Here are a couple of examples:

1. The (chiral) contribution of the non-zero modes of a single real scalar field to the torus partition function is $\frac1{\eta(\tau)}$. A derivation is as follows: Recall that the non-zero modes are equivalent to oscillators with frequency a positive non-zero integer (in suitable units) -- call them $\alpha_{-n}$ for $n=1,2,\ldots$. Computing Tr$(q^{L_0-(1/24)})$ over the Fock space of the oscillator mode is $(1-q^{n})^{-1}$. The total partition function is the product for all $n$. Along with the $q^{-1/24}$, one obtains the inverse of the Dedekind eta function. (Couldn't find a decent reference and hence am writing out the computation.)
2. Many, not all,  multiplicative eta products that appear in the work of Dummit, Kisilevsky and McKay (Cont. Math. Vol. 45 (1985) 89-98) as well as the article  by Geoff Mason in the same volume make an appearance as the generating function of $\tfrac12$-BPS states in four-dimensional $\mathcal{N}=4$ string theory.

Comments

Thanks. Could you give a reference for the chiral contribution?