Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I tried to start an entry theta function, but it’s hard to tell for me if anything of it has been saved. The $n$Lab is too busy doing something else than serving pages.
If you are interested, and in connection to our recent conversation, Perelomov has in his book
an example of treatment of coherent states related to the theta function case (in my memory somewhere early in the book). I may later try to dig page number.
This could be due to the problem with page-saving, but I notice that the links for Riemann theta function and Jacobi theta function (the only ones I checked) direct back to theta function, which I suspect was not intentional, unless one intended to write articles on them right there on the theta function page.
re #3: I have now split off various examples as stand-alone entries (minimal ones, though, at the moment)
have added a general Definition to theta function.
That definition makes immediate that theta-functions are just secitons of line bundles on complex tori, but I haven’t added that statement yet.
Now somebody edits a page so I can not do: put a link to elliptic function. Theta functions are a cornerstone of the theory of elliptic functions.
Okay, done.
I have expanded the Idea-section at theta function in an attempt to clarify that/why there is typically dependence on two different kinds of coordinates (“Jacobi form”)
$\theta(\mathbf{z},\mathbf{\tau}) = \theta\left(gauge\;field\;configuration\;on\;\Sigma\;, \; complex\;structure\;on\;\Sigma\right) \,.$In the process I have given Riemann theta function a brief entry of its own.
Curiously, this means that the number theoretic Jacobi theta function with its trivial dependence on the first variably “$z$” ($\theta(0,\tau)$) is not actually a theta function in the sense of “holomorphic section of holomorphic line bundle on complex torus in canonical covering coordinates”.
I have added to the Idea-seciton of theta function pointers to remark (4.12) in
where it is discussed how the Riemann theta functions in their dependence both on $\mathbf{z}$ and $\mathbf{\tau}$ are the local coordinate expressions of the covariantly constant sections of the Hitchin connection on the moduli space of Riemann surfaces (for circle gauge group).
Added similar brief pointers also to Riemann theta function and to Jacobi theta function.
At theta function and at conformal block I have edited slightly to make more explicit the fact that in the nonabelian case the elements of spaces of conformal blocks deserve to be thought of as “generalized theta functions”, following the terminology in Beauville-Laszlo 93.
I have added pointer to
Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)
Razvan Gelca, Alejandro Uribe, Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$ (arXiv:1007.2010)
at appropriate places to theta function and Chern-Simons theory. These are possibly the best (most explicit/detailed/useful) discussion of theta functions in Chern-Simons that I have seen in the literature so far.
I’ll record here more references that should be added to the “via quantization”-section at theta function once the $n$Lab comes back:
Fernando Falceto, Krzystof Gawedzki, Chern-Simons states at genus one, Comm. Math. Phys. Volume 159, Number 3 (1994), 549-579. (Euclid)
Kazuhiro Hikami, Mock (False) Theta Functions as Quantum Invariants (arXiv:math-ph/0506073)
Kazuhiro Hikami, Quantum Invariants, Modular Forms, and Lattice Points II, J. Math. Phys. 47, 102301-32pages (2006) (arXiv:math/0604091)
Razvan Gelca, Theta Functions and Knots, World Scientific 2014 (prologue pdf, publisher page)
1 to 12 of 12