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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJul 5th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIf you are interested, and in connection to our recent conversation, Perelomov has in his book * A. Perelomov, Generalized coherent states and their applications, Springer 1986 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.

   If you are interested, and in connection to our recent conversation, Perelomov has in his book

   • A. Perelomov, Generalized coherent states and their applications, Springer 1986

   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.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 2nd 2012
   • (edited Aug 2nd 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThis 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 21st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexre #3: I have now split off various examples as stand-alone entries (minimal ones, though, at the moment)

   re #3: I have now split off various examples as stand-alone entries (minimal ones, though, at the moment)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 21st 2014
   • (edited Aug 21st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added a general _[Definition](http://ncatlab.org/nlab/show/theta+function#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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeAug 21st 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNow 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 21st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, done.

   Okay, done.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 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”.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 9th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to the Idea-seciton of _[[theta function]]_ pointers to remark (4.12) in * [[Nigel Hitchin]], _Flat connections and geometric quantization_, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841)) 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]]_.

   I have added to the Idea-seciton of theta function pointers to remark (4.12) in

   • Nigel Hitchin, Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)

   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.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexAt _[[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](http://ncatlab.org/nlab/show/theta+function#BeauvilleLaszlo93).

    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.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added pointer to * [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_ ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf)) * [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/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 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.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2015
    • (edited Jul 19th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI'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](https://projecteuclid.org/euclid.cmp/1104254732)) * Kazuhiro Hikami, _Mock (False) Theta Functions as Quantum Invariants_ ([arXiv:math-ph/0506073](http://arxiv.org/abs/math-ph/0506073)) * Kazuhiro Hikami, _Quantum Invariants, Modular Forms, and Lattice Points II_, J. Math. Phys. 47, 102301-32pages (2006) ([arXiv:math/0604091](http://arxiv.org/abs/math/0604091)) * [[Razvan Gelca]], _Theta Functions and Knots_, World Scientific 2014 ([prologue pdf](http://www.worldscientific.com/doi/suppl/10.1142/8872/suppl_file/8872_chap01.pdf), [publisher page](http://www.worldscientific.com/worldscibooks/10.1142/8872))

    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)