Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2012

    I tried to start an entry theta function, but it’s hard to tell for me if anything of it has been saved. The nnLab is too busy doing something else than serving pages.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 5th 2012

    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.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 2nd 2012
    • (edited Aug 2nd 2012)

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2014

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

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2014
    • (edited Aug 21st 2014)

    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.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 21st 2014

    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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2014

    Okay, done.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2014

    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”)

    θ(z,τ)=θ(gaugefieldconfigurationonΣ,complexstructureonΣ). \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 “zz” (θ(0,τ)\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”.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2014

    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 z\mathbf{z} and τ\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.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2014

    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.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2015

    I have added pointer to

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2015
    • (edited Jul 19th 2015)

    I’ll record here more references that should be added to the “via quantization”-section at theta function once the nnLab comes back: