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Atrium > Mathematics, Physics & Philosophy: 0-dimensional TQFT - a divertissement

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  1. I'm planning to write a divertissement on 0-dimensional TQFT and singular cohomology. The idea is that is well known (at least after Segal-Stolz-teichner approach to elliptic cohomology) that the "space" of all 1-dimensional Euclidean (or Riemannian) field theories is the spectrum of K-theory. In particular Riemannian 1-dimensional bordism with a target manifold X describes the K-theory of X. This is usually takes as the starting point towards the spectrum of elliptic cohomology, i.e., one goes from 1-dimensional to 2-dimensional field theories. In the note I'm planning I'd like to go a step back, i.e, to investigate the apparently trivial space of 0-dimensional field theories and to show how one obtains the spectrum of singular cohomology from it. As an illustrative example, considering 0-dimensional bordism with a target X amounts to considering infinite symmetric powers of X, and the Dold-Thom theorem comes into the TQFT picture.

    A second -more serious- step would consist in going from 1-dimensional to 0-dimensional field theories and read the Chern character in this framework.

    I'm pretty sure all of this is well known, but I've been unable to find any satisfactory reference (any suggestion?) so I decided to write a note (on the 0-dimensional case, for the moment). Any comment/criticism/exchange of thoughts/collaboration is welcome and encouraged.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2009
    • (edited Dec 18th 2009)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> I'm planning to write a divertissement on 0-dimensional TQFT and singular cohomology. </blockquote> <p>That would be awesome!</p> <p>A tiny little bit of mostly unpolished material on this already exists on the lab, starting at <a href="https://ncatlab.org/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a>.</p> <p>in particular there is</p> <ul> <li><a href="https://ncatlab.org/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></li> </ul> </div>
    • CommentRowNumber3.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 18th 2009
    • (edited Dec 18th 2009)
    Fine! Then I can add a (0,1)-dimensional TQFT and singular cohomology subsection there (this will take at least few days), and also try to polish a bit here and there on the geometric model for elliptic cohomology entry, if that will be the case.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2009

    Thanks, great.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2009

    maybe one remark:

    please don't try to produce something perfect, only to never finish it.

    The more imperfect your entry will be, the better. The rest of us wants to have soemthing to work on, too! :-)

  6. You'll agree that writing something more imperfect than 0-dimensional TQFT would be an hard task.. :)
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> ... <a href="https://ncatlab.org/nlab/show/0-dimensional+TQFT">0-dimensional TQFT</a> ... </blockquote> <p>Thanks! Nice.</p> <p>I just went through it and edited a bit, adding hyperlinks and headlines and a toc. Have a look to check if you are happy with what I did.</p> </div>

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