Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some knowledge as to the basic heuristics. I know this is a really trivial question, but again I am a not very experienced yet, can someone sketch the basic technique,. ..
2 Answers
$\begingroup$
$\endgroup$
(The following is likely not the answer you are looking for, but it is an answer to what you actually ask.)
The case can be made that all of quantization is via cohomology, namely that quantization is fundamentally pull-push in some generalized cohomology theory, hence is index theory.
For geometric quantization in mechanics this insight is usually attributed to Bott: geometric quantization is really push-forward of the pre-quantum data in complex K-theory, see here on the nLab for how this works and see at
for more references. The basics of this insight are old, but still not very widely appreciated.
One can further conceptualize this and observe that the mechanical system encoded by some Poisson manifold is the boundary field theory of the corresponding non-perturbative 2d Poisson-Chern-Simons theory (this is discussed here) and that it is the natural "holographic" quantization of the boundary of that 2d QFT which is given by push-forward in K-theory. This is spelled out fairly comprehensively in
- Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, Thesis, Utrecht 2013
One can then ask how to quantize not just mechanics (1-dimensional field theory) but also higher dimension field theory cohomologically. Some comments on how to quantize 2-dimensional field theory by pull-push in eliptic cohomology/tmf are at the end of the above thesis, but much less is known here.
The basic mechanism of the cohomological quantization discussed by Nuiten is remarkably "fundamental". In a note I am writing
it is observed that at the heart of it cohomological quantization is a fundamental operation in Grothendieck's "yoga of six functors". This is, as discussed there, essentially the cohomological integration mechanism that also appears in section 4.1 of the recent
- Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory
To feel one's way as to the cohomological quantization of higher dimensional field theories, there is for instance the article
- Daniel S. Freed, Michael J. Hopkins, Constantin Teleman, Consistent Orientation of Moduli Spaces (arXiv:0711.1909)
which discusses quantization of something like a simplified version of Chern-Simons theory via pull-push in complex K-theory. By the close relation of Chern-Simons theory to Yang-Mills theory, you may imagine this getting a tad closer to the remote goal of quantization of Yang-Mills theory via pull-push in generalized cohomology.
answered Jan 6, 2014 at 1:42
$\begingroup$
$\endgroup$
John Baez's introductory lectures can be found at http://math.ucr.edu/home/baez/qg-fall2006/qg-fall2006.html
answered Jan 6, 2014 at 5:59