I've recently been given a book called Quantum Topology by Louis H. Kauffman from a friend of mine.

I was wondering what branches of mathematics do I need to be able to read this? What branches of mathematics do I need to understand quantum topology? I understand that a knowledge of Hilbert spaces, algebraic topology and obviously QM is needed. I should have read Hatcher Algebraic Topology book by summer of 2012 as doing a course in it next year.

I would like to be in the position to full understand the book in early 2013.

Also, anyone got any recommended books on quantum algebra?

To make it more concrete. What books should I read so I can understand quantum topology?

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    $\begingroup$ I took a knot theory class related to this in Moscow. I recommend you to finish basic subjects first. Also counting one's progress by time is not a good criterion. $\endgroup$ – Kerry Dec 5 '11 at 2:50

I'm assuming that by "Hatcher Algebraic topology book" you actually mean Algebraic Topology instead of one of his other books (say, Vector Bundles and K Theory), and that this is Kauffman's Quantum Topology. Do bear in mind that this is a book intended for professional mathematicians, so when you ask what background you need, you're essentially outlining a graduate career! What follows are some quick recommendations on my part for introductory texts to the general field in which the Kauffman text seems to fall.

A working knowledge of basic algebraic topology is certainly necessary to understand the material in this book (really, necessary to understand much of modern mathematics), but it appears to me by no means sufficient. Since the subject appears to be applications of the mathematics of quantum theory to low-dimensional topology, you'll probably also want to study low-dimensional topology and quantum mechanics.

For a quick introduction to ideas in low-dimensional topology, you might consider picking up Thurston's book Three-Dimensional Geometry and Topology (much broader but less polished lecture notes are available online) and Scorpan's Wild World of 4 Manifolds. It looks like you'll also probably want an introduction to knots as well; you might check out Lickorish, An Introduction to Knot Theory, or Rolfsen, Knots and Links. For a deeper guide to four-dimensional theory (very different in flavor from three-dimensional theory!) you might check out 4-Manifolds and Kirby Calculus by Stipsicz and Gompf.

At the same time, you'll want to study the mathematical formalism of quantum field theories. I don't know that I can help you much here, but from time to time I hear buzzwords like "geometric quantization" and " For a classical start, I have been recommended Reed and Simon, Methods in Modern Mathematical Physics.

Mind, this doesn't mean you shouldn't read your book as well! You should never try to learn mathematics linearly (even though we often teach our undergraduates that way), and the first article, at least, looks like a reasonably clean survey.

  • $\begingroup$ Thanks for the book recommendation. I'm make sure I pick up Thurston's book. Originally planned to get that book as I wanted to understand Geometrization conjecture now will definitely get it. Will plan to read all the book mention by 2013. Thanks. $\endgroup$ – simplicity Dec 5 '11 at 3:23

Quantum topology does tend to be a bit interdisciplinary. You do certainly need to know quite a bit about quantum groups, for instance through Chari-Pressley or Kassel's books. From the topology side, one should know some knot theory, from Rolfsen for example; and classical 3-manifold theory doesn't hurt either. All of the physics stuff is needed for Kauffman's book, but not for some other quantum topology textbooks. For example, good introductions to quantum topology which don't factor through physics include Ohtsuki's Quantum Invariants and Turaev's Quantum invariants of knots and 3-manifolds.


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