- Cold Neutrons and Topological Knots
- The Jones Polynomial and Khovanov Homology
- Pari/GP – (tips on) Using Pari/GP
I cover background material for Chern-Simons Theory and other areas for quantum super-A-polynomials. I’m also taking this opportunity to write up accompanying (Python) scripts (github:qSApoly).
Notes as a LaTeX file, and accompanying scripts are all on this github repository: github:qSApoly.
Also note, starting 20160423, I’m writing on the relation between Cold neutrons (in beta decay) to Topological Gauge Theory and Knot Polynomials (knot homologies) and attempting to write actively (i.e. every week turn around), though I should warn that for me right now, Propulsion, and “seeking opportunities in propulsion development” takes precedence.
The full, comprehensive report on “Cold neutrons and Topological Knots” is on my github repository qsApoly under subdirectory LaTeXandpdf and will be, first in precedence, updated there: in no way do I guarantee that the following on my wordpress blog here is updated, so you should check my github.
Why cold neutrons?
I was standing outside on a warm, sunny day on campus, right in front of the GALCIT building, sipping on San Pellegrino soda, when a colleague ran into me and asked what I had been up to. I mentioned that I was trying to switch over to aerospace engineering from theoretical physics and I was finding it difficult to explain my excitement for Chern-Simons theory to engineers I talked to (which I can understand).
“You wrote your thesis on Chern-Simons theory?”
“Yeah, well, I wrote my Masters thesis on exact solutions for a Chern-Simons theory, which end up being polynomials.
“I need someone to talk to about Chern-Simons theory; let’s go talk in my office right now.”
And off we went.
Clarify Manifold setup ; explore various manifold setups; pdf or equation of motion out of and Euler-Lagrange equation and compare those equations to instanton equations of Gaiotto and Witten (2011); understand Virasoro algebra and conformal blocks for the quantum “states” that we can act upon; read more: Gaiotto and Witten (2011); Gukov (2007), and of course classic Witten (1988).
What I’ll focus on here are implementations of the Jones polynomial and Khovanov Homology for Torus knots, up to less than 14 crossings, in SnapPy, knotkit, Sage Math. All the material and further documentation is in the folder JonesPoly_and_KH of my github repository ernestyalumni: qSApoly.
I brushed up on these necessary topics that follows.
Groups, Lie Groups, Lie Algebra
In the pdf qsuperApoly.pdf, I review some representation theory to try to go beyond the usual presentation of “Group Theory for Physicists.”
The book I’ve been reading and highly recommend for Lie Groups, Lie Algebra from a mathematician’s standpoint, wanting to help physicists, is
- Yvette Kosmann-Schwarzbach, Groups and Symmetries: From Finite Groups to Lie Groups, Springer, 2010. e-ISBN 978-0-387-78866-1
I’m also reading the concurrently with
- John Baez, Javier P Muniain. Gauge Fields, Knots and Gravity (Series on Knots and Everything), World Scientific Publishing Company (October 24, 1994), ISBN-13: 978-9810220341
to make the link or application to physics.
Note on Mathematical Preliminaries
One should know Differential Geometry from the point of mathematicians. The best course for this is the Differenzierbare Mannigfaltigkeiten (Differential Manifolds) course offered by the Mathematics department at LMU, Munich; it changes every year with a different instructor and so check for the course offering on their department page. The level of mathematical rigor is high.
Werner Ballman had a lucid and straightforward exposition on Vector Bundles and Connections that helped with my reading of Taubes (2011).
Topological Quantum Field Theory
The Simons Center for Geometry and Physics (SCGP) has very excellent, high-quality (in terms of video and audio quality), high-quality (in terms of having the best researchers in pure mathematics and theoretical physics speak, AND having them communicate very lucidly the absolute latest in math and physics) videos of lectures and workshops and they organize the videos very well and make all the material available, making the research very accessible and transparent. It’s a fantastic resource that I highly recommend.
This is a great introduction to Topological Quantum Field Theory (TQFT) from Ingo Runkel:
Graduate Workshop on Topological Quantum Field Theory – Ingo Runkel Part 1
Of course, if you’re watching this and you don’t know what a principal-G bundle is or a holonomy, or a connection, then your foundation is shaky since TQFT is built upon first a principal-G bundle and how it has a connection. I’m reading Clifford Taubes’ Differential Geometry: Bundles, Connections, Metrics and Curvature (Oxford Graduate Texts in Mathematics, Vol. 23, 2011) to try to understand principal-G bundles and connections from his point of view. Right now (20151017), I’m struggling with the connection as a splitting of the short exact sequence. I’ve updated my notes (pdf here and LaTeX file on github) accordingly:
This showed that vector fields are isomorphic to derivations, and explicitly gives the construction of the isomorphism: 1300Y Geometry and Topology 1300Y Geometry and Topology
This clarifies Taubes, 11.4.4 Part 3-4 connections
Since the Snake Lemma and 5-lemma are needed to show, in a civilized manner, the splitting lemma, could we use the Snake Lemma and 5-lemma for connections on a principal-G bundle? chapter-4