© | Dror Bar-Natan: Classes: 2013-14: AKT: < >

# AKT-140214 Video

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Gauge fixing, the beginning of Feynman diagrams.

# Week of... Notes and Links
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
6 Feb 10 Homework Assignment 5.
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
: A Monday class: back to expansions.
E Apr 7 : A Friday class on what we mostly didn't have time to do.

Dror's Notebook
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0:08:40  The set of differential $k$-forms on a manifold $M$ (example $\mathbb{R}^3$) is a vector space $\Omega^k(M)$ and when $k=0$ then $\Omega^0(M)$ is the set of smooth functions. Thus smooth functions are 0-forms. Now $k$-forms are integrated on $k$-manifolds. For example, a 1-form $f(x,y) \mathrm{d}x + g(x,y) \mathrm{d}y$ can be integrated on a curve $\gamma$. Also differential forms can be differentiated using the operator d called the exterior operator where $d$ acts on a $k$-form to produce a $k+1$-form and that $\mathrm{d}\circ \mathrm{d} =0$.

Now

1. if $f \in \Omega^0(\mathbb{R}^3)$, then $\mathrm{d}f = \sum_i^3 \frac{\partial{f}}{\partial{x_i}}\mathrm{d}x_i$ is a 1-form so that $\mathrm{d}f \in \Omega^1(M)$. Thus $d: \mathrm{d} \Omega^0(\mathbb{R}^3) \rightarrow \Omega^1(\mathbb{R}^3)$ is the gradient operator $\mathrm{grad}$.

2. If we have a 1-form $v = v_x\mathrm{d}x + v_y\mathrm{d}y + v_z\mathrm{d}z$, then $\mathrm{d}v = \left( \frac{\partial{v_z}}{\partial{y}}- \frac{\partial{v_y}}{\partial{z}}\right)\mathrm{d}y\wedge \mathrm{d}z + \left( \frac{\partial{v_x}}{\partial{z}}- \frac{\partial{v_z}}{\partial{x}}\right)\mathrm{d}x\wedge \mathrm{d}z + \left( \frac{\partial{v_x}}{\partial{y}} - \frac{\partial{v_y}}{\partial{x}}\right)\mathrm{d}x\wedge \mathrm{d}y$ which is a two form. In this case we have $d: \mathrm{d} \Omega^1(\mathbb{R}^3) \rightarrow \Omega^2(\mathbb{R}^3)$ is the $\mathrm{curl}$ operator.

3. If we have 2-form $\omega = (\omega_x, \omega_y, \omega_z)$ then again get a 3-form $\mathrm{d}\omega = \left( \frac{\partial{\omega_x}}{\partial{x}} + \frac{\partial{\omega_y}}{\partial{y}} + \frac{\partial{\omega_z}}{\partial{z}} \right)\mathrm{d}x\wedge\mathrm{d}y\wedge \mathrm{d}z$. If we think of $\mathrm{d}x\wedge\mathrm{d}y\wedge \mathrm{d}z$ as a function $f$, then again we get $d: \mathrm{d} \Omega^2(\mathbb{R}^3) \rightarrow \Omega^3(\mathbb{R}^3)$ is the divergence operator $\mathrm{div}$.

0:30:35  The handout we used for this part is at pensieve/Classes/0102-FeynmanDiagrams/index.html?im=Notebook-014.jpg.