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

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Videography by Iva Halacheva | troubleshooting |

# | Week of... | Notes and Links |
---|---|---|

1 | Jan 6 | About This Class (PDF). Monday: Course introduction, knots and Reidemeister moves, knot colourings. Tricolourability without Diagrams Wednesday: The Gauss linking number combinatorially and as an integral. Friday: The Schroedinger equation and path integrals. Friday Introduction (the quantum pendulum) |

2 | Jan 13 | Homework Assignment 1. Monday: The Kauffman bracket and the Jones polynomial. Wednesday: Self-linking using swaddling. Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres. |

3 | Jan 20 | Homework Assignment 2. Monday: The definition of finite-type and some examples. Wednesday: The self-linking number and framings. Friday: Integrating a polynomial times a Gaussian. Class Photo. |

4 | Jan 27 | Homework Assignment 3. Monday: Chord diagrams and weight systems. Wednesday: Swaddling maps and framings, general configuration space integrals. Friday: Some analysis of . |

5 | Feb 3 | Homework Assignment 4. Monday: 4T, the Fundamental Theorem and universal finite type invariants. The Fulton-MacPherson Compactification (PDF). Wednesday: The Fulton-MacPherson Compactification, Part I. Friday: More on pushforwards, , and . |

6 | Feb 10 | Homework Assignment 5. Monday: The bracket-rise theorem and the invariance principle. Wednesday: The Fulton-MacPherson Compactification, Part II. Friday: Gauge fixing, the beginning of Feynman diagrams. |

R | Feb 17 | Reading Week. |

7 | Feb 24 | Monday: A review of Lie algebras. Wednesday: Graph cohomology and . Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF). |

8 | Mar 3 | Homework Assignment 6 (PDF) Monday: Lie algebraic weight systems. Wednesday: Graph cohomology and the construction of . Graph Cohomology and Configuration Space Integrals (PDF) Friday: Gauge invariance, Chern-Simons, holonomies. Mar 9 is the last day to drop this class. |

9 | Mar 10 | Homework Assignment 7 (PDF) Monday: The weight system. Wednesday: The universal property, hidden faces. Friday: Insolubility of the quintic, naive expectations for CS perturbation theory. |

10 | Mar 17 | Homework Assignment 8 (PDF) Monday: and PBW. Wednesday: The anomaly. Friday: Faddeev-Popov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF). |

11 | Mar 24 | Homework Assignment 9 (PDF) Monday: is a bi-algebra. Wednesday: Understanding and fixing the anomaly. Friday: class cancelled. |

12 | Mar 31 | Monday, Wednesday: class cancelled. Friday: A Monday class: back to expansions. |

E | Apr 7 | Monday: A Friday class on what we mostly didn't have time to do. |

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Dror's Notebook | ||

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Let $so(N) = \{Q \in gl(N) | Q^TQ = QQ^T = I, detQ = 1\}$, with the commutator as its bracket (i.e. $[A,B] = AB - BA$), and the metric $\langle A, B \rangle = tr(AB)$.

Let $\{\pm M_{ij}\}_{i < j}$ be a basis for $so(N)$, where $(M_{ij})_{kl} = \delta_{ij}\delta_{jl} - \delta_{il}\delta_{jk}$.

With this, compute $$t_{(ij)(kl)} = \langle M_{ij}, M_{kl} \rangle = tr(M_{ij}M_{kl}) = const\cdot\delta_{ik}\delta_{jl}$$

Note that this also gives us the inverses $t^{(ij)(kl)} = const\cdot\delta^{ik}\delta^{jl}$.

Now the structure constants: $$f_{(ij)(kl)(mn)} = \langle[M_{ij}, M_{kl}], M_{mn} \rangle = \langle M_{ij}M_{kl}, M_{mn} \rangle - \langle M_{kl}M_{ij}, M_{mn} \rangle$$ $$f_{(ij)(kl(mn)} = tr(M_{ij}M_{kl}M_{mn}) - tr(M_{kl}M_{ij}M_{mn}) = const\cdot\epsilon_{(ij)(kl)(mn)}$$

(With an appropriate choice of signs and ordering of the basis. In $so(3)$, an appropriate ordering and choice of signs is $\mathcal{B} = \{M_{12}, M_{23}, -M_{13}\}$.

If we order the basis, we can associate an integer lying somewhere from 1 to $N(N-1)/2$ (the dimension of $so(N)$) to each pair of indices $(ij)$, so the expression $\epsilon_{(ij)(kl)(mn)}$ makes sense - namely, let $a$, $b$, and $c$ correspond to $(ij)$, $(kl)$, and $(mn)$, respectively, and let $\epsilon_{(ij)(kl)(mn)} = \epsilon_{abc}$, the usual totally antisymmetric tensor.

Thus, up to a constant, $t^{(ij)(kl)} = \delta^{ik}\delta^{jl}$ and $f_{(ij)(kl)(mn)} = \epsilon_{(ij)(kl)(mn)}$.

As in the $gl(N)$ case, we can represent the result $t^{(ij)(kl)} = \delta^{ik}\delta^{jl}$ as a splitting of two lines in the diagram, as in the image below. In addition, we can represent the result $f_{(ij)(kl)(mn)} = \epsilon_{(ij)(kl)(mn)}$ as a trivalent vertex becoming a sum of diagrams, over transpositions of certain lines.

The diagram illustrated below goes to the following expression: $$I = \sum_{i,...,n,i',...,n'}f_{(ij)(kl)(mn)}t^{(ij)(i'j')}t^{(kl)(k'l')}t^{(mn)(m'n')}$$ $$I = \sum_{i,...,n,i',...,n'}\epsilon_{(ij)(kl)(mn)}\delta^{ii'}\delta^{jj'}\delta^{kk'}\delta^{ll'}\delta^{mm'}\delta^{nn'}$$ $$I = \sum_{i,...,n}\epsilon_{(ij)(kl)(mn)}$$

The last line exactly corresponds with the last illustration.