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AKT-140307 Video

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Gauge invariance, Chern-Simons, holonomies.


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# Week of... Notes and Links
1 Jan 6 About This Class (PDF).
dbnvp Monday: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
dbnvp Wednesday: The Gauss linking number combinatorially and as an integral.
dbnvp Friday: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
dbnvp Monday: The Kauffman bracket and the Jones polynomial.
dbnvp Wednesday: Self-linking using swaddling.
dbnvp Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
dbnvp Monday: The definition of finite-type and some examples.
dbnvp Wednesday: The self-linking number and framings.
dbnvp Friday: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
dbnvp Monday: Chord diagrams and weight systems.
dbnvp Wednesday: Swaddling maps and framings, general configuration space integrals.
dbnvp Friday: Some analysis of d^{-1}.
5 Feb 3 Homework Assignment 4.
dbnvp Monday: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part I.
dbnvp Friday: More on pushforwards, d^{-1}, and d^\ast.
6 Feb 10 Homework Assignment 5.
dbnvp Monday: The bracket-rise theorem and the invariance principle.
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part II.
dbnvp Friday: Gauge fixing, the beginning of Feynman diagrams.
R Feb 17 Reading Week.
7 Feb 24 dbnvp Monday: A review of Lie algebras.
dbnvp Wednesday: Graph cohomology and \Omega_{dR}^\ast(\Gamma).
dbnvp Friday: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
dbnvp Monday: Lie algebraic weight systems.
dbnvp Wednesday: Graph cohomology and the construction of Z_0.
Graph Cohomology and Configuration Space Integrals (PDF)
dbnvp Friday: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
dbnvp Monday: The gl(N) weight system.
dbnvp Wednesday: The universal property, hidden faces.
dbnvp Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
dbnvp Monday: W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g}) and PBW.
dbnvp Wednesday: The anomaly.
dbnvp Friday: Faddeev-Popov, part I.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
dbnvp Monday: {\mathcal A} is a bi-algebra.
dbnvp Wednesday: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
dbnvp Friday: A Monday class: back to expansions.
E Apr 7 dbnvp Monday: A Friday class on what we mostly didn't have time to do.
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0:07:50 [add] Some formulas for gauge transformations.
0:29:29 [add] Motivation from physics, economics, and mathematics.
0:30:30 [edit] The Chern-Simon form of a connection.
0:37:42 [edit] The holonomy of a connection along a path.
0:41:01 [edit] Proposition: $CS(A^g) = CS(A)+\frac13 \int_\mathbb{R^3}Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$ $$CS(A^g)=\int_\mathbb{R^3} Tr(A^g \wedge d A^g + \frac23 A^g \wedge A^g \wedge A^g)$$ $$Tr(A^g \wedge d A^g + \frac23 A^g \wedge A^g \wedge A^g) =$$ $$Tr((g^{-1} A g + g^{-1} dg)\wedge d(g^{-1} A g + g^{-1} dg)+\frac23((g^{-1} A g + g^{-1} dg)\wedge(g^{-1} A g + g^{-1} dg)\wedge(g^{-1} A g + g^{-1} dg)))=$$ $$Tr(g^{-1} A \wedge (d A) g + g^{-1} A g \wedge d g^{-1} \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} (d A) g - g^{-1} d g \wedge g^{-1} A \wedge d g +$$ $$g^{-1} d g \wedge d g^{-1} \wedge A g - g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge d g^{-1} \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ $$\frac23 Tr( g^{-1} A \wedge A \wedge A g + g^{-1} A \wedge A \wedge d g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} A g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +$$ $$g^{-1} d g \wedge g^{-1} A \wedge A g + g^{-1} d g \wedge g^{-1} A g \wedge g^{-1} d g + g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} A g + g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) $$

Now $0 = d (g^{-1} g) = (d g) g^{-1} + g d g^{-1}$

So $(dg) g^{-1} = - g d g^{-1}$

Applying this to the fifth and seventh terms of the equation above yields $$ g^{-1} d g \wedge d g^{-1} \wedge A g = g^{-1} d g \wedge d g^{-1} g \wedge g^{-1} A g = - g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} A g$$ and $$g^{-1} A g \wedge d g^{-1} \wedge d g = g^{-1} A g \wedge d g^{-1} g \wedge g^{-1} d g = - g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g $$

Combining this with the fact that the trace is invariant under cyclic permutations show that

$$Tr(g^{-1} A \wedge (d A) g - 2 g^{-1} A \wedge A \wedge d g - 2 g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g + g^{-1} d g \wedge d g^{-1} \wedge d g) +$$ $$ \frac23 Tr( g^{-1} A \wedge A \wedge A g + 3 g^{-1} A \wedge A \wedge d g + 3 g^{-1} A g \wedge g^{-1} d g \wedge g^{-1} d g +g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) =$$ $$Tr(g^{-1} A \wedge d A g + g^{-1} d g \wedge g^{-1} d A g - g^{-1} d g \wedge g^{-1} A \wedge d g+ g^{-1} d g \wedge d g^{-1} \wedge d g) + \frac23 Tr( g^{-1} A \wedge A \wedge A g + g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$ Now deal with the extra terms $$Tr(g^{-1} d g \wedge g^{-1} (d A) g ) = Tr(g^{-1} d g \wedge d(g^{-1} A) g - g^{-1} d g \wedge d g^{-1} \wedge A g) = Tr(g^{-1} d g \wedge d(g^{-1} A) g + g^{-1} d g \wedge g^{-1} A \wedge d g)$$ Finally $$Tr(g^{-1} d g \wedge d(g^{-1} A) g) = Tr(d g \wedge d(g^{-1} A)) = Tr(d (gd(g^{-1} A))) = d Tr(g d(g^{-1} A))$$ This shows that $$CS(A^g)=\int_\mathbb{R^3}Tr(g^{-1} A \wedge (d A) g + \frac23 g^{-1} A \wedge A \wedge A g + \frac13 g^{-1} d g \wedge d g^{-1} \wedge d g) +\int_\mathbb{R^3} d Tr(g d(g^{-1} A))$$

If we assuming that A and g are compactly supported then by Stokes' theorem $$\int_\mathbb{R^3}d Tr(g d(g^{-1} A)) = 0$$

So

$$CS(A^g)=\int_\mathbb{R^3}Tr(g^{-1} A \wedge (d A) g + \frac23 g^{-1} A \wedge A \wedge A g + \frac13 g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$ $$=\int_\mathbb{R^3}Tr(g^{-1} A \wedge (d A) g + \frac23 g^{-1} A \wedge A \wedge A g) +\frac13 \int_\mathbb{R^3}Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$ $$=\int_\mathbb{R^3}Tr(A \wedge d A + \frac23 A \wedge A \wedge A) +\frac13 \int_\mathbb{R^3}Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$ $$=CS(A)+\frac13 \int_\mathbb{R^3}Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g)$$

If we define $CS_1(A,g)=CS(A^g)-CS(0^g)$ Then $$CS_1(A,g) = CS(A^g) - \frac13 \int_\mathbb{R^3}Tr(g^{-1} d g \wedge g^{-1} d g \wedge g^{-1} d g) = CS_1(A,I) = CS(A)$$

0:42:09 [add] The Chern-Simons form, holonomies.
0:43:22 [edit] An explicit formula for \mathrm{hol}_{\gamma}(A).
0:51:30 [add] An iterated integral formula for the holonomy.