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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{MonteVerita-2604}
\def\title{Chern-Simons-Witten Theory Near The Co-Commutative Limit}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to Monte Varit\`a!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/mv26}{http://drorbn.net/mv26}}}
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\def\eps{\epsilon}
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%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 7 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in
{\red\bf Abstract.} Perhaps every algebra meeting should have one analysis talk (and vice versa),
lest we forget that the other exists. In my role as the outsider, I will tell you today about the
other -- perturbative -- evaluation of path integrals, where instead of hoping that nature will
help us compute faster, we approximate nature by things we already can compute quickly.

Specifically I will tell you how in the Chern-Simons-Witten theory you can perturb the base Lie
algebra from where it's easy towards where it's strong, leading to the strongest genuinely
computable knot invariant we presently have.

I wish I could give my talk in the language of the Kabbalah, but I ain't smart enough for
that. So I'll highlight the Kabbalistic points that we're still missing, and then stick to the
Talmud. 
}}}}

\def\Acknowledgement{{\raisebox{2.25mm}{\parbox[t]{3.25in}{\footnotesize
{\bf\red Acknowledgement.} Work supported by NSERC grant RGPIN-2025-06718
and by the Chu Family Foundation (NYC).
}}}}

\def\Speculation{{\raisebox{2.25mm}{\parbox[t]{3.95in}{
{\bf\red Non-Apology.}
This talk touches the disreputable subject of {\em speculation}. But it's speculation in
which the results are already there; we merely speculate on where they may be coming from.
}}}}


\def\CSW{{\raisebox{1mm}{\parbox[t]{2.6in}{
Chern-Simons-Witten with Lie algebra $\frakg$ and representation $V$:
\[
  \int_{\mathrlap{A\in\Omega^1(\bbR^3;\frakg)}}\calD A\, \bbe^{
    \frac{i}{4\pi}\int_\bbR A\wedge dA+\frac{2\sqrt{\hbar}}{3}A\wedge A\wedge A
  }
  \operatorname{tr}_V\calP\!\exp_\gamma(\sqrt{\hbar}A)
\]
}}}}

\def\RT{{\raisebox{1mm}{\parbox[t]{1.85in}{
Reshetikhin-Turaev for $(\frakg,V)$ at $q=\bbe^{\hbar}$
}}}}

\def\hbarexp{{\raisebox{2mm}{\parbox[t]{1.65in}{
{\centering $\hbar$-expansion}
}}}}

\def\SolvApp{{\raisebox{0mm}{\parbox[t]{1.55in}{\centering
Solvable Approximation: $\frakg\to\frakg_\eps$
}}}}

\def\CSWeps{{\raisebox{1mm}{\parbox[t]{2.6in}{
Chern-Simons-Witten with Lie algebra $\frakg_\eps$:
\[
  \int_{\mathrlap{A\in\Omega^1(\bbR^3;\frakg_\eps)}}\calD A\, \bbe^{
    \frac{i}{4\pi}\int_\bbR A\wedge dA+\frac{2\sqrt{\hbar}}{3}A\wedge A\wedge A
  }
  \calP\!\exp_\gamma(\sqrt{\hbar}A)
\]
}}}}

\def\URTeps{{\raisebox{1mm}{\parbox[t]{1.85in}{
Universal Reshetikhin-Turaev for $\frakg_\eps$ with $q=\bbe^{\hbar\eps}$
}}}}

\def\Invariants{{\raisebox{1mm}{\parbox[t]{2.6in}{
Today's invariants:
\[ \left(\begin{array}{c|c|c|c}
  \eps=0 & \eps^2=0 &\eps^3=0 & \ldots \\
  \text{Alexander's }\Delta & \rho_1,\theta & \rho_2,\ldots & \ldots
\end{array}\right) \]
}}}}

\def\epsexp{{\raisebox{2mm}{\parbox[t]{1.65in}{
{\centering
  $\eps$-expansion: perturbation theory, Feynman diagrams
  \vskip 3mm
  \includegraphics[width=\linewidth]{../Kyoto-230727/DoPeGDO-thumb.pdf}
}
\vskip -10mm \rightline{\footnotesize In Kyoto, \web{k23}}
}}}}

\def\MissingKabbalah{{\raisebox{0mm}{\parbox[t]{2.4in}{
Missing Kabbalah:
\par \quad 1.~Exact evaluation at $\eps=0$ giving $\Delta$.
\par \qquad 2.~Then perturbation theory.
}}}}

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TBW.

\newpage

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\hfill{\normalsize\red\bf References.}

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