\documentclass[11pt,notitlepage]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{dbnsymb, amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic, enumitem, bbm, pdfpages} \usepackage{tensor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} \usepackage[textwidth=8.5in,textheight=11in,centering]{geometry} \parindent 0in % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref} \hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor={blue!50!black} } % Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph \usepackage[framemethod=tikz]{mdframed} \usepackage[T1]{fontenc} \def\myurl{http://www.math.toronto.edu/~drorbn} \def\thistalk{McGill-1702} \def\title{What else can you do with solvable approximations?} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for the invitation!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/\thistalk}{http://drorbn.net/\thistalk/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \def\blue{\color{blue}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \def\red{\color{red}} \def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}} \def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\ad{\operatorname{ad}} \def\Ad{\operatorname{Ad}} \def\aft{$\overrightarrow{\text{4T}}$} \def\AS{\mathit{AS}} \def\bbZZ{{\mathbb Z\mathbb Z}} \def\CW{\text{\it CW}} \def\diag{\operatorname{diag}} \def\FL{\text{\it FL}} \def\IHX{\mathit{IHX}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\STU{\mathit{STU}} \def\TC{\mathit{TC}} \def\tr{\operatorname{tr}} \def\vT{{\mathit v\!T}} \def\bbe{\mathbbm{e}} \def\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calG{{\mathcal G}} \def\calI{{\mathcal I}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calP{{\mathcal P}} \def\calR{{\mathcal R}} \def\calS{{\mathcal S}} \def\calT{{\mathcal T}} \def\calU{{\mathcal U}} \def\fraka{{\mathfrak a}} \def\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakh{{\mathfrak h}} \def\tilE{\tilde{E}} % From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes \DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}} %%% \def\credits{{Joint with Roland van der Veen}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Abstract.} Recently, Roland van der Veen and myself found that there are sequences of solvable Lie algebras ``converging'' to any given semi-simple Lie algebra (such as $sl_2$ or $sl_3$ or $E8$). Certain computations are much easier in solvable Lie algebras; in particular, using solvable approximations we can compute in polynomial time certain projections (originally discussed by Rozansky) of the knot invariants arising from the Chern-Simons-Witten topological quantum field theory. This provides us with the first strong knot invariants that are computable for truly large knots. But $sl_2$ and $sl_3$ and similar algebras occur in physics (and in mathematics) in many other places, beyond the Chern-Simons-Witten theory. Do solvable approximations have further applications? }}}} \def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang} \to{\red\uppertriang}$} \def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang} \to{\red\uppertriang}\otimes\!{\red\uppertriang}$} \def\Recomposing{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Recomposing $gl_n$.} Half is enough! $gl_n\oplus\fraka_n = \calD(\uppertriang,b,\delta)$: \vskip 14mm Now define $gl^\epsilon_n\coloneqq\calD(\uppertriang,b,\epsilon\delta)$. Schematically, this is $[\uppertriang,\uppertriang]=\uppertriang$, $[\lowertriang,\lowertriang]=\epsilon\lowertriang$, and $[\uppertriang,\lowertriang]=\lowertriang+\epsilon\uppertriang$. In detail, it is }}}} \def\glne{{\raisebox{0mm}{\parbox[t]{3.0625in}{ $[e_{ij},e_{kl}]\!=\!\delta_{jk}e_{il}-\delta_{li}e_{kj}$ \hfill$[f_{ij},f_{kl}]\!=\!\epsilon\delta_{jk}f_{il}-\epsilon\delta_{li}f_{kj}$ \newline $[e_{ij},f_{kl}] \!=\! \delta_{jk}(\epsilon\delta_{jl}f_{il})$ \newline\null\hfill $-\delta_{li}(\epsilon\delta_{kj}f_{kj})$ \newline$[g_i,e_{jk}] \!=\! (\delta_{ij}-\delta_{ik})e_{jk}$ \hfill$[h_i,e_{jk}] \!=\! \epsilon(\delta_{ij}-\delta_{ik})e_{jk}$ \newline$[g_i,f_{jk}] \!=\! (\delta_{ij}-\delta_{ik})f_{jk}$ \hfill$[h_i,f_{jk}] \!=\! \epsilon(\delta_{ij}-\delta_{ik})f_{jk}$ }}}} \def\SolvApp{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Solvable Approximation.} At $\epsilon=1$ and modulo $h=g$, the above is just $gl_n$. By rescaling at $\epsilon\neq 0$, $gl_n^\epsilon$ is independent of $\epsilon$. We let $gl_n^k$ be $gl_n^\epsilon$ regarded as an algebra over $\bbQ[\epsilon]/\epsilon^{k+1}=0$. It is the ``$k$-smidgen solvable approximation'' of $gl_n$! Recall that $\frakg$ is ``solvable'' if iterated commutators in it ultimately vanish: $\frakg_2\coloneqq[\frakg,\frakg],\ \frakg_3\coloneqq[\frakg_2,\frakg_2],\ \ldots,\ \frakg_d=0$. Equivalently, if it is a subalgebra of some large-size $\uppertriang$ algebra. {\red Note.} This whole process makes sense for arbitrary semi-simple Lie algebras. }}}} \def\WhySolv{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Why are ``solvable algebras'' any good?} Contrary to common beliefs, computations in semi-simple Lie algebras are just awful: \vskip 10mm Yet in solvable algebras, exponentiation is fine and even BCH, $z=\log(\bbe^x\bbe^y)$, is bearable: }}}} \def\R{$R^{\pm 1}$} \def\C{$C^{\pm 1}$} \def\CSW{{\raisebox{2mm}{\parbox[t]{2.75in}{ {\red Chern-Simons-Witten.} Given a knot $\gamma(t)$ in $\bbR^3$ and a metrized Lie algebra $\frakg$, set $Z(\gamma)\coloneqq$ \[ \int_{A\in\Omega^1(\bbR^3,\frakg)}\calD A\,\bbe^{ik\,cs(A)}\text{\it PExp}_\gamma(A), \] where $cs(A)\coloneqq\frac{1}{4\pi}\int_{\bbR^3}\tr\left(AdA+\frac23A^3\right)$ and \[ \text{\it PExp}_\gamma(A) \coloneqq \prod_0^1 \exp(\gamma^\ast A) \in\calU=\hat{\calU}(\frakg), \] and $\calU(\frakg)\coloneqq\langle\text{words in $\frakg$}\rangle/(xy-yx=[x,y])$. In a favourable gauge, one may hope that this computation will localize near the crossings and the bends, and all will depend on just two quantities, \[ R=\sum a_i\otimes b_i\in \calU\otimes \calU \quad\text{and}\quad C\in \calU. \] This was never done formally, yet $R$ and $C$ can be ``guessed'' and all ``quantum knot invariants'' arise in this way. So for the trefoil, \[ Z=\sum_{i,j,k}Ca_ib_ja_kC^2b_ia_jb_kC. \] }}}} \def\extract{{\raisebox{0mm}{\parbox[t]{3.95in}{ But $Z$ lives in $\calU$, a complicated space. How do you extract information out of it? {\red Solution 1, Representation Theory.} Choose a finite dimensional representation $\rho$ of $\frakg$ in some vector space $V$. By luck and the wisdom of Drinfel'd and Jimbo, $\rho(R)\in V^\ast\otimes V^\ast\otimes V\otimes V$ and $\rho(C)\in V^\ast\otimes V$ are computable, so $Z$ is computable too. But in exponential time! \vskip 23mm {\red Solution 2, Solvable Approximation.} Work directly in $\hat\calU(\frakg_k)$, where $\frakg_k=sl_2^k$ (or a similar algebra); everything is expressible using low-degree polynomials in a small number of variables, hence everything is poly-time computable! }}}} \def\GST{\parbox{0.5in}{\tiny Gompf, Scharlemann, Thompson }} \def\gzero{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Example 0.} Take $\frakg_0=sl_2^0=\bbQ\langle h,e,l,f\rangle$, with $h$ central and $[f,l]=f$, $[e,l]=-e$, $[e,f]=h$. In it, using normal orderings, \[ R=\bbO\left(\exp\left(hl+\frac{\bbe^h-1}{h}ef\right)\mid e\otimes\text{\it lf}\right), \quad\text{and,} \] \[ \bbO\left(\bbe^{\delta ef}\mid\text{\it fe}\right) = \bbO\left(\nu \bbe^{\nu\delta\text{\it ef}}\mid\text{\it ef}\right) \quad\text{with }\nu=(1+h\delta)^{-1}. \] }}}} \def\gone{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Example 1.} Take $R=\bbQ[\epsilon]/(\epsilon^2=0)$ and $\frakg_1=sl_2^1=R\langle h,e,l,f\rangle$, with $h$ central and $[f,l]=f$, $[e,l]=-e$, $[e,f]=h-2\epsilon l$. In it, \[ \bbO\left(\bbe^{\delta ef}\mid\text{\it fe}\right) = \bbO\left(\nu(1+\epsilon\nu\delta\Lambda/2)\bbe^{\nu\delta\text{\it ef}}\mid\text{\it elf}\right), \quad\text{where $\Lambda$ is} \] \[ 4\nu^3\delta^2e^2f^2 + 3\nu^3\delta^3he^2f^2 + 8\nu^2\delta ef + 4\nu^2\delta^2hef + 4\nu\delta elf - 2\nu\delta h + 4l. \] {\red Fact.} Setting $h_i=h$ (for all $i$) and $t=\bbe^h$, the $\frakg_1$ invariant of any tangle $T$ can be written in the form \[ Z_{\frakg_1}(T)=\bbO\left(\omega^{-1}\bbe^{hL+\omega^{-1}Q}(1+\epsilon\omega^{-4}P)\mid \midotimes_ie_il_if_i\right), \] where $L$ is linear, $Q$ quadratic, and $P$ quartic in the $\{e_i,l_i,f_i\}$ with $\omega$ and all coefficients polynomials in $t$. Furthermore, everything is poly-time computable. }}}} \def\WhatElse{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Question.} What else can you do with solvable approximation? Chern-Simons-Witten theory is often ``solved'' using ideas from conformal field theory and using quantization of various moduli spaces. Does it make sense to use solvable approximation there too? Elsewhere in physics? Elsewhere in mathematics? {\red See Also.} Talks at George Washington University [\web{gwu}], Indiana [\web{ind}], and Les Diablerets [\web{ld}], and a University of Toronto ``Algebraic Knot Theory'' class [\web{akt}]. }}}} \pagestyle{empty} \begin{document} \latintext \setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} \begin{center} \null\vfill\input{SolvApp1.pdftex_t}\vfill\null \end{center} \AtEndDocument{\includepdf[pages={1-3}, noautoscale=true, offset=0in 0in]{../GWU-1612/Elves.pdf}} \end{document} \endinput