\documentclass[11pt,notitlepage]{article} \usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, amscd, dbnsymb, ../picins, needspace, colortbl, tensor, import, bbm} \usepackage[all]{xy} % Following http://tex.stackexchange.com/questions/69901/how-to-typeset-greek-letters: \usepackage[LGR,T1]{fontenc} \newcommand{\textgreek}[1]{\begingroup\fontencoding{LGR}\selectfont#1\endgroup} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} %\usepackage[textwidth=197mm,textheight=284mm,headsep=0.15in,centering]{geometry} \usepackage[a4paper,margin=5mm]{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 } \usepackage[]{qrcode} \def\myurl{http://www.math.toronto.edu/~drorbn} \def\thistalk{Sydney-191002} \def\thistalkshortcut{syd4} \def\title{Some Feynman Diagrams in Algebra} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\webdef{{\textgreek{web}$\coloneqq$\url{http://drorbn.net/\thistalkshortcut/}}} \def\web#1{{\href{http://drorbn.net/\thistalkshortcut/#1}{\textgreek{web}/#1}}} \def\blue{\color{blue}} \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}}} % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#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\CW{\text{\it CW}} \def\diag{\operatorname{diag}} \def\FL{\text{\it FL}} \def\Hom{\operatorname{Hom}} \def\IHX{\mathit{IHX}} \def\mor{\operatorname{mor}} \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\wt{\operatorname{wt}} \def\bbe{\mathbbm{e}} \def\bbH{{\mathbb H}} \def\bbN{{\mathbb N}} \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\calE{{\mathcal E}} \def\calF{{\mathcal F}} \def\calG{{\mathcal G}} \def\calI{{\mathcal I}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calS{{\mathcal S}} \def\calT{{\mathcal T}} \def\calU{{\mathcal U}} \def\frakg{{\mathfrak g}} \def\tilE{\tilde{E}} \def\cellscale{0.645} \begin{document} \setlength{\jot}{0.5ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} %\setlength{\intextsep}{0ex} \parpic[r]{\qrcode[height=2.5em,level=L,nolink,tight]{drorbn.net/\thistalkshortcut}} \parbox[b]{3.4in}{{\footnotesize\navigator}\newline{\Large\bf\red \title}} \hfill\parbox[b]{0.7in}{\footnotesize With Roland\newline van der Veen }\ \includegraphics[height=8mm]{../../Projects/Gallery/VanDerVeen.jpg}\hfill \hfill\parbox[b]{2.4in}{\footnotesize \null\hfill\webdef \newline\null\hfill Slides w/ no handout/URL should be banned! } \vskip -3mm \rule{\dimexpr\linewidth-3em}{1pt} \vspace{-8mm} \begin{multicols*}{2} %\raggedcolumns {\red\bf Abstract.} I will explain how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams. Possibly in a later seminar I will explain how this technology can be used to construct a tangle invariant whose need was explained in a talk I gave in the topology seminar on Sep 16 (\web{top}). \vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm} {\red{\bf Secret Slide} (must not be shown).} Il y a beaucoup de $Z\colon\{\text{Noeuds}\} \to \left(\calU(\frakg) \text{ ou }\calU_q(\frakg)\right) \cong \calS(\frakg) \cong \bbQ[z_1,z_2,\ldots]$. \vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm} {\red\bf Conventions.} 1.~For a set $A$, let $z_A\coloneqq\{z_i\}_{i\in A}$ and let ${\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}}$. 2.~Everything converges! \vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm} {\bf\red The Generating Series $\calD\colon\Hom(\bbQ[z_A]\!\to\!\bbQ[z_B]) \to \bbQ\llbracket\zeta_A,z_B \rrbracket$.} {\bf Claim.} $F\in\Hom(\bbQ[z_A]\to\bbQ[z_B]) \xrightarrow[\calD]{\raisebox{-0.75ex}[0ex][0ex]{$\sim$}} \bbQ[z_B]\llbracket\zeta_A\rrbracket\ni \calF$ via \[ \calD(F) \coloneqq \sum_{n\in\bbN^A}\frac{\zeta_A^n}{n!}F(z_A^n) = F \left(\bbe^{\sum_{a\in A}\zeta_a z_a}\right) = \calF, \] \[ \calD^{-1}(\calF)(p) = \left(\left.p\right|_{z_a\to\partial_{\zeta_a}}\calF\right)_{\zeta_a=0} \quad\text{for $p\in\bbQ[ z_A]$}. \] {\bf Claim.} If $F\in\Hom(\bbQ[ z_A]\to\bbQ[ z_B])$, $G\in\Hom(\bbQ[ z_B]\to\bbQ[ z_C])$, $\calF=\calD(F)$, and $\calG=\calD(G)$, then \[ \calD(F\act G) = \left( \calF|_{z_b\to\partial_{\zeta_b}}\calG \right)_{\zeta_b=0}. \] \vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm {\red\bf Basic Examples.} {\bf 1.} $\calD(id\colon\bbQ[y,a,x]\to\bbQ[y,a,x]) = \bbe^{\eta y+\alpha a+\xi x}$. \needspace{1.25cm} \parpic[r]{$\xymatrix@C=5mm@R=5mm{ \bbQ[ z]_i\otimes\bbQ[ z]_j \ar[r]^<>(0.5){m^{ij}_k} \ar@{=}[d] & \bbQ[ z]_k \ar@{=}[d] \\ \bbQ[ z_i,z_j] \ar[r]^<>(0.5){m^{ij}_k} & \bbQ[ z_k] }$} {\bf 2.} The standard commutative product $m^{ij}_k$ of polynomials is given by $z_i,z_j\to z_k$. Hence $\calD(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_i z_i+\zeta_j z_j}) = \bbe^{(\zeta_i+\zeta_j)z_k}$. \needspace{1.25cm} \parpic[r]{$\xymatrix@C=5mm@R=5mm{ \bbQ[ z]_i \ar[r]^<>(0.5){\Delta^i_{jk}} \ar@{=}[d] & \bbQ[ z]_j\otimes\bbQ[ z]_k \ar@{=}[d] \\ \bbQ[ z_i] \ar[r]^<>(0.5){\Delta^i_{jk}} & \bbQ[ z_j,z_k] }$} {\bf 3.} The standard co-commutative co-product $\Delta^i_{jk}$ of polynomials is given by $z_i\to z_j+z_k$. Hence $\calD(\Delta^i_{jk}) = \Delta^i_{jk}(\bbe^{\zeta_i z_i}) = \bbe^{\zeta_i(z_j+z_k)}$. \vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm {\red\bf Heisenberg Algebras.} Let $\bbH=\langle x,y\rangle/[x,y]=\hbar$ (with $\hbar$ a scalar), let $\bbO_i\colon\bbQ[x_i,y_i]\to\bbH_i$ is the ``$x$ before $y$'' PBW ordering map and let $hm^{ij}_k$ be the composition \[ \begin{CD} \bbQ[x_i,y_i,x_j,y_j] @>\bbO_i\otimes\bbO_j>> \bbH_i\otimes\bbH_j @>m^{ij}_k>> \bbH_k @>\bbO_k^{-1}>> \bbQ[x_k,y_k]. \end{CD} \] Then $\calD(hm^{ij}_k) = \bbe^{\Lambda_\hbar}$, where $\Lambda_\hbar=-\hbar\eta_i\xi_j+(\xi_i+\xi_j)x_k+(\eta_i+\eta_j)y_k$. {\bf Proof 1.} Recall the ``Weyl form of the CCR'' $\bbe^{\eta y}\bbe^{\xi x} = \bbe^{-\hbar\eta\xi}\bbe^{\xi x}\bbe^{\eta y}$, and compute \begin{multline*} \calD(hm^{ij}_k) = \bbe^{\xi_ix_i+\eta_iy_i+\xi_jx_j+\eta_jy_j} \act \bbO_i\otimes\bbO_j \act m^{ij}_k \act \bbO_k^{-1} \\ = \bbe^{\xi_ix_i}\bbe^{\eta_iy_i}\bbe^{\xi_jx_j}\bbe^{\eta_jy_j} \act m^{ij}_k \act \bbO_k^{-1} = \bbe^{\xi_ix_k}\bbe^{\eta_iy_k}\bbe^{\xi_jx_k}\bbe^{\eta_jy_k} \act \bbO_k^{-1} \\ = \bbe^{-\hbar\eta_i\xi_j}\bbe^{(\xi_i+\xi_j)x_k}\bbe^{(\eta_i+\eta_j)y_k} \act \bbO_k^{-1} = \bbe^{\Lambda_\hbar}. \end{multline*} {\bf Proof 2.} We compute in a faithful 3D representation $\rho$ of $\bbH$: \newline\null\hfill\text{(\web{hm})} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Examples/hm-\the\snip.pdf} \ifnum \snip<4 \repeat %\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm \columnbreak {\red\bf A real DoPeGDO Example} (DoPeGDO$\coloneqq$Docile Perturbed Gaussian Differential Operators). Let $sl_{2+}^\epsilon \coloneqq L\langle y,b,a,x\rangle$ subject to ${[a,x]=x}$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $[x,y]=\epsilon a+b$. So $t\coloneqq\epsilon a-b$ is central and if $\exists \epsilon^{-1}$, $sl_{2+}^\epsilon/\langle t\rangle\cong sl_2$. Let $CU\coloneqq\calU(sl_{2+}^\epsilon)$, and let $cm^{ij}_k$ be the composition below, where $\bbO_i\colon\bbQ[y_i,b_i,a_i,x_i]\to CU_i$ be the PBW ordering map in the order $ybax$: \[ \xymatrix@C=9mm@R=5mm{ CU_i\otimes CU_j \ar[r]^<>(0.5){m^{ij}_k} & CU_k \\ \bbQ[ y_i,b_i,a_i,x_i,y_j,b_j,a_j,x_j ] \ar[r]^<>(0.5){cm^{ij}_k} \ar[u]_<>(0.4){\bbO_{i,j}} & \bbQ[ y_k,b_k,a_k,x_k ] \ar[u]_<>(0.4){\bbO_k} & } \] {\bf Claim.} Let \begin{multline*} \Lambda = \left(\eta _i+\frac{e^{-\alpha _i-\epsilon \beta _i} \eta _j}{1+\epsilon \eta _j \xi _i}\right) y_k+\left(\beta _i+\beta _j+\frac{\log \left(1+\epsilon \eta _j \xi _i\right)}{\epsilon }\right) b_k+ \\ \left(\alpha _i+\alpha _j+\log \left(1+\epsilon \eta _j \xi _i\right)\right) a_k+\left(\frac{e^{-\alpha _j-\epsilon \beta _j} \xi _i}{1+\epsilon \eta _j \xi _i}+\xi _j\right) x_k \end{multline*} Then $\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i + \eta_jy_j+\beta_jb_j+\alpha_ja_j+\xi_jx_j} \act \bbO_{i,j} \act cm^{ij}_k = \bbe^\Lambda \act \bbO_k$, and hence $\calD(cm^{ij}_k)=\bbe^\Lambda$. \needspace{3\baselineskip} {\bf Proof.} We compute in a faithful 2D representation $\rho$ of $CU$: \newline\null\hfill\text{(\web{cm})} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Examples/cm-\the\snip.pdf} \ifnum \snip<6 \repeat {\bf Note 1.} If the lower half of the alphabet ($a,b,\alpha,\beta$) is regarded as constants, then $\Lambda=C+Q+\sum_{k\geq 1}\epsilon^kP^{(k)}$ is a docile perturbed Gaussian relative to the upper half of the alphabet ($x,y,\xi,\eta$): $C$ is a scalar, $Q$ is a quadratic, and $\deg P^{(k)}\leq 2k+2$. {\bf Note 2.} $\wt(x,y,\xi,\eta,a,b,\alpha,\beta,\epsilon)=(1,1,1,1,2,0,0,2,-2)$. \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} {\bf\red Quadratic Casimirs.} If $t\in\frakg\otimes\frakg$ is the quadratic Casimir of a semi-simple Lie algebra $\frakg$, then $\bbe^t$, regarded by PBW as an element of $\calS^{\otimes 2}=\Hom\left(\calS(\frakg)^{\otimes 0}\to\calS(\frakg)^{\otimes 2}\right)$, has a latin-latin dominant Gaussian factor. Likewise for $R$-matrices. \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} {\red\bf DoPeGDO} $\coloneqq$ The category with objects finite sets$^{\dagger 1}$ and \[ \mor(A\to B) = \left\{\calF =\omega\exp(Q+P) \right\} \subset \bbQ\llbracket\zeta_A,z_B,\epsilon\rrbracket, \] where: $\bullet$~$\omega$ is a scalar.$^{\dagger 2}$ $\bullet$~$Q$ is a ``small'' $\epsilon$-free quadratic in $\zeta_A\cup z_B$.$^{\dagger 3}$ $\bullet$~$P$ is a ``docile perturbation'': $P=\sum_{k\geq 1}\epsilon^kP^{(k)}$, where $\deg P^{(k)}\leq 2k+2$.$^{\dagger 4}$ $\bullet$~Compositions:$^{\dagger 6}$ \[ \calF\act \calG = \calG\circ \calF \coloneqq \left( \calG|_{\zeta_i\to\partial_{z_i}}\calF \right)_{z_i=0} \!\! = \left( \calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}. \] %\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} \columnbreak {\red\bf So What?} $\bullet$~If $V$ is a representation, then $V^{\otimes n}$ explodes as a function of $n$, while in {\bf DoPeGDO} and up to a fixed power of $\epsilon$, the ranks of $\mor(A\to B)$ grow polynomially as a function of $|A|$ and $|B|$. $\bullet$~Approximating $sl^\epsilon_{2+}$ retains more of its structure then representing it! \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} {\bf\red Compositions (1).} In $\mor(A\!\to\!B)$, \[ Q = \sum_{i\in A,j\in B}E_{ij}\zeta_iz_j + \frac12\sum_{i,j\in A}F_{ij}\zeta_i\zeta_j + \frac12\sum_{i,j\in B}G_{ij}z_iz_j, \] and so \[ \import{.}{Compositions.pdftex_t} \] \parpic[r]{\scalebox{0.9}{\import{.}{FeynmanDiagrams.pdftex_t}}} where $\bullet$\ $E=E_1(I-F_2G_1)^{-1}E_2$. \newline$\bullet$\ $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$. \newline$\bullet$\ $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$. \newline$\bullet$\ $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$. \newline$\bullet$\ $P$ is computed as the solution of a messy PDE or using ``connected Feynman diagrams'' (yet we're still in algebra!). \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} {\bf\red DoPeGDO Footnotes.} Each variable has a ``weight''$\in\{0,1,2\}$, and always $\wt z_i+\wt\zeta_i=2$. \par$\dagger 1$.~Really, ``weight-graded finite sets'' $A=A_0\sqcup A_1\sqcup A_2$. \par$\dagger 2$.~Really, a power series in the weight-0 variables$^{\dagger 5}$. \par\hangindent=6mm$\dagger 3$.~The weight of $Q$ must be 2, so it decomposes as $Q=Q_{20}+Q_{11}$. The coefficients of $Q_{20}$ are rational numbers while the coefficients of $Q_{11}$ may be weight-0 power series$^{\dagger 5}$. \par\hangindent=6mm$\dagger 4$.~Setting $\wt\epsilon=-2$, the weight of $P$ is $\leq 2$ (so the powers of the weight-0 variables are not constrained)$^{\dagger 5}$. \par\hangindent=6mm$\dagger 5$.~In the knot-theoretic case, all weight-0 power series are rational functions of bounded degree in the exponentials of the weight-0 variables. \par\hangindent=6mm$\dagger 6$.~There's also an obvious product \[ \mor(A_1\to B_1)\times\mor(A_2\to B_2)\to\mor(A_1\sqcup A_2\to B_1\sqcup B_2). \] \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} \parpic[r]{\import{.}{FullDoPeGDO.pdftex_t}} {\bf\red Full DoPeGDO.} Compute compositions in two phases: $\bullet$~A 2-0 phase over $\bbQ$, in which the weight-1 variables are spectators. $\bullet$~A 1-1 phase over the ring of power series in the weight-0 variables, in which the weight-2 variables are spectators. \vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} {\bf\red Questions.} $\bullet$~Are there QFT precedents for ``two-step Gaussian integration''? $\bullet$~In QFT, one saves even more by considering ``one-particle-irreducible'' diagrams and ``effective actions''. Does this mean anything here? $\bullet$~Understanding $\Hom(\bbQ[z_A]\to\bbQ[z_B])$ seems like a good cause. Can you find other applications for the technology here? %\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} \columnbreak \par{\bf\red Compositions (2).} Recall that with all indices $i$ running in some set $B$, \[ \calF\act \calG = \left(\calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0} \overset{(1)}{=} \left.\bbe^{\sum\partial_{z_i}\partial_{\zeta_i}}(\calF\calG)\right|_{z_i=\zeta_i=0}, \quad\parbox{0.7in}{\tiny (1) Strictly speaking,\newline true only when\newline $B\cap(A\cup C)=\emptyset$.} \] so in general we wish to understand \[ [F\colon\calE]_B\coloneqq \bbe^{\frac12\sum_{i,j\in B} F_{ij}\partial_{z_i}\partial_{z_j}}\calE \quad\text{and}\quad \langle F\colon\calE\rangle_B\coloneqq \left.[F\colon\calE]_B\right|_{z_B\to 0}, \] where $\calE$ is a docile perturbed Gaussian. The following lemma allows us to restrict to the case where $\calE$ has no $B$-$B$ quadratic part: {\bf Lemma 1.} With convergences left to the reader, \[ \left\langle F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right\rangle_B = \det(1-GF)^{-1/2}\left\langle F(1-GF)^{-1} \colon \calE \right\rangle_B. \] The next lemma dispatches the case where $\calE$ has a $B$-linear part: {\bf Lemma 2.} $\displaystyle \left\langle F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right\rangle_B = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j} \left\langle F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right\rangle_B$. %\needspace{4\baselineskip} Finally, we deal with the docile perturbation case: {\bf Lemma 3.} With an extra variable $\lambda$, $Z_\lambda\coloneqq\log[\lambda F\colon \bbe^P]_B$ satisfies and is determined by the following PDE / IVP: \[ Z_0=P \quad\text{and}\quad \partial_\lambda Z_\lambda = \frac12 \sum_{i,j\in B}F_{ij}\left( \partial_{z_i}\partial_{z_j}Z_\lambda + (\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda) \right). \] \resizebox{\linewidth}{!}{\import{../QMUL-1908/}{Lemmas.pdftex_t}} %\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm} % %{{\red References.}\footnotesize %\par\vspace{-3mm} %\renewcommand{\section}[2]{}% %\begin{thebibliography}{} %\setlength{\parskip}{0pt} %\setlength{\itemsep}{0pt plus 0.3ex} % %\bibitem[BN]{KBH} D.~Bar-Natan, % {\em Balloons and Hoops and their Universal Finite Type Invariant, BF % Theory, and an Ultimate Alexander Invariant,} % \web{KBH}, \arXiv{1308.1721}. % %\end{thebibliography}} \end{multicols*} \end{document} \endinput