From: Ruben L. Mkrtchyan mrl55 [snail] list.ru Oct 8, 2012 N -> -N duality and universality ================================ Dear Predrag, you will remember that we met about 30 years ago when you visited Moscow and I have been a post-graduate student at Landau institute. You told us about your work on invariant tensors and exceptional groups in supergravity, and I had just established an equivalence of SO(2n) and Sp(-2n) gauge theories by loop equations technique. Recently we have succeeded in extending, partially, this result to all other simple groups. Namely, we show that for Chern-Simons gauge theory its perturbative partition function is expressible through Vogel's universal parameters, as well as some Wilson observables. I assume that there is no need to explain you what are the Vogel's parameters since people (and me also) often mention you and birdtracks in their works on Universal Lie Algebras. Moreover, on this way it become possible, in a reasonable sense, to create a kind of classification of simple Lie algebras, which leads naturally to all known simple Lie algebras, three algebras with similar properties (including E(7+1/2), and also to a number of negative-dimension objects. Ref's are "On the map of Vogel's plane" arXiv.org/abs/1209.5709 "Universality in Chern-Simons theory" arXiv.org/abs/1203.0766 "Casimir eigenvalues for universal Lie algebra" arXiv.org/abs/1105.0115 "On duality and negative dimensions in the theory of Lie groups and symmetric spaces" arXiv.org/abs/1011.0151 The open question is to present in a universal form the partition function of Yang-Mills theory, and to find out what observables can be presented in a similar universal form. Ruben Nov 17, 2012 The recent result is that non-perturbative part of Chern-Simons theory partition function (which is a group volume, very interesting object) is also universal, so complete partition function of Chern-Simons is universal. As a puzzle with unknown answer can suggest a question – why E8, E7 and F4 lines on magic square (triangle) are straight lines on Vogel plane, but E6 is not? Correspondingly, your equations on dimensions on that lines for former three lines can be obtained from Vogel's dimension formula, but that for E6 doesn't. Ruben Feb 17, 2013 Please note my recent work "Non-perturbative universal Chern-Simons theory" arXiv.org/abs/1302.1507 There a non-perturbative part of partition function of Chern-Simons is proved to be universal. Together with result from previous paper on a universality of perturbative part of Chern-Simons this means that total partition function is universal. Mathematically nonperturbative part of partition function is simply an invariant volume of compact simple (gauge) group, so this classical object is proved to be universal. For SU(N) it is essentially so called Barnes G-function, so we get some Lie-algebraic generalization of that. This is the first universal object which is not rational function of universal parameters. ========================================================= check this statement in arXiv.org/abs/1105.0115: "Note that these calculations allow us to predict the eigenvalues of quartic Casimirs for exceptional Lie algebras as well. The comparison with the calculations from birdtracks.eu shows the agreement in all cases except E7, which is probably due to a misprint in birdtracks.eu."