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."