# Geometry and Mathematical Physics Seminar 2011/2012

**Thursday 1 March 2012. Joint Seminar: Geometry and Algebra. **

*The Frank Adams Room (Room 1.212), the Alan Turing Building. 3pm and 4.10pm*
##
Universality and $n\to-n$ Duality in Lie Algebras and Gauge Theories

Ruben Mkrtchyan (Alikhanian National Science Laboratory/ Yerevan Physics Institute)

`mrl@web.am`

Universal, in the sense of Vogel's *Universal Lie Algebra*, expression for the generating function of the spectra of Casimir's operators on adjoint representation is presented, for two choices of Casimir operators. Generalized universal Freudenthal-de Vries strange relations are derived. They lead to a universal representation of (the perturbative part of) the $S_{00}$ element of the modular transformations matrix (= the partition function of the Chern-Simons theory on a $3$-sphere) of characters of Kac-Moody algebra. Extension of universality to the untwisted Kac-Moody algebras is discussed and the "extended Vogel space" $P^3/S_3$ is introduced. The $n\to -n$ duality of the Lie algebras $so(2n)$ and $sp(2n)$ is extended to the classical symmetric spaces using the Macdonald duality for the Jack and Jacobi symmetric functions. The explicit formulae for the Popov-Perelomov generating function of Casimir's spectra are shown to satisfy the duality.

The seminar will consist of two independent parts:

- 3pm-3.50pm: introductory lecture accessible for students;
- 4.10pm-5.10pm: main talk (independent of the introductory part).