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Igor Salom and Đorđe Šijački

Generalization of the  Gell-Mann decontraction formula for sl(n,R) and its applications in affine gravity. Igor Salom and Đorđe Šijački. Generalization of the Gell-Mann formula for sl(n,R) and applications in affine gravity - Talk outline -.

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Igor Salom and Đorđe Šijački

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  1. Generalization of the Gell-Mann decontraction formula for sl(n,R) and its applications in affine gravity Igor Salom and Đorđe Šijački

  2. Generalization of the Gell-Mann formula for sl(n,R) and applications in affine gravity-Talk outline - sl(n,R)algebra in theory of gravity – what is specific? What is the Gell-Mann decontraction formula an why is it important in this context? Validity domain and need for generalization Generalization of the Gell-Mann formula Illustration: application of the formula in affine theory of gravity

  3. sl(n,R)algebra in gravity and HEP Affine models of gravity in n space-time dimensions (gauging RnΛGL(n,R) symmetry) “World spinors” in n space-time dimensions Algebra of M-theory is often extended tor528 Λgl(32,R) Systems with conserved n-dimensional volume (strings, pD-branes...) Effective QCDin terms of Regge trajectories

  4. In these context we need to know how to represent SL(n,R) generators… in some simple, “easy to use” form if possible, in SO(n)(orSO(1,n-1)) subgroup basis, for infinite-dimensional unitary representations, and, in particular, for infinite-dimensional spinorial representations: SL(n,R) is double cover of SL(n,R)!

  5. How to findSL(n,R) generators? Induction from parabolic subgroups Construct generators as differential operators in the space of group parameters Analytical continuation of complexified SU(n) representations ... Using the Gell-Mann decontraction formula

  6. Now, what is the Gell-Mann decontraction formula? Loosely speaking: it is formula inverse to theInönü-Wigner contraction.

  7. The Gell-Mann decontraction formula Inönü-Wigner contraction ? Gell-Mann formula Gell-Mann formula (as named by R. Hermann)

  8. Example: Poincare to de Sitter Define function of Poincare generators: Check: • …unfortunately, this works so nicely only for so(m,n) cases. Not for sl(n,R).

  9. SL(n,R) group Definition: group of unimodular n x n real matrices (with matrix multiplication) Algebra relations: so(n) irrep. of traceless symmetric matrices

  10. SL(n,R) Inönü-Wigner contraction of SL(n,R) Find representations of the contracted semidirect product and apply Gell-Mann formula to get sl(n,R) representations. • Rn(n+1)/2-1 Λ Spin(n) Representations of this group are easy to find

  11. Space of square integrable functions over Spin(n) manifold Space of square integrable functions is rich enough to contain representatives from all equivalence classes of irreps. of both SL(n,R) and Tn(n+1)/2-1 Λ Spin(n) groups (Haris Chandra). As a basis we choose Wigner D functions: k indices label SL(n,R)  SО(n) multiplicity

  12. Contracted algebra representations Contracted abelian operators U represent as multiplicative Wigner D functions: Action of spin(n) subalgebra is “natural” one: Matrix elements are simply products of Spin(n) CG coefficients

  13. Try to use Gell-Mann formula Take and plug it in the Gell-Mann formula, i.e.: and then check commutation relations. works only in spaces over SO(n)/(SO(p)×SO(q)), q+p=n no spinorial representations here no representations with multiplicity w.r.t. Spin(n) → Insufficient for most of physical applications! (“Conditions for Validity of the Gell-Mann Formula in the Case of sl(n,R) and/or su(n) Algebras”, Igor Salom and Djordje Šijački, in Lie theory and its applications in physics, American Institute of Physics Conference Proceedings, 1243 (2010) 191-198.)

  14. Learning from the solved cases • All irreducible representations of SL(3,R)andSL(4,R)are known – Dj. Šijačkifound using different approach • Matrix elements of SL(3,R)representations with multiplicity indicate an expression of the form: • This is a correct, “generalized” formula! • Similarly in SL(4,R)case. ! Additional label, overall 2, matching the group rank!

  15. Spin(n) left action generators

  16. Generalized formula in SL(5,R) case 4 labels, matching the group rank. new terms Not easy even to check that this is correct (i.e. closes algebra relations). Generalization of the Gell-Mann formula for sl(5,R) and su(5) algebras,Igor Salom and Djordje Šijački, International Journal of Geometric Methods in Modern Physics, 7 (2010) 455-470.

  17. Can we find the generalized formula for arbitraryn? Idea: rewrite all generalized formulas (n=3,4,5) in Cartesian coordinates. All formulas fit into a general expression, now valid for arbitratry n: Using a D-functions identity: direct calculation shows that the expression satisfies algebra relations. Overalln-1parameters, matching the group rank! They determine Casimir values.

  18. Matrix elements for arbitrary SL(n,R) irreducible representation • Matrix elements: • All required properties met: • simple expression • in Spin(n) basis • valid for arbitrary representation (including infinite dimensional ones, and spinorial ones, and with nontrivial multiplicity)!

  19. Collateral result for su(n) Multiplying shear generators T → iT turns algebra into su(n) All results applicable tosu(n): su(n)matrices inso(n)basis – a nontrivial result.

  20. Application – affine theory of gravity • A generic affine theory Lagrangian in n space-time dimensions : • A symmetry breaking mechanism is required. What kind of fields are these?

  21. sl(n,R) matrix elements appear in vertices

  22. Example: n=5, multiplicity free • Similarly for the term: Vector component of infinite-component bosonic multifield, transforming as a multiplicity free SL(5,R) representation labelled by

  23. Example: n=5, nontrivial multiplicity From the form of the generalized Gell-Mann formula we deduce that all component can not belong to the same irreducible representation Due to multiplicity, there are , a priori, 5 different 5-dimensional vector components, i.e. Lorentz subfields, of the infinite-component bosonic multifield – one vector field for each valid combination of left indices k.

  24. Example: n=5, nontrivial multiplicity Sheer connection transforms these fields one into another. Interaction terms are:

  25. Conclusion Not much use in gravity for the original Gell-Mann formula (for sl(n,R) case), but we generalized it, in the case of arbitrary n. New formula is of a simple form and applicable to allirreducible representations. If you ever need expressions for SL(n,R) or SU(n) generators in SO(n) basis, you can find them in Generalization of the Gell-Mann decontraction formula for sl(n,R) and su(n) algebras, Igor Salom and Djordje Šijački, International Journal of Geometric Methods in Modern Physics, 8 (2011), 395-410.

  26. Relation to the max weight labels • Labels are weights of max weight vector : Using: one obtains:

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