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 -

• 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

sl(n,R) algebra in gravity and HEP

• Affine models of gravity in n space-time dimensions (gauging Rn Λ GL(n,R) symmetry)

• “World spinors” in n space-time dimensions

• Algebra of M-theory is often extended to r528 Λ gl(32,R)

• Systems with conserved n-dimensional volume (strings, pD-branes...)

• Effective QCD in terms of Regge trajectories

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) (or SO(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)!

How to find SL(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

Now, what is the Gell-Mann decontraction formula?

Loosely speaking: it is formula inverse to the Inönü-Wigner contraction.

The Gell-Mann decontraction formula

Gell-Mann formula

(as named by R. Hermann)

Gell-Mann formula?

Inönü-Wigner contraction

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).

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

Rn(n+1)/2-1 Λ Spin(n)SL(n,R)

Representations of this group are easy to find

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.

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

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

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.)

• All irreducible representations of SL(3,R) and SL(4,R) are known – Dj. Šijački found 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.

Learning from the solved cases

!Additional label,

overall 2, matching the group rank!

Spin(n) left action generators

Generalized formula in SL(5,R) case

new terms

Not easy even to check that this is correct (i.e. closes algebra relations).

4 labels, matching the group rank.

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.

Can we find the generalized formula for arbitrary n?

• 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.

Overall n-1 parameters, matching the group rank! They

determine Casimir values.

• 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)!

Matrix elements for arbitrary SL(n,R) irreducible representation

Collateral result for su(n)

• Multiplying shear generators T → iT turns algebra into su(n)

• All results applicable to su(n): su(n) matrices in so(n) basis – a nontrivial result.

• A generic affine theory Lagrangian in n space-time dimensions :

• A symmetry breaking mechanism is required.

Application – affine theory of gravity

What kind of fields are these?

sl(n,R) matrix elements appear in vertices

Example: n=5, multiplicity free

• Vector component of infinite-component bosonic multifield, transforming as a multiplicity free SL(5,R) representation labelled by

• Similarly for the term:

Example: n=5, nontrivial multiplicity

• 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.

From the form of the generalized Gell-Mann formula we deduce

that all component can not belong to the same irreducible

representation

Example: n=5, nontrivial multiplicity

• Sheer connection transforms these fields one into another. Interaction terms are:

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 all irreducible 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.

Relation to the max weight labels

• Labels are weights of max weight vector :

Using:

one obtains: