SL(n,R) and Diff(n,R) - Decontraction formula and

Unitary Irreducible Representations

Djordje Sijacki and Igor SalomInstitute of Physics, University of Belgrade

Quantum Integrable Systems and Geometry, September 2012, Olhao, Portugal

sl(n,R) and diff(n,R) algebras in gravity and HEP

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

• “World spinors” - infinite-component Diff(n,R) spinors in n space-time dimensions

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

• SL(3,R), SL(4,R) unirreps and Regge trajectories of hadrons (3-quark and quark-antiquark bound states)

• Chromogravity as an effective IR region QCD

Cromogravity – Effective QCD in the IR region

SU(3) color gauge – B is dressed gluon field, g is SU(3) Cartan metric

SU(3) variation:

Expand B around a constant vacuum solution N(pure gauge) of the instanton type:

In the IR region (variation of pseudo-metric G):

Generalization to colorless n-gluon fields:

d is the color SU(3) totally symmetric 8x8x8 → 1 tensor

QCD variation in the IR region:

Operators of the Chromo-Diffeomorphism algebra

- Diff(n,R) linearly realized- Diff(n,R) realized nonlineary over SL(n,R)

Spinorial matter fields of SL(n,R), GL(n,R) and/or Diff(n,R) nonlinearly realized w.r.t. SL(n,R), are infinite-component fields coupled to GL(n,R) connections:

Q are the SL(n,R), GL(n,R) group generators (infinite matrices!)

Frame fields:

Infinite pseudo-frame(alephzeroads)

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

• A symmetry breaking mechanism is required.

Affine theory of gravity

DOUBLE COVERING of SL(n,R), GL(n,R) and Diff(n,R)

Iwasawa decomposition of a semisimple Lie algebra:

g=k+a+n;

G=KxAxN is any connected group with Lie algebra g,

K (compact), A (Abelian) and N (nilpotent) are its corresponding analytic subgroups

– the groups A and N are simply connected.

There exists a universal covering group

where Is the universal covering of K.

For SL(n,R), GL(n,R), the universal covering group of

K=SO(n) is a double covering given by Spin(n)

For the group of diffeomorphisms Diff(n,R)Stewart proved:

where the subgroup H is contractible to a point.

As a result, as O(n) is the compact subgroup of GL(n,R),there is a double cover:

For SL(4,R)

A finite dimensional covering of SL(n,R), i.e. Diff(n,R) exists provided one can embed their covering into a

group of finite complex matrices that contain Spin(n) as a subgroup.The natural candidate for SL(n,R) covering, from the set of

Cartan's classical Lie groups, is SL(n,C), however there is no match of the group dimensionalities, i.e.

dim(SL(n,C)) = n < dim(Spin(n)) = 2^[(n-1)/2]

except for n=8, but than the maximal compact subgroup of

SL(8,C) is SO(8) and not Spin(8)!

The universal (double) coverings of

SL(n,R), GL(n,R) and Diff(n,R), for n>2, are

groups of infinite complex matrices.

I is necessary to know, for various (math) physics applications, how to represent the SL(n,R) generators, i.e. to find their infinite-dimensional unirreps ...

… in some simple, “easy to use” form,

… 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) generetors representations

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• ...• By making use of 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 rather 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 (Harish Chandra).

• As a basis we choose Wigner D functions:

k indices label SL(n,R) SО(n)

multiplicity

Chain of groups: Spin(n); Spin(n-1), Spin(n-2), …, Spin(2)

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, 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.

Igor Salom and Djordje Šijački, International Journal of Geometric Methods in Modern Physics, 8 (2011), 395-410

• Matrix elements:

• All required properties met: Simple closed 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 (relevant in various nuclear physics applications).

UNITARY IRREDUCIBLE REPRESENTATIONS

Harish-Chandra proved that all unitary irreducible representations of a noncompact group G can be obtained in Hilbert spaces H of square-integrable functions over the maximal compact subgroup K.

Unitarity means that:

and the additional conditions that the bilinear form is ascalar product are hermiticity and positive definiteness:

In order to obtain ALL unitary irreducible representations we consider the most general scalar product of functions over K=Spin(n), with an invariant measure dk and an arbitrary kernel function over KxK

Now we combine:

(1) SL(n,R) generators matrix elements expressions for an arbitrary representation, as given by the generalized decontraction formula,

(2) representation unitarity requirement,

(3) scalar product hermiticity and positive definiteness,

(4) irreducibility requirement, as given by invariant lattices of the Spin(n) labels weight space.

Simplest case:

Principal series of (infite-dimensional) unitary irreducible (spinorial and tensorial) representations of the SL(n,R) double covering group:

representation labels – purely imaginary

kernel function – Dirac's delta function

All matrix elements of the noncompact generators for an arbitrary unirrep given by generalized decontraction formula

• 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: