Black HolesTits -Satake Universality classesand Nilpotent OrbitsPietro FrèUniversity of Torino and Italian Embassy in Moscow
Based on common work with Aleksander S. Sorin & Mario Trigiante
Stekhlov Institute June 30th 2011
A well defined mathematical problem
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Our goal is just to find and classify all spherical symmetric solutions of Supergravity with a static metric of Black Hole type
The solution of this problem is found by reformulating it into the context of a very rich mathematical framework which involves:1. The Geometry of COSET MANIFOLDS2. The theory of Liouville Integrable systems constructed on Borel-
type subalgebras of SEMISIMPLE LIE ALGEBRAS3. The addressing of a very topical issue in conyemporary
ADVANCED LIE ALGEBRA THEORY namely:1. THE CLASSIFICATION OF ORBITS OF NILPOTENT
OPERATORS
The N=2 Supergravity Theory
We have gravity andn vector multiplets
2 n scalars yielding n complex scalars zi
and n+1 vector fields AThe matrix N encodes together with the metric hab Special
Geometry
Special Kahler Geometry
symplectic section
Special Geometry identities
The matrix N
When the special manifold is a symmetric coset ..
Symplectic embedding
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The main point
Dimensional Reduction to D=3
D=4 SUGRA with SKn
D=3 -model on Q4n+4
4n + 4 coordinates
Gravity
From vector fields
scalars
Metric of the target manifold
THE C-MAP
Space red. / Time red.Cosmol. / Black Holes
SUGRA BH.s = one-dimensional Lagrangian model
Evolution parameter
Time-like geodesic = non-extremal Black HoleNull-like geodesic = extremal Black HoleSpace-like geodesic = naked singularity
A Lagrangian model can always be turned into a Hamiltonian one by means of standard procedures.
SO BLACK-HOLE PROBLEM = DYNAMICAL SYSTEM
FOR SKn = symmetric coset space THIS DYNAMICAL SYSTEM is LIOUVILLE INTEGRABLE, always!
When homogeneous symmetric manifolds
C-MAPGeneral Form of the Lie algebra decomposition
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Relation between
One just changes the sign of the scalars coming from W(2,R) part in:
Examples
The solvable parametrization
There is a fascinating theorem which provides an identification of the geometry
of moduli spaces with Lie algebras for (almost) all supergravity theories.
THEOREM: All non compact (symmetric) coset manifolds are metrically equivalent to a solvable group manifold
• There are precise rules to construct Solv(U/H)• Essentially Solv(U/H) is made by
• the non-compact Cartan generators Hi 2 CSA K and
• those positive root step operators E which are not orthogonal to the non compact Cartan subalgebra CSA K
Splitting the Lie algebra U into the maximal compact subalgebra H plus the orthogonal complement K
The simplest example G2(2)
One vector multiplet
Poincaré metric
Symplectic section
Matrix N
OXIDATION 1The metric
where Taub-NUT charge
The electromagnetic charges
From the -model viewpoint all these first integrals of the motion
Extremality parameter
OXIDATION 2
The electromagnetic field-strenghts
U, a, » z, ZA parameterize in the G/H case the coset representative
Coset repres. in D=4
Ehlers SL(2,R)
gen. in (2,W)
Element of
From coset rep. to Lax equation
From coset representative
decomposition
R-matrix
Lax equation
Integration algorithm
Initial conditions
Building block
Found by Fre & Sorin 2009 - 2010
Key property of integration algorithm
Hence all LAX evolutions occur within distinct orbits of H*
Fundamental Problem: classification of ORBITS
The role of H*
Max. comp. subgroup
Different real form of H
COSMOL.
BLACK HOLES
In our simple G2(2) model
The algebraic structure of Lax
For the simplest model ,the Lax operator, is in the representation
of
We can construct invariants and tensors with powers of L
Invariants & Tensors
Quadratic Tensor
Tensors 2
BIVECTOR
QUADRATIC
Tensors 3
Hence we are able to construct quartic tensors
ALL TENSORS, QUADRATIC and QUARTIC are symmetric
Their signatures classify orbits, both regular and nilpotent!
Tensor classification of orbits
How do we get to this classification? The answer is the following: by choosing a new Cartan subalgebra inside H* and recalculating the step operators associated with roots in the new Cartan Weyl basis!
Relation between old and new Cartan Weyl bases
Hence we can easily find nilpotent orbits
Every orbit possesses a representative of the form
Generic nilpotency 7. Then imposereduction of nilpotency
The general pattern
The method of standard triplets
Angular momenta
Partitions
(j=3) The largest orbit NO5
(j=1, j=1/2, j=1/2) The orbit NO2
(j=1, j=1,j=0) Splits into NO3 and NO4 orbits
(j=1/2, j=1/2, j=0, j=0, j=0) The smallest orbit NO1
Tits Satake Theory
To each non maximally non-compact real form U (non split) of a Lie algebra of rank r1 is associated a unique subalgebra UTS ½ U which is maximally split.
UTS has rank r2 < r1
The Cartan subalgebra CTS ½ UTS is the non compact part of the full cartan subalgebra
root systemof rank r1
ProjectionSeveral roots of the higher system have the same projection.
These are painted copies of the same wall.
The Billiard dynamics occurs in the rank r2 system
Two type of roots
1
2
3
Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
Let us distinguish the roots that have
a non-zero
z-component,
from those that have
a vanishing
z-component
INGREDIENT 3
Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
Let us distinguish the roots that have
a non-zero
z-component,
from those that have
a vanishing
z-component
Now let us project all the root vectors onto the
plane z = 0
Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
Let us distinguish the roots that have
a non-zero
z-component,
from those that have
a vanishing
z-component
Now let us project all the root vectors onto the
plane z = 0
Tits Satake Projection: an example
The D3 » A3 root system contains 12 roots:
Complex Lie algebra SO(6,C)
We consider the real section SO(2,4)
The Dynkin diagram is
The projection creates
new vectors
in the plane z = 0
They are images of
more than one root
in the original system
Let us now consider the
system of 2-dimensional vectors obtained from the projection
Tits Satake Projection: an example
This system of
vectors is actually
a new root system in rank r = 2.
1
2
1 2
2 1 2
It is the root system B2 » C2 of the Lie Algebra Sp(4,R) » SO(2,3)
Tits Satake Projection: an example
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2
1 2
2 1 2The root system
B2 » C2
of the Lie Algebra
Sp(4,R) » SO(2,3)
so(2,3) is actually a
subalgebra of so(2,4).
It is called the
Tits Satake subalgebra
The Tits Satake algebra is maximally
split. Its rank is equal to the non compact
rank of the original algebra.
Universality Classes
One example
Tits-Satake Projection SO(4,5)
The orbits are the same for all members of the universality class (still unpublished result)
Спосибо за внимание
Thank you for your attention