pietro frè university of torino and italian embassy in moscow based on common work with aleksander...
TRANSCRIPT
![Page 1: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/1.jpg)
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
![Page 2: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/2.jpg)
A well defined mathematical problem
3
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
![Page 3: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/3.jpg)
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
![Page 4: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/4.jpg)
Special Kahler Geometry
symplectic section
![Page 5: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/5.jpg)
Special Geometry identities
![Page 6: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/6.jpg)
The matrix N
![Page 7: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/7.jpg)
When the special manifold is a symmetric coset ..
Symplectic embedding
![Page 8: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/8.jpg)
9
The main point
![Page 9: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/9.jpg)
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
![Page 10: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/10.jpg)
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!
![Page 11: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/11.jpg)
When homogeneous symmetric manifolds
C-MAPGeneral Form of the Lie algebra decomposition
![Page 12: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/12.jpg)
13
Relation between
One just changes the sign of the scalars coming from W(2,R) part in:
Examples
![Page 13: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/13.jpg)
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
![Page 14: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/14.jpg)
The simplest example G2(2)
One vector multiplet
Poincaré metric
Symplectic section
Matrix N
![Page 15: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/15.jpg)
OXIDATION 1The metric
where Taub-NUT charge
The electromagnetic charges
From the -model viewpoint all these first integrals of the motion
Extremality parameter
![Page 16: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/16.jpg)
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
![Page 17: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/17.jpg)
From coset rep. to Lax equation
From coset representative
decomposition
R-matrix
Lax equation
![Page 18: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/18.jpg)
Integration algorithm
Initial conditions
Building block
Found by Fre & Sorin 2009 - 2010
![Page 19: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/19.jpg)
Key property of integration algorithm
Hence all LAX evolutions occur within distinct orbits of H*
Fundamental Problem: classification of ORBITS
![Page 20: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/20.jpg)
The role of H*
Max. comp. subgroup
Different real form of H
COSMOL.
BLACK HOLES
In our simple G2(2) model
![Page 21: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/21.jpg)
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
![Page 22: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/22.jpg)
Invariants & Tensors
Quadratic Tensor
![Page 23: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/23.jpg)
Tensors 2
BIVECTOR
QUADRATIC
![Page 24: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/24.jpg)
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!
![Page 25: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/25.jpg)
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!
![Page 26: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/26.jpg)
Relation between old and new Cartan Weyl bases
![Page 27: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/27.jpg)
Hence we can easily find nilpotent orbits
Every orbit possesses a representative of the form
Generic nilpotency 7. Then imposereduction of nilpotency
![Page 28: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/28.jpg)
The general pattern
![Page 29: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/29.jpg)
The method of standard triplets
![Page 30: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/30.jpg)
Angular momenta
![Page 31: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/31.jpg)
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
![Page 32: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/32.jpg)
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
![Page 33: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/33.jpg)
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
![Page 34: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/34.jpg)
Two type of roots
1
2
3
![Page 35: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/35.jpg)
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
![Page 36: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/36.jpg)
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
![Page 37: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/37.jpg)
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
![Page 38: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/38.jpg)
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
![Page 39: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/39.jpg)
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)
![Page 40: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/40.jpg)
Tits Satake Projection: an example
1
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.
![Page 41: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/41.jpg)
![Page 42: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/42.jpg)
Universality Classes
![Page 43: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/43.jpg)
One example
Tits-Satake Projection SO(4,5)
![Page 44: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/44.jpg)
The orbits are the same for all members of the universality class (still unpublished result)
![Page 45: Pietro Frè University of Torino and Italian Embassy in Moscow Based on common work with Aleksander S. Sorin & Mario Trigiante Stekhlov Institute June 30th](https://reader038.vdocuments.mx/reader038/viewer/2022110304/5519d587550346047c8b4d78/html5/thumbnails/45.jpg)
Спосибо за внимание
Thank you for your attention