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@. G. /. H. exp[. Solv. ]. Hidden Symmetries, Solvable Lie Algebras, Reduction and Oxidation in Superstring Theory. Pietro Fré Dubna July 2003. An algebraic characterization of superstring dualities. In D < 10 the structure of Superstring Theory is governed.
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Hidden Symmetries, Solvable Lie Algebras, Reduction and Oxidation in Superstring Theory
Pietro Fré
Dubna July 2003
An algebraic characterization of superstring dualities
DUALITIES
Special Geometries
The second method is more general, the first knows more about superstrings, but the two must be consistent
In the exceptional cases the scalar coset is not necessarily but
can be chosen to be a non compact coset. Namely Special
Geometries include classes of non compact coset manifolds
Heterotic Superstring
E8 x E8 in D=10
Heterotic Superstring SO(32) in D=10
M Theory
D=11 Supergravity
Type I Superstring
in D=10
This is the parameter space of the
theory. In peninsulae it becomes
similar to a string theory
Type II B superstring in D=10
Type IIA superstring in D=10
Type II B
Type II A
Heterotic SO(32)
Type I SO(32)
Heterotic E8xE8
M theory
D=11
D=10
D=9
Rather then by number of supersymmetries we can go by dimensions
at fixed number of supercharges. This is what we have done above
for the maximal number of susy charges, i.e. 32.
These scalar geometries can be derived by sequential
toroidal compactifications.
The scalar potential V(f) is introduced by the gauging. Prior to that we have invariance underduality rotationsofelectric and magnetic field strengths
D=4,8
D=6,10
This embedding is the key point in the construction
of Nextended supergravity lagrangians in even
dimensions. It determines the form of the kinetic matrix of the selfdualp+1 forms and later controls the gauging procedures.
This is the basic object entering susy rules and later fermion shifts and the scalar potential
A general expression for the vector kinetic matrix in terms of the symplectically embedded coset representatives.
This matrix is also named the period matrix because when we have Calabi Yau compactifications the scalar manifold is no longer a coset manifold and the kinetic matrix of vectors can instead be determined form algebraic geometry as the period matrix of the Calabi Yau 3fold
We have:
Scalar fields are associated with positive roots or Cartan generators
Given the Real form of the algebra U, for each positive root
there is an appropriate step operator belonging to such a real form
The sequential toroidal compactification has an algebraic counterpart in the embedding of subalgebras
ST algebra
W is a nilpotent algebra including no Cartan
Ramond scalars
Dilaton
The dilaton
U duality in D=10
Indeed the bosonic Lagrangian of both Type IIA and Type IIB reduces to the gravity coupled sigma model
With target manifold
Type II B painting
+
Spinor
weight

Type IIA painting

SO(7,7) Dynkin diagram
Neveu Schwarz sector
Spinor weight =
Ramond Ramond sector
Parametrizes both metrics Gijand Bfields Bij on the Torus
Internal dilaton
Bfield
Metric moduli space
The extra dimensions are compactified on circles of various radii
Number of vector fields in SUGRA in D+1 dimensions
From
Since all fields are chosen to depend only on one coordinate, t = time, then we can just reduce everything to D=3
E8
D=10 SUGRA (superstring theory)
D=10 SUGRA (superstring theory)
E8 maps D=10 backgrounds
into D=10 backgrounds
dimensional
reduction
dimensional oxidation
E8
D=3 sigma model
D=3 sigma model
K is a constant by means of the field equations of scalar fields.
The orbits of geodesics contain as many parameters as that normal form!!!
Orthogonal decomposition
Non orthogonal decomposition
Indeed we have the following identification of the representationK to which the tangent vectors belong:
We can conclude that any tangent vector can be brought to have only CSA components by means of H transformations
The cosmological solutions in D=10 are therefore parametrized by 8 essential parameters. They can be obtained from an 8 parameter generating solution of the sigma model by means of SO(16) rotations.
The essential point is to study these solutions and their oxidations
This solution can be OXIDED in many different ways to a complete solution of D=10 Type IIA or Type IIB supergravity. This depends on the various ways of embedding the A2 Lie algebra into the E8 Lie algebra.
The physical meaning of the various oxidations is very much different, but they are related by HIDDEN SYMMETRY transformations.
Where the field strengths are:
Chern Simons
term
Note that the Chern Simons term couples the RR fields to the NS fields !!
PROBLEM:
There are several inequivalent ways, due to the following graded structure of the Solvable Lie algebra of E8
where
Note that
B34 = 0 ; C89= 0 since in our particular solution the tangent vector fields associated with the roots a1,2 are zero. Yet we have also the second Cartan swtiched on and this remembers that the system contains not only the D3 brane but also the 5branes. This memory occurs through the behaviour of the dilaton field which is not constant rather it has a non trivial evolution.
The rolling of the dilaton introduces a distinction among the directions pertaining to the D3 brane which have now different evolutions.
In this context, the two parameters of the A2 generating solution of the following interpretation:
For our choice of oxidation the field equations of type IIB supergravity reduce to
5 brane contribution to the stress energy tensor
D3 brane contribution to the stress energy tensor
and one can easily check that they are explicitly satisfied by use of the A2 model
solution with the chosen identifications
Non vanishing
components
We observe the phenomenon of cosmological billiard
of Damour, Nicolai, Henneaux
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