Solvable Lie Algebras in Supergravity and Superstrings Pietro Fré Bonn February 2002 An algebraic characterization of superstring dualities

In D < 10 Superstring In D < 10 the structure of Superstring Theory is governed... The geometry of the scalar manifold M M = G/H is mostly a non compact coset manifold Non compact cosets admit an algebraic description in terms of solvable Lie algebras

Before the gauging

By compactification from higher dimensions. In this case the scalar manifold is identified as the moduli space of the internal compact manifold By compactification from higher dimensions. In this case the scalar manifold is identified as the moduli space of the internal compact manifold By direct construction of each supergravity in the chosen dimension. In this case one uses the a priori constraints provided by supersymmetry. In particular holonomy and the need to reconcile p+1 forms with scalars By direct construction of each supergravity in the chosen dimension. In this case one uses the a priori constraints provided by supersymmetry. In particular holonomy and the need to reconcile p+1 forms with scalars Two ways to determine G/H or anyhow the scalar manifold DUALITIES Special Geometries The second method is more general, the first knows more about superstrings, but the two must be consistent

The scalar manifold of supergravities is necessarily a non compact G/H, except: 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

Scalar cosets in d=4

Scalar manifolds by dimensions in maximal supergravities 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.

How to determine the scalar cosets G/H from supersymmetry

.....and symplectic or pseudorthogonal representations

How to retrieve the D=4 table

Essentials of Duality Rotations The scalar potential V(  is introduced by the gauging. Prior to that we have invariance under duality rotations of electric and magnetic field strengths

Duality Rotation Groups

The symplectic or pseudorthogonal embedding in D=2r

continued This embedding is the key point in the construction of N-extended supergravity lagrangians in even dimensions. It determines the form of the kinetic matrix of the self-dual p+1 forms and later controls the gauging procedures. D=4,8 D=6,10

The symplectic case D=4,8 This is the basic object entering susy rules and later fermion shifts and the scalar potential

The Gaillard and Zumino master formula We have: 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 3-fold

Summarizing: The scalar sector of supergravities is “mostly” a non compact coset U/H The scalar sector of supergravities is “mostly” a non compact coset U/H The isometry group U acts as a duality group on vector fields or p-forms The isometry group U acts as a duality group on vector fields or p-forms U includes target space T-duality and strong/weak coupling S-duality. U includes target space T-duality and strong/weak coupling S-duality. For non compact U/H we have a general mathematical theory that describes them in terms of solvable Lie algebras..... For non compact U/H we have a general mathematical theory that describes them in terms of solvable Lie algebras.....

Solvable Lie algebra description...

Differential Geometry = Algebra

Maximal Susy implies E r+1 series Scalar fields are associated with positive roots or Cartan generators

The relevant Theorem

How to build the solvable algebra Given the Real form of the algebra U, for each positive root  there is an appropriate step operator belonging to such a real form

String interpretation of scalar fields

...in the sequential toroidal compactification The sequential toroidal compactification has an algebraic counterpart in the embedding of subalgebras

Sequential Embeddings of Subalgebras and Superstrings

The type IIA chain of subalgebras W is a nilpotent algebra with including no Cartan ST algebra

Type IIA versus Type IIB decomposition of the Dynkin diagram Dilaton Ramond scalars The dilaton

The Type IIB chain of subalgebras U duality in D=10

A decomposition that mixes NS and RR states (= U duality)

The electric subalgebra In the symplectic representation this is the only off diagonal root that mixes electric and magnetic field strengths

Understanding type IIA / type IIB T-duality algebraically Two ways of choosing the simple roots Chirality with respect to T algebra

The two realizations of E 7(7) The two realizations correspond to attaching the 7 th root in two different positions

Dilaton and radii are in the CSA The extra dimensions are compactified on circles of various radii

Outer Automorphisms of the S T subalgebra

An explicit example of T duality

The Maximal Abelian Ideal From Number of vector fields in SUGRA in D+1 dimensions

Conclusions: (thanks to my collaborator, Mario Trigiante) (thanks to my collaborator, Mario Trigiante) The Solvable Lie algebra representation has many applications, in particular: The Solvable Lie algebra representation has many applications, in particular: Classification of p-brane and black hole solutions Classification of p-brane and black hole solutions Classification and construction of supergravity gaugings Classification and construction of supergravity gaugings Study of supersymmetry breaking patterns and super Higgs phenomena Study of supersymmetry breaking patterns and super Higgs phenomena