Solvable Lie Algebras in Supergravity and Superstrings

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

2. In D < 10the structure ofSuperstringTheory 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

3. Before the gauging

4. Two ways to determine G/H or anyhow the scalar 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 reconcilep+1 forms with scalars DUALITIES Special Geometries The second method is more general, the first knows more about superstrings, but the two must be consistent

5. 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

6. Scalar cosets in d=4

7. 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.

8. How to determine the scalar cosets G/H from supersymmetry

9. .....and symplectic or pseudorthogonal representations

10. How to retrieve the D=4 table

11. Essentials of Duality Rotations The scalar potential V(f) is introduced by the gauging. Prior to that we have invariance underduality rotationsofelectric and magnetic field strengths

12. Duality Rotation Groups

13. The symplectic or pseudorthogonal embedding in D=2r

14. .......continued D=4,8 D=6,10 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-dualp+1 forms and later controls the gauging procedures.

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

16. 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 The Gaillard and Zumino master formula We have:

17. Summarizing: • 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 • 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.....

18. Solvable Lie algebra description...

19. Differential Geometry = Algebra

20. Maximal Susy implies Er+1 series Scalar fields are associated with positive roots or Cartan generators

21. The relevant Theorem

22. 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

23. String interpretation of scalar fields

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

25. Sequential Embeddings of Subalgebras and Superstrings

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

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

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

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

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

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

32. The two realizations of E7(7) The two realizations correspond to attaching the 7th root in two different positions

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

34. Outer Automorphisms of the S T subalgebra

35. An explicit example of T duality

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

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