Hidden Symmetries, Solvable Lie Algebras, Reduction and Oxidation in Superstring Theory

1. @ 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

2. 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 termsof solvable Lie algebras

3. For instance, the Bose Lagrangian of any SUGRA theory in D=4 is of the form:

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. In D=10 there are 5 consistent Superstring Theories. They are perturbative limits of just one theory 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

8. Type II B Type II A Heterotic SO(32) Type I SO(32) Heterotic E8xE8 The 5 string theories in D=10 and the M Theory in D=11 are different perturbative faces of the same non perturbative theory. M theory D=11 D=10 D=9

9. Table of Supergravities in D=10

10. The Type II Lagrangians in D=10

11. 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.

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

13. .....and symplectic or pseudorthogonal representations

14. How to retrieve the D=4 table

15. 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

16. Duality Rotation Groups

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

18. .......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.

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

20. 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:

21. 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.....

22. Solvable Lie algebra description...

23. Differential Geometry = Algebra

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

25. The relevant Theorem

26. 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

27. The Nomizu Operator

28. Explicit Form of the Nomizu connection

29. Definition of the cocycle N

30. String interpretation of scalar fields

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

32. Sequential Embeddings of Subalgebras and Superstrings

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

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

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

36. If we compactify down to D=3we have E8(8) Indeed the bosonic Lagrangian of both Type IIA and Type IIB reduces to the gravity coupled sigma model With target manifold

37. Painting the Dynkin diagram = constructing a suitable basis of simple roots Type II B painting + Spinor weight

38. - A second painting possibility Type IIA painting

39. Surgery on Dynkin diagram - SO(7,7) Dynkin diagram Neveu Schwarz sector Spinor weight = Ramond Ramond sector

40. String Theory understanding of the algebraic decomposition Parametrizes both metrics Gijand B-fields Bij on the Torus Internal dilaton B-field Metric moduli space

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

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

43. An application: searching for cosmological solutions in D=10 via D=3 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

44. What follows next is a report on work to be next published • Based on the a collaboration: • P. F. , F. Gargiulo, K. Rulik (Torino, Italy) • M. Trigiante (Utrecht, The Nederlands) • V. Gili (Pavia, Italy) • A. Sorin (Dubna, Russian Federation)

45. Decoupling of 3D gravity

46. Decoupling 3D gravity continues... K is a constant by means of the field equations of scalar fields.

47. The matter field equations are geodesic equations in the target manifold U/H • Geodesics are fixed by initial conditions • The starting point • The direction of the initial tangent vector • SinceU/H is a homogeneous space all initial points are equivalent • Initial tangent vectors span a representation ofHand by means of H transformations can be reduced to normal form. The orbits of geodesics contain as many parameters as that normal form!!!

48. Orthogonal decomposition Non orthogonal decomposition The orbits of geodesics are parametrized by as many parameters as the rank of U Indeed we have the following identification of the representationK to which the tangent vectors belong:

49. and since 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

50. Let us consider the geodesics equation explicitly