1. Hidden Symmetries, Solvable Lie Algebras, Reduction and Oxidation in Superstring Theory
Pietro Fré
Dubna July 2003 ]
exp[ /
Solv H G @
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 terms of 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 reconcile p+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
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
Ora che abbiamo vagamente intuito che cosa è una stringa, possiamo dare uno sguardo alla seguente figura che è divenuta celeberrima dopo l’introduzione della M teoria nel Non è facile da spiegare, ma era un risultato noto da circa 15 anni che di teoriequantisticamente consistenti delle superstringhe ne esistessero cinque e solo cinque. Esse hanno buffi nomi come la tipo IIA, la tipo IIB, la tipo I, l’eterotica SO(32) ovvero l’eterotica E8 x E8. Nonostante i buffi nomi queste teorie si distinguono per proprietà piuttosto semplici quale l’essere fatte di lacci chiusi ovvero sia di lacci che di corde aperte. Altro elemento di distinzione è il permettere che certe vibrazioni circolino sul foglio di mondo sia in senso destrorso che sinistrorso ovvero solo in uno dei due sensi. A prima vista le proprietà di queste cinque teorie sono radicalmente differenti. La grande scoperta del 1995 fu che questa diversità dipende solo dal fatto che esse sono studiate in un regime di deboli interazioni mutue tra stringhe. Quando le cinque teorie vengono analizzate in un regime di accoppiamento forte le differenze scompaiono ed anzi si scopre che ciascuna di esse descrive ad accoppiamento debole la fase di acoppiamento forte di un’altra teoria dello stesso insieme. Sorprendentement, il fatto stesso che sitratti di teorie di corde è un artefatto dell’acoppiamento debole. Man mano che ci inoltra nella fase di acoppiamento forte si scopre che vi sono altri oggetti estesi nella teoria, oltre quelli unidimensionali. Tali oggetti vengono chiamati brane e si parla di una p-brana per indicarne uno di dimensione p. Questo viaggio si fa muovendosi in uno spazio di parametri che controllano la teoria e che i fisici chiamano moduli. E’ la regione a forma di stella di questo disegno. Nella parte interna dello spazio dei moduli abbiamo la vera e propria teoria M e le eccitazioni sono p-brane per tutti i possibili valori di p da 0 a 9. Nelle regioni peninsulari la teoria M assomiglia sempre ad una teoria di stringa in dieci dimensioni, con una sola eccezione. Vi è una penisola in cui essa assomiglia ad una teoria di membrane o 2-brane in undici dimensioni. E’ questa penisola una delle possibili interpretazioni della M di teoria M.
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. 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
Heterotic E8xE8
Type I SO(32)
Heterotic SO(32)
Type II B
Type II A
D=10
D=9

9. Table of Supergravities in D=10
Theories
BOSE STATES
FERMI STATES
NS - NS
R - R
Left handed
Right handed
Type II A
Type II B
Heterotic SO(32)
Heterotic E8 x E8
Type I SO(32)
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 under duality rotations of electric 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-dual p+1 forms and later controls the gauging procedures.

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

20. 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

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

32. Sequential Embeddings of Subalgebras and Superstrings

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

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

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

36. If we compactify down to D=3 we 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 Gij and 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. The Maximal Abelian Ideal
From
Number of vector fields in SUGRA in D+1 dimensions

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
Since U/H is a homogeneous space all initial points are equivalent
Initial tangent vectors span a representation of H and 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
The orbits of geodesics are parametrized by as many parameters as the rank of U
Indeed we have the following identification of the representation K to which the tangent vectors belong:
Orthogonal decomposition
Non orthogonal decomposition

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

51. and turn them to the anholonomic basis
The strategy to solve the differential equations consists now of two steps:
First solve the first order differential system for the tangent vectors
Then solve for the coset representative that reproduces such tangent vectors

52. The Main Differential system:

53. Summarizing:
If we are interested in time dependent backgrounds of supergravity/superstrings we dimensionally reduce to D=3
In D=3 gravity can be decoupled and we just study a sigma model on U/H
Field equations of the sigma model reduce to geodesics equations. The Manifold of orbits is parametrized by the dual of the CSA.
Geodesic equations are solved in two steps.
First one solves equations for the tangent vectors. They are defined by the Nomizu connection.
Secondly one finds the coset representative
Finally we oxide the sigma model solution to D=10, namely we embed the effective Lie algebra used to find the solution into E8. Note that, in general there are several ways to oxide, since there are several, non equivalent embeddings.

54. The paradigma of the A2 Lie Algebra

55. The A2 differential system

56. Searching the normal form for the J=2 representation

57. The normal form is a diagonal traceless matrix, obviously!!!

58. Fixing the normal tangent vector

59. NORMAL FORM of the 5-vector

60. Explicit solution for the tangent vectors

61. Which are solved by:

62. This is the final solution for the scalar fields, namely the parameters in the Solvable Lie algebra representation
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.

63. Type II B Action and Field equations in D=10
Where the field strengths are:
Chern Simons
term
Note that the Chern Simons term couples the RR fields to the NS fields !!

64. The type IIB field equations

65. Inequivalent embeddings
PROBLEM:
There are several inequivalent ways, due to the following graded structure of the Solvable Lie algebra of E8
where

66. 5 physically inequivalent embeddings

67. Choosing an example of type 4 embedding
Physically this example corresponds to a superposition of three extended objects:
An euclidean NS 1-brane in directions 34 or NS5 in directions
An euclidean D1-brane in directions 89 or D5 in directions
An euclidean D3-brane in directions 3489

68. If we oxide our particular solution...
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 5-branes. 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:

69. The effective field equations for this oxidation
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

70. Explicit Oxidation: The Metric and the Ricci tensor
Non vanishing
components

71. Plots of the Radii for the case with
We observe the phenomenon of cosmological billiard
of Damour, Nicolai, Henneaux

72. Energy density and equations of state
P in 567
P in 89
P in 12
P in 34

73. Plots of the Radii for the case with this is a pure D3 brane case

74. Energy density and equations of state
P in 567
P in 89
P in 12
P in 34