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The Geometric Approach to Supergravity and Free Differential Algebras

The Geometric Approach to Supergravity and Free Differential Algebras. Pietro Fré @ 30 years of Supergravity Paris October 2006. The Rheonomic approach to SUGRA theories. Ne’eman & Regge 1978 D’Auria & Fré 1979 Castellani, D’Auria, Fré, van Nieuwenhuizen, Pilch, Townsend 1980 - 1982.

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The Geometric Approach to Supergravity and Free Differential Algebras

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  1. The Geometric Approach to Supergravity and Free Differential Algebras Pietro Fré @ 30 years of Supergravity Paris October 2006

  2. The Rheonomic approach to SUGRA theories • Ne’eman & Regge 1978 • D’Auria & Fré 1979 • Castellani, D’Auria, Fré, van Nieuwenhuizen, Pilch, Townsend 1980 - 1982 A simple idea, similar to analyticity, provided by equations analogous to Cauchy Riemann equations determines all possible supergravity theories encoding their symmetries and their dynamics into a single constructive principle

  3.  Rheonomy of superspace Vertical fermionic directions = tangent space to space-time x PRINCIPLE OF RHEONOMY = space-time

  4. Flow chart to costruct a SUGRA

  5. Let us illustrate these ideas • Starting from D=11 supergravity • Typically higher dimension SUGRAS involve p-form gauge fields • The main question are: • What are the underlying algebras of which sugras are the gauge theories? • How do we construct the gauge theories of such “algebras”?

  6. The answers are: • The underlying algebras are not Lie algebras nor super Lie algebras • They are FDAs (free differential algebras) • FDAs enlarge the category of (super) Lie algebras and are the proper structures to include p-forms • FDAs differently from Lie algebras enclude their own gauging • Merging Rheonomy with FDAs one constructs SUGRAS

  7. Free Differential Algebrasand Sullivan structural Theorems String Theory leads to p-form gauge fields that are an essential part of all supergravity multiplets in higher dimensions. The algebraic structure underlying SUGRA in D>4 is that of an FDA = Free Differential Algebra

  8. Contractible Algebras Minimal Algebras Theorem: The most general FDA is the semidirect sum of a contractible algebra with a minimal algebra Sullivan’s Structural Theorems The analogue of Levi’s Theorem: minimal versus contractible algebras. Sullivan’s first theorem

  9. Physical interpretation of Sullivan’s first theorem Differently from normal Lie algebras, FDAs already contain their own gauging. The important point is to identify the structure of the underlying minimal algebra. The FDA is non trivial only if the minimal algebra is non trivial. The structure of this latter is obtained by considering the zero curvature equations, namely by modding out the contractible subalgebra P. Fré Class. Quan. Grav. 1, (1984) L81

  10. Definition of the Lie algebra A p-cochain is an exterior polynomial with constant coefficients constant Chevalley cohomology The algebraic action of the coboundary : induces a sequence of maps: Cohomology groups as usual...!

  11. Sullivan’s second Theorem

  12. SUPERPOINCARE’ ALGEBRAin D=11 Let us illustrate these ideas with M-theory, namely with SUGRA in D=11 • E. Cremmer & B. Julia 1979 • R. D’Auria & P. Fré 1981 Central charges

  13. Dual generators Cohomology class 4-cocycle FDA extension with A[3] generator Sullivan’s Theorems at work inD=11 The Maurer Cartan formulation of the super Poincaré Lie algebra

  14. Minimal FDA with A[3] extension Cohomology class 7-cycle A[6] form extension Second Round

  15. D=11 Super Poincaré 3-form extension M2 - branes 6-form extension M5 - branes The complete FDA in D = 11 • Black generators = super Poincaré algebra • Blue generators = Cohomological extension yields Minimal FDA • Red generators = Curvatures =contractible subalgebra of FDA

  16. The Rheonomic Parametrization • Black forms = outer directions • Red curvature components = inner components • Blue forms = inner directions • Outer components = linear expression in terms of inner ones • THIS IS THE PRINCIPLE OF RHEONOMY • IT ENCODES ALL THE DYNAMICS OF SUPERGRAVITY

  17. The field equations are justconsistency conditions for…. THE RHEONOMIC PARAMETRIZATION

  18. FDAs in recent (and less recent) work and Open Problems. • Relation between FDAs and ordinary (super) Lie algebras (At the end do we deal with ordinary superalgebras?): • Frè & D’Auria (1982) • Azcarraga et al (2004) • Castellani 2005 • FDAs in compactifications of M - theory • Dall’Agata, Ferrara, D’Auria,Trigiante (2004-2005) • Fré & Trigiante 2005 • FDAs, the problem of gauge completion and supermanifolds (2006 work in fieri)

  19. Super Poincaré Reduction of FDA to a super Lie algebra Can we enlarge the original super Lie algebra sL ½ FDA in such a way that the non trivial cohomology classes become trivial? Example in D=11: ENLARGE by adding three new generators Bab , Ba1..a5,  Shows that B.s are dual to the central charges Z.s is dual to a new necessary spinor charge Q0

  20. Provided the parameters are : On this algebra A[3] is trivial • Frè & D’Auria 1981 (two solutions s= -1 , s=3/2) • Azcarraga et al 2004-2005 (one parameter solution)

  21. In this way: • The minimal FDA subalgebra M (vacuum symmetry) can be replaced by an ordinary (non semisimple) superalgebra sL0. • It would be nice if the gauging of the minimal subalgebra M (adding the contractible generators = curvatures) were just the gauging of sL0. • Unfortunately it looks that this is not the case. There is no rhenomic parametrization of the sL0 curvatures that yields the rheonomic parametrization of the complete FDA! • The FDA remains (so far) the proper algebraic structure encoding both the dynamics and the symmetries of SUGRA

  22. Setting the proposal and the problem The no go theorem • Becker & Becker hep-th 0010282 • Hull & Reid-Edwards hep-th 0503114 • Dall’Agata Ferrara hep-th 0502066 • Dall’Agata, D’Auria Ferrara, hep-th 0503122 • D’Auria Ferrara Trigiante hep-th 0507225 • Frè Class. Quant. Grav. 1(1984) L81 • Frè hep-th 0510068 • Frè & Trigiante 2006 Flux compactifications and FDA.s • There is a lot interest in Flux compactifications both of String Theory and M-theory • Relation between fluxes and gauge algebras Ggauge in D=4 effective SUGRA.s • There were claims that Ggauge could be a non trivial FDA, rather than a normal Lie algebra • Special interest in so named twisted tori, namely in compactifications on 7-dim group manifolds

  23. Kaluza Klein vectors 3-form expansion M-theory compactified on twisted tori = group-manifolds

  24. Covariant derivative The 3-form and a double elliptic complex D=4 space-time q-forms valued in Chevalley p-forms

  25. Expansion of F[4] and fluxes The analysis of closure dF[4] =0 relates to the theory of FDA.s, but let us consider first the solution ansatz in order to find Lorentz invariant vacua Three possibilities

  26. How the D=4 FDA could be non trivial Furthermore: But the flux should also be a solution of M-theory field equations!! This is the PROBLEM !!!!!!!!!!!!!!!

  27. Englert solution Three type of solutions • The flat type • M11 = Mink4£ M7 where M7 = Ricci flat • The Freund Rubin type • M11 = AdS4£ M7 where M7 = Einstein manifold • The Englert type • M11 = AdS4£ M7 where M7 = Einstein manifold with weak G2 holonomy and associated flux

  28. Einstein eq. Englert Equation Englert Manifolds i.e. Weak G2 Holonomy 7-manifolds admitting such a structure were named Englert spaces. This is equivalent to SO(7)+ de Sitter holonomy in 1984 parlance to weak G2 holonomy in modern parlance

  29. Weak G2 = SO(7)+ de Sitter holonomy The spin connection AB plus the vielbein eC define on any non Ricci flat 7-manifold a connection which is actually SO(8) Lie algebra valued and the spinors are sections of such an SO(8) associated bundle. The stability subgroup of the 8s of SO(8) is SO(7)+ SO(7)+ de Sitter or weak G2 holonomy implies that the manifold is Einstein The Flux

  30. Main negative result • There are no G7 Lie groups such that the above ansatz leads to a bona fide solution of M-theory with non vanishing flux gIKJL. • The token to prove the result is the reduction of the problem to a question of holonomy on the internal manifold M7 • Solutions with fluxes exist iff M7 has weak G2 holonomy (= modern jargon), if it is an Englert manifold (=old jargon) • There are no 7-dimensional Lie Groups of Englert type, i.e.of weak G2 holonomy. • (THEOREM in LIE ALGEBRA THEORY)

  31. where and + Killing spinors There are no groups, but there are several cosets of weak G2 holonomy The symmetry of these vacua is: The FDA can be reconstructed from the Maurer Cartan 1-forms of the supercoset Where are the missing 32 – 4 £ N theta.s corresponding to broken supersymmetries ? PROBLEM of SUPERGAUGECOMPLETION

  32. For instance for the compactification on which preserves 3 supersymmetries we conjectured: Conjecture under study:(A.P. Grassi & P.F.) and try to reconstruct FDA from the Maurer Cartan forms of the above Supercoset.

  33. Conclusions • The algebraic structure underlying Supergravity and M theory is after 30 years still not completely clear. • FDAs and the compact subalgebras of Kac Moody algebras (talk by H. Nicolai) are possibly the two sides of one medal that has still to be clarified. • Such clarification cannot fail to open new perspectives and tell us what really is the symmetry all of us have been gauging these last 30 years

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