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Gaugings and other Supergravity Tools for p brane Physics. Pietro Fré Lectures at the RTN School, Paris 2001 IHS. p-Brane Actions. The parameter D and the harmonic function H(y). Electric and magnetic p-branes. “Elementary”. Conformal branes and AdS space.
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Gaugings and other Supergravity Tools for p brane Physics Pietro Fré Lectures at the RTN School, Paris 2001 IHS
Electric and magnetic p-branes “Elementary”
These two forms are related by a coordinate transformation AdS is a special case of a Domain Wall ELECTRIC BRANE
Conformal brane a=0 Coordinate patches and the conformal gauge
Randall Sundrum gravity trapping Kaluza Klein expansion in non compact space These potentials have a Volcano shape that allows the existence of normalizable zero mode describing the graviton in D-1 dimensions. The continuum Kaluza Klein spectrum contributes only a small correction to the D-1 dimensional Newton’s law Randall Sundrum
The “dual frame” of Boonstra, Skenderis and Townsend We learn that although the AdS x S8-p is not a solution of supergravity, we can notheless compactify on the sphere S8-p, or other compact manifold X8-p !!!
The transverse cone (D-p-1) - Cone An X8-p compact manifold is the base of the transverse cone C( X8-p ) p-brane In some sense In D=10 the p-brane splits the space into a d=p+1 world volume and a transverse cone C( X8-p ) that has the compact manifold X8-pas base.
This raises some basic questions and we have some partial answers: • Which supergravity is it that accommodates the Domain Wall solution after the “sphere” reduction? • It is a “gauged supergravity” • But which “gauging” ? • Typically a non compact one. It is compact for AdS branes! • What are the possible gaugings? • These are classifiable and sometimes classified • How is the gauging determined and how does it reflect microscopic string dynamics? • ??? This is the research frontier!
Supergravity bosonic items The interesting structures are produced by the gauging. This a superstructure imposed on the geometric structure of “ungauged “ supergravity
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
Scalar manifolds by dimensions.... 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 geoemtries can be derived by sequential toroidal compactifications.
and some comments on it Let us now have a closer inspection at the role of symplectic embeddings and duality transformations. They exist in D=4 and do not exist in D=5. Yet in D=5 there is a counterpart of this provided by the mechanisms of very special geometry that have a common origine: how to reconcile p-forms with scalars!
.......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.
This is the basic object entering susy rules and later fermion shifts and the scalar potential The symplectic caseD=4,8
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
Supergravity on D=5 and more scalar geometries We consider now the scalar geometries of D=5 Supergravity. Their structure is driven by a typical five dimensional feature The coupling of the multi CHERN SIMONS TERM All vector fields participate
The D=5, N=2 vector multiplets Real Geometry The pseudo Majorana condition is responsible for the holonomy Usp(N) of the scalar manifold (N=# of supersymmetries)
Hypermultiplets As it always happens, the conjugation properties of the fermions determine the restricted holonomy of the scalar manifold. We already know that the holonomy must have a factor Usp(2)=SU(2). Now we also learn about a factor Sp(2m,R).The result is Quaternionic manifolds
Gunaydin, Sierra and Townsend discovered in 1985.... Real Geometry defined in terms of these This is the graviphoton
N=8, D=5 Supergravity The ordinary Maurer Cartan 1-forms are replaced by gauged ones, when gauging
Triplet of HyperKahler 2-forms Identification of the SU(2) curvatures with the HyperKahler forms
Implications of Restricted Holonomy The restricted SU(2) x Sp(2m,R) holonomy implies this decomposition of the Riemann tensor and this stronger identity. They are essential for supersymmetry
The general concept of Killing spinor The fermion shifts contain a crucial informations about vacua. How are the fermion shifts determined? In terms of coset representatives (or analogues in the special geometries) and gauge group structure constants!
