Gaugings and other Supergravity Tools for p brane Physics Pietro Fré Lectures at the RTN School, Paris 2001 IHS

p-Brane Actions

The parameter  and the harmonic function H(y)

Electric and magnetic p-branes “Elementary”

Conformal branes and AdS space

AdS is a special case of a Domain Wall These two forms are related by a coordinate transformation ELECTRIC BRANE

Coordinate patches and the conformal gauge Conformal brane a=0

Randall Sundrum gravity trapping 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 Kaluza Klein expansion in non compact space

Positivity of the Wall Tension

The “dual frame” of Boonstra, Skenderis and Townsend We learn that although the AdS x S 8-p is not a solution of supergravity, we can notheless compactify on the sphere S 8-p, or other compact manifold X 8-p !!!

“Near brane” factorization in the dual frame

p-brane (D-p-1) - Cone An X 8-p compact manifold is the base of the transverse cone C( X 8-p ) In D=10 the p-brane splits the space into a d=p+1 world volume and a transverse cone C( X 8-p ) that has the compact manifold X 8-p as base. In some sense The transverse cone

Domain wall supergravity from “sphere reduction”

The DW/QFT correspondence of Boonstra Skenderis & Townsend

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? Which supergravity is it that accommodates the Domain Wall solution after the “sphere” reduction? It is a “gauged supergravity” It is a “gauged supergravity” But which “gauging” ? But which “gauging” ? Typically a non compact one. It is compact for AdS branes! Typically a non compact one. It is compact for AdS branes! What are the possible gaugings? What are the possible gaugings? These are classifiable and sometimes classified These are classifiable and sometimes classified How is the gauging determined and how does it reflect microscopic string dynamics? How is the gauging determined and how does it reflect microscopic string dynamics? ??? This is the research frontier! ??? 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

Before the gauging

The only exceptions

By compactification from higher dimensions. In this case the scalar manifold is identified as the moduli space of the internal compact 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 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 Two ways to determine G/H or anyhow the scalar manifold DUALITIES Special Geometries The second method is more general, the first knows more about superstrings, but the two must be consistent

Scalar cosets in d=4

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.

How to determine the scalar cosets G/H from supersymmetry

.....and symplectic or pseudorthogonal representations

How to retrieve the D=4 table

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!

Duality rotations 1

Duality Rotations 2

Duality Rotations 3

Duality Rotation Groups

The symplectic or pseudorthogonal embedding in D=2r

continued 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. D=4,8 D=6,10

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

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 The pseudo Majorana condition is responsible for the Usp(N) holonomy Usp(N) of the scalar manifold (N=# of supersymmetries) RealGeometry

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 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

Supergravity items in N=8, D=5

General Form of the Gravitino SUSY RULE

Very Special Geometry

The section X(  is the crucial item

Hypergeometry: Quaternionic or HyperKahler manifolds

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

General aspects of gaugings and susy breaking

The potential is determined by the fermion shifts

Integrability in D=4

Integrability in D dimensions

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!

1 st Example N=8,D=4 SUGRA Of this theory de Wit and Nicolai wrote the compact gauging in Hull introduced many non compact gaugings. In 1998 an exhaustive classification was shown. It coincides with Hull models and contraction thereof

Structure of the coset (Cremmer and Julia 1980)

and the Usp(28,28) structure..... Properties of the coset representative

The Susy Rules in this case

The electric Group SL(8,R) This is the essential algebraic datum

The gauged Cartan Maurer forms

The T tensors

The allowed irreducible tensors

Who is in, who is out in T = Of the many representations only two remain, for example the 28 is deleted in the above decomposition of the tensor product

The fermion shifts

The Result of the Classification It turns out that the T-identities can be reduced to algebraic identities t-identities on the embedding matrix. It turns out that the T-identities can be reduced to algebraic identities t-identities on the embedding matrix. These identities can be completely solved and once finds all possible gauge algebras These identities can be completely solved and once finds all possible gauge algebras These algebras are compact, non compact and also there are non compact non semisimple. They are classified by a signature These algebras are compact, non compact and also there are non compact non semisimple. They are classified by a signature

Let us define the CSO(p,q,r) algebras Singular signature

....continued In terms of these algebras we have all possible N=8 gaugings with their associated embedding matrix in SL(8,R)

The complete list Cordaro, Gualtieri,P.F., Termonia,Trigiante (1998) Boonstra, Skenderis & Townsend identify this with the near brane sugra of the D2 brane

Triholomorphic isometries

The triholomorphic moment map

The triholomorphic Poisson bracket This identity plays a crucial role in the construction of the gaugings both in D=4 and D=5 D’Auria, Ferrara, Fré (1991)

Conditions on the choice of the gauge group in N=2,D=5

.....continued

Gauging of the composite connections Since the scalar manifold is not a coset G/H we cannot apply the gauging via Maurer Cartan equation. Yet it has enough geoemtric structure to define a gauging for each connection in the game.

Ceresole & Dall’Agata’s result The study of this potential is in fieri. Non semisimple gaugings might be a corner to explore more carefully