Solution-generating techniques in supergravity

and their applications to AdS/CFT

Jerome Gaillard

University of Wisconsin-Madison

Based on collaboration with E. Conde, D. Elander, D. Martelli,C. Nunez, I. Papadimitriou, M. Piai and A.V. Ramallo

Outline

1 Motivations

2 Different classes of gravity duals

3 Solution-generating techniques

4 G-structures

5 Adding brane sources in supergravity

6 Chain of dualities

7 Non-abelian T-duality

8 Conclusions

Motivations

Studying strongly coupled field theories through gauge/gravitycorrespondence

Going beyond the best-understood cases like AdS5 × S5 andN = 4 SYM for example

Exploring the possibilities offered by brane sources

Deepening our understanding of the relation between geometryand field theory

Constructing the gravity dual of gauge theories with interestingnew features

non-trivial dynamics

fundamental matter

Different classes of gravity duals

True AdS spaces: dual to CFT

AdS5 × S5 dual to N = 4 Super-Yang-Mills Maldacena ’97

Asymptotically AdS spaces: dual to the RG flow of somefundamental theory

Describing the RG flow from N = 4 to N = 1∗ Yang-MillsPolchinski and Strassler ’00

Logarithmic UV deformation from asymptotically AdS: the RGflow does not terminate in the UV

Cascading theories Klebanov and Tseytlin, Klebanov and Strassler ’00

Wrapped-brane models: Kaluza-Klein compactification of ahigher-dimensional theory

Duals to effective field theories, require a UV completion toreach a CFT Witten ’98, Maldacena and Nunez ’00

Solution-generating techniques

Necessary because supergravity has many fields, and theequations of motion are non-linear

Generates complicated solutions starting from simpler ones

Relates apparently very different solutions

Apply known string dualities to solutions: T-duality, S-duality

Use geometric properties of supersymmetry: G-structures

Combine techniques to create many interesting new families ofsolutions

G-structures

Geometric formulation of supersymmetry

Supersymmetry implies the existence of some globally definedspinor

Preserving a given number of supersymmetry is equivalent toimposing a particular G-structure on the solution manifold

The system of first order equations obtained by requiring thesupersymmetry variations to be satisfied can be translated ingeometric terms

The presence of a G-structure comes with the existence ofglobally defined differential forms that obey some first orderequations

G -structures

All tensors and spinors can be decomposed in representations ofG

Existence of G -invariant tensors and spinors

Example: SU(3)-structure in six dimensions

Two-forms in adjoint representation 15 of SO(6)

15 = 1+ 3+ 3+ 8

Three-forms in representation 20 of SO(6)

20 = 1+ 1+ 3+ 3+ 6+ 6

Singlets in decomposition mark the existence of G -invariantforms

SU(3)-structure

Type IIB supergravity

Dual to N = 1 field theory in four dimensions

Warped background with metric (Einstein frame)

ds2 = e

2∆ds

21,3 + ds

26

Fluxes consistent with 4d Poincare invariance

Type IIB spinors of positive chirality

ǫ1 = ξ+ ⊗ aη+ + cc

ǫ2 = ξ+ ⊗ bη+ + cc

η†+η+ = 1

Calibrated cycles

Preserving some supersymmetry gives more control over thesetup

A way to characterise supersymmetric cycles is with calibrationforms

A calibration form φ verifies

dφ = 0 and φ|Σ ≤ Vol|Σ

for any cycle Σ

Calibrated cycles are the ones preserving supersymmetry

A cycle is calibrated if it verifies

φ|Σ = Vol|Σ

Calibrated cycles

In backgrounds with fluxes, one needs to consider generalisedcalibrations

They are not closed, but verify

dφ ∝ Fluxes

Generalised calibrated cycles are defined in the same way asbefore

If the background has a G -structure, generalised calibrations arerelated to G -invariant forms

Smearing

Large number of source branes requires backreaction

Need to solve for the action:

S = SSUGRA + Ssources

where Ssources corresponds to the DBI and WZ action for thesource branes

It can be written as

Ssources = −

∫

M10

(

φ− C ∧ eB)

∧ Ξ

where Ξ parameterises the distribution of the source branes

Smearing

This action results in the violation of the Bianchi identities

The violation is proportional to Ξ

In general, the distribution of sources will break the symmetriesof the original system

Such cases are very complicated to solve (coupled system ofnon-linear PDEs)

One simplification is taking the distribution Ξ to respect thesymmetries: it is called smearing

Using dualities

Goal: applying a series of string dualities to a known solution toget a new one

Combination of T-dualities in Minkowski space directions, lift toM-theory and boost with the eleventh direction

Determining step is the boost, that creates or modifies fields

T-dualities are just used to enforce Poincare invariance

Only effect on the manifold is a warping between the Minkowskiand internal parts

Modification of the RR and NS fluxes, as well as the dilaton

Duality transformations

Such a chain of duality transformations relates theMaldacena-Nunez (MN) solution to the Klebanov-Strassler (KS)one through the baryonic branch of KS Maldacena and Martelli ’09

MN background: D5-branes wrapped on a two-cycle inside theresolved conifold

KS solution: fractional D3-branes deforming the conifold

Baryonic branch background interpolates between MN and KS

Very similar story in 3+7 dimensions:

Interpolation between Maldacena-Nastase and a warpedG2-holonomy manifold JG, Martelli ’10

Field theory unknown

Rotation of the G-structure

Transformation not possible when adding sources:Problem with lift to M-theory

Can be reformulated in terms of G-structuresMinasian, Petrini, Zaffaroni ’09

To see the effect of this chain of dualities on the geometricstructure, look at the effect on the spinors

Creation of a phase difference between the two supergravityspinors

Introduces a new function in the G-structure related to thatphase

The description in terms of G-structures stays ten-dimensional,so is fully compatible with brane sourcesJG, Martelli, Nunez, Papadimitriou ’10

Field theory interpretation of this transformation

In the 4+6 dimensional case, move from MN to the baryonicbranch of KS, and all the way to KS

When considering D5-sources as flavours in MN, modifies KS tothe quiver:

SU(Nc + n + ns)× SU(n + ns)

where n and ns are the numbers of bulk and source D3-branesrespectively

This solution is undergoing both a cascade of Seiberg dualities,and a Higgsing process when going from the UV to the IR

Interpretation of the interpolating theory is more tricky

Field theory interpretation of this transformation

We can also think of starting by adding D3-brane sources in KS

They seem to blow up in the interpolating solution

In the interpolating solution, the nature of those sources as D5or D3-branes is unclear

They become D5-branes in the MN limit, but do not have thecorrect behaviour for interpretation as flavours

Multi-scale theory

Possibility of engineering multi-scale theories

Distribute sources in a finite range of the radial direction

In the 4+6 dimensional example, combination of Seibergdualities and Higgsing along the RG flow

Still a logarithmic deformation of AdS in the UV

Model for a field theory with tumbling dynamics, which is veryhard to study at strong coupling

Step towards models possibly interesting for phenomenology

Non-abelian T-duality

Abelian T-duality when the theory has a U(1) isometry

Goes from type IIA to type IIB supergravity and vice-versa

Non-abelian T-duality possible when the background has anon-abelian isometry group, e.g. SU(2) Sfetsos and Thompson ’10

Depending on the dimensionality (odd or even) of the isometrygroup, the type of supergravity changes or stays the same

Can lead to a solution of massive type IIA supergravity

Contrary to the previous generating technique, non-abelianT-duality radically changes the manifold of the solution

Non-abelian T-duality

Because of the big changes on the gravity side, big changesexpected on the field theory side

Could show a relation between extremely different field theories

Not a duality of the full string theory, so should not produce anexact duality between field theories, but can relate them in somemeaningful way

Can preserve supersymmetry, at least partially

Only requirement is a symmetry group, so potentially many newsolutions

Non-abelian T-duality

Again not applicable directly to backgrounds with sources

Need a geometric formulation

Important transformation of the spinors, which likely breaks theG-structure

If supersymmetry is preserved, existence of a G-structure

Has been shown in some cases to go from SU(3) to SU(2)structure

Not yet a general understanding of the effect of the non-abelianT-duality on the geometric structure

Conclusion

Using solution-generating techniques to find new supergravitysolutions

Understanding relations between different theories, on thegravity and the field theory side

Can be applied to theories with brane sources

Big potential for the non-abelian T-duality, because it can beapplied to many different solutions, and its effects on thebackgrounds are important

Link between non-abelian T-duality and geometry

Need a better understanding of the effect of thosetransformations on the sources