Solution-generating techniques in supergravity and their ... · All tensors and spinors can be...
Transcript of Solution-generating techniques in supergravity and their ... · All tensors and spinors can be...
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