A SUPERGRAVITY SOLUTIONS TOOL BOX Neil Lambert King’s College London Indian String Meeting HRI October 15, 2007

Table of Contents Introduction Supergravity basics Stability arguments Geometry of (generalized) Killing spinors Conclusions

Introduction Supergravity Solutions

Introduction Supergravity Solutions Bosonic Solutions

Introduction Supergravity Solutions Bosonic Solutions SupersymmetricSolutions

adS/CFT landscape gauge theory

Penrose limit Near horizon limit BMN adS/CFT landscape gauge theory

Many, many solutions (just think of GR) A least one 600+ page book [Stephani, Kramer, MacCallum, Hoensalaers, Herlt] Even many supersymmetric solutions: branes, brane intersections, branes wrapped on cycles,… Quite a few separate developments over the past few years:

pp-waves with 32, 30,29,28,… supersymmetries BMN - Penrose limits [Figueroa O’Farrill, Papadopoulos,…] Black rings and black Saturn in 5 dimensions horizons have components of topology S3 and S2x S1 [Emparan, Reall, Gutowski, Figuroas, Elvang,…] New adS x X solutions dual to to non-trivial CFT’s [Gauntlett, Sparks, Mateos,…] Hitchin’s Generalized geometry and N=1 mirror symmetry Calabi-Yau’s generalized to include effects of fluxes [Tomasello, Minasian, Waldram, Petrini, Grana,…] Stability of ‘fake’ supersymmetric solutions Adapt Witten-Nester arguments to non-supersymmetric solutions [Freedman, Townsend,…] Classification program General form of supersymmetric solution known in some theories (e.g. D=5 supergravity) [Gauntlett et al.; Papadopoulos et al.,…]

Too much to discuss here! We will review supergravity theory and its basic tools that are used in these studies Supersymmetry algebra and it’s closure Stability arguments of supersymmetric (and non-supersymmetric) solutions Geometry generalized Killing spinors G-structures and classification of solutions

Supergravity Basics The study of supergravity has been central to string theory development The maximal supergravities are low energy effective actions for the massless fields of string theory and M-theory Uniquely determined by supersymmetry and hence they contain all effects, peturbative and non-perturbative, to lowest order in derivatives. However they are not generally trusted in regions of high curvature or at high energy (here one needs String-Theory)

Supergravities are classical field theories which contain a spin-2 veilbien and its spin-3/2 superpartner: the gravitino Supergravities are classical field theories which contain a spin-2 veilbien and its spin-3/2 superpartner: the gravitino e a ¹ Ã ®

Supergravities are classical field theories which contain a spin-2 veilbien and its spin-3/2 superpartner: the gravitino Supergravities are classical field theories which contain a spin-2 veilbien and its spin-3/2 superpartner: the gravitino e a ¹ Ã ® tangent index: 0,…,D-1 spinor index: 1,…,2[D/2] Bose-Fermi degeneracy typically requires additional Bosons (and sometimes Fermions) Not possible beyond D=11

Thus one usually needs to include additional Bosonic fields: Next modify the supersymmetry rules: C ¹ 1 ; : p + F 2 = d ± e a ¹ = i 2 ² ¡ Ã C 1 : p + ? D X F ( ¢ ) r

Thus one usually needs to include additional Bosonic fields: Next modify the supersymmetry rules: C ¹ 1 ; : p + F 2 = d ± e a ¹ = i 2 ² ¡ Ã C 1 : p + » [ ] D X F ( ¢ ) r Essentially fixed Generalized connection on the spinor bundle

Start to construct an action by considering, at lowest order in Fermions, Taking the variation gives: Some further manipulations give: S F = 1 2 R d D x ¹ Ã ¡ º ¸ r ± S F = R d D x ¹ Ã ¡ º ¸ r ² ¡ ¹ º ¸ r ² = 1 2 [ ; ] ( R g ) + X D F ¢ f r o m [ D ¹ ; º ]

± Ã ± C You need to compare this against a Bosonic variation With some hard work and modifications of the anstaz you can find the action and the explicit form for the extra terms in as well as and any other Fermions. ± S B = 1 2 p ¡ g ( R ¹ º T ) ² Ã + F t e r m s ± Ã ¹ ± C ¹ 1 : p + :

For example consider 11D supergravity [Cremmer, Julia, Scherk] ± e a ¹ = i 2 ² ¡ Ã C º ¸ p 8 [ ] D + ( ½ ¾ ) F S = Z d 1 x 2 p ¡ g R F 4 ^ ? 6 C 3 ¹ Ã º ¸ D 9 ( ½ ¾ ¿ + ) :

For example consider 11D supergravity [Cremmer, Julia, Scherk] ± e a ¹ = i 2 ² ¡ Ã C º ¸ p 8 [ ] D + ( ½ ¾ ) F S = Z d 1 x 2 p ¡ g R F 4 ^ ? 6 C 3 ¹ Ã º ¸ D 9 ( ½ ¾ ¿ + ) : 1 2 ¹ Ã ¡ º ¸ r

Another important issue/tool arises because generically the supersymmetry algebra of supergravity doesn’t close off-shell Imposing closure therefore gives the (Fermionic) equations of motion Can be used to determine the all equations of motion from the supervariations, e.g. type IIB supergravity [Howe, Schwarz, West] More recently used to give information on higher derivative terms such as automorphic forms including exact non-perturbative corrections [Green, Sethi] [ ± 1 ; 2 ] X = t r a n s l i o + c ¡ L e z g u

Stability Arguments Many years ago Witten provided a proof of the positive mass theorem for pure gravity that was inspired by supergravity Subsequently refined by Nester Applied to obtain BPS bounds on supergravity solitions (p-branes) that are saturated by supersymmetric solutions [Gibbons, Hull] [Izquierdo, NL, Papadopoulos, Townsend ]

^ E = ² ¡ r One starts by defining the Nester tensor In a supergravity one finds that, on-shell, where are the supervariations of the additional Fermions Conversely one can determine the form of and the equations of motion by demanding that there is such a relation ^ E ¹ º = ² ¡ ¸ r D ¹ ^ E º = r ² ¡ ¸ P A Â Â A r ; Â A

The upshot of this is that You can always solve Witten condition Thus one finds with equality iff there is supersymmetry D ¹ ^ E = ¡ j i r ² 2 + P A Â ¡ i r ² = spatial slice H d S ¹ ^ E = R § D ¸ sphere at infinity r i ² = Â A

So what? One can write It turns out that and Thus we find a bound ^ E ¹ º = + P ² ( F ¢ ¡ ) ¹ ² ¡ º ¸ D H d S ¹ º E = ² b r y P ¡ ADM momentum P H ¹ ² ( F ¢ ¡ ) = b d r y Q charges H d S ¹ º ^ E = M ¡ p P Q 2 ¸

for which these arguments apply (at least for a limit set of fields) ‘Fake’ supersymmetry uses the fact that in non-supersymmetric theories one may still be able to find suitable definitions of for which these arguments apply (at least for a limit set of fields) Even in some supersymmetric theories there are alternative choices for which the argument goes through Thus non-supersymmetric solutions can be argued to be stable non-perturbatively (but of course assuming a supergravity appoximation). r ; Â A

V = ¡ ¤ For a simple example consider domain walls in D dimensions There can be more than one choice of W e.g.: no ‘supersymmetric’ domain walls but all of values of are ‘supersymmetric’ or only is ‘supersymmetric’ but there is a ‘supersymmetric’ domain wall V = ³ d W Á ´ 2 ¡ D 1 ( ) V = ¡ ¤ 2 W = q D ¡ 2 1 ¤ Á W = q D ¡ 2 1 ¤ c o s h ( Á ) Á = d Á r = W

Geometry of Generalized Killing Spinors A key point has been that the gravitino supervariation more or less controls the whole show. Closure gives equations of motion Controls the Nestor tensor and forms the centre of stability arguments Supersymmetric solutions are found by setting the Fermion supervariations to zero This gives a natural generalisation of the Levi-Civita connection to include other fields Understanding this connection gives a natural structure to classify and describe supersymmetric solutions

Ultimately both concern solutions to the same equation: So we are led to introduce the methods used to attempt to classify all supersymmetric solutions. Two approaches: Killing Spinors [Papadopoulos, Gran, Roost] G-structures [Gauntlett et al.] Ultimately both concern solutions to the same equation: ± Ã ¹ = r ² Kiilling spinor

Killing Spinor approach: One literally chooses a basis of spinors and proceeds to solve the Killing spinor equations Exploits the fact that these are linear equations Classify supersymmetric spaces by giving their Killing spinors Spinors have smaller Isotropy groups G inside Spin(D) and matches up with G-structures

T = ¹ ² ¡ d x ^ X F r ² = [ r ; ] ² = ! ( R ¡ + P d F ¢ ) G-Structures: Supersymmetry implies a solution to nowhere vanishing Integrability: Structure group of the Spinor bundle is reduced These spinors give rise to special tensors on spacetime r ¹ ² = [ r ¹ ; º ] ² = ! ( 1 4 R ¸ ½ ¡ + P d F ¢ ) T p = ¹ ² ¡ 1 : d x ^ X F q

Reduces the structure group of the frame bundle SO(n) G These preferred tensors are invariant under G give rise to a ‘G-structure’ on the manifold All tensors can then be classified by their representation under G Fluxes also decompose under G and must take on ‘preferred’ forms. These place constrains on the form of the metric Given a (non-null) Killing spinor the only equation of motion that one is required to solve are the Bianchi identities Classify supersymmetric solutions by listing their G-structures and the resulting constraints on the field strengths etc.. Also enables some explicit forms for metric

Mathematically a G-structure is a G sub-bundle of the frame bundle [Bryant;Joyce] In general the frame bundle is a GL(n) bundle existence of metric GL(n) O(n) existence of orientation O(n) SO(n) existence of a complex structure SO(2n) SU(n) N.B. the existence of these structure does not imply that the Levi-Civita holonomy is reduced to G. This only happens if there is no torsion

D ´ = The classic example of this is a Calabi-Yau ´ ´ ° = ! ! Spacetime is of the form Mink x X6 Where X6 is a 6-dimensional manifold which admits a single Killing Spinor (Levi-Civita connection): Here is a six-dimensional commuting spinor We can construct covariantly constant tensors So we find a constant (that we set to one) and a covariantly constant 3-form These are the ingredients of an SU(3) holonomy manifold D i ´ = ´ ´ T ° i j k = ! ! 3

D ´ = The classic example of this is a Calabi-Yau ´ ´ ° = ! ! Spacetime is of the form Mink x X6 Where X6 is a 6-dimensional manifold which admits a single Killing Spinor (Levi-Civita connection): Here is a six-dimensional commuting spinor We can construct covariantly constant tensors So we find a constant (that we set to one) and a covariantly constant 3-form These are the ingredients of an SU(3) holonomy manifold D i ´ = ´ ´ T ° i j k = ! ! 3

d ! = ¡ N ? ´ ´ ° = ! r ´ = D ¡ N ° F = N d x ^ p 2 3 1 Another simple example consists of Freund-Rubin solutions in M-theory Here spacetime is adS4 x X7 Where is a seven-dimensional commuting spinor Again we construct the tensors: Now we find a constant and a 3-form such that This is called a weak G2 structure; examples include S7 If N=0 one has a G2 holonomy manifold explicitly determines the metric F = N d x ^ 1 2 3 r i ´ = D ¡ p 2 4 N ° ´ ´ T ° i j k = ! d ! 3 = ¡ p 2 1 N ? ! 3

What has been classified? All supersymmetric solutions of minimal five-dimensional supergravity: Similar to M-theory From a single spinor one finds one scalar, one 1-form and three 2-forms Define an SU(2) structure (non-null case) For the null case one finds an R3 structure S = R d 5 x 1 2 p ¡ g 4 F ^ ? + 6 3 A Symplectic spinor a,b=1,2 ¹ ´ a b " ° º

One finds an explicit form for the fields With the equations Hyper-Kahler d s 2 = ¡ f ( t + ! ) 1 M 4 F p 3 [ ] G § ? d G + = D 2 f 4 9 ( )

In 11 dimensions one knows the classification with a single spinor SU(5) structure in the non-null case Kμ is a Killing vector Unfortunately M10 and F4 are not particularly constrained K ¹ = ² ¡ º ¸ ½ ¿ d s 2 = f ( t + ! ) ¡ 1 M

Maximal (32) supersymmetries are fully classified For example in 11 dimensions: Mink11, adS4 x S7, adS7 x S4, pp-wave For example in type IIB: Mink10, adS5 x S5, pp-wave Papadopoulos et al. have a classification of all type I supersymmetric backgrounds (gμν,B, φ) Cases with less supersymmetry not fully classified No solution (at least locally) with 31 supersymmetries! More than 24 supersymmetries implies the spacetime is homogeneous

Conclusions There has been a variety of fruitful developments and suprises in supersymmetric supergravity solutions We have reviewed some elementary aspects of supergravity that have played a central role: Construction: how does a supergravity work? Role of the Fermionic supervariation Stability proofs Classification of supersymmetric solutions We hope these ‘tools’ will be useful for further developments