Homogeueooskirewaticalspace.tn# ( Bologna , 6/6/2018 ) based on 1¥!! : off}%I 1802.04048 with Tomasz Andrzejewski

work. impugners with Stefan Pvohazka and Ross Grassi e

Several motivations : . explore new holographic limits of ADSICFT. generalisation of supersymmetry and/or supergravity beyond the arena of lorentzian geometry

Sowedraeuatispersonae

we start with the deEHerspacetiwI in Dtl dimensions : Lorentzian manifolds with nonzero constant sectional

curvature :

• Adspt I C R "

soft . . . + xp - xD +2, . xp +22 = - R2 s o C anti deserter )

• dSD+ ,C IRD " '

'x,'t . - - xx ,5txD+ ,

- x ⇐ = + R2 > O C de Sitter )

The lie algebra of isometries is so CD, 22 for Ads

,z+ ,and CD -11,1 ) for dSp+, .

Taking the curvature → 0 limit ( equivalently ,R -7 A ) of Adsit , or dSD+ , results in Miukowsluspa@tin#,

which is an affine spaceIAD "

with a flat metric g = deft - - - tdxp - idk ,5+,

C introducing the speed of light c)Ads•

C M We can now take limits in which c→ a ( uonnelativistic ) or c → o ( ultra relativistic ) .

• 5-

as/

If PEA" "

, (Tp IAD"

, gp ) = ( IR" "

, ? ) > L c light come )( Lorentzian

✓ inner product00

q L Rft - - . txp - ERDE, = O

NI : C and G are not,

- - - -. I

Lorentzian manifolds .

. -

asinine 've 'IEI4 EH -

a carnelian structure-

-- -

- .

-

Carroll - M I N K O W s K I - Galilean

If ( M , g) is a loreutgian manifold , ( Tpm , gp) a ( RM , y ) and so we

Ads C Adshave a ligature and we can take the non - and ultra - relativistic

• •

° M limits .lu particular for (A) dSD+ ,

we obtain Carroll & galilean• •

ds C ds versions of (A) ds .

••

•

Ads G(A) ds C & (A) ds G are reductive homogeneous spaaetimes of kinewatical

•

G lie groups .

Their canonical connections ( those with zero Nomiau maps )•

are not flat and the flat limit gives G & C.

ds G

All spacetime in the above diagram are symmetric homogeneous spaces of kinewatical lie groups . The canonical

connection is torsion - free , but only flat for M,C

,G

.

We want to clarify homogeneous space times of khewatical

he groups .we will work locally in terms of their lie algebraic data

.

Kinewatical lie algebras and their homogeneous spacetime

Def A kinewatical lie algebra ( with D - dim 'l space isotropy ) Is a real tie algebra 9 of dim ICDTDCDTZ )

satisfying 0 g > r = I (D)← 1- dinil

trivial Classification : D= 0 F ! I - dim 'l LAand @ g I-modr +0 2 Yes r . mod

D= 1 Bianchi ( 1898 )D - dimtlvector nep AT -7 S ← D= 2 JMFT Andrzejewski (

'18 )

of r FIFTY { nz ✓ → ✓ ← D= z Baoytnuyb + ( Levy - Leblond ) ( ' 86 )rise

to more KLAS.

D > 3 JMF ('17 )

An aristotelian lie algebra Is a neal LA of dim'

ZDCD +13 + 1 satisfying 0 9 > r = see (D)

Typically ,aristotelian lie algebras are quotients of KLAS by a uectouae ideal

.

% 9 Fmod" * ✓ • S

The clarifications in D 72 can be done via deformation theory . Every KLA is a deformation of the

static KLA : the unique KLA where ZVTOS Is an abelian ideal ( and hence S is central. )

For generic D, one fends the following KLAS : • static • Newton - Hooke ( 72+-1 • a family yefi ,

I ] where

• galilean • se CD , 2) j→ -1 Is 72 -

• Carroll • SI ( Dtl , 1) • a family X 7,0 where

• Porn call'

• Sg ( D -12 ) X → 0 is Nt• euclidean

• a KLA where adfs ,- ] is notdiagonal is able

We are interested in the"

binewatical spacetime"

: @+17. dimensional homogeneous spaces of hinewatecal lie groupsG of the form M=G/µ where the LA g of G Is hinewatical and the tie sobalgebra tysgcorresponding to the ( closed ) subgroup H is of the form Y = r @ V ( as r - mod ) .

we work Infinitesimally interns of the lie pair ( 9 ,b ) and we clarify these lie pairs up

to

the natural equivalence : ( 9 . ,b .) ~ ( 92

,bz ) if there exists Y : 9. ⇒ 92 with %

,

: hp # 52 .

in practice , me use the clarification of KL As to fix a KLA g and clarify tie pains ( 9,4 ) subjectto the equivalence ( 9

,b.) ~C 9. bz ) If F YEAH (g) at

. 4/4,

:b,⇒ ba .

A lie pain ( g , b ) is effective if y does not contain any nontrivial ideal of g.Equivalence dames of effective lie pairs are in bijectiue correspondence with local iso . dames of homogeneousspacetime

.

Notice that there may be won - equivalent lie pairs ( 9,4 .)

,(9,52) with the same 9 .

For example , g = @ C Dti,1 ) has at least two

, because dspt , and HDH have g as LA of Isometrics.

Non - effective brnewatical lie pairs have a rectorial ideal I and ( G/I , YI ) describes an aristotelian" "

space time.

roves r

The result of the clarification ( for generic D) is obtained by adding some more spacetime to our

carneliandramatis personae .

NR : non - reductive homogeneous spacetime of se ( DH ,i ) NR Ads C

woontsiadsm•

• •

s : static aristotelian ( all limit To it ) C M × >,°• •

TS : torsional static aristotelian dsc ds•

•x > 0@DSG)× : one - parameter family X 70 . ( ADSG is X⇒ ) s

•

Ads G torsional• ( X=o )all torsional except for X=o •

• G( ds G)

y: one . parameter family VEEI , 1] . ( DSG is r= - I )

tasnstoteliot

.

all torsional except for y= -1 .

ds

¥( r= - i ) )

Newton-

CartedVEG ,

1]ZEEI ,

1 ] torsional

Some details not in the talk

In writing down KLAS,it is convenient to choose a standard basis Jab= - Jba for r

,and Ba

, Pa for ZV and H

for S.

The [ J ,* ] brackets are the same for all KLAS and hence they are distinguished by the remainingbrackets

.

When describing he pairs ( 9 , b ) we always choose a basis for the KLA in which 4 is spanned

by (Jab ,Ba ) for binewatical space times or ( Jab ) for aristotelian spacetime .

the homogeneous spacetime

are then described by uniting the brackets of Ba,

Pa,H in this basis

.

The homogeneous spacetime in the

talk can be described locally as follows :

Aristotelian Newton - Cartan

static F ,- ] =°

• galleon [ H , B) = - P

• torsional static [ H ,P ] =P • ( Ads G)× [ H ,B]= - P [ H ,

P ]= (1+2/2) B + ZXP

• ( ds G)y [ H ,

B ] = - P TH , P ] = VB + ( 1 - V ) P

famollian• Carroll [ B ,P]=H

• Ads C [ B. P ]=H [ H ,P]=B [ p ,P]=J

• ds C [ B ,p]=H [ H ,P ] = - B [ P ,P]= - J

• NR [ B ,P]=H + J [ H ,P ] = - P [ H ,B]=B

Loneutgian

• Miuhowskie [ H ,B]= - P [ B. B) =J [ Bcp ]=H

• de Sater [ H ,B]= - P [ B ,B]=J [ B. P ]=H [ H , P ]= - B [ P ,P]= - J

• Antide Siter [ it ,B]= - P [ B ,B]=J [ B. P ]=H [ H ,P]= B [ P ,p ] = J