Introduction Conformal Bargmann structures & Schrödinger ... › ~bekaert › Conference_2010 ›...

Schrödinger manifolds

Christian DuvalCPT & UM

Non-commutative structures & non-relativistic (super)symmetriesTours, 22-24 June 2010

Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 1 / 27

1 Introduction

2 Reminder: Conformal structures & Poincaré metrics

3 Conformal Bargmann structures & Schrödinger manifolds

4 Homogeneous Schrödinger manifolds

5 References


Holography

AdS/CFT correspondence [Maldacena et al.] in nonrelativistic fieldtheory [Balasubramanian-McGreevy, Son] highlights the metric

g =1r2

(d∑

i=1

(dx i)2 + dr2 + 2dtds

)− dt2

r4 (?)

on a (d + 2,1)-dimensional Lorentzian spacetime.Key property: its isometries form the Schrödinger group ofnonrelativistic Galilei spacetime, coordinatized by (x1, . . . , xd , t).Our purpose: spell out the geometric origin of (?) in an entirelynonrelativistic framework.[C. Duval, P. Horváthy, S. Lazzarini, “Schrödinger manifolds”] (inpreparation)


Bargmann structures: a reminder

Lorentzian extension of Newtonian spacetime goes back to Eisenhart.

Definition [DBKP]A Bargmann manifold is a triple (M, g, ξ) where M smooth manifoldwith Lorentz metric g of signature (d + 1,1) & lightlike, covariantlyconstant, nowhere zero vector field ξ.

The quotient M = M/dir(ξ) becomes a Newton-Cartan spacetime.The closed 1-form

θ = g(ξ)

is the “clock” descending to absolute time axis T = M/ ker(θ).Newton’s field equations read

Ric(g) = (4πG%+ Λ)θ ⊗ θ

yielding a vanishing scalar curvature, R(g) = 0.


Examples of Bargmann manifolds

1 The canonical flat structure: M = Rn, with n = d + 2 coordinatizedby (x1, . . . , xd , t , s) with

g =d∑

i=1

(dx i)2 + 2dt ds & ξ =∂

∂s

2 The various pp-wave solutions where

g =d∑

i,j=1

gij(x, t)dx i dx j + 2dt ds − 2U(x, t)dt2 & ξ =∂

∂s

with canonical wave covector: θ = dt .3 Etc.


The “Schrödinger” group?


The Schrödinger equation

Let Fλ(M) be Diff(M)-module of λ-densities, loc. Ψ = f |Vol|λ withf ∈ C∞(M,C), & Lie derivative LλX f = X (f ) + λDiv(X )f , ∀X ∈ Vect(M).

Let (M, g) be a n-dimensional Ψ-Riemannian manifold. The Yamabeoperator is the conformally invariant ∆conf

g : F n−22n→ F n+2

2n, viz

∆confg = ∆g −

n − 24(n − 1)

R(g)

PropositionGiven Bargmann manifold (M, g, ξ) of dimension n = d + 2, the system

∆confg Ψ = 0 &

~iL

d2d+4ξ Ψ = m Ψ (1)

descends as the Schrödinger equation of mass m on associatedNewton-Cartan spacetime.


The Schrödinger symmetries

Denote by [Φ 7→ Φλ] the action of Diff(M) on Fλ(M). A symmetry ofthe Schrödinger equation is a Φ ∈ Diff(M) sth

∆confg Φλ = Φµ ∆conf

g & Lλξ Φλ = Φλ Lλξ

with λ = d2d+4 , and µ = d+4

2d+4 .

PropositionThe symmetries of the Schrödinger equation form the Schrödingergroup Sch(M, g, ξ) = Conf(M, g) ∩ Aut(M, ξ) of those Φ ∈ Diff(M) sth

Φ∗g = Ω2Φ g & Φ∗ξ = ξ (ΩΦ ∈ C∞(M,R∗+))

which hence permute the solutions of the Schrödinger equation (1).

This group descends to NC spacetime M as centerless Schrödingergroup. In the flat case, the latter may be identified with

Sch(Rd+1,1, ξ)/Centre ∼= (O(d)× SL(2,R)) n (Rd × Rd )


Conformal Bargmann structures

Clearly, the fundamental structure is conformal class [g] of Bargmannmetrics on extended spacetime M.

For any g, g ∈ [g], one duly has g(ξ, ξ) = g(ξ, ξ) = 0. To further insure∇ξ = ∇ξ = 0, one finds1

g ∼ g ⇐⇒ g = Ω2g & dΩ ∧ θ = 0 (2)

DefinitionA conformal Bargmann structure is an equivalence class (M, [g], ξ) ofBargmann manifolds for the equivalence relation (2).

It basically involves a conformal class of Lorentz metrics on a principalfibre bundle π : M → M with structural group (R,+) or U(1) withlightlike, parallel, fundamental vector field.

1The conformal factor must be a function of the time axis, T .Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 9 / 27

Formal theory of Poincaré metrics


Fefferman & Graham (1985, . . . , 2008)

In their quest of conformal invariants of a conformal structure ofsignature (p,q), FG have devised two equivalent constructs:

1 The ambient metric on a pseudo-Riemannian manifold ofsignature (p + 1,q + 1)

2 The Poincaré metric on a pseudo-Riemannian manifold ofsignature (p + 1,q)

Let us recall (and stick to) item 2.


The Poincaré metrics

Start with mfld M & conformal class [g] of metrics of signature (p,q),sth n = p + q > 1. Consider M+ sth M = ∂M+. Let r ∈ C∞(M+) verifyr > 0 in Int(M+), and r = 0 & dr 6= 0 on ∂M+ (defining function for M).

DefinitionWe say that (M+, g+) has (M, [g]) as conformal infinity if

r2 g+|TM ∈ [g]

DefinitionA Poincaré metric for (M, [g]) is a pair (M+, g+) where M+ = M × R+

(with ∂M+ = M × 0) sthg+ has [g] as conformal infinityRic(g+) = −kg+ (normalization condition: k = n))


The Poincaré metrics (II)

Poincaré metrics admit the local expression

g+ =1r2

n∑i,j=1

g+ij (x , r)dx idx j + dr2

It has been proved [FG] that a Poincaré metric exists and is unique2 forn odd provided

g+ij are even functions of r .

Some weaker conditions are needed to prove existence if n even.

2Up to diffeomorphisms fixing M.Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 13 / 27

Example: Einstein space Einn−1,1 = ∂(AdSn+1)

Archetype of Poincaré metric: Anti de-Sitter (AdS) metric associatedwith Einstein conformal structure (compactified Minkowski space).

Start with Rn+2, where n = d + 2, with metric of signature (n,2):

G =d∑

i=1

(dx i)2 + 2dxd+1dxd+2 + 2dxd+3dxd+4 (3)

Consider unit hyperboloid3

AdSn+1 = X ∈ Rn,2 |XX = −1

theng+ = G|AdSn+1

is Lorentzian, of constant sectional curvature.

3Shorthand notation: XX ≡ G(X ,X ).Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 14 / 27

Example: Einstein space Einn−1,1 = ∂(AdSn+1) Cont’d

r

RPn+1

AdSn+1

r=0

Einn−1,1

Figure: AdS and Einstein

One often viewsAdSn+1 ⊂ PRn+2 asprojectivization of ballB = X ∈ Rn,2 |XX < 0.Boundary of AdSn+1 is theEinstein space

Einn−1,1 = PQ

projectivization ofQ = Q ∈ Rn,2\0 |QQ = 0:conformal∞ of (AdSn+1, g+):

Einn−1,1∼= (Sn−1 × S1)/Z2


The Schrödinger group & null geodesics of Eind+1,1

Einstein space Eind+1,1 is compactified Minkowski spacetime Rd+1,1.It is a homogeneous space of O(d + 2,2) % metric G in (3). ProjectionO(d + 2,2)→ Eind+1,1 is given by4

A 7→ dir(Q) & Q = Ad+3

We haveEind+1,1 = O(d + 2,2)/CE(d + 1,1)

Think now of null geodesics of Eind+1,1 endowed with its canonical flatconformal structure.

4Notation: Ai = Aei , where (e1 · · · ed+4) is a basis of Rd+2,2 with Gram matrix G.Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 16 / 27

The Schrödinger group & null geodesics of Eind+1,1 (II)

Start with null slit tangent space NTQ \ Q of P,Q ∈ Rd+2,2 \0 sthPP = QQ = PQ = 0 & presymplectic 2-form σ = d(PdQ). Space ofnull geodesics of Eind+1,1 is (NTQ \ Q)/ ker(σ) symplectomorphic to5

OZ0 = O(d + 2,2)/H (4)

nilpotent (co)adjoint orbit of Z0 = P0 ∧Q0 (sth Z 20 = 0). The projection

O(d + 2,2)→ OZ0 is given by

A 7→ P ∧Q where P = Ad+2,Q = Ad+3

O(d + 2,2)dir(•Q0)

yyssssssssss Ad(•)Z0

%%KKKKKKKKKK

PQ OZ0

FactThe stabilizer is H = Sch(d + 1,1), the Schrödinger group

5Two copies of the slit tangent bundle of Sd+1.Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 17 / 27

The Schrödinger group Sch(d + 1,1) ⊂ O(d + 2,2)

The origin Z0 = P0 ∧Q0 of the orbit OZ0 chosen so as P0 = ed+2,Q0 = ed+3, i.e.,

Z0 =

0 0 ξ−ξ∗ 0 0

0 0 0

∈ o(d + 2,2) (5)

where ξ ∈ Rd+1,1 is null wrt the Bargmann metric6

g =d∑

i=1

(dx i)2 + 2dxd+1dxd+2

PropositionThere holds Sch(d + 1,1) = A ∈ O(d + 2,2) |AZ0 = Z0A

6We have put ξ∗ = g(ξ).Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 18 / 27

Do Poincaré metrics specialize to the case of

conformal Bargmann structures?


Defining Schrödinger manifolds

DefinitionA (Poincaré-)Schrödinger manifold for a given conformal Bargmannstructure (M, [g], ξ) is a triple (M, g, ξ) with g a Lorentz metric, ξ anowhere vanishing lightlike Killing vector field, and M = ∂M sth

1 ξ∣∣T∗M = ξ

2 g−1∣∣T∗M = µ ξ ⊗ ξ (normalization condition: µ = 1)

3 g+ = g + µ θ ⊗ θ (where θ = g(ξ)) is a Poincaré metric for (M, [g])

New geometrical object: the null Killing vector field ξ. ConformalBargmann structures: conformal infinity of Schrödinger manifolds.

Look for some examples!


Homogeneous Schrödinger manifolds

In a basis of Rd+2,2 = Rd+1,1 ⊕ R1,1 with Gram matrix

G =

g 0 00 0 10 1 0

any A ∈ O(d + 2,2) sth AZ0 = Z0A is of the form

A =

L aξ CB∗ b d−aξ∗ 0 e

∈ Sch(d + 1,1) (6)

where L ∈ L(Rd+2), B,C ∈ Rd+2, and a,b,d ,e ∈ R are constrained.

Examine now the last two column vectors!


Homogeneous Schrödinger manifolds (II)

PropositionFor all λ 6= 0, the manifolds

Mλ = Q = X + λY ∈ Rd+2,2 |XX = YY = XY − 1 = 0,Z0Y = 0

are connected, (d + 3)-dimensional, homogeneous spaces

Mλ∼= Sch(d + 1,1)/(E(d)× R)

The projection Sch(d + 1,1)→ Mλ is given by

A 7→ Q = Ad+4 + λAd+3

From Eq. (6) we get, specifically, Ad+3 ∈ im(Z0).

Remark

If λ < 0, then Mλ ⊂ AdSd+3(√−2λ) is an open submanifold.


Homogeneous Schrödinger manifolds (III)

Consider on Mλ (i) the induced metric gλ, viz

gλ(δQ, δ′Q) = δQ δ′Q, ∀δQ, δ′Q ∈ TQMλ

as well as (ii) the 1-form θ defined by

θ(δQ) = −Q Z0 δQ, ∀δQ ∈ TQMλ

These tensor fields of Mλ are clearly Sch(d + 1,1)-invariant.

Proposition

The following metrics on Mλ, namely

gλ,µ = gλ − µ θ ⊗ θ with µ ∈ R (7)

are Lorentzian if λ < 0. They furthermore admit a nowhere vanishingnull Killing vector field

ξ = g−1λ,µ(θ) (8)

The group Sch(d + 1,1) acts isometrically on (Mλ, gλ,µ) & preserves ξ.


Homogeneous Schrödinger manifolds (III)

Consider on Mλ (i) the induced metric gλ, viz

gλ(δQ, δ′Q) = δQ δ′Q, ∀δQ, δ′Q ∈ TQMλ

as well as (ii) the 1-form θ defined by

θ(δQ) = −Q Z0 δQ, ∀δQ ∈ TQMλ

These tensor fields of Mλ are clearly Sch(d + 1,1)-invariant.

Proposition

The following metrics on Mλ, namely

gλ,µ = gλ − µ θ ⊗ θ with µ ∈ R (7)

are Lorentzian if λ < 0. They furthermore admit a nowhere vanishingnull Killing vector field

ξ = g−1λ,µ(θ) (8)

The group Sch(d + 1,1) acts isometrically on (Mλ, gλ,µ) & preserves ξ.Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 23 / 27

Homogeneous Schrödinger manifolds (IV)

Let us investigate the limit λ→ 0 as a route to “null infinity”.

Indeed QQ = 2λ→ 0, i.e., Q ∈ Q, in the limit. Hence M ⊂ PQ. Nowξ = limλ→0 ξ remains nowhere vanishing. So7

M = PQ ∈ Q |Z0Q 6= 0

is canonically endowed with a conformal Bargmann structure (M, [g], ξ)and has topology

M ∼= (Rd+1 × S1)/Z2

At last, one proves

Theorem

The triple (Mλ, gλ,µ, ξ) is a Schrödinger manifold with (M, [g], ξ) asconformal infinity. Normalization conditions fix λ = −1

2 and µ = 1.

7Remember that ξ(Q) = Z0Q.Christian Duval CPT & UM (Aix-Marseille II) Schrödinger manifolds Tours, 23-06-10 24 / 27

Homogeneous Schrödinger manifolds (V)

Figure: Sch = Mλ & Barg = M


Homogeneous Schrödinger manifolds (VI)

We finish with the local expressions of homogeneous Schrödingerstructures. We have, locally,

Q =1r

x−1

2g(x , x) + λr2

1

∈ Mλ

where x ∈ Rd+2 & r ∈ R\0, and g is flat Bargmann metric of Rd+2.

The Schrödinger metric is therefore for arbitrary λ < 0, and µ ∈ R:

gλ,µ =1r2

(g− 2λdr ⊗ dr

)+ µ

θ ⊗ θr4 (9)

with g =∑d

i=1 dx i ⊗ dx i + dt ⊗ ds + ds ⊗ dt and θ = dt , wheret = xd+1 & s = xd+2. The distinguished null Killing vector field reads

ξ =∂

∂s(10)


References

Fefferman, Graham, “Conformal invariants”, Astérisque, 1985; “The ambientmetric”, arXiv:0710.0919v2 [math-DG]

Gibbons, Horváthy, ChD, “Celestial Mechanics, Conformal Structures andGravitational Waves”, PRD (1991)

Maldacena, Martelli, Tachikawa, “Comments on string theory backgrounds withnon-relativistic conformal symmetry”, arXiv:0807.1100 [hep-th]

Son, “Toward and AdS/cold atom correspondence: a geometric realization of theSchrödinger symmetry”, PRD (2008)

Balasubramanian, McGreevy, “Gravity duals for non-relativistic CFTs”, PRL(2008)

Leistner, Nurowski, “Ambient metrics of the n-dimensional pp-waves”,arXiv:0810.2903 [math.DG]

Hassaïne, Horváthy, ChD, “The geometry of Schrödinger symmetry innon-relativistic CFT”, Ann Phys (2009)

Blau, Hartong, Rollier, “Geometry of Schrödinger space-times I, II”,arXiv:0904.3304v2 [hep-th], arXiv:1005.0760v1 [hep-th]


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