Gravity and the planar spin-2 Schr odinger equation Eric Bergshoeff - - PowerPoint PPT Presentation

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Gravity and the planar spin-2 Schr odinger equation Eric Bergshoeff - - PowerPoint PPT Presentation

Dec 13, 2022 295 likes •705 views

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Gravity and the planar spin-2 Schr odinger equation Eric Bergshoeff Groningen University work done in collaboration with Jan Rosseel and Paul Townsend workshop

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Gravity and the planar spin-2 Schr¨

  • dinger equation

Eric Bergshoeff

Groningen University

work done in collaboration with Jan Rosseel and Paul Townsend

workshop on Exceptional Field Theories, Strings and Holography

Mitchell Institute, College Station April 25 2018

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Motivation

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Holography

Gravity is not only used to describe the gravitational force!

Christensen, Hartong, Kiritsis, Obers and Rollier (2013-2015)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Supersymmetry

supersymmetry allows to apply powerful localization techniques to exactly calculate partition functions of (non-relativistic) supersymmetric field theories

Pestun (2007); Festuccia, Seiberg (2011), Pestun, Zabzine (2016)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Condensed Matter

Effective Field Theory (EFT) coupled to NC background fields serve as response functions and lead to restrictions on EFT compare to Coriolis force

Luttinger (1964), Greiter, Wilczek, Witten (1989), Son (2005, 2012), Can, Laskin, Wiegmann (2014) Jensen (2014), Gromov, Abanov (2015), Gromov, Bradlyn (2017)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

NC Geometry in a Nutshell

  • Inertial frames: Galilean symmetries
  • Constant acceleration: Newtonian gravity/Newton potential Φ(x)
  • no frame-independent formulation

(needs geometry!)

Riemann (1867)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Galilei Symmetries

  • time translations :

δt = ξ0 but not δt = λixi !

  • space translations :

δxi = ξi i = 1, 2, 3

  • spatial rotations :

δxi = λi j xj

  • Galilean boosts :

δxi = λit

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

‘Gauging’ Poincare

symmetry generators gauge field parameters curvatures space-time transl. PA EµA – RµνA(P) Lorentz transf. MAB ΩµAB ΛAB(xµ) RµνAB(M) Imposing the constraint Rµν

A(P) ≡ 2∂[µ Eν] A − Ω[µ AB Eν] B = 0

(‘zero torsion’) and assuming that EµA is invertible the spin-connection field ΩµAB becomes a dependent field : ΩµAB → ΩµAB(E)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

‘Gauging’ Galilei

symmetry generators gauge field curvatures time translations H τµ τµν = ∂[µτν] space translations Pa eµa Rµνa(P) Galilean boosts G a ωµa Rµνa(G) spatial rotations Jab ωµab Rµνab(J) Imposing Constraints Rµν a(P) = 0 : does only solve for part of ωµa , ωµab

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Absolute Time

τµν ≡ ∂[µτν] = 0 → τµ = ∂µτ ∆T =

  • C

dxµτµ =

  • C

dτ is path-independent

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

From Galilei to Bargmann

the zero commutator [Ga, Pb] = 0 implies that a massive particle with non-zero spatial momentum Pb cannot by any boost transformation Ga be brought to a rest frame ⇒ [Ga, Pb] = δabM → extra gauge field mµ

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The NC Transformation Rules

The independent NC fields {τµ, eµa, mµ} transform as follows: δτµ = 0 , δeµa = λab eµb + λaτµ , δmµ = ∂µσ + λa eµa The spin-connection fields ωµab and ωµa are functions of τµ , eµa and mµ

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

What about the dynamics ?

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

From General Relativity to NC gravity

Poincare ⊗ U(1)

‘gauging’

= ⇒ GR plus ∂µMν − ∂νMµ = 0

contraction

⇓ ⇓

the NC limit

Bargmann

‘gauging’

= ⇒ Newton-Cartan gravity

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Contraction Poincare

  • PA, MBC
  • = 2 ηA[B PC] ,
  • MAB, MCD
  • = 4 η[A[C MD]B]

P0 = 1 2ω H , Pa = Pa , A = (0, a) Mab = Jab , Ma0 = ω Ga Taking the limit ω → ∞ gives the Galilei algebra:

  • Pa, Mb0
  • = δabP0

⇒

  • Pa, Gb
  • = 0
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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Contraction Poincare ⊗ U(1)

  • PA, MBC
  • = 2 ηA[B PC] ,
  • MAB, MCD
  • = 4 η[A[C MD]B]

plus Z P0 = 1 2ω H + ω Z , Z = 1 2ω H − ω Z , A = (0, a) Pa = Pa , Mab = Jab , Ma0 = ω Ga Taking the limit ω → ∞ gives the Bargmann algebra including Z:

  • Pa, Mb0
  • = δabP0

⇒

  • Pa, Gb
  • = δab Z
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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The NC Limit I

Dautcourt (1964); Rosseel, Zojer + E.B. (2015)

STEP I: express relativistic fields {EµA, Mµ} in terms of non-relativistic fields {τµ, eµa, mµ} Eµ0 = ω τµ + 1 2ω mµ , Mµ = ω τµ − 1 2ω mµ , Eµa = eµa constraint : ∂[µτν] = 1 2ω2 ∂[µmν]

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The NC Limit II

STEP II: substitute the expressions into the transformation rules and the e.o.m. and take the limit ω → ∞ ⇒

  • the NC transformation rules are obtained and agree with

the gauging procedure

  • the NC equations of motion are obtained

Note: the standard textbook limit gives Newton gravity

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The NC Equations of Motion

´ Elie Cartan 1923

The NC equations of motion are given by τ µeν

cRµν c(G)

= R0c

c(G) = 0

1 τ µeν

cRµν ca(J)

= R0c

ca(J) = 0

a eµ(aeνcRµν cb)(J) = R(accb)(J) = 0 (ab)

  • there is no known action that gives rise to these equations of motion
  • after gauge-fixing τµ = δµ,0, eµa = δµa and m0 = Φ the 4D NC

e.o.m. reduce to △Φ = 0

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

what about non-relativistic matter?

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Motivation

special feature FQH Effect: existence of a gapped collective non-rel. parity non-invariant helicity-2 excitation, known as the GMP mode

Girvin, MacDonald and Platzman (1985)

recent proposal for a non-relativistic spatially covariant bimetric EFT describing non-linear dynamics of this massive spin-2 GMP mode

Haldane (2011); Gromov and Son (2017)

in a linearized approximation around a flat background this gives rise to a single spin-2 Planar Schr¨

  • dinger Equation

i ˙ Ψ + 2 2m∇2Ψ = 0

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Key Question

Rosseel, Townsend + E.B., to appear in PRL

has this single helicity 2 Planar Schr¨

  • dinger Equation

a (massive) gravity origin?

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The ‘force limit’ of spin 0

1 c2 ¨ Φ − ∇2Φ + mc

  • 2

Φ = 0 Take the non-relativistic limit c → ∞ keeping λ = /mc fixed → ∇2Φ = 1 λ2 Φ no massive spin 0 particle !

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

The ‘particle limit’ of complex spin 0

1 c2 ¨ Φ − ∇2Φ + mc

  • 2

Φ = 0 To avoid infinities we redefine Φ = e− i

(mc2)tΨ

so that the Klein-Gordon equation becomes − 1 2mc2

  • i d

dt 2 Ψ − i ˙ Ψ − 2 2m∇2Ψ = 0 and the c → ∞ limit yields the Schr¨

  • dinger equation

i ˙ Ψ + 2 2m∇2Ψ = 0

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

General Feature

  • ne complex massive helicity mode

⇔

  • ne Schr¨
  • dinger Equation
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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Alternative Particle Limit of 3D Real Proca

  • make time-space decomposition Aµ =
  • A0,

A

  • eliminate auxiliary field A0
  • rescale

A → B and define B =

1 √ 2

  • B1 + iB2
  • ⇒

L = 1 c2 ˙ B⋆ ˙ B + B⋆∇2B − mc

  • 2

B⋆B redefine B = e−imc2tΨ[1] : breaks parity ⇒ i ˙ Ψ[1] + 2 2m∇2Ψ[1] = 0 single planar spin-1 Schroedinger equation

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

From Spin-1 to Spin-2

Aµ: 3 = 1+2 under spatial SO(2): A0 and A1 + iA2 fµν with ηµνfµν = 0: 5=1+2+2 under spatial SO(2): f11 + f22, f01 + if02 and 1 2

  • f11 − f22
  • + if12

⇒ single planar spin-2 Schroedinger equation

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Towards Interactions: special features of 3D

  • J. Rosseel, P. Townsend + E.B., work in progress
  • ‘taking the square-root’:

− m2 = O(m)O(−m) with [O(m)]µ

ρ = ǫµ τρ∂τ + mδµ ρ

  • ‘boosting up the derivatives’:

∂µAµ = 0 → Aµ = ǫµνρ∂ρBσ

  • ‘CS-like’ formulation:

L = 1

2grsar · das + 1 6frstar ·

  • as × at

r = 1, · · · , N

  • take real limit or complex limit followed by self-duality truncation?
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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Non-relativistic 3D Chern-Simons Like Gravity

  • The 3D Galilei and Bargmann algebras do not allow an invariant

bilinear form

  • Precisely in 3D there exists a so-called Extended Bargmann Algebra

with two central extensions and an invariant bilinear from. The second central extension is related to spin

Jackiw, Nair (2000)

  • can one use two such algebras to construct a CS-like bi-metric

gravity theory describing the non-linear dynamics of a massive spin 2 particle instead of a massive deformation of Poisson’s equation?

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Outline

Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

This Talk

  • Does the non-relativistic limit of some 3D Relativistic Gravity model
  • r the direct construction of a CS-like gravity theory based upon

some non-relativistic algebra give the fully-covariant completion of the EFT proposal for the GMP mode in the FQE Effect?

Gromov and Son (2017)

  • this may lead to interesting connections between 3D gravity and the

FQHE concerning

  • higher derivatives
  • higher spins

Golkar, Dung Xuan Nguyen, Roberts, Son (2016)

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Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments

Take Home Message

Applied Newton-Cartan Geometry leads to fruitful interactions between Condensed Matter Physics, Mathematics, Gravity, String Theory and even Engineering !