String Theory and Nonrelativistic Gravity

Transcription

1 String Theory and Nonrelativistic Gravity Eric Bergshoeff Groningen University work done in collaboration with Ceyda Şimşek, Jaume Gomis, Kevin Grosvenor and Ziqi Yan A topical conference on elementary particles, astrophysics, and cosmology Miami, December 18, 2018

2 A Motivation

3 Holography Gravity is not only used to describe the gravitational force!

4 Non-relativistic Holography two approaches Keep general relativity in the bulk but take background geometry with non-relativistic isometries Christensen, Hartong, Kiritsis, Obers and Rollier ( ) Take non-relativistic gravity in the bulk Gomis, Ooguri (2001); Gopakumar, Bagchi (2009)

5 Outline Newton-Cartan Gravity

6 Outline Newton-Cartan Gravity String NC Gravity

7 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String

8 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality

9 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

10 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

11 4D Galilei Symmetries time translations: δt = ξ 0 but not δt = λ i x i! space translations: δx i = ξ i i = 1,2,3 spatial rotations: δx i = λ i j x j Galilean boosts: δx i = λ i t

12 Conventional Constraints symmetry generators gauge field # curvatures # time translations H τ µ τ µν = [µ τ ν] 6 space translations P A A E µ R A µν (P) 18 Galilean boosts G A A Ω µ 12 R A µν (G) spatial rotations J AB AB Ω µ 12 R AB µν (J) µ = 0,1,2,3;A = 1,2,3 R µν A (P) = 0 (18) : does only solve for part of Ω µ A,Ω µ AB

13 From Galilei to Bargmann (NC) the zero commutator [G A,P B ] = 0 implies that a massive particle with non-zero spatial momentum P B cannot by any boost transformation G A be brought to a rest frame [G A,P B ] = δ AB Z extra gauge field M µ and additional 6 conventional constraints: R µν (Z) = 0

14 Newton-Cartan Geometry The independent NC fields {τ µ,e µ A,M µ } transform as follows: δτ µ = 0, δe µ A = λ A B E µ B +λ A τ µ, δm µ = µ σ +λ A E µ A The spin-connection fields Ω µ AB and Ω µ A are functions of τ µ,e µ A and M µ

15 Two Degenerate Metrics Instead of one non-degenerate metric ĝ µν = ʵ Â Ê ν ˆB ηâˆb with  = 0,1,2,3, like in general relativity, NC gravity has two degenerate metrics, one for the time directions and one τ µν = τ µ τ ν h µν = E µ AE ν Bδ AB with τ µ E µ A = τ µ E µ A = 0 for the spatial directions. We can make a boost-invariant metric H µν at the price of giving up invariance under the central charge transformations.

16 NC Gravity couples to particles what about strings?

17 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

18 Comparing Limits The NR limit of a particle coupled to general relativity is finite provided that the particle is coupled to a (zero flux) gauge field ˆM µ which gives with ω in the NR limit ˆM µ = ωτ µ The string should be coupled to a (zero flux) 2-form gauge field ˆM µν with ˆM µν = ω 2 ǫ AB τ µ A τ ν B, A = 0,1 defining a string NC geometry with generalized clock functions τ µ A

19 Extended Objects General Relativity Every extended object couples to the same general relativity Nonrelativistic Gravity particles couple to NC gravity strings couple to string NC gravity extra circle is needed to obtain wound strings! Gomis, Ooguri (2001) Nonrelativistic gravity is not unique!

20 String Galilei Symmetries Gomis, Ooguri (2001); Andringa, Gomis, de Roo + E.B. (2012) D +1 flat indices { 2 longitudinal indices A D 1 transverse indices A longitudinal translations H A transverse translations P A string Galilei boosts G AB longitudinal Lorentz rotations M AB transverse spatial rotations J A B

21 Noncentral Extension [G AA,P B ] = 0 [G AA,P B ] = δ A B Z A The independent string NC fields are {τ µ A,E µ A,M µ A } longitudinal metric: transverse metric : τ µν τ µ A τ ν B η AB H µν E µ A E ν B δ A B +( τ µ A M ν B +τ ν A M µ B ) η AB

22 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

23 NR Limit Nambu-Goto Particle S NG = M } dτ { ẋ µ ẋ νĝ µν ˆM µ ẋ µ Ê µ 0 = ωτ µ + 1 ω M µ, Ê µ A = E µ A, ˆMµ = ωτ µ, M = ωm S NG (N.R.) = m 2 dτ ẋµ ẋ ν τ ρ ẋ ρ H µν

24 Nambu-Goto relativistic NG is non-linear in longitudinal and transverse embedding coordinates nonrelativistic NG is nonlinear in longitudinal embedding coordinates only

25 NR Limit Polyakov Particle S Pol. = 1 2 { dτ 1 Âẋ µ Ê ˆBẋ ν ν eêµ ηâˆb +M 2 e 2M ˆM µ ẋ µ} S Pol. (ω 2 ) = 1 2 = 1 2 dτ 1 e ω2[ τ µ ẋ µ me ] 2 dτ 1 e { λ(τ µ ẋ µ me) 1 4ω 2λ2} S Pol. (N.R.) = 1 2 dτ 1 e {ẋ µ ẋ ν H µν +λ ( τ µ ẋ µ me )} Solving for the Lagrange multiplier E.O.M. brings us back to the Nambu-Goto particle.

26 The Nonrelativistic Polyakov String h αβ = e α a e β b η ab e α a τ α A e α e α 0 +e α 1, ē α e α 0 e α 1 τ µ τ µ 0 +τ µ 1, τ µ τ µ 0 τ µ 1 S Pol. = T 2 [ hh d 2 σ αβ α x µ β x ν H µν +ǫ αβ( λe α τ µ + λē ) α τ µ β x µ] T 2 d 2 σǫ αβ α x µ β x ν B µν + 1 4π d 2 σ hr(h)φ This is the generalization of Gomis, Ooguri to a string NC background

27 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

28 Relativistic T-duality Buscher (1987,1988); Roček, Verlinde (1992) adapted coordinates: x µ = (y,x i ) k µ µ = y S parent = S Pol. ( α y v α ) T }{{} quadratic in v α! d 2 σǫ αβ ỹ α v β δs parent δv α = 0 v α is solved for in terms of dual coordinate ỹ G yy = 1 G yy R 1 R

29 Non-relativistic Longitudinal T-duality adapted coordinates: x µ = (y,x i ) with y longitudinal S parent = S Pol. ( α y v α ) T d 2 σǫ αβ ỹ α v β δs parent δv α = 0 v α is solved for in terms of dual coordinate ỹ and the Lagrange multipliers λ, λ λ, λ are no Lagrange multipliers anymore!

30 Dual Action S long. = T 2 ( hh ) d 2 σ αβ α x µ β x ν Gµν +ǫ αβ α x µ β x ν Bµν with x µ = (ỹ,x i ) and Gỹỹ = 0 : lightlike direction The longitudinal spatial T-dual of the NR string is the Polyakov string moving in a GR background with a lightlike direction The transverse spatial T-dual of the NR string is again a NR string with a transverse spatial isometry direction à la Buscher Buscher (1987,1988); Roček, Verlinde (1992)

31 Remarks T-duality rules are consistent with ambiguity nonrelativistic background general relativity with a null isometry direction is reminiscent to DLCQ relativistic string Rigid susy in Lorentzian curved spacetimes See, e.g., Cassani, Klare, Martelli, Tomasiello, Zaffaroni (2014)

32 Outline Newton-Cartan Gravity String NC Gravity The NR Polyakov String T-duality Comments

33 Nonrelativistic String Theory String NC Geometry is to NR string theory what Riemannian geometry is to relativistic string theory Nonrelativistic String Theory can be defined independent of any limit of relativistic string theory in this talk we discussed non-relativistic T-duality

34 Open Issues target space interpretation T-duality Does β-function calculation leads to consistent backgrounds? Target Space Action, Anyonic strings? Şimşek, Jaume Gomis, Grosvenor, Yan + E.B. (2018) cp. to Duval, Horvathy (2000); Jackiw, Nair (2000); Mezincescu, Townsend (2010) See also, Gußmann, Sarkar amd Wintergerst (2018) Double Field Theory? S. M. Ko, C. Melby-Thompson, R. Meyer and J.-H. Park (2015) Nonrelativistic holography?

35 Take Home Message Nonrelativistic String Theory can be studied on its own! Gomis, Ooguri (2001)