O(D,D) Covariant Noether Currents and Global Charges in Double Field - - PowerPoint PPT Presentation

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O(D,D) Covariant Noether Currents and Global Charges in Double Field - - PowerPoint PPT Presentation

Nov 17, 2022 160 likes •368 views

O(D,D) Covariant Noether Currents and Global Charges in Double Field Theory Yuho Sakatani Seoul National University (SNU), Institute for Basic Sciences (IBS) based on arXiv:1507.07545, to appear in JHEP , with J-H. Park (Sogang U.), S-J.

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Yuho Sakatani

Seoul National University (SNU), Institute for Basic Sciences (IBS) based on arXiv:1507.07545, to appear in JHEP , with J-H. Park (Sogang U.), S-J. Rey, W. Rim (SNU, IBS) YITP workshop, 12 Nov. 2015

O(D,D) Covariant Noether Currents and Global Charges in Double Field Theory

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Motivation

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Double Field Theory (DFT)

Formal aspects of DFT have been studied in detail, but the applications are not studied well. We study Noether currents in DFT. DFT: manifestly O(D,D)-covariant formulation of Supergravity Standard coords. in D-dims. Dual coords. also have D-dims. T-dual “Gravitational” theory in 2D-dimemsions.

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Asymptotically Flat spacetime

ADM momenta in General Relativity

(asympt.) translational sym. [Iyer, Wald, 1994]

The Noether charge has the form,

“Komar potential” (general covariant) ADM momenta : (asympt.) const.

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ADM momenta in DFT (1/2)

■ Objective：

in the Double Field Theory. Asymptotically Flat Doubled Spacetime

(asympt.) translational sym. (const.) (const.)

translational sym. ! Find the expression for ADM momenta, like

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ADM momenta in DFT (2/2)

■ Expectation: ADM momenta & F1 charges will be unified by treating and on an equal footing. T-duality

winding momenta

DFT

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Double Field Theory

  • C. Hull and B. Zwiebach, JHEP 0909, 099 (2009),
  • O. Hohm, C. Hull and B. Zwiebach, JHEP 1008 008 (2010),…,
  • I. Jeon, K. Lee, J-H. Park, JHEP 1104, 014 (2011), ………
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Double Field Theory (1/3)

DFT action:

Manifestly O(D,D)-covariant formulation of Supergravity 2D-dimensional “Doubled Space”

  • Fund. fields: Generalized metric :

T-duality inv. dilaton: density

[Hohm, Hull, Zwiebach, 2010]

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Double Field Theory (2/3)

■ Gauge symmetry : generalized diffeo.

Gauge parameter:

■ Consistency of the theory In our work, we choose the canonical section.

  • Gen. Lie derivative:

(Indices are raised or lowered)

(strong constraint)

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Double Field Theory (3/3)

Gauge symmetry of DFT

D-dim Diffeo. B-field gauge transf.

DFT Conventional Supergravity (NS-NS sector)

■ 。

  • Diffeo. + B-field gauge transf.
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Stringy Differential Geometry

[I. Jeon, K. Lee, J-H. Park, 2011; Hohm, Zwiebach, 2012]

Covariant derivative : (tensor!) Projection:

Generalization of Einstein-Hilbert action

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O(D,D) Covariant Noether Currents and Global Charges in Double Field Theory

arXiv:1507.07545, to appear in JHEP , J-H. Park (Sogang U.), S-J. Rey, W. Rim, YS (SNU, IBS)

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Noether current (standard procedure)

Pre-symplectic potential

DFT action:

EOM (traceless part) EOM In particular, under a global sym. (Noether current) (“Einstein tensor”) EOM

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Boundary term

Our action: To make the Dirichlet problem well-defined, we add a boundary term to the DFT action: (Identically conserved) Noether current

No !

O(D,D)-covariant generalization of the Komar potential

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Noether Charge

Namely, the dual components of the ADM momenta are F1 charges !

■ Noether charge :

[see also C. Blair, arXiv:1507.07541, “Conserved currents of double field theory”] July 28th

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Extension

■ DFT + Cosmological const. + YM

[I. Jeon, K. Lee, J-H. Park, 2011]

gauge field: ■(Identically) conserved current

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Applications

  • 1. Pure Einstein gravity

(Our formula reproduces the well-known ADM energy.)

  • 2. Null wave background
  • 3. F1 background
  • 4. Non-Riemannian background (previous talk)
  • 5. Other examples

(R-N black hole, linear-dilaton background,…)

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Non-Riemannian background

F1-background : T-dualities along the -directions Non-Riemannian BG. In the conventional (super)gravity, we cannot calculate ADM momenta [K. Lee, J-H Park, ‘13]

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Summary

  • DFT is a manifestly O(D,D)-covariant formulation
  • f supergravity.
  • From the strong constraint,

the background fields are independent of . (i.e., there are isometries in the dual directions)

  • We calculated O(D,D)-covariant Noether currents

associated with the translational symmetries in asymptotically flat doubled spacetimes.

  • We showed that the Noether charges associated

with translations along the dual directions coincide with the string winding charges.

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Future directions

  • Extension to the Exceptional Field Theory

(U-duality covariant generalization), In EFT, there are many additional dual coordinates. We can unify All brane charges as the generalized ADM momenta

  • Black hole thermodynamics

Hawking temperature, Bekenstein-Hawking entropy in DFT/EFT ?