Non-associative Deformations of Geometry in Double Field Theory - - PowerPoint PPT Presentation

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Non-associative Deformations of Geometry in Double Field Theory - - PowerPoint PPT Presentation

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Max Planck Institut f ur Physik M unchen based on

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Non-associative Deformations of Geometry in Double Field Theory

Michael Fuchs

Max Planck Institut f¨ ur Physik M¨ unchen based on JHEP 04(2014)141 by R. Blumenhagen, MF, F. Haßler, D. L¨ ust, R. Sun

29th IMPRS Workshop July 7th, 2014

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Outline

  • Deformation Quantization
  • T-Duality and non-associativity in String Theory
  • Non-associative Deformations of Geometry in DFT
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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Canonical Quantization

Replace Poisson-bracket by commutator: {x, p}PB = 1 → [x, p] = i Fulfilled for instance by operators: ˆ p = −i ∂

∂x

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Deformation Quantization:

No operators, instead change multiplication law: Replace f · g by f ⋆ g := f · g + i 2 ∂xf ∂pf T 0 1 −1 ∂xg ∂pg

  • Insert coordinate and momentum:

x ⋆ p = x · p + i

2

p ⋆ x = p · x − i

2

   [x, p] = x ⋆ p − p ⋆ x = i

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Take home message: Commutation relations realized by deformed product 9709040 f ⋆ g := f · g + i 2 ωij ∂if ∂jg + O(2)

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

String Theory

Fundamental objects not points, but strings Strings must live in 10D → compactify!

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

T-Duality

Closed strings wind around compactified dimensions: momenta pi

T-Duality

← → winding momenta ˜ pi

  • coordinate xi

T-Duality

← → winding coordinate ˜ xi

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Non-geometric Fluxes

T-Duality mixes G and B ⇒ change of geometry Hijk

Tk

← → Fij k

Tj

← → Qi jk

Ti

← → Rijk

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Non-associative Geometry

Blumenhagen, L¨ ust, Plauschinn et alii: [xa, xb] ∼ = Rabcpc “fuzzy” geometry due to Heisenberg uncertainty: ∆xa∆xb ∼ = [xa, xb] = 0

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

non-vanishing Jacobi identity! ˆ = non-associative operators! Not possible in ordinary quantum mechanics! Deformed product vanishes for observables by momentum conservation!

1106.0316 by Blumenhagen, Deser, L¨ ust, Plauschinn, Rennecke

Our work: Investigate in double field theory how non-associativity vanishes!

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Double Field Theory

Combine normal

winding

  • in 2D vector

PM = pi ˜ pi

  • ∂M =

∂i ˜ ∂i

  • XM =

xi ˜ xi

  • ⇒ Coordinates and winding on equal footage!

BUT: Constraints needed for consistency!

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Non-associative Deformations of Geometry in DFT

Translate deformed product into DFT: f △g △h = f · g · h + FABC

contains H,f,Q,R

∂Af ∂Bg ∂Ch We found: Deformation vanishes by consistency constraints! Deformation in physical situations (action) ˆ = integration:

  • DFT

FABC DAf DBg DCh PI = −

  • DFT

ZAB

  • Bianchi ZAB=0!

f DAg DBh

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Another deformation:

DFT allows for another deformation: f △g △h = f · g · h + ˘ FABC ∂Af ∂Bg ∂Ch ⇒ Generalization of open strings in B-field background 9812219 No reason to vanish! Integral:

  • DFT

˘ FABC DAf DBg DCh PI = −

  • DFT

GAB

  • eom: GAB=0!

f DAg DBh

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Conclusion

associativity of observables preserved by crucial ingredients

  • f double field theory

˘ FABC ∂Af ∂Bg∂Ch FABC ∂Af ∂Bg∂Ch equation of motion Bianchi identity continuity equation closure of algebra

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Deformation Quantization T-Duality and Non-associativity Non-associative Deformations of Geometry in DFT

Outlook

Future research directions:

  • Derive higher orders of the product (ongoing)
  • Non-associativity in Hamiltionian formalism