Double Field Theory and the Geometry of Duality CMH & Barton - PowerPoint PPT Presentation

Double Field Theory and the Geometry of Duality CMH & Barton Zwiebach arXiv:0904.4664, 0908.1792 CMH & BZ & Olaf Hohm arXiv:1003.5027, 1006.4664

Closed String Field Theory • Captures exotic and complicated structure of interacting string • Non-polynomial, algebraic structure not Lie algebra, cocycles. • On torus: winding modes, T -duality. • Seek subsector capturing exotic structure & duality, simple enough to analyse explicitly

Strings on d-Torus • Supergravity limit: symmetry O(d,d) • String: Perturbative T -duality O(d,d;Z) • Kaluza-Klein theory: includes momentum modes on torus • Include string winding or brane wrapping modes? Duality symmetry?

Strings on T d ˜ X = X L ( σ + τ ) + X R ( σ − τ ) , X = X L − X R conjugate to momentum, to winding no. ˜ X X ∂ a X = � ab ∂ b ˜ dX = ∗ d ˜ X X

Strings on T d ˜ X = X L ( σ + τ ) + X R ( σ − τ ) , X = X L − X R conjugate to momentum, to winding no. ˜ X X ∂ a X = � ab ∂ b ˜ dX = ∗ d ˜ X X Need “auxiliary” for interacting theory ˜ X i) Vertex operators e ik L · X L , e ik R · X R ii) String field Kugo & Zwiebach Φ [ x, ˜ x, a, ˜ a ]

Strings on T d ˜ X = X L ( σ + τ ) + X R ( σ − τ ) , X = X L − X R conjugate to momentum, to winding no. ˜ X X ∂ a X = � ab ∂ b ˜ dX = ∗ d ˜ X X Need “auxiliary” for interacting theory ˜ X i) Vertex operators e ik L · X L , e ik R · X R ii) String field Kugo & Zwiebach Φ [ x, ˜ x, a, ˜ a ] Doubled Torus 2d coordinates � ˜ � x i X ≡ Transform linearly under O ( d, d ; Z ) x i Doubled sigma model CMH 0406102

String Field Theory on Minkowski Space String field Φ [ X ( σ ) , c ( σ )] X i ( σ ) → x i , oscillators Expand to get infinite set of fields g ij ( x ) , b ij ( x ) , φ ( x ) , . . . , C ijk...l ( x ) , . . . Integrating out massive fields gives field theory for g ij ( x ) , b ij ( x ) , φ ( x )

String Field Theory on a torus String field Φ [ X ( σ ) , c ( σ )] X i ( σ ) → x i , ˜ x i , oscillators Expand to get infinite set of double fields g ij ( x, ˜ x ) , b ij ( x, ˜ x ) , φ ( x, ˜ x ) , . . . , C ijk...l ( x, ˜ x ) , . . . Seek double field theory for g ij ( x, ˜ x ) , b ij ( x, ˜ x ) , φ ( x, ˜ x )

Double Field Theory • Construct from SFT, DFT of “massless” fields g ij ( x, ˜ x ) , b ij ( x, ˜ x ) , φ ( x, ˜ x ) • Double field theory on doubled torus • Novel symmetry, reduces to diffeos + B-field trans. in any half-dimensional subtorus • Backgrounds depending on seen by { x a } particles, on seen by winding modes. { ˜ x a } Backgrounds with both: unfamiliar. Earlier versions: Siegel, Tseytlin

• Restriction to “massless” fields NOT a low- energy limit • Lowest terms in level expansion • T -duality a manifest symmetry • General solution of SFT: double fields • DFT needed for non-geometric backgrounds • Real dependence on full doubled geometry, dual dimensions not auxiliary or gauge artifact. Double geom. physical and dynamical

Strings on a Torus R n − 1 , 1 × T d • Coordinates x i = ( y µ , x a ) x a ∼ x a + 2 π • Momentum p i = ( k µ , p a ) • Winding ( p a , w a ) ∈ Z 2 d w a • Fourier transform ( k µ , p a , w a ) → ( y µ , x a , ˜ x a ) • Doubled Torus R n − 1 , 1 × T 2 d x a ∼ ˜ ˜ x a + 2 π • String Field Theory gives infinite set of fields φ ( y µ , x a , ˜ x a ) n + d = D = 26 or 10

T -Duality • Interchanges momentum and winding • Equivalence of string theories on dual backgrounds with very different geometries • String field theory symmetry, provided fields depend on both Kugo, Zwiebach x, ˜ x • For fields not Buscher ψ ( y ) ψ ( y, x, ˜ x ) • Aim: generalise to fields ψ ( y, x, ˜ x ) Generalised T -duality Dabholkar & CMH

Free field equn, M mass in D dimensions M 2 ≡ − ( k 2 + p 2 + w 2 ) = 2 α ′ ( N + ¯ N − 2) Constraint N − p a w a = 0 L 0 − ¯ L 0 = N − ¯ p a w a = 0 M 2 = 0 Massless states N = ˜ N = 1 Constrained fields φ ( y, x, ˜ x ) ∂ ∂ ∆ ≡ ∆ φ = 0 ∂ x a ∂ ˜ x a h ij ( y µ , x a , ˜ x a ) , b ij ( y µ , x a , ˜ x a ) , d ( y µ , x a , ˜ x a ) h ij → { h µ ν , h µa , h ab }

Constrained fields φ ( y, x, ˜ x ) ∂ ∂ ∆ ≡ ∆ φ = 0 ∂ x a ∂ ˜ x a Momentum space φ ( k µ , p a , w a ) ∆ = p a w a Momentum space: Dimension n+2d p a w a = 0 Cone: dimension n+2d-1

Constrained fields φ ( y, x, ˜ x ) ∂ ∂ ∆ ≡ ∆ φ = 0 ∂ x a ∂ ˜ x a Momentum space φ ( k µ , p a , w a ) ∆ = p a w a Momentum space: Dimension n+2d p a w a = 0 Cone: dimension n+2d-1 DFT: fields on this cone, with discrete p,w Problem: naive product of fields on cone do not lie on cone. Vertices need projectors

Constrained fields φ ( y, x, ˜ x ) ∂ ∂ ∆ ≡ ∆ φ = 0 ∂ x a ∂ ˜ x a Momentum space φ ( k µ , p a , w a ) ∆ = p a w a Momentum space: Dimension n+2d p a w a = 0 Cone: dimension n+2d-1 DFT: fields on this cone, with discrete p,w Problem: naive product of fields on cone do not lie on cone. Vertices need projectors Restricted fields: Fields that depend on d of 2d torus momenta, e.g. or φ ( k µ , w a ) φ ( k µ , p a ) Simple subsector, no projectors needed, easier

α ′ = 1 Torus Backgrounds � η µ ν � � 0 � 0 0 E ij ≡ G ij + B ij G ij = , B ij = G ab B ab 0 0 x i = { y µ , x a } x i = { ˜ x a } = { 0 , ˜ x a } ˜ y µ , ˜ Left and Right Derivatives ∂ ∂ ∂ ∂ ¯ D i = ∂ x i − E ik , D i = ∂ x i + E ki ∂ ˜ ∂ ˜ x k x k ∆ = 1 D 2 ) = − 2 ∂ ∂ 2( D 2 − ¯ ∂ ˜ ∂ x i x i � = 1 2( D 2 + ¯ D 2 = G ij D i D j D 2 )

Quadratic Action � 1 2 e ij � e ij + 1 D j e ij ) 2 + 1 � S (2) = 4( ¯ 4( D i e ij ) 2 [ dxd ˜ x ] − 2 d D i ¯ � D j e ij − 4 d � d Invariant under D j λ i + D i ¯ ¯ δ e ij = λ j , δ d = − 1 4 D · λ − 1 D · ¯ ¯ λ 4 ∆ λ = ∆ ¯ using constraint λ = 0 Discrete Symmetry e ij → e ji , D i → ¯ D i , ¯ D i → D i , d → d

Comparison with Conventional Actions ∂ ˜ Take ∂ i ≡ G ik B ij = 0 ∂ ˜ x k D i = ∂ i − ˜ D i = ∂ i + ˜ ¯ ∂ i , ∂ i � = ∂ 2 + ˜ ∆ = − 2 ∂ i ˜ ∂ 2 ∂ i e ij = h ij + b ij � Usual quadratic action dx L [ h, b, d ; ∂ ] 1 4 h ij ∂ 2 h ij + 1 2( ∂ j h ij ) 2 − 2 d ∂ i ∂ j h ij L [ h, b, d ; ∂ ] = − 4 d ∂ 2 d + 1 4 b ij ∂ 2 b ij + 1 2( ∂ j b ij ) 2

Double Field Theory Action � S (2) = � L [ h, b, d ; ∂ ] + L [ − h, − b, d ; ˜ [ dxd ˜ x ] ∂ ] ∂ k h ik )( ∂ j b ij ) − 4 d ∂ i ˜ � + ( ∂ k h ik )(˜ ∂ j b ij ) + (˜ ∂ j b ij Action + dual action + strange mixing terms ∂ i ǫ j + ∂ j ǫ i + ˜ ǫ j + ˜ δ h ij = ∂ i ˜ ∂ j ˜ ǫ i , δ b ij = − (˜ ∂ i ǫ j − ˜ ∂ j ǫ i ) − ( ∂ i ˜ ǫ j − ∂ j ˜ ǫ i ) , δ d = − ∂ · ǫ + ˜ ∂ · ˜ ǫ . Diffeos and B-field transformations mixed. Cubic action found for full DFT

T -Duality Transformations of Background � a � b T -duality g = ∈ O ( d, d ; Z ) c d E ′ = ( aE + b )( cE + d ) − 1 � ˜ � x i transforms as a vector X ≡ x i � ˜ � � a � � ˜ � x ′ b x X ′ = = gX = x ′ c d x

T -Duality is a Symmetry of the Action Fields e ij ( x, ˜ x ) , d ( x, ˜ x ) Background E ij E ′ = ( aE + b )( cE + d ) − 1 � ˜ � � a � � ˜ � x ′ b x X ′ = = gX = x ′ c d x Action invariant if: k ¯ l e ′ M ≡ d t − E c t kl ( X ′ ) e ij ( X ) = M i M j M ≡ d t + E t c t ¯ d ( X ) = d ′ ( X ′ ) With general momentum and winding dependence!

Projectors and Cocycles Naive product of constrained fields does not satisfy constraint 0 Ψ 2 = 0 but 0 ( Ψ 1 Ψ 2 ) � = 0 L − 0 Ψ 1 = 0 , L − L − but ∆ ( AB ) � = 0 ∆ A = 0 , ∆ B = 0 String product has explicit projection Double field theory requires projections, novel forms Leads to a symmetry that is not a Lie algebra, but is a homotopy lie algebra SFT has non-local cocycles in vertices, DFT should too Cocycles and projectors not needed in cubic action

General fields ψ ( x, ˜ x ) Fields on Spacetime M ψ ( x ) ψ ( x ′ ) Restricted Fields on N, T -dual to M M,N null wrt O(D,D) metric ds 2 = 2 dx i dx i Subsector with fields and parameters all restricted to M or N • Constraint satisfied on all fields and products of fields • No projectors or cocycles • T -duality covariant: independent of choice of N • Can find full non-linear form of gauge transformations • Full gauge algebra, full non-linear action

Background Independence? Action S ( E, e, d ) is background independent: δ χ S = 0 Constant shift δ E ij = − χ ij δ e ij = χ ij − 1 k e kj − χ k + O ( e 2 ) � � χ i j e ik 2

Background Independence? Action S ( E, e, d ) is background independent: δ χ S = 0 Constant shift δ E ij = − χ ij δ e ij = χ ij − 1 k e kj − χ k + O ( e 2 ) � � χ i j e ik 2 Background independent field: δ χ E ij = 0 E ij ≡ E ij + e ij + 1 k e kj + O ( e 3 ) 2 e i