Double Field Theory, String Field Theory and T -Duality CMH & - PowerPoint PPT Presentation

Double Field Theory, String Field Theory and T -Duality CMH & Barton Zwiebach

What is string theory? • Supergravity limit - misses stringy features • Winding modes, T -duality, cocycles, algebraic structure not Lie algebra,... • On torus extra dual coordinates { ˜ x a } { x a } • String field theory: interactions, T -duality • Double field theory on doubled torus g ab ( x a , ˜ x a ) , b ab ( x a , ˜ x a ) , φ ( x a , ˜ x a ) Earlier versions: Siegel, Tseytlin

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 ) ∂ ∂ ∆ ≡ − 2 ∆ φ = 0 α ′ ∂ 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 }

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 )

Kinetic Operator � = 1 2( D 2 + ¯ D 2 ) = ∂ t H ( E ) ∂ � ∂ � ∂ ˜ x i ∂ = ∂ ∂ x j D × D E ij ≡ G ij + B ij Generalised Metric 2 D × 2 D � G − BG − 1 B BG − 1 � H ( E ) = − G − 1 B G − 1

Closed String Field Theory Matter CFT + Ghost CFT: General State � � � ψ I ( k, p, w ) V I | k, p, w � | Ψ � = dk p,w I or in position space � � x ) V I | y, x, ˜ | Ψ � = [ dydxd ˜ x ] ψ I ( y, x, ˜ x � I Vertex operators, ghost number 2 V I Infinite set of fields ψ I ( y, x, ˜ x ) SFT gives action for component fields

Closed String Field Theory Zwiebach S = 1 0 Q | Ψ � + 1 3! { Ψ , Ψ , Ψ } + 1 2 � Ψ | c − 4! { Ψ , Ψ , Ψ , Ψ } + · · · Symmetry δ Ψ = Q Λ + [ Λ , Ψ ] + . . . String fields ghost number 2, parameters | Λ � ghost number 1 are constrained: ( b 0 − ¯ ( L 0 − ¯ L 0 ) | Ψ � = 0 , b 0 ) | Ψ � = 0 , ( b 0 − ¯ ( L 0 − ¯ L 0 ) | Λ � = 0 , b 0 ) | Λ � = 0 , String Products [ A, B ] , [ A, B, C ] , [ A, B, C, D ] , ... { Ψ , Ψ , ..., Ψ } = � Ψ | c − 0 [ Ψ , ..., Ψ ] �

� d θ 2 π e i θ ( L 0 − ¯ L 0 ) b − 0 [ Ψ 1 , Ψ 2 ] ′ [ Ψ 1 , Ψ 2 ] ≡ [A,B]’ inserts the states A,B in 3-punctured sphere that defines the vertex [ A, B ] = ( − ) AB [ B, A ] Graded, like a super-Lie bracket [ A, [ B, C ]] ± [ B, [ C, A ]] ± [ C, [ A, B ]] = Q [ A, B, C ] ± [ QA, B, C ] ± [ A, QB, C ] ± [ A, B, QC ] Failure of graded Jacobi = failure of Q to be a derivation Homotopy Lie Alegebra

Massless Fields − 1 � � α j 2 e ij ( p ) α i | Ψ � = [ dp ] − 1 ¯ − 1 c 1 ¯ c 1 � + d ( p ) ( c 1 c − 1 − ¯ c 1 ¯ c − 1 ) + ... | p � � � � − 1 c 1 − i ¯ c 1 + µ ( p ) c + i λ i ( p ) α i α i | Λ � = [ dp ] λ i ( p ) ¯ − 1 ¯ | p � 0 • Use in action, gauge transformations • Fix symmetry, eliminate auxiliary fields µ • Gives action and symmetries for e ij = h ij + b ij , d • Background E ij = G ij + B ij

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

Dilaton ∂ i ǫ j + ∂ j ǫ i + ˜ ǫ j + ˜ δ h ij = ∂ i ˜ ∂ j ˜ ǫ i , δ d = − ∂ · ǫ + ˜ ∂ · ˜ ǫ . φ = d + 1 4 η ij h ij invariant under transformation ǫ In non-linear theory d is a density, dilaton scalar is φ e − 2 d = e − 2 φ √− g φ = d − 1 ˜ 4 η ij h ij invariant under transformation ˜ ǫ -duality d is invariant, φ → ˜ Dual dilaton. Under T φ

Cubic Terms in Action � � 4 e ij ( D i ¯ D j d ) d + 4 d 2 � d [ dxd ˜ x ] + 1 � � ( D i e kl )( ¯ D j e kl ) − ( D i e kl ) ( ¯ D l e kj ) − ( D k e il )( ¯ D j e kl ) 4 e ij + 1 D k e ik + D i D k e kj ) + 1 2( D k e ij ) 2 + 1 � 2 e ij ( ¯ D j ¯ 2( ¯ D k e ij ) 2 2 d + ( D i e ij ) 2 + ( ¯ D j e ij ) 2 � � D j λ i + 1 � � δ λ e ij = ¯ ( D i λ k ) e kj − ( D k λ i ) e kj + λ k D k e ij 2 action invariant δ λ d = − 1 4 D · λ + 1 to this order 2( λ · D ) d

Linearised Symmetries: diffeos on doubled space? ∂ 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 . Non-linear terms & algebra NOT doubled diffeos ⇒ Diffeos after field redefs ij ≡ e ij ± 1 e ± k e kj + O ( e 3 ) 2 e i For fields independent of gives diffeos x, δ e + ˜ ǫ ij For fields independent of gives diffeos ˜ x, δ e − ǫ ij No field redef can give both kinds of diffeo

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 ′ )

Conjecture for full non-linear transformations: E ′ ( X ′ ) = ( a E ( X ) + b )( c E ( X ) + d ) − 1 E = E + e d ′ ( X ′ ) = d ( X ) Linearising in gives previous result e ij

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 � d θ 2 π e i θ ( L 0 − ¯ L 0 ) b − 0 [ Ψ 1 , Ψ 2 ] ′ [ Ψ 1 , Ψ 2 ] ≡ Double field theory requires projections, novel forms SFT has non-local cocycles in vertices, DFT should too Cocycles and projectors not needed in cubic action

Double Field Theory • New limit of strings, captures some of the magic of string theory • Constructed cubic action, quartic will have new stringy features • T -duality, cocycles, homotopy Lie, constraints • Simpler than SFT, can address stringy issues in simpler setting • Generalised Geometry doubles Tangent space, DFT doubles coordinates. Geometry?