University of Belgrade Faculty of Physics Noncommutativity and - PowerPoint PPT Presentation

University of Belgrade Faculty of Physics Noncommutativity and Nonassociativity of Closed Bosonic String on T-dual Toroidal Backgrounds Presenter: Danijel Obrić September, 2018 2. septembar 2018 1 / 15

Table of Contents 1 Introduction 2 Bosonic strings Bosonic strings in background fields 1 Choice of background fields 2 3 T-dualization T-dualization along x direction 1 T-dualization along y direction 2 T-dualization along z direction 3 4 Noncommutativity and nonassociativity relations Noncommutativity relations 1 Nonassociativity relations 2 5 Conclusion 2. septembar 2018 2 / 15

Introduction 1 Noncommutativity of coordiantes 2 T-duality 3 Generalized Buscher procedure 2. septembar 2018 3 / 15

Bosonic strings Bosonic strings in background fields The action of the closed bosonic string in the presence of background fields is given by d 2 ξ √− g 2 g αβ G µν + ǫ αβ � �� 1 Φ � ∂ α x µ ∂ β x ν + 4 k π R ( 2 ) � S = k √− g B µν . (1) Σ Keeping conformal symmetry at quantum level demands that background fields obey following equations. R µν − 1 ρσ + 2 D µ ∂ ν Φ = 0 , (2) 4 B µρσ B ν D ρ B ρ µν − 2 ∂ ρ Φ B ρ µν = 0 , (3) 4 ( ∂ Φ) 2 − 4 D µ ∂ µ Φ + 1 12 B µνρ B µνρ − R = 0 . (4) B µνρ = ∂ µ B νρ + ∂ ν B ρµ + ∂ ρ B µν is field strength tensor of Kalb Ramond field. 2. septembar 2018 4 / 15

Bosonic strings Bosonic strings in background fields We will work in D = 3 dimensions with following background fields:     0 0 1 0 0 Hz  ,  . B µν = − Hz 0 0 G µν = 0 1 0 (5)   0 0 0 0 0 1 Final form of our action, in world-sheet light-cone coordinates ξ ± = 1 2 ( τ ± σ ) , ∂ ± = ∂ τ ± ∂ σ . is then: � d 2 ξ ± � S = k Hz ( ∂ + x ∂ − y − ∂ + y ∂ − x ) Σ + 1 � 2 ( ∂ + x ∂ − x + ∂ + y ∂ − y + ∂ + z ∂ − z ) (6) . 2. septembar 2018 5 / 15

T-dualization T-dualization along x direction Replacing partial with covariant derivatives (7) ∂ ± x → D ± x = ∂ ± x + v ± . Removal of unphysical degrees of freedom S add = k � d 2 ξ y 1 ( ∂ + v − − ∂ − v + ) . (8) 2 Σ Gauge fixing, x = const . � d 2 ξ [ 1 S fix = k 2 ( v + v − + ∂ + y ∂ − y + ∂ + z ∂ − z ) + v + Hz ∂ − y Σ − ∂ + yHzv − + 1 2 y 1 ( ∂ + v − − ∂ − v + )] . (9) Equations of motion are: F + − = ∂ + v − − ∂ − v + = 0 → v ± = ∂ ± x (10) v ± = ± ∂ ± y 1 ± 2 Hz ∂ ± y (11) 2. septembar 2018 6 / 15

T-dualization T-dualization along x direction Obtaining T-dual action � � 1 2 ( ∂ + y 1 ∂ − y 1 + ∂ + y ∂ − y + ∂ + z ∂ − z ) d 2 ξ x S = k Σ + ∂ + y 1 Hz ∂ − y + ∂ + yHz ∂ − y 1 � . (12) Getting T-dual transformation laws ∂ ± x ∼ x ∼ = y ′ 1 + 2 Hzy ′ , = ± ∂ ± y 1 ± 2 Hz ∂ ± y → ˙ (13) π x = δ S π x ∼ x − 2 Hzy ′ ) = ky ′ x = k ( ˙ → 1 . (14) δ ˙ 2. septembar 2018 7 / 15

T-dualization T-dualization along y direction Repeating previous procedure for y coordinate we obtain � � 1 d 2 ξ S fix = k 2 ( ∂ + y 1 ∂ − y 1 + v + v − + ∂ + z ∂ − z ) + ∂ + y 1 Hzv − Σ + v + Hz ∂ − y 1 + 1 � 2 y 2 ( ∂ + v − − ∂ − v + ) . (15) Equation of motion are: ∂ + v − − ∂ − v + = 0 → v ± = ∂ ± y , (16) v ± = ± ∂ ± y 2 − 2 Hz ∂ ± y 1 . (17) 2. septembar 2018 8 / 15

T-dualization T-dualization along y direction Twice dualized action is given as � � 1 d 2 ξ xy S = k 2 ( ∂ + y 1 ∂ − y 1 + ∂ + y 2 ∂ − y 2 + ∂ + z ∂ − z ) Σ � + ∂ + y 2 Hz ∂ − y 1 − ∂ + y 1 Hz ∂ − y 2 . (18) T-dual transformation law: ∂ ± y ∼ y ∼ = y ′ = ± ∂ ± y 2 − 2 Hz ∂ ± y 1 → ˙ 2 − 2 Hz ˙ y 1 , (19) π y = δ S π y ∼ y + 2 Hzx ′ ) = ky ′ y = k ( ˙ → 2 . (20) δ ˙ 2. septembar 2018 9 / 15

T-dualization T-dualization along z direction Introduction of covariant derivative and invariant coordinate � z inv = d ξ α D α z = z ( ξ ) − z ( ξ 0 ) + ∆ V , ∂ ± z → D ± z = ∂ ± z + v ± ; (21) P where � � ( d ξ + v + + d ξ − v − ) . d ξ α v α = ∆ V = (22) P P After removing additional degrees of freedom and fixing gauge with z ( ξ ) = z ( ξ 0 ) we have following action: � � d 2 ξ S fix = k − H ∆ V ( ∂ + y 1 ∂ − y 2 − ∂ + y 2 ∂ − y 1 ) Σ + 1 2 ( ∂ + y 1 ∂ − y 1 + ∂ + y 2 ∂ − y 2 + ∂ + z ∂ − z ) + 1 � 2 y 3 ( ∂ + v − − ∂ − v + ) . (23) 2. septembar 2018 10 / 15

T-dualization T-dualization along z direction Equations of motion are: ∂ + v − − ∂ − v + = 0 (24) → v ± = ∂ ± z , v ± = ± ∂ ± y 3 − 2 β ∓ , (25) where β ± = ± 1 2 H ( y 1 ∂ ∓ y 2 − y 2 ∂ ∓ y 1 ) . (26) Fully T-dualized action is: � 1 � d 2 ξ xyz S = k 2 ( ∂ + y 1 ∂ − y 1 + ∂ + y 2 ∂ − y 2 + ∂ + y 3 ∂ − y 3 ) Σ � − ∂ + y 1 H ∆ ¯ y 3 ∂ − y 2 + ∂ + y 2 H ∆ ¯ y 3 ∂ − y 1 . (27) T-dual transformation laws are: ∂ ± z ∼ z ∼ = ± ∂ ± y 3 − 2 β ∓ = y ′ 3 + H ( y 1 y ′ 2 − y 2 y ′ → ˙ 1 ) , (28) π z = δ S 3 + kH ( xy ′ − yx ′ ) . π z ∼ = ky ′ z = k ˙ z → (29) δ ˙ 2. septembar 2018 11 / 15

Noncommutativity and nonassociativity relations Noncommutativity relations Transformation laws in canonical form are: ′ = 1 2 = 1 3 = 1 k π z − H ( xy ′ − yx ′ ) . y ′ y ′ y 1 k π x , k π y , (30) Only non-trivial Poisson brackets are: = − H � � σ ) } ∼ { y 1 ( σ ) , y 3 (¯ 2 y ( σ ) − y (¯ σ ) θ ( σ − ¯ σ ) , (31) k = H � � σ ) } ∼ { y 2 ( σ ) , y 3 (¯ 2 x ( σ ) − x (¯ σ ) θ ( σ − ¯ σ ) . (32) k For σ − ¯ σ = 2 π we have = − H � � { y 1 ( σ + 2 π ) , y 3 ( σ ) } ∼ 4 π N y + y ( σ ) (33) , k = H � � { y 2 ( σ + 2 π ) , y 3 ( σ ) } ∼ 4 π N x + y ( σ ) , (34) k where N x and N y are winding numbers defined as x ( σ + 2 π ) − x ( σ ) = 2 π N x , y ( σ + 2 π ) − y ( σ ) = 2 π N y (35) 2. septembar 2018 12 / 15

Noncommutativity and nonassociativity relations Nonassociativity relations Nonassociativity relation is defined as { y 1 ( σ 1 ) , y 2 ( σ 2 ) , y 3 ( σ 3 ) } ≡ { y 1 ( σ 1 ) , { y 2 ( σ 2 ) , y 3 ( σ 3 ) }} + { y 2 ( σ 2 ) , { y 3 ( σ 3 ) , y 1 ( σ 1 ) }} + { y 3 ( σ 3 ) , { y 1 ( σ 1 ) , y 2 ( σ 2 ) }} = − 2 H � ∼ θ ( σ 1 − σ 2 ) θ ( σ 2 − σ 3 ) k 2 � + θ ( σ 2 − σ 1 ) θ ( σ 1 − σ 3 ) + θ ( σ 1 − σ 3 ) θ ( σ 3 − σ 2 ) (36) . For σ 2 = σ 3 = σ and σ 1 = σ + 2 π we get: = 2 H { y 1 ( σ + 2 π ) , y 2 ( σ ) , y 3 ( σ ) } ∼ (37) k . 2. septembar 2018 13 / 15

Conclusion After two T-dualizations we had that Q flux theory is commutative. Closed string noncommuatativity and nonassociativity are consequence of the fact that Kalb-Ramond field is coordinate dependent. Parameters of noncommutativity and nonassociativity are proportional to the field strength H. In ordinary space, coordinate dependent background is a sufficient condition for closed string noncommutativity. 2. septembar 2018 14 / 15

THANKS FOR YOUR ATTENTION 2. septembar 2018 15 / 15