T-dualization of type II pure spinor superstring in double space - PowerPoint PPT Presentation

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks T-dualization of type II pure spinor superstring in double space Bojan Nikoli´ c and Branislav Sazdovi´ c Institute of Physics Belgrade, Serbia 9th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics 18.-23. September 2017, Belgrade, Serbia Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Outline of the talk T-duality 1 Model 2 Bosonic T-duality 3 Fermionic T-duality 4 Concluding remarks 5 Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Superstrings There are five consistent superstring theories. They are connected by web of T and S dualities. There are three approaches to superstring theory: NSR (Neveu-Schwarz-Ramond), GS (Green-Schwarz) and pure spinor formalism (N. Berkovits, hep-th/0001035). T-duality transformation does not change the physical content of the theory. Well known bosonic and recently discovered fermionic T-duality. Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Idea od double space Double space = initial coordinates plus T-dual partners - Siegel, Duff, Tseytlin about 25 years ago. Interest for this subject emerged again (Hull, Berman, Zwiebach) in the context of T-duality as O ( d , d ) transformation. The approach of Duff has been recently improved when the T-dualization along some subset of the initial and corresponding subset of the T-dual coordinates has been interpreted as permutation of these subsets in the double space coordinates (arXiv:1505.06044, 1503.05580). All calculations are made in full double space. In double space T-duality is a symmetry transformation. Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks General pure spinor action for type II superstring We start from the general pure spinor action for type II superstring (arXiv: 0405072) � θ β + ∂ + θ α A αµ Π µ ∂ + θ α A αβ ∂ − ¯ + A µα ∂ − ¯ d 2 ξ − + Π µ θ α � S = θ β + d α E αµ Π µ − + ∂ + θ α E αβ ¯ + E µβ ¯ Π µ − + d α E αβ ∂ − ¯ d β + Π µ + A µν Π ν + d β d β + 1 θ β + 1 − + 1 d α P αβ ¯ 2 ∂ + θ α Ω α,µν ¯ 2 N µν + Ω µν,β ∂ − ¯ 2 N µν + Ω µν,ρ Π ρ N µν + − 1 − + 1 d β + 1 C µν β ¯ + Ω µ,νρ ¯ + ¯ 2 d α C αµν ¯ 2 Π µ N νρ 2 N µν N µν + − � 1 + S µν,ρσ ¯ 4 N µν N ρσ + + S λ + S ¯ (1) λ . − Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Bosonic T-duality - assumptions and approximations Bosonic T-dualization - we assume that background fields are independent of x µ . In mentioned reference, expressions for background fields as well as action are obtained in an iterative procedure as an expansion in powers of θ α and ¯ θ α . Every step in iterative procedure depends on previuous one, so, for mathematical simplicity, we consider only basic ( θ and ¯ θ independent) components. Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Fermionic T-duality - assumptions and consistency check Fermionic T-dualization - we assume that θ α and ¯ θ α are Killing directions. Consequently, auxiliary superfirlds are zero according to arXiv: 0405072. If we assume that rest of background fields are constant then their curvatures are zero. Using space-time field equations we confirmed the consistency of the choice of constant P αβ . Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Action In both cases, under introduced assumptions, action gets the form � � 1 � ∂ + x µ Π + µν ∂ − x ν + d 2 ξ 4 πκ Φ R ( 2 ) S = κ (2 Σ � Φ � � π α + e − π α ∂ − ( θ α + Ψ α θ α + ¯ 2 µ x µ ) + ∂ + (¯ d 2 ξ Ψ α µ x µ )¯ 2 κ π α F αβ ¯ + π β Σ µ and ¯ Definitions: Π ± µν = B µν ± 1 2 G µν , Φ is dilaton field, Ψ α Ψ α µ are NS-R fields and F αβ is R-R field strength. Momenta π α and π α are canonically conjugated to θ α and ¯ θ α . All spinors are ¯ Majorana-Weyl ones. All background fields are constant. Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Busher bosonic T-duality Global shift symmetry exists x a → x a + b , where index a is subset of µ . We introduce gauge fields v a ± and make change in the action ∂ ± x a → ∂ ± x a + v a ± . Additional term in the action S gauge ( y , v ± ) = 1 � d 2 ξ v a + ∂ − y a − ∂ + y a v a � � 2 κ , − Σ where y a is Lagrange multiplier. It makes v a ± to be unphysical degrees of freedom. On the equations of motion for y a we get initial action, while, fixing x a to zero, on th equations of motion for v a ± we get T-dual action. Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Transformation laws Solution of the equation of motion for y a is v a ± = ∂ ± x a . Combining this solution with equations of motion for gauge fields v a ± we obtain T-dual transformation laws ∂ ± x a ∼ ± Π ∓ bi ∂ ± x i − κ ˆ = − 2 κ ˆ θ ab θ ab ± ( ∂ ± y b − J ± b ) , (3) = − 2 Π ∓ ab ∂ ± x b − 2 Π ∓ ai ∂ ± x i + J ± a . ∂ ± y a ∼ (4) Here J ± µ = ± 2 κ Ψ α ± µ π ± α and θ ac 2 κ δ ab , where 1 ± Π ∓ cb = Ψ α + µ ≡ Ψ α Ψ α − µ ≡ ¯ Ψ α π − α ≡ ¯ µ , µ , π + α ≡ π α , π α . (5) Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Transformation laws in double space In double space spanned by Z M = ( x µ , y µ ) T they are of the form ∂ ± Z M ∼ = ± Ω MN � H NP ∂ ± Z P + J ± N � , (6) where G E − 2 B µρ ( G − 1 ) ρν � � µν H MN = , (7) 2 ( G − 1 ) µρ B ρν ( G − 1 ) µν is so called generalized metric, while � 0 � 2 (Π ± G − 1 ) µν J ± ν � � 1 D Ω MN = , J ± M = (8) . − ( G − 1 ) µν J ± ν 1 D 0 Ω MN is constant symmetric matrix and it is known as SO ( D , D ) invariant metric. Here G E µν = G µν − 4 ( BG − 1 B ) µν . Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks T-duality as permutation in double space T-dualization in double space is represented by permutation x a  y a   0 0 1 a 0    x i x i 0 1 i 0 0 a Z M ≡ N Z N ≡  = ( T a ) M        .       x a 1 a 0 0 0 y a     y i 0 0 0 1 i y i Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks T-duality as permutation in double space Demanding that a Z M has the transformation law of the same form as initial coordinates Z M , we find the T-dual generalized metric K H KL ( T a ) L a H MN = ( T a ) M N , (9) and T-dual current a J ± M = ( T a ) M N J ± N . (10) Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks NS-NS background fields From (9) we obtain the T-dual NS-NS background fields which are in full agreement with those obtained by Buscher procedure 2 ˆ ± i = κ ˆ a Π ab ± = κ θ ab a Π a θ ab ∓ , ∓ Π ± bi , a Π ± ia = − κ Π ± ib ˆ a Π ± ij = Π ± ij − 2 κ Π ± ia ˆ θ ba θ ab ∓ , ∓ Π ± bj . Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space

T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks NS-R background fields From (10) we obtain the form of the T-dual NS-R fields a Ψ α a = κ ˆ Ψ α a = κ a Ω αβ ˆ θ ab a ¯ θ ab − ¯ Ψ β + Ψ α b , b . (11) i − 2 κ Π − ib ˆ a ¯ i = a Ω αβ (¯ Ψ β i − 2 κ Π + ib ˆ − ¯ Ψ β a Ψ α i = Ψ α θ ba + Ψ α Ψ α θ ba a , a ) . (12) From transformation laws we see that two chiral sectors transform differently. Consequently, there are two sets of vielbeins in T-dual picture as well two sets of gamma matrices. This T-dual vielbeins are connected by Lorentz transformation, while spinorial representation of this Lorentz transformation, a Ω αβ , relates two sets of gamma matrices. In order to have unique set of gamma matrices, we have to multiply one fermionic index by a Ω αβ . Bojan Nikoli´ c T-dualization of type II pure spinor superstring in double space