Poisson-Lie T-Duality and supermanifolds

Transcription

1 Poisson-Lie T-Duality and supermanifolds L. Hlavatý, I. Petr, V. Štěpán, J. Vysoký Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague IPM String School and Workshop 2010 Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

2 Contents 1 Motivation 2 Calculus 3 What is it good for? Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

3 2-dimensional σ-model Field with action S ϕ := ϕ : (Σ, η, ω) (M, G, B) 1 [ η ab (ϕ G) ab + ω ab (ϕ ] B) ab ω = 2 L ω η = dτ dτ dσ dσ & ω = dτ dσ Euler & Lagrange equations: S ϕ S ϕ+tψ d dt S ϕ+tψ(0) = 0 L ϕ µ L a ( a ϕ µ) = 0 If it is not possible to solve it...? Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

4 PLT-Duality 1 Rewrite the Lagrangian in lightcone coordinates x ± = σ ± τ L = F µν ϕ µ + ϕ ν F = G + B 2 If there is an action of some Lie group G on M s.t. M G M G 3 Try to find a Drinfel d double D = (G G) it s Lie algebra d = g + g g, g maximally isotropic w.r.t. ad-invariant form d 4 Check the condition (L Vi F) µν = F µρ V ρ j jk f i V λ k F λν V i left invariant fields on G structure constants of g f jk i 5 Then we can construct the σ-model ϕ on G M G and map the solution ϕ the solution ϕ Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

5 Are we able to super it? What s hidden behind the words? superfield supermanifold superspace superalgebra And how to work with it? Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

6 1 Motivation 2 Calculus 3 What is it good for? Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

7 References (F.) A. Rogers: A global theory of supermanifolds, J. Math. Phys. 21(6) (June 1980) (F.) A. Rogers: Super Lie groups: global topology and local structure, J. Math. Phys. 22(5) (May 1981) (F.) A. Rogers: Supermanifolds: theory and apllications, World Scientific Publishing Co. Pte. Ltd., Singapore c 2007 J. Cook & R. Fulp: Infinite dimensional super Lie groups, Differ.Geom.Appl.26: ,2008 А. Ю. Хренников: Суперанализ, Физматлит, Москва 2005 A. Yu. Khrennikov: Superanalysis, Kluwer Academic Publishers, 1999 Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

8 Supernumbers Grassmann algebra Λ = { λ = with generators and a condition k N 0 } λ i1...i k q i1...q ik λ i1...i k R i 1 <...<i k 1 {q n } n N, k, n N (q n q k = q k q n ), 1 q k = q k 1 λ := k N 0 is Z 2 -graded Banach algebra i 1 <...<i k λ i1...i k < + Λ = Λ 0 Λ 1 l 1 (R), which is supercommutative (α = α 0 + α 1 ) α i β j = ( 1) ij β j α i Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

9 Superspace Superspace R p π Λ := Λ p 0 Λπ 1 with a norm x := a x a is a Banach space. G-differentiability of a mapping f : R p π Λ Λ at point x: a {1,..., p + π} G a f (x) Λ ω : R p π Λ Λ f (x + h) = f (x) + h a G a f (x) + h ω(h), ω(0) = 0 Fréchet: f (x + h) = f (x) + f (x) h + h ω(h) f (x) h = h 1 G 1 f (x) h p+π G p+π f (x) Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

10 Supermanifolds & supergroups Supermanifold M R p π Λ ( G C ) = ( Supermanifolds Banach manifolds ) ( Supergroups Banach Lie groups ) G φ H exp exp g 0 h 0 G e φ g0 Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

11 1 Motivation 2 Calculus 3 What is it good for? Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

12 N = (1, 1) SUSY σ-model ϕ : R 2 2 Λ Gn O dθ r θ s = δ rs, d 2 θ = dθ dθ +, D ± = ±i ± θ ± θ S ϕ = 1 d 2 xd 2 θ [ G µν (ϕ)dϕ 2i µ Dφ ν + B µν (ϕ)dϕ µ (γ 5 D)ϕ ν] PLT-Duality Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

13 Target space is not purely even supermanifold ϕ : R 2 0 Λ Gp π & PLT-Duality S ϕ = [(η, ϕ G) (ω, ϕ B) ] ω A. Eghbali & A. Rezaei-Aghdam: Poisson-Lie T-dual sigma models on supermanifolds, JHEP 0909:094, 2009 Action G M M M G M G Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14

14 Thank you for your attention Vojtěch Štěpán (FNSPE CTU) PLT-Duality & supermanifolds April / 14