Poisson sigma models as constrained Hamiltonian systems

Transcription

1 1 / 17 Poisson sigma models as constrained Hamiltonian systems Ivo Petr Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague 6th Student Colloquium and School on Mathematical Physics Stará Lesná 2012

2 2 / 17 Outline 1 Poisson manifolds and σ-model 2 Poisson-Lie group 3 σ-models on a Poisson-Lie group and PLT-Duality

3 3 / 17 Poisson manifold Differentiable manifold (M, {, }) with Poisson bracket {, } : C (M) C (M) C (M) bilinear, antisymmetric {f, g} = {g, f } Jacobi identity {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 Leibnitz rule {fg, h} = f {g, h} + {f, h}g Poisson bivector Π Π = Π ij x i x j Λ 2 TM Π ij C (M) ij f g {f, g} = Π(df, dg) = Π Π ij (x) = {x i, x j } x i x j Π ij = Π ji ij Πkl kj Πli lj Πik Π + Π + Π = 0 x j x j x j Casimir functions f is called Casimir function if {f, g} = 0 g C (M)

4 Poisson σ-model [Schaller & Strobl, 1994],[Calvo & Falceto, 2006] Building blocks Σ - differentialble manifold, dimσ = 2, coordinates σ µ (M, Π) - Poisson manifold dimm = n, coordinates x i Dynamical fields vector bundle map (X, Ã) : T Σ T M base manifold map X : Σ M, in coordinates X i = x i X (σ) total spaces map à : T Σ T M or A : T Σ Γ(Σ, X (T M)) A, V (σ) := Ã(Vσ) T X (σ)m A(σ) = A i (σ)dx i X (σ) σ-model action and field equations Action S X,A = Σ A i dx i Πij (X )A i A j where dx i = X (dx i ) Equations of Motion dx i + Π ij (X )A j = 0 da i + 1 Π jk (X )A 2 x i j A k = 0 4 / 17

5 5 / 17 First order string action Action for a bosonic string S str = 1 4πα d 2 σ det(h) ( G ij (X )h µν + ɛ µν B ij (X ) ) µx i νx j Σ where h is a metric on Σ, G is a metric on M, B is a 2-form on M. Seiberg-Witten limit [Baulieu et al., 2002] Sstr fo = A i dx i + πα E ij (X )A i A j + 1 Σ 2 Πij (X )A i A j using Equations of Motion for A i dx i + Π ij (X )A j + E ij (X ) A j = 0 classical equivalence if (G + B) 1 = E + 1 2πα Π Seiberg-Witten limit α 0 while (G, B) fixed, we get the action of PSM.

6 6 / 17 Gauge theory for SU(2) Yang-Mills action su(2)-valued 1-form A = A i T i curvature 2-form F = (da i cjk i S YM = 1 4g A j A k )T i, where c jk i = ɛ ijk Tr(F F ) Σ Action for a linear PSM with a Casimir function R 3 with Poisson bivector Π ij = 3 k=1 ɛijk x k Casimir function R 2 = 3 k=1 x k x k on M, volume form ω on Σ SYM fo = X i (da i + 1 Σ 2 cjk i A j A k ) g 3 2 ω X k X k Equations of Motion X i = 1 g F i give classical equivalence with S YM. For g 0 describes action of a PSM. k=1

7 7 / 17 Local symmetry [Ikeda, 1994], [Schaller & Strobl, 1994] Local symmetry of the action of PSM ɛ j = ɛ j (σ) δ ɛx i = ɛ j Π ji (X ) δ ɛa i = dɛ i + Πjk (X )A x i j ɛ k generalization of Y-M SU(2) symmetry - linear Π S X +δx,a+δa S X,A = Σ d(dx i ɛ i ) For Π fulfilling Jacobi identity and Σ = or ɛ Σ = 0 Algebra of these transformations is closed only on-shell [δ ɛ, δ ɛ ]X i = ɛ j ɛ k [δ ɛ, δ ɛ ]A i = d(ɛ j ɛ k Π jk Π li x l Π jk x i ) + Πjk A x i j (ɛ oɛ Π op p x ) ɛ jɛ 2 Π jk k k x i x (dx l + Π lm A m) l

8 8 / 17 Constrained system formalism With A i (τ, σ) = A iτ dτ + A iσ dσ, action gives τ2 ( ) S X,A = dτ dσ A iσ X,τ i A iτ (X,σ i + Π ij A jσ ) Hamiltonian formalism τ 1 σ coordinates X i (τ, σ), momenta P i (τ, σ) = A iσ (τ, σ) and canonical Hamiltonian ( ) H = dσ P i (σ)x,τ i (σ) L(σ) σ ( ) = dσa iτ (σ) X,σ i (σ) + Πij (X (σ))a jσ (σ) = dσλ i (σ)φ i (σ) σ σ Equations of Motion - constrained system, H 0 = 0 Ẋ i (σ) = δh 0 δp i (σ) + δ dσ λ j (σ )Φ j (σ ), δp i (σ) Ṗ i (σ) = δh 0 δx i (σ) δ δx i dσ λ j (σ )Φ j (σ ), (σ) Φ i (σ) = 0.

9 Constraints and gauge transformations Constraints Φ i (σ) = X i (σ) + Π ik (X (σ))p k (σ) are first class Gauge transformations δ ɛx i (σ) = δ ɛp i (σ) = {Φ i (σ ), Φ j (σ )} = δ(σ σ )Π ij,n(x (σ ))Φ n (σ ) 0 {Φ i (σ), H} = λ j (σ)π ji,n (X (σ))φn (σ) 0 dσ ɛ j (σ ){X i (σ), Φ j (σ )} = ɛ j (σ)π ji (X (σ)) dσ ɛ j (σ ){P i (σ), Φ j (σ )} = ɛ i (σ) + P k(σ)π kj,i (X (σ))ɛ j (σ) δ ɛλ i (σ) = ɛ i (σ) + λ k (σ)π kj,i (X (σ))ɛ j (σ) Algebra of gauge transformations [δ ɛ, δ ɛ ]X i = ɛ j ɛ Π jk k x l Π li = δ ɛ X i [δ ɛ, δ ɛ ]A i = δ ɛ A i ɛ j ɛ 2 Π jk k x i x l (dx l + Π lm A m) 9 / 17

10 10 / 17 Poisson-Lie group Poisson map (M, {, }), (N, {, }) Poisson manifolds, F : N M is Poisson map if ( f, g C (M))({f, g} M F = {f F, g F } N ) Multiplicativity of Π compatibility of group and Poisson structure - µ : G G G is Poisson map with {f, g} M N (x, y) = {f (., y), g(., y)} M (x) + {f (x,.), g(x,.)} N (y) Poisson structure on G G Π(gg ) = (L g L g )Π(g ) + (R g R g )Π(g) Relation to bialgebra structures for (G, Π) connected and simply connected P-L group with algebra g exists unique bialgebra structure (g, δ) such that δ = DΠ and vice versa.

11 11 / 17 Construction of Π on the group G Manin triple (d, g, g) d = g g, g, g are subalgebras of d, isotropic w.r.t. non-degenerate symmetric ad-invariant scalar product, d [T a, T b ] = fc ab T c [ T a, T b ] = f ab c T c [T a, T b ] = fb ca T c + f bct a c ( a(g) T b(g) T adjoint representation of G on d Ad g 1 = 0 d(g) T ) Poisson-Lie structure on G in the basis of right-invariant fields Π(g) = b(g).a 1 (g) Sklyanin bracket gives the same result G connected Lie group with g, δ given by an r-matrix r g g, then Π(g) = (L g L g )(r) (R g R g )(r)

12 12 / 17 Example - R 2 gravity su(2)-yang-mills is formulated on the double (1 9), what about others? 4 and 6 dimensional Drinfeld doubles classified [Šnobl & Hlavatý, 2002] R 2 gravity [Schaller & Strobl, 1994] L R 2 = 1 d 2 x det g ( 1 4 M 4 R2 + 1) in Einstein-Cartan variables (e 1, e 2 ) zweibein, ω connection 1-form L R 2 = X (de 1 ω e 2 ) + Y (de 2 + ω e 1 ) + Zdω + ( 1 M 4 Z 2 )e 1 e 2 for X i = (X, Y, Z), A i = (e 1, e 2, ω) we have PSM with {X, Y } = Z {Y, Z} = X {Z, X } = Y induces Lie algebra structure of Bianchi9 (so(3)) so it can be built on (1 9) as {X, Y } = 1 z {Y, Z} = X {Z, X } = Y 2 or Bianchi8 (sl(2, R)), however using (1 7 0 ), (2i 7 0 ), (2ii 7 0 ) etc. needs simpler coordinate transformation

13 13 / 17 Poisson-Lie T-Duality σ-model Φ : Σ G, Φ C Σ R 2 with Minkowski metric d-dimensional Lie-group G with F = G + B action S F(Φ) = dσ +dσ Φ µ F µν(φ) +Φ ν Σ Dualizable σ-models generalized symmetry condition [Klimčík & Ševera, 1995] (L V af) µν = F µκv κb f a bc v λc F λν self-consistency leads to the Drinfeld double - D (G G) is a Lie group whose Lie algebra d admits a decomposition d = g g into a pair of subalgebras, maximally isotropic with respect to a symmetric, ad-invariant, nondegenerate bilinear form.,. d

14 14 / 17 Construction of dualizable σ-models σ-model on G given orthogonal subspaces E ± d - graph of E 0, dual decomposition l = g h = gh in ±ll 1, E = 0 E(g) = (a(g) + E 0 b(g)) 1 E 0 d(g) in right-invariant fields setting E(g) = ( E Π(g) ) 1 with Π(g) = b(g)a(g) 1 F(g) = e E(g) e T ( ±gg 1 ) a = ±φ µ e µa(g) F(g) solves the generalized symmetry condition and we have σ-model with Equations of Motion +( h h 1 ) c ( + h h 1 ) c + f c ab( h h 1 ) a ( + h h 1 ) b = 0. ( h h 1 ) a + E ab (g)(g 1 g) b = 0 ( + h h 1 ) a (g 1 +g) b E ba (g) = 0

15 Dual σ-model Dual σ-model on G model on G E + = Span(T i + E ij 0 T j ) g 1 E + g = Span(T i + E ij (g) T j ) model on G for invertible E 0 E + = Span( T j + (E 1 0 ) ji T i ) g 1 E + g = Span( T j + Ẽ( g) ji T i ) model on G for non-invertible E 0 E + = Span(T i + E ij 0 T j ) g 1 E + g = Span(T i + Ẽ( g)ij Tj ) Ẽ( g) = d( g) 1 (E 0 a( g) + b( g)) Equations of Motion and Lagrangian L = λ +i Ẽ ij ( g)λ j + λ +i ( g 1 g) i ( g 1 + g) i λ i for λ ±i = ( ±hh 1 ) i null vectors of Ẽ act as Lagrange multipliers +λ c λ +c f ab c λ +aλ b = 0 ( g 1 g) a + Ẽ ab ( g)λ b = 0 ( g 1 + g) a λ +b Ẽ ba ( g) = 0 15 / 17

16 16 / 17 References Baulieu, L., Losev, A. S., & Nekrasov, N. A. (2002). JHEP, 02, 021. Calvo, I. & Falceto, F. (2006). JHEP, 04, 058. Ikeda, N. (1994). Ann. Phys. 235, Klimčík, C. & Ševera, P. (1995). Physics Letters B, 351 (4), Schaller, P. & Strobl, T. (1994). Mod. Phys. Lett. A9, Šnobl, L. & Hlavatý, L. (2002). International Journal of Modern Physics A, 17,

17 17 / 17 Thank you for your attention.