Branes in Poisson sigma models

Branes in Poisson sigma models Fernando Falceto ´ Depto. F´ısica Teorica Universidad de Zaragoza

XVIII Int. Fall Workshop on Geometry and Physics – p.1/21

Branes in Poisson sigma models Fernando Falceto ´ Depto. F´ısica Teorica Universidad de Zaragoza


Poisson sigma model. N. Ikeda: Non linear gauge theories, applications to 2d gravity.


Poisson sigma model. N. Ikeda: Non linear gauge theories, applications to 2d gravity. P. Schaller and T. Strobl: put the model into the right geometric setup and coined its name, applications to gravity, BF theory, Yang-Mills...


Poisson sigma model. N. Ikeda: Non linear gauge theories, applications to 2d gravity. P. Schaller and T. Strobl: put the model into the right geometric setup and coined its name, applications to gravity, BF theory, Yang-Mills... A. Cattaneo and G. Felder: relation to Kontsevich’s deformation quantization, to symplectic groupoids, AKSZ formalism...


Poisson sigma model. N. Ikeda: Non linear gauge theories, applications to 2d gravity. P. Schaller and T. Strobl: put the model into the right geometric setup and coined its name, applications to gravity, BF theory, Yang-Mills... A. Cattaneo and G. Felder: relation to Kontsevich’s deformation quantization, to symplectic groupoids, AKSZ formalism... Cattaneo and Felder: coisotropic branes.


Poisson sigma model. N. Ikeda: Non linear gauge theories, applications to 2d gravity. P. Schaller and T. Strobl: put the model into the right geometric setup and coined its name, applications to gravity, BF theory, Yang-Mills... A. Cattaneo and G. Felder: relation to Kontsevich’s deformation quantization, to symplectic groupoids, AKSZ formalism... Cattaneo and Felder: coisotropic branes. More general branes?


Plan Poisson sigma model Examples Boundary conditions Quantization Final Remarks


Plan Poisson sigma model Examples Boundary conditions Quantization Final Remarks

With: Iván Calvo David García-Álvarez Krzysztof Gawe¸dzki


Poisson sigma model The target: (M, { , }) - { , } a Poisson bracket in C ∞ (M ).


Poisson sigma model The target: (M, { , }) - { , } a Poisson bracket in C ∞ (M ). - In coordinates X = (X 1 , . . . , X n ) for M Πij (X) = {X i , X j }(X)


Poisson sigma model The target: (M, { , })

Πij (X) = {X i , X j }(X)

The fields: - Σ two dimensional space-time (worldsheet). - The fields are given by the bundle map (X, η) : T Σ → T ∗ M

i.e.

X : Σ → M,

η ∈ Ω1 (Σ, X ∗ T ∗ M )

with coordinates σ = (σ 1 , σ 2 ) for Σ η = ηκi (σ)dσ κ dX i = ηi dX i



Πij (X) = {X i , X j }(X)

η = ηκi (σ)dσ κ dX i = ηi dX i

The fields: X : Σ → M , The action:

1 ij S(X, η) = ηi ∧ dX + Π (X)ηi ∧ ηj 2 Σ Z

i



Πij (X) = {X i , X j }(X)

η = ηκi (σ)dσ κ dX i = ηi dX i

The fields: X : Σ → M , The action:

1 ij S(X, η) = ηi ∧ dX + Π (X)ηi ∧ ηj 2 Σ Z

i

The equations of motion: dX − Π♯ (X)η = 0 (Π♯ η)j = Πij ηi 1 dηi + ∂i Πjk (X)ηj ∧ ηk = 0 2

i. e. (X, η) : T Σ → T ∗ M Lie algebroid homomorphism. XVIII Int. Fall Workshop on Geometry and Physics – p.5/21

The Gauge symmetry: Under the transformations δβ X = Π♯ (X)β δβ ηi = dβi + ∂i Πjk ηj βk ,

β = βi (σ)dX i ∈ Γ(X ∗ T ∗ M )


The Gauge symmetry: Under the transformations β = βi (σ)dX i ∈ Γ(X ∗ T ∗ M ) δβ X = Π♯ (X)β δβ ηi = dβi + ∂i Πjk ηj βk , Z δβ S = d(dX i βi ). Σ


The Gauge symmetry: Under the transformations β = βi (σ)dX i ∈ Γ(X ∗ T ∗ M ) δβ X = Π♯ (X)β δβ ηi = dβi + ∂i Πjk ηj βk , Z δβ S = d(dX i βi ). Σ

[δβ , δβ ′ ]X i = δ[β,β ′ ] X i [δβ , δβ ′ ]ηi = δ[β,β ′ ] ηi −βk βl′ ∂i ∂j Πkl (dX j − Πsj ηs )

With [β, β ′ ]k = βi βj′ ∂k Πij (X) XVIII Int. Fall Workshop on Geometry and Physics – p.6/21

Examples R2 gravity in two dimensions

- dim(M )=3 - (η1 , η2 , η3 ) ≡ (e1 , e2 , ω)

(zweibein and connection)



- dim(M )=3 - (η1 , η2 , η3 ) ≡ (e1 , e2 , ω)


Π23 (X) = X 1 , Π31 (X) = X 2 Π12 (X) = −(X 3 )2 + Λ



- dim(M )=3 - (η1 , η2 , η3 ) ≡ (e1 , e2 , ω)


Π23 (X) = X 1 , Π31 (X) = X 2 Π12 (X) = −(X 3 )2 + Λ

- Then the Poisson sigma model in (M, {., .}) upon integration of X -fields, leads to 2-d R2 gravity. Z 1 2 √ 2 gd σ R +Λ SR2 = Σ 4


Examples BF theories.

- (g, [., .]) any Lie algebra, M = g∗ .



- (g, [., .]) any Lie algebra, M = g∗ . - A, B ∈ g viewed as linear functions on g∗ .

- Then {A, B} = [A, B] defines the Kostant-Kirillov-Souriau Poisson bracket in g∗



- (g, [., .]) any Lie algebra, M = g∗ . - A, B ∈ g viewed as linear functions on g∗ .

- Then {A, B} = [A, B] defines the Kostant-Kirillov-Souriau Poisson bracket in g∗ The action in this case is equivalent to Z SBF = X i Fi Σ

with X(σ) ∈ g∗ and F = dη + [η, η] ∈ Ω2 (M ) ⊗ g


Examples Poisson-Lie sigma models. For any Poisson-Lie group (G, {., .}) we can define its Poisson-sigma model. It has several interesting properties:


Examples Poisson-Lie sigma models. For any Poisson-Lie group (G, {., .}) we can define its Poisson-sigma model. It has several interesting properties: Generalizes the BF-theory.


Examples Poisson-Lie sigma models. For any Poisson-Lie group (G, {., .}) we can define its Poisson-sigma model. It has several interesting properties: Generalizes the BF-theory. The gauge group is the dual Poisson-Lie group (G∗ , {., .}∗ ), acting by dressing transformation.


Examples Poisson-Lie sigma models. For any Poisson-Lie group (G, {., .}) we can define its Poisson-sigma model. It has several interesting properties: Generalizes the BF-theory. The gauge group is the dual Poisson-Lie group (G∗ , {., .}∗ ), acting by dressing transformation. Duality in the Hamiltonian formulation.


Examples Poisson-Lie sigma models. For any Poisson-Lie group (G, {., .}) we can define its Poisson-sigma model. It has several interesting properties: Generalizes the BF-theory. The gauge group is the dual Poisson-Lie group (G∗ , {., .}∗ ), acting by dressing transformation. Duality in the Hamiltonian formulation. With group (G∗ , {., .}∗ ) it is equivalent to G/G WZW model.


Boundary conditions Take a worldsheet with boundary. ι : ∂Σ ֒→ Σ Put a brane N ⊂ M . i.e. X : Σ → M s.t. ι∗ X : ∂Σ → N


Boundary conditions Take a worldsheet with boundary. ι : ∂Σ ֒→ Σ Put a brane N ⊂ M . i.e. X : Σ → M s.t. ι∗ X : ∂Σ → N Z Z 1 i i δX S = − δX ηi + δX (dηi + ∂i Πjk ηj ∧ ηk ) 2 ∂Σ Σ We must have (ι∗ X, ι∗ η) : T ∂Σ → T N ◦

T N ◦ ⊂ TN∗ M

T Np◦ = {ξ ∈ Tp∗ M |ξ(v) = 0 ∀v ∈ Tp N }


Boundary conditions Take a worldsheet with boundary. ι : ∂Σ ֒→ Σ Put a brane N ⊂ M . i.e. X : Σ → M s.t. ι∗ X : ∂Σ → N Z Z 1 i i δX S = − δX ηi + δX (dηi + ∂i Πjk ηj ∧ ηk ) 2 ∂Σ Σ We must have (ι∗ X, ι∗ η) : T ∂Σ → T N ◦

T N ◦ ⊂ TN∗ M

From the equations of motion, ( dX − Π♯ η = 0 ) ι∗ dX = Π♯ (ι∗ X)ι∗ η ⇒ ⇒ Π♯ (ι∗ X)ι∗ η ∈ Ω1 (∂Σ, ι∗ X ∗ T N ) XVIII Int. Fall Workshop on Geometry and Physics – p.10/21

Boundary conditions The boundary conditions (B.C.) for a brane N ⊂ M are: ∗

∗

(ι X, ι η) : T ∂Σ → AN

◦

AN := T N ∩ Π

♯ −1

(T N ) ⊂ TN∗ M



∗

(ι X, ι η) : T ∂Σ → AN

◦

AN := T N ∩ Π

♯ −1

(T N ) ⊂ TN∗ M

We assume AN of constant rank. (regular brane or pre-Poisson)



∗

(ι X, ι η) : T ∂Σ → AN

◦

AN := T N ∩ Π

♯ −1

(T N ) ⊂ TN∗ M

We assume AN of constant rank. (regular brane or pre-Poisson) Then: - AN is a Lie subalgebroid of T ∗ M .



∗

(ι X, ι η) : T ∂Σ → AN

◦

AN := T N ∩ Π

♯ −1

(T N ) ⊂ TN∗ M

We assume AN of constant rank. (regular brane or pre-Poisson) Then: - AN is a Lie subalgebroid of T ∗ M . - The gauge transformation δβ subject to the same B. C. ι∗ β ∈ Γ(ι∗ X ∗ AN )

preserves B.C. and is a symmetry. XVIII Int. Fall Workshop on Geometry and Physics – p.11/21

Pre-Poisson branes (examples) Free Boundary Conditions: N = M then AN = 0.

AN := T N ◦ ∩ Π♯

−1

(T N )



AN := T N ◦ ∩ Π♯

−1

(T N )

Coisotropic brane: Dirac’s first class constraints. Π♯ (T N ◦ ) ⊂ T N ⇔ AN = T N ◦ .



AN := T N ◦ ∩ Π♯

−1

(T N )


Constant rank Poisson-Dirac: AN ⊂ Ker(Π♯ ). Reduction of symplectic groupoids (M. Crainic, R. L. Fernandes)



AN := T N ◦ ∩ Π♯

−1

(T N )


Constant rank Poisson-Dirac: AN ⊂ Ker(Π♯ ). Reduction of symplectic groupoids (M. Crainic, R. L. Fernandes) Cosymplectic brane: Dirac’s second class constraints. AN = 0.



AN := T N ◦ ∩ Π♯

−1

(T N )


Constant rank Poisson-Dirac: AN ⊂ Ker(Π♯ ). Reduction of symplectic groupoids (M. Crainic, R. L. Fernandes) Cosymplectic brane: Dirac’s second class constraints. AN = 0. Theorem: Every pre-Poisson submanifold can be embedded coisotropically in a cosymplectic submanifold. XVIII Int. Fall Workshop on Geometry and Physics – p.12/21

Quantization Batalin-Vilkoviski quantization Poisson sigma model has a gauge symmetry of the open type (its algebra closes only on-shell).


Quantization The fields X i , ηi the original fields. βi , γ i the ghost and antighosts. λi the auxiliary field (Lagrange multiplier) ∗ Hodge star operator Lorenz gauge d ∗ ηi = 0


Quantization The fields X i , ηi the original fields. βi , γ i the ghost and antighosts. λi the auxiliary field (Lagrange multiplier) ∗ Hodge star operator Lorenz gauge d ∗ ηi = 0 The gauge fixed action Z 1 ij i Sgf = ηi ∧ dX + Π (X)ηi ∧ ηj − λi d ∗ ηi − 2 Σ − ∗ dγ i ∧ (dβi + ∂i Πkl (X)ηk βl )− 1 − ∗ dγ i ∧ ∗dγ j ∂i ∂j Πkl (X)βk βl 4


Quantization The fields X i , ηi the original fields. βi , γ i the ghost and antighosts. λi the auxiliary field (Lagrange multiplier) ∗ Hodge star operator Lorenz gauge d ∗ ηi = 0 The gauge fixed action Z 1 ij i Sgf = ηi ∧ dX + Π (X)ηi ∧ ηj − λi d ∗ ηi − 2 Σ − ∗ dγ i ∧ (dβi + ∂i Πkl (X)ηk βl )− 1 − ∗ dγ i ∧ ∗dγ j ∂i ∂j Πkl (X)βk βl 4 Perturbative expansion. XVIII Int. Fall Workshop on Geometry and Physics – p.13/21

Free B. C. N = M

8

Σ = D the unit disk. Pick three points at the boundary 0, 1, ∞.

0

1


Free B. C. N = M

Then the perturbative expansion of R iS e ~ gf f (X(0))g(X(1))δ(X(∞) − x)

gives the Kontsevich’s star product. f ⋆ g(x) = f (x)g(x) + i ~2 {f, g}(x) + . . .

8

Σ = D the unit disk. Pick three points at the boundary 0, 1, ∞. δx

f

0

1 g


Free B. C. N = M

8

Σ = D the unit disk. Pick three points at the boundary 0, 1, ∞. δx

Then the perturbative expansion of R iS e ~ gf f (X(0))g(X(1))δ(X(∞) − x)

f

gives the Kontsevich’s star product.

0

f ⋆ g(x) = f (x)g(x) + i ~2 {f, g}(x) + . . . δx

f

g

δx

h

=

δx

f

h f

g

(f ⋆ g) ⋆ h

1

=

h

g

f ⋆ (g ⋆ h) g XVIII Int. Fall Workshop on Geometry and Physics – p.14/21

Coisotropic brane. Adapted coordinates X = (X a , X µ ),

N = {(X a , X µ = 0)}


Coisotropic brane. Adaptedcoordinates X = (X a , X µ ), ι∗ X a = f ree, ι∗ ηa = 0 B. C. 

N = {(X a , X µ = 0)} a


Coisotropic brane. Adaptedcoordinates X = (X a , X µ ), ι∗ X a = f ree, ι∗ ηa = 0 B. C. ι∗ X µ = 0, ι∗ ηµ = f ree

N = {(X a , X µ = 0)} a

µ



The perturbative expansion of Z i e ~ Sgf f (X(0))g(X(1))δ(X(∞) − x) ι∗ X∈N

N = {(X a , X µ = 0)} a

µ

δx

f

0

1 g



The perturbative expansion of Z i e ~ Sgf f (X(0))g(X(1))δ(X(∞) − x) ι∗ X∈N

N = {(X a , X µ = 0)} a

µ

δx

f

0

defines an associative ⋆ product in A~N

≡ {f ∈

C ∞ (N )[[~]]

if anomaly vanishes.

s.t.

δ ~ (N )f

= 0},

1 g

δ ~ (N )X i = Πµi (X)βµ + ...


Coisotropic branes. Bimodules. N0 , N1 coisotropic branes with vanishing anomaly. δ ~ (N0 , N1 )X i = Πiν βν + . . .

(dX ν ) a basis of T N0◦ ∩ T N1◦ .

A~N0 N1 ≡ {f ∈ C ∞ (N0 ∩ N1 )[[~]] s.t. δ ~ (N0 , N1 )f = 0}




A~N0 N1 ≡ {f ∈ C ∞ (N0 ∩ N1 )[[~]] s.t. δ ~ (N0 , N1 )f = 0}

We define the action of A~N0 and A~N1 on A~N0 N1

δx

δx

f ◮ Ψ(x) =

Ψ ◭ g(x) =

f N0

Ψ

N1

g N0

Ψ

N1

Which makes A~N0 N1 a A~N0 -bimodule-A~N1 .




A~N0 N1 ≡ {f ∈ C ∞ (N0 ∩ N1 )[[~]] s.t. δ ~ (N0 , N1 )f = 0}

We define the action of A~N0 and A~N1 on A~N0 N1

δx

δx

f ◮ Ψ(x) =

Ψ ◭ g(x) =

f N0

Ψ

N1

g N0

Ψ

N1

Which makes A~N0 N1 a A~N0 -bimodule-A~N1 . Quantization of Poisson maps. XVIII Int. Fall Workshop on Geometry and Physics – p.16/21

Cosymplectic brane Adapted coordinates X = (X a , X A ),

N = {(X a , X A = 0)}


Cosymplectic brane Adapted coordinates X = (X a , X A ),  ∗ X a = f ree, ∗η = 0  ι ι a  B. C.  

N = {(X a , X A = 0)} a


Cosymplectic brane Adapted coordinates X = (X a , X A ),  ∗ X a = f ree, ∗η = 0  ι ι a  B. C.  ι∗ X A = 0, ι ∗ ηA = 0

N = {(X a , X A = 0)} a

A


Cosymplectic brane Adapted coordinates X = (X a , X A ),  ∗ X a = f ree, ∗η = 0  ι ι a  B. C.  ι∗ X A = 0, ι ∗ ηA = 0 1st

idea.

Sgf

N = {(X a , X A = 0)} a

A

1 ij = ηi ∧ dX + Π (X)ηi ∧ ηj − λi d ∗ ηi − 2 Σ − ∗ dγ i ∧ (dβi + ∂i Πkl (X)ηk βl )− 1 − ∗ dγ i ∧ ∗dγ j ∂i ∂j Πkl (X)βk βl 4 Z

i

det ΠAB (x) 6= 0, perform the Gaussian integration in ηA


Cosymplectic brane Adapted coordinates X = (X a , X A ),  ∗ X a = f ree, ∗η = 0  ι ι a  B. C.  ι∗ X A = 0, ι ∗ ηA = 0 1st

idea.

Sgf

N = {(X a , X A = 0)} a

A

1 ij = ηi ∧ dX + Π (X)ηi ∧ ηj − λi d ∗ ηi − 2 Σ − ∗ dγ i ∧ (dβi + ∂i Πkl (X)ηk βl )− 1 − ∗ dγ i ∧ ∗dγ j ∂i ∂j Πkl (X)βk βl 4 Z

i

det ΠAB (x) 6= 0, perform the Gaussian integration in ηA

Effective action has a well defined pert. expansion. It is hard to compute and relate to star product.


Cosymplectic brane 2nd idea Change gauge fixing: d ∗ ηa = 0, X A = 0 for cosymplectic branes only: δβ X A = ΠAB βB + ΠAa βa . λa and λA new Lagrange multipliers.


Cosymplectic brane 2nd idea Change gauge fixing: d ∗ ηa = 0, X A = 0 for cosymplectic branes only: δβ X A = ΠAB βB + ΠAa βa . λa and λA new Lagrange multipliers. Z 1 ij i Sgf = ηi ∧ dX + Π (X)ηi ∧ ηj − λa d ∗ ηa − λA X A − 2 Σ − ∗ dγ a ∧ (dβa + ∂a Πij (X)ηi βj ) − γA ΠAi (X)βi − 1 − ∗ dγ a ∧ ∗dγ b ∂a ∂b Πij (X)βi βj 4


Cosymplectic brane 2nd idea Change gauge fixing: d ∗ ηa = 0, X A = 0 for cosymplectic branes only: δβ X A = ΠAB βB + ΠAa βa . λa and λA new Lagrange multipliers. Z 1 ij i Sgf = ηi ∧ dX + Π (X)ηi ∧ ηj − λa d ∗ ηa − λA X A − 2 Σ − ∗ dγ a ∧ (dβa + ∂a Πij (X)ηi βj ) − γA ΠAi (X)βi − 1 − ∗ dγ a ∧ ∗dγ b ∂a ∂b Πij (X)βi βj 4

Integrating in λA , γA (linear) and in ηA (quadratic). One obtains the effective action XVIII Int. Fall Workshop on Geometry and Physics – p.18/21

Cosymplectic brane eff Sgf

1 ab = ηa ∧ dX + ΠD (X)ηa ∧ ηb − λa d ∗ ηa 2 Σ − ∗ dγ a ∧ (dβa + ∂a Πcd D (X)ηc βd )− 1 − ∗ dγ a ∧ ∗dγ b ∂a ∂b Πcd D (X)βc βd 4 Z

a

ab − ΠaA Π Bb , the Dirac bracket in N . Πab = Π Π AB D


Cosymplectic brane eff Sgf

1 ab = ηa ∧ dX + ΠD (X)ηa ∧ ηb − λa d ∗ ηa 2 Σ − ∗ dγ a ∧ (dβa + ∂a Πcd D (X)ηc βd )− 1 − ∗ dγ a ∧ ∗dγ b ∂a ∂b Πcd D (X)βc βd 4 Z

a

ab − ΠaA Π Bb , the Dirac bracket in N . Πab = Π Π AB D Z i ~ e Sgf f (X(0))g(X(1))δ(X(∞) − x) = f ⋆D g ι∗ X∈N

defines an associative product in A~N = C ∞ (N )[[h]].


Pre-Poisson brane Adaptedcoordinates X ∗ X a = f ree,  ι  B. C. ι∗ X µ = 0,  ∗ A ι X = 0,

= (X a , X µ , X A ) = (X p , X A ). brane ι∗ ηa = 0. ι∗ ηµ = f ree. 1st class ι∗ ηA = 0. 2nd class

Gauge fixing: d ∗ ηp = 0, X A = 0.




Gauge fixing: d ∗ ηp = 0, X A = 0. Z 1 pq eff p Sgf = ηp ∧ dX + ΠD (X)ηp ∧ ηq − λp d ∗ ηp 2 Σ − ∗ dγ p ∧ (dβp + ∂p Πqr D (X)ηq βr )− 1 − ∗ dγ p ∧ ∗dγ q ∂p ∂q Πrs D (X)βr βs 4




Gauge fixing: d ∗ ηp = 0, X A = 0. Z 1 pq eff p Sgf = ηp ∧ dX + ΠD (X)ηp ∧ ηq − λp d ∗ ηp 2 Σ − ∗ dγ p ∧ (dβp + ∂p Πqr D (X)ηq βr )− 1 − ∗ dγ p ∧ ∗dγ q ∂p ∂q Πrs D (X)βr βs 4

i. e. it defines an effective Poisson sigma model in M ′ = {(X a , X µ , X A = 0)} with brane N ′ = {(X a , X µ = 0)}. XVIII Int. Fall Workshop on Geometry and Physics – p.20/21

Final remarks Unlike the coisotropic branes, the cosymplectic ones do always lead to associative star products.


Final remarks Unlike the coisotropic branes, the cosymplectic ones do always lead to associative star products. Cosymplectic branes redefine the perturbative expansion.


Final remarks Unlike the coisotropic branes, the cosymplectic ones do always lead to associative star products. Cosymplectic branes redefine the perturbative expansion. Deformation of branes, from cosymplectic to coisotropic.


Final remarks Unlike the coisotropic branes, the cosymplectic ones do always lead to associative star products. Cosymplectic branes redefine the perturbative expansion. Deformation of branes, from cosymplectic to coisotropic. Quantization with several branes?




