Dirac structures Henrique Bursztyn, IMPA Geometry, mechanics and dynamics: the legacy of J. Marsden Fields Institute, July 2012
Outline: 1. Mechanics and constraints (Dirac s theory) 2. Degenerate symplectic geometry: two viewpoints 3. Origins of Dirac structures 4. Properties of Dirac manifolds 5. Recent developments and applications
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)...
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)... Dirac bracket on second class C = {x, ϕ i (x) = 0}: {f, g} Dirac := {f, g} {f, ϕ i }c ij {ϕ j, g}
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)... Dirac bracket on second class C = {x, ϕ i (x) = 0}: {f, g} Dirac := {f, g} {f, ϕ i }c ij {ϕ j, g} Geometric meaning of relationship between brackets?
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)... Dirac bracket on second class C = {x, ϕ i (x) = 0}: {f, g} Dirac := {f, g} {f, ϕ i }c ij {ϕ j, g} Geometric meaning of relationship between brackets? Intrinsic geometry of constraint submanifold = presymplectic
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)... Dirac bracket on second class C = {x, ϕ i (x) = 0}: {f, g} Dirac := {f, g} {f, ϕ i }c ij {ϕ j, g} Geometric meaning of relationship between brackets? Intrinsic geometry of constraint submanifold = presymplectic Questions: Intrinsic geometry of constraints in Poisson phase spaces?
1. Phase spaces and constraints Symplectic phase space with constraint submanifold C M First class (coisotropic), second class (symplectic)... Dirac bracket on second class C = {x, ϕ i (x) = 0}: {f, g} Dirac := {f, g} {f, ϕ i }c ij {ϕ j, g} Geometric meaning of relationship between brackets? Intrinsic geometry of constraint submanifold = presymplectic Questions: Intrinsic geometry of constraints in Poisson phase spaces? Global structure behind presymplectic foliations?
2. Two viewpoints to symplectic geometry nondegenerate ω Ω 2 (M) nondegenerate π Γ( 2 T M) dω = 0 [π, π] = 0 i Xf ω = df X f = π (df) {f, g} = ω(x g, X f ) {f, g} = π(df, dg)
2. Two viewpoints to symplectic geometry nondegenerate ω Ω 2 (M) nondegenerate π Γ( 2 T M) dω = 0 [π, π] = 0 i Xf ω = df X f = π (df) {f, g} = ω(x g, X f ) {f, g} = π(df, dg) Going degenerate: presymplectic and Poisson geometries...
3. Origins of Dirac structures T. Courant s thesis (1990): Unified approach to presymplectic / Poisson structures
3. Origins of Dirac structures T. Courant s thesis (1990): Unified approach to presymplectic / Poisson structures Dirac structure: subbundle L TM = T M T M such that L = L
3. Origins of Dirac structures T. Courant s thesis (1990): Unified approach to presymplectic / Poisson structures Dirac structure: subbundle L TM = T M T M such that L = L [Γ(L), Γ(L)] Γ(L) (integrability)
3. Origins of Dirac structures T. Courant s thesis (1990): Unified approach to presymplectic / Poisson structures Dirac structure: subbundle L TM = T M T M such that L = L [Γ(L), Γ(L)] Γ(L) (integrability) Courant bracket on Γ(TM): [(X, α), (Y, β)] = ([X, Y ], L X β L Y α 1 d(β(x) α(y ))). 2
3. Origins of Dirac structures T. Courant s thesis (1990): Unified approach to presymplectic / Poisson structures Dirac structure: subbundle L TM = T M T M such that L = L [Γ(L), Γ(L)] Γ(L) (integrability) Courant bracket on Γ(TM): [(X, α), (Y, β)] = ([X, Y ], L X β L Y α 1 d(β(x) α(y ))). 2 Non-skew bracket: [(X, α), (Y, β)] = ([X, Y ], L X β i Y dα).
initial examples... Examples
Examples initial examples... Another example... M = R 3, coordinates (x, y, z) L = span (, zdx), (, zdy), (0, dz) y x
Examples initial examples... Another example... M = R 3, coordinates (x, y, z) L = span (, zdx), (, zdy), (0, dz) y x For z 0, this is graph of π = 1 z : x y {x, y} = 1, {x, z} = 0, {y, z} = 0. z singular Poisson versus smooth Dirac...
Lie algebroid... Presymplectic foliation 4. Properties of Dirac manifolds
4. Properties of Dirac manifolds Lie algebroid... Presymplectic foliation Nondegenerate Poisson structure Symplectic structure Poisson structure Symplectic foliation Dirac structure Presymplectic foliation
4. Properties of Dirac manifolds Lie algebroid... Presymplectic foliation Nondegenerate Poisson structure Symplectic structure Poisson structure Symplectic foliation Dirac structure Presymplectic foliation Hamiltonian vector fields
4. Properties of Dirac manifolds Lie algebroid... Presymplectic foliation Nondegenerate Poisson structure Symplectic structure Poisson structure Symplectic foliation Dirac structure Presymplectic foliation Hamiltonian vector fields Poisson algebra of admissible functions
4. Properties of Dirac manifolds Lie algebroid... Presymplectic foliation Nondegenerate Poisson structure Symplectic structure Poisson structure Symplectic foliation Dirac structure Presymplectic foliation Hamiltonian vector fields Poisson algebra of admissible functions Quotient Poisson manifolds...
4. Properties of Dirac manifolds Lie algebroid... Presymplectic foliation Nondegenerate Poisson structure Symplectic structure Poisson structure Symplectic foliation Dirac structure Presymplectic foliation Hamiltonian vector fields Poisson algebra of admissible functions Quotient Poisson manifolds... Dirac structures = pre-poisson
Inducing Dirac structures on submanifolds ϕ : C (M, L),
Inducing Dirac structures on submanifolds ϕ : C (M, L), L C := L (T C T M) L T C T C T C.
Inducing Dirac structures on submanifolds ϕ : C (M, L), L C := L (T C T M) L T C T C T C. Smoothness issue Try pulling back π = x x y to x-axis...
Inducing Dirac structures on submanifolds ϕ : C (M, L), L C := L (T C T M) L T C T C T C. Smoothness issue Try pulling back π = x x y to x-axis... Transversality condition: Enough that L T C has constant rank.
Poisson-Dirac submanifolds of Poisson manifolds (M, π). Pull-back of π to C is smooth and Poisson (T C π (T C ) = 0)
Poisson-Dirac submanifolds of Poisson manifolds (M, π). Pull-back of π to C is smooth and Poisson (T C π (T C ) = 0) Leafwise symplectic submanifolds : generalizes symplectic submanifolds to Poisson world...
Poisson-Dirac submanifolds of Poisson manifolds (M, π). Pull-back of π to C is smooth and Poisson (T C π (T C ) = 0) Leafwise symplectic submanifolds : generalizes symplectic submanifolds to Poisson world... induced bracket extends Dirac bracket
Poisson-Dirac submanifolds of Poisson manifolds (M, π). Pull-back of π to C is smooth and Poisson (T C π (T C ) = 0) Leafwise symplectic submanifolds : generalizes symplectic submanifolds to Poisson world... induced bracket extends Dirac bracket Moment level sets J : M g Poisson map (=moment map), C = J 1 (0) M Transversality ok e.g. if 0 is regular value, g-action free. Moment level set inherits Dirac structure.
Poisson-Dirac submanifolds of Poisson manifolds (M, π). Pull-back of π to C is smooth and Poisson (T C π (T C ) = 0) Leafwise symplectic submanifolds : generalizes symplectic submanifolds to Poisson world... induced bracket extends Dirac bracket Moment level sets J : M g Poisson map (=moment map), C = J 1 (0) M Transversality ok e.g. if 0 is regular value, g-action free. Moment level set inherits Dirac structure. Dirac geometry = intrinsic geometry of constraints...
5. Recent developments and applications Courant algebroids, twist by closed 3-forms Lie algebroids/groupoids, equivariant cohomology Generalized symmetries and moment maps (e.g. G-valued...) Spinors and generalized complex geometry Supergeometric viewpoint Back to mechanics: Lagrangian systems with constraints (nonholonomic), implicit Hamiltonian systems (e.g. electric circuits); generalizations to field theory (multi-dirac)... Geometry of nonholonomic brackets... among others...
Twists by closed 3-forms
Twists by closed 3-forms Consider closed 3-form φ Ω 3 cl(m):
Twists by closed 3-forms Consider closed 3-form φ Ω 3 cl(m): φ-twisted Courant bracket: [(X, α), (Y, β)] φ = [(X, α), (Y, β)] + i Y i X φ.
Twists by closed 3-forms Consider closed 3-form φ Ω 3 cl(m): φ-twisted Courant bracket: [(X, α), (Y, β)] φ = [(X, α), (Y, β)] + i Y i X φ. Then Dirac structures: modified integrability conditions, but similar properties... Twisted Poisson structure: 1 2 [π, π] = π (φ)
The Cartan-Dirac structure on Lie groups G Lie group,, g : g g R Ad-invariant.
The Cartan-Dirac structure on Lie groups G Lie group,, g : g g R Ad-invariant. Cartan-Dirac structure: L G := {(u r u l, 1 u r + u l, 2 ) u g}. g This is φ G -integrable, where φ G Ω 3 (M) is the Cartan 3-form.
The Cartan-Dirac structure on Lie groups G Lie group,, g : g g R Ad-invariant. Cartan-Dirac structure: L G := {(u r u l, 1 2 u r + u l, g ) u g}. This is φ G -integrable, where φ G Ω 3 (M) is the Cartan 3-form. Singular foliation: Conjugacy classes Leafwise 2-form (G.H.J.W. 97): ω(u G, v G ) g := Adg Ad g 1 2 u, v g
The Cartan-Dirac structure on Lie groups G Lie group,, g : g g R Ad-invariant. Cartan-Dirac structure: L G := {(u r u l, 1 2 u r + u l, g ) u g}. This is φ G -integrable, where φ G Ω 3 (M) is the Cartan 3-form. Singular foliation: Conjugacy classes Leafwise 2-form (G.H.J.W. 97): ω(u G, v G ) g := Compare with Lie-Poisson on g... Adg Ad g 1 2 u, v g
Supergeometric viewpoint
Supergeometric viewpoint (E,, ) (M, {, }) deg. 2, symplectic N-manifold [, ], ρ Θ C 3 (M), {Θ, Θ} = 0 L E, L = L L M Lagrangian submanifold Dirac structure L, [Γ(L), Γ(L)] Γ(L) Lagrangian submf. L, Θ L cont.
Supergeometric viewpoint (E,, ) (M, {, }) deg. 2, symplectic N-manifold [, ], ρ Θ C 3 (M), {Θ, Θ} = 0 L E, L = L L M Lagrangian submanifold Dirac structure L, [Γ(L), Γ(L)] Γ(L) Lagrangian submf. L, Θ L cont. After all, everything is a Lagrangian submanifold...