Moduli spaces of flat connections on colored surfaces David - PowerPoint PPT Presentation

Moduli spaces of flat connections on colored surfaces David Li-Bland joint work with Pavol ˇ Severa Tuesday, July 31, 2012

Introduction Moduli space of flat connections Towards a finite dimensional construction Flat connections on the 1-simplex Flat connections on the 2-simplex Main Theorem Examples Coloring Edges Domain Walls Coloring n -edges Poisson Structures on Moduli spaces

Moduli space of flat connections Let ( g , �· , ·� ) be a quadratic Lie algebra, Σ = Theorem (Atiyah-Bott) The moduli space M (Σ) = A flat (Σ) / C ∞ (Σ , G ) of flat connections over Σ carries a symplectic structure. Proof. Infinite dimensional symplectic reduction...

Introduction Moduli space of flat connections Towards a finite dimensional construction Flat connections on the 1-simplex Flat connections on the 2-simplex Main Theorem Examples Coloring Edges Domain Walls Coloring n -edges Poisson Structures on Moduli spaces

Finite Dimensional Construction Triangulate the surface: a 2 b 1 a 1 b 2 A flat connection assigns an element of G (the holonomy) to each edge. M (Σ) is collection of possible (coherent) assignments.

Finite Dimensional Construction Triangulate the surface: b 2 a 1 a 2 b 1 b 2 a 1 a 2 b 1 A flat connection assigns an element of G (the holonomy) to each edge. M (Σ) is collection of possible (coherent) assignments.

Finite Dimensional Construction Triangulate the surface: b 2 a 1 a 2 b 1 b 2 a 1 a 2 b 1 A flat connection assigns an element of G (the holonomy) to each edge. M (Σ) is collection of possible (coherent) assignments.

Finite Dimensional Construction Triangulate the surface: g g 1 g 2 A flat connection assigns an element of G (the holonomy) to each edge. M (Σ) is collection of possible (coherent) assignments.

Finite Dimensional Construction Triangulate the surface: g g 1 g 2 A flat connection assigns an element of G (the holonomy) to each edge. M (Σ) is collection of possible (coherent) assignments.

Flat connections and triangulations A triangulation breaks our surface into ◮ vertices (0-dimensional simplex) ◮ edges (1-dimensional simplex) ◮ faces (2-dimensional simplex) What does a flat connection look like over simplices of these dimensions?

Flat connections and triangulations A triangulation breaks our surface into ◮ vertices (0-dimensional simplex) ◮ edges (1-dimensional simplex) ◮ faces (2-dimensional simplex) What does a flat connection look like over simplices of these dimensions?

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } hol : G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G hol : G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G hol : G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G Holonomy = g − 1 0 g 1 g 0 g 1 hol : A flat ([0 , 1]) → G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G Holonomy = g − 1 0 g 1 g 0 g 1 hol : A flat ([0 , 1]) → G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G Holonomy = g − 1 0 g 1 g 0 g 1 hol : A flat ([0 , 1]) → G

Flat connections on the 1-simplex ◮ A flat ([0 , 1]) := flat connections over [0 , 1] ◮ C ∞ based ([0 , 1] , G ) := { f such that f (0) = f (1) = id } G G Holonomy = g − 1 0 g 1 g 0 g 1 ∼ = hol : A flat ([0 , 1]) / C ∞ based ([0 , 1] , G ) − − − − → G

Residual gauge transformations G G Holonomy = g − 1 0 g 1 g 0 g 1 C ∞ ([0 , 1] , G ) A flat ([0 , 1])

Residual gauge transformations G G Holonomy = g − 1 0 g 1 g 0 g 1 C ∞ ([0 , 1] , G ) / C ∞ based ([0 , 1] , G ) A flat ([0 , 1]) / C ∞ based ([0 , 1] , G )

Residual gauge transformations G G Holonomy = g − 1 0 g 1 g 0 g 1 C ∞ ([0 , 1] , G ) / C ∞ based ([0 , 1] , G ) G × G ∼ = A flat ([0 , 1]) / C ∞ based ([0 , 1] , G ) G hol

Residual gauge transformations G G Holonomy = g − 1 0 g 1 g 0 g 1 ¯ g ⊕ g ∼ = A flat ([0 , 1]) / C ∞ based ([0 , 1] , G ) G hol

Pictoral Notation

Pictoral Notation G

Pictoral Notation ¯ g � G � g

Flat connections on the 2-simplex ◮ A flat (∆) := flat connections over ∆ ◮ C ∞ based (∆ , G ) := { f ∈ such that f (0) = f (1) = f (2) = id } 1 0 2 ⊆ G 3

Flat connections on the 2-simplex ◮ A flat (∆) := flat connections over ∆ ◮ C ∞ based (∆ , G ) := { f ∈ such that f (0) = f (1) = f (2) = id } 1 0 2 ⊆ G 3

Flat connections on the 2-simplex ◮ A flat (∆) := flat connections over ∆ ◮ C ∞ based (∆ , G ) := { f ∈ such that f (0) = f (1) = f (2) = id } G G G A flat (∆) / C ∞ based (∆ , G ) ⊆ G 3

Flat connections on the 2-simplex ◮ A flat (∆) := flat connections over ∆ ◮ C ∞ based (∆ , G ) := { f ∈ such that f (0) = f (1) = f (2) = id } g 2 g 1 g 3 A flat (∆) / C ∞ based (∆ , G ) = { g 1 g 2 g 3 = id } ⊆ G 3

Flat connections on the 2-simplex ◮ A flat (∆) := flat connections over ∆ ◮ C ∞ based (∆ , G ) := { f ∈ such that f (0) = f (1) = f (2) = id } g 2 g 1 g 3 M (∆) = { g 1 g 2 g 3 = id } ⊆ G 3

Residual gauge transformations hol : M (∆) → G 3 g 2 hol G g 1 G ( g 1 g 2 ) − 1 G g 3 ∼ g ⊕ g ) 3 preserves M (∆) = g ∆ ⊆ (¯

Residual gauge transformations hol : M (∆) → G 3 g � G � g ¯ g � G � g g 2 hol g 1 ¯ ( g 1 g 2 ) − 1 ¯ g � G � g g 3 ∼ g ⊕ g ) 3 preserves M (∆) = g ∆ ⊆ (¯

Residual gauge transformations hol : M (∆) → G 3 ξ � G � η ζ � G � ξ g 2 hol g 1 g ∆ -equivariant ( g 1 g 2 ) − 1 ζ � G � η g 3 ∼ g ⊕ g ) 3 preserves M (∆) = g ∆ ⊆ (¯

. . . � �

2-simplices hol . . . . . . Still need to take quotient by gauge transformations.

2-simplices hol . . . . . . 1-skeleton Still need to take quotient by gauge transformations.

2-simplices hol . . . . . . hol − 1 (1-skeleton) 1-skeleton Still need to take quotient by gauge transformations.

2-simplices hol . . . . . . hol − 1 (1-skeleton) 1-skeleton Still need to take quotient by gauge transformations.

2-simplices hol . . . . . . Moment Map hol − 1 (1-skeleton) 1-skeleton Still need to take quotient by gauge transformations.

Moment Map (Bursztyn,Iglesias-Ponte,ˇ Severa) ξ � G � η ζ � G � ξ g 2 hol g 1 g ∆ -equivariant ( g 1 g 2 ) − 1 ζ � G � η ◮ (¯ g ⊕ g ) × G is a Courant algebroid. g ⊕ g ) 3 defines a Dirac structure. ◮ g ∆ ⊆ (¯ ◮ The action of g ∆ on M (∆) is Hamiltonian for the unique 2-form ω = 1 2 � g − 1 1 dg 1 , dg 2 g − 1 2 � ∈ Ω 2 � � M (∆) .

Moment Map (Bursztyn,Iglesias-Ponte,ˇ Severa) ξ � G � η ζ � G � ξ g 2 hol g 1 g ∆ -equivariant ( g 1 g 2 ) − 1 ζ � G � η ◮ (¯ g ⊕ g ) × G is a Courant algebroid. g ⊕ g ) 3 defines a Dirac structure. ◮ g ∆ ⊆ (¯ ◮ The action of g ∆ on M (∆) is Hamiltonian for the unique 2-form ω = 1 2 � g − 1 1 dg 1 , dg 2 g − 1 2 � ∈ Ω 2 � � M (∆) .

Moment Map (Bursztyn,Iglesias-Ponte,ˇ Severa) ξ � G � η ζ � G � ξ g 2 hol g 1 g ∆ -equivariant ( g 1 g 2 ) − 1 ζ � G � η ◮ (¯ g ⊕ g ) × G is a Courant algebroid. g ⊕ g ) 3 defines a Dirac structure. ◮ g ∆ ⊆ (¯ ◮ The action of g ∆ on M (∆) is Hamiltonian for the unique 2-form ω = 1 2 � g − 1 1 dg 1 , dg 2 g − 1 2 � ∈ Ω 2 � � M (∆) .

Moment Map (Bursztyn,Iglesias-Ponte,ˇ Severa) ξ � G � η ζ � G � ξ g 2 hol g 1 g ∆ -equivariant ( g 1 g 2 ) − 1 ζ � G � η ◮ (¯ g ⊕ g ) × G is a Courant algebroid. g ⊕ g ) 3 defines a Dirac structure. ◮ g ∆ ⊆ (¯ ◮ The action of g ∆ on M (∆) is Hamiltonian for the unique 2-form ω = 1 2 � g − 1 1 dg 1 , dg 2 g − 1 2 � ∈ Ω 2 � � M (∆) .

Reduction, part1 hol M (∆) n G 3 n (moment map = holonomy) g n g n g ⊕ g ) 3 n ∆ ⊆ (¯ ∆ g ⊕ g ) 3 n . Choose l ⊆ (¯ hol hol − 1 ( l · id) (moment level l · id ⊆ G 3 n ) l · id l ∩ g n l ∩ g n ∆ ∆

Reduction, part1 hol M (∆) n G 3 n (moment map = holonomy) g n g n g ⊕ g ) 3 n ∆ ⊆ (¯ ∆ g ⊕ g ) 3 n . Choose l ⊆ (¯ hol hol − 1 ( l · id) (moment level l · id ⊆ G 3 n ) l · id l ∩ g n l ∩ g n ∆ ∆

Reduction, part2 Theorem (Li-Bland, ˇ Severa) g ⊕ g ) 3 n is a Lagrangian Lie subalgebra. Suppose l ⊆ (¯ Then, under suitable transversality assumptions, the restriction of the 2-form to hol − 1 ( l · id) ⊆ M (∆) n descends to define a symplectic form on hol − 1 ( l · id) / ( g n ∆ ∩ l ) .