Introduction to Higgs bundles Steve Bradlow Department of Mathematics University of Illinois at Urbana-Champaign July 23-27, 2012 Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 1 / 1
Disclaimer These slides are precisely as they were during the lectures on July 23, 25, 27, 2012. As such, they contain several omissions and inaccuracies, in both the mathematics and the attributions. Some of these, it must be admitted, are blemishes which reflect the author’s limitations, but others reflect the fact that: The slides formed but one part of the lectures. They were accompanied by verbal commentary designed to explain and embellish the contents of the slides This is not a paper. Any talk has to strike a balance between accuracy and accessibility. This balance inevitably involves the inclusion of some half-truths and/or white lies. The author apologizes to anyone who is in any way led astray by the inaccuracies or slighted by the omissions. Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 2 / 1
Goals and plan for this mini-course What are Higgs bundles? How do they relate to surface group representations? What do we gain by taking the Higgs bundle point of view? Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 3 / 1
Goals and plan for this mini-course What are Higgs bundles? How do they relate to surface group representations? What do we gain by taking the Higgs bundle point of view? The Plan: 1 (Lectures I and II)Description of surface group representations from a bundle perspective, with necessary background to define Higgs bundles and to see their relation to the representations 2 (Lecture III) Examples and properties of Higgs bundles Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 3 / 1
The main dramatis personae S a closed surface of genus g G a Lie group (mostly GL ( n , C ) for us) Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 4 / 1
The main dramatis personae S a closed surface of genus g G a Lie group (mostly GL ( n , C ) for us) Representations ρ : π 1 ( S ) → G i ( a i b i a − 1 b − 1 π 1 ( S ) = < a 1 , . . . , a g , b 1 , . . . , b g | � ) = 1 > i i � a i �→ α i i ( α i β i α − 1 β − 1 ρ : such that � ) = 1 i i b i �→ β i Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 4 / 1
The main dramatis personae S a closed surface of genus g G a Lie group (mostly GL ( n , C ) for us) Representations ρ : π 1 ( S ) → G i ( a i b i a − 1 b − 1 π 1 ( S ) = < a 1 , . . . , a g , b 1 , . . . , b g | � ) = 1 > i i � a i �→ α i i ( α i β i α − 1 β − 1 ρ : such that � ) = 1 i i b i �→ β i Higgs bundles on Σ = ( S , J ), i.e. pairs ( E , ϕ ) E → Σ a rank n holomorphic bundle ϕ : E → E ⊗ ( T 1 , 0 Σ) ∗ , i.e. ϕ ∈ H 0 ( End ( E ⊗ K Σ ) Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 4 / 1
� � � � From ρ : π 1 ( s ) → G to ( E , ϕ ) (with G = GL ( n , C )) � local system ρ : π 1 ( S ) → G on S bundle with ( E , ϕ ) flat connection on Σ on S Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 5 / 1
� Step 1: from ρ : π 1 ( S ) → G to a G -Local Systems ˜ c S × ρ G � S Take the universal cover ˜ S path lifting defines local sections π π 1 ( S ) acts on ˜ S preserving fibers of c S Use ρ : π 1 ( S ) → G to construct ˜ S × G /π 1 ( S ) = ˜ S × ρ G [A local system] Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 6 / 1
� � � � Structure of ˜ S × ρ G ˜ S × ρ G | U α ≃ U α × G Over U α [ σ α ( x ) , g ] ↔ ( x , g ) Over U α ∩ U β [ σ β ( x ) , g ] ( x , g ) [ γ αβ ] σ β ( x ) ρ [ γ αβ ] σ α ( x )= [ σ α ( x ) , ρ [ γ αβ ] g ] ( x , ρ [ γ αβ ] g ) ˜ S × ρ G is a G -Local System described by: { U α } α ∈ I (open cover of S ) { g αβ = ρ ([ γ αβ ]) } (transition data satisfying { g αβ g βδ g δα = 1 } ) Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 7 / 1
Monodromy Any G -Local System defines a represen- tation ρ : π 1 ( S ) → G by monodromy: cover loop γ by U 1 , U 2 , . . . , U k define ρ ([ γ ]) = g N ( N − 1) g ( N − 1)( N − 2) . . . g 21 Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 8 / 1
Monodromy Any G -Local System defines a represen- tation ρ : π 1 ( S ) → G by monodromy: cover loop γ by U 1 , U 2 , . . . , U k define ρ ([ γ ]) = g N ( N − 1) g ( N − 1)( N − 2) . . . g 21 Coming up: G -Local System = bundle with flat connection monodromy = holonomy Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 8 / 1
Vector Bundles Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 9 / 1
Vector Bundles Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 10 / 1
� Bundle Basics: I. Vector bundles over M � E V M a cover { U α } α ∈ I for M local trivializations E | U α ≃ U α × V transition functions (gluing data): g αβ : U α ∩ U β → GL ( V ) Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 11 / 1
� Bundle Basics: I. Vector bundles over M � E V M a cover { U α } α ∈ I for M local trivializations E | U α ≃ U α × V transition functions (gluing data): g αβ : U α ∩ U β → GL ( V ) � E = ( U α × V ) / ∼ where ( x , v α ) ∼ ( x , g αβ ( x ) v β ) α ∈ I Cocycle condition on triple overlaps: g αβ g βγ g γα = 1 Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 11 / 1
� Bundle Basics: I. Vector bundles over M � E V M a cover { U α } α ∈ I for M local trivializations E | U α ≃ U α × V transition functions (gluing data): g αβ : U α ∩ U β → GL ( V ) � E = ( U α × V ) / ∼ where ( x , v α ) ∼ ( x , g αβ ( x ) v β ) α ∈ I Cocycle condition on triple overlaps: g αβ g βγ g γα = 1 Example If g αβ = I for all U α ∩ U β � = ∅ then E = M × V Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 11 / 1
Bundle Basics: Principal and associated bundles { U α } + { g αβ : U α ∩ U β → G } = E G Principal G -bundle � E G = ( U α × G ) / ∼ where ( x , g β ) ∼ ( x , g αβ ( x ) g β ) α ∈ I Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 12 / 1
Bundle Basics: Principal and associated bundles { U α } + { g αβ : U α ∩ U β → G } = E G Principal G -bundle � E G = ( U α × G ) / ∼ where ( x , g β ) ∼ ( x , g αβ ( x ) g β ) α ∈ I E G + { r : G → GL ( V ) } = E G ( V )( or E V ) Associated V -bundle � E G ( V ) = ( U α × V ) / ∼ where ( x , v α ) ∼ ( x , r ( g αβ ( x )) v β ) α ∈ I Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 12 / 1
Bundle Basics: Principal and associated bundles { U α } + { g αβ : U α ∩ U β → G } = E G Principal G -bundle � E G = ( U α × G ) / ∼ where ( x , g β ) ∼ ( x , g αβ ( x ) g β ) α ∈ I E G + { r : G → GL ( V ) } = E G ( V )( or E V ) Associated V -bundle � E G ( V ) = ( U α × V ) / ∼ where ( x , v α ) ∼ ( x , r ( g αβ ( x )) v β ) α ∈ I Example ( G = GL ( n , C )) V = gl ( n , C ); r = adjoint = ⇒ E G ( V ) = End ( E ) Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 12 / 1
G -Local Systems as Principal Bundles A G -Local System is the same thing as a Principal G -bundle described by transition functions that are locally constant, i.e. dg αβ = 0 Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 13 / 1
G -Local Systems as Principal Bundles A G -Local System is the same thing as a Principal G -bundle described by transition functions that are locally constant, i.e. dg αβ = 0 Definition (Flat bundles) For a bundle E , a choice of local trivializations for which dg αβ = 0 is called a flat structure on the bundle. A bundle together with a flat structure is called a flat bundle Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 13 / 1
G -Local Systems as Principal Bundles A G -Local System is the same thing as a Principal G -bundle described by transition functions that are locally constant, i.e. dg αβ = 0 Definition (Flat bundles) For a bundle E , a choice of local trivializations for which dg αβ = 0 is called a flat structure on the bundle. A bundle together with a flat structure is called a flat bundle (With M = S ) Representations correspond to flat principal π 1 ( S ) → G G -bundles Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 13 / 1
� � � � The next step.... � local system = flat bundle ρ : π 1 ( S ) → G on S bundle with ( E , ϕ ) flat connection on Σ on S Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 14 / 1
π � M Connections on vector bundles... E ..provide the solution to the following: 1 At a point q ∈ E which directions are “horizontal” or “parallel to the base” Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 15 / 1
π � M Connections on vector bundles... E ..provide the solution to the following: 1 At a point q ∈ E which directions are “horizontal” or “parallel to the base” 2 How to compare fibers over different points in the base? Steve Bradlow (UIUC) Higgs bundles Urbana-Champaign, July 2012 15 / 1
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