Meromorphic connections and the Stokes groupoids Brent Pym (Oxford) Based on arXiv:1305.7288 ( Crelle 2015) with Marco Gualtieri and Songhao Li 1 / 34
Warmup Exercise Find the flat sections of the connection � 1 � dz − z ∇ = d − 0 0 z 2 on the trivial bundle E = O ⊕ 2 over the curve X = C . X i.e. find a fundamental matrix solution of the ODE � z − 2 � − z − 1 d ψ dz = ψ 0 0 NB: Pole of order two, i.e. ∇ : E → Ω 1 X ( D ) ⊗ E , where D = 2 · { 0 } ⊂ X . 2 / 34
Solution method Goal: flat sections of � 1 � dz − z ∇ = d − 0 0 z 2 Strategy: Find a gauge transformation φ taking ∇ to the simpler diagonal connection � 1 � dz 0 ∇ 0 = φ − 1 ∇ φ = d − 0 0 z 2 Solutions of ∇ 0 are easily found: � � e − 1 / z 0 ψ 0 = . 0 1 Then we can write ψ = φψ 0 . 3 / 34
The gauge transformation Want: � � 1 � dz � � 1 � dz − z 0 φ − 1 d − φ = d − 0 0 z 2 0 0 z 2 Guess form for φ : � 1 � z 2 df f ( z ) φ = a solution ⇐ ⇒ dz = f − z . 0 1 Solution has series expansion � n ! z n +1 . f ( z ) = n ≥ 0 DIVERGES!!!!! 4 / 34
All is not lost Borel summation/multi-summation: recover solutions from divergent series (´ E. Borel, ´ Ecalle, Ramis, Sibuya, ...) The essential idea: � 1 � ∞ � � ∞ � ∞ a n z n +1 = t n e − t / z dt a n n ! 0 n =0 n =0 � ∞ � � ∞ � a n t n e − t / z dt = n ! 0 n =0 and the new series (the Borel transform) is more likely to converge. z 5 / 34
Our example � ∞ � � ∞ � ∞ � ∞ � e − t / z n ! z n +1 = t n e − t / z dt = 1 − t dt 0 0 n =0 n =0 Stokes phenomenon: sums for Im ( z ) > 0 and Im ( z ) < 0 differ: z = 2 π i Res = − 2 π ie − 1 / z − z NB: this comes from the other solution of ODE. 6 / 34
Resummation, cont. (Nearly) equivalent: Weight the partial sums: � � n � ∞ � ∞ � µ n a n z n +1 = lim µ →∞ e − µ a k z k +1 n ! n =0 n =0 k =0 7 / 34
Pros and cons Success: solution of the ODE with the divergent series as an asymptotic expansion ; truncating the series gives a good approximation for small z The Stokes phenomenon: “correct” sum of the series varies from sector to sector (wall crossing) — patched by “generalized monodromy data” Drawbacks: the procedure is a bit ad hoc: Correct weights depend on order of pole and “irregular type” Not directly applicable to related and important situations ◮ WKB approximation (aka λ -connections) ◮ Normal forms in dynamical systems ◮ Perturbative QFT Leads to even more complicated theory of “resurgence” (´ Ecalle) 8 / 34
The problem Question What is the geometry of these resummation procedures? Answer (Gualtieri–Li–P.) It is governed by a very natural Lie groupoid . 9 / 34
Viewpoint A holomorphic flat connection ∇ : E → Ω 1 X ⊗ E gives an action of vector fields by derivations T X × E → E ( η, ψ ) �→ ∇ η ψ, compatible with Lie brackets: ∇ η ∇ ξ − ∇ ξ ∇ η = ∇ [ η,ξ ] Slogan: { holomorphic flat connections } = { representations of T X } . 10 / 34
Parallel transport s γ ′ γ t Solve the ODE ∇ ψ = 0 along a path γ : [0 , 1] → X from s to t Get the parallel transport Ψ( γ ) : E| s → E| t If γ, γ ′ are homotopic, then Ψ( γ ) = Ψ( γ ′ ). 11 / 34
The fundamental groupoid Domain for parallel transport is the fundamental groupoid: Π 1 ( X ) = { paths γ : [0 , 1] → X } / (end-point-preserving homotopies) Source and target s , t : Π 1 ( X ) → X s ( γ ) = γ (0) t ( γ ) = γ (1) Product: concatenation of paths, defined when endpoints match Identities: constant paths, one for each x ∈ X Inverses: reverse directions Lemma Π 1 ( X ) has a unique manifold structure such that ( s , t ) : Π 1 ( X ) → X × X is a local diffeomorphism. Thus Π 1 ( X ) is a (complex) Lie groupoid . 12 / 34
Example: the fundamental groupoid of C ∗ = C \ { 0 } We have an isomorphism C × C ∗ ∼ = Π 1 ( C ∗ ) ( λ, z ) �→ [ γ λ, z ] C ∗ ◮ Source and target: e λ z t ( λ, z )= e λ z s ( λ, z )= z ◮ Identities: z i ( z ) = (0 , z ) ◮ Product: γ λ, z ( t ) = exp( t λ ) · z ( λ, z )( λ ′ , z ′ ) = ( λ + λ ′ , z ′ ) defined whenever z = e λ ′ z ′ . 13 / 34
Parallel transport as a representation Parallel transport of holomorphic connection ∇ is an isomorphism of bundles on Π 1 ( X ): Ψ : s ∗ E → t ∗ E If ψ is a fundamental solution, then Ψ = t ∗ ψ · s ∗ ψ − 1 It’s a representation of Π 1 ( X ): Ψ( γ − 1 ) = Ψ( γ ) − 1 Ψ( γ 1 γ 2 ) = Ψ( γ 1 )Ψ( γ 2 ) Ψ(1 x ) = 1 Version of the Riemann–Hilbert correspondence : Integration { representations of T X } { representations of Π 1 ( X ) } Differentiation 14 / 34
Meromorphic connections p 1 p 3 p 2 D = k 1 · p 1 + · · · + k n · p n an effective divisor ( k i ∈ N ) Meromorphic connection ∇ : E → Ω 1 X ( D ) ⊗ E In a local coordinate z near p i � d ψ � dz − A ( z ) ∇ ψ = dz ⊗ z k i ψ . Can’t define parallel transport for paths that intersect D 15 / 34
Lie-theoretic perspective T X ( − D ) the sheaf of vector fields vanishing on D . ◮ Locally free (a vector bundle). Near a point p ∈ D , we have � � T X ( − D ) ∼ z k ∂ z = ◮ Anchor map a : T X ( − D ) → T X ◮ Closed under Lie brackets Thus, T X ( − D ) is a very simple example of a Lie algebroid Pairing with ∇ : E → Ω 1 X ( D ) ⊗ E gives a holomorphic action T X ( − D ) × E → E . 16 / 34
Lie-theoretic perspective Slogan: { flat connections on X with poles ≤ D } = { representations of T X ( − D ) } Consequence: The correct domain for the solutions is the Lie groupoid that “integrates” T X ( − D ). 17 / 34
Lie groupoids (Ehresmann, Pradines 50–60s) A Lie groupoid G ⇒ X is G 1 A manifold X of objects s t gh 2 A manifold G of arrows • 3 Maps s , t : G → X g • h • • indicating the source and target X 4 Composition of arrows a c b whose endpoints match • h − 1 5 An identity arrow for → G each object i : X ֒ 6 Inversion · − 1 : G → G . satisfying associativity, etc. 18 / 34
Infinitesimal counterpart: Lie algebroids G i ( X ) A Vector bundle A = N i ( X ) , G with Lie bracket [ · , · ] : A × A → A on sections and anchor map a : A → T X satisfying the Leibniz rule [ ξ, f η ] = ( L a ( ξ ) f ) η + f [ ξ, η ] . 19 / 34
Examples G A G ⇒ { pt } a Lie group g its Lie algebra H × X ⇒ X group action h → T X infinitesimal action Π 1 ( X ) T X Pair ( X ) = X × X ⇒ X T X 20 / 34
Algebroid representations A representation of A is a flat A -connection, i.e. an operator ∇ : E → A ∨ ⊗ E satisfying ∇ ( f ψ ) = ( a ∨ df ) ⊗ ψ + f ∇ ψ and having zero curvature in � 2 A ∨ ⊗ End E . Examples: 1 For X = {∗} and A = g : finite-dimensional g -reps 2 For A = T X : have A ∨ = Ω 1 X and ∇ a usual flat connection 3 For T X ( − D ) : have A ∨ = Ω 1 X ( D ) , and ∇ a meromorphic flat connection with poles bounded by D 4 Logarithmic connections, λ -connections, connections with central curvature, Poisson modules (= “semi-classical” bimodules), ... 21 / 34
Parallel transport for algebroid connections An A -path is a Lie algebroid homomorphism Γ : T [0 , 1] → A A -connections pull back to usual connections on [0 , 1]. Thus, parallel transport is defined on the fundamental groupoid of A : {A -paths } Π 1 ( A ) = {A -homotopies } Examples: For A = g a Lie algebra, get Π 1 ( g ) = G , the simply-connected group For A = T X , get Π 1 ( T X ) = Π 1 ( X ). 22 / 34
Integrability of algebroids (analogue of Lie III) The Crainic–Fernandes theorem (Annals 2003) gives necessary and sufficient conditions for Π 1 ( A ) to have a smooth structure, making it a Lie groupoid. Parallel transport of A -connections along A -paths gives: Integration { representations of A} { representations of Π 1 ( A ) } Differentiation Theorem (Debord 2001) If A → T X is an embedding of sheaves, then A is integrable. 23 / 34
Applied to T X ( − D ) Get a Lie groupoid Π 1 ( X , D ), functorial in X and D Two types of algebroid paths: ◮ Usual paths in X \ D , so we have open dense Π 1 ( X \ D ) ֒ → Π 1 ( X , D ) ◮ Boundary: a one-dimensional Lie group of loops at each p ∈ D � p X ) mult ( p ) − 1 ֒ ( T ∗ → Π 1 ( X , D ) p ∈ D e.g., mult ( p ) = 1 limiting X = C γ r procedure r → 0 log γ r (1) r → 0 [ γ r ] = lim lim γ r (0) r D ∈ C giving “loops” at D . 24 / 34
Parallel transport Meromorphic connection ∇ : E → Ω 1 X ( D ) ⊗ E Usual parallel transport defined on Π 1 ( X \ D ) extends to Ψ : s ∗ E → t ∗ E globally defined and holomorphic on Π 1 ( X , D ). Caveat: Π 1 ( X , D ) was constructed as an infinite-dimensional quotient—not very explicit. 25 / 34
Our paper With M. Gualtieri and S. Li we give Explicit local normal forms, the Stokes groupoids Finite-dimensional global construction using the uniformization theorem ◮ analytic open embedding in a P 1 -bundle X , D Ω 1 / 2 → P ( J 1 Π 1 ( X , D ) ֒ X ) ◮ groupoid structure maps given by solving the uniformizing ODE ◮ e.g. groupoid for X = P 1 and D = 0 + 1 + ∞ involves hypergeometric functions and the elliptic modular function λ ( τ ) Constructions of Pair ( X , D ) by iterated blowups Application to divergent series 26 / 34
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