Ends of the moduli space of Higgs bundles. Frederik Witt

Transcription

1 Universiy of Münser The Geomery, Topology and Physics of Moduli Spaces of Higgs Bundles, Singapore, NUS 6h of Augus 2014

2 based on arxiv: [mah.dg] join wih R. Mazzeo (Sanford) J. Swoboda (Bonn) H. Weiß (Kiel)

3 (X, J) Riemann surface

4 (X, J) Riemann surface closed oriened surface of genus γ 2

5 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1)

6 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π

7 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π (E, H) X hermiian vecor bundle

8 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π (E, H) X hermiian vecor bundle rank r and degree d µ = d/r slope of E

9 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π (E, H) X hermiian vecor bundle rank r and degree d µ = d/r slope of E uniary connecion A covarian derivaive d A = A + A wih curvaure F A = d A d A

10 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π (E, H) X hermiian vecor bundle rank r and degree d µ = d/r slope of E uniary connecion A covarian derivaive d A = A + A wih curvaure F A = d A d A i X Tr F A = 2πd, A defines a holomorphic srucure on E.

11 (X, J) Riemann surface closed oriened surface of genus γ 2 K X X canonical line bundle, deg(k ) = 2(γ 1) compaible Kähler form ω wih X ω = 2π (E, H) X hermiian vecor bundle rank r and degree d µ = d/r slope of E uniary connecion A covarian derivaive d A = A + A wih curvaure F A = d A d A i X Tr F A = 2πd, A defines a holomorphic srucure on E. Conversely, a holomorphic srucure on E gives Chern connecion A H.

12 For A {uniary connecions}, affine space over Ω 1 (u(e))

13 For A {uniary connecions}, affine space over Ω 1 (u(e)) Φ Γ(End(E) K X ) Higgs field

14 For A {uniary connecions}, affine space over Ω 1 (u(e)) Φ Γ(End(E) K X ) Higgs field consider Hichin s self-dualiy equaions F A + [Φ Φ ] = µiω Id A Φ = 0 } (H)

15 For A {uniary connecions}, affine space over Ω 1 (u(e)) Φ Γ(End(E) K X ) Higgs field consider Hichin s self-dualiy equaions F A + [Φ Φ ] = µiω Id A Φ = 0 } (H) Soluions aced on by uniary gauge ransformaions Γ(U(E)).

16 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)).

17 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)). Fix A 0 uniary such ha Tr F A0 = d iω.

18 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)). Fix A 0 uniary such ha Tr F A0 = d iω. Self-dualiy equaions of he fixed deerminan case F A + [Φ Φ ] = 0 A Φ = 0 } (H )

19 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)). Fix A 0 uniary such ha Tr F A0 = d iω. Self-dualiy equaions of he fixed deerminan case F A + [Φ Φ ] = 0 A Φ = 0 } (H ) where A = A 0 + α, α Ω 1 (su(e)).

20 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)). Fix A 0 uniary such ha Tr F A0 = d iω. Self-dualiy equaions of he fixed deerminan case F A + [Φ Φ ] = 0 A Φ = 0 } (H ) where A = A 0 + α, α Ω 1 (su(e)). Φ Γ(End 0 (E) K ), i.e. Φ is race-free.

21 Decompose F A = F A + 1 r Tr(F A) Id E, F A Ω2 (su(e)). Fix A 0 uniary such ha Tr F A0 = d iω. Self-dualiy equaions of he fixed deerminan case F A + [Φ Φ ] = 0 A Φ = 0 } (H ) where A = A 0 + α, α Ω 1 (su(e)). Φ Γ(End 0 (E) K ), i.e. Φ is race-free. Soluions are aced on by Γ(SU(E)).

22 From now on: r = 2, d = odd.

23 From now on: r = 2, d = odd. Theorem (Hichin) M d = {(A, Φ) soluions of H }/Γ(SU(E)) is a smooh non-compac manifold of dimension 12(γ 1). Furhermore, M d carries a complee hyperkähler meric.

24 From now on: r = 2, d = odd. Theorem (Hichin) M d = {(A, Φ) soluions of H }/Γ(SU(E)) is a smooh non-compac manifold of dimension 12(γ 1). Furhermore, M d carries a complee hyperkähler meric. Asympoics of he meric ( e.g. L 2 -cohomology)?

25 From now on: r = 2, d = odd. Theorem (Hichin) M d = {(A, Φ) soluions of H }/Γ(SU(E)) is a smooh non-compac manifold of dimension 12(γ 1). Furhermore, M d carries a complee hyperkähler meric. Asympoics of he meric ( e.g. L 2 -cohomology)? Geomeric compacificaion?

26 From now on: r = 2, d = odd. Theorem (Hichin) M d = {(A, Φ) soluions of H }/Γ(SU(E)) is a smooh non-compac manifold of dimension 12(γ 1). Furhermore, M d carries a complee hyperkähler meric. Asympoics of he meric ( e.g. L 2 -cohomology)? Geomeric compacificaion? Degeneraion behaviour of soluions near he end?

27 Wha do we expec? We know (Hichin):

28 Wha do we expec? We know (Hichin): (E, Φ) sable Higgs bundle soluion of (H ) in he complex gauge orbi of (A H, Φ)

29 Wha do we expec? We know (Hichin): (E, Φ) sable Higgs bundle soluion of (H ) in he complex gauge orbi of (A H, Φ) µ : M d [0, ), [A, Φ] Φ L 2 proper Morse-Bo

30 Wha do we expec? We know (Hichin): (E, Φ) sable Higgs bundle soluion of (H ) in he complex gauge orbi of (A H, Φ) µ : M d [0, ), [A, Φ] Φ L 2 proper Morse-Bo Ansaz: Consider soluions (A, Φ ), 1, of F A + 2 [Φ Φ ] = 0 A Φ = 0 } (H )

31 Wha do we expec? We know (Hichin): (E, Φ) sable Higgs bundle soluion of (H ) in he complex gauge orbi of (A H, Φ) µ : M d [0, ), [A, Φ] Φ L 2 proper Morse-Bo Ansaz: Consider soluions (A, Φ ), 1, of F A + 2 [Φ Φ ] = 0 A Φ = 0 } (H ) Assume (A, Φ ) (A, Φ ) as. Then F A = 0, [Φ Φ ] = 0, A Φ = 0.

32 Φ normal, hence uniarily diagonalisable

33 Φ normal, hence uniarily diagonalisable Assume ha Φ is simple, i.e. q = de Φ H 0 (X, K 2 X ) has only simple zeroes.

34 Φ normal, hence uniarily diagonalisable Assume ha Φ is simple, i.e. q = de Φ H 0 (X, K 2 X ) has only simple zeroes. Conradicion if Φ is holomorphic on q 1 (0)!

35 Φ normal, hence uniarily diagonalisable Assume ha Φ is simple, i.e. q = de Φ H 0 (X, K 2 X ) has only simple zeroes. Conradicion if Φ is holomorphic on q 1 (0)! Guess: under hese assumpions (cf. also Taubes) A is singular in q 1 (0) F A concenraes near q 1 (0) as.

36 q simple de 1 (q) M d is a Prym variey and in paricular a 6(γ 1) dimensional orus (Hichin).

37 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable.

38 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable. To obain a (parial) compacificaion of M d, we will

39 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable. To obain a (parial) compacificaion of M d, we will formalise he noion of a limiing configuraion (A, Φ ).

40 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable. To obain a (parial) compacificaion of M d, we will formalise he noion of a limiing configuraion (A, Φ ). prove heir exisence.

41 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable. To obain a (parial) compacificaion of M d, we will formalise he noion of a limiing configuraion (A, Φ ). prove heir exisence. Desingularisaion of limiing configuraions = Soluions for (E, Φ), large

42 From now on: Φ simple, q = de Φ. In paricular, (E, Φ) is sable. To obain a (parial) compacificaion of M d, we will formalise he noion of a limiing configuraion (A, Φ ). prove heir exisence. Desingularisaion of limiing configuraions = Soluions for (E, Φ), large Ansaz: Glue local soluions near q 1 (0) o soluions on X = X \ q 1 (0) given by limiing configuraions.

43 Firs sep: Local soluions of (H ) near q 1 (0)

44 Firs sep: Local soluions of (H ) near q 1 (0) Le Φ be normal over D = D \ {0}, q = de Φ = zdz 2.

45 Firs sep: Local soluions of (H ) near q 1 (0) Le Φ be normal over D = D \ {0}, q = de Φ = zdz 2. There exiss g Γ(D, SU(E)) wih ( ) 0 z Φ g = Φ fid := z dz, 0 z he fiducial Higgs field wih respec o a fixed uniary smooh fiducial frame, coninuous on D, smooh on D.

46 Firs sep: Local soluions of (H ) near q 1 (0) Le Φ be normal over D = D \ {0}, q = de Φ = zdz 2. There exiss g Γ(D, SU(E)) wih ( ) 0 z Φ g = Φ fid := z dz, 0 z he fiducial Higgs field wih respec o a fixed uniary smooh fiducial frame, coninuous on D, smooh on D. Lemma Le A be a uniary connecion over D wih F A = 0, A Φ fid = 0. There exiss a unique g Γ(D, SU(E)) such ha ( ) (dz A g = A fid := z d z ), (Φ fid z ) g = Φ fid.

47 Proposiion (Folklore) Can desingularise (A fid, Φ fid ) by soluions (A fid, Φ fid ) of (H )

48 Proposiion (Folklore) Can desingularise (A fid, Φ fid ) by soluions (A fid, Φ fid ) of (H ) which are smooh across z = 0.

49 Proposiion (Folklore) Can desingularise (A fid, Φ fid ) by soluions (A fid, Φ fid ) of (H ) which are smooh across z = 0. as, (A fid, Φ fid ) (A fid, Φ fid ) exponenially fas in and uniformly in C on any exerior region r r 0 > 0.

50 Proposiion (Folklore) Can desingularise (A fid, Φ fid ) by soluions (A fid, Φ fid ) of (H ) which are smooh across z = 0. as, (A fid, Φ fid ) (A fid, Φ fid ) exponenially fas in and uniformly in C on any exerior region r r 0 > 0.

51 Ansaz: ( ) A fid 1 0 ( (z) = f ( z ) dz z 0 1 ( Φ fid (z) = ϕ fid (z)dz = d z ) z, ) 0 z e h ( z ) z e h ( z ) dz 0 z

52 Ansaz: ( ) A fid 1 0 ( (z) = f ( z ) dz z 0 1 ( Φ fid (z) = ϕ fid (z)dz = d z ) z, ) 0 z e h ( z ) z e h ( z ) dz 0 z (A fid, Φ fid ) solves (H ) r f (r) = 2 2 r 2 sinh 2h f (r) = r r h (r)

53 Ansaz: ( ) A fid 1 0 ( (z) = f ( z ) dz z 0 1 ( Φ fid (z) = ϕ fid (z)dz = d z ) z, ) 0 z e h ( z ) z e h ( z ) dz 0 z (A fid, Φ fid ) solves (H ) r f (r) = 2 2 r 2 sinh 2h f (r) = r r h (r) Simplify: Se ρ = 8 3 r 3/2 and wrie h (r) = ψ(ρ) for some ψ (ρ ρ ) 2 ψ = 1 2 ρ2 sinh 2ψ.

54 Ansaz: ( ) A fid 1 0 ( (z) = f ( z ) dz z 0 1 ( Φ fid (z) = ϕ fid (z)dz = d z ) z, ) 0 z e h ( z ) z e h ( z ) dz 0 z (A fid, Φ fid ) solves (H ) r f (r) = 2 2 r 2 sinh 2h f (r) = r r h (r) Simplify: Se ρ = 8 3 r 3/2 and wrie h (r) = ψ(ρ) for some ψ (ρ ρ ) 2 ψ = 1 2 ρ2 sinh 2ψ. Painlevé III equaion [Cecoi & Vafa / Gaioo, Moore & Neizke] [Mason & Woodhouse / Biquard & Boalch]

55 Ansaz: ( ) A fid 1 0 ( (z) = f ( z ) dz z 0 1 ( Φ fid (z) = ϕ fid (z)dz = d z ) z, ) 0 z e h ( z ) z e h ( z ) dz 0 z (A fid, Φ fid ) solves (H ) r f (r) = 2 2 r 2 sinh 2h f (r) = r r h (r) Simplify: Se ρ = 8 3 r 3/2 and wrie h (r) = ψ(ρ) for some ψ (ρ ρ ) 2 ψ = 1 2 ρ2 sinh 2ψ. Painlevé III equaion [Cecoi & Vafa / Gaioo, Moore & Neizke] [Mason & Woodhouse / Biquard & Boalch] Pick soluion wih suiable asympoics. [McCoy, Tracy & Wu / Widom]

56 Second sep: Consrucion of limiing configuraions

57 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0).

58 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0). Choose uniary frames over D(p j ), p j q 1 (0), and holomorphic coordinaes such ha q = zdz 2.

59 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0). Choose uniary frames over D(p j ), p j q 1 (0), and holomorphic coordinaes such ha q = zdz 2. (A, Φ ) is a limiing configuraion if

60 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0). Choose uniary frames over D(p j ), p j q 1 (0), and holomorphic coordinaes such ha q = zdz 2. (A, Φ ) is a limiing configuraion if F A = 0, [Φ Φ ] = 0, A Φ = 0 over X.

61 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0). Choose uniary frames over D(p j ), p j q 1 (0), and holomorphic coordinaes such ha q = zdz 2. (A, Φ ) is a limiing configuraion if F A = 0, [Φ Φ ] = 0, A Φ = 0 over X. (A, Φ ) is of fiducial form near q 1 (0).

62 Second sep: Consrucion of limiing configuraions Le q H 0 (X, K 2 ) wih simple zeros, X = X \ q 1 (0). Choose uniary frames over D(p j ), p j q 1 (0), and holomorphic coordinaes such ha q = zdz 2. (A, Φ ) is a limiing configuraion if F A = 0, [Φ Φ ] = 0, A Φ = 0 over X. (A, Φ ) is of fiducial form near q 1 (0). Theorem [Exisence of limiing configuraions] (E, Φ) Higgs bundle wih Φ simple, q = de Φ, A = A H. There exiss g Γ(X, SL(E)) unique up o uniary ransformaions such ha (A, Φ) g is a limiing configuraion.

63 Ouline of he proof:

64 Ouline of he proof: Sar wih a Higgs bundle (E, Φ), Φ simple. Pu X in = D (p j ), X ex = X \ X in. p j q 1 (0)

65 Ouline of he proof: Sar wih a Higgs bundle (E, Φ), Φ simple. Pu X in = D (p j ), X ex = X \ X in. p j q 1 (0) We can choose a complex coordinae z and a holomorphic frame near p j q 1 (0) such ha ( ) 0 1 Φ = dz, de Φ = zdz 2. z 0

66 Ouline of he proof: Sar wih a Higgs bundle (E, Φ), Φ simple. Pu X in = D (p j ), X ex = X \ X in. p j q 1 (0) We can choose a complex coordinae z and a holomorphic frame near p j q 1 (0) such ha ( ) 0 1 Φ = dz, de Φ = zdz 2. z 0 Define a hermiian meric H by declaring hese frames o be uniary and exend H o all of X.

67 Ouline of he proof: Sar wih a Higgs bundle (E, Φ), Φ simple. Pu X in = D (p j ), X ex = X \ X in. p j q 1 (0) We can choose a complex coordinae z and a holomorphic frame near p j q 1 (0) such ha ( ) 0 1 Φ = dz, de Φ = zdz 2. z 0 Define a hermiian meric H by declaring hese frames o be uniary and exend H o all of X. Connecion marix of associaed Chern connecion A = A H vanishes in hese frames

68 Lemma Wih respec o hese given fiducial frames, we have for g = ( z z 1 4 (A fid, Φ fid ) = (A, Φ) g ) Γ(X in, SL(E)).

69 Lemma Wih respec o hese given fiducial frames, we have for g = ( z z 1 4 (A fid, Φ fid ) = (A, Φ) g ) Γ(X in, SL(E)). Remark Explici consrucion yields also g Γ( p j q 1 (0) D(p j), SL(E)) such ha (A fid, Φ fid ) = (A, Φ) g. In paricular, all fiducial soluions (A fid, Φ fid ) of (H ) are muually complex gauge equivalen.

70 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)).

71 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)). Summarising, we have consruced a

72 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)). Summarising, we have consruced a Chern connecion A

73 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)). Summarising, we have consruced a Chern connecion A g 1 Γ(X, SL(E))

74 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)). Summarising, we have consruced a Chern connecion A g 1 Γ(X, SL(E)) such ha (A 1, Φ 1 ) := (A, Φ) g 1 is of fiducial form near q 1 (0)

75 Elemenary obsrucion heory shows Lemma Any normalising local secion g Γ(U, SL(E)), U X open, can be exended o a normalising secion g Γ(X, SL(E)). Summarising, we have consruced a Chern connecion A g 1 Γ(X, SL(E)) such ha (A 1, Φ 1 ) := (A, Φ) g 1 is of fiducial form near q 1 (0) [Φ 1 Φ 1 ] = 0 on X.

76 Consider hen he A 1 -parallel, infiniesimal Φ 1 -sabiliser il Φ1 = {γ sl(e) γ = γ, [γ, Φ 1 ]} = 0.

77 Consider hen he A 1 -parallel, infiniesimal Φ 1 -sabiliser il Φ1 = {γ sl(e) γ = γ, [γ, Φ 1 ]} = 0. The connecion Laplacian A1 = da 1 d A1 : Γ(isu(E)) Γ(isu(E)) resrics o Γ(iL Φ ) Γ(iL Φ ).

78 Consider hen he A 1 -parallel, infiniesimal Φ 1 -sabiliser il Φ1 = {γ sl(e) γ = γ, [γ, Φ 1 ]} = 0. The connecion Laplacian A1 = da 1 d A1 : Γ(isu(E)) Γ(isu(E)) resrics o Γ(iL Φ ) Γ(iL Φ ). A local compuaion yields ifa 1 Ω 2 (il Φ1 ) and he

79 Consider hen he A 1 -parallel, infiniesimal Φ 1 -sabiliser il Φ1 = {γ sl(e) γ = γ, [γ, Φ 1 ]} = 0. The connecion Laplacian A1 = da 1 d A1 : Γ(isu(E)) Γ(isu(E)) resrics o Γ(iL Φ ) Γ(iL Φ ). A local compuaion yields ifa 1 Ω 2 (il Φ1 ) and he Lemma For γ Γ(X, il Φ1 ), F A exp(γ) 1 = 0 if and only if A1 γ = i F A 1.

80 Soluions o he Poisson equaion on X via heory of ellipic conic operaors. [Mazzeo & Moncouquiol]

81 Soluions o he Poisson equaion on X via heory of ellipic conic operaors. [Mazzeo & Moncouquiol] Fix polar coordinaes and hink of D (p j ) as a cone wih S 1 link. Then A1 γ = ( r r r + 1 r 2 T )γ wih T he r-independen angenial operaor.

82 Soluions o he Poisson equaion on X via heory of ellipic conic operaors. [Mazzeo & Moncouquiol] Fix polar coordinaes and hink of D (p j ) as a cone wih S 1 link. Then A1 γ = ( r r r + 1 r 2 T )γ wih T he r-independen angenial operaor. Involved funcion spaces: For l N, δ R, and V b = he C span of {r r, θ }, se H l b (X) = {u L2 (X) V 1... V j u L 2, j l, V i V b }, r δ H l b (X) = {r δ u u H l b (X)}.

83 ν is an indicial roo of A1 if and only if ν 2 is an eigenvalue of T.

84 ν is an indicial roo of A1 if and only if ν 2 is an eigenvalue of T. Explici compuaion indicial roos of A1 Γ(iLΦ1 ) are given by Z and are simple.

85 ν is an indicial roo of A1 if and only if ν 2 is an eigenvalue of T. Explici compuaion indicial roos of A1 Γ(iLΦ1 ) are given by Z and are simple. Knowing he indicial roos see when A1 is Fredholm on suiable funcion spaces r δ Hb l.

86 ν is an indicial roo of A1 if and only if ν 2 is an eigenvalue of T. Explici compuaion indicial roos of A1 Γ(iLΦ1 ) are given by Z and are simple. Knowing he indicial roos see when A1 is Fredholm on suiable funcion spaces r δ Hb l. For insance, A1 : r δ H l+2 b (il Φ ) r δ 2 H l b (il Φ) is Fredholm and inverible when 1/2 < δ < 3/2.

87 A secion is polyhomogeneous a p j if is coefficiens c have an asympoic expansion near p j, c j N j r ν j (log r) k a j,k (θ), k=0 wih a corresponding expansion for is derivaives.

88 A secion is polyhomogeneous a p j if is coefficiens c have an asympoic expansion near p j, c j N j r ν j (log r) k a j,k (θ), k=0 wih a corresponding expansion for is derivaives. If A1 γ = 0 γ polyhomogeneous wih exponens deermined by he indicial roos.

89 A secion is polyhomogeneous a p j if is coefficiens c have an asympoic expansion near p j, c j N j r ν j (log r) k a j,k (θ), k=0 wih a corresponding expansion for is derivaives. If A1 γ = 0 γ polyhomogeneous wih exponens deermined by he indicial roos. Since A 1 fla near p j There exiss polyhomogeneous γ wih leading erm r 1/2. Furher, (A 2, Φ 2 ) = (A 1, Φ 1 ) exp(γ) saisfies F A 2 = 0, Φ 2 = Φ 1.

90 A secion is polyhomogeneous a p j if is coefficiens c have an asympoic expansion near p j, c j N j r ν j (log r) k a j,k (θ), k=0 wih a corresponding expansion for is derivaives. If A1 γ = 0 γ polyhomogeneous wih exponens deermined by he indicial roos. Since A 1 fla near p j There exiss polyhomogeneous γ wih leading erm r 1/2. Furher, (A 2, Φ 2 ) = (A 1, Φ 1 ) exp(γ) saisfies F A 2 = 0, Φ 2 = Φ 1. Apply final uniary ransformaion o pu (A 2, Φ 2 ) back in fiducial form near p j.

91 Third sep: Glueing

92 Third sep: Glueing So far we consruced from (E, Φ), Φ simple, q = de Φ, a

93 Third sep: Glueing So far we consruced from (E, Φ), Φ simple, q = de Φ, a limiing configuraion in he complex gauge orbi of (A, Φ). In paricular, his solves (H ) over X.

94 Third sep: Glueing So far we consruced from (E, Φ), Φ simple, q = de Φ, a limiing configuraion in he complex gauge orbi of (A, Φ). In paricular, his solves (H ) over X. (A fid, Φ fid ) solving (H ) on he disc and desingularising (A fid, Φ fid ).

95 Third sep: Glueing So far we consruced from (E, Φ), Φ simple, q = de Φ, a limiing configuraion in he complex gauge orbi of (A, Φ). In paricular, his solves (H ) over X. (A fid, Φ fid ) solving (H ) on he disc and desingularising (A fid, Φ fid ). Theorem [Exisence heorem for (H )] For 1 here exiss a soluion (A, Φ ) of (H ) in he complex gauge orbi of (A, Φ) such ha (A, Φ ) converges exponenially fas o (A, Φ ) in C loc (X ) as.

96 Theorem is based on a fixed poin argumen. Consider H : H 2 (Herm(E)) L 2 (Λ 2 su(e)), g F (A appr + 2 [(Φ appr ) g ) g (Φ appr ) g ]. for a suiably defined approximae soluion (A appr, Φ appr ) defined over X:

97 Theorem is based on a fixed poin argumen. Consider H : H 2 (Herm(E)) L 2 (Λ 2 su(e)), g F (A appr + 2 [(Φ appr ) g ) g (Φ appr ) g ]. for a suiably defined approximae soluion (A appr, Φ appr ) defined over X: Take γ Γ(X in, isu(e)) wih (A fid, Φ fid ) = (A fid, Φ fid ) exp(γ ).

98 Theorem is based on a fixed poin argumen. Consider H : H 2 (Herm(E)) L 2 (Λ 2 su(e)), g F (A appr + 2 [(Φ appr ) g ) g (Φ appr ) g ]. for a suiably defined approximae soluion (A appr, Φ appr ) defined over X: Take γ Γ(X in, isu(e)) wih (A fid, Φ fid ) = (A fid, Φ fid ) exp(γ ). Choose smooh cu-off funcion χ: X [0, 1] wih supp χ X in and χ(z) 1 for z p p D 1/2(p).

99 Theorem is based on a fixed poin argumen. Consider H : H 2 (Herm(E)) L 2 (Λ 2 su(e)), g F (A appr + 2 [(Φ appr ) g ) g (Φ appr ) g ]. for a suiably defined approximae soluion (A appr, Φ appr ) defined over X: Take γ Γ(X in, isu(e)) wih (A fid, Φ fid ) = (A fid, Φ fid ) exp(γ ). Choose smooh cu-off funcion χ: X [0, 1] wih supp χ X in and χ(z) 1 for z p p D 1/2(p). For g appr := exp(χγ ), he new pair (A appr, Φ appr ) := (A, Φ ) gappr is smooh across he puncures and coincides wih (A, Φ ) on X ex and (A fid, Φ fid ) near q 1 (0).

100 Lemma There exis C, δ > 0 such ha for 1, H (Id E ) L 2 Ce δ.

101 Lemma There exis C, δ > 0 such ha for 1, H (Id E ) L 2 Ce δ. Remark (A appr, Φ appr ) is complex gauge equivalen o (A, Φ) over X.

102 On he oher hand, L = D IdE H : H 2 (isu(e)) L 2 (isu(e)) is a posiive operaor of Schrödinger ype which is inverible and L 1 L(L 2,H 2 ) = O(m ) is polynomially bounded in.

103 On he oher hand, L = D IdE H : H 2 (isu(e)) L 2 (isu(e)) is a posiive operaor of Schrödinger ype which is inverible and L 1 L(L 2,H 2 ) = O(m ) is polynomially bounded in. Lipschiz consan of higher order erm Q in Taylor expansion H (exp(γ)) = H (Id E ) + L γ + Q (γ) also polynomially bounded.

104 On he oher hand, L = D IdE H : H 2 (isu(e)) L 2 (isu(e)) is a posiive operaor of Schrödinger ype which is inverible and L 1 L(L 2,H 2 ) = O(m ) is polynomially bounded in. Lipschiz consan of higher order erm Q in Taylor expansion H (exp(γ)) = H (Id E ) + L γ + Q (γ) also polynomially bounded. Consider T : B ρ H 2 (isu(e)) H 2 (isu(e)), γ L 1 (H (Id E ) + Q (γ)) on he ball of radius ρ around he zero secion. For suiable ρ, T : B ρ B ρ will be a conracion mapping.

105 On he oher hand, L = D IdE H : H 2 (isu(e)) L 2 (isu(e)) is a posiive operaor of Schrödinger ype which is inverible and L 1 L(L 2,H 2 ) = O(m ) is polynomially bounded in. Lipschiz consan of higher order erm Q in Taylor expansion H (exp(γ)) = H (Id E ) + L γ + Q (γ) also polynomially bounded. Consider T : B ρ H 2 (isu(e)) H 2 (isu(e)), γ L 1 (H (Id E ) + Q (γ)) on he ball of radius ρ around he zero secion. For suiable ρ, T : B ρ B ρ will be a conracion mapping. There exiss a unique γ Γ(isu(E)) such ha (A, Φ ) = (A appr, Φ appr ) exp(γ) solves (H ). Convergence follows by design.

106 Thank you for your aenion!