# The Higgs bundle moduli space

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1 The Higgs bundle moduli space Nigel Hitchin Mathematical Institute University of Oxford Hamburg September 8th 2014 Add title here 00 Month

2

3 re ut on

4 INSTANTONS

5 connection on R 4, A i (x) n n skew Hermitian covariant derivative i = x i + A i curvature F ij =[ i, j ] equations F 12 + F 34 =0=F 13 + F 42 =0=F 14 + F 23

6 anti-self-dual Yang-Mills F + F = 0 boundary conditions F R 4 2 < gauge equivalence g : R 4 U(n) i g 1 i g 1 M.F.Atiyah,N.J.Hitchin,V.G.Drinfeld & Yu.I.Manin: Construc tion of instantons, Phys. Lett. A 65 (1978)

7 MONOPOLES

8 connection on R 3 A i (x) + Higgs field φ, n n skew Hermitian curvature F ij =[ i, j ] equations F 12 +[ 3, φ] =0=F 23 +[ 1, φ] =0=F 31 +[ 2, φ]

9 connection on R 3 A i (x) + Higgs field φ, n n skew Hermitian curvature F ij =[ i, j ] equations F 12 +[ 3, φ] =0=F 23 +[ 1, φ] =0=F 31 +[ 2, φ]

10

11 anti-self-dual Yang-Mills, invariant under x 4 -translation φ A 4 boundary conditions R 3 F 2 + φ 2 < N.J.Hitchin On the construction of monopoles, Comm.Math.Phys. 83 (1982)

12

13 TWO DIMENSIONS...

14 connection on R 2 A i (x) + Higgs fields φ 1, φ 2, n n skew Hermitian curvature F ij =[ i, j ] equations F 12 +[φ 1, φ 2 ]=0=[ 1, φ 2 ]+[ 2, φ 1 ]=[ 1, φ 1 ]+[ 2, φ 2 ]

15 Volume 70B, number 3 PHYSICS LETTERS 10 October 1977 TWO- AND THREE-DIMENSIONAL INSTANTONS M.A. LOHE Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2BZ, U.K. Received 6 June 1977 The four-dimensional Yang-Mills Lagrangian implies corresponding structures in lower dimensions. Instantons, characterized by a zero energy-momentum tensor as well as finite action, emerge as the solutions of coupled first order equations. For the Abelian case all such solutions are determined by the nonqinear Poisson-Boltzmann equation.

16 term have acquired special values. One can obtain first order equations which imply the field equations, and these are deduced from eqs. (6): F~ = +-e eijeabc~b c, Dit) a +- ei/d/4 a = O. (1 I) Unfortunately, in this case the model is not interesting Unfortunately, in this case the mo because the action is always zero.

17 [ 1 + i 2, φ 1 + iφ 2 ] = 0 Cauchy-Riemann z = x 1 + ix 2 C Φ =(φ 1 + iφ 2 )d(x 1 + ix 2 ) F +[Φ, Φ ] = 0 conformally invariant define on any Riemann surface

18 C Hermitian vector bundle E connection A+ Higgs field Φ Ω 1,0 (End E) Equations: A Φ =0, F A +[Φ, Φ ]=0 moduli space M = all solutions modulo C unitary automorphisms of E

19 THE CASE OF U(1) E = line bundle L, End L trivial bundle Φ = holomorphic 1-form A = flat connection on L moduli space = Jac(Σ) H 0 (Σ,K)=T Jac(Σ)

20 M finite-dimensional non-compact hyperkähler metric

21 SOLUTIONS

22 HOLOMORPHIC VECTOR BUNDLES Riemann surface Σ + holomorphic vector bundle E + holomorphic section Φ of End E K

23 HOLOMORPHIC VECTOR BUNDLES Riemann surface Σ + holomorphic vector bundle E + holomorphic section Φ of End E K Hermitian metric on E compatible connection... find a metric such that F A +[Φ, Φ ]=0

24 (E,Φ) stable there exists a unique solution NJH, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), (modelled on Donaldson s proof of Narasimhan-Seshadri) C.Simpson, Higgs bundles and local systems, Inst. Études Sci. Publ. Math. 75 (1992), 595. Hautes (modelled on Uhlenbeck-Yau s theorem on Hermitian Yang- Mills)

25 V E subbundle Φ-invariance = Φ(V ) V K E K stable = for each Φ-invariant subbundle V deg(v ) rk V < deg(e) rk E

26 FLAT VECTOR BUNDLES Riemann surface Σ flat connection on vector bundle E

27 FLAT VECTOR BUNDLES Riemann surface Σ flat connection on vector bundle E Hermitian metric on E π 1 -equivariant map f : Σ GL(n, C)/U(n)... find an equivariant harmonic map f.

28 irreducible (irreducible representation π 1 (Σ) GL(n, C)) there exists a unique solution S.K.Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987) K.Corlette, Flat G-bundles with canonical metrics, J.Differential Geom. 28 (1988),

29 irreducible (irreducible representation π 1 (Σ) GL(n, C)) there exists a unique solution S.K.Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987) K.Corlette, Flat G-bundles with canonical metrics, J.Differential Geom. 28 (1988), Equations 0,1 Φ =0, F A +[Φ, Φ ]=0 A + Φ + Φ is a flat connection

30 ANOTHER VIEWPOINT

31 charge one SU(2)-instanton on R 4 SO(4)-invariant solution quaternion x = x 0 + ix 1 + jx 2 + kx 3 connection A =Im xd x 1+ x 2

32 curvature F A = dx d x (1 + x 2 ) 2 translate from centre 0 to a i A A k approximate charge k solution if a i far apart Taubes grafting exact solution

33 charge one SU(2)-monopole on R 3 SO(3)-invariant solution x = ix 1 + jx 2 + kx 3 1 connection A = sinh r 1 r 1 r x dx Higgs field φ = 1 tanh r 1 r 1 r x

34 translate from centre 0 to a i A A k approximate charge k solution if a i far apart Taubes grafting exact solution A.Jaffe & C.Taubes, Vortices and monopoles, Birkhäuser, Boston (1980)

35 SU(2) Higgs bundle equations on R 2 SO(2)-invariant? = anti-self-dual Yang-Mills on R 4 invariant by R 2 SO(2)

36 L.Mason & N.Woodhouse, Self-duality and the Painlevé transcendents, Nonlinearity 6 (1993), further reductions to PI and PE. Pm: Two translations and a rotation: x=ai Y=tag-{af z=a,. The reduction by Y and X + Z gives the Emst equation, which is known to have a further reduction to Pm (as well as to Pv): see the papers cited in [l, p Prv: A null translation, a rotation combined with a dilatation, and an anti-self-dual Ernst equation: stationary axisymmetric spacetimes Painlevé III: ψxx = 1 2 e2x sinh 2ψ

37 L.Mason & N.Woodhouse, Self-duality and the Painlevé transcendents, Nonlinearity 6 (1993), further reductions to PI and PE. Pm: Two translations and a rotation: x=ai Y=tag-{af z=a,. The reduction by Y and X + Z gives the Emst equation, which is known to have a further reduction to Pm (as well as to Pv): see the papers cited in [l, p Prv: A null translation, a rotation combined with a dilatation, and an anti-self-dual Ernst equation: stationary axisymmetric spacetimes Painlevé III: ψxx = 1 2 e2x sinh 2ψ

38 R.Mazzeo, J.Swoboda, H.Weiss & F.Witt, Ends of the moduli space of Higgs bundles, arxiv v2 D.Gaiotto, G.Moore, & A.Neitzke Wall crossing, Hitchin systems and the WKB approximation Ad in Math. 234 (2013)

39 A SINGULAR SOLUTION connection A = 1 8 dz z d z z U(1)-connection F A = da =0 Higgs field Φ, need [Φ, Φ ]=0 Φ = 0 r 1/2 zr 1/2 0 dz normal using the constant Hermitian metric

40 A REGULAR SOLUTION connection A t = f t (r) dz z d z z Higgs field Φ = 0 r 1/2 e h t(r) e iθ r 1/2 e h t(r) 0 dz h t (r) =ψ(log(8tr 3/2 /3)) f t (r) = rdh t dr ψ special solution to Painlevé III

41 Thm (MSWW) (A t, Φ t ) is smooth at the origin and converges exponentially in t, uniformly in r>r 0 to the singular solution. (A t, Φ t ) satisfy the equations A Φ t =0,F At + t 2 [Φ t, Φ t ]=0 (A t,tφ t ) satisfies the Higgs bundle equations.

42 COMPACT RIEMANN SURFACE g>1

43 compact Riemann surface Σ can t widely separate approximate solutions but...if (V,Φ) is stable, so is (V,tΦ) t = 0 (stability condition on Φ-invariant subbundles) if (A, Φ) is a solution, then there is also a solution (A t,tφ)

44 study solutions (A t,tφ) ast Note 1: (A, e iθ Φ) is a solution if (A, Φ) is ([e iθ Φ, (e iθ Φ) ]=[Φ, Φ ]) Note 2: Consider V = L L Φ = 0 a 0 0 tφ = t 1/2 0 0 t 1/2 0 a 0 0 t 1/2 0 0 t 1/2 defines same point in moduli space

45 Higgs field Φ = 0 r 1/2 e h t(r) e iθ r 1/2 e h t(r) 0 dz det Φ = zdz 2 in general, for SU(2), det Φ is a holomorphic quadratic differential on the Riemann surface Σ invariant by gauge transformations Φ g 1 Φg defined by equivalence class in the moduli space.

46 quadratic differential = holomorphic section of K 2 deg K 2 =4g 4, g = genus generically det Φ has 4g 4 simple zeros.

47 Definition: A limiting configuration for a Higgs bundle (A, Φ) where det Φ has simple zeros is (A, Φ ) satisfying F A =0, [Φ, Φ ]=0 and behaving like the model singular solution near each zero of det Φ.

48 Φ = 0 r 1/2 zr 1/2 0 dz put z = w 2, det Φ = zdz 2 = (2w 2 dw) 2 eigenvalues of Φ ±2w 2 dw holomorphic 1-form α =2w 2 dw σ(w) = w, σ α = α

49 GLOBALLY... det Φ = q, q holomorphic section of K 2 K defines a double covering S branched over the zeros of q S compact Riemann surface genus g S =4g 3

50 GLOBALLY... det Φ = q, q holomorphic section of K 2 K defines a double covering S branched over the zeros of q S compact Riemann surface genus g S =4g 3 x = q defines a holomorphic 1-form α with double zeros on the ramification points involution σ : S S exchanges sheets, σ α = α

51 on S let L be a flat line bundle with σ L = L rank 2 vector bundle E = L L Higgs field Φ =(α, α) satisfies equations with [Φ, Φ ]=0 (E,Φ) isσ-invariant

52 σ-invariant Higgs bundle on S = orbifold Higgs bundle on Σ parabolic Higgs bundle, singular connection...

53 Thm (MSWW) Take a limiting configuration (A, Φ ). Then there exists a family (A t,tφ t ) of solutions to the Higgs bundle equations for large t where (A t, Φ t ) (A, Φ ) If (A, Φ ) is associated to a solution (A 0, Φ 0 ) then (A t, Φ t ) is gauge-equivalent to (A 0, Φ 0 ).

54 Thm (MSWW) Take a limiting configuration (A, Φ ). Then there exists a family (A t,tφ t ) of solutions to the Higgs bundle equations for large t where (A t, Φ t ) (A, Φ ) If (A, Φ ) is associated to a solution (A 0, Φ 0 ) then (A t, Φ t ) is gauge-equivalent to (A 0, Φ 0 ). describes [A 0,tΦ 0 ] in moduli space as t

55 GEOMETRY OF THE MODULI SPACE

56 M has a natural complete hyperkähler metric complex structures I,J,K Kähler forms ω 1, ω 2, ω 3 circle action fixing ω 1, rotating ω 2, ω 3

57 HIGGS BUNDLES Σ compact Riemann surface V smooth vector bundle with Hermitian metric A = infinite-dimensional affine space of -operators on V A A : Ω 0 (V ) Ω 0,1 (V ) A (fs) = f A (s) + fs A B Ω 0,1 (End(V ))

58 M = A Ω 1,0 (End(V )) T A M = Ω 0,1 (End(V )) Ω 1,0 (End(V )) Hermitian form ω 1 Σ (tr aa + tr φφ ) ω 2 + iω 3 Σ tr aφ flat hyperkähler manifold

59 G = group of U(n) gauge transformations g = {ψ Ω 0 (End(V )), ψ = ψ} g = {ω Ω 2 (End(V )), ω = ω}

60 G = group of U(n) gauge transformations g = {ψ Ω 0 (End(V )), ψ = ψ} g = {ω Ω 2 (End(V )), ω = ω} moment map µ( A, Φ) = (F A + [Φ, Φ ], A Φ) F A = curvature of Hermitian connection with 0,1 = A

61 µ( A, Φ) = (F A + [Φ, Φ ], A Φ) hyperkähler quotient = µ 1 (0)/G = moduli space A Φ = 0 = holomorphic Higgs field Φ End(V ) K

62 µ( A, Φ) = (F A + [Φ, Φ ], A Φ) hyperkähler quotient = µ 1 (0)/G = moduli space A Φ = 0 = holomorphic Higgs field Φ End(V ) K + Φ + Φ connection F = [ 1,0 + Φ, 0,1 + Φ ] = 0 flat GL(n, C) connection

63 µ( A, Φ) = (F A + [Φ, Φ ], A Φ) hyperkähler quotient = µ 1 (0)/G = moduli space A Φ = 0 = holomorphic Higgs field Φ End(V ) K + Φ + Φ connection complex structure I F = [ 1,0 + Φ, 0,1 + Φ ] = 0 flat GL(n, C) connection complex structure J

64 R.Mazzeo, J.Swoboda, H.Weiss & F.Witt, Ends of the moduli space of Higgs bundles, arxiv v2

65 R.Mazzeo, J.Swoboda, H.Weiss & F.Witt, Ends of the moduli space of Higgs bundles, arxiv v2 R.Bielawski, Asymptotic behaviour of SU(2) monopole metrics, J. Reine Angew. Math. 468 (1995), (well separated monopoles simple zeros of det Φ.)

66 R.Mazzeo, J.Swoboda, H.Weiss & F.Witt, Ends of the moduli space of Higgs bundles, arxiv v2 R.Bielawski, Asymptotic behaviour of SU(2) monopole metrics, J. Reine Angew. Math. 468 (1995), (well separated monopoles simple zeros of det Φ.) R.Bielawski, Asymptotic metrics for SU(N)-monopoles with maximal symmetry breaking, Comm. Math. Phys. 199 (1998), (higher rank groups)

67 R.Mazzeo, J.Swoboda, H.Weiss & F.Witt, Ends of the moduli space of Higgs bundles, arxiv v2 R.Bielawski, Asymptotic behaviour of SU(2) monopole metrics, J. Reine Angew. Math. 468 (1995), (well separated monopoles simple zeros of det Φ.) R.Bielawski, Asymptotic metrics for SU(N)-monopoles with maximal symmetry breaking, Comm. Math. Phys. 199 (1998), (higher rank groups) R.Bielawski, Monopoles and clusters., Comm. Math. Phys. 284 (2008), (clusters multiple zeros of det Φ.)

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