Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 1 / 32

Papers Joint work with L. Schaposnik Higgs bundles and (A, B, A)-branes arxiv:1305.4638 (to appear in Comm. Math. Phys.) Real structures on moduli spaces of Higgs bundles arxiv:1309.1195 (to appear in Adv. Theor. Math. Phys.) David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 2 / 32

Character Varieties Let Σ be a compact Riemann surface of genus g > 1, G a complex semi-simple Lie group. Definition The character variety of reductive representations: M Rep (Σ, G) = Hom + (π 1 (Σ), G)/G is the space of reductive representations ρ : π 1 (Σ) G modulo the action of G by conjugation. A representation ρ : π 1 (Σ) G is reductive if the induced representation of π 1 (Σ) on the Lie algebra g is completely reducible. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 3 / 32

Flat connections Definition M F lat (G, Σ) is the moduli space of flat G-connections with reductive holonomy, up to gauge equivalence. As is well known, a representation π 1 (Σ) G up to conjugacy is equivalent to a flat G-connection up to gauge equivalence. Thus we have a homeomorphism M Rep (Σ, G) M F lat (Σ, G) between the Betti moduli space M Rep (Σ, G) and the de-rham moduli space M F lat (Σ, G). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 4 / 32

Character Varieties There are several related character varieties one might consider: The character variety Hom + (π 1 (Σ), G R )/G R of representations into a real form G R of G. The character variety Hom + (π 1 (Σ ), G)/G of a non-oriented surface Σ with oriented double cover Σ. The character variety Hom + (π 1 (M), G)/G of a 3-manifold M with boundary Σ. The character variety Hom + (π1 orb (Σ), G)/G of orbifold representations, where Σ is now an orbifold Riemann surface. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 5 / 32

Character Varieties As we will recall, the character variety M Rep (Σ, G) can be identified with the moduli space M Higgs (Σ, G) of G-Higgs bundles on Σ. M Higgs (Σ, G) can be studied from various points of view: algebraic geometry, Morse theory, spectral data,... Under this correspondence the related character varieties Hom + (π 1 (Σ), G R )/G R, Hom + (π 1 (M), G)/G, Hom + (π 1 (Σ ), G)/G, Hom + (π orb 1 (Σ), G)/G, correspond to complex Lagrangians (or branes) in M Higgs (Σ, G). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 6 / 32

Character Varieties The interpretation of these character varieties as branes has some potential applications: We can use what we know about the Higgs bundle moduli spaces to spaces to say something about the character varieties According to geometric Langlands/Homological mirror symmetry, each brane in M Higgs (Σ, G) should have a corresponding brane in the M Higgs (Σ, L G), the Langlands dual moduli space We may be able to quantize these character varieties using the notion of brane quantization (Gukov, Witten). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 7 / 32

Higgs bundles Let K be the canonical bundle of Σ. Definition A Higgs bundle on Σ is a pair (E, Φ), where E is a holomorphic vector bundle on Σ, Φ is a holomorphic K-valued endomorphism Φ : E E K. For a bundle E of rank r, degree d, the slope of E is µ(e) = d/r. Definition A Higgs bundle (E, Φ) is semi-stable if for each holomorphic subbundle F E which is Φ-invariant ( Φ(F ) F K), we have µ(f ) µ(e). (E, Φ) is stable if the inequality is strict (whenever F E). (E, Φ) is polystable if it is a sum of stable Higgs bundles of the same slope. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 8 / 32

Higgs bundles One can construct a moduli space M Higgs (Σ, GL(n, C)) of semi-stable Higgs bundles. Points of M Higgs (Σ, GL(n, C)) are in bijection with polystable Higgs bundles. Similarly, for a complex semisimple Lie group G, we can define G-Higgs bundles (P, Φ): Definition A G-Higgs bundle is a pair (P, Φ), where P is a holomorphic principal G-bundle, Φ is a holomorphic K-valued section of the adjoint bundle. We can likewise form a moduli space M Higgs (Σ, G) of semi-stable G-Higgs bundles. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 9 / 32

The Hitchin equations Let P be a principal G-bundle, with no holomorphic structure given. Fix a reduction of structure of P to the maximal compact (given by a Hermitian structure). Let ( A, Φ) be a consist of: A -operator on ad(p ) (i.e., a holomorphic structure on P ) A K-valued section Φ of ad(p ). Definition The pair ( A, Φ) satisfies the Hitchin equations if where F A is the curvature of A. A Φ = 0, F A + [Φ, Φ ] = 0, David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 10 / 32

The Hitchin equations Theorem (Hitchin, Simpson) A Higgs bundle (P, Φ) is gauge equivalent to a solution of the Hitchin equations if and only if (P, Φ) is polystable. It follows that the moduli space M Hitchin (Σ, G) of solutions to the Hitchin equations is homeomorphic to the Higgs bundle moduli space: M Hitchin (Σ, G) M Higgs (Σ, G) This is the Hitchin-Kobayashi correspondence for Higgs bundles. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 11 / 32

The Hitchin equations Let ( A, Φ) satisfy the Hitchin equations. Let A be the unitary connection associated to A (the Chern connection). Then = A + Φ + Φ is a flat connection. Theorem (Donaldson, Corlette) A flat G-connection arises from a solution of the Hitchin equations if and only if the holonomy of is reductive. Thus we have a homeomorphism of moduli spaces: M Hitchin (Σ, G) M F lat (Σ, G). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 12 / 32

Summary We have several descriptions of the same moduli space: M Rep (Σ, G) M F lat (Σ, G) M Hitchin (Σ, G) M Higgs (Σ, G). M Rep (Σ, G) M F lat (Σ, G) is the correspondence between flat connections and their holonomy M F lat (Σ, G) M Hitchin (Σ, G) is the Donaldson-Corlette theorem M Hitchin (Σ, G) M Higgs (Σ, G) is the Hitchin-Simpson theorem David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 13 / 32

The Hitchin moduli space Theorem (Hitchin) The Hitchin moduli space M Hitchin (Σ, G) has a natural hyper-kähler structure (on its smooth points). In fact, M Hitchin (Σ, G) is constructed by an infinite dimensional hyper-kähler quotient. Let g be the hyper-kähler metric, I, J, K the complex structures, ω I, ω J, ω K the corresponding Kähler forms. Recall that a hyper-kähler manifold also carries natural holomorphic symplectic forms: Ω I = ω J + iω K, Ω J = ω K + iω I, Ω K = ω I + iω J. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 14 / 32

The Hitchin moduli space The moduli space M Rep (Σ, G) has a natural holomorphic symplectic structure (Atiyah-Bott, Goldman). Under the identification M Rep (Σ, G) M Hitchin (Σ, G) the holomorphic symplectic structure is (J, Ω J ). The moduli space M Higgs (Σ, G) also has a natural holomorphic symplectic structure. Under the identification M Higgs (Σ, G) M Hitchin (Σ, G) the holomorphic symplectic structure is (I, Ω I ). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 15 / 32

Branes in the Hitchin moduli space In a Kähler manifold we can speak of A- and B-branes, which roughly speaking are: A-brane = a Lagrangian submanifold, supporting a flat bundle. B-brane = a complex submanifold supporting a holomorphic bundle. In a hyper-kähler manifold there are four cases: (BBB)-brane = a hypercomplex submanifold supporting a hyperholomorphic bundle. (BAA)-brane = a complex Lagrangian with respect to (I, Ω I ), supporting a flat bundle. similarly we have (ABA)- and (AAB)-branes. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 16 / 32

Branes in the Hitchin moduli space These definitions are not the whole story for multiple reasons: Homological mirror symmetry suggests B-branes live in the derived category of coherent sheaves and A-branes live in the Fukaya category The Higgs bundle moduli spaces carry gerbes, so our vector bundles should really be twisted vector bundles (and our sheaves should be twisted sheaves) There are more general A-branes, the coisotropic branes (Kapustin-Orlov) David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 17 / 32

Real representations Let G R be a real form of G, σ : G G the corresponding Cartan involution. This induces an involution i 1 : M Higgs (Σ, G) M Higgs (Σ, G) given by i 1 (P, Φ) = (σ(p ), σ(φ)). Representations with holonomy in G R lie in the fixed point set of i 1. i 1 is holomorphic in I, anti-holomorphic in J, K. Hence, the fixed point set is a (BAA)-brane. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 18 / 32

Non-orientable surfaces Let Σ be a compact non-orientable surface with χ(σ ) < 0 and let Σ Σ be the oriented double cover (Σ has genus g > 1). Since Σ is the orientation double cover, it has an orientation reversing involution f : Σ Σ. This induces an involution i 2 : M Higgs (Σ, G) M Higgs (Σ, G) given by i 2 (P, Φ) = (f (ρ(p )), f (ρ(φ))), where ρ : G G is the anti-linear involution corresponding to the compact real form. Representations of Σ lie in the fixed point set of i 2. i 2 is holomorphic in J, anti-holomorphic in I, K. The fixed point set is an (ABA)-brane. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 19 / 32

3-manifolds with boundary Let M be a compact oriented 3-manifold with boundary M = Σ. Consider the flat connections on Σ which extend to flat connections on M. This defines a subspace L M M Higgs (Σ, G). Under some technical assumptions, L M is a complex Lagrangian submanifold with respect to (J, Ω J ), an (ABA)-brane. This construction is related to the Casson invariant and quantization of Chern-Simons theory. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 20 / 32

Orbifold representations Let Γ be a finite group which acts on Σ holomorphically. We get an induced action of Γ on M Higgs (Σ, G) which is holomorphic in I, J, K. The fixed point set of the action is therefore a (BBB)-brane. Flat orbifold connections on Σ/Γ are fixed points of the action of Γ. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 21 / 32

Anti-holomorphic involutions Let f : Σ Σ be an anti-holomorphic involution. Recall that f induces an involution i 2 : M Higgs (Σ, G) M Higgs (Σ, G) by i 2 (P, Φ) = (f (ρ(p )), f (ρ(φ))). In terms of flat connections, this is just the pullback: f ( ). We will consider in detail the fixed points of i 2. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 22 / 32

Fixed points of i 2 Consider the case G = SL(n, C). We are looking at pairs (E, ) where E is a rank n complex vector bundle and is a flat connection on E. If (E, ) is stable as Higgs bundle, then the holonomy representation π 1 (Σ) SL(n, C) is irreducible (acting on C n ). Theorem If has irreducible holonomy on C n, then (E, ) is a fixed point of i 2 if and only if f : Σ Σ lifts to a linear involution f : E E that preserves. Can think of (E,, f) as a flat Z 2 -equivariant bundle. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 23 / 32

Z 2 -equivariant K-theory To (irreducible) fixed points of i 2 we have a topological invariant [E] K Z2 (Σ) in the (reduced) Z 2 -equivariant K-theory of Σ. The fixed point set of f will be a disjoint union of some number m of embedded circles in Σ. Then K Z2 (Σ) = { Z2 if m = 0, Z m if m > 0. Over each fixed circle C i of f, we get a representation of Z 2 on the fibre C n. Let n + i be the number of +1 eigenvalues and n i the number of 1 eigenvalues. Then [E] (n + 1 n 1, n+ 2 n 2,..., n+ m n m) Z m gives the isomorphism in the m > 0 case. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 24 / 32

Sign factor Note: the lift f : E E is only unique up to sign f f. Therefore the topological invariant (n 1,..., n m ) Z m is only defined up to an overall sign ±(n 1,..., n m ) Z m /{±}. Question: how is this invariant reflected in the spectral data? David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 25 / 32

Spectral data A generic GL(n, C) Higgs bundle (E, Φ) can be constructed from spectral data, consisting of: A branched n-fold cover p : S Σ (the spectral curve) A line bundle U on S (the spectral line) Moreover the projection p : S Σ must factor through an inclusion S K of S into the total space of the canonical bundle. The inclusion S K yields a tautological section λ H 0 (S, p (K)). Given (E, Φ), the spectral curve S is given by the characteristic equation for λ: det(λi Φ) = 0. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 26 / 32

Spectral data Given a spectral curve S and spectral line U S, we recover (E, Φ) as: E = p (U) is the push-forward. Φ : E E K is obtained by pushing down the map λ : U U K David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 27 / 32

Fixed points If (E, Φ) is a fixed point of i 2 then the involution f : Σ Σ lifts to the spectral curve f : S S giving a commutative diagram S p f S Σ f Σ Assume that f has fixed points. Then Σ has a spin structure K 1/2 such that there is an anti-linear involution γ : K 1/2 K 1/2 for which (γ γ)(a) = f (a). p David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 28 / 32

Fixed points Write the spectral line U in the form U = L p (K) (n 1)/2. E = p (U) has trivial determinant if and only if L P rym(s, Σ). In particular, deg(l) = 0. Since L has degree 0, we can think of it as a flat line bundle. Then (E, Φ) is a fixed point of i 2 if and only if f : S S lifts to a linear involution f : L L preserving the flat structure. So L is a Z 2 -equivariant line bundle on S. Note that the lift f is only defined up to f f. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 29 / 32

Fixed points The class [E] K Z2 (Σ) depends only on the isomorphism class of L as a Z 2 -equivariant line bundle. Let C i be a fixed circle of f and C i,1,..., C i,k the fixed circles of f lying over C i (note that k can be zero). For each C i,j the Z 2 -action on L Ci,j has an eigenvalue s i,j = ±1. The restriction of γ : K 1/2 K 1/2 to C i defines a real structure on K 1/2 Ci. Squaring we obtain a non-vanishing real section of K Ci, defined up to positive rescaling. This is an orientation on C i. Similarly we have orientations on each C i,j. Theorem Let d i,j be the degree of the projection C i,j C i. Then n + i n i = k s i,j d i,j. j=1 David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 30 / 32

Associated 3-manifold Let σ : Σ [ 1, 1] Σ [ 1, 1] be given by and let σ(z, t) = (f(z), t) M = Σ [ 1, 1]/σ. Through this construction M is naturally a 3-dimensional orbifold with boundary. However, M is topologically a 3-manifold with boundary M = Σ. David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 31 / 32

Associated 3-manifold Representations of π 1 (Σ) which are obtained by restriction from representations of π 1 (M) are fixed points of i 2. Conversely: Theorem Suppose f has fixed points. If (E, Φ) has irreducible holonomy then the corresponding flat connection on Σ extends to a flat connection on M if and only if [E] K Z2 (Σ) is given by [E] = ±(n, n,..., n) If this is not the case then E can instead be viewed as a flat orbifold bundle on M, indeed we have K orb (M) K Z2 (Σ [ 1, 1]) K Z2 (Σ). David Baraglia (ADL) Higgs Bundles and Character Varieties 29 May 32 / 32