Mirror symmetry, Langlands duality and the Hitchin system I

Transcription

1 Mirror symmetry, Langlands duality and the Hitchin system I Tamás Hausel Royal Society URF at University of Oxford hausel/talks.html April 200 Simons lecture series Stony Brook /

2 Talk with same title in RIMS, Kyoto 6 September /

3 Mirror Symmetry phenomenon first arose in various forms in string theory mathematical predictions (Candelas-de la Ossa-Green-Parkes 99) mathematically it relates the symplectic geometry of a Calabi-Yau manifold X d to the complex geometry of its mirror Calabi-Yau Y d first aspect is the topological mirror test h p,q (X ) = h d p,q (Y ) compact hyperkähler manifolds satisfy h p,q (X ) = h d p,q (X ) (Strominger-Yau-Zaslow 996) suggests a geometrical construction how to obtain Y from X many predictions of mirror symmetry have been confirmed - no general understanding yet 3 /

4 Hodge numbers X n non-singular complex algebraic variety H k (X ; C) := {α Ωk (X ) dα=0} {α Ω k (X ) α=dβ} H i,j (X ) closed forms α of type (i, j) i.e. locally α = I =i, J =j a I,Jdz I d z J modulo exact forms by Hodge theory of harmonic forms H k (X ; C) = i+j=k H i,j (X ) Hodge numbers h i,j := dim H i,j (X ) Betti numbers b k := dim H k (X ) = i+j=k h i,j symmetries: h i,j = h j,i Hodge symmetry 2 h i,j = h n j,n i Poincaré duality example: Hodge diamond of a non-singular elliptic curve h, = h,0 = h 0, = h 0,0 = 4 /

5 Hodge diamonds of mirror Calabi-Yaus Fermat quintic X ˆX := X /(Z 5 ) K3 surface X ˆX mirror K /

6 Langlands duality the Langlands program aims to describe Gal(Q/Q) via representation theory G reductive group, L G its Langlands dual e.g L GL n = GL n ; L SL n = PGL n, L PGL n = SL n [Laumon 987, Beilinson Drinfeld 995] Geometric Langlands conjecture on a complex curve X {G-local systems on X } {Hecke eigensheaves on BunL G(X )} [Kapustin Witten 2006] deduces this from reduction of S-duality (electro-magnetic duality) in N = 4 SUSY YM in 4d 6 /

7 Diffeomorphic spaces in non-abelian Hodge theory C genus g curve; G = GL n (C) or SL n (C) M d Dol (G) := M d DR(G) := { moduli space of stable rank n degree d G-Higgs bundles (E, φ) i.e. E rank n degree d bundle on C φ H 0 (C, ad(e) K) Higgs field } moduli space of flat G-connections on C \ {p}, with holonomy e 2πid n Id around p when (d, n) = these are smooth non-compact varieties Γ = Jac C [n] = Z 2g n acts on M d (SL n ) by tensoring M d (PGL n ) := M d (SL n )/Γ is an orbifold Theorem (Non-Abelian Hodge Theorem) M d Dol (G) diff = M d DR (G) 7 /

8 Hitchin map the characteristic polynomial of φ H 0 (C, End(E) K) χ(φ) H 0 (C, K) H 0 (C, K 2 ) H 0 (C, K n ) defines Hitchin map χ GLn : M d Dol (GL n) A GLn = n i=h 0 (C, K i ) χ SLn : M d Dol (SL n) A SLn = n i=2h 0 (C, K i ) χ PGLn : M d Dol (PGL n) A PGLn = n i=2h 0 (C, K i ) Theorem (Hitchin 987, Nitsure 99, Faltings 993) χ is proper and a completely integrable Hamiltonian system. (ω(x χi, X χj ) = 0) Over a generic point a A the fibre χ (a) is a torsor for an Abelian variety. 8 /

9 Theorem (Hausel,Thaddeus 2003) For a generic a A SLn = APGLn the fibres χ SL n (a) and χ are torsors for dual Abelian varieties. M d Dol (PGL n) M d Dol (SL n) χ PGLn χsln A PGLn = ASLn. PGL n (a) M d DR (PGL n) and M d DR (SL n) satisfy the SYZ construction for a pair of mirror symmetric Calabi-Yau manifolds. 9 /

10 Topological mirror tests (Deligne 97) weight filtration for any complex algebraic variety X : W 0 W k W 2d = H d c (X ; Q), plus a pure Hodge structure on W k /W k of weight k define E(X ; x, y) = ( ) d x i y j h i,j ( W k /W k (H d c (X, C)) ) E B st (M/Γ) = [γ] [Γ] E(Mγ ; L B γ ) C(γ) (uv) F (γ) if Y X /Γ is crepant then (Kontsevich 996) E st (X /Γ; x, y) = E(Y ; x, y) Conjecture (Hausel Thaddeus 2003, DR-TMS, Dol-TMS ) For all d, e Z, satisfying (d, n) = (e, n) =, we have ( ) ( ) Est Be M d DR(SL n (C)); x, y = E ˆB d st M e DR(PGL n (C)); x, y ( ) ( ) Est Be M d Dol (SL n(c)); x, y = E ˆB d st M e Dol (PGL n(c)); x, y 0 /

11 Example let ˇM be moduli space of SL 2 parabolic Higgs bundles on elliptic curve E with one parabolic point Z 2 acts on E and C as additive inverse x x ˇM E C/Z 2 blowing up; χ : ˇM C/Z 2 = C is elliptic fibration with ˆD 4 singular fiber over 0 Γ = E[2] = Z 2 2 acts on ˇM by multiplying on E ˆM the PGL 2 moduli space is ˇM/Γ an orbifold, elliptic fibration over C with A singular fiber with three C 2 /Z 2 -orbifold points on one of the components blowing up the three orbifold singularities is crepant gives ˇM the topological mirror test: E st ( ˆM; x, y) Kontsevich = E( ˇM; x, y) /