Mirror symmetry, Langlands duality and the Hitchin system

Transcription

1 Mirror symmetry, Langlands duality and the Hitchin system Tamás Hausel Royal Society URF at University of Oxford hausel/talks.html June 2009 Workshop on mirror symmetry MIT, Cambridge

2 Diffeomorphic spaces in non-abelian Hodge theory C genus g curve; G = GL n or SL n M d Dol (G) := moduli space of semistable 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 { } M d moduli space of flat G-connections DR(G) := on C \ {p}, with holonomy e 2πid n Id around p g M d B(G) := {A 1, B 1,.., A g, B g G A 1 i B 1 i A i B i = e 2πid n Id} /G i=1 when (d, n) = 1 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) diff = M d B (G)

3 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=1h 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 1987) χ is proper and a completely integrable Hamiltonian system. Over a generic point a A the fiber χ 1 (a) is a torsor for an Abelian variety.

4 Strominger-Yau-Zaslow Theorem (Hausel,Thaddeus 2003) For a generic a A SLn = APGLn the fibers χ 1 SL n (a) and χ 1 SL n (a) are torsors for dual Abelian varieties. M d Dol (PGL n) M d Dol (SL n) χ PGLn χsln A PGLn = ASLn. M d DR (PGL n) and M d DR (SL n) satisfy the SYZ construction for a pair of mirror symmetric Calabi-Yau manifolds.

5 Homological Mirror Symmetry - Geometric Langlands (Kontsevich 1994) s homological mirror symmetry proposal D b (Coh(M d DR(SL n ))) D b (Fuk(M d DR(PGL n ))) D b (Fuk(M d DR(SL n ))) D b (Coh(M d DR(PGL n ))) further hope D b (Fuk(M d DR (G)) Db (Dmod(Bun G )) above HMS D b (Dmod(Bun G )) D b (Coh(M d DR(G L ))) Geometric Langlands program of (Beilinson-Drinfeld 1995) for G = SL n and G L = PGL n (Kapustin-Witten 2007) above from reduction of S-duality (electro-magnetic duality) in N = 4 SUSY YM in 4d in the semi-classical limit HMS D b (Coh(M d Dol (SL n))) = D b (Coh(M d Dol (PGL n))) fibrewise Fourier-Mukai transform?

6 Topological mirror tests (Deligne 1972) constructs weight filtration W 0 W i W 2k = H k c (X ; Q) for any complex algebraic variety X, plus a pure Hodge structure on W k /W k 1 of weight k define E(X ; x, y) = ( 1) d x i y j H i,j ( W k /W k 1 (H d c (X, C)) ) Conjecture (Hausel Thaddeus 2003, DR-TMS, Dol-TMS ) For all d, e Z, satisfying (d, n) = (e, n) = 1, we have ( ) ( ) Est Be M d DR(SL n (C)); x, y = E ˆB d st M e DR(PGL n (C)); x, y. ( ) ( ) M d Dol (SL n(c)); x, y = E ˆB d st M e Dol (PGL n(c)); x, y. E Be st Conjecture ( Hausel Villegas 2004, B-TMS ) ( ) Est Be M d B(SL n (C)); x, y = E ˆB d st ( M e B(PGL n (C)); x, y ).

7 Results (Hausel Thaddeus 2003) Dol-TMS ( DR-TMS) for n = 2, 3 and (d, n) = 1 using description of H (M 1 Dol (SL n)) of (Hitchin 1987) for n = 2 and of (Gothen 1994) for n = 3 (Hausel Villegas 2004, Mereb 2009) B-TMS for n is prime and n = 4 using arithmetic techniques and character tables of GL n (F q ) and SL n (F q ) (de Cataldo Hausel Migliorini, 2008) finds that Dol,DR and B-TMS are closely related surprising connection to Ngô s work on the Fundamental Lemma in the Langlands program

8 Weight filtration X variety defined over Z Frob q : H k (X (F q ); Q l ) H k (X (F q ); Q l ) Frobenius automorphism (Deligne 1974) proved Weil s Riemann hypothesis: eigenvalues of Frob q have absolute value q i/2 for i N Jordan decomposition of Frob q on H k weight filtration W l H k containing all Jordan blocks of eigenvalue with modulus q i/2 i l comparison theorem: H (X (C); C) = H (X (F q ), Q l ) C (Deligne 1972) proved the existence of W 0 W i W 2k = H k (X ; Q) for any complex algebraic variety X, which is functorial compatible with cup-product

9 Arithmetic of character variety For a smooth complex variety X define E(X ; q) = W i /W i 1 (H k (X ))( 1) k q d i 2 (Katz 2008) proves that if E(q) := X (F q ) is polynomial in q then E(X (C); q) = E(q) (Hausel-Villegas 2008) calculates E(M B ; q) = M B (F q ) = GL n(f q) 2g 2 χ Irr(GL n(f q)) χ(ξ χ(1) 2g 1 n ) we find E(M B ; 1/q) = q d E(M B ; q) palindromic by Alvis-Curtis duality q n(n 1) 2 χ(1)(1/q) = χ (1)(q) for dual pair χ, χ Irr(GL n (F q )) Curious Hard Lefschetz Conjecture (theorem when n = 2): L l : Gr W d 2l (Hi l (M B )) Gr W d+2l Hi+l (M B ) x x α l, where α W 4 H 2 (M B )

10 Perverse filtration f : X Y a proper map between complex algebraic varieties of relative dimension d (de Cataldo-Migliorini 2005) introduce perverse filtration P i P i+1... P k (X ) = H k (X ) from the study of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for Rf (Q X ) into perverse sheaves recipe (de Cataldo-Migliorini, 2008) for perverse filtration when X smooth and Y affine: take Y 0 Y i... Y d = Y s.t. Y i generic with dim(y i ) = i then P k i 1 H k (X ) = ker(h k (X) H k (f 1 (Y i ))) the Relative Hard Lefschetz Theorem holds: L l : Gr P d l (H (X )) Gr P d+l H +2l (X ) x x α l where α H 2 (X ) is a relative ample class

11 Main conjecture recall Hitchin map χ : M Dol A (E, φ) charpol(φ) thus induces perverse filtration on H (M Dol ) is proper, Conjecture ( P=W, de Cataldo-Hausel-Migliorini 2008) P k (M Dol ) = W 2k (M B ) under the isomorphism H (M Dol ) = H (M B ) from non-abelian Hodge theory. Slogan: Topology of the Hitchin map reflects the arithmetic of the character variety Theorem (de Cataldo-Hausel-Migliorini 2009) P = W when G = GL 2, PGL 2 or SL 2. Proof is inspired by the proof of (Ngô, 2008) of the geometrical stabilization of the trace formula

12 Outlook Define PE(M Dol ; x, y, q) := q k E(Gr P k (H (M Dol )); x, y) PE(M Dol ; x, y, 1) = E(M Dol ; x, y) = E(M DR ; x, y) Conjecture P = W PE(M Dol ; 1, 1, q) = E(M B ; q) Conjecture (Topological Mirror test) ( ) ( ) PEst Be M d Dol (SL n); x, y, q = PE ˆB d st M e Dol (PGL n); x, y, q. Dol-TMS, DR-TMS of (Hausel-Thaddeus 2003), and B-TMS of (Hausel-Villegas 2004) When combined with RHL this says h i,j;p st,e (M d Dol (SL n)) = h i+dn/2 p,j+dn/2 p;dn p (M e Dol (PGL n)) st,d expected from fibrewise Fourier-Mukai and so from HMS Theorem when n = 2 reflects Ngô s geometric stabilization of the trace formula Fundamental Lemma for the (semi)stable nilpotent cone?