S-duality in hyperk¨ ahler Hodge theory Tam´ as Hausel Royal Society URF at University of Oxford & University of Texas at Austin http://www.math.utexas.edu/ ∼ hausel/talks.html September 2006 Geometry Conference in Honour of Nigel Hitchin Madrid 1
Problem Problem 1 (Hitchin, 1995) . What is the space of L 2 harmonic forms on the moduli space of Higgs bundles on a Riemann surface? 2
HyperK¨ ahLeR quotients • Construction of (Hitchin-Karlhede-Lindstr¨ om-Roˇ cek, 1987): • M hyperk¨ ahler manifold � • G Lie group, G M preserving the hyperk¨ ahler structure • hyperk¨ ahler moment map: µ H = ( µ I , µ J , µ K ) : M → R 3 ⊗ g ∗ • For ξ ∈ R 3 ⊗ ( g ∗ ) G the hyperk¨ ahler quotient M //// ξ G := µ − 1 H ( ξ ) /G, has a natural hyperk¨ ahler metric at its smooth points 3
Moduli of Yang-Mills instantons on R 4 • P → R 4 a U ( n )-principal bundle over R 4 • M = � { A connection on P ; | R 4 tr ( F A ∧ ∗ F A ) | < ∞} • A = A 1 dx 1 + A 2 dx 2 + A 3 dx 3 + A 4 dx 4 in a fixed gauge, where A i ∈ V = Ω 0 ( R 4 , adP) • g ∈ G = Ω( R 4 , Ad ( P )) acts on A ∈ M by g ( A ) = g − 1 Ag + g − 1 dg , preserving the hyperk¨ ahler structure • µ H ( A ) = 0 ⇔ F A = ∗ F A , self-dual Yang-Mills equation • M ( R 4 , P ) = µ − 1 H (0) / G, the moduli space of finite energy self-dual Yang-Mills instantons on P , has a natural hyperk¨ ahler metric • same story for X 4 ALE gravitational instanton ⇒ M ( X 4 ALE , P ) Naka- jima quiver variety 4
Moduli space of magnetic monopoles • Assume that A i are independent of x 4 • A = A 1 dx 1 + A 2 dx 2 + A 3 dx 3 connection on R 3 • A 4 = φ ∈ Ω 0 ( R 3 , ad P ) the Higgs field • G = Ω( R 3 , Ad P ) � M = { ( A, φ ) + boundary cond. } preserving the nat- ural hyperk¨ ahler metric on M • µ H ( A, φ ) = 0 ⇔ F A = ∗ d A φ Bogomolny equation • M ( R 3 , P ) = µ − 1 H (0) / G, the moduli space of magnetic monopoles on R 3 , has a natural hyperk¨ ahler metric • Atiyah-Hitchin 1985 finds the metric explicitly on M 2 ( R 3 , P SU (2) ) ⇒ describe scattering of two monopoles 5
Moduli space of Higgs bundles • Assume that A i are independent of x 3 , x 4 • A = A 1 dx 1 + A 2 dx 2 connection on R 2 • Φ = ( A 3 − A 4 i ) dz ∈ Ω 1 , 0 ( R 2 , ad P ⊗ C ) complex Higgs field • G = Ω( R 2 , Ad P ) � M = { ( A, Φ) of finite energy } preserving the nat- ural hyperk¨ ahler metric on M • the moment map equations µ H ( A, Φ) = 0 ⇔ F ( A ) = − [Φ , Φ ∗ ] , d ′′ A Φ = 0 . equivalent with Hitchin’s self-duality equations • replacing R 2 with a genus g compact Riemann surface C ; M ( C, P ) = µ − 1 H (0) / G has a natural hyperk¨ ahler metric 6
L 2 harmonic forms on complete manifolds • M complete Riemannian manifold, α ∈ Ω k ( M ) is harmonic iff dα = � d ∗ α = 0; it is L 2 iff M α ∧∗ α < ∞ ; H ∗ ( M ) is the space of L 2 harmonic forms • Hodge (orthogonal) decomposition: cpt ) ⊕ H ∗ ⊕ δ (Ω ∗ Ω ∗ L 2 = d (Ω ∗ cpt ) , • H ∗ cpt ( M ) → H ∗ ( M ) → H ∗ ( M ) is the forgetful map • Thus im( H ∗ cpt ( M ) → H ∗ ( M )) ”topological lower bound” for H ∗ ( M ) • H ∗ cpt ( M ) → H ∗ ( M ) is equivalent with the intersection pairing on H ∗ cpt ( M ), by Poincar´ e duality 7
S-duality conjectures on L 2 harmonic forms Conjecture 1 (Sen,1994) . � 0 ( R 3 , P SU (2) )) � H ∗ ( � M k SL (2 , Z ) k ⇓ � 0 d � = mid dim( H d ( � 0 ( R 3 , P SU (2) ))) = M k φ ( k ) d = mid Conjecture 2 (Vafa-Witten,1994) . Let M k,c 1 = M k,c 1 ( X 4 ALE , P U ( n ) ) . φ φ For d � = mid , dim( H d ( M )) = 0 , while dim( H mid ( M )) = dim(im( H mid cpt ( M ) → H mid ( M ))) . ⇓ � q k − c/ 24 dim( H mid ( M k,c 1 Z φ ( q ) = )) is a modular form . φ c 1 ,k 8
Results on L 2 harmonic forms • Sen 1994 ⇒ L 2 harmonic 2-form on � M 2 0 ( R 3 , P SU (2) ) ∼ = • Segal-Selby 1996 ⇒ dim(im( H mid → H mid ( M ))) = φ ( k ) for M = cpt ( M ) � 0 ( R 3 , P SU (2) ) M k • Hausel 1998 ⇒ dim(im( H mid cpt ( M ) → H mid ( M ))) = 0 for g > 1 and M = M 1 Dol ( SL (2 , C )) • Hitchin 2000 ⇒ H d ( M ) = 0 unless d = mid ; for a complete hy- perk¨ ahler manifold of linear growth; proves Sen’s conjecture for k = 2 • Hausel-Hunsicker-Mazzeo 2002 proves for fibered boundary manifolds M (like ALE, ALF or ALG gravitational instantons) H mid ( M ) = im( IH mid ( M ) → IH mid ( M )) m m ¯ • Carron 2005 proves for a QALE space M : H mid ( M ) = im( H mid cpt ( M ) → H mid ( M )) 9
Mixed Hodge Structure of Deligne • � p,q H p,q ; k ( M ) is the associated graded to the weight and Hodge filtrations on the cohomology H k ( M, C ) of a complex algebraic variety M • h p,q ; k = dim( H p,q ; k ( M )) , the mixed Hodge numbers • H ( M ; x, y, t ) = � p,q,k h p,q ; k ( M ) x p y q t k , the mixed Hodge polynomial • P ( M ; t ) = H ( M ; 1 , 1 , t ), the Poincar´ e polynomial • E ( M ; x, y ) = x n y n H (1 /x, 1 /y, − 1), the E-polynomial of a smooth va- riety M . 10
Arithmetic and topological content of the E-polynomial Theorem 2 (Katz 2005) . If M is a smooth quasi-projective variety de- fined over Z and # { M ( F q ) } = E ( q ) is a polynomial in q , then E ( M ; x, y ) = E ( xy ) . • MHS on H ∗ ( M, C ) is pure if h p,q ; k = 0 unless p + q = k ⇔ H ( M ; x, y, t ) = ( xyt 2 ) n E ( − 1 xt , − 1 yt ) ⇒ P ( M ; t ) = H ( M ; 1 , 1 , t ) = t 2 n E ( − 1 t , − 1 t ); exam- ples of varieties with pure MHS: smooth projective varieties, M Dol , M DR , Nakajima’s quiver varieties � H ( M ; xT, yT, tT − 1 ) • the pure part of H ( M ; x, y, t ) is PH ( M ; x, y ) = Coeff T 0 for a smooth M , it is always the image of the cohomology of a smooth compactification 11
Nakajima quiver varieties • Γ quiver with vertex set I and edges E ⊂ I × I ; v , w ∈ N I two dimension vectors; V i and W i corresponding vector spaces • V v , w = � a ∈ E Hom( V t ( a ) , V h ( a ) ) ⊕ � i ∈ I Hom( V i , W i ), action GL( v ) = � i ∈ I GL( V i ) → GL( V v ) • for ξ = 1 v ∈ gl ( v ) GL( v ) define the (always smooth) Nakajima quiver variety by M ( v , w ) = V v , w × V ∗ v , w //// ξ GL( v ) Theorem 3 (Nakajima 1998) . There is an irreducible representation of the Kac-Moody algebra g (Γ) of highest weight w on ⊕ v H mid ( M ( v , w )) . In particular the Weyl-Kac character formula gives the middle Betti numbers of Nakajima quiver varieties. When Γ affine Dynkin diagram M ( v , w ) is a component of M ( X Γ , U ( n ) . In the affine case the Weyl-Kac character formula is known to have modular properties ⇒ Vafa-Witten. 12
Theorem 4 (Hausel 2005) . For any quiver Γ , the Betti numbers of the Nakajima quiver varieties are: �� ( i,j ) ∈ E t − 2 � λi,λj � ��� i ∈ I t − 2 � λi, (1 w i ) � � � � � � T v � t − 2 � λi,λi � � � mk ( λi ) (1 − t 2 j ) v ∈ N I λ ∈P ( v ) � i ∈ I k j =1 P t ( M ( v , w )) t − d ( v , w ) T v = , � ( i,j ) ∈ E t − 2 � λi,λj � � � v ∈ N I � � T v � t − 2 � λi,λi � � � mk ( λi ) (1 − t 2 j ) v ∈ N I λ ∈P ( v ) i ∈ I k j =1 Corollary 5. The RHS is a deformation of the Weyl-Kac character for- mula ⇒ A Γ ( v , 0) = m v proving Kac’s conjecture (1982) , where � �� � � � abs. indec. reps of Γ over F q � � A Γ ( v , q ) := � � of dimension v , modulo isomorphism � � Corollary 6. When the quiver is affine ADE the RHS becomes an infinite product ⇒ ”elementary” proof of the modularity in the Vafa-Witten S- duality conjecture 13
Spaces diffeomorphic to M ( C, P U ( n ) ) � � moduli space of semistable rank n M d Dol ( GL ( n, C )) := degree d Hitchin pairs on C � � moduli space of flat GL ( n, C )-connections M d DR ( GL ( n, C )) := 2 πid n Id around p on C \ { p } , with holonomy e M d B ( GL ( n, C )) := { A 1 , B 1 , . . . , A g , B g ∈ GL ( n, C ) | A − 1 1 B − 1 1 A 1 B 1 . . . A − 1 B − 1 A g B g = ξ n Id } /GL ( n, C ) g g 14
Topological Mirror Test Theorem 7 (Hausel–Thaddeus 2002) . In the following diagram M d M d Dol ( PGL ( n )) − → Dol ( SL ( n )) ↓ χ PGL ( n ) ↓ χ SL ( n ) ∼ H PGL ( n ) = H SL ( n ) . the generic fibers of the Hitchin maps χ PGL ( n ) and χ SL ( n ) are dual Abelian varieties. ⇒ M d DR ( PGL ( n )) and M d DR ( SL ( n )) satisfy the SYZ construction for a pair of mirror symmetric Calabi-Yau manifolds. Conjecture 3 (Hausel–Thaddeus 2002) . For all d, e ∈ Z , satisfying ( d, n ) = ( e, n ) = 1 , � � � , � x, y ; M e E B e B d = E ˆ x, y ; M d DR ( SL ( n, C )) DR ( PGL ( n, C )) st st which morally should be related to S-duality in the recent work (Kapustin- Witten 2006) about a physical interpration of the Geometric Langlands programme. 15
Mirror symmetry for finite groups of Lie type Conjecture 4 (Hausel–R-Villegas 2004) . � � � x, y, M e � E B e B d = E ˆ x, y, M d B ( SL ( n, C )) B ( PGL ( n, C )) st st Theorem 8 (Hausel–R-Villegas, 2004) . G = SL ( n ) or GL ( n ) G ( F q ) finite group of Lie type E ( √ q, √ q, M d B ( G ( F q )) } = � | G ( F q ) | 2 g − 2 B ( G )) = # {M d χ (1) 2 g − 1 χ ( ξ d n ) χ ∈ Irr ( G ( F q )) ⇓ ” differences between the character tables of PGL ( n, F q ) and its Langlands dual SL ( n, F q ) are governed by mirror symmetry” 16
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