Topological mirror symmetry via p -adic integration Dimitri Wyss ´ Ecole Polytechnique F´ ed´ erale de Lausanne Institue of Science and Technology Austria dimitri.wyss@epfl.ch June 18, 2017 Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 1 / 15
Overview Joint with Michael Groechening and Paul Ziegler Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 2 / 15
Overview Joint with Michael Groechening and Paul Ziegler Mirror symmetry considerations led [Hausel-Thaddeus, 2003] to conjecture an equality of string-theoretic Hodge numbers of SL n and PGL n Higgs moduli spaces. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 2 / 15
Overview Joint with Michael Groechening and Paul Ziegler Mirror symmetry considerations led [Hausel-Thaddeus, 2003] to conjecture an equality of string-theoretic Hodge numbers of SL n and PGL n Higgs moduli spaces. Their conjecture can be reformulated in terms of counting points over finite fields. This in turn can be done by computing p -adic volumes. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 2 / 15
Overview Joint with Michael Groechening and Paul Ziegler Mirror symmetry considerations led [Hausel-Thaddeus, 2003] to conjecture an equality of string-theoretic Hodge numbers of SL n and PGL n Higgs moduli spaces. Their conjecture can be reformulated in terms of counting points over finite fields. This in turn can be done by computing p -adic volumes. We can compare the p -adic volumes of the two moduli spaces, since ”singular Hitchin fibers have measure 0”. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 2 / 15
Moduli space of SL n Higgs bundles. Let C be a smooth projective curve of genus g and K = K C the canonical bundle. A Higgs bundle on C is a pair ( E , φ ), where E is a rank n vector bundle on C and φ ∈ H 0 ( C , End( E ) ⊗ K ). Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 3 / 15
Moduli space of SL n Higgs bundles. Let C be a smooth projective curve of genus g and K = K C the canonical bundle. A Higgs bundle on C is a pair ( E , φ ), where E is a rank n vector bundle on C and φ ∈ H 0 ( C , End( E ) ⊗ K ). Definition For an integer d coprime to n and a line bundle L of degree d on C define the moduli space of (twisted) SL n -Higgs bundles as M d SL n ( C ) = { Stable Higgs bundles ( E , φ ) , with det E ∼ = L , tr φ = 0 } / ∼ M d SL n ( C ) is a smooth quasi-projective variety. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 3 / 15
Moduli space of PGL n Higgs bundles The n -torsion points Γ = Jac C [ n ] ∼ = ( Z / n Z ) 2 g act on M d SL n ( C ) by tensoring: γ · ( E , φ ) = ( E ⊗ γ, φ ) , for γ ∈ Γ . Definition The moduli space of (twisted) PGL n Higgs bundles is M d PGL n ( C ) = M d SL n ( C ) / Γ . Remark: More generally one can construct moduli space of G -Higgs bundles for any reductive G , it is however unclear how to ”twist” in general. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 4 / 15
� � Hitchin Fibration Given a Higgs bundle ( E , φ ) ∈ H 0 ( C , End( E ) ⊗ K ) we can consider its characteristic polynomial h ( φ ) ∈ � n i =1 H 0 ( C , K ⊗ i ). This gives morphisms M d M d SL n PGL n h SL n h PGL n A = � n i =2 H 0 ( C , K ⊗ i ) Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 5 / 15
� � Hitchin Fibration Given a Higgs bundle ( E , φ ) ∈ H 0 ( C , End( E ) ⊗ K ) we can consider its characteristic polynomial h ( φ ) ∈ � n i =1 H 0 ( C , K ⊗ i ). This gives morphisms M d M d SL n PGL n h SL n h PGL n A = � n i =2 H 0 ( C , K ⊗ i ) Theorem (Hitchin, Simpson) The Hitchin maps h SL n , h PGL n are proper and their generic fibers are complex Lagrangian torsors for abelian varieties P SL n and P PGL n respectively. Furthermore P SL n and P PGL n are dual abelian varieties. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 5 / 15
� � Hitchin Fibration Given a Higgs bundle ( E , φ ) ∈ H 0 ( C , End( E ) ⊗ K ) we can consider its characteristic polynomial h ( φ ) ∈ � n i =1 H 0 ( C , K ⊗ i ). This gives morphisms M d M d SL n PGL n h SL n h PGL n A = � n i =2 H 0 ( C , K ⊗ i ) Theorem (Hitchin, Simpson) The Hitchin maps h SL n , h PGL n are proper and their generic fibers are complex Lagrangian torsors for abelian varieties P SL n and P PGL n respectively. Furthermore P SL n and P PGL n are dual abelian varieties. If it weren’t for the torsor structure , M d SL n and M d PGL n would be mirror partners in the sense of Strominger-Yau-Zaslow. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 5 / 15
Twisted SYZ Mirror Symmetry The correct duality between the fibrations h SL n , h PGL n should take the torsor structure into account [Hitchin 2001]: � Z / n Z -Gerbes B , ¯ B on M d SL n , M d PGL n . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 6 / 15
Twisted SYZ Mirror Symmetry The correct duality between the fibrations h SL n , h PGL n should take the torsor structure into account [Hitchin 2001]: � Z / n Z -Gerbes B , ¯ B on M d SL n , M d PGL n . Theorem (Hausel-Thaddeus 2003) PGL n , ¯ The pairs ( M d SL n , B ) and ( M d B ) are SYZ mirror partners i.e. for a generic a ∈ A we have isomorphisms of P SL n and P PGL n torsors SL n ( a ) ∼ h − 1 = Triv ( h − 1 PGL n ( a ) , ¯ B ) h − 1 PGL n ( a ) ∼ = Triv ( h − 1 SL n ( a ) , B ) . Remark: [Donagi-Pantev, 2012] prove a similar statement for any pair of Langlands dual groups. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 6 / 15
Topological Mirror Symmetry Because of the lack of properness and the presence of singularities, one cannot hope for the usual symmetry of the Hodge diamond. Instead Hausel-Thaddeus ’argue’ that there should be an equality of stringy Hodge numbers . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 7 / 15
Topological Mirror Symmetry Because of the lack of properness and the presence of singularities, one cannot hope for the usual symmetry of the Hodge diamond. Instead Hausel-Thaddeus ’argue’ that there should be an equality of stringy Hodge numbers . Definition For any complex variety X define the E -polynomial by � ( − 1) i h p , q ; i ( X ) x p y q , E ( X ; x , y ) = p , q , i ≥ 0 where h p , q ; i ( X ) = dim C ( Gr Ho p Gr w p + q H i c ( X )) denote the compactly supported mixed Hodge numbers of X . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 7 / 15
Topological Mirror Symmetry Because of the lack of properness and the presence of singularities, one cannot hope for the usual symmetry of the Hodge diamond. Instead Hausel-Thaddeus ’argue’ that there should be an equality of stringy Hodge numbers . Definition For any complex variety X define the E -polynomial by � ( − 1) i h p , q ; i ( X ) x p y q , E ( X ; x , y ) = p , q , i ≥ 0 where h p , q ; i ( X ) = dim C ( Gr Ho p Gr w p + q H i c ( X )) denote the compactly supported mixed Hodge numbers of X . The compactly supported cohomology of M d SL n and M d PGL n is pure i.e. h p , q ; i = 0 unless i = p + q . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 7 / 15
Topological Mirror Symmetry Conjecture (Hausel-Thaddeus 2003) There is an equality ¯ E ( M d st ( M d B SL n ; x , y ) = E PGL n ; x , y ) . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 8 / 15
Topological Mirror Symmetry Conjecture (Hausel-Thaddeus 2003) There is an equality ¯ E ( M d st ( M d B SL n ; x , y ) = E PGL n ; x , y ) . The right hand side takes into account the orbifold structure and can be written as ¯ ¯ st ( M d B � ( xy ) F ( γ ) E B γ (( M d SL n ) γ / Γ; x , y ) , SL n / Γ; x , y ) = E γ ∈ Γ where E ¯ B γ denotes the E -polynomial with coefficients in the local system ¯ B γ → ( M d SL n ) γ / Γ and F ( γ ) the Fermionic shift. The Conjecture is true for n = 2 , 3 [HT 2003]. Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 8 / 15
Reduction to finite fields The point count analogue of E ( X ; x , y ) is # X ( F q ). Consequently we define ¯ B st M d � q F ( γ ) � tr(Fr , ( ¯ # PGL n ( F q ) = B γ ) x ) . γ ∈ Γ x ∈ ( M d SL n ) γ / Γ( F q ) Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 9 / 15
Reduction to finite fields The point count analogue of E ( X ; x , y ) is # X ( F q ). Consequently we define ¯ B st M d � q F ( γ ) � tr(Fr , ( ¯ # PGL n ( F q ) = B γ ) x ) . γ ∈ Γ x ∈ ( M d SL n ) γ / Γ( F q ) Essentially by a theorem of Katz, the conjecture then follows from Theorem (Groechening-W.-Ziegler) ¯ # M d st M d B SL n ( F q ) = # PGL n ( F q ) . (1) Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 9 / 15
Reduction to p -adic integration Let F be a finite extension of Q p with ring of integers O F and residue field k F ∼ = F q . Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 10 / 15
Reduction to p -adic integration Let F be a finite extension of Q p with ring of integers O F and residue field k F ∼ = F q . One can integrate differential forms on p -adic manifolds in a similar way as on real manifolds. In particular for any O F -variety X we can integrate top forms on the manifold X ◦ = X ( O F ) ∩ X sm ( F ). Dimitri Wyss (EPFL/IST) SWAGP 2017 June 18, 2017 10 / 15
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