Topological mirror symmetry for Hitchin systems Tamás Hausel IST Austria http://hausel.ist.ac.at Higgs bundles in mathematics and physics summer school University of Hamburg September 2018 1 / 29
Institute of Science and Technology Austria 22 minutes from Vienna in the Vienna Woods PhD granting research institute in the basic sciences hiring at all levels: ist.ac.at/about-ist-austria/open-positions/ 2 / 29
Contents Lecture 1: topological mirror symmetry for SL n vs PGL n Higgs bundles with stringy Hodge numbers Lecture 2: topological mirror symmetry by Morse theory for C action and by p-adic integration Lecture 3: topological mirror symmetry with branes by equivariant K-theory 3 / 29
Talk on the same subject, RIMS, Kyoto 6 September 2001 4 / 29
Mirror Symmetry phenomenon first arose in various forms in string theory mathematical predictions(candelas-de la Ossa-Green-Parkes 1991) 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) (Kontsevich 1994) suggests homological mirror symmetry D b (Fuk(X,!)) D b (Coh(Y, I)) (Strominger-Yau-Zaslow 1996) suggests a geometrical construction how to obtain Y from X many predictions of mirror symmetry have been confirmed - no general understanding yet 5 / 29
Hodge diamonds of mirror Calabi-Yaus Fermat quintic X ˆX := X/(Z 5 ) 3 1 0 0 0 1 0 1 101 101 1 0 1 0 0 0 1 K3 surface X 1 0 0 1 20 1 0 0 1 1 0 0 0 101 0 1 1 1 1 0 101 0 0 0 1 ˆX mirror K3 1 0 0 1 20 1 0 0 1 6 / 29
Hitchin systems for SL n and PGL n Higgs bundles degree 1 line bundle on smooth complex projective curve C -twisted SL n -Higgs bundle (E, ) 1 rank(e) =n det(e) = 2 2 H 0 (C; End 0 (E) K) ˇM moduli space of stable SL n -Higgs bundles, non-singular and hyperkähler Z 2g =Pic C [n] n acts on ˇM by tensoring ) ˆM := ˇM/ PGL n -Higgs moduli space is an orbifold ȟ : ˇM! A := H 0 (C, K 2 ) H 0 (C, K n ) (E, ) 7! charpol( )=t n + a 2 t n 2 + + a n { ĥ : ˆM!A Theorem (Hitchin 1987) The Hitchin systems ȟ, ĥ are completely integrable, proper, Hamiltonian systems. 7 / 29
SYZ for SL n and PGL n Higgs bundles y y y y C T C A!Auniversal spectral curve for ȟ : ˇM!A ˇP := Prym A (C)!Auniversal Prym variety ˇP ˇM over A, when a 2A sm ˇP a ˇM a is a ˇP a -torsor =Pic(C)[n] acts on ˇP ˆP := ˇP/!A ˆP ˆM over A, when a 2A sm ˆP a ˆM a is a ˆP a -torsor ˇB 2 H 2 ( ˇM, Z(SL n )) H 2 ( ˇM, U(1)) gerbe given by liftings of the universal projective bundle to an SL n bundle ˆB 2 H 2 ( ˇM, U(1)) = H 2 ( ˆM, U(1)) - equivariant liftings Theorem (Hausel Thaddeus 2003) For a 2A sm 1 ˇP _ a ˆP a are dual Abelian varieties 2 ˇB ˇMa and ˆB ˆMa trivial 3 Triv U(1) ( ˇM a, ˇB) ˆMa and Triv U(1) ( ˆM a, ˆB) ˇMa 8 / 29
Topological mirror test ˇM @ @@@@@@@ ˆM ȟ ~~ ~~~~~~~ ĥ A [Hausel Thaddeus 2003] { generic fibers torsors for dual abelian varieties { hyperkähler rotation satisfies SYZ In the first two lectures we will discuss the mirror symmetry proposal of [Hausel Thaddeus 2003]: Hitchin systems for Langlands dual groups satisfy Strominger-Yau-Zaslow, so could be considered mirror symmetric; in particular they should satisfy the topological mirror tests: Conjecture (Hausel Thaddeus 2003, Topological mirror test ) h p,q ( ˇM) = h p,q st ( ˆM, ˆB) = h p,q st ( ˇM/ X, ˆB) := h p F( ),q F( ) ( ˇM /, LˆB) 2 9 / 29
E-polynomials (Deligne 1972) constructs weight filtration W 0 W k W 2d = Hc d (X; Q) for any complex algebraic variety X, plus a pure Hodge structure on W k /W k 1 of weight k we say that the weight filtration is pure when W k /W k 1 (Hc(X)) i, 0 ) k = i; examples include smooth projective varieties, X ˇM and ˆM define E(X; x, y) := ( 1) d x i y j h i,j W k /W k 1 (Hc d (X, C)) i,j,d basic properties: additive - if X i X locally closed s.t. [X i = X then E(X; x, y) = P E(X i ; x, y) multiplicative - F! E! B locally trivial in the Zariski topology E(E; x, y) =E(B; x, y)e(f; x, y) (X)) and E(X; x, y) = P p,q h p,q (X)x p y q is the Hodge E(X; t, t) is the Poincaré polynomial when weight filtration is pure then h p,q (X) :=h p,q (H p+q c 10 / 29
Stringy E-polynomials let finite group act on a non-singular complex variety M E st (M/ ; x, y) := P [ ]2[ ] E(M /C( ); x, y)(xy) F( ) stringy E-polynomial F( ) is the fermionic shift, defined as F( )= P w i, where acts on TX X with eigenvalues e 2 iw i, w i 2 [0, 1) F( ) is an integer when M is CY and acts trivially on K M motivating property [Kontsevich 1995] if f : X! M/ crepant resolution, K X = f K M/ then E(X; x, y) =E st (M/ ; x, y) if B 2 H 2 (M, U(1)) is a -equivariant flat U(1)-gerbe on M, then on each M we get an automorphism of B M { C( )-equivariant local system L B, we can define E B st (M/ ; x, y) := P [ ]2[ ] E(M, L B, ; x, y) C( ) (xy) F( ) stringy E-polynomial twisted by a gerbe 11 / 29
Topological mirror symmetry for the toy model let ˇM be moduli space of SL2 parabolic Higgs bundles on elliptic curve E with one parabolic point Z 2 acts on E and C as additive inverse x 7! x ˇM!(E C)/Z 2 blowing up; h : ˇM!C/Z2 C is elliptic fibration with ˆD4 singular fiber over 0 =E[2] Z 2 2 acts on ˇM by multiplying on E ˆM the PGL 2 moduli space is ˇM/ an orbifold, elliptic fibration over C with A 1 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 topological mirror test: E st ( ˆM; x, y) Kontsevich = E( ˇM; x, y) 12 / 29
Toy model 13 / 29
Topological mirror symmetry conjecture - unravelled Conjecture (Hausel Thaddeus, 2003) E( ˇM) =E ˆB st ( ˆM) Proved for n = 2, 3 using [Hitchin 1987] and [Gothen 1994]. as acts on H ( ˇM) we have { H ( ˇM) apple2ˆhapple( ˇM) { E( ˇM) = P apple2ˆ E apple ( ˇM) = E 0 ( ˇM) + k E B st ( ˆM) = P 2 E( ˇM, L B, ) = E( ˇM) + X H 1 (C, Z n ) and wedge product induces w : refined Topological Mirror Test for w( )=apple: E apple ( ˇM) =E( ˇM /, L B, ) variant z } { X apple2ˆ E apple ( ˇM) E( ˇM /, L B, ) 2 {z } stringy ˆ 14 / 29
Example SL 2 fix n = 2 T := C acts on ˇM by (E, ) 7! (E, ) Morse { H ( ˇM) = L F i ˇM T H +µ i(f i ) as -modules F 0 = Ň where = 0; then [Harder Narasimhan 1975] ) H (F 0 ) is trivial -module for i = 1,...,g 1! F i = {(E, ) E L 1 L 2, = 0 0 ' 0,'2 H 0 (L 1 1 L 2K)} { F i! S 2g 2i 1 (C) Galois cover with Galois group Theorem (Hitchin 1987) The action on H (F i ) is only non-trivial in the middle degree 2g 2i 1. For apple 2 ˆ we have! 2g 2i 1 2g 2 dim Happle (F i ) =. 2g 2i 1 15 / 29
Example PGL 2 2 = Pic 0 (C)[2] { C 2:1! C with Galois group Z 2 M 1 (GL 1, C ) k T Jac 1 (C ) push-forward! M 1 (GL 2, C) ˇM1 # det N m (C /C)! T Jac 1 (C) 3 (, 0) let ˇM(GL1, C ):=N m (C /C) 1 (, 0) endoscopic H -Higgs moduli space after [Narasimhan Ramanan, 1975] ˇM = ˇM(GL 1, C )/Z 2 T Prym 1 (C /C)/Z 2 can calculate dim H 2g 2i+1 ( ˇM /, LˆB, )= 2g 2 2g 2i 1 and 0 otherwise Theorem (Hausel Thaddeus, 2003) when n = 2 and apple = w( ) E apple ( ˇM) = E( ˇM / ; L B, ) 16 / 29
p-adic integration p 2 Z prime; Q p := { P n n 0 a n p n : a n 2{0,...,p 1}} the field of p-adic numbers Q p completion of Q w.r.t. the p-adic absolute value a b p = p v p(b) v p (a) non-archimedian: a p apple 1 for every integer a 2 Z Q Z p := {x 2 Q p : x p apple 1} p-adic integers compact subring local, with residue field F p := Z p /pz p Q p locally compact { Haar measure µ, normalized µ(z p )=1 { integration R Q p fdµ 2 C for f 2 C c (Q p, C) generalizes to integration using degree n differential forms on Q p -analytic manifolds compact example: Z p -points of the smooth locus of a variety defined over Z p dµ! measure given by the Q p analytic n-form! on an n-dimensional Q p manifold crucial fact: X Y such that codim(x) 1, and! top dimensional form on Y then R X(Z p ) dµ! = 0 17 / 29
p-adic integration and Hodge numbers Theorem (Weil 1961) Let X be a scheme over Z p of dimension n and! 2 H 0 (X, n X ) nowhere vanishing (called gauge form). Then Z X(Z p ) dµ! = X(F p) p n. example: X = {(x, y) 2 Z 2 p : xy = 1} then X(Z p )=Z p = { P n 0 a n p n : a 0, 0} = Z p \ pz p, thus 1 µ(x(z p )) = µ(z p ) µ(pz p )=1 p = p 1 p = F p p = X(F p) p but this is, µ(z p \{0}) =1 [Deligne 1974] ) if X and Y defined over R C finitely generated over Z, and if for any : R! F q we have X (F q ) = Y (F q ) then E(X; x, y) =E(Y; x, y). this implies a theorem of Batyrev and Kontsevich: birational smooth projective CY varieties have the same Hodge numbers 18 / 29
p-adic integration and stringy Hodge numbers Theorem (Gröchenig Wyss Ziegler 2017) X = Y/ a CY orbifold, with gauge form! and B 2 H 2 (Y, Q/Z) then there exists a function f B : X(Z p )! C such that = p dim X X [ ]2[ ]p F( ) X d Z X(Z p ) f B dµ! = #B st X(F p) p dim X Tr (Fr p, H d ét (Y, L B) C( ) )( 1) d { for CY orbifolds X i = Y i / i with gauge form! i and gerbe B i Z Z f B1 dµ!1 = f B2 dµ!2 8p ) E B 1 st (X 1(C)) = E B 2 st (X 2(C)) X 1 (Z p ) X 2 (Z p ) when B i = 0 { [Batyrev 1997] [Kontsevich 1995]: X! Y/ crepant resolution of CY orbifold: E st (Y/ ) = E(X) 19 / 29
Proof of TMS via p-adic integration strategy of proof [Gröchenig-Wyss-Ziegler 2017]: recall ˇM SLn Higgs moduli space, smooth, ˇB 2 H 2 ( ˇM, Q/Z) ˆM := ˇM/ PGL n moduli space, orbifold, ˆB 2 H 2 ( ˇM, Q/Z) consider them over Z p for a 2A(Z p ) \A sm (Q p ) when ˇM a (Z p ), ; and ˆM a (Z p ), ; then fˇb and fˆb are 1 on them and R ˇM a (Z p ) fˇb = vol( ˇM a (Z p )) = vol( ˆM a (Z p )) = R ˆMa (Z p ) fˆb when ˇM a (Z p )=; and ˆM a (Z p ), ; then Tate duality { fˇb is a non-trivial character on Abelian variety ˆM a (Z p ) ) R ˆM a (Z p ) fˆb = 0 R ˇM(Z p ) fˇbdµˇ! = R ȟ 1 (A(Z p )\A sm (Q p )) fˇbdµˇ! = R ĥ 1 (A(Z p )\A sm (Q p )) fˆbdµˆ! = R ˆM(Zp ) fˆbdµˆ! { E st ( ˇM(C)) = E ˆB st ( ˆM(C)) topological mirror symmetry! { E apple ( ˇM) =E( ˇM / ; LˆB) refined TMS holds along singular fibers of Hitchin map: new proof of Ngô s geometric stabilisation theorem 20 / 29
Comments on proof proof was accomplished by comparing p-adic volumes of smooth fibers of ȟ and ĥ over Q p such fibers could behave badly over Z p example: : X = Z 2 p :! Z p s.t. (x, y) =xy all fibers of X(Z p )! Z p over z, 0 are smooth Q p manifolds e.g. solutions of xy = p ) either x 2 pz p and y 2 Z p or x 2 Z p and y 2 pz p : is two copies of Z p, smooth over Q p but singular over Z p! F p another comment: the same proof should show that R ˇM 0 (Z p ) dµˇ! = R ˆM(Zp ) dµˆ! where ˇM 0 is, singular, moduli of semistable SL n Higgs bundles of trivial determinant can we define E st ( ˇM 0 ) such that it will agree with E st ( ˆM)? 21 / 29
Mirror symmetry for Higgs bundles & G complex reductive group G L its Langlands dual e.g. GL L n = GL n, SL L n = PGL n C smooth complex projective curve M Dol (G) (or M(G)) moduli of semi-stable G-Higgs bundles; h G : M Dol (G)!A G Hitchin map M DR (G) moduli space flat G connections M Dol (G) diff M DR (G) non-abelian Hodge theorem (Hitchin-Donaldson-Simpson-Corlette) M DR (G) MMMMMMMMMMM h G A G M DR (G L ) h G L wppppppppppp A G L special Lagrangian fibration, with dual generic fibres (Strominger-Yau-Zaslow 1995) mirror symmetry (Kapustin-Witten 2005) S : Fuk(M DR (G)) D coh (M DR (G L )) homological mirror symmetry of (Kontsevich 1994) 22 / 29
Semi-classical limit & (Donagi Pantev 2012) semi-classical limit S sc : D coh (M(G)) D coh (M(G L )) holds along generic fibers of the Hitchin map by fiberwise Fourier-Mukai transform M(G) LLLLLLLLLL h G A G M(G L ) h G L xqqqqqqqqqqq A G L recall FM : D coh (hg 1(a)) D coh(hg 1(a)_ ) D coh (h 1 (a)) G L e.g. FM swaps skycraper sheaves with line bundles cohomological shadow should be (Hausel-Thaddeus 2003) for G = SL n { E st (M(SL n )) = E st (M(PGL n )) topological mirror symmetry conjecture proved SL 2, SL 3 recently proved SL n (Gröchenig Wyss Ziegler 2017) using p-adic integration 23 / 29
BAA branes from now on G = GL 2 U(1, 1) GL 2 real form; U(1, 1)-Higgs bundle rank 2 E! E K C such that, 0 and E! E K C E! E K C M(U(1, 1)) M(GL 2 ) has g components L 0,...,L g 1 (Schaposnik 2015, Hitchin 2016) L i locus of Higgs bundles M 1 M 2! (M 1 M 2 ) K C with deg(m 1 )=i g + 1 for i = 0,...,g 2 L i M(GL 2 ) s smooth, total space of vector bundle E i on F i = Jac(C) C 2i =(M 1, H 0 (M 1 2 M 1K C )) L i M(GL 2 ) holomorphic Lagrangian subvarieties { BAA branes L = det(rpr 1 E) determinant line bundle on M(GL 2 ) from 0 B@ 1 0 1 2 0 CA universal bundle E! E K C on M(GL 2 ) C L i := L 2 O Li 2 D coh (M(GL 2 )); note L 2 i = K Li { L _ i L i 24 / 29
BBB branes what is S sc (L i )? first clue: S sc generically Fourier-Mukai transform (Schaposnik 2015) L i \ h 1 (a) 4g 4 is 2i copies of Jac(C) { S sc (L i ) \ h 1 (a) 4g 4 rank 2i vector bundle supported on M(SL 2 ) second clue: physics { mirror of BAA brane should be BBB brane i.e. triholomorphic sheaf (Hitchin 2016) { S sc (L i )= 2i (V) := i should be exterior power of the Dirac-Yang-Mills bundle V on M(SL 2 ) V := Rpr 1 (E! E K C )[1] triholomorphic rank 4g 4 vector bundle on M(SL 2 ) s note i has an orthogonal structure { i_ i can we find computational evidence for S sc (L i )= i outside generic locus of h? Yes! compute Hom Dcoh (M(GL 2 ))(L i, j ) 2 C[[t]] using T = C action (E, )! (E, ) 25 / 29
Main computation X semi-projective T-variety and F2Coh T (X) denote T(F ) = P dim(h k (X; F ) l )( 1) k l t 2 C((t)) T action (E, )! (E, ) { M(GL 2 ) semi-projective fixed point components F i = Jac(C) C 2i for i = 0,...,g 2 and N(GL 2 ) where =0 L i M(GL 2 ) is the total space E i of vector bundle on F i s 2 := L 2g 2 i=0 s2i 2i V Grothendieck-Riemann-Roch T(L i ; L i s 2) = T (F i ; L i s 2 Sym t 2(E i )) = R ȟ(l F i )ȟ( i s 2)ȟ(Sym t 2(E i ))td(t F i ) Theorem (Hausel Mellit Pei 2017) for i apple g 2 T(M(GL 2 ); L i s 2) = 1 2 i H z 2 t = z4(g 1 i) ((1+ s zt )(1+sz))2g 2 i ((1+ s sz z )(1+ t )) i (1 tz 2 ) 3g 3 i (1 t/z 2 ) 2i g+1 4 + s z 1+ s z + sz 1+sz + 4tz2 1 tz 2 + 4t z 2 1 t z 2 s zt 1+ s zt sz t 1+ sz t! g dz z 2 C[[t, s]] 26 / 29
Symmetry from mirror symmetry we expect T(M(GL 2 ); L i j ) = T (M(GL 2 ); L j i ) f(z, s) := z4 (1 t/z 2 ) 2 (1+ s zt )(1+sz) h(z, s) :=z @f @z Theorem { w = (1 tz 2 ) 2 (1+ s sz z )(1+ t ) (1+ s zt )(1+sz)(1+ s sz z )(1+ t (1 z 2 )( tz2 +1) (sz+t)(tz+s) (sz+1)(s+z) t ) T (M(GL 2 ); L i j )= H s = 1 2. H f = f(w, f 1 2 )=s 2 ; s(w, f 1 2 )=f 1 2 and h(w, f 1 2 )=h(z, s) Theorem (Hausel Mellit Pei 2017) T(M(GL 2 ); L i j )= T (M(GL 2 ); L j i ) h g 1 df ds f i s 2j f s 27 / 29
Reflection of mirror symmetry semi-classical limit S sc : D coh (M(GL 2 )) D coh (M(GL 2 )) S sc (L i )= i should imply Hom Dcoh (M(GL 2 ))(L i, j ) Hom Dcoh (M(GL 2 ))( i, L j ) more generally define equivariant Euler form T(L i, j ):= P k,l dim(h k (RHom(L i, j )) l )( 1) k t l 2 C[[t]] T( j, L i )= T ( j_ L i ) = T ( j L i ) T(L j, i )= T (L _ j j ) = T (L 2 O Lj K Lj i ) = T (L j i ) Theorem (Hausel Mellit Pei) We have T ( j, L i )= T (L j, i ) 28 / 29
Problems Maple indicates T (L i, L j )= T ( i, j ), can we prove it? another conjectured mirror (Neitzke) is S sc (L) =L 1, this should imply T (L ±1 L j )= T (L 1 j ). Maple agrees (Hitchin 2016) proposes similar story for U(m, m) GL 2m with mirror being the Dirac bundle supported on M(Sp(m)). Can our computations generalize to that case? Can we find computational evidence that S sc (L H ), i.e. H-Higgs bundles for a real form H G has support in M Dol (N H ) M Dol (G L ) where N H G L Nadler group of H as in (Baraglia, Schaposnik 2016)? Do these observations fit into the TQFT framework for T(M(G), L k )? What is S sc (L k )? How do Wilson and t Hooft operators act on L i, j? Could we guess FM transform on K T (h 1 (0)) for the nilpotent cone? 29 / 29