Periodic monopoles and difference modules

Periodic monopoles and difference modules Takuro Mochizuki RIMS, Kyoto University 2018 February

Introduction In complex geometry it is interesting to obtain a correspondence between objects in differential geometry and objects in algebraic geometry. Non-abelian Hodge theory (rough) On complex algebraic manifold, we have the following correspondences: ( ) ( ) Harmonic bundles Higgs bundles (Differential geometry) (Algebraic geometry) ( ( Harmonic bundles (Differential geometry) Harmonic bundles (Differential geometry) ) ) ( ( Flat bundles (Algebraic geometry) Ð-Flat bundles (Algebraic geometry) ) ) Remark Ð-connection Ð ( )=(Ð ò + ò ) + Ð ( ) (Higgs bundle: Ð = ¼, Flat bundle: Ð = ½)

Rough statement of the main result Theorem We have a natural correspondence of the following objects: Irreducible periodic monopoles of GCK type Stable parabolic difference modules of degree ¼

Explanation of the main result Monopoles (Å, ) : an oriented -dimensional Riemannian manifold. (, ) : a vector bundle with a Hermitian metric on Å. : a unitary connection of (, ). : an anti-self adjoint endomorphism of (, ) (called Higgs field). Definition (,,, ) is called monopole on Å if ( ) = ¼ (Bogomolny equation). Here, denote the Hodge star operator.

Periodic monopoles of generalized Cherkis-Kapustin type In this talk, we are particularly interested in monopoles (,,, ) on (Ë ½ R ¾ )\ =(Ë ½ C)\ ( finite subset). Condition around Any point of should be Dirac type singularity of (,,, ). Dirac type singularity was introduced by Kronheimer. Proposition (M-Yoshino) ( È is Dirac type singularity É = Ç (È, É) ½) for any É close to È. Condition around Ë ½ { } = Ç ( ÐÓ Û ) and ( ) ¼ as Û (Û: the coordinate of C). Definition (,,, ) is a periodic monopole of GCK (generalized Cherkis- Kapustin) type if the above conditions are satisfied.

Difference modules Let Ð be any complex number (twistor parameter). Let be the automorphism of C[Ý] determined by (Ý)=Ý+¾ ½Ð. (The pull back of C C given by (Ý)=Ý+¾ ½Ð.) Definition A (¾ ½Ð-)difference module is a C[Ý]-module Î with a C-linear automorphism such that ( )= ( ) ( ) ( C[Ý], Î). A = Ò ZC[Ý]( ) Ò is an algebra by (Ý+¾ ½Ð) = (Ý). Difference modules are clearly equivalent to A-modules. Definition Î is torsion-free torsion-free C[Ý]-module. Î is of finite type finitely generated over A, and Ñ C(Ý) Î C(Ý)<.

Parabolic structure at finite place Let Î be a torsion-free difference module of finite type. Definition Parabolic structure at finite place of Î consists of A free C[Ý]-module Î Î s.t. A Î = Î and Î C(Ý)= Î C(Ý). A function Ñ C Z ¼ s.t. C Ñ( )<. We assume Î C[Ý] =( ) ½ (Î) C[Ý], where ={ C Ñ( )>¼}, and C[Ý] denotes the localization of C[Ý] with respect to Ý ( ). For each C, Ø =(¼ Ø (½) < Ø (¾) < <Ø (Ñ( )) Lattices Ä, of Î C((Ý )) ( =½,...,Ñ( ) ½). < ½). We also set Ä,¼ = Î C[[Ý ]] and Ä,Ñ(Ü) =( ) ½ Î C[[Ý ]]

Parabolic structure at infinity Let Î be a torsion-free difference module of finite type. Let Î denote the formal completion at, Î = Î C[Ý]C((Ý ½ )) (C((Ý ½ ))-vector space). Parabolic structure at of Î is a filtered bundle P Î = ( ) P Î R over Î, i.e., Each P Î is a C[[Ý ½ ]]-lattice of Î, i.e., P Î are free C[[Ý ½ ]]-submodules of Î s.t. P Î C((Ý ½ ))= Î. P Î P Î ( ). P +Ò Î = Ý Ò P Î ( R, Ò Z). R, > ¼ such that P Î = P + Î.

Formal difference modules We have the automorphism of C((Ý ½ )) given by (Ý ½ )=Ý ½ (½+¾ ½Ð Ý ½ ) ½. Î Î induces Î Î s.t. ( )= ( ) ( ). For any Õ Z ½, set Ë(Õ) = { Õ ½ =½ b Ý /Õ b C }. Theorem (Turrittin) Ô Z >¼ and a decomposition compatible with Î C((Ý ½/Ô ))= Û Ô ½ Z C b Ë(Ô) Î Û,,b such that C[[Ý ½/Ô ]]-lattices Ä Û,,b Î Û,,b for which ( ½ Ý Û (½+b) ) ÄÛ,,b (Ý ½/Ô ) Ô Ä Û,,b. Set Ö(Û) = C b Ë(Ô) Ñ C((Ý ½/Ô )) Î Û,,b (well defined for Û Q).

Good parabolic structure at infinity A filtered bundle P Î over Î induces a filtered bundle P ( Î C((Ý ½/Ô )) ) over Î C((Ý ½/Ô )). Definition P Î is called good if we have a decomposition P ( Î C((Ý ½/Ô )) ) = Û,,b P Î Û,,b such that ( ½ Ý Û (½+b) ) P Î Û,,b (Ý ½/Ô ) Ô P Î Û,,b ( R). Parabolic difference module Parabolic difference module means a torsion-free difference module of finite type with a parabolic structure at finite place and a good parabolic structure at infinity. ( Î,(Î, Ñ,{Ø, Ä } C),P Î )

Degree Let (Î,(Î, Ñ,{Ø, Ä } C),P Î) be a parabolic difference module. We have the O P ½-module P ¼ V induced by free C[Ý]-module Î and a lattice P ¼ Î of Î. We obtain (P ¼ V )= P ½ ½(P ¼ V ). For each ½< ¼, we have the C-vector space Ö P ( Î) = P Î/ P< Î. Ö(Û) =,b Ñ C((Ý ½/Ô )) Î Û,,b for Û ½ Ô Z. If Ñ( )>¼, for each, we set ( (Ä, +½, Ä, ) = ) ( Ð Ò Ø Ä, +½/(Ä, +½ Ä, ) ) Ð Ò Ø Ä, /(Ä, +½ Ä, ). Definition The degree of the parabolic difference module is ( ) Î,(Î, Ñ,{Ø, Ä } C),P Î = (P ¼ F Î ) ÑC Ö P ½< ¼ ( Î) Û ¾ Ö(Û) + C (½ Ø ( ) ) (Ä,, Ä, ½).

Stability condition For any C(Ý)-subspace Î Î = Î C(Ý) such that ( Î )= Î, we have the induced parabolic difference module Î : Î = Î Î, Î = A Î, Ä, = Ä, Î Î = Î C((Ý ½ )), P Î = P Î Î. Definition (Î,(Î, Ñ,{Ø, Ä } C)P Î) is stable if (Î,(Î, Ñ,{Ø, Ä } C),P Î ) Ñ C(Ý) Î for any subspace ¼ Î Î with ( Î )= Î. < (Î,(Î, Ñ,{Ø, Ä } C),P Î) Ñ C(Ý) Î

Main theorem Theorem For each Ð, we have a natural bijective correspondence of the following objects: Irreducible periodic monopoles of GCK type. Stable parabolic ¾ ½Ð-difference modules of degree ¼. This is an analogue of the correspondence of harmonic bundles and Ð-flat bundles. Charbonneau and Hurtubise studied the Kobayashi-Hitchin correspondence of Dirac type singular monopoles Ë ½ Ë and holomorphic vector bundles with an automorphism (Ð = ¼). Recently, Kontsevich and Soibelman are developing the non-abelian Hodge theory of doubly periodic monopoles.

From monopoles to parabolic difference modules Case Ð = ¼ We regard Ë ½ C Û as the quotient space of R Ø C Û by the Z-action Ò (Ø, Û)=(Ø+ Ò, Û) (Ò Z). Let (,,, ) be a monopole on Ë ½ C Û. {Ø} C Û are holomorphic vector bundles by the operator ò,û = Û. We obtain {¼} C Û {½} C Û induced by the parallel transport along the paths {(Ø, Û) ¼ Ø ½} with respect to ò,ø = Ø ½. is holomorphic by [ò,ø,ò,û]=¼ ( = Bogomolny equation). {¼} C Û = {½} C Û. We obtain a holomorphic bundle {¼} C Û with an automorphism, and hence a ¼-difference module À ¼ (C Û, {¼} C Û ) (too large).

Case Ð ¼ We introduce two coordinates (Ø ¼, ¼ ), (Ø ½, ½ ) on R Ø C Û : (Ø ¼, ¼ )= ½ ( (½ Ð ¾ )Ø+ ¾ ÁÑ(Ð Û), ) Û+¾ ½ÐØ+ ¾ Ð Û, ½+ Ð ¾ (Ø ½, ½ )= ( Ø ¼ + ÁÑ(Ð ¼ ), (½+ Ð ¾ ) ¼ ) = ( Ø+ ÁÑ(Ð Û), Û+¾ ½ÐØ+ Ð ¾Û ). The Z-action is described as Ò (Ø ½, ½ )=(Ø ½ + Ò, ½ + ¾ ½Ð Ò) (Ò Z). We may identify Ë ½ Ø C Û as the quotient of R ؽ C ½ by the action. Note Ø ¼ Ø ¼ + ¼ ¼ = Ø Ø+ Û Û. By the Bogomolny equation, ò,ø ¼ = ؼ ½ and ò, = ¼ are commutative. ¼

We have the relation ò ؽ = ò ؼ, ò = ½ ½+ Ð ¾ Ð ¾ ½ ½Ð ò ؼ + ½ ½+ Ð ¾ ò ¼. We define the differential operators ò,ø ½ and ò, ½ on as follows: ò,ø ½ = ò,ø ¼, ò, = ½ ½+ Ð ¾ Then, [ ò,ø ½, ò, ½ ] = ¼. Ð ¾ ½ ½Ð ò,ø ¼ + {ؽ } C ½ are holomorphic vector bundles by ò, ½. ½ ½+ Ð ¾ ò, ¼. holomorphic isomorphism {¼} C ½ {½} C ½ by ò,ø ½. We obtain a holomorphic bundle {¼} C ½ with an isomorphism {¼} C ½ ( ½ ) ( {¼} C ½ ), ( ( ½ )= ½ + ¾ ½Ð). We obtain a ¾ ½Ð-difference module À ¼ (C, ½ {¼} C ½ ) (too large).

Meromorphic extension We regard {Ø ½ } C ½ P ½ ½. For any open subset Í P ½ ½ with Í, we set P( {ؽ } C ½ )(Í) = { (Í\{ }, ) = Ç( ½ Æ ) Æ R }, P ( {ؽ } C ½ )(Í) = { (Í\{ }, ) = Ç( ½ + ) > ¼}. = O P ½( )-module P( {ؽ } C ½ ) and O P ½-modules P ( {ؽ } C ½ ) ( R). Theorem Suppose (,,, ) is a monopole of GCK-type. P( {ؽ } C ½ ) is a locally free O P ½( )-module. P ( {ؽ } C ½ ) ( R) are locally free O P ½-modules. induces an isomorphism P( {¼} C ½ ) P( {½} C ½ ). (We remark P ( {¼} C ½ ) P ( {½} C ½ ) in general.) We need to study the asymptotic behaviour of periodic monopoles around Ë ½ { }, and to obtain the estimate of the curvature.

From periodic monopoles to parabolic difference modules We obtain a C[ ½ ]-free module. Î = Î = À ¼ ( P ½,P( {¼} C ½ ) ). It is naturally a ¾ ½Ð-difference module: ( À ¼ P ½,P( {¼} C ½ ) ) ( À ¼ P ½,P( {½} C ½ ) ) ( = À ¼ P ½,( ½ ) P( {¼} C ½ ) ) ( = À ¼ P ½,P( {¼} C ½ ) ). (P ( {¼} C ½ R) induces a filtered bundle P Î over Î. If (,,, ) has Dirac type singularities at finite points, then is meromorphic. So, Î = Î C( ½ ) is a difference module. We set Î = A Î. We naturally obtain a parabolic structure at finite place. Theorem The above construction induces an equivalence of irreducible periodic monopoles of GCK type and stable parabolic difference modules of degree ¼.

Analogy with the non-abelian Hodge theory C ¾ ={(Þ, Û)} with the Kähler metric Þ Þ+ Û Û is a hyperkähler manifold. Let Í be an open subset of C ¾ Let (,, ) be an instanton on Í, i.e., is a vector bundle with a Hermitian metric and a unitary connection satisfying ( )+ ( )=¼ (ASD equation). For each twistor parameter Ð, we obtain the complex manifold Í Ð holomorphic vector bundle Ð on Í Ð. and the Let C ¾ be a closed subgroup. If Í is -invariant, and if (,, ) is -equivariant, then Ð is also -equivariant. If =C Þ {¼} and Í =C Þ Í Û, then -equivariant instantons are equivalent to harmonic bundles on Í Û, and the -equivariant holomorphic bundles on Í Ð are equivalent to Ð-flat bundles on Í Û. (non-abelian Hodge theory on P ½ ). If =(R ½Z) {¼} and Í =C Þ Í Û, then -equivariant instantons are equivalent to monopoles on Ë ½ Í Û, and -equivariant holomorphic bundles are equivalent to (, ò,ø ¼, ò, ¼ ).

-dimensional meromorphic Ð-flat bundle We may regard (, ò,ø ½, ò, ½ ) as an -dimensional Ð-flat bundle. M Ð =(R ؽ C ½ )/Z Ë ½ Ø C Û C Û is given by (Ø ½, ½ )= Ì ÌM Ð ½ (ÌC Û ) is described as Ì (ò ؽ )= ½ ½ ½+ Ð ¾( ¾ ½Ð ؽ ) ( ½ ) ½ ¾ ½Ð ½+ Ð ¾ òû + ¾ ½Ð òû, Ì (ò )= ½ ½+ Ð ¾ ò Û. We define the operator D Ð on by D Ð ( ) =(½+ Ð ¾ ) ( ½ ) ò Ø ½ + ò ( ) Û+(½+ Ð ¾ )ò ½ ( ) Û. ½ ¾ For an open subset Í C Û, for local sections of on ½ (Í), and for -functions on Í, D Ð ( ½ ( ) )= ½ ( )D Ð ( )+ ½( (Ð ò Û + ò Û ) ) Monopoles are harmonic bundles on Ð-flat bundles.

KH-correspondence for analytically stable bundles Hermitian-Einstein metric (, Û): Kähler manifold (, ò ): holomorphic vector bundle on. For a Hermitian metric of, let ( ) denote the curvature of the Chern connection. Definition is called a Hermitian-Einstein metric of (, ò ) if Ä ( )= ÌÖ Ä ( ) Ö Ò.

Analytically stable bundles Let be a compact Lie group. Suppose that and (, ò, ¼ ) are equipped with -action. Suppose Ä ( ¼ ) <. For any saturated O -submodule, we have with codimension ¾ such that is a subbundle of \ \, and let ¼, denote the induced metric of. We define \ Ñ(, ¼ ) = ½ Ö Ò ÌÖ ( Ä ( )) ¼, ÚÓÐ R { } Definition (, ò, ¼ ) is analytically stable with respect to the -action if Ñ(, ¼ )<Ñ(, ¼ ) for any -invariant saturated subsheaf with ¼<Ö Ò < Ö Ò.

Theorem of Simpson (A1) The volume of (, Û) is finite. (A2) R ¼ such that - {Ü (Ü) } is compact for any, - is bounded. (A3) R ¼ R ¼ with (¼)=¼ and (Ü)=Ü (Ü ½) such that the following holds for any bounded function R ¼ : - If for >¼, then ÙÔ ( ) ( ). Moreover, if ¼ then = ¼. Theorem (Simpson) Under these assumptions, if (, ò, ¼ ) is analytically stable w.r.t - action, then there exists a -invariant Hermitian-Einstein metric of (, ò ) such that (i) and ¼ are mutually bounded, (ii) Ø( )= Ø( ¼ ), (iii) ò ( ½ ¼ ) is ľ.

A generalization (A3) ½, ¾ > ¼, R >¼ with <, such that the following holds for any bounded function R ¼ : - If for >¼, then ÙÔ ½ + ¾. Moreover, if ¼ then = ¼. Theorem Under this assumption, if (, ò, ¼ ) is analytically stable w.r.t -action, then there exists a -invariant Hermitian-Einstein metric of (, ò ) such that (i) and ¼ are mutually bounded, (ii) Ø( )= Ø( ¼ ), (iii) ò ( ½ ¼ ) is ľ.

Application to monopoles (Ð = ¼) We consider the action of Z ¾ and R-action Ö on C ¾ Þ,Û given by (Ò ½,Ò ¾ )(Þ, Û)=(Þ+Ò ½ + ½Ò ¾, Û), Ö (Þ, Û)=(Þ+, Û). Set =C ¾ /Z ¾ = Ë ½ (Ë ½ C Û ). We have the induced Ë ½ -action Ö on. Map Õ ¼ Ë ½ C is induced by (Þ, Û) (ÁÑ(Þ), Û)=(Ø, Û). Lemma Let Ë ½ C be a finite subset. Ë ½ -equivariant (, ò ) on \ Õ ½ ( ) ¼ (, ò,ø,ò,û) on (Ë ½ C)\ Ë ½ -equivariant (, ò, ) on \ Õ ½ ¼ ( ) with Ä ( )=¼ monopoles on (Ë ½ C)\.

Application to monopoles (Ð ¼) Let Ð denote the complex manifold with the complex structure given by ¼ = ½ ½ ), ½+ Ð ¾(Þ+ÐÛ ÐÛ+ Ð ¾ ¼ = Û). ½+ Ð ¾(Û+Ð(Þ Þ)+о Note ¼ ¼ + ¼ ¼ = Þ Þ+ Û Û. We have (Ø ¼, ¼ )=(ÁÑ( ¼ ), ¼ ). Lemma Ë ½ -equivariant (, ò ) on Ð \ Õ ½ ( ) ¼ (, ò,ø ¼, ò, ) on (Ë ½ C)\. ¼ Ë ½ -equivariant (, ò, ) with Ä ( )=¼ on Ð \ Õ ½ ( ) ¼ monopoles on (Ë ½ C)\. We may apply the Kobayashi-Hitchin correspondence for analytically stable bundles to the construction of monopoles on Ë ½ C\.

Examples Example 1 Take a finite set Ë C and l Ë Z >¼. Consider È(Ý)= Ë(Ý ) l( ) C(Ý). We set Î È =C(Ý) ½ C(Ý) ¾ with the automorphism : ( ) ¼ È(Ý) ( ½, ¾ )=( ½, ¾ ) Set Î È =C[Ý] ½ C[Ý] ¾ and Î È = A Î È in Î È. Proposition Take any (Ø ) Ë {¼ Ü<½} Ë. Set l = Ël( )= Ý È. l odd There exists a monopole on (Ë ½ C)\{(Ø, ) Ë} whose underlying ¼-difference module is Î È. l even There exist monopoles on (Ë ½ C)\{(Ø, ) Ë} parametrized by { } ( ½, ¾ ) R ¾ ½ + ¾ + (½ Ø )l( )=¼ Ë whose underlying ¼-difference modules are Î È. ½ ¼

If Î is a C(Ý)-subspace of Î È = Î È C(Ý) such that ( Î )= Î, then Î is Î È or ¼. We consider the parabolic structure at finite place. We define Ñ C Z ¼ by Ñ( )=½ ( Ë) and Ñ( )=¼ ( Ë). For Ë, we take ¼ Ø < ½. It is enough to classify the good parabolic structure at over PÎ È such that the parabolic degree is ¼.

l is odd Let Ø C((Ý ½/¾ )) be determined by Ø ¾l = È(Ý) and Ý ½/¾ Ø C[[Ý ½/¾ ]]. We set Ú ½ = Ø l ½ + ¾ and Ú ¾ = Ø l ½ ¾, then (Ú ½ )=Ø l Ú ½ and (Ú ¾ )= Ø l Ú ¾. A good filtered bundle P Î is determined by P (Ú ½ )= P (Ú ¾ )=. Because ½ =(¾Ø l ) ½ (Ú ½ + Ú ¾ ) ¾ = ¾ ½ (Ú ½ Ú ¾ ), we have P ( ½ )= l ¾ and P ( ¾ )= ¾. The degree of the parabolic ¼-difference module is ¼ l ¾ + (½ Ø )( l( )) ½ (l/¾)¾= ¾ ¾ Ë Ë (½ Ø )l( ). is uniquely determined by (Ø ) Ë by the vanishing condition of the degree. Proposition (l odd) For (Ø ) Ë {¼ Ü<½} Ë, we have a monopole on (Ë ½ C)\{(Ø, ) Ë} whose underlying ¼-difference module is Î È.

l=¾ even We have Ø C((Ý ½ )) determined by Ø ¾ = È(Ý) and Ý ½ Ø C[[Ý ½ ]]. We set Ú ½ = Ø ½ + ¾, Ú ¾ = Ø ½ ¾, then (Ú ½ )=Ø Ú ½ and (Ú ¾ )= Ø Ú ¾. A good filtered bundle P Î is determined by P (Ú )= ( =½, ¾). The parabolic degree is ½ ¾ (½ Ø )l( ) ½ ¾ ( ) ¾= ½ ¾ (½ Ø )l( ). Ë Ë Proposition (l even) For (Ø ) Ë {¼ { Ü < ½} Ë, we have monopoles on (Ë ½ C)\{(Ø }, ) Ë} parametrized by ( ½, ¾ ) R ¾ ½ + ¾ + Ë(½ Ø )l( ) = ¼ whose underlying ¼-difference module is Î È.

Example 2 Take a polynomial É(Ý) C[Ý]. Consider Î = Î =C[Ý] ½ C[Ý] ¾ with the action ( ) ( ½, ¾ )=( ½, ¾ ) Set = É+É ½ É ¾ ¾ C((Ý ½ )), then and ½ are the roots of É. ¼ ½ Let Ñ C Z ¼ be given by Ñ( )=¼ for any. Set Ú ½ = ½ + ¾ and Ú ¾ = ½ + ½ ¾, then ( ½ )= ½ and ( ¾ )= ½ ¾. A good parabolic filtered bundle P Î over Î is determined by = P (Ú ). The parabolic ¼-difference module (Î,(Î, Ñ),P Î) is stable of degree (É) ½ ¾. Proposition Wehave monopoles on Ë ½ C parametrizedby { ( ½, ¾ ) R ¾ ½ + ¾ + (É)=¼ } whose underlying ¼-difference module is Î. ½ É.