Periodic Monopoles and qopers

Transcription

1 Periodic Monopoles and qopers Vasily Pestun Institut des Hautes Études Scientifiques 28 July 2017 Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

2 Geometric Langlands Correspondence Let C be Riemann surface. Then there is a conjectured equivalence geometric Langlands Dmod(Bun G (C)) QCoh (LocL G (C)) Beilinson,Deligne,Drinfeld,Laumon Arinkin,Gaitsgory,Frenkel,Lafforgue,Lurie,Mircovic,Vilonen talk by Donagi geometric Langlands as a mirror symmetry A ɛ 1 Ω I (Hit G (C)) B Jɛ (HitL G (C)) Hitchin, Hausel, Thaddeus, Donagi, Pantev, Arinkin, Bezrukavnikov, Braverman Bershadsky,Johansen,Sadov,Vafa Kapustin,Witten Teschner Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

3 Canonical Coisotropic Brane and Opers geometric Langlands as a A-B mirror In particular A ɛ 1 Ω I (Hit G (C)) B Jɛ (HitL G (C)) B cc-brane (A ɛ 1 Ω I ) B opers (B Jɛ ) B cc-brane is the A-model canonical space-filling brane [Kapustin-Orlov] = quantized algebra of functions on T Bun G = the sheaf D in Dmod(Bun G ) B opers := mirror(b cc-brane ) The B cc-brane is holomorphic Lagrangian brane in HitL G,J ɛ Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

4 In the limit ɛ = 0 (see talk by Donagi) we have baby geometric Langlands as B-B model mirror B I (Hit G (C)) B I (HitL G (C)) B space-filling brane B Hitchin section (1) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

5 B opers is HyperKahler rotation of B Hitchin section in the limit ζ twistor = ɛr, R 0 where R R + rescales the Higgs field in the real moment map ω I F A R 2 φ φ = 0 Gaoitto,Moore,Neitzke Gaiotto Dumitrescu, Fredrickson, Rydonakis, Mazzeo, Mulase, Neitzke In this limit HitL G,J ɛ as a complex space is the space of flat ɛ-holomorphic connections LocL G (C) Jɛ = {G-bundle on C, ɛ-connection ɛ z + A z } (2) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

6 quantum geometric Langlands Feigin, Frenkel, Gaitsgory, Kapustin, Witten quantum geometric Langlands as a A-A mirror A ɛ 1 1 Ω Jɛ 2 (Hit G (C)) A ɛ 1 2 Ω Jɛ 1 (HitL G (C)) W β (g) WL β( L g) where L β = ɛ 1 ɛ 2 = r( L k + L h ) = 1 k + h = r β W β= ( L g) O(BL g,opers(c )) Z(U h (ĝ)) W β ( L g) O β 1(BL g,opers(c )) W g -algebra is quantization of the Poisson algebra of functions on B g,opers (C ) with quantization parameter β 1 Kostant, Drinfeld-Sokolov, Feigin-Frenkel Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

7 geometric Langlands Depending on parameters ɛ 1, ɛ 2 we are dealing with different level of complexity of geometric Langlands duality: baby geometric Langlands: ɛ 1 = ɛ 2 = 0 classical Hitchin integrable system ordinary geometric Langlands: ɛ 1 0, ɛ 2 = 0 quantum Hitchin integrable system / critical level / classical monodromy / classical commutative W-algebra quantum geometric Langlands: ɛ 1 0, ɛ 2 0 2d CFT / chiral vertex algebra / associative W-algebra Dealing with representation theory of Uĝ, differential equations, KZ equations, CFT conformal blocks, 4d Nekrasov partition functions Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

8 Class S-theory in Ω-background 10d IIB strings on X 4 C C 2 /Γ ADE 6d N = (0, 2) theory of type g ADE on X 4 C 4d N = 2 theory of class S G (C) on X 4 Hit G (C) is the Coulomb branch of vacua for S G (C) theory on X 4 = R 3 S 1 The class S theory on Ω-background X 4 = C 2 ɛ 1,ɛ 2 gives us a microscope to nail down geometric Langlands Nekrasov,Shatashivili Alday,Gaiotto,Tachikawa Gaiotto,Moore,Neitzke Teschner Kapustin, Witten Nekrasov,Witten Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

9 Proposal for q-geometric Langlands (c.f. Feigin,Frenkel,Reshetikhin, more recently Aganagic, Frenkel, Okounkov and Saponov) G L G a compact Langlands self-dual Lie group C a flat Riemann surface, e.g. C is C, C or E Mon G (C S 1 ) is the moduli space of G-monopoles on C S 1 with prescribed singularities at a coweight colored divisor on C flat action of abelian group C on C, so that ɛ C acts by shift z z + ɛ where C is identified with C, C/Z, or C/(Z + τz) a constant section dz of K C ; it induces holomorphic symplectic form Ω I on Mon G (C S 1 ) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

10 Proposal for q-geometric Langlands Conjecturally there are equivalences of the (derived) categories associated to the hyperkahler moduli space of periodic monopoles Mon G (C S 1 ) baby q-geometric Langlands is B-B mirror B I (Mon G (C S 1 )) B I (MonL G (C ɛ1 S 1 )) ordinary q-geometric Langlands is A-B mirror A ɛ 1 1 Ω I (Mon G (C S 1 )) B(MonL G (C ɛ1 S 1 )) quantum q-geometric Langlands is A-A mirror A ɛ 1 1 Ω Jɛ 2 (Mon G (C ɛ2 S 1 )) A ɛ 1 2 Ω Jɛ 1 (MonL G (C ɛ1 S 1 )) Here C ɛ1 S 1 is C fibered over S 1 with a twist ɛ. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

11 rational/trigonometric/elliptic Type of C = Jac(Č) Č Ě cusp Ě nod Ě C C C E 4d 5d 6d rational trigonometric elliptic cohomology K-theory elliptic cohomology Yangian algebra quantum affine algebra elliptic quantum group difference q-difference elliptic q-difference q-lift / categorification / K-theory version of various objects of (quantum) geometric Langlands and CFT. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

12 Integrable system of periodic monopoles Phase space The space Mon G (C S 1 ) I is the holomorphic phase space of integrable system of group valued Higgs bundles on C Hurtubuise,Markman Kapustin, Cherkis Charbonneau, Hurtubise Nekrasov, VP Group Higgs bundles Mon G (C S 1 ) I = {G-bundle on C, meromorphic section g of Ad G bundle} Let G be simply connected simple Lie group. Hamiltonians The ring of commuting Hamiltonians is generated by fundamental characters χ Ri (g(z)) where R i denotes the irreducible representation with fundamental highest weight λ i. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

13 Mon G (C SR 1 ) is hyperkahler. There is a family of holomorphic structures on Mon G (C S 1 ) fibered over CP 1 twistor. There is a convenient description of Mon G (C S 1 R ) ζ in the limit ζ 0, R with fixed ɛ = ζr aka conformal limit Gaiotto, and talks by Holland and Bridgeland The resulting complex space Mon(C S 1 ) Jɛ is equivalent to the complex space of ɛ-difference connections Finite difference ɛ-connection MonL G (C S 1 ) Jɛ = { L G-bundle P on C, mero morphism ɛ P P} The space MonL G (C S 1 ) Jɛ is q-geometric Langlands version of the space of flat ɛ-connections LocL G (C) Jɛ of the B-side of the ordinary geometric Langlands. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

14 Main question In the ordinary geometric Langlands the key role plays the question of finding the image B opers of the quantization brane B cc-brane under the mirror map. The answer leads to effective solution of quantum integrable system. Beilinson,Drinfeld, Kapustin,Witten, Gukov, Witten, Nekrasov,Rosly,Shatashvili Teschner, Gaiotto, Neitzke-Holland q-geometric Langlands A-B mirror Can we compute the q-oper brane, that is can we find the image of the canonical coisotropic quantization brane B cc-brane B q-oper under the q-geometric Langlands equivalence A ɛ 1 1 Ω I (Mon G (C S 1 )) B(MonL G (C ɛ1 S 1 ))? Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

15 What does it mean to compute? (α, β) coordinates and the generating function W (α) of q-opers To compute Lagrangian brane of q-opers means for us to find a suitable system of Darboux coordinates (α i, β i ) in MonL G (C ɛ1 S 1 ) and the generating function W (α) such that β i = W α i In the case of ordinary sl 2 opers on P 1 with 4 regular singularities the problem has been solved by Nekrasov, Rosly, Shatashvili. The generalization to sl n opers and irregular punctures has been addressed by Neitzke-Hollands, presented in the talk by Hollands, also see talk by Bridgeland. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

16 q-oper solution In this talk we will find the solution of q-oper problem. 1 Give geometric definition of the space of q-difference connections that are q-opers B q-opers MonL G (C ɛ S 1 ) 2 Define Darboux coordinates (α i, β i ) in MonL G (C ɛ S 1 ) 3 Present the generating function W (α) such that is the graph of B q-opers. β i = W α i The presented technic applies to any compact Lie group G, and any flat curve C, in the talk we consider concrete examples. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

17 q-oper definition From now on fix the q-difference case C = C, denote by z C the multiplicative coordinate, and set to be multiplicative shift q = e ɛ z qz A q-connection A(z) defines the parallel transport, i.e. q-difference equation on a trivializing section s(qz) = A(z)s(z) where A G(C ) is a G-valued function of z c.f. talk by Okounkov gauge transformation: A(z) g(qz)a(z)g(z) 1 Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

18 What does it mean for G-group valued q-connection A(z) to be a q-oper? Recall that A g-oper for g valued connection z + a z comes from Kostant section (Kostant 59-63) s K : g/g g of the adjoint Lie algebra quotient π : g g/g, see talk by Ben-Zvi. After fixing a Borel subalgebra, Kostant section yields an element a g by its conjugacy class. Example of Kostant section for sl 2 Let g = sl 2 and fix a conjugacy class u = 1 2 tr x 2 of a regular a g. Then Kostant section is ( ) 0 u a = 1 0 ( ) 0 u The sl 2 -oper is a connection of the form z +, talk Hollands, Ben-Zvi 1 0 Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

19 Parallel to the work of Kostant for the Lie algebra, there is construction of Steinberg 65 section for the Lie group. Example of Steinberg section for SL 2 Let G = SL 2 and fix a conjugacy class t 1 = trg of a regular g G. Then Steinberg section is ( ) t1 1 g = 1 0 Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

20 Example of Steinberg section for SL n Fix a conjugacy class of a regular g G by the fundamental characters t k = χ Rk (g) where χ Rk = tr Λ k Cng. Then Steinberg section is t 1 t 2 t g = Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

21 Definition of Steinberg section for simple G Fix g = g h g + and let e i g + be the standard generators of weight α i where α i are simple roots with i = 1... r. Let s i N(T ) be the Weyl reflections in simple roots α i, in particular s 1... s r is Coxeter element. Then an element r g(t) = s i exp( e i t i ), i=1 t i C is Steinberg section: there is an isomorphism (i.e. polynomial map in both directions) between affine spaces of the parameters (t 1,..., t r ) and the affine space of the fundamental characters (χ 1,..., χ r ) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

22 Caution In the SL r+1 example we find t i = χ i, i = 1... r but in general the map between χ i and t i is not identity. For example for SO(8) with Dynkin graph Y we find t 1 = χ 1 t 2 = χ t 3 = χ 3 t 4 = χ 4 where χ 1 is vector, χ 2 is adjoint, χ 3, χ 4 are spinors. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

23 Definition of q-oper A q-oper in Mon G (C q S 1 ) on C is the following data a reduction of the structure group of the G-bundle to a Borel subgroup B a q-connection A(z) in the form of Steinberg section A(z) = r s i exp( e i t i (z)) i=1 Frenkel, Semenov-Tian-Shansky, Sevostyanov Example of SL 2 q-oper A(z) = ( t1 (z) ) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

24 Example of SL n q-oper t 1 (z) t 2 (z) t 3 (z) A(z) = Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

25 Example of SO 8 q-oper Pick a basis in the fundamental representation of SO 8 such that the metric has the form and choose the conventional basis of simple roots. Then SO 8 q-oper is t 1 (z) t 2 (z) t 3 (z)t 4 (z) t 4 (z) t 3 (z) A(z) = 0 0 t 4 (z) t 1 (z) t 2 (z) t 3 (z) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

26 Isomorphism to the base The Steinberg affine map from the parameters (t 1 (z),..., t r (z)) of Steinberg section to the space of adjoint invariants defined by the fundamental characters (χ 1 (z),..., χ r (z)) provides canonical holomorphic isomorphism between the brane of q-opers B q-opers and the base of Mon G (C S 1 ) integrable system. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

27 Darboux coordinates in the q-char variety The power of Nekrasov, Rosly, Shatashvili construction comes from certain system of distinguished coordinates (α, β) in the character variety Char G (C) (representation of the fundamental group π 1 (C) in G). Riemann-Hilbert The character variety Char G (C) is isomorphic to the space Loc G (C) of the pairs (holomorphic G-bundle, holomorphic flat connection z + a z ) but the isomorphism is complex analytic, rather than algebraic. This isomorphism requires to compute the monodromies of the flat connection ( z, z + a z ) and is called Riemann-Hilbert correspondence. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

28 Unwrap the spirals Similarly, for the Mon G (C q S 1 ) we need to introduce the coordinates (α, β) in the space qchar G (C ) the analogue of the character variety. To construct qchar variety we look at space C q S 1 as the family of spirals R fibered over the elliptic curve C = C /q Z. The qchar variety qchar G (C ) = Mon G (E q R t ) The holomorphic description is given along the rays R t from t = to t = +. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

29 Monopole scattering data So we shall look for canonical coordinates (α, β) in the space Mon G ( C R). This space is well-understood after the work of Hitchin on monoles in R 3. In fact, if C were C Mon G (C R) n Maps n (P 1, G/B) where the monopole charge n takes values in the coroot lattice of G. The key idea is that we can filter the solutions to the parallel transport equation along the rays R D t s = 0 and construct two flags according to the asymptotics of growth as t + or as t. Birkhoff,Stokes,Hitchin,Hurtubise,Jarvis, c.f. talk by Hollands. To specify a flag is equivalent to specify a reduction of G-bundle structure to B-bundle. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

30 Example of qchar for SL 2 For SL 2 charge k monopoles we expect Maps( C, P 1 ) k, i.e. degree k rational rational functions. Suppose that scattering monodromy from t = to t = modulo B transformations is ( ) ( ) a(z) b(z) 0 c(z) d(z) The invariant data is the ratio b(z) a(z) = n i=1 β i z α i where α i C are locally flat relative to dz, and β i are the residues. The system (α, β) provides canonical coordinates for SL 2 qchar {α i, β j } = δ ij β j Hitchin, Donaldson, Hurtubise, Jarvis, Gerasimov, Harchev, Lebedev, Oblezin, Finkelberg, Kuznetsov, Markarian, Mirkovic, Braverman Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

31 Hitchin scattering For SL 2 monopoles on C R there is two-dimensional space of solutions of parallel transport along R parametrize by the points z C. D t s = 0 Let s ± (z, t) be the two solutions of minimal growth as t ±, they specify two lines L + C 2 and L C 2. 0 L + (z) C 2 0 L (z) C 2 For generic z the two lines L + (z) and L (z) are in generic position with L + L = 0. Still it could happen that at some point z C the lines L + (z ) and L (z ) coincide. The set of such points z are α i coordinates. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

32 From this special solution s(α i, t) of minimal growth at t ± we find the conjugated coordinate β i as the abelian monodromy, that is ratio β i = lim t s(α i, t)e λ+t lim t s(α i, t)e λ t in situation when the minimal growth solution has regular asymptotics with fixed values of λ ± coming from the boundary data of monopoles Mon at infinity. For non-regular growth we use more general suitable basis of normalizing coefficents. The GL n -difference equations and difference Hilbert-Riemann correspondence have been adressed since the ancient times Birkhoff 1913, and more recent work by Baranovsky-Ginzburg, Jimbo-Sakai, Borodin, Krichever,Ramis, Sauloy, Zhang, Etingof, Singer, Vizio, Kontsevich, Soibelman, and c.f. talk by Okounkov The twistor geometry of the periodic monopoles provides a new perspective on this ancient story. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

33 The construction (α, β) coordinates for monopole scattering problem of Maps( C, G/B) has natural generalization for arbitrary simple Lie algebra G, and introduction of singularities. In generic case we have r i=1 n i pairs of coordinates (α i,j, β i,j ) where ni α i is monopole charge, with j = 1... n i. The coordinates α i,j are the points on C in which the map to G/B lands in the divisor colored by the simple root α i. Some versions of qchar-varieties for Mon G ( C R), have appeared under different names such as rational/trigonometric/elliptic Zastava Finkelberg et.al, Braverman et.al, Beilinson-Drinfeld Grassmanian Gerasimov et.al, or the fiber of Hecke correspondence Kapustin-Witten. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

34 Separation of Variables / Abelianization / q-miura transformation By gauge transformation Ã(z) = g(qz)a(z)g 1 (z) of the q-connection A(z) in the equation s(qz) = A(z)s(z) the q-oper ( ) t1 (z) 1 A(z) = 1 0 can be converted into the lower triangular form with ( Y Ã(z) = 1 ) (z) 0 1 Y (z) and t 1 (z) = Y (qz) + Y 1 (z) The variables Y i (z) can be thought as generalized eigenvalues of Kac-Moody group element represented by the Steinberg section t 1 (z),..., t r (z). Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

35 Now we can integrate abelianized equation. Define Q i (z) such that Q i (qz) = Y i (qz)q i (z) and take the solution Q i (z) 1 at z 0 (assuming that 0 is regular singuarity with generalized eigenvalues Y i (z) 1). The Q i (z) generically blows up along the ray z/q n as n +. But for certain rays α i,j q Z the function Q i (α i,j q k ) has the asymptotics of minimal growth, say Q i (zq k ) β i,j q i k where q i is the minimal eigenvalue of generalized root type eigenvalue Y i (z) at z. This gives canonical coordinates (α i,j, β i,j ) Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

36 Generating function Proposition The 5d K-theoretic ADE quiver gauge theory partition function Z on C 2 q 1,q 2 S 1 is the generating function of the B q oper in the qchar G in coordinates (α, β) in a sense that B q oper is Lagrangian defined by the graph β i,j = lim q2 1 Z(q 2a i,j,..., )/Z(a i,j,... ) The expresion of t i (z) in terms of the generalized eigenvalues Y i (z) is called q-character Frenkel, Reshetikhin, Semenov-Tian-Shansky, Sevostyanov. It coincides with the q-character coming from the quiver gauge theory construction Nekrasov, VP, Shatashvili 13. Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

37 Quiver gauge theory in Ω-background Nakajima Douglas, Moore Kapustin, Cherkis VP, Nekrasov 12 6d N = (0, 1) Γ ADE -quiver gauge theory on X 4 Č 4d N = 2 Γ ADE (C) quiver gauge theory on X 4 The Γ-quiver gauge theory Ω-background X 4 = C 2 ɛ 1,ɛ 2 for C = C opens the window into the q-geometric quantum Langlands Mon G (C S 1 ) is the Coulomb branch of vacua for Γ ADE (C) theory on X 4 = R 3 S 1 Nekrasov,VP, Shatashivili 13 Nekrasov VP, Kimura Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38

38 Dankeschön! Vasily Pestun (IHES) Periodic Monopoles and qopers 28 July / 38