Periodic Monopoles and qopers
|
|
- Felix Dorsey
- 5 years ago
- Views:
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
Vasily Pestun! Strings 2016 Beijing, August 4, 2016
Moduli spaces, instantons, monopoles! and Quantum Algebras Vasily Pestun!! Strings 2016 Beijing, August 4, 2016 1 4d gauge theories What can we do beyond perturbation theory?! Are there hidden algebraic
More informationarxiv: v1 [math.ag] 13 Dec 2018
MULTIPLICATIVE HITCHIN SYSTEMS AND SUPERSYMMETRIC GAUGE THEORY CHRIS ELLIOTT AND VASILY PESTUN arxiv:1812.05516v1 [math.ag] 13 Dec 2018 Abstract. Multiplicative Hitchin systems are analogues of Hitchin
More informationR-matrices, affine quantum groups and applications
R-matrices, affine quantum groups and applications From a mini-course given by David Hernandez at Erwin Schrödinger Institute in January 2017 Abstract R-matrices are solutions of the quantum Yang-Baxter
More informationRepresentation theory of W-algebras and Higgs branch conjecture
Representation theory of W-algebras and Higgs branch conjecture ICM 2018 Session Lie Theory and Generalizations Tomoyuki Arakawa August 2, 2018 RIMS, Kyoto University What are W-algebras? W-algebras are
More informationGeometry of Conformal Field Theory
Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction
More informationHitchin s system and the Geometric Langlands conjecture
Hitchin s system and the Geometric Langlands conjecture Ron Donagi Penn Kobe, December 7, 2016 Introduction This talk has two goals. Introduction This talk has two goals. The first goal of this talk is
More informationClassical Geometry of Quantum Integrability and Gauge Theory. Nikita Nekrasov IHES
Classical Geometry of Quantum Integrability and Gauge Theory Nikita Nekrasov IHES This is a work on experimental theoretical physics In collaboration with Alexei Rosly (ITEP) and Samson Shatashvili (HMI
More informationNew worlds for Lie Theory. math.columbia.edu/~okounkov/icm.pdf
New worlds for Lie Theory math.columbia.edu/~okounkov/icm.pdf Lie groups, continuous symmetries, etc. are among the main building blocks of mathematics and mathematical physics Since its birth, Lie theory
More informationWhat elliptic cohomology might have to do with other generalized Schubert calculi
What elliptic cohomology might have to do with other generalized Schubert calculi Gufang Zhao University of Massachusetts Amherst Equivariant generalized Schubert calculus and its applications Apr. 28,
More informationStability data, irregular connections and tropical curves
Stability data, irregular connections and tropical curves Mario Garcia-Fernandez Instituto de iencias Matemáticas (Madrid) Barcelona Mathematical Days 7 November 2014 Joint work with S. Filippini and J.
More informationHiggs Bundles and Character Varieties
Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character
More informationLanglands duality from modular duality
Langlands duality from modular duality Jörg Teschner DESY Hamburg Motivation There is an interesting class of N = 2, SU(2) gauge theories G C associated to a Riemann surface C (Gaiotto), in particular
More informationHitchin s system and the Geometric Langlands conjecture
Hitchin s system and the Geometric Langlands conjecture Ron Donagi Penn String Math Conference, Hamburg, 25 July, 2017 Introduction The goal of this talk is to explain some aspects of the Geometric Langlands
More informationOn the geometric Langlands duality
On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:
More informationREPORT DOCUMENTATION PAGE ^RL-SR-AR-TR-10-0' 78 08/01/ /31/2009 FA GEOMETRIC LANGLANDS PROGRAM AND DUALITIES IN QUANTUM PHYSICS
REPORT DOCUMENTATION PAGE ^RL-SR-AR-TR-10-0' 78 The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for revie' maintaining the
More information3d Gauge Theories, Symplectic Duality and Knot Homology I. Tudor Dimofte Notes by Qiaochu Yuan
3d Gauge Theories, Symplectic Duality and Knot Homology I Tudor Dimofte Notes by Qiaochu Yuan December 8, 2014 We want to study some 3d N = 4 supersymmetric QFTs. They won t be topological, but the supersymmetry
More informationDEFECTS IN COHOMOLOGICAL GAUGE THEORY AND DONALDSON- THOMAS INVARIANTS
DEFECTS IN COHOMOLOGICAL GAUGE THEORY AND DONALDSON- THOMAS INVARIANTS SISSA - VBAC 2013 Michele Cirafici CAMGSD & LARSyS, IST, Lisbon CAMGSD @LARSyS based on arxiv: 1302.7297 OUTLINE Introduction Defects
More informationBPS states, Wall-crossing and Quivers
Iberian Strings 2012 Bilbao BPS states, Wall-crossing and Quivers IST, Lisboa Michele Cirafici M.C.& A.Sincovics & R.J. Szabo: 0803.4188, 1012.2725, 1108.3922 and M.C. to appear BPS States in String theory
More informationANNOTATED BIBLIOGRAPHY. 1. Integrable Systems
ANNOTATED BIBLIOGRAPHY DAVID BEN-ZVI 1. Integrable Systems [BF1] D. Ben-Zvi and E. Frenkel, Spectral Curves, Opers and Integrable Systems. Publications Mathèmatiques de l Institut des Hautes Études Scientifiques
More informationInstanton calculus for quiver gauge theories
Instanton calculus for quiver gauge theories Vasily Pestun (IAS) in collaboration with Nikita Nekrasov (SCGP) Osaka, 2012 Outline 4d N=2 quiver theories & classification Instanton partition function [LMNS,
More informationarxiv: v3 [hep-th] 7 Sep 2017
Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory arxiv:1607.08330v3 [hep-th] 7 Sep 2017 Meng-Chwan Tan Department of Physics, National University of
More informationΩ-deformation and quantization
. Ω-deformation and quantization Junya Yagi SISSA & INFN, Trieste July 8, 2014 at Kavli IPMU Based on arxiv:1405.6714 Overview Motivation Intriguing phenomena in 4d N = 2 supserymmetric gauge theories:
More informationSurface Defects and the BPS Spectrum of 4d N=2 Theories
Surface Defects and the BPS Spectrum of 4d N=2 Theories Solvay Conference, May 19, 2011 Gregory Moore, Rutgers University Davide Gaiotto, G.M., Andy Neitzke Wall-crossing in Coupled 2d-4d Systems: 1103.2598
More informationRefined Donaldson-Thomas theory and Nekrasov s formula
Refined Donaldson-Thomas theory and Nekrasov s formula Balázs Szendrői, University of Oxford Maths of String and Gauge Theory, City University and King s College London 3-5 May 2012 Geometric engineering
More informationMirror symmetry, Langlands duality and the Hitchin system I
Mirror symmetry, Langlands duality and the Hitchin system I Tamás Hausel Royal Society URF at University of Oxford http://www.maths.ox.ac.uk/ hausel/talks.html April 200 Simons lecture series Stony Brook
More informationDONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/ v2 [math.ag] 24 Oct 2006
DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/0607060v2 [math.ag] 24 Oct 2006 DAVID BALDUZZI Abstract. The Donagi-Markman cubic is the differential of the period map for algebraic completely integrable
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationAbstract. G.Williamson. M.Watanabe. I.Losev
Abstract G.Williamson Title : Representations of algebraic groups and Koszul duality Abstract : My talk will be about representations of algebraic groups and the Hecke category. I will describe a conjectural
More informationThe Langlands dual group and Electric-Magnetic Duality
The Langlands dual group and Electric-Magnetic Duality DESY (Theory) & U. Hamburg (Dept. of Math) Nov 10, 2015 DESY Fellows Meeting Outline My hope is to answer the question : Why should physicists pay
More informationPeriodic 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
More informationLoop Groups, Characters and Elliptic Curves
Loop Groups, Characters and Elliptic Curves David Ben-Zvi University of Texas at Austin ASPECTS of Topology in Geometry and Physics Oxford, December 2012 Overview I will discuss ongoing joint work with
More informationEDWARD FRENKEL BIBLIOGRAPHY. Books
EDWARD FRENKEL BIBLIOGRAPHY Books 1. Vertex Algebras and Algebraic Curves (with D. Ben-Zvi), Mathematical Surveys and Monographs 88, AMS, First Edition, 2001; Second Edition, 2004 (400 pp.). 2. Langlands
More informationW-algebras, moduli of sheaves on surfaces, and AGT
W-algebras, moduli of sheaves on surfaces, and AGT MIT 26.7.2017 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory
More informationTowards a modular functor from quantum higher Teichmüller theory
Towards a modular functor from quantum higher Teichmüller theory Gus Schrader University of California, Berkeley ld Theory and Subfactors November 18, 2016 Talk based on joint work with Alexander Shapiro
More informationVertex Algebras and Algebraic Curves
Mathematical Surveys and Monographs Volume 88 Vertex Algebras and Algebraic Curves Edward Frenkei David Ben-Zvi American Mathematical Society Contents Preface xi Introduction 1 Chapter 1. Definition of
More informationTitles and Abstracts
29 May - 2 June 2017 Laboratoire J.A. Dieudonné Parc Valrose - Nice Titles and Abstracts Tamas Hausel : Mirror symmetry with branes by equivariant Verlinde formulae I will discuss an agreement of equivariant
More informationElements of Topological M-Theory
Elements of Topological M-Theory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a Calabi-Yau threefold X is (loosely speaking) an integrable spine of
More informationMoment map flows and the Hecke correspondence for quivers
and the Hecke correspondence for quivers International Workshop on Geometry and Representation Theory Hong Kong University, 6 November 2013 The Grassmannian and flag varieties Correspondences in Morse
More informationSupersymmetric gauge theory, representation schemes and random matrices
Supersymmetric gauge theory, representation schemes and random matrices Giovanni Felder, ETH Zurich joint work with Y. Berest, M. Müller-Lennert, S. Patotsky, A. Ramadoss and T. Willwacher MIT, 30 May
More informationFactorization Algebras Associated to the (2, 0) Theory IV. Kevin Costello Notes by Qiaochu Yuan
Factorization Algebras Associated to the (2, 0) Theory IV Kevin Costello Notes by Qiaochu Yuan December 12, 2014 Last time we saw that 5d N = 2 SYM has a twist that looks like which has a further A-twist
More informationAFFINE KAC-MOODY ALGEBRAS, INTEGRABLE SYSTEMS AND THEIR DEFORMATIONS
AFFINE KAC-MOODY ALGEBRAS, INTEGRABLE SYSTEMS AND THEIR DEFORMATIONS EDWARD FRENKEL Representation theory of affine Kac-Moody algebras at the critical level contains many intricate structures, in particular,
More informationGauge Theory and Mirror Symmetry
Gauge Theory and Mirror Symmetry Constantin Teleman UC Berkeley ICM 2014, Seoul C. Teleman (Berkeley) Gauge theory, Mirror symmetry ICM Seoul, 2014 1 / 14 Character space for SO(3) and Toda foliation Support
More informationarxiv: v2 [hep-th] 31 Jan 2018
February 1, 2018 Prepared for submission to JHEP Vortices and Vermas arxiv:1609.04406v2 [hep-th] 31 Jan 2018 Mathew Bullimore 1 Tudor Dimofte 2,3 Davide Gaiotto 2 Justin Hilburn 2,4,5 Hee-Cheol Kim 2,6,7
More informationRESEARCH STATEMENT NEAL LIVESAY
MODULI SPACES OF MEROMORPHIC GSp 2n -CONNECTIONS RESEARCH STATEMENT NEAL LIVESAY 1. Introduction 1.1. Motivation. Representation theory is a branch of mathematics that involves studying abstract algebraic
More informationCasimir elements for classical Lie algebras. and affine Kac Moody algebras
Casimir elements for classical Lie algebras and affine Kac Moody algebras Alexander Molev University of Sydney Plan of lectures Plan of lectures Casimir elements for the classical Lie algebras from the
More informationREPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS.
REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS. 1.0.1. Berkeley team. 1. Research Proposal (1) Tim Cramer Postdoc. I am interested in derived algebraic geometry and its application to
More informationOutline of mini lecture course, Mainz February 2016 Jacopo Stoppa Warning: reference list is very much incomplete!
Outline of mini lecture course, Mainz February 2016 Jacopo Stoppa jacopo.stoppa@unipv.it Warning: reference list is very much incomplete! Lecture 1 In the first lecture I will introduce the tropical vertex
More informationLinear connections on Lie groups
Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationSpectral Networks and Their Applications. Caltech, March, 2012
Spectral Networks and Their Applications Caltech, March, 2012 Gregory Moore, Rutgers University Davide Gaiotto, o, G.M., Andy Neitzke e Spectral Networks and Snakes, pretty much finished Spectral Networks,
More informationA wall-crossing formula for 2d-4d DT invariants
A wall-crossing formula for 2d-4d DT invariants Andrew Neitzke, UT Austin (joint work with Davide Gaiotto, Greg Moore) Cetraro, July 2011 Preface In the last few years there has been a lot of progress
More informationVertex Algebras and Geometry
711 Vertex Algebras and Geometry AMS Special Session on Vertex Algebras and Geometry October 8 9, 2016 Denver, Colorado Mini-Conference on Vertex Algebras October 10 11, 2016 Denver, Colorado Thomas Creutzig
More information2d-4d wall-crossing and hyperholomorphic bundles
2d-4d wall-crossing and hyperholomorphic bundles Andrew Neitzke, UT Austin (work in progress with Davide Gaiotto, Greg Moore) DESY, December 2010 Preface Wall-crossing is an annoying/beautiful phenomenon
More informationTopological reduction of supersymmetric gauge theories and S-duality
Topological reduction of supersymmetric gauge theories and S-duality Anton Kapustin California Institute of Technology Topological reduction of supersymmetric gauge theories and S-duality p. 1/2 Outline
More informationRecent Advances in SUSY
Recent Advances in SUSY Nathan Seiberg Strings 2011 Thank: Gaiotto, Festuccia, Jafferis, Kapustin, Komargodski, Moore, Rocek, Shih, Tachikawa We cannot summarize thousands of papers in one talk We will
More informationTwistorial Topological Strings and a tt Geometry for N = 2 Theories in 4d
Twistorial Topological Strings and a tt Geometry for N = Theories in 4d arxiv:141.4793v1 [hep-th] 15 Dec 014 Sergio Cecotti, Andrew Neitzke, Cumrun Vafa International School of Advanced Studies (SISSA,
More informationDERIVED HAMILTONIAN REDUCTION
DERIVED HAMILTONIAN REDUCTION PAVEL SAFRONOV 1. Classical definitions 1.1. Motivation. In classical mechanics the main object of study is a symplectic manifold X together with a Hamiltonian function H
More informationGenerators of affine W-algebras
1 Generators of affine W-algebras Alexander Molev University of Sydney 2 The W-algebras first appeared as certain symmetry algebras in conformal field theory. 2 The W-algebras first appeared as certain
More informationDel Pezzo surfaces and non-commutative geometry
Del Pezzo surfaces and non-commutative geometry D. Kaledin (Steklov Math. Inst./Univ. of Tokyo) Joint work with V. Ginzburg (Univ. of Chicago). No definitive results yet, just some observations and questions.
More informationHilbert Series and Its Applications to SUSY Gauge Theories
Hilbert Series and Its Applications to SUSY Gauge Theories NOPPADOL MEKAREEYA Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Progress in Quantum Field Theory and String Theory April 2012 Noppadol
More informationQuadratic differentials as stability conditions. Tom Bridgeland (joint work with Ivan Smith)
Quadratic differentials as stability conditions Tom Bridgeland (joint work with Ivan Smith) Our main result identifies spaces of meromorphic quadratic differentials on Riemann surfaces with spaces of stability
More informationSEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009)
SEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009) DENNIS GAITSGORY 1. Hecke eigensheaves The general topic of this seminar can be broadly defined as Geometric
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationRepresentations Are Everywhere
Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another.
More informationChristmas Workshop on Quivers, Moduli Spaces and Integrable Systems. Genoa, December 19-21, Speakers and abstracts
Christmas Workshop on Quivers, Moduli Spaces and Integrable Systems Genoa, December 19-21, 2016 Speakers and abstracts Ada Boralevi, Moduli spaces of framed sheaves on (p,q)-toric singularities I will
More informationFrom Schur-Weyl duality to quantum symmetric pairs
.. From Schur-Weyl duality to quantum symmetric pairs Chun-Ju Lai Max Planck Institute for Mathematics in Bonn cjlai@mpim-bonn.mpg.de Dec 8, 2016 Outline...1 Schur-Weyl duality.2.3.4.5 Background.. GL
More informationRESEARCH PROPOSAL FOR P. DELIGNE CONTEST
RESEARCH PROPOSAL FOR P. DELIGNE CONTEST SERGEY OBLEZIN Over the recent decades representation theory has gained a central role in modern mathematics, linking such areas as number theory, topology, differential
More informationSéminaire BOURBAKI Juin ème année, , n o 1010
Séminaire BOURBAKI Juin 2009 61ème année, 2008-2009, n o 1010 GAUGE THEORY AND LANGLANDS DUALITY arxiv:0906.2747v1 [math.rt] 15 Jun 2009 INTRODUCTION by Edward FRENKEL In the late 1960s Robert Langlands
More informationEDWARD FRENKEL BIBLIOGRAPHY. Books
EDWARD FRENKEL BIBLIOGRAPHY Books 1. Vertex Algebras and Algebraic Curves (with D. Ben-Zvi), Mathematical Surveys and Monographs 88, AMS, First Edition, 2001; Second Edition, 2004 (400 pp.). 2. Langlands
More informationMoy-Prasad filtrations and flat G-bundles on curves
Moy-Prasad filtrations and flat G-bundles on curves Daniel Sage Geometric and categorical representation theory Mooloolaba, December, 2015 Overview New approach (joint with C. Bremer) to the local theory
More informationKnots and Mirror Symmetry. Mina Aganagic UC Berkeley
Knots and Mirror Symmetry Mina Aganagic UC Berkeley 1 Quantum physics has played a central role in answering the basic question in knot theory: When are two knots distinct? 2 Witten explained in 88, that
More informationarxiv:q-alg/ v3 23 Sep 1999
AFFINE ALGEBRAS, LANGLANDS DUALITY AND BETHE ANSATZ EDWARD FRENKEL arxiv:q-alg/9506003v3 23 Sep 1999 In memory of Claude Itzykson 1. Introduction. By Langlands duality one usually understands a correspondence
More informationt Hooft loop path integral in N = 2 gauge theories
t Hooft loop path integral in N = 2 gauge theories Jaume Gomis (based on work with Takuya Okuda and Vasily Pestun) Perimeter Institute December 17, 2010 Jaume Gomis (Perimeter Institute) t Hooft loop path
More informationM-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems
M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems Meng-Chwan Tan National University of Singapore January 1,
More informationSpaces of quasi-maps into the flag varieties and their applications
Spaces of quasi-maps into the flag varieties and their applications Alexander Braverman Abstract. Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps
More informationRepresentation Theory as Gauge Theory
Representation Theory as Gauge Theory David Ben-Zvi University of Texas at Austin Clay Research Conference Oxford, September 2016 Themes I. Harmonic analysis as the exploitation of symmetry 1 II. Commutative
More informationSymplectic varieties and Poisson deformations
Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the
More informationKnot Homology from Refined Chern-Simons Theory
Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones
More informationLanglands Duality and Topological Field Theory
Langlands Duality and Topological Field Theory Anton Kapustin California Institute of Technology, Pasadena, CA 91125, U.S.A. February 14, 2010 2 0.1 Introduction The Langlands Program is a far-reaching
More informationGravitating vortices, cosmic strings, and algebraic geometry
Gravitating vortices, cosmic strings, and algebraic geometry Luis Álvarez-Cónsul ICMAT & CSIC, Madrid Seminari de Geometria Algebraica UB, Barcelona, 3 Feb 2017 Joint with Mario García-Fernández and Oscar
More informationLanglands Duality and Topological Field Theory
Langlands Duality and Topological Field Theory Anton Kapustin California Institute of Technology, Pasadena, CA 91125, U.S.A. February 15, 2010 2 0.1 Introduction The Langlands Program is a far-reaching
More informationQuantum Integrability and Gauge Theory
The Versatility of Integrability Celebrating Igor Krichever's 60th Birthday Quantum Integrability and Gauge Theory Nikita Nekrasov IHES This is a work on experimental theoretical physics In collaboration
More information16.2. Definition. Let N be the set of all nilpotent elements in g. Define N
74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the
More informationTechniques for exact calculations in 4D SUSY gauge theories
Techniques for exact calculations in 4D SUSY gauge theories Takuya Okuda University of Tokyo, Komaba 6th Asian Winter School on Strings, Particles and Cosmology 1 First lecture Motivations for studying
More informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationOn the Virtual Fundamental Class
On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants
More informationRefined Chern-Simons Theory, Topological Strings and Knot Homology
Refined Chern-Simons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: 1105.5117 arxiv: 1202.4456 Chern-Simons theory played
More informationHecke modifications. Aron Heleodoro. May 28, 2013
Hecke modifications Aron Heleodoro May 28, 2013 1 Introduction The interest on Hecke modifications in the geometrical Langlands program comes as a natural categorification of the product in the spherical
More informationThe geometry of Landau-Ginzburg models
Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror
More informationContents. Preface...VII. Introduction... 1
Preface...VII Introduction... 1 I Preliminaries... 7 1 LieGroupsandLieAlgebras... 7 1.1 Lie Groups and an Infinite-Dimensional Setting....... 7 1.2 TheLieAlgebraofaLieGroup... 9 1.3 The Exponential Map..............................
More informationKhovanov Homology And Gauge Theory
hep-th/yymm.nnnn Khovanov Homology And Gauge Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract In these notes, I will sketch
More informationSheaves on Fibered Threefolds and Quiver Sheaves
Commun. Math. Phys. 278, 627 641 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0408-y Communications in Mathematical Physics Sheaves on Fibered Threefolds and Quiver Sheaves Balázs Szendrői
More informationFundamental Lemma and Hitchin Fibration
Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental
More informationGeneralized Tian-Todorov theorems
Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:
More informationGauge Theory, Ramification, And The Geometric Langlands Program
arxiv:hep-th/0612073v2 8 Oct 2007 Gauge Theory, Ramification, And The Geometric Langlands Program Sergei Gukov Department of Physics, University of California Santa Barbara, CA 93106 and Edward Witten
More informationCONFORMAL FIELD THEORIES
CONFORMAL FIELD THEORIES Definition 0.1 (Segal, see for example [Hen]). A full conformal field theory is a symmetric monoidal functor { } 1 dimensional compact oriented smooth manifolds {Hilbert spaces}.
More informationSpectral networks at marginal stability, BPS quivers, and a new construction of wall-crossing invariants
Spectral networks at marginal stability, BPS quivers, and a new construction of wall-crossing invariants Pietro Longhi Uppsala University String-Math 2017 In collaboration with: Maxime Gabella, Chan Y.
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More informationDeformations of logarithmic connections and apparent singularities
Deformations of logarithmic connections and apparent singularities Rényi Institute of Mathematics Budapest University of Technology Kyoto July 14th, 2009 Outline 1 Motivation Outline 1 Motivation 2 Infinitesimal
More informationRiemann-Hilbert problems from Donaldson-Thomas theory
Riemann-Hilbert problems from Donaldson-Thomas theory Tom Bridgeland University of Sheffield Preprints: 1611.03697 and 1703.02776. 1 / 25 Motivation 2 / 25 Motivation Two types of parameters in string
More information