W-algebras, moduli of sheaves on surfaces, and AGT
|
|
- Kenneth Henry
- 5 years ago
- Views:
Transcription
1 W-algebras, moduli of sheaves on surfaces, and AGT MIT
2 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory
3 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r.
4 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot,
5 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot, who defined an action of the W algebra on the cohomology group H of the moduli space M of sheaves/instantons on A 2.
6 The AGT correspondence Alday-Gaiotto-Tachikawa found a connection between: [ ] [ ] 4D N = 2 gauge theory for Ur) A r 1 Toda field theory Specifically, the Nekrasov partition function for the LHS should match conformal blocks for the W algebra of type gl r. In the pure gauge theory, the mathematical connection was established by Maulik-Okounkov and Schiffmann-Vasserot, who defined an action of the W algebra on the cohomology group H of the moduli space M of sheaves/instantons on A 2. Then the Nekrasov partition function is 1, 1, where 1 is the unit cohomology class, which one can uniquely determine based on how the W algebra acts on it the Gaiotto state).
7 The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 )
8 The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2
9 The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H
10 The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H We will prove this by q deforming everything, redefining the W algebra action, and proving that A m commutes with it.
11 The Ext operator Things are even more interesting in the presence of matter. Mathematically, matter in the bifundamental representation: C r 1 C r 2 Ur 1 ) Ur 2 ) is encoded in the Ext operator of Carlsson-Okounkov: ) ] A m : H H, A m = p 1 [c Ext F, F), m p2 Therefore, AGT for linear quiver gauge theories follows if one presents A m as an intertwiner of the W algebra action on H We will prove this by q deforming everything, redefining the W algebra action, and proving that A m commutes with it. Since our construction is purely geometric, it makes sense for sheaves on surfaces S more general than the affine plane A 2
12 Deformation Mathematically, deformation refers to two related processes:
13 Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory.
14 Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory. Replacing Lie algebras by quantum groups, e.g. g U q g) or: Heisenberg q Heisenberg = Z[q 1 ±1, q±1 2 ] a n n Z\0 [a m, a n ] = δ n m n1 q n 1 )1 qn 2 )1 qrn ) 1 q n
15 Deformation Mathematically, deformation refers to two related processes: Replacing the cohomology of M by its algebraic K theory: H K = K equiv M) = K equiv M n ) n=0 which entails replacing 4D N = 2 by 5D N = 1 gauge theory. Replacing Lie algebras by quantum groups, e.g. g U q g) or: Z[q 1 ±1 Heisenberg q Heisenberg =, q±1 2 ] a n n Z\0 [a m, a n ] = δm n n1 q n 1 )1 qn 2 )1 qrn ) 1 q n More generally, we will consider the deformed W algebra of Feigin-Frenkel and Awata-Kubo-Odake-Shiraishi. In type gl 1, this algebra is q Heisenberg, while for sl 2, it deforms Virasoro.
16 The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j )
17 The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j ) q W r is defined as the subalgebra generated by W d,k given by: [ k b s ] [ i ) ] n b s i ) n W k x) = : u si exp nx n exp nx n : 1 s 1... s k r i=1 n=1 n=1
18 The deformed W algebra The original definition of the q W algebra is via free fields: qw r H r = Z[q 1 ±1, q±1 2 ] bi) n 1 i r n Z\0 [b i) n, bj) m ] = δmn1 n q1 n)1 qn 2 )1 δ i jq nδ i<j ) q W r is defined as the subalgebra generated by W d,k given by: [ k b s ] [ i ) ] n b s i ) n W k x) = : u si exp nx n exp nx n : 1 s 1... s k r i=1 n=1 n=1 More intrinsically, q W r can be described as the algebra generated by symbols W d,k modulo relations such as: ) ) x y W k x)w 1 y)ζ yq k W 1 y)w k x)ζ = xq = q [ ) ) ] 1 1)q 2 1) x y δ q 1 yq k W k+1 x) δ W k+1 y) xq
19 The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i.
20 The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M
21 The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M as it isn t required to commute with q W r on the nose. Instead: ) [Φ m, W k x)] m k 1 mr u 1...u r q r k x u 1 = 0 1)...u r for all k 1, where [A, B] s = AB sba.
22 The Verma module and the vertex operator Let u 1,..., u r C. Define the Verma module of q W r as: M = q W r / right ideal W d,k, W 0,k e k u 1,..., u r )) d>0 1 k r and similarly define M with the parameters u i replaced by u i. For m C, the vertex operator will almost be an intertwiner: Φ m : M M as it isn t required to commute with q W r on the nose. Instead: ) [Φ m, W k x)] m k 1 mr u 1...u r q r k x u 1 = 0 1)...u r for all k 1, where [A, B] s = AB sba. Lemma: the endomorphism Φ m is uniquely determined, up to constant multiple, by property 1).
23 The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential.
24 The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part.
25 The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part. We define the action of W x, y) = k>0 d Z W d,k x d y) k W x, yd x ) = Lx, yd x ) Ey) Ux, yd x ) where D x is the difference operator f x) f xq) on K as:
26 The main Theorem Theorem N) There exists an action q W r K, for which the latter is isomorphic to the Verma module M. Moreover, the Ext operator A m : K K is equal to the vertex operator Φ m : M M, up to a simple exponential. The second part is an intersection-theoretic computation, once we give a geometric definition of the action in first part. We define the action of W x, y) = k>0 d Z W d,k x d y) k W x, yd x ) = Lx, yd x ) Ey) Ux, yd x ) where D x is the difference operator f x) f xq) on K as: and L, E, U are geometric endomorphisms of K that are lower triangular, diagonal, and upper triangular, respectively.
27 The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r
28 The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ.
29 The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ. Let K denote the equivariant K theory of M, whose elements are formal differences of T equivariant vector bundles on M.
30 The moduli space of sheaves Explicitly, our M is the moduli space of framed sheaves: ) F torsion-free sheaf on P 2 φ, F = O r There is an action of T = C C C ) r on M, where the first two factors act on P 2 and the third factor acts on φ. Let K denote the equivariant K theory of M, whose elements are formal differences of T equivariant vector bundles on M. Since vector bundles can be tensored with T representations, K is a module over the ring Z[q 1 ±1, q±1 2, u±1 1,..., u±1 r ].
31 K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k).
32 K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ.
33 K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ. A convenient basis of a localization of) K is given by: ) λ = skyscraper skeaf of F λ K
34 K theory and partitions An r partition will be a collection of r usual partitions: λ = λ 1),..., λ r) ) where λ k) = λ k) 1 λ k) 2...) and we will write λ for the sum of the sizes of the λ k). T fixed points of M are indexed by r partitions, specifically: F λ = I λ 1)... I λ r) where I λ C[x, y] is the monomial ideal of shape λ. A convenient basis of a localization of) K is given by: ) λ = skyscraper skeaf of F λ K Given a box at coordinates i, j) in the Young diagram of the constituent partition λ k) of an r partition λ, we call: z = u k q i 1q j 2 the weight of.
35 The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z y )
36 The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y )
37 The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y ) This space comes endowed with maps π d), πd) + : Z d M that remember only F 0 and F d, respectively,
38 The correspondences The operator Ey) is multiplication by the exterior algebra of the universal sheaf, and therefore its matrix coefficients are: λ Ey) µ = δ µ λ r i=1 1 u ) i ζ y λ z For any d 1, let us consider the following locus: Z d = {F 0... F d sheaves, x A 2, s.t. F k 1 /F k = Ox k} y ) This space comes endowed with maps π d), πd) + : Z d M that remember only F 0 and F d, respectively, and with line bundles L 1,..., L d on Z d that keep track of the length one quotients F 0 /F 1,..., F d 1 /F d.
39 The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0
40 The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0 Therefore, the matrix coefficients of Lx, y) are given by: λ Lx, y) µ = 1 i<j λ\µ ζ T a standard Young tableau of shape λ\µ zj z i ) λ\µ i=1 µ ζ zi z x λ\µ 1 y z 1 ) λ\µ 1 i=1 ) r j=1 1 z ) iq u j ) 1 qz i z i+1
41 The operators Lx, y) and Ux, y) are adjoint, so let s focus on the first one: ) Lx, y) := π d) x d + 1 y π d) : K K[[x, y 1 ]] L 1 d=0 Therefore, the matrix coefficients of Lx, y) are given by: λ Lx, y) µ = 1 i<j λ\µ ζ T a standard Young tableau of shape λ\µ zj z i ) λ\µ i=1 µ ζ zi z x λ\µ 1 y z 1 ) λ\µ 1 i=1 ) r j=1 1 z ) iq u j ) 1 qz i z i+1 where we recall that a standard Young tableau is a labeling of the boxes of λ\µ with the numbers 1, 2,... such that the numbers increase as we go up and to the right. Also z i = z i.
42 Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S.
43 Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S.
44 Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S
45 Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S To be completely precise, the generators W d,k q W r will give rise not to endomorphisms of K M, but to maps K M K M S
46 Replacing A 2 with a general surface S Up to some conjectures, everything can be made sense of when our sheaves live over a general smooth surface S. The moduli space M will parametrize stable sheaves on S. The q W algebra that acts on K M still has generators W d,k, but the structure constants Z[q ±1 ] now depend on: 1, q±1 2 {q 1, q 2 } = Chern roots of the cotangent bundle of S To be completely precise, the generators W d,k q W r will give rise not to endomorphisms of K M, but to maps K M K M S Finally, the parameter m that defines the Ext bundle and the operator A m, will now be a K theory class on S.
Supersymmetric 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 informationExtended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function
CFT and integrability in memorial of Alexei Zamolodchikov Sogan University, Seoul December 2013 Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function Yutaka Matsuo (U. Tokyo)
More informationStable bases for moduli of sheaves
Columbia University 02 / 03 / 2015 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons Moduli of sheaves on
More informationHeriot-Watt University. AGT relations for abelian quiver gauge theories on ALE spaces Pedrini, Mattia; Sala, Francesco; Szabo, Richard Joseph
Heriot-Watt University Heriot-Watt University Research Gateway AGT relations for abelian quiver gauge theories on ALE spaces Pedrini, Mattia; Sala, Francesco; Szabo, Richard Joseph Published in: Journal
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 informationModuli spaces of sheaves and the boson-fermion correspondence
Moduli spaces of sheaves and the boson-fermion correspondence Alistair Savage (alistair.savage@uottawa.ca) Department of Mathematics and Statistics University of Ottawa Joint work with Anthony Licata (Stanford/MPI)
More informationTopological Matter, Strings, K-theory and related areas September 2016
Topological Matter, Strings, K-theory and related areas 26 30 September 2016 This talk is based on joint work with from Caltech. Outline 1. A string theorist s view of 2. Mixed Hodge polynomials associated
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 informationWe can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle
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 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 informationGeometric Realizations of the Basic Representation of ĝl r
Geometric Realizations of the Basic Representation of ĝl r Joel Lemay Department of Mathematics and Statistics University of Ottawa September 23rd, 2013 Joel Lemay Geometric Realizations of ĝl r Representations
More informationDifference Painlevé equations from 5D gauge theories
Difference Painlevé equations from 5D gauge theories M. Bershtein based on joint paper with P. Gavrylenko and A. Marshakov arxiv:.006, to appear in JHEP February 08 M. Bershtein Difference Painlevé based
More informationVertex algebras, chiral algebras, and factorisation algebras
Vertex algebras, chiral algebras, and factorisation algebras Emily Cliff University of Illinois at Urbana Champaign 18 September, 2017 Section 1 Vertex algebras, motivation, and road-plan Definition A
More informationPERVERSE SHEAVES ON INSTANTON MODULI SPACES PRELIMINARY VERSION (July 13, 2015)
PERVERSE SHEAVES ON INSTANTON MODULI SPACES PRELIMINARY VERSION (July 13, 2015) HIRAKU NAKAJIMA Contents Introduction 1 1. Uhlenbeck partial compactification in brief 7 2. Heisenberg algebra action 8 3.
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 informationCohomological Hall algebra of a preprojective algebra
Cohomological Hall algebra of a preprojective algebra Gufang Zhao Institut de Mathématiques de Jussieu-Paris Rive Gauche Conference on Representation Theory and Commutative Algebra Apr. 24, 2015, Storrs,
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 informationHilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants
Contemporary Mathematics Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants Wei-Ping Li 1, Zhenbo Qin 2, and Weiqiang Wang 3 Abstract. Some connections of the ordinary intersection
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationI. Why Quantum K-theory?
Quantum groups and Quantum K-theory Andrei Okounkov in collaboration with M. Aganagic, D. Maulik, N. Nekrasov, A. Smirnov,... I. Why Quantum K-theory? mathematical physics mathematics algebraic geometry
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 informationModuli spaces of sheaves on surfaces - Hecke correspondences and representation theory
Moduli spaces of sheaves on surfaces - Hecke correspondences and representation theory Andrei Neguț In modern terms, enumerative geometry is the study of moduli spaces: instead of counting various geometric
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 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 informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 1.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 1. 1. Algebras. Derivations. Definition of Lie algebra 1.1. Algebras. Let k be a field. An algebra over k (or k-algebra) is a vector space A endowed with a bilinear
More informationCoherent sheaves on elliptic curves.
Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of
More informationD-MODULES: AN INTRODUCTION
D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing
More informationThe Hitchin map, local to global
The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present
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 informationCohomology theories on projective homogeneous varieties
Cohomology theories on projective homogeneous varieties Baptiste Calmès RAGE conference, Emory, May 2011 Goal: Schubert Calculus for all cohomology theories Schubert Calculus? Cohomology theory? (Very)
More informationarxiv: v3 [math.rt] 7 Nov 2016
IAS/Park City Mathematics Series Volume 00, Pages 000 000 S 1079-5634(XX)0000-0 Lectures on perverse sheaves on instanton moduli spaces arxiv:1604.06316v3 [math.rt] 7 Nov 2016 Hiraku Nakajima Contents
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 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 informationArithmetic and physics of Higgs moduli spaces
Arithmetic and physics of Higgs moduli spaces Tamás Hausel IST Austria http://hausel.ist.ac.at Geometry working seminar March, 2017 Diffeomorphic spaces in non-abelian Hodge theory C genus g curve; fix
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 informationLanglands parameters of quivers in the Sato Grassmannian
Langlands parameters of quivers in the Sato Grassmannian Martin T. Luu, Matej enciak Abstract Motivated by quantum field theoretic partition functions that can be expressed as products of tau functions
More informationQuantum toroidal symmetry and SL(2,Z) covariant description of AGT and qq-character
MATRIX Creswick 2017 Quantum toroidal symmetry and SL(2,Z) covariant description of AGT and qq-character Yutaka Matsuo The University of Tokyo July 5th, 2017 based on works arxiv:1705.02941 (FHMZh), 1703.10759
More informationBraid group actions on categories of coherent sheaves
Braid group actions on categories of coherent sheaves MIT-Northeastern Rep Theory Seminar In this talk we will construct, following the recent paper [BR] by Bezrukavnikov and Riche, actions of certain
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 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 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 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 informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationThe BRST complex for a group action
BRST 2006 (jmf) 23 Lecture 3: The BRST complex for a group action In this lecture we will start our discussion of the homological approach to coisotropic reduction by studying the case where the coisotropic
More information3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).
3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)
More informationPoisson and Hochschild cohomology and the semiclassical limit
Poisson and Hochschild cohomology and the semiclassical limit Matthew Towers University of Kent http://arxiv.org/abs/1304.6003 Matthew Towers (University of Kent) arxiv 1304.6003 1 / 15 Motivation A quantum
More informationA p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationHigher representation theory in algebra and geometry: Lecture II
Higher representation theory in algebra and geometry: Lecture II Ben Webster UVA ebruary 10, 2014 Ben Webster (UVA) HRT : Lecture II ebruary 10, 2014 1 / 35 References or this lecture, useful references
More informationCategorification of quantum groups and quantum knot invariants
Categorification of quantum groups and quantum knot invariants Ben Webster MIT/Oregon March 17, 2010 Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, 2010 1 / 29 The big picture
More informationHILBERT SCHEMES: GEOMETRY, COMBINATORICS, AND REPRESENTATION THEORY. 1. Introduction
HILBERT SCHEMES: GEOMETRY, COMBINATORICS, AND REPRESENTATION THEORY DORI BEJLERI Abstract. This survey will give an introduction to the theory of Hilbert schemes of points on a surface. These are compactifications
More informationSheaf cohomology and non-normal varieties
Sheaf cohomology and non-normal varieties Steven Sam Massachusetts Institute of Technology December 11, 2011 1/14 Kempf collapsing We re interested in the following situation (over a field K): V is a vector
More informationFourier Mukai transforms II Orlov s criterion
Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and
More informationTHE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationAlgebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.
Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a
More informationHitchin fibration and endoscopy
Hitchin fibration and endoscopy Talk given in Kyoto the 2nd of September 2004 In [Hitchin-Duke], N. Hitchin proved that the cotangent of the moduli space of G-bundle over a compact Riemann surface is naturally
More informationLECTURE 11: SOERGEL BIMODULES
LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing
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 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 informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 7.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 7. 7. Killing form. Nilpotent Lie algebras 7.1. Killing form. 7.1.1. Let L be a Lie algebra over a field k and let ρ : L gl(v ) be a finite dimensional L-module. Define
More informationSymplectic algebraic geometry and quiver varieties
Symplectic algebraic geometry and quiver varieties Victor Ginzburg Lecture notes by Tony Feng Contents 1 Deformation quantization 2 1.1 Deformations................................. 2 1.2 Quantization..................................
More informationOn algebraic index theorems. Ryszard Nest. Introduction. The index theorem. Deformation quantization and Gelfand Fuks. Lie algebra theorem
s The s s The The term s is usually used to describe the equality of, on one hand, analytic invariants of certain operators on smooth manifolds and, on the other hand, topological/geometric invariants
More informationNoncommutative compact manifolds constructed from quivers
Noncommutative compact manifolds constructed from quivers Lieven Le Bruyn Universitaire Instelling Antwerpen B-2610 Antwerp (Belgium) lebruyn@wins.uia.ac.be Abstract The moduli spaces of θ-semistable representations
More informationPreliminary Exam Topics Sarah Mayes
Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition
More informationNOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES
NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES SHILIN U ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of
More informationNef line bundles on M 0,n from GIT
Nef line bundles on M 0,n from GIT David Swinarski Department of Mathematics University of Georgia November 13, 2009 Many of the results here are joint work with Valery Alexeev. We have a preprint: arxiv:0812.0778
More informationQuantum Cohomology and Symplectic Resolutions
Quantum Cohomology and Symplectic Resolutions Davesh Maulik Lecture notes by Tony Feng Contents 1 Introduction 2 1.1 The objects of interest............................. 2 1.2 Motivation...................................
More informationInstanton counting, Donaldson invariants, line bundles on moduli spaces of sheaves on rational surfaces. Lothar Göttsche ICTP, Trieste
Instanton counting, Donaldson invariants, line bundles on moduli spaces of sheaves on rational surfaces Lothar Göttsche ICTP, Trieste joint work with Hiraku Nakajima and Kota Yoshioka AMS Summer Institute
More informationNOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE
NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the
More informationFree fermion and wall-crossing
Free fermion and wall-crossing Jie Yang School of Mathematical Sciences, Capital Normal University yangjie@cnu.edu.cn March 4, 202 Motivation For string theorists It is called BPS states counting which
More informationVector Bundles on Algebraic Varieties
Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general
More informationColoured Kac-Moody algebras, Part I
Coloured Kac-Moody algebras, Part I Alexandre Bouayad Abstract We introduce a parametrization of formal deformations of Verma modules of sl 2. A point in the moduli space is called a colouring. We prove
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
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 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 informationAFFINE LAUMON SPACES AND A CONJECTURE OF KUZNETSOV arxiv: v1 [math.ag] 2 Nov 2018
AFFINE LAUMON SPACES AND A CONJECTURE OF KUZNETSOV arxiv:1811.01011v1 [math.ag] 2 Nov 2018 ANDREI NEGUȚ Abstract. We prove a conjecture of Kuznetsov stating that the equivariant K theory of affine Laumon
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 2.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined
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 informationDEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE
DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint
More informationTHE NONCOMMUTATIVE TORUS
THE NONCOMMUTATIVE TORUS The noncommutative torus as a twisted convolution An ordinary two-torus T 2 with coordinate functions given by where x 1, x 2 [0, 1]. U 1 = e 2πix 1, U 2 = e 2πix 2, (1) By Fourier
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 informationDefinition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).
9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson
More informationMath 249B. Nilpotence of connected solvable groups
Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C
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 informationarxiv: v1 [hep-th] 22 Dec 2015
W-symmetry, topological vertex and affine Yangian Tomáš Procházka 1 arxiv:1512.07178v1 [hep-th] 22 Dec 2015 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians University Theresienstr.
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 informationConformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G.
Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means
More informationCATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY ABSTRACTS
CATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY Alexei Bondal (Steklov/RIMS) Derived categories of complex-analytic manifolds Alexender Kuznetsov (Steklov) Categorical resolutions of singularities
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 informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
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 informationDarboux theorems for shifted symplectic derived schemes and stacks
Darboux theorems for shifted symplectic derived schemes and stacks Lecture 1 of 3 Dominic Joyce, Oxford University January 2014 Based on: arxiv:1305.6302 and arxiv:1312.0090. Joint work with Oren Ben-Bassat,
More informationSEMI-GROUP AND BASIC FUNCTIONS
SEMI-GROUP AND BASIC FUNCTIONS 1. Review on reductive semi-groups The reference for this material is the paper Very flat reductive monoids of Rittatore and On reductive algebraic semi-groups of Vinberg.
More informationTHE HITCHIN FIBRATION
THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus
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 informationW -Constraints for Simple Singularities
W -Constraints for Simple Singularities Bojko Bakalov Todor Milanov North Carolina State University Supported in part by the National Science Foundation Quantized Algebra and Physics Chern Institute of
More informationADVANCES IN R-MATRICES AND THEIR APPLICATIONS [AFTER MAULIK-OKOUNKOV, KANG-KASHIWARA-KIM-OH...]
ADVANCES IN R-MATRICES AND THEIR APPLICATIONS [AFTER MAULIK-OKOUNKOV, KANG-KASHIWARA-KIM-OH...] DAVID HERNANDEZ This is an English translation of the Bourbaki Seminar no. 1129 (March 2017). INTRODUCTION
More information