V = 1 2 (g ijχ i h j ) (2.4)
|
|
- Kerry Morris
- 5 years ago
- Views:
Transcription
1 4 VASILY PESTUN 2. Lecture: Localization 2.. Euler class of vector bundle, Mathai-Quillen form and Poincare-Hopf theorem. We will present the Euler class of a vector bundle can be presented in the form of an integral over fermionic variables. This presentation connects to Mathai-Quillen formalism for localization in topological or supersymmetric field theories. Let E be an oriented real vector bundle E of rank 2n over a manifold. Let x µ be local coordinates on the base, and let their differentials be denoted ψ µ = dx µ. Let h i be local coordinates on the fibers of E. Let ΠE denote the superspace obtained from the total space of the bundle E by inverting the parity of the fibers, so that the coordinates in the fibers of ΠE are odd variables χ i. Let g ij be the matrix of a Riemannian metric on the bundle E. Let A i µ be the matrix valued -form on representing a connection on the bundle E. Using the connection A we can define an odd vector field δ on the superspace ΠT (ΠE), or, equivalently, a de Rham differential on the space of differential forms Ω (ΠE). In local coordinates (x µ, ψ µ ) and (χ i, h i ) the definition of δ is δx µ = ψ µ δψ µ = 0 δχ i = h i A i jµψ µ χ j δh i = δ(a i jµψ µ χ j ) (2.) Here h i = Dχ i is the covariant de Rham differential of χ i, so that under the change of framing on E given by χ i = s i j χ j the h i transforms in the same way, that is h i = s i j h j. The odd vector field δ is nilpotent δ 2 = 0 (2.2) and is called de Rham vector field on ΠT (ΠE). Consider an element Φ Ω (ΠE), i.e. Φ is a function on ΠT (ΠE), defined by the equation where t R >0 and Φ = exp( tδv ) (2.3) (2π) 2n V = 2 (g ijχ i h j ) (2.4) Notice that since h i has been defined as Dχ i the definition (2.3) is coordinate independent. To expand the definition of Φ (2.3) we compute δ(χ, h) = (h Aχ, h) (χ, daχ A(h Aχ)) = (h, h) (χ, F A χ) (2.5) where we suppresed the indices i, j, the d denotes the de Rham differential on and F A the curvature 2-form on the connection A F A = da + A A (2.6) The Gaussian integration of the form Φ along the vertical fibers of ΠE gives [dh][dχ] exp( δ(χ, h)) = (2π) 2n 2 (2π) Pf(F A) (2.7) n which agrees with definition of the integer valued Euler class (.8). The representation of the Euler class in the form (2.3) is called the Gaussian Mathai-Quillen representation of the Thom class.
2 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY The Euler class of the vector bundle E is an element of H 2n (, Z). If dim = 2n, the number obtained after integration of the fundamental cycle on e(e) = Φ (2.8) ΠT (ΠE) is an integer Euler characterstic of the vector bundle E. If E = T the equation (2.8) provides the Euler characteristic of the manifold in the form e() = exp( tδv ) t 0 = (2π) dim ΠT (ΠT ) (2.9) (2π) dim ΠT (ΠT ) Given a section s of the vector bundle E, we can deform the form Φ in the same δ- cohomology class by taking V s = 2 (χ, h + s) (2.0) After integrating over (h, χ) the the resulting differential form on has factor exp( 2t s2 ) (2.) so it is concentraited in a neigborhood of the locus s (0) of zeroes of the section s. exercise: write down the computation more precisely In this way the Poincare-Hopf theorem is proven: given an oriented vector bundle E on an oriented manifold, with rank E = dim, the Euler characteristic of E is equal to the number of zeroes of a generic section s of E counted with orientation e(e) = (2π) n Pf(F A ) = (2π) dim ΠT (ΠT ) exp( tδv s ) = x s (0) sign det ds x (2.2) where ds x : T x E x is the differential of the section s at a zero x s (0). The assumption that s is a generic section implies that det ds x is non-zero. More generally, let r = rank E and d = dim, with r d, take a generic section s of E and consider its set of zeroes F s (0). Then F is a subvariety of of dimension d r. Let α Ω d r () be a closed form on, equivalently α is a function on ΠT. Then the integral α, Φ E,s := α exp( tδv (2π) r s ) (2.3) ΠT (ΠE) does on deformations of section s to λs with a parameter λ R. Then by scaling the section s to λs and sending λ 0 we find α, Φ E,s = α exp( tδv (2π) r λs ) λ 0 = α e(e) (2.4) while sending λ we find α, Φ E,s = (2π) r ΠT (ΠT ) ΠT (ΠT ) α exp( tδv λs ) λ = F α (2.5)
3 6 VASILY PESTUN The equality of two expressions for α, Φ E,s can be interpreted as a localization formula α e(e) = F α (2.6) In this way we proved that cohomology class [e(e)] H r () is Poincare dual to the homology class [F ] H d r () where F is the zero set of generic section of bundle E Equivariant Atiyah-Bott-Berline-Vergne localization formula. Suppose that a compact abelian Lie group T acts equivariantly on the oriented vector bundle E, and that α Ω G () is a closed equivariant differential form on in Cartan model, that is d T α = 0. Then equivariant version of (2.6) holds α e T (E) = F α (2.7) exercise: prove (2.7) in Cartan model for equivariant cohomology replacing Euler class by equivariant Euler class Now let be an oriented real even-dimensional Riemannian manifold, E = T be the tangent bundle, and T be a compact group acting on, and suppose that the set F = T of T -fixed points has dimension 0, i.e. F is a union of discrete points. A section s of tangent bundle E is a vector field. Assume that there is a circle subgroup S T that generates a vector field s on whose set of zeroes coincide with T, i.e. F = S = T. Let α be d T -closed T -equivariant differential form on in Cartan model. Then equivariant Euler class localization formula (2.7) α e T (T ) = x T α x (2.8) Equivariant cohomologies H T () form a ring. Formally, we can consider the field of fractions of this ring, and multiply α on the left and right side of the above equality by a cohomology class which is inverse to e T (T ), then we arrive to the equation α = α x (2.9) e T (T x ) x T where e T (T x ) := e T (T ) x is equivariant Euler class of the tangent bundle to evaluated at the point x. Since x is a discrete fixed point of T -action on, the fiber T x of the tangent bundle at point x forms a T -module. Since T is compact real abelian Lie group, a real T -module splits into a direct sum of dim 2 R T x irreducible real two-dimensional modules (L i R 2 C ) i=...n on which the weights of the T action are all non-zero. Then by (.63), (.8) and we find that the equivariant Euler class is e T (T x ) = (2π) 2 dim 2 dim i= w i (2.20)
4 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 207 where w i t are weights. In basis (ɛ α ) α=... dim T of linear coordinate functions on t we can write explicitly e T (T x ) = (2π) 2 dim 2 dim i= w iα ɛ α (2.2) 2.3. Duistermaat-Heckman localization. A particular example where the Atiyah-Bott- Berline-Vergne localization formula can be applied is a symplectic space on which a Lie group T acts in a Hamiltonian way. Namely, let (, ω) be a real symplectic manifold of dim R = 2n with symplectic form ω and let compact connected Lie group T act on in Hamiltonian way, which means that there exists a function, called moment map or Hamiltonian µ : t (2.22) such that dµ a = i a ω (2.23) in some basis (T a ) of t where i a is the contraction operation with the vector field generated by the T a action on. The degree 2 element ω T Ω () St defined by the equation is a d T -closed equivariant differential form: ω T = ω + ɛ a µ a (2.24) d T ω T = (d + ɛ a i a )(ω + ɛ b µ b ) = ɛ a dµ a + ɛ a i a ω = 0 (2.25) This implies that the mixed-degree equivariant differential form α = e ω T (2.26) is also d T -closed, and we can apply the Atiyah-Bott-Berline-Vergne localization formula to the integral exp(ω T ) = ω n exp(ɛ a µ a ) (2.27) n! For T = SO(2) so that Lie(SO(2)) R the integral (2.27) is the typical partition function of a classical Hamiltonian mechanical system in statistical physics with Hamiltonian function µ : R and inverse temperature parameter ɛ. Suppose that T = SO(2) and that the set of fixed points T is discrete. Then the Atiyah-Bott-Berline-Vergne localization formula (2.9) implies n! ω n exp(ɛ a µ a ) = exp(ɛa µ a ) e T (ν x ) x T (2.28) where ν x is the normal bundle to a fixed point x T in and e T (ν x ) is the T -equivariant Euler class of the bundle ν x. The rank of the normal bundle ν x is 2n and the structure group is SO(2n). In notations of section.9 we evaluate the T -equivariant characteristic Euler class of the principal G- bundle for T = SO(2) and G = SO(2n) by equation (.62) for the invariant polynomial on g = so(2n) given by p = (2π) n Pf according to definition (.8).
5 8 VASILY PESTUN 2.4. Gaussian integral example. To illustrate the localization formula (2.28) suppose that = R 2n with symplectic form n ω = dx i dy i (2.29) and SO(2) action ( xi i= ) ( ) ( ) cos wi θ sin w i θ xi y i sin w i θ cos w i θ y i (2.30) where θ R/(2πZ) parametrizes SO(2) and (w,..., w n ) Z n. The point 0 is the fixed point so that T = {0}, and the normal bundle ν x = T 0 is an SO(2)-module of real dimension 2n and complex dimension n that splits into a direct sum of n irreducible SO(2) modules with weights (w,..., w n ). We identify Lie(SO(2)) with R with basis element {} and coordinate function ɛ Lie(SO(2)). The SO(2) action (2.30) is Hamiltonian with respect to the moment map µ = µ 0 + n w i (x 2 i + yi 2 ) (2.3) 2 i= Assuming that ɛ < 0 and all w i > 0 we find by direct Gaussian integration ω n (2π) n exp(ɛµ) = n! ( ɛ) n n i= w exp(ɛµ 0 ) (2.32) i and the same result by the localization formula (2.28) because e T (ν x ) = Pf(ɛρ()) (2.33) (2π) n according to the definition of the T -equivariant class (.62) and the Euler characteristic class (.8), and where ρ : Lie(SO(2)) Lie(SO(2n)) is the homomorphism in (.6) with 0 w w ρ() = (2.34) w n w n 0 according to (2.30) Example of a two-sphere. Let (, ω) be the two-sphere S 2 with coordinates (θ, α) and symplectic structure ω = sin θdθ dα (2.35) Let the Hamiltonian function be so that H = cos θ (2.36) ω = dh dα (2.37) and the Hamiltonian vector field be v H = α. The differential form ω T = ω + ɛh = sin θdθ dα ɛ cos θ
6 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY is d T -closed for d T = d + ɛi α (2.38) Let α = e tω T (2.39) Locally there is a degree form V such that ω T = d T V, for example V = (cos θ)dα (2.40) but globally V does not exist. The d T -cohomology class [α] of the form α is non-zero. The localization formula (2.27) gives exp(ω T ) = 2π ɛ exp( ɛ) + 2π exp(ɛ) (2.4) ɛ where the first term is the contribution of the T -fixed point θ = 0 and the second term is the contribution of the T -fixed point θ = π.
k=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 informationLECTURES ON EQUIVARIANT LOCALIZATION
LECTURES ON EQUIVARIANT LOCALIZATION VASILY PESTUN Abstract. These are informal notes of the lectures on equivariant localization given at the program Geometry of Strings and Fields at The Galileo Galilei
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationTHE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES
THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES Denis Bell 1 Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu This
More informationSupersymmetry and Equivariant de Rham Theory
Supersymmetry and Equivariant de Rham Theory Bearbeitet von Victor W Guillemin, Shlomo Sternberg, Jochen Brüning 1. Auflage 1999. Buch. xxiii, 232 S. Hardcover ISBN 978 3 540 64797 3 Format (B x L): 15,5
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationarxiv: v3 [hep-th] 15 Oct 2016
Review of localization in geometry Vasily Pestun Institut des Hautes Études Scientifique, France pestun@ihes.fr ariv:608.02954v3 [hep-th] 5 Oct 206 Abstract Review of localization in geometry: equivariant
More informationLecture Notes on Equivariant Cohomology
Lecture Notes on Equivariant Cohomology atvei Libine April 26, 2007 1 Introduction These are the lecture notes for the introductory graduate course I taught at Yale during Spring 2007. I mostly followed
More informationLECTURE 26: THE CHERN-WEIL THEORY
LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if
More informationLECTURE 4: SYMPLECTIC GROUP ACTIONS
LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is
More informationRes + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X
Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the H-equivariant Euler class of the normal bundle ν(f), c is a non-zero constant 2, and is defined below in (1.3).
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationThe Dirac-Ramond operator and vertex algebras
The Dirac-Ramond operator and vertex algebras Westfälische Wilhelms-Universität Münster cvoigt@math.uni-muenster.de http://wwwmath.uni-muenster.de/reine/u/cvoigt/ Vanderbilt May 11, 2011 Kasparov theory
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationLecture on Equivariant Cohomology
Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove
More informationQualifying Examination HARVARD UNIVERSITY Department of Mathematics Tuesday, January 19, 2016 (Day 1)
Qualifying Examination HARVARD UNIVERSITY Department of Mathematics Tuesday, January 19, 2016 (Day 1) PROBLEM 1 (DG) Let S denote the surface in R 3 where the coordinates (x, y, z) obey x 2 + y 2 = 1 +
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationDonaldson Invariants and Moduli of Yang-Mills Instantons
Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationE 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?
. Characteristic Classes from the viewpoint of Operator Theory. Introduction Overarching Question: How can you tell if two vector bundles over a manifold are isomorphic? Let X be a compact Hausdorff space.
More informationIGA Lecture I: Introduction to G-valued moment maps
IGA Lecture I: Introduction to G-valued moment maps Adelaide, September 5, 2011 Review: Hamiltonian G-spaces Let G a Lie group, g = Lie(G), g with co-adjoint G-action denoted Ad. Definition A Hamiltonian
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationChern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action,
Lecture A3 Chern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action, S CS = k tr (AdA+ 3 ) 4π A3, = k ( ǫ µνρ tr A µ ( ν A ρ ρ A ν )+ ) 8π 3 A µ[a ν,a ρ
More informationGeometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry
Geometry and Dynamics of singular symplectic manifolds Session 9: Some applications of the path method in b-symplectic geometry Eva Miranda (UPC-CEREMADE-IMCCE-IMJ) Fondation Sciences Mathématiques de
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationA Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds
arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn
More informationLECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory
More informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten
More informationExercises on characteristic classes
Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives
More informationTRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES
TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES KEN RICHARDSON Abstract. In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationQuaternionic Complexes
Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534
More informationPatrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationGeneralized Topological Index
K-Theory 12: 361 369, 1997. 361 1997 Kluwer Academic Publishers. Printed in the Netherlands. Generalized Topological Index PHILIP A. FOTH and DMITRY E. TAMARKIN Department of Mathematics, Penn State University,
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 20, 2015 (Day 1)
Tuesday January 20, 2015 (Day 1) 1. (AG) Let C P 2 be a smooth plane curve of degree d. (a) Let K C be the canonical bundle of C. For what integer n is it the case that K C = OC (n)? (b) Prove that if
More information12 Geometric quantization
12 Geometric quantization 12.1 Remarks on quantization and representation theory Definition 12.1 Let M be a symplectic manifold. A prequantum line bundle with connection on M is a line bundle L M equipped
More informationThe Hopf Bracket. Claude LeBrun SUNY Stony Brook and Michael Taylor UNC Chapel Hill. August 11, 2013
The Hopf Bracket Claude LeBrun SUY Stony Brook and ichael Taylor UC Chapel Hill August 11, 2013 Abstract Given a smooth map f : between smooth manifolds, we construct a hierarchy of bilinear forms on suitable
More informationarxiv:alg-geom/ v1 29 Jul 1993
Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic
More informationNOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY
NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say
More informationEquivariant Toeplitz index
CIRM, Septembre 2013 UPMC, F75005, Paris, France - boutet@math.jussieu.fr Introduction. Asymptotic equivariant index In this lecture I wish to describe how the asymptotic equivariant index and how behaves
More informationt, H = 0, E = H E = 4πρ, H df = 0, δf = 4πJ.
Lecture 3 Cohomologies, curvatures Maxwell equations The Maxwell equations for electromagnetic fields are expressed as E = H t, H = 0, E = 4πρ, H E t = 4π j. These equations can be simplified if we use
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 informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say
More informationMorse Theory and Applications to Equivariant Topology
Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and
More informationWHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014
WHAT IS K-HOMOLOGY? Paul Baum Penn State Texas A&M University College Station, Texas, USA April 2, 2014 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, 2014 1 / 56 Let X be a compact C manifold without
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 informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationAtiyah-Singer Revisited
Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator
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 informationSpin(10,1)-metrics with a parallel null spinor and maximal holonomy
Spin(10,1)-metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in
More informationEquivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions
Equivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions Jia-Ming (Frank) Liou, Albert Schwarz February 28, 2012 1. H = L 2 (S 1 ): the space of square integrable complex-valued
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationReminder on basic differential geometry
Reminder on basic differential geometry for the mastermath course of 2013 Charts Manifolds will be denoted by M, N etc. One should think of a manifold as made out of points (while the elements of a vector
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 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 informationInstantons and Donaldson invariants
Instantons and Donaldson invariants George Korpas Trinity College Dublin IFT, November 20, 2015 A problem in mathematics A problem in mathematics Important probem: classify d-manifolds up to diffeomorphisms.
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 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 informationSOME EXERCISES IN CHARACTERISTIC CLASSES
SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined
More informationMorse theory and stable pairs
Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction
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 informationMcGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA
McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper
More informationContents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39.
Preface xiii Chapter 1. Selected Problems in One Complex Variable 1 1.1. Preliminaries 2 1.2. A Simple Problem 2 1.3. Partitions of Unity 4 1.4. The Cauchy-Riemann Equations 7 1.5. The Proof of Proposition
More informationChern forms and the Fredholm determinant
CHAPTER 10 Chern forms and the Fredholm determinant Lecture 10: 20 October, 2005 I showed in the lecture before last that the topological group G = G (Y ;E) for any compact manifold of positive dimension,
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, February 25, 1997 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday, February 25, 1997 (Day 1) 1. Factor the polynomial x 3 x + 1 and find the Galois group of its splitting field if the ground
More informationCup product and intersection
Cup product and intersection Michael Hutchings March 28, 2005 Abstract This is a handout for my algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely,
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationExact results in AdS/CFT from localization. Part I
Exact results in AdS/CFT from localization Part I James Sparks Mathematical Institute, Oxford Based on work with Fernando Alday, Daniel Farquet, Martin Fluder, Carolina Gregory Jakob Lorenzen, Dario Martelli,
More informationIntersection of stable and unstable manifolds for invariant Morse functions
Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and
More informationCharacteristic classes and Invariants of Spin Geometry
Characteristic classes and Invariants of Spin Geometry Haibao Duan Institue of Mathematics, CAS 2018 Workshop on Algebraic and Geometric Topology, Southwest Jiaotong University July 29, 2018 Haibao Duan
More informationSection 2. Basic formulas and identities in Riemannian geometry
Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l
More informationarxiv: v1 [math.sg] 6 Nov 2015
A CHIANG-TYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiang-type lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one
More informationGEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS
GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action
More informationAn Invitation to Geometric Quantization
An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to
More informationTopological DBI actions and nonlinear instantons
8 November 00 Physics Letters B 50 00) 70 7 www.elsevier.com/locate/npe Topological DBI actions and nonlinear instantons A. Imaanpur Department of Physics, School of Sciences, Tarbiat Modares University,
More informationStable complex and Spin c -structures
APPENDIX D Stable complex and Spin c -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.
More informationGENERALIZING THE LOCALIZATION FORMULA IN EQUIVARIANT COHOMOLOGY
GENERALIZING THE LOCALIZATION FORULA IN EQUIVARIANT COHOOLOGY Abstract. We give a generalization of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action.
More informationThe symplectic structure on moduli space (in memory of Andreas Floer)
The symplectic structure on moduli space (in memory of Andreas Floer) Alan Weinstein Department of Mathematics University of California Berkeley, CA 94720 USA (alanw@math.berkeley.edu) 1 Introduction The
More informationThe Canonical Sheaf. Stefano Filipazzi. September 14, 2015
The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationDIFFERENTIAL FORMS AND COHOMOLOGY
DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the
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 informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 18, 2011 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 18, 2011 (Day 1) 1. (CA) Evaluate 0 x 2 + 1 x 4 + 1 dx Solution. We can consider the integration from to instead. For
More informationINTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY
INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY YOUNG-HOON KIEM 1. Definitions and Basic Properties 1.1. Lie group. Let G be a Lie group (i.e. a manifold equipped with differentiable group operations mult
More informationTorus actions and Ricci-flat metrics
Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF - 6108-00358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a-12294
More informationThe transverse index problem for Riemannian foliations
The transverse index problem for Riemannian foliations John Lott UC-Berkeley http://math.berkeley.edu/ lott May 27, 2013 The transverse index problem for Riemannian foliations Introduction Riemannian foliations
More informationBACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More informationQuantising proper actions on Spin c -manifolds
Quantising proper actions on Spin c -manifolds Peter Hochs University of Adelaide Differential geometry seminar Adelaide, 31 July 2015 Joint work with Mathai Varghese (symplectic case) Geometric quantization
More informationHeat Kernels, Symplectic Geometry, Moduli Spaces and Finite Groups
Heat Kernels, Symplectic eometry, Moduli Spaces and Finite roups Kefeng Liu 1 Introduction In this note we want to discuss some applications of heat kernels in symplectic geometry, moduli spaces and finite
More informationCOMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD
COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD MELINDA LANIUS 1. introduction Because Poisson cohomology is quite challenging to compute, there are only very select cases where the answer is
More informationDUISTERMAAT HECKMAN MEASURES AND THE EQUIVARIANT INDEX THEOREM
DUISTEAAT HECKAN EASUES AND THE EQUIVAIANT INDEX THEOE Let N be a symplectic manifold, with a Hamiltonian action of the circle group G and moment map µ : N. Assume that the level sets of µ are compact
More informationLecture 8: More characteristic classes and the Thom isomorphism
Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More information