B-FIELDS, GERBES AND GENERALIZED GEOMETRY
|
|
- Bryce Kennedy
- 5 years ago
- Views:
Transcription
1 B-FIELDS, GERBES AND GENERALIZED GEOMETRY Nigel Hitchin (Oxford) Durham Symposium July 29th
2 PART 1 (i) BACKGROUND (ii) GERBES (iii) GENERALIZED COMPLEX STRUCTURES (iv) GENERALIZED KÄHLER STRUCTURES 2
3 BACKGROUND 3
4 BASIC SCENARIO manifold M n replace T by T T inner product of signature (n, n) (X + ξ, X + ξ) = i X ξ skew adjoint transformations: End T Λ 2 T Λ 2 T... in particular B Λ 2 T 4
5 BASIC SCENARIO manifold M n replace T by T T inner product of signature (n, n) (X + ξ, X + ξ) = i X ξ skew adjoint transformations: End T Λ 2 T Λ 2 T... in particular B Λ 2 T 5
6 BASIC SCENARIO manifold M n replace T by T T inner product of signature (n, n) (X + ξ, X + ξ) = i X ξ skew adjoint transformations: End T Λ 2 T Λ 2 T... in particular B Λ 2 T 6
7 TRANSFORMATIONS exponentiate B: X + ξ X + ξ + i X B B Ω 2... the B-field natural group Diff(M) Ω 2 (M) 7
8 SPINORS Take S = Λ T S = S ev S od Define Clifford multiplication by (X + ξ) ϕ = i X ϕ + ξ ϕ (X + ξ) 2 ϕ = i X ξϕ = (X + ξ, X + ξ)ϕ exp B(ϕ) = (1 + B B B +...) ϕ 8
9 DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 9
10 DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 10
11 DERIVATIVES Lie bracket: 2i [X,Y ] α = d([i X, i Y ]α) + 2i X d(i Y α) 2i Y d(i X α) + [i X, i Y ]dα A = X + ξ, B = Y + η use Clifford multiplication A α to define a bracket [A, B]: 2[A, B] α = d((a B B A) α) + 2A d(b α) 2B d(a α) + (A B B A) dα COURANT bracket [X + ξ, Y + η] = [X, Y ] + L X η L Y ξ 1 2 d(i Xη i Y ξ) 11
12 Apply a 2-form B... d e B de B = d + db [X + ξ, Y + η] [X + ξ, Y + η] 2i X i Y db Diff(M) Ω 2 closed (M) preserves inner product, exterior derivative and Courant bracket. 12
13 GENERALIZED GEOMETRIC STRUCTURES SO(n, n) compatibility integrability d or Courant bracket transform by Diff(M) Ω 2 closed (M) 13
14 RIEMANNIAN METRIC Riemannian metric g ij X g(x, ) : g : T T graph of g = V T T T T = V V 14
15 T * graph of g T graph of g 15
16 GENERALIZED RIEMANNIAN METRIC V T T positive definite rank n subbundle = graph of g + B : T T g + B T T : g symmetric, B skew 16
17 GERBES 17
18 GERBES: ČECH 2-COCYCLES g αβγ : U α U β U γ S 1 (g αβγ = g 1 βαγ =...) δg = g βγδ g 1 αγδ g αβδg 1 αβγ = 1 on U α U β U γ U δ This defines a gerbe. 18
19 CONNECTIONS ON GERBES Connective structure: A αβ + A βγ + A γα = g 1 αβγ dg αβγ Curving: B β B α = da αβ db β = db α = H Uα global three-form H J.-L. Brylinski, Characteristic classes and geometric quantization, Progr. in Mathematics 107, Birkhäuser, Boston (1993) 19
20 TWISTING T T da αβ + da βγ + da γα = d[g 1 αβγ dg αβγ] = 0 identify T T on U α with T T on U β by X + ξ X + ξ + i X da αβ defines a vector bundle E 0 T E T 0 with... an inner product and a Courant bracket. 20
21 TWISTED COHOMOLOGY identify Λ T on U α with Λ T on U β by ϕ e da αβϕ defines a vector bundle S = spinor bundle for E d : C (S) C (S) well-defined ker d/ im d = twisted cohomology. 21
22 ... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 22
23 ... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 23
24 ... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 24
25 ... WITH A CURVING ϕ α = e da αβϕ β dϕ α = 0 Curving: B β B α = da αβ e B αϕ α = e B βϕ β = ψ dψ + H ψ = 0 25
26 Definition: A generalized metric is a subbundle V E such that rk V = dim M and the inner product is positive definite on V. V T = 0 splitting of the sequence 0 T E T 0 V E another splitting difference Hom(T, T ) = T T = Riemannian metric 26
27 SPLITTINGS IN LOCAL TERMS splitting: C α C (U α, T T ) : C β C α = da αβ Sym(C α ) = Sym(C β ) = metric Alt(C α ) = B α = curving of the gerbe H = db α closed 3-form 27
28 two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 28
29 two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 29
30 two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] is a one-form 30
31 two splittings V and V of 0 T E T 0 X vector field, lift to X + C (V ) and X C (V ) in E Courant bracket [X +, Y ], Lie bracket [X, Y ] [X +, Y ] [X, Y ] + is a one-form 31
32 [X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X, Y + ] [X, Y ] + = 2g( XY ) has skew torsion H 32
33 [X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X +, Y ] [X, Y ] + = 2g( XY ) has skew torsion H 33
34 [X +, Y ] [X, Y ] + = 2g( + X Y ) + Riemannian connection with skew torsion H [X, Y + ] [X, Y ] = 2g( XY ) has skew torsion H 34
35 T * graph of g T graph of g 35
36 EXAMPLE: the Levi-Civita connection [ x i g ik dx k, ] + g jk dx k x j [ x i, x j ] + = = ( gjk + g ik g ) ij dx k = 2g lk Γ l ij x i x j x k 36
37 GENERALIZED COMPLEX STRUCTURES 37
38 A generalized complex structure is: J : T T T T, J 2 = 1 (JA, B) + (A, JB) = 0 if JA = ia, JB = ib then J[A, B] = i[a, B] (Courant bracket) U(m, m) SO(2m, 2m) structure on T T M. Gualtieri:Generalized complex geometry math.dg/
39 EXAMPLES complex manifold J = ( I 0 0 I ) J = i : [... / z i...,... d z i...] symplectic manifold J = ( 0 ω 1 ω 0 ) J = i : [..., / x j + i ω jk dx k,...] 39
40 EXAMPLE: HOLOMORPHIC POISSON MANIFOLDS σ = σ ij z i z j [σ, σ] = 0 J = i :..., z j,..., d z k + l σ kl,... z l 40
41 GENERALIZED KÄHLER MANIFOLDS 41
42 Kähler complex structure + symplectic structure complex structure J 1 on T T symplectic structure J 2 on T T compatibility (ω Ω 1,1 ): J 1 J 2 = J 2 J 1 42
43 A generalized Kähler structure: two commuting generalized complex structures J 1, J 2 such that (J 1 J 2 (X + ξ), X + ξ) is definite. GUALTIERI S THEOREM A generalized Kähler manifold is: two integrable complex structures I +, I on M a metric g, hermitian with respect to I +, I a 2-form B U(m) connections +, with skew torsion ±H = ±db 43
44 (J 1 J 2 ) 2 = 1 eigenspaces V, V, metric on V positive definite connections +, torsion ±db... also define on 0 T E T 0 44
45 J 1 = J 2 on V ±I + J 1 = J 2 on V ±I {J 1 = i} {J 2 = i} = V 1,0 is Courant integrable [X 1,0 + ξ, Y 1,0 + η] = Z 1,0 + ζ I + integrable 45
46 V Apostolov, P Gauduchon, G Grantcharov, Bihermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), S. J. Gates, C. M. Hull and M. Roček, Twisted multiplets and new supersymmetric nonlinear σ-models. Nuclear Phys. B 248 (1984),
47 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 47
48 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 48
49 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 49
50 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 50
51 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 51
52 [J 1, J 2 ] = 0, [I +, I ] 0 in general g([i +, I ]X, Y ) = Φ(X, Y ) 2-form Φ Λ 2,0 + Λ 0,2 for both complex structures σ Λ 0,2 = Λ 2 T σ is holomorphic σ is a holomorphic Poisson structure 52
53 EXAMPLES T 4 and K3, σ = holomorphic 2-form (D. Joyce) CP 2, σ = / z 1 / z 2 CP 1 CP 1 = projective quadric. σ = Q/ z 3 [ / z 1 / z 2 ] moduli space of ASD connections on above. 53
54 J. Bogaerts, A. Sevrin, S. van der Loo, S. Van Gils, Properties of Semi-Chiral Superfields, Nucl.Phys. B562 (1999) V Apostolov, P Gauduchon, G Grantcharov, Bihermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), NJH, Instantons, Poisson structures and generalized Kaehler geometry, math.dg/ C. Bartocci and E. Macrì, Classification of Poisson surfaces, Communications in Contemporary Mathematics, 7 (2005),
THE GEOMETRY OF B-FIELDS. Nigel Hitchin (Oxford) Odense November 26th 2009
THE GEOMETRY OF B-FIELDS Nigel Hitchin (Oxford) Odense November 26th 2009 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified
More informationGeneralized complex geometry and topological sigma-models
Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models
More informationNigel Hitchin (Oxford) Poisson 2012 Utrecht. August 2nd 2012
GENERALIZED GEOMETRY OF TYPE B n Nigel Hitchin (Oxford) Poisson 2012 Utrecht August 2nd 2012 generalized geometry on M n T T inner product (X + ξ,x + ξ) =i X ξ SO(n, n)-structure generalized geometry on
More informationLectures on generalized geometry
Surveys in Differential Geometry XVI Lectures on generalized geometry Nigel Hitchin Preface These notes are based on six lectures given in March 2010 at the Institute of Mathematical Sciences in the Chinese
More informationDedicated to the memory of Shiing-Shen Chern
ASIAN J. MATH. c 2006 International Press Vol. 10, No. 3, pp. 541 560, September 2006 002 BRACKETS, FORMS AND INVARIANT FUNCTIONALS NIGEL HITCHIN Dedicated to the memory of Shiing-Shen Chern Abstract.
More informationA note about invariant SKT structures and generalized Kähler structures on flag manifolds
arxiv:1002.0445v4 [math.dg] 25 Feb 2012 A note about invariant SKT structures and generalized Kähler structures on flag manifolds Dmitri V. Alekseevsky and Liana David October 29, 2018 Abstract: We prove
More informationGeneralized complex geometry and supersymmetric non-linear sigma models
hep-th/yymmnnn UUITP-??/04 HIP-2004-??/TH Generalized complex geometry and supersymmetric non-linear sigma models Talk at Simons Workshop 2004 Ulf Lindström ab, a Department of Theoretical Physics Uppsala
More informationHAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY
HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY TIMOTHY E. GOLDBERG These are notes for a talk given in the Lie Groups Seminar at Cornell University on Friday, September 25, 2009. In retrospect, perhaps
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 informationOn the holonomy fibration
based on work with Alejandro Cabrera and Marco Gualtieri Workshop on Geometric Quantization Adelaide, July 2015 Introduction General theme: Hamiltonian LG-spaces q-hamiltonian G-spaces M Ψ Lg /L 0 G /L
More informationDIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013
DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 16th October 1843 ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIING-SHEN CHERN Introduction.
More informationMulti-moment maps. CP 3 Journal Club. Thomas Bruun Madsen. 20th November 2009
Multi-moment maps CP 3 Journal Club Thomas Bruun Madsen 20th November 2009 Geometry with torsion Strong KT manifolds Strong HKT geometry Strong KT manifolds: a new classification result Multi-moment maps
More informationContact manifolds and generalized complex structures
Contact manifolds and generalized complex structures David Iglesias-Ponte and Aïssa Wade Department of Mathematics, The Pennsylvania State University University Park, PA 16802. e-mail: iglesias@math.psu.edu
More informationPoisson modules and generalized geometry
arxiv:0905.3227v1 [math.dg] 20 May 2009 Poisson modules and generalized geometry Nigel Hitchin May 20, 2009 Dedicated to Shing-Tung Yau on the occasion of his 60th birthday 1 Introduction Generalized complex
More informationC-spaces, Gerbes and Patching DFT
C-spaces, Gerbes and Patching DFT George Papadopoulos King s College London Bayrischzell Workshop 2017 22-24 April Germany 23 April 2017 Material based on GP: arxiv:1402.2586; arxiv:1412.1146 PS Howe and
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 informationarxiv:math/ v1 [math.dg] 6 Jul 1999 LECTURES ON SPECIAL LAGRANGIAN SUBMANIFOLDS
arxiv:math/9907034v1 [math.dg] 6 Jul 1999 LECTURES ON SPECIAL LAGRANGIAN SUBMANIFOLDS Nigel Hitchin Abstract. These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and
More informationOrientation transport
Orientation transport Liviu I. Nicolaescu Dept. of Mathematics University of Notre Dame Notre Dame, IN 46556-4618 nicolaescu.1@nd.edu June 2004 1 S 1 -bundles over 3-manifolds: homological properties Let
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Calabi-Yau manifolds with B-fields Frederik Witt Preprint Nr. 31/2008 Calabi Yau manifolds with B fields F. Witt September 19, 2008 Abstract In recent work N. Hitchin
More informationGeneralized N = 1 orientifold compactifications
Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006
More informationNonabelian Gerbes, Differential Geometry and Stringy Applications
Nonabelian Gerbes, Differential Geometry and Stringy Applications Branislav Jurčo, LMU München Vietri April 13, 2006 Outline I. Anomalies in String Theory II. String group and string structures III. Nonabelian
More informationTWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY
TWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY Andrei Moroianu CNRS - Ecole Polytechnique Palaiseau Prague, September 1 st, 2004 joint work with Uwe Semmelmann Plan of the talk Algebraic preliminaries
More informationPublished as: J. Geom. Phys. 10 (1993)
HERMITIAN STRUCTURES ON HERMITIAN SYMMETRIC SPACES F. Burstall, O. Muškarov, G. Grantcharov and J. Rawnsley Published as: J. Geom. Phys. 10 (1993) 245-249 Abstract. We show that an inner symmetric space
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationInfinitesimal Einstein Deformations. Kähler Manifolds
on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds
More informationHomogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky
Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey
More informationGeneralized complex geometry
Generalized complex geometry Marco Gualtieri Abstract Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry,
More informationModuli Space of Higgs Bundles Instructor: Marco Gualtieri
Moduli Space of Higgs Bundles Instructor: Lecture Notes for MAT1305 Taught Spring of 2017 Typeset by Travis Ens Last edited January 30, 2017 Contents Lecture Guide Lecture 1 [11.01.2017] 1 History Hyperkähler
More informationSpecial holonomy and beyond
Clay athematics Proceedings Special holonomy and beyond Nigel Hitchin Abstract. anifolds with special holonomy are traditionally associated with the existence of parallel spinors, but Calabi-Yau threefolds
More informationLectures on Dirac Operators and Index Theory. Xianzhe Dai
Lectures on Dirac Operators and Index Theory Xianzhe Dai January 7, 2015 2 Chapter 1 Clifford algebra 1.1 Introduction Historically, Dirac operator was discovered by Dirac (who else!) looking for a square
More informationGeneralized almost paracontact structures
DOI: 10.1515/auom-2015-0004 An. Şt. Univ. Ovidius Constanţa Vol. 23(1),2015, 53 64 Generalized almost paracontact structures Adara M. Blaga and Cristian Ida Abstract The notion of generalized almost paracontact
More informationElementary Deformations HK/QK. Oscar MACIA. Introduction [0] Physics [8] Twist [20] Example [42]
deformations and the HyperKaehler/Quaternionic Kaehler correspondence. (Joint work with Prof. A.F. SWANN) Dept. Geometry Topology University of Valencia (Spain) June 26, 2014 What do we have 1 2 3 4 Berger
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 informationInstantons, Poisson structures and generalized Kähler geometry
arxiv:math/0503432v1 [math.dg] 21 ar 2005 Instantons, Poisson structures and generalized Kähler geometry Nigel Hitchin athematical Institute 24-29 St Giles Oxford OX1 3LB UK hitchin@maths.ox.ac.uk October
More informationGeneralized complex geometry
Generalized complex geometry Marco Gualtieri Abstract Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry,
More informationGeneralized complex geometry
Generalized complex geometry arxiv:math/0401221v1 [math.dg] 18 Jan 2004 Marco Gualtieri St John s College University of Oxford A thesis submitted for the degree of Doctor of Philosophy November 2003 Acknowledgements
More informationConstructing compact 8-manifolds with holonomy Spin(7)
Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.
More informationTopic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016
Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces
More informationTransverse geometry. consisting of finite sums of monomials of the form
Transverse geometry The space of leaves of a foliation (V, F) can be described in terms of (M, Γ), with M = complete transversal and Γ = holonomy pseudogroup. The natural transverse coordinates form the
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 informationSome brief notes on the Kodaira Vanishing Theorem
Some brief notes on the Kodaira Vanishing Theorem 1 Divisors and Line Bundles, according to Scott Nollet This is a huge topic, because there is a difference between looking at an abstract variety and local
More informationarxiv: v5 [math.dg] 17 Oct 2013
Hodge theory and deformations of SKT manifolds arxiv:1203.0493v5 [math.dg] 17 Oct 2013 Gil R. Cavalcanti Department of Mathematics Utrecht University Abstract We use tools from generalized complex geometry
More informationGeneralized CRF-structures
arxiv:0705.3934v1 [math.dg] 27 May 2007 Generalized CRF-structures by Izu Vaisman ABSTRACT. A generalized F-structure is a complex, isotropic subbundle E of T c M T c M (T cm = TM R C and the metric is
More informationHolonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012
Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel
More informationEigenvalue estimates for Dirac operators with torsion II
Eigenvalue estimates for Dirac operators with torsion II Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Metz, March 2012 1 Relations between different objects on a Riemannian manifold (M n,
More informationHard Lefschetz Theorem for Vaisman manifolds
Hard Lefschetz Theorem for Vaisman manifolds Antonio De Nicola CMUC, University of Coimbra, Portugal joint work with B. Cappelletti-Montano (Univ. Cagliari), J.C. Marrero (Univ. La Laguna) and I. Yudin
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 informationREDUCTION OF METRIC STRUCTURES ON COURANT ALGEBROIDS. Gil R. Cavalcanti
JOURNAL OF SYMPLECTIC GEOMETRY Volume 4, Number 3, 317 343, 2007 REDUCTION OF METRIC STRUCTURES ON COURANT ALGEBROIDS Gil R. Cavalcanti We use the procedure of reduction of Courant algebroids introduced
More informationSupersymmetric Sigma Models and Generalized Complex Geometry
UNIVERSITY OF UTRECHT MASTER THESIS Supersymmetric Sigma Models and Generalized Complex Geometry Author: Joey VAN DER LEER-DURÁN Supervisors: Dr. G.R. CAVALCANTI Prof.dr. S. VANDOREN July 24, 2012 Contents
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
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 informationarxiv: v1 [math.dg] 14 Dec 2015
ALGEBROIDS AND TWISTOR SPACES arxiv:1512.04517v1 [math.dg] 14 Dec 2015 STEVEN GINDI Contents 1. Introduction 2 2. Preliminaries 3 2.1. Twistor Spaces 3 2.2. Background on a Pair of Complex Structures 4
More informationThe Geometry of two dimensional Supersymmetric Nonlinear σ Model
The Geometry of two dimensional Supersymmetric Nonlinear σ Model Hao Guo March 14, 2005 Dept. of Physics, University of Chicago Abstract In this report, we investigate the relation between supersymmetry
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 informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
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 informationDirac operators with torsion
Dirac operators with torsion Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Golden Sands / Bulgaria, September 2011 1 Relations between different objects on a Riemannian manifold (M n, g):
More informationThe Calabi Conjecture
The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely
More informationTopological Solitons from Geometry
Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv:1111.2934.
More informationCompact Riemannian manifolds with exceptional holonomy
Unspecified Book Proceedings Series Compact Riemannian manifolds with exceptional holonomy Dominic Joyce published as pages 39 65 in C. LeBrun and M. Wang, editors, Essays on Einstein Manifolds, Surveys
More informationIntroduction to supersymmetry
Introduction to supersymmetry Vicente Cortés Institut Élie Cartan Université Henri Poincaré - Nancy I cortes@iecn.u-nancy.fr August 31, 2005 Outline of the lecture The free supersymmetric scalar field
More informationThe Strominger Yau Zaslow conjecture
The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex
More informationAtiyah classes and homotopy algebras
Atiyah classes and homotopy algebras Mathieu Stiénon Workshop on Lie groupoids and Lie algebroids Kolkata, December 2012 Atiyah (1957): obstruction to existence of holomorphic connections Rozansky-Witten
More informationGeneralized Calabi-Yau manifolds
arxiv:math/0209099v1 [math.dg] 10 Sep 2002 Generalized Calabi-Yau manifolds Nigel Hitchin athematical Institute 24-29 St Giles Oxford OX1 3LB UK hitchin@maths.ox.ac.uk February 1, 2008 Abstract A geometrical
More informationTwo simple ideas from calculus applied to Riemannian geometry
Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University
More informationThe Elements of Twistor Theory
The Elements of Twistor Theory Stephen Huggett 10th of January, 005 1 Introduction These are notes from my lecture at the Twistor String Theory workshop held at the Mathematical Institute Oxford, 10th
More informationBundle gerbes. Outline. Michael Murray. Principal Bundles, Gerbes and Stacks University of Adelaide
Bundle gerbes Michael Murray University of Adelaide http://www.maths.adelaide.edu.au/ mmurray Principal Bundles, Gerbes and Stacks 2007 Outline 1 Informal definition of p-gerbes 2 Background 3 Bundle gerbes
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationHow to recognise a conformally Einstein metric?
How to recognise a conformally Einstein metric? Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014).
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 informationThe Higgs bundle moduli space
The Higgs bundle moduli space Nigel Hitchin Mathematical Institute University of Oxford Hamburg September 8th 2014 Add title here 00 Month 2013 1 re ut on 1977... INSTANTONS connection on R 4, A i (x)
More informationGeneralized Kähler Geometry
Commun. Math. Phys. 331, 297 331 (2014) Digital Object Identifier (DOI) 10.1007/s00220-014-1926-z Communications in Mathematical Physics Generalized Kähler Geometry Marco Gualtieri University of Toronto,
More informationThe Fundamental Gerbe of a Compact Lie Group
The Fundamental Gerbe of a Compact Lie Group Christoph Schweigert Department of Mathematics, University of Hamburg and Center for Mathematical Physics, Hamburg Joint work with Thomas Nikolaus Sophus Lie
More informationclass # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]
More informationContact pairs (bicontact manifolds)
Contact pairs (bicontact manifolds) Gianluca Bande Università degli Studi di Cagliari XVII Geometrical Seminar, Zlatibor 6 September 2012 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds)
More informationHeterotic Flux Compactifications
Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,
More informationPHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH
PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH 1. Differential Forms To start our discussion, we will define a special class of type (0,r) tensors: Definition 1.1. A differential form of order
More informationHamiltonian flows, cotangent lifts, and momentum maps
Hamiltonian flows, cotangent lifts, and momentum maps Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Symplectic manifolds Let (M, ω) and (N, η) be symplectic
More information1 Introduction and preliminaries notions
Bulletin of the Transilvania University of Braşov Vol 2(51) - 2009 Series III: Mathematics, Informatics, Physics, 193-198 A NOTE ON LOCALLY CONFORMAL COMPLEX LAGRANGE SPACES Cristian IDA 1 Abstract In
More informationGeneralized Kähler Geometry from supersymmetric sigma models
UUITP-01/06 HIP-006-11/TH Generalized Kähler Geometry from supersymmetric sigma models arxiv:hep-th/0603130v 10 Jul 006 Andreas Bredthauer a, Ulf Lindström a,b, Jonas Persson a, and Maxim Zabzine a a Department
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 informationGeneralized Contact Structures
Generalized Contact Structures Y. S. Poon, UC Riverside June 17, 2009 Kähler and Sasakian Geometry in Rome in collaboration with Aissa Wade Table of Contents 1 Lie bialgebroids and Deformations 2 Generalized
More informationGravity theory on Poisson manifold with R-flux
Gravity theory on Poisson manifold with R-flux Hisayoshi MURAKI (University of Tsukuba) in collaboration with Tsuguhiko ASAKAWA (Maebashi Institute of Technology) Satoshi WATAMURA (Tohoku University) References
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 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 informationA RESULT ON NIJENHUIS OPERATOR
A RESULT ON NIJENHUIS OPERATOR Rashmirekha Patra Dept. of Mathematics Sambalpur University,Odisha,India 17 December 2014 Some History In 1984 Magri and Morosi worked on Poisson Nijenhuis manifolds via
More informationSolvable Lie groups and the shear construction
Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016 1 Swann s twist 2 The shear construction The
More informationOn the BCOV Conjecture
Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called
More informationMarsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds
Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds Chenchang Zhu Nov, 29th 2000 1 Introduction If a Lie group G acts on a symplectic manifold (M, ω) and preserves the
More informationGeometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type
Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint
More informationDirac structures. Henrique Bursztyn, IMPA. Geometry, mechanics and dynamics: the legacy of J. Marsden Fields Institute, July 2012
Dirac structures Henrique Bursztyn, IMPA Geometry, mechanics and dynamics: the legacy of J. Marsden Fields Institute, July 2012 Outline: 1. Mechanics and constraints (Dirac s theory) 2. Degenerate symplectic
More informationTHE HODGE DECOMPOSITION
THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional
More informationSymplectic geometry of deformation spaces
July 15, 2010 Outline What is a symplectic structure? What is a symplectic structure? Denition A symplectic structure on a (smooth) manifold M is a closed nondegenerate 2-form ω. Examples Darboux coordinates
More informationString Theory Compactifications with Background Fluxes
String Theory Compactifications with Background Fluxes Mariana Graña Service de Physique Th Journées Physique et Math ématique IHES -- Novembre 2005 Motivation One of the most important unanswered question
More informationSelf-dual conformal gravity
Self-dual conformal gravity Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014). Dunajski (DAMTP, Cambridge)
More informationMinimal surfaces in quaternionic symmetric spaces
From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences
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 informationarxiv:math/ v1 [math.dg] 29 Sep 1998
Unknown Book Proceedings Series Volume 00, XXXX arxiv:math/9809167v1 [math.dg] 29 Sep 1998 A sequence of connections and a characterization of Kähler manifolds Mikhail Shubin Dedicated to Mel Rothenberg
More informationThe exceptional holonomy groups and calibrated geometry
Proceedings of 12 th Gökova Geometry-Topology Conference pp. 110 139 Published online at GokovaGT.org The exceptional holonomy groups and calibrated geometry Dominic Joyce Dedicated to the memory of Raoul
More informationDonaldson and Seiberg-Witten theory and their relation to N = 2 SYM
Donaldson and Seiberg-Witten theory and their relation to N = SYM Brian Williams April 3, 013 We ve began to see what it means to twist a supersymmetric field theory. I will review Donaldson theory and
More information