Generalized Complex Geometry and Blow-Ups
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Arxiv:1704.04175V1 [Math.DG] 13 Apr 2017 Umnfls[ Submanifolds Mty[ Ometry Eerhr Ncmlxgoer.I at H Loihshe Algorithms to the Aim Fact, the in with Geometry
SAGEMATH EXPERIMENTS IN DIFFERENTIAL AND COMPLEX GEOMETRY DANIELE ANGELLA Abstract. This note summarizes the talk by the author at the workshop “Geometry and Computer Science” held in Pescara in February 2017. We present how SageMath can help in research in Complex and Differential Geometry, with two simple applications, which are not intended to be original. We consider two "classification problems" on quotients of Lie groups, namely, "computing cohomological invariants" [AFR15, LUV14], and "classifying special geo- metric structures" [ABP17], and we set the problems to be solved with SageMath [S+09]. Introduction Complex Geometry is the study of manifolds locally modelled on the linear complex space Cn. A natural way to construct compact complex manifolds is to study the projective geometry n n+1 n of CP = C 0 /C×. In fact, analytic submanifolds of CP are equivalent to algebraic submanifolds [GAGA\ { }]. On the one side, this means that both algebraic and analytic techniques are available for their study. On the other side, this means also that this class of manifolds is quite restrictive. In particular, they do not suffice to describe some Theoretical Physics models e.g. [Str86]. Since [Thu76], new examples of complex non-projective, even non-Kähler manifolds have been investigated, and many different constructions have been proposed. One would study classes of manifolds whose geometry is, in some sense, combinatorically or alge- braically described, in order to perform explicit computations. In this sense, great interest has been deserved to homogeneous spaces of nilpotent Lie groups, say nilmanifolds, whose ge- ometry [Bel00, FG04] and cohomology [Nom54] is encoded in the Lie algebra. -
Symplectic Geometry
Vorlesung aus dem Sommersemester 2013 Symplectic Geometry Prof. Dr. Fabian Ziltener TEXed by Viktor Kleen & Florian Stecker Contents 1 From Classical Mechanics to Symplectic Topology2 1.1 Introducing Symplectic Forms.........................2 1.2 Hamiltonian mechanics.............................2 1.3 Overview over some topics of this course...................3 1.4 Overview over some current questions....................4 2 Linear (pre-)symplectic geometry6 2.1 (Pre-)symplectic vector spaces and linear (pre-)symplectic maps......6 2.2 (Co-)isotropic and Lagrangian subspaces and the linear Weinstein Theorem 13 2.3 Linear symplectic reduction.......................... 16 2.4 Linear complex structures and the symplectic linear group......... 17 3 Symplectic Manifolds 26 3.1 Definition, Examples, Contangent bundle.................. 26 3.2 Classical Mechanics............................... 28 3.3 Symplectomorphisms, Hamiltonian diffeomorphisms and Poisson brackets 35 3.4 Darboux’ Theorem and Moser isotopy.................... 40 3.5 Symplectic, (co-)isotropic and Lagrangian submanifolds.......... 44 3.6 Symplectic and Hamiltonian Lie group actions, momentum maps, Marsden– Weinstein quotients............................... 48 3.7 Physical motivation: Symmetries of mechanical systems and Noether’s principle..................................... 52 3.8 Symplectic Quotients.............................. 53 1 From Classical Mechanics to Symplectic Topology 1.1 Introducing Symplectic Forms Definition 1.1. A symplectic form is a non–degenerate and closed 2–form on a manifold. Non–degeneracy means that if ωx(v, w) = 0 for all w ∈ TxM, then v = 0. Closed means that dω = 0. Reminder. A differential k–form on a manifold M is a family of skew–symmetric k– linear maps ×k (ωx : TxM → R)x∈M , which depends smoothly on x. 2n Definition 1.2. We define the standard symplectic form ω0 on R as follows: We 2n 2n 2n 2n identify the tangent space TxR with the vector space R . -
Complex Cobordism and Almost Complex Fillings of Contact Manifolds
MSci Project in Mathematics COMPLEX COBORDISM AND ALMOST COMPLEX FILLINGS OF CONTACT MANIFOLDS November 2, 2016 Naomi L. Kraushar supervised by Dr C Wendl University College London Abstract An important problem in contact and symplectic topology is the question of which contact manifolds are symplectically fillable, in other words, which contact manifolds are the boundaries of symplectic manifolds, such that the symplectic structure is consistent, in some sense, with the given contact struc- ture on the boundary. The homotopy data on the tangent bundles involved in this question is finding an almost complex filling of almost contact manifolds. It is known that such fillings exist, so that there are no obstructions on the tangent bundles to the existence of symplectic fillings of contact manifolds; however, so far a formal proof of this fact has not been written down. In this paper, we prove this statement. We use cobordism theory to deal with the stable part of the homotopy obstruction, and then use obstruction theory, and a variant on surgery theory known as contact surgery, to deal with the unstable part of the obstruction. Contents 1 Introduction 2 2 Vector spaces and vector bundles 4 2.1 Complex vector spaces . .4 2.2 Symplectic vector spaces . .7 2.3 Vector bundles . 13 3 Contact manifolds 19 3.1 Contact manifolds . 19 3.2 Submanifolds of contact manifolds . 23 3.3 Complex, almost complex, and stably complex manifolds . 25 4 Universal bundles and classifying spaces 30 4.1 Universal bundles . 30 4.2 Universal bundles for O(n) and U(n).............. 33 4.3 Stable vector bundles . -
Pure Spinor Helicity Methods
Pure Spinor Helicity Methods Rutger Boels Niels Bohr International Academy, Copenhagen R.B., arXiv:0908.0738 [hep-th] Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 1 / 25 in books? Why you should pay attention, an experiment The greatest common denominator of the audience? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in books? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in books: Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment The greatest common denominator of the audience in science books? Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 for all legs simultaneously pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers Why you should pay attention subject: calculation of scattering amplitudes in D > 4 with many legs Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 for all legs simultaneously Why you should pay attention subject: calculation of scattering amplitudes in D > 4 with many legs pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 Why you -
Clifford Algebras and Spinors∗
Bulg. J. Phys. 38 (2011) 3–28 This paper is dedicated to the memory of Matey Mateev, a missing friend Clifford Algebras and Spinors∗ I. Todorov1,2, 1Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria (permanent address) 2Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland Received 9 March 2011 Abstract. Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a re- view of background material, needed, in particular, in now fashionable theoret- ical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-six-year-old Introduction to Majorana masses [1] and includes historical notes and biographical data on past participants in the story. PACS codes: 03.65.Fd, 04.20.Gz 1 Quaternions, Grassmann and Clifford Algebras Clifford’s1 paper [2] on “geometric algebra” (published a year before his death) had two sources: Grassmann’s2 algebra and Hamilton’s3 quaternions whose ∗Lectures presented at the University of Sofia in October-November, 2010. Lecture notes pre- pared together with Dimitar Nedanovski (e-mail: [email protected]) and completed dur- ing the stay of the author at the ICTP, Trieste and at CERN, Geneva. 1William Kingdon Clifford (1845-1879) early appreciated the work of Lobachevsky and Rie- mann; he was the first to translate into English Riemann’s inaugural lecture On the hypotheses which lie at the bases of geometry. His view of the physical world as variation of curvature of space antici- pated Einstein’s general theory of relativity. -
Hidden Exceptional Symmetry in the Pure Spinor Superstring
PHYSICAL REVIEW D 101, 026006 (2020) Hidden exceptional symmetry in the pure spinor superstring † R. Eager,1,* G. Lockhart,2, and E. Sharpe 3 1Kishine Koen, Yokohama 222-0034, Japan 2Institute for Theoretical Physics, University of Amsterdam, 1098 XH Amsterdam, The Netherlands 3Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA (Received 12 November 2019; published 6 January 2020) The pure spinor formulation of superstring theory includes an interacting sector of central charge cλ ¼ 22, which can be realized as a curved βγ system on the cone over the orthogonal Grassmannian þ OG ð5; 10Þ. We find that the spectrum of the βγ system organizes into representations of the g ¼ e6 affine algebra at level −3, whose soð10Þ−3 ⊕ uð1Þ−4 subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine e6 characters. We interpret this as an instance of a more general pattern of enhancements in curved βγ systems, which also includes the cases g ¼ soð8Þ and e7, corresponding to target spaces that are cones over the complex Grassmannian Grð2; 4Þ and the complex Cayley plane OP2. We identify these curved βγ systems with the chiral algebras of certain two-dimensional (2D) (0,2) conformal field theories arising from twisted compactification of 4D N ¼ 2 superconformal field theories on S2. DOI: 10.1103/PhysRevD.101.026006 I. INTRODUCTION Nevertheless, we find that the ghost sector partition function [2] can be expressed as a linear combination of The pure spinor formalism [1] is a reformulation of ðeˆ6Þ affine characters, superstring theory which has the virtue that it can be −3 ðˆ Þ ðˆ Þ quantized while preserving manifest covariance with Z ¼ χˆ e6 −3 − χˆ e6 −3 : ð2Þ respect to ten-dimensional super-Poincar´e symmetry. -
Almost Complex Structures and Obstruction Theory
ALMOST COMPLEX STRUCTURES AND OBSTRUCTION THEORY MICHAEL ALBANESE Abstract. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018. Let X be a CW complex and Y a simply connected space. Last time we discussed the obstruction to extending a map f : X(n) ! Y to a map X(n+1) ! Y ; recall that X(k) denotes the k-skeleton of X. n+1 There is an obstruction o(f) 2 C (X; πn(Y )) which vanishes if and only if f can be extended to (n+1) n+1 X . Moreover, o(f) is a cocycle and [o(f)] 2 H (X; πn(Y )) vanishes if and only if fjX(n−1) can be extended to X(n+1); that is, f may need to be redefined on the n-cells. Obstructions to lifting a map p Given a fibration F ! E −! B and a map f : X ! B, when can f be lifted to a map g : X ! E? If X = B and f = idB, then we are asking when p has a section. For convenience, we will only consider the case where F and B are simply connected, from which it follows that E is simply connected. For a more general statement, see Theorem 7.37 of [2]. Suppose g has been defined on X(n). Let en+1 be an n-cell and α : Sn ! X(n) its attaching map, then p ◦ g ◦ α : Sn ! B is equal to f ◦ α and is nullhomotopic (as f extends over the (n + 1)-cell). -
Complex Structures and 2 X 2 Matrices
COMPLEX STRUCTURES AND 2 2 MATRICES × Simon Salamon University of Warwick, 14 May 2004 Part I Complex structures on R4 Part II SKT structures on a 6-manifold Part I The standard complex structure A (linear) complex structure on R4 is simply a linear map J: R4 R4 satisfying J 2 = 1. For example, ! − 0 1 0 1 0 J0 = 0 − 1 0 0 1 B 1 0 C B − C @ A In terms of a basis (dx1; dx2; dx3; dx4) of R4 , Jdx1 = dx2; Jdx3 = dx4 − − The choices of sign ensure that, setting z1 = x1 + ix2; z2 = x3 + ix4; the elements dz1; dz2 C4 satisfy 2 Jdz1 = J(dx1 + idx2) = i(dx1 + idx2) = idz1 Jdz2 = idz2: 2 Spaces of complex structures Any complex structure J is conjugate to J0 , so the set of complex structures is 1 = A− J0A : A GL(4; R) C f 2 g GL(4; R) = ; ∼ GL(2; C) a manifold of real dim 8, the same dim as GL(2; C). + Taking account of orientation, = − . C C t C Theorem + S2 . C ' Proof. Any J + has a polar decomposition 2 C 1 1 T 1 1 J = SO = O− S− = (O S− O)( O− ): − − 1 1 Thus, O = O− = P − J P defines an element − 0 SO(4) U(2)P = S2: 2 U(2) ∼ 3 Deformations and 2 2 matrices × A complex structure J is determined by its i-eigenspace 1;0 4 E = Λ = v C : Jv = iv ; f 2 g i on E since E E = C4 and J = ⊕ i on E: − For example, E = dz1; dz2 and E = dz1; dz2 . -
Twistors, Generalizations and Exceptional Structures
Twistors, Generalizations and Exceptional Structures Rold˜ao da Rocha∗ Jayme Vaz, Jr.† Abstract This paper is intended to describe twistors via the paravector model of Clifford algebras and to relate such description to conformal maps in the Clifford algebra over R4,1, besides pointing out some applications of the pure spinor formalism. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac-Clifford algebra C ⊗ Cℓ1,3 using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose the Clifford algebra over R4,1 is also used to describe conformal maps, instead of R2,4. It is possible to identify, via the pure spinor formalism, the twistor fiber in four, six and eight dimensions, respectively, with the coset 1 3 spaces SO(4)/(SU(2) × U(1)/Z2) ≃ CP , SO(6)/(SU(3) × U(1)/Z2) ≃ CP and SO(8)/(Spin(6) × Spin(2)/Z2). The last homogeneous space is closely re- lated to the SO(8) spinor decomposition preserving SO(8) symmetry in type IIB superstring theory. Indeed, aside the IIB superstring theory, there is no SO(8) spinor decomposition preserving SO(8) symmetry and, in this case, one can introduce distinct coordinates and conjugate momenta only if the Spin(8) symmetry is broken by a Spin(6)× Spin(2) subgroup of Spin(8). Also, it is re- viewed how to generalize the Penrose flagpole, illustrating the use of the pure arXiv:math-ph/0412037v2 25 Feb 2005 spinor formalism to construct a flagpole that is more general than the Pen- rose one, which arises when a defined parameter goes to zero. -
Generalized Geometry, an Introduction Assignment 1
Generalized Geometry, an introduction Assignment 1 Universitat Aut`onomade Barcelona Summer course, 8-19 July 2019 Problem 1. Consider the vector space R4 with the symplectic form 2 0 0 0 0 X 0 0 !((x1; y1; x2; y2); (x1; y1; x2; y2)) = (xiyi − yixi): i=1 • Find a subspace U such that U = U !. • Find a plane U, i.e., a subspace isomorphic to R2, such that U +U ! = R4. Problem 2. A subspace U of a symplectic vector space (V; !) such that U ! ⊆ U is called a coisotropic subspace. Prove that the quotient U=U ! naturally inherits a symplectic structure. This is called the coisotropic reduction. Problem 3. Given a symplectic form, we have an isomorphism V ! V ∗. We invert this isomorphism to get a map V ∗ ! V , which we can see as a map π : V ∗ × V ∗ ! k: Prove that the map π is bilinear, non-degenerate and skew-symmetric. Problem 4. Consider a real vector space with a linear complex structure (V; J) and its complexification VC. Prove that iJ = Ji. When does the map ai + bJ; for a; b 2 R, define a linear complex structure on VC, seen as a real vector space? Problem 5. Consider a real vector space with a linear complex structure (V; J). Prove that the map J ∗ : V ∗ ! V ∗; given by J ∗α(v) = α(Jv); for α 2 V ∗ and v 2 V , defines a linear complex structure on V ∗. Given a basis i (vi) with dual basis (v ), prove that ∗ i ∗ J v = −(Jvi) : Problem 6. The invertible linear transformations of Rn are called the general linear group and denoted by GL(n; R). -
Special Homogeneous Almost Complex Structures on Symplectic
Special homogeneous almost complex structures on symplectic manifolds Alberto Della Vedova∗ October 17, 2018 Abstract Homogeneous compatible almost complex structures on symplec- tic manifolds are studied, focusing on those which are special, mean- ing that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint or- bits. Conversely, compact isotropy co-adjoint orbits of semi-simple Lie groups are shown to admit special compatible almost complex struc- tures whenever they satisfy a necessary topological condition. Some classes of examples including twistor spaces of hyperbolic manifolds and discrete quotients of Griffiths period domains of weight two are discussed. 1 Introduction The aim of the present paper is provide some new examples of symplec- tic manifolds endowed with compatible almost complex structures having special curvature properties. More specifically, on a symplectic manifold (M,ω) we will call special a compatible almost complex structure J whose Chern-Ricci form ρ is a multiple of the symplectic form ω (cfr. definition 7). This condition has been previously considered by Apostolov and Dr˘aghici [1, Section 7], and should be thought of as a natural extension of the K¨ahler- Einstein condition to non-integrable compatible almost complex structures. arXiv:1706.06401v2 [math.SG] 19 Oct 2017 As in the integrable case, the existence of a special J forces the first Chern class c1 of (M,ω) to be a multiple of [ω]. Since ρ represents 4πc1, the ex- istence of a special J satisfying ρ = λω makes (M,ω) a symplectic Fano, symplectic Calabi-Yau or symplectic general type depending on λ is posi- tive, zero or negative respectively. -
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 ********************************************************************************* Information: Course Name: Symplectic Geometry Instructor: Zuoqin Wang Time/Room: Wed. 2:00pm-6:00pm @ 1318 Reference books: • Lectures on Symplectic Geometry by A. Canas de Silver • Symplectic Techniques in Physics by V. Guillemin and S. Sternberg • Lectures on Symplectic Manifolds by A. Weinstein • Introduction to Symplectic Topology by D. McDuff and D. Salamon • Foundations of Mechanics by R. Abraham and J. Marsden • Geometric Quantization by Woodhouse Course webpage: http://staff.ustc.edu.cn/˜ wangzuoq/Symp15/SympGeom.html ********************************************************************************* Introduction: The word symplectic was invented by Hermann Weyl in 1939: he replaced the Latin roots in the word complex, com-plexus, by the corresponding Greek roots, sym-plektikos. What is symplectic geometry? • Geometry = background space (smooth manifold) + extra structure (tensor). { Riemannian geometry = smooth manifold + metric structure. ∗ metric structure = positive-definite symmetric 2-tensor { Complex geometry = smooth manifold + complex structure. ∗ complex structure = involutive endomorphism ((1,1)-tensor) { Symplectic geometry = smooth manifold + symplectic structure 1 2 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY ∗ Symplectic structure = closed non-degenerate 2-form · 2-form = anti-symmetric