DOCSLIB.ORG

Spectral Networks - a Story of Wall-Crossing in Geometry and Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Networks - a Story of Wall-Crossing in Geometry and Physics University of Heidelberg Master Thesis Spectral Networks - A story of Wall-Crossing in Geometry and Physics Author: Supervisor: Sebastian Schulz Prof. Dr. Anna Wienhard A thesis submitted in fulfilment of the requirements for the degree of Master of Science in the Faculty of Mathematics and Computer Sciences Mathematical Institute November 24, 2015 Declaration of Authorship I, Sebastian Schulz, declare that this thesis titled, ’Spectral Networks - A story of Wall-Crossing in Geometry and Physics’ and the work presented in it are my own. I confirm that: • This work was done wholly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly at- tributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. Signed: Date: iii UNIVERSITY OF HEIDELBERG Abstract Faculty of Mathematics and Computer Sciences Mathematical Institute Differential Geometry Research Group Master of Science Spectral Networks - A story of Wall-Crossing in Geometry and Physics by Sebastian Schulz This thesis deals with the phenomenon of wall-crossing for BPS indices in d = 4 N = 2 supersymmetric gauge theories with gauge group SU(K). Compactification over S1 yields a three-dimensional σ-model with target space M a fiber bundle over the Coulomb branch B of the four-dimensional theory. We demonstrate how the wall- crossing is captured by smoothness conditions on the Hyperkähler metric of M. Three ways of determining the 4d BPS spectrum are explained, drawing on the work of Gaiotto, Moore and Neitzke. Firstly, a twistor space construction reduces the problem to finding holomorphic Darboux coordinates which are obtained as solutions to a Riemann-Hilbert problem for large radii R of the circle. Secondly, for a subclass of theories obtained by compactifying a six-dimensional theory over a surface C, the Darboux coordinates can be computed from Fock-Goncharov coordinates on certain triangulations of C for gauge group SU(2). Thirdly, a codimension one sublocus of C called a Spectral Network cap- tures the BPS degeneracies in a more efficient way. Die vorliegende Abschlussarbeit beschäftigt sich mit dem Phänomen des Wall-Crossing für BPS Indizes vierdimensionaler Eichtheorien mit N = 2 Supersymmetrie und Eich- gruppe SU(K). Kompaktifizierung über einer S1 liefert ein dreidimensionales σ-Modell, dessen Zielraum M ein Faserbündel über dem Coulombzweig B der vierdimensionalen Theorie ist. Wir zeigen auf, wie das Wall-Crossing geometrisch aufgefasst werden kann als Glattheit der Hyperkählermetrik von M. Das Kernstück ist die Beschreibung dreier Wege das BPS Spektrum der 4d-Theorie zu bestimmen, was maßgeblich die Arbeit von Gaiotto, Moore und Neitzke war. Erstens kann durch eine Twistorraum-Konstruktion das Problem dazu reduziert werden, holomorphe Darboux-Koordinaten für das Faser- bündel M → B zu finden. Diese lassen sich aus der Lösung eines Riemann-Hilbert- Problems bestimmen, die zumindest für große Radien des Kompaktifizierungskreises ex- istiert. Zweitens beschreiben wir eine Unterklasse von Theorien, die ihrerseits als Kom- paktifzierung einer sechsdimensionalen Theorie über einer gepunkteten Riemannschen Fläche entsteht. Die holomorphen Darboux-Koordinaten können für den SU(2)-Fall aus Fock-Goncharov-Koordinaten bestimmter Triangulierungen von C berechnet werden. Drittens und letztens erläutern wir, wie bestimmte Weben von Linien auf C, genannt Spektrale Netzwerke, die BPS Indizes sehr effizient kodieren. v Acknowledgements I would like to thank my advisors Anna Wienhard and Daniel Roggenkamp for intro- ducing me to this fascinating topic, along with their endless patience and the guidance I have received both academically and personally. My thanks go to the Differential Geometry Research Group for providing a warm and welcoming atmosphere. I am particularly grateful to Daniele Alessandrini for the helpful discussions. I would like to thank my friends and family for helping me to keep my head where it belongs: Above this thesis. My particular thanks go to my friends Beatriz Tapia and Sebastian Nill for the careful proofreading and the positive criticism. vii Contents Declaration of Authorship iii Abstract v Acknowledgements vii 1 Introduction 1 2 Functorial Quantum Field Theories 3 2.1 Classical TQFT’s . .3 2.2 Higher categories and the Cobordism Hypothesis . .7 2.3 Towards Functorial Quantum Field Theories . 13 2.4 Compactification of QFTs . 19 3 Four-dimensional N = 2 Gauge theories 21 3.1 Gauge theory and σ-models . 21 3.2 BPS states . 24 3.3 The BPS index and the Wall-Crossing Formula . 25 3.4 Seiberg-Witten theory . 29 3.5 Dimensional reduction to three dimensions . 30 3.6 Construction of the Hyperkähler metric . 32 4 Theories of class S 35 4.1 The Theory X ................................. 35 4.2 QFTs with defects . 37 4.3 Compactify to 4d: Theories of class S ................... 39 4.4 Connection to the Hitchin integrable system . 41 4.5 String webs: The geometry of BPS states . 43 4.6 The sl(2, C) case . 46 5 Spectral Networks 51 5.1 2d-4d Wall-Crossing . 52 5.2 Surface operators and line operators . 55 5.3 The Formal Parallel Transport Theorem . 59 5.4 Spectral Networks at generic ϑ ....................... 62 5.5 Spectral Networks at critical ϑ ....................... 65 6 Conclusion and outlook 67 A Some category theory 69 A.1 Symmetric monoidal categories . 69 A.2 Internal categories . 71 A.3 Pseudo-categories . 73 ix B Supermathematics 75 B.1 Supermanifolds . 75 B.2 Super Lie algebras . 76 B.3 Representations of the super Poincaré algebra . 81 Bibliography 87 x Chapter 1 Introduction Supersymmetric gauge theories enjoy an intimate relationship to geometry, a fact that has led to a fruitful interplay of mathematics and physics. A particular interesting class of gauge theories are those in four dimensions and with N = 2 supersymmetry. The physical content of this thesis deals with certain properties of these theories. The main factors contributing are the Low Energy Effective Action (LEEA) and their BPS spectrum, the spectrum of distinguished states, called BPS states, that are very stable. While the former can be computed explicitely through the acclaimed Seiberg-Witten theory, the latter have only been determined in very particular cases. We aim at de- scribing different methods to derive this spectrum for a large class of theories, drawing on powerful methods from complex geometry. Let us explain the structure of this thesis. We start in chapter 2 by explaining a functorial approach to Quantum Field Theories. Its spirit is to circumvent the direct notion of path integrals by indirectly encoding its properties in a (higher) functor, a method that has been put to great success for Topological and Conformal Field Theories. Our use for these methods are to describe two important concepts in modern theoretical physics: Compactification and defects. While the former is a technique to obtain a lower dimensional QFT from a higher dimensional one, the latter encode ways of embedding lower dimensional QFTs inside higher dimensional ones. The concepts of this chapter depend crucially on abstract notions from category theory which we have gathered in appendix A. The main content of this thesis can be found in chapters 3-5.Each of these follow a similar pattern: The early sections introduce novel structures from physics through ax- iom systems that we aim to make as precise as possible. The mathematical implications are derived in the consecutive sections. Chapter 3 introduces the notions of four-dimensional gauge theories with N = 2 supersymmetry required for further understanding. We have attached a primer to su- persymmetry in appendix B which some readers might find useful. We review the notion of BPS states: Those are distinguished states of the theory which are very stable but can form or decay at certain walls in a manner governed by a wall-crossing formula. An index Ω is introduced in order to count these states. However, its computation can be infeasible due to the lack of proper tools in the strong coupling regime. A key insight is that Ω is locally constant over the moduli space of vacua B but jumps at the walls at which BPS states are generated or annihilated. Hence knowing the walls and how the index jumps, one can determine Ω globally by computing its value in a single region together with the jumps it is subject to. The question then remains how to determine these jumps. We start in section 3.5 by presenting one way in which geometry can enter this pic- 1 ture. Upon compactification over a circle SR of radius R, one obtains an N = 4 theory in three-dimensions with moduli space a Hyperkähler manifold M in the IR limit. M 1 2 Chapter 1. Introduction is naturally a fiber bundle over B with generic fiber an abelian variety, but these fibers degenerate over the singular loci of B. The wall-crossing formula is intimately related to the Hyperkähler metric g of M: It is equivalent to saying that g is smooth. Hence, the problem of determining the indices Ω has been translated into computing the Hyperkäh- ler metric g. We review a twistor space construction of this metric from [GMN10] in 3.6 in which holomorphic Darboux coordinates arise as solutions to a certain Riemann- Hilbert problem. The downside of this approach is that it only works for large radii R of the circle. A different approach is possible for a subclass of theories that arise as compactifica- tion limits from a six-dimensional theory X. The theory X is not very well-understood due to its lack of a proper action functional, hence compactifications are an important way to shed some light on it.
Load more

Recommended publications
  • TWAS Fellowships Worldwide
    CDC Round Table, ICTP April 2016 With science and engineering, countries can address challenges in agriculture, climate, health TWAS’s and energy. guiding principles 2 Food security Challenges Water quality for a Energy security new era Biodiversity loss Infectious diseases Climate change 3 A Globally, 81 nations fall troubling into the category of S&T- gap lagging countries. 48 are classified as Least Developed Countries. 4 The role of TWAS The day-to-day work of TWAS is focused in two critical areas: •Improving research infrastructure •Building a corps of PhD scholars 5 TWAS Research Grants 2,202 grants awarded to individuals and research groups (1986-2015) 6 TWAS’ AIM: to train 1000 PhD students by 2017 Training PhD-level scientists: •Researchers and university-level educators •Future leaders for science policy, business and international cooperation Rapidly growing opportunities P BRAZIL A K I N D I CA I RI A S AF TH T SOU A N M KENYA EX ICO C H I MALAYSIA N A IRAN THAILAND TWAS Fellowships Worldwide NRF, South Africa - newly on board 650+ fellowships per year PhD fellowships +460 Postdoctoral fellowships +150 Visiting researchers/professors + 45 17 Programme Partners BRAZIL: CNPq - National Council MALAYSIA: UPM – Universiti for Scientific and Technological Putra Malaysia WorldwideDevelopment CHINA: CAS - Chinese Academy of KENYA: icipe – International Sciences Centre for Insect Physiology and Ecology INDIA: CSIR - Council of Scientific MEXICO: CONACYT– National & Industrial Research Council on Science and Technology PAKISTAN: CEMB – National INDIA: DBT - Department of Centre of Excellence in Molecular Biotechnology Biology PAKISTAN: ICCBS – International Centre for Chemical and INDIA: IACS - Indian Association Biological Sciences for the Cultivation of Science PAKISTAN: CIIT – COMSATS Institute of Information INDIA: S.N.
    [Show full text]
  • Mirror Symmetry for Honeycombs
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 373, Number 1, January 2020, Pages 71–107 https://doi.org/10.1090/tran/7909 Article electronically published on September 10, 2019 MIRROR SYMMETRY FOR HONEYCOMBS BENJAMIN GAMMAGE AND DAVID NADLER Abstract. We prove a homological mirror symmetry equivalence between the A-brane category of the pair of pants, computed as a wrapped microlocal sheaf category, and the B-brane category of its mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the A-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. Contents 1. Introduction 71 2. Combinatorial A-model 78 3. Mirror symmetry 95 4. Symplectic geometry 100 Acknowledgments 106 References 106 1. Introduction This paper fits into the framework of homological mirror symmetry, as introduced in [23] and expanded in [19, 20, 22]. The formulation of interest to us relates the A-model of a hypersurface X in a toric variety to the mirror Landau-Ginzburg B- model of a toric variety X∨ equipped with superpotential W ∨ ∈O(X∨). Following Mikhalkin [27], a distinguished “atomic” case is when the hypersurface is the pair of pants ∗ n ∼ ∗ 1 n Pn−1 = {z1 + ···+ zn +1=0}⊂(C ) = T (S ) n+1 with mirror Landau-Ginzburg model (A ,z1 ···zn+1).
    [Show full text]
  • SUPERSYMMETRY 1. Introduction the Purpose
    SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1.
    [Show full text]
  • Proofs JHEP 092P 1215 Xxxxxxx ???, 2016 Springer : Doi: March 1, 2016 : Y February 25, 2016 December 11, 2015 : : Setting
    Published for SISSA by Springer Received: December 11, 2015 Revised: February 25, 2016 Accepted: March 1, 2016 Published: ???, 2016 proofs JHEP_092P_1215 Notes on super Killing tensors P.S. Howea and U. Lindstr¨omb,c aDepartment of Mathematics, King’s College London, The Strand, London WC2R 2LS, U.K. bDepartment of Physics and Astronomy, Theoretical Physics, Uppsala University, SE-751 20 Uppsala, Sweden cTheoretical Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, U.K. E-mail: [email protected], [email protected] Abstract: The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the even Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed. Keywords: Extended Supersymmetry, Superspaces, Higher Spin Symmetry ArXiv ePrint: 1511.04575 Open Access, c The Authors. doi:xxxxxxx Article funded by SCOAP3. Contents 1 Introduction 1 2 Killing tensors in superspace 3 3 Superparticles 6 4 SCKTs in flat superspaces 10 proofs JHEP_092P_1215 5 Analytic superspace 18 6 Components of superconformal Killing tensors 24 6.1 D = 3 25 6.2 D = 4 26 6.3 D = 6 27 6.4 Algebras 28 7 Comments on higher spin 29 8 Concluding remarks 30 A Supersymmetric SN bracket 31 1 Introduction In ordinary Lorentzian spacetime a Killing vector, K, is a symmetry of the metric, i.e.
    [Show full text]
  • Supersymmetric Lagrangians
    1 Section 1 Setup Supersymmetric Lagrangians Chris Elliott February 6, 2013 1 Setup Let's recall some notions from classical Lagrangian field theory. Let M be an oriented supermanifold: our spacetime, of dimension njd. For our purposes, we will most often think about the case where M is either Minkowski space Mˇ n = R1;n−1, or Super Minkowski space M = Mˇ n × ΠS where S is a representation of Spin(1; n − 1). Definition 1.1. Let E ! M be a smooth (super) fibre bundle. The associated space of fields is the space of smooth sections φM ! E.A Lagrangian density on F is a function L: F! Dens(M) { where Dens(M) denotes the bundle of densities on M { that satisfies a locality condition. Precisely, we form the pullback bundle Dens(M)F / Dens(M) M × F / M π1 and require L to be a section of this bundle such that, for some k, for all m 2 M, L(m; φ) only depends on the first k derivatives of φ. We can phrase this in terms of factoring through the kth jet bundle of E. As usual, we define the action functional to be the integral Z S(φ) = L(φ): M A Lagrangian system generally includes additional information, namely a choice of variational 1-form γ. In general I won't use this data, but I should mention when it might play a role. 1.1 Symmetries In this talk, we will be discussing Lagrangian systems with certain kinds of symmetry called supersymmetry. As such, it'll be important to understand what it actually means to be a symmetry of such a system.
    [Show full text]
  • [Math.RA] 14 Jun 2007 the Minkowski and Conformal Superspaces
    IFIC/06-28 FTUV-05/0928 The Minkowski and conformal superspaces R. Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5. 40126 Bologna. Italy. e-mail: fi[email protected] M. A. Lled´o Departament de F´ısica Te`orica, Universitat de Val`encia and IFIC C/Dr. Moliner, 50, E-46100 Burjassot (Val`encia), Spain. e-mail: Maria.Lledo@ific.uv.es and V. S. Varadarajan Department of Mathematics, UCLA. Los Angeles, CA, 90095-1555, USA e-mail: [email protected] Abstract We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it arXiv:math/0609813v2 [math.RA] 14 Jun 2007 as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra. 1Investigation supported by the University of Bologna, funds for selected research top- ics. 1 Contents 1 Introduction 3 2 The complex Minkowski space time and the Grassmannian G(2,4) 6 2.1 The Pl¨ucker embedding and the Klein quadric . 7 2.2 Therealform ........................... 10 3 Homogeneous spaces for Lie supergroups 11 3.1 The supermanifold structure on X = G/H ........... 12 3.2 The action of G on X ...................... 15 3.3 The functor of points of X = G/H ............... 16 4 The super Poincar´eand the translation superalgebras 17 4.1 Thetranslationsuperalgebra.
    [Show full text]
  • Lectures on Supersymmetry and Superstrings 1 General Superalgebra
    Lectures on supersymmetry and superstrings Thomas Mohaupt Abstract: These are informal notes on some mathematical aspects of su- persymmetry, based on two ‘Mathematics and Theoretical Physics Meetings’ in the Autumn Term 2009.1 First super vector spaces, Lie superalgebras and su- percommutative associative superalgebras are briefly introduced together with superfunctions and superspaces. After a review of the Poincar´eLie algebra we discuss Poincar´eLie superalgebras and introduce Minkowski superspaces. As a minimal example of a supersymmetric model we discuss the free scalar superfield in two-dimensional N = (1, 1) supersymmetry (this is more or less copied from [1]). We briefly discuss the concept of chiral supersymmetry in two dimensions. None of the material is original. We give references for further reading. Some ‘bonus material’ on (super-)conformal field theories, (super-)string theories and symmetric spaces is included. These are topics which will play a role in future meetings, or which came up briefly during discussions. 1 General superalgebra Definition: A super vector space V is a vector space with a decomposition V = V V 0 ⊕ 1 and a map (parity map) ˜ : (V V ) 0 Z · 0 ∪ 1 −{ } → 2 which assigns even or odd parity to every homogeneous element. Notation: a V a˜ =0 , ‘even’ , b V ˜b = 1 ‘odd’ . ∈ 0 → ∈ 1 → Definition: A Lie superalgebra g (also called super Lie algebra) is a super vector space g = g g 0 ⊕ 1 together with a bracket respecting the grading: [gi , gj] gi j ⊂ + (addition of index mod 2), which is required to be Z2 graded anti-symmetric: ˜ [a,b]= ( 1)a˜b[b,a] − − (multiplication of parity mod 2), and to satisfy a Z2 graded Jacobi identity: ˜ [a, [b,c]] = [[a,b],c] + ( 1)a˜b[b, [a,c]] .
    [Show full text]
  • Relative Quantum Field Theory 3
    RELATIVE QUANTUM FIELD THEORY DANIEL S. FREED AND CONSTANTIN TELEMAN Abstract. We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions. 1. Introduction The (0, 2)-superconformal field theory in six dimensions, which we term Theory X for brevity, was discovered as a limit of superstring theories [W1, S]. It is thought not to have a lagrangian description, so is difficult to access directly, yet some expectations can be deduced from the string theory description [W2, GMN]. Two features are particularly relevant: (i) it is not an ordinary quantum field theory, and (ii) the theory depends on a Lie algebra, not on a Lie group. A puzzle, emphasized by Greg Moore, is that the dimensional reduction of Theory X to five dimensions is usually understood to be an ordinary quantum field theory—contrary to (i)—and it is a supersym- metric gauge theory so depends on a particular choice of Lie group—contrary to (ii). In this paper we spell out the modified notion indicated in (i), which we call a relative quantum field theory, and use it to resolve this puzzle about Theory X by pointing out that the dimensional reduction is also a relative theory. Relative gauge theories are not particular to dimension five. In fact, the possibility of studying four-dimensional gauge theory as a relative theory was exploited in [VW] and [W3].
    [Show full text]
  • Breakthrough Prize in Mathematics Awarded to Terry Tao
    156 Breakthrough Prize in Mathematics awarded to Terry Tao Our congratulations to Australian Mathematician Terry Tao, one of five inaugu- ral winners of the Breakthrough Prizes in Mathematics, announced on 23 June. Terry is based at the University of California, Los Angeles, and has been awarded the prize for numerous breakthrough contributions to harmonic analysis, combi- natorics, partial differential equations and analytic number theory. The Breakthrough Prize in Mathematics was launched by Mark Zuckerberg and Yuri Milner at the Breakthrough Prize ceremony in December 2013. The other winners are Simon Donaldson, Maxim Kontsevich, Jacob Lurie and Richard Tay- lor. All five recipients of the Prize have agreed to serve on the Selection Committee, responsible for choosing subsequent winners from a pool of candidates nominated in an online process which is open to the public. From 2015 onwards, one Break- through Prize in Mathematics will be awarded every year. The Breakthrough Prizes honor important, primarily recent, achievements in the categories of Fundamental Physics, Life Sciences and Mathematics. The prizes were founded by Sergey Brin and Anne Wojcicki, Mark Zuckerberg and Priscilla Chan, and Yuri and Julia Milner, and aim to celebrate scientists and generate excitement about the pursuit of science as a career. Laureates will receive their trophies and $3 million each in prize money at a televised award ceremony in November, designed to celebrate their achievements and inspire the next genera- tion of scientists. As part of the ceremony schedule, they also engage in a program of lectures and discussions. See https://breakthroughprize.org and http://www.nytimes.com/2014/06/23/us/ the-multimillion-dollar-minds-of-5-mathematical-masters.html? r=1 for further in- formation..
    [Show full text]
  • NEWSLETTER No
    NEWSLETTER No. 458 May 2016 NEXT DIRECTOR OF THE ISAAC NEWTON INSTITUTE In October 2016 David Abrahams will succeed John Toland as Director of the Isaac Newton Institute for Mathematical Sciences and NM Rothschild and Sons Professor of Mathematics in Cambridge. David, who is a Royal Society Wolfson Research Merit Award holder, has been Beyer Professor of Applied Mathematics at the University of Man- chester since 1998. From 2014-16 he was Scientific Director of the International Centre for Math- ematical Sciences in Edinburgh and was President of the Institute of Mathematics and its Applica- tions from 2007-2009. David’s research has been in the broad area of applied mathematics, mainly focused on the theoretical understanding of wave processes including scattering, diffraction, localisation and homogenisation. In recent years his research has broadened somewhat, to now cover topics as diverse as mathematical finance, nonlinear vis- He has also been involved in a range of public coelasticity and glaciology. He has close links with engagement activities over the years. He a number of industrial partners. regularly offers mathematics talks of interest David plays an active role within the internation- to school students and the general public, al mathematics community, having served on over and ran the annual Meet the Mathematicians 30 national and international working parties, outreach events for sixth form students with panels and committees over the past decade. Chris Howls (Southampton). With Chris Budd This has included as a Member of the Applied (Bath) he has organised a training conference Mathematics sub-panel for the 2008 Research As- in 2010 on How to Talk Maths in Public, and in sessment Exercise and Deputy Chair for the Math- 2014 co-chaired the inaugural Festival of Math- ematics sub-panel in the 2014 Research Excellence ematics and its Applications.
    [Show full text]
  • M-Theory from the Superpoint
    M-Theory from the Superpoint John Huerta∗, Urs Schreibery March 21, 2017 Abstract The \brane scan" classifies consistent Green{Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The \brane bouquet" generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5- brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive maximal invariant central extensions, and show that it discovers the super- Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet. Contents 1 Introduction 2 2 Automorphisms of super-Minkowski spacetimes9 3 The consecutive maximal invariant central extensions of the superpoint 12 4 Outlook 17 A Background 17 A.1 Super Lie algebra cohomology............................... 17 A.2 Super Minkowski spacetimes................................ 20 ∗CAMGSD, Instituto Superior T´ecnico,Av. Ravisco Pais, 1049-001 Lisboa, Portugal yMathematics Institute of the Academy, Zitnaˇ 25, 115 67 Praha 1, Czech Republic, on leave at Bonn 1 1 Introduction In his \vision talk" at the annual string theory conference in 2014, Greg Moore highlighted the following open question in string theory [41, section 9]: Perhaps we need to understand the nature of time itself better.
    [Show full text]
  • On The\'Etale Homotopy Type of Higher Stacks
    ON THE ETALE´ HOMOTOPY TYPE OF HIGHER STACKS DAVID CARCHEDI Abstract. A new approach to ´etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the ´etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if X is an arbitrary higher stack on the ´etale site of affine schemes of finite type over C, then the ´etale homotopy type of X agrees with the homotopy type of the underlying stack Xtop on the topological site, after profinite completion. In particular, if X is an Artin stack locally of finite type over C, our definition of the ´etale homotopy type of X agrees up to profinite completion with the homotopy type of the underlying topological stack Xtop of X in the sense of Noohi [30]. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of -categories which we believe to be of independent interest. ∞ 1. Introduction Given a complex variety V, one can associate topological invariants to this variety by computing invariants of its underlying topological space Van, equipped with the complex analytic topology. However, for a variety over an arbitrary base ring, there is no good notion of underlying topological space which plays the same role.
    [Show full text]