DOCSLIB.ORG

Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories by Artan Sheshmani*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories by Artan Sheshmani* Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories by Artan Sheshmani* Abstract. This article provides the summary of 1. Introduction [GSY17a] and [GSY17b] where the authors studied the In recent years, there has been extensive math- enumerative geometry of “nested Hilbert schemes” ematical progress in enumerative geometry of sur- of points and curves on algebraic surfaces and faces deeply related to physical structures, e.g. their connections to threefold theories, and in around Gopakumar-Vafa invariants (GV); Gromov- particular relevant Donaldson-Thomas, Vafa-Witten Witten (GW), Donaldson-Thomas (DT), as well as and Seiberg-Witten theories. Pandharipande-Thomas (PT) invariants of surfaces; and their “motivic lifts”. There is also a tremen- dous energy in the study of mirror symmetry of sur- Contents faces from the mathematics side, especially Homo- logical Mirror Symmetry. On the other hand, phys- 1 Introduction . 25 ical dualities in Gauge and String theory, such as Acknowledgment . 26 Montonen-Olive duality and heterotic/Type II du- 2 Nested Hilbert Schemes on Surfaces . 26 ality have also been a rich source of spectacular 2.1 Special Cases . 27 predictions about enumerative geometry of moduli 2.2 Nested Hilbert Scheme of Points . 27 spaces on surfaces. For instance an extensive re- 3 Nested Hilbert Schemes and DT Theory of search activity carried out during the past years was Local Surfaces . 28 to prove the modularity properties of GW or DT in- References . 30 variants as suggested by the heterotic/Type II dual- ity. The first prediction of this type, the Yau-Zaslow conjecture [YZ96], was proven by Klemm-Maulik- Pandharipande-Scheidegger [KMPS10]. Further recent developments in this particularly fruitful direction * Center for Mathematical Sciences and Applications, Har- include: Pandharipande-Thomas proof [PT16] of the vard University, Department of Mathematics, 20 Garden Street, Room 207, Cambridge, MA, 02139 Katz-Klemm-Vafa conjecture [KKV99] for K3 sur- Centre for Quantum Geometry of Moduli Spaces, Aarhus faces, Maulik-Pandharipande proof of modularity of University, Department of Mathematics, Ny Munkegade 118, GW invariants for K3 fibered threefolds [MP13], building 1530, 319, 8000 Aarhus C, Denmark Gholampour-Sheshmani-Toda proof of modularity of National Research University Higher School of Economics, PT and Gholampour-Sheshmani proof of modular- Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow, Russia, 119048 ity of DT invariants of stable sheaves on K3 fibra- E-mail: [email protected] tions [GST17, GS18] (moreover, the generalizations DECEMBER 2019 NOTICES OF THE ICCM 25 of the latter in [BCDDQS16]), and finally Gholampur- Their main exciting realization was when L = KS, the Sheshmani-Thomas [GS14] proof of modular prop- canonical bundle of S. In this case the above DT in- erty of the counting of curves on surfaces deforming variants of X recover some well known invariants freely (in a fixed linear system) in ambient CY three- in physics, the Vafa-Witten (VW) invariants [VW94], folds. which are known to have modularity property, fol- SU(2)-Seiberg-Witten (SW) / DT correspondence. lowing the work of Vafa-Witten [VW94] (and verified The introduction of other versions of gauge theo- by Tanaka-Thomas [TT17), TT18(2)]). On the other ries in dimensions 6 started numerous exciting de- hand the authors also showed that in some cases the velopments in physical mathematics and mathemat- nested Hilbert scheme invariants also have modular ical physics involving enumerative geometry, mirror property [GSY17a, Theorem 7]. Therefore, we were symmetry, and related physics of 4 dimensional man- able to show that in some cases, the SW invariants ifolds, realized as a complex two dimensional sub- can be described in terms of modular forms. variety which could exhibit deformations inside am- bient higher dimensional target varieties, such as CY threefolds. The SW/DT correspondence is one of Acknowledgment such platforms to find interesting structural sym- The author was partially supported by NSF metries between theory of surfaces and theory of DMS-1607871, NSF DMS-1306313 and Laboratory of threefolds: having fixed a spin structure on a com- Mirror Symmetry NRU HSE, RF Government grant, ag. plex surface S the SW invariant of S roughly counts No 14.641.31.0001. The author would like to further with sign the number of points in the parametriz- sincerely thank the Center for Mathematical Sciences ing space (the moduli space) of solutions to SW and Applications at Harvard University, the center equations defined on S [D96, Section 1]. The fo- for Quantum Geometry of Moduli Spaces at Aarhus cus of SW theory in physics is the study of mod- University, and the Laboratory of Mirror Symmetry in uli space of vacua in N = 2, D = 4 super Yang-Mills Higher School of Economics, Russian federation, for theory, and in particular certain dualities of the the great help and support. theory such as electric-magnetic duality (Montonen- Olive duality). In [GLSY17] Gukov-Liu-Sheshmani-Yau conjectured a relation between SW invariants (asso- 2. Nested Hilbert Schemes on ciate to SU(2) gauge invariant theory) of a projec- Surfaces tive surface S and DT invariants of a noncompact Hilbert schemes of points and curves on a non- threefold X obtained by total space of a line bun- singular surface S have been vastly studied. Their dle L on S. In a later work [GSY17a, GSY17b], which rich geometric structures have proved to have many will be elaborated below, Gholampour-Sheshmani- applications in mathematics and physics (see [N99] Yau proved the conjecture in [GLSY17, Equation for a survey). The current article is a survey of the 56]. articles [GSY17a] and [GSY17b], where the authors The key object which bridges the geometry of studied the enumerative geometry of “nested Hilbert complex surface to the ambient noncompact com- schemes” of points and curves on algebraic surfaces. plex threefold is the “nested Hilbert scheme of sur- Given the sequence face”. This space parametrizes a nested chain of configurations of curves and points in the surface. n := n1;n2;:::;nr;r ≥ 1 Let us assume that S is a projective simply con- nected complex surface and let L be a line bundle of nonnegative integers, and β := β1;:::;βr−1, a se- on S. In [GSY17a], Gholampour-Sheshmani-Yau an- 2 quence of classes in H (S;Z) such that βi ≥ 0, we de- alyzed the moduli space of stable compactly sup- note the corresponding “nested” Hilbert scheme by ported sheaves of modules on the noncompact three- [n] [n] Sβ . A closed point of Sβ corresponds to fold X. In physics terms these are D4-D2-D0 branes wrapping the zero section of X (i.e. sheaves are sup- (Z1;Z2;:::;Zr); (C1;:::;Cr−1) ported on S or a fat neighborhood of S in X). The au- thors showed in [GSY17a] that the DT invariants of where Zi ⊂ S is a 0-dimensional subscheme of length X satisfy an equation in terms of SW invariants of S ni, and Ci ⊂ S is a divisor with [Ci] = βi, and Zi+1 is a and certain correction terms governed by invariants subscheme of Zi [Ci for any i < r, or equivalently of nested Hilbert schemes: (1) IZi (−Ci) ⊂ IZi+1 : DT invariants of X = (SW invariants of S) · (Combinatorial coefficients) + Invariants of nested In [GSY17a] and [GSY17b], in order to define in- Hilbert scheme of S variants for the nested Hilbert schemes (see [GSY17a, 26 NOTICES OF THE ICCM VOLUME 7,NUMBER 2 [n] Definitions 2.13, 2.14]), the authors constructed a vir- where ωS is the rank n tautological vector bundle [n] vir [n] 1 tual fundamental class [Sβ ] and then integrate ap- over S associated to the canonical bundle ωS of S. propriate cohomology classes against it. More pre- 3. If β = 0 and n = n2 = n1 − 1, then it is known that [n+1;n] ∼ [n] [n] cisely, they constructed a natural perfect obstruction Sβ=0 = P(I ) is nonsingular, where I is the uni- [n] [n] theory over Sβ . This is done by studying the defor- versal ideal sheaf over S × S [L99, Section 1.2]. mation/obstruction theory of the maps of coherent Then, sheaves given by the natural inclusions (1) following [S[n+1;n]]vir = −[S[n+1;n]] \ c (O (1) ): Illusie. It turned out that this construction in partic- β=0 β=0 1 P ωS ular provides a uniform way of studying all known 4. If n = n = 0 and β 6= 0, then S[0;0] is the Hilbert obstruction theories of the Hilbert schemes of points 1 2 β [0;0] vir and curves, as well as the stable pair moduli spaces scheme of divisors in class β, and [Sβ ] coincides on S. The first main result of [GSY17a] is as fol- with virtual cycle used to define Poincaré invariants lows (see also [GSY17a, Proposition 2.5 and Corollary in [DKO07]. [0;n2] 2.6]): 5. If n1 = 0 and β 6= 0, then Sβ is the relative Hilbert scheme of points on the universal divisor over S[0;0], Theorem 1 ([GSY17a, Theorem 1]). Let S be a nonsin- β which as shown in [PT10], is a moduli space of sta- gular projective surface over C and ωS be its canonical [0;n ] [n] ble pairs and [S 2 ]vir is the same as the virtual fun- bundle. The nested Hilbert scheme S with r ≥ 2 car- β β damental class constructed in [KT14] in the context ries a natural virtual fundamental class of stable pair theory. If Pg(S) = 0 this class was used r−1 in [KT14] to define stable pair invariants. [n] vir [n] 1 [Sβ ] 2 Ad(Sβ ); d = n1 + nr + ∑ βi · (βi − c1(ωS)): 2 i=1 In certain cases, the authors constructed a re- [n1;n2] duced virtual fundamental class for Sβ by reducing the perfect obstruction theory leading to Theorem 1 2.1 Special Cases ([GSY17a, Propositions 2.10, 2.12]): In the simplest special case, i.e.
Load more

Recommended publications
  • Introduction
    Proceedings of Symposia in Pure Mathematics Volume 81, 2010 Introduction Jonathan Rosenberg Abstract. The papers in this volume are the outgrowth of an NSF-CBMS Regional Conference in the Mathematical Sciences, May 18–22, 2009, orga- nized by Robert Doran and Greg Friedman at Texas Christian University. This introduction explains the scientific rationale for the conference and some of the common themes in the papers. During the week of May 18–22, 2009, Robert Doran and Greg Friedman orga- nized a wonderfully successful NSF-CBMS Regional Research Conference at Texas Christian University. I was the primary lecturer, and my lectures have now been published in [29]. However, Doran and Friedman also invited many other mathe- maticians and physicists to speak on topics related to my lectures. The papers in this volume are the outgrowth of their talks. The subject of my lectures, and the general theme of the conference, was highly interdisciplinary, and had to do with the confluence of superstring theory, algebraic topology, and C∗-algebras. While with “20/20 hindsight” it seems clear that these subjects fit together in a natural way, the connections between them developed almost by accident. Part of the history of these connections is explained in the introductions to [11] and [17]. The authors of [11] begin as follows: Until recently the interplay between physics and mathematics fol- lowed a familiar pattern: physics provides problems and mathe- matics provides solutions to these problems. Of course at times this relationship has led to the development of new mathematics. But physicists did not traditionally attack problems of pure mathematics.
    [Show full text]
  • Heterotic String Compactification with a View Towards Cosmology
    Heterotic String Compactification with a View Towards Cosmology by Jørgen Olsen Lye Thesis for the degree Master of Science (Master i fysikk) Department of Physics Faculty of Mathematics and Natural Sciences University of Oslo May 2014 Abstract The goal is to look at what constraints there are for the internal manifold in phe- nomenologically viable Heterotic string compactification. Basic string theory, cosmology, and string compactification is sketched. I go through the require- ments imposed on the internal manifold in Heterotic string compactification when assuming vanishing 3-form flux, no warping, and maximally symmetric 4-dimensional spacetime with unbroken N = 1 supersymmetry. I review the current state of affairs in Heterotic moduli stabilisation and discuss merging cosmology and particle physics in this setup. In particular I ask what additional requirements this leads to for the internal manifold. I conclude that realistic manifolds on which to compactify in this setup are severely constrained. An extensive mathematics appendix is provided in an attempt to make the thesis more self-contained. Acknowledgements I would like to start by thanking my supervier Øyvind Grøn for condoning my hubris and for giving me free rein to delve into string theory as I saw fit. It has lead to a period of intense study and immense pleasure. Next up is my brother Kjetil, who has always been a good friend and who has been constantly looking out for me. It is a source of comfort knowing that I can always turn to him for help. Mentioning friends in such an acknowledgement is nearly mandatory. At least they try to give me that impression.
    [Show full text]
  • Arxiv:1404.2323V1 [Math.AG] 8 Apr 2014 Membranes and Sheaves
    Membranes and Sheaves Nikita Nekrasov and Andrei Okounkov April 2014 Contents 1 A brief introduction 2 1.1 Overview.............................. 2 1.2 MotivationfromM-theory . 3 1.3 Planofthepaper ......................... 7 1.4 Acknowledgements ........................ 7 2 Contours of the conjectures 8 2.1 K-theorypreliminaries ...................... 8 2.2 Theindexsheaf.......................... 10 2.3 Comparison with Donaldson-Thomastheory . 15 2.4 Fields of 11-dimensional supergravity and degree zero DT counts 19 3 The DT integrand 24 3.1 Themodifiedvirtualstructuresheaf. 24 3.2 The interaction term Φ ...................... 28 arXiv:1404.2323v1 [math.AG] 8 Apr 2014 4 The index of membranes 33 4.1 Membranemoduli......................... 33 4.2 Deformationsofmembranes . 36 5 Examples 37 5.1 Reducedlocalcurves ....................... 37 5.2 Doublecurves........................... 39 5.3 Single interaction between smooth curves . 42 5.4 HigherrankDTcounts. .. .. 43 5.5 EngineeringhigherrankDTtheory . 45 1 6 Existence of square roots 49 6.1 Symmetricbundlesonsquares . 49 6.2 SquarerootsinDTtheory . 50 6.3 SquarerootsinM-theory. 52 7 Refined invariants 54 7.1 Actionsscalingthe3-form . 54 7.2 Localization for κ-trivialtori................... 56 7.3 Morsetheoryandrigidity . 60 8 Index vertex and refined vertex 61 8.1 Toric Calabi-Yau 3-folds . 61 8.2 Virtualtangentspacesatfixedpoints . 63 8.3 Therefinedvertex ........................ 66 A Appendix 70 A.1 Proofofthebalancelemma . 70 1 A brief introduction 1.1 Overview Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×- actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem.
    [Show full text]
  • Contemporary Mathematics 310
    CONTEMPORARY MATHEMATICS 310 Orbifolds in Mathematics and Physics Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory May 4-8~ 2001 University of Wisconsin I Madison I Wisconsin Alejandro Adem Jack Morava Yongbin Ruan Editors http://dx.doi.org/10.1090/conm/310 Orbifolds in Mathematics and Physics CoNTEMPORARY MATHEMATICS 310 Orbifolds in Mathematics and Physics Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory May 4-8, 2001 University of Wisconsin, Madison, Wisconsin Alejandro Adem Jack Morava Yongbin Ruan Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius 2000 Mathematics Subject Classification. Primary 81 T30, 55N91, 17B69, 19M05. Library of Congress Cataloging-in-Publication Data Orbifolds in mathematics and physics : proceedings of a conference on mathematical aspects of orbifold string theory, May 4-8, 2001, University of Wisconsin, Madison, Wisconsin / Alejandro Adem, Jack Morava, Yongbin Ruan, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 310) Includes bibliographical references. ISBN 0-8218-2990-4 (alk. paper) 1. Orbifolds-Congresses. 2. Mathematical physics-Congresses. I. Adem, Alejandro. II. Morava, Jack, 1944- III. Ruan, Yongbin, 1963- IV. Contemporary mathematics (Ameri- can Mathematical Society) : v. 310. QA613 .0735 2002 539. 7'2--dc21 2002034264 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale.
    [Show full text]
  • Arxiv:1512.05388V2 [Hep-Th] 2 Jan 2016 1
    BPS/CFT CORRESPONDENCE: NON-PERTURBATIVE DYSON-SCHWINGER EQUATIONS AND qq-CHARACTERS NIKITA NEKRASOV Abstract. We study symmetries of quantum field theories involving topologically dis- tinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the qq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger- type relations. These relations imply that the supersymmetric partition functions in the presence of Ω-deformation and defects obey the Ward identities of two dimen- sional conformal field theory and its q-deformations. The details will be discussed in the companion papers. Contents 1. Introduction 4 1.1. Dyson-Schwinger equations 4 1.2. Non-perturbative Dyson-Schwinger identities 6 1.3. Organization of the presentation 8 1.4. Acknowledgements 10 2. The BPS/CFT correspondence 12 2.1. = 2 partition functions 12 2.2. DefectN operators and lower-dimensional theories 13 2.3. The Y- and X-observables 14 2.4. The physics of X-observables 14 arXiv:1512.05388v2 [hep-th] 2 Jan 2016 2.5. Hidden symmetries 17 2.6. Some notations. 17 2.7. Equivariant virtual Chern polynomials 22 3. Supersymmetric gauge theories 22 3.1. Quivers 22 3.2. Quivers with colors 22 3.3. The symmetry groups 22 3.4. The parameters of Lagrangian 24 3.5. The group H 25 3.6. Perturbative theory 26 3.7. Realizations of quiver theories 28 4. Integration over instanton moduli spaces 30 1 2 NIKITA NEKRASOV 4.1. Instanton partition function 30 4.2.
    [Show full text]
  • VITA Sheldon Katz Urbana, IL 61801 Phone
    VITA Sheldon Katz Urbana, IL 61801 Phone: (217) 265-6258 Education S.B. 1976 Massachusetts Institute of Technology (Mathematics) Ph.D. 1980 Princeton University (Mathematics) Professional Experience University of Illinois Professor 2001{ Dean's Special Advisor, College of LAS 2018{ Interim Chair, Department of Mathematics Fall 2017 Chair, Department of Mathematics 2006{2011 Oklahoma State University Regents Professor 1999{2002 Southwestern Bell Professor 1997{1999 Professor 1994{2002 Associate Professor 1989{1994 Assistant Professor 1987{1989 University of Oklahoma Assistant Professor (tenure 1987) 1984{1987 University of Utah Instructor 1980{1984 Visiting Positions MSRI Simons Visiting Professor 2018 Simons Center 2012 Mittag-Leffler Institute, Sweden Visiting Professor 1997 Duke University Visiting Professor 1991{1992 University of Bayreuth, West Germany Visiting Professor 1989 Institute for Advanced Study Member 1982{1983 Publications 1. Degenerations of quintic threefolds and their lines. Duke Math. Jour. 50 (1983), 1127{1135. 2. Lines on complete intersection threefolds with K=0. Math. Z. 191 (1986), 297{302. 3. Tangents to a multiple plane curve. Pac. Jour. Math. 124 (1986), 321{331. 4. On the finiteness of rational curves on quintic threefolds. Comp. Math. 60 (1986), 151{162. 5. Hodge numbers of linked surfaces in P4. Duke Math. Jour. 55 (1987), 89{95. 6. The cubo-cubic transformation of P3 is very special. Math. Z. 195 (1987), 255{257. 7. Iteration of Multiple point formulas and applications to conics. Proceed- ings of the Algebraic Geometry Conference, Sundance, UT 1986, SLN 1311, 147{155. 8. Varieties cut out by quadrics: Scheme theoretic versus homogeneous gen- eration of ideals (with L.
    [Show full text]
  • Mirror Symmetry Is a Phenomenon Arising in String Theory in Which Two Very Different Manifolds Give Rise to Equivalent Physics
    Clay Mathematics Monographs 1 Volume 1 Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has Mirror Symmetry Mirror significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differ- ential equations obtained from a “mirror” geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar Vafa invariants. This book aims to give a single, cohesive treatment of mirror symmetry from both the mathematical and physical viewpoint. Parts 1 and 2 develop the neces- sary mathematical and physical background “from scratch,” and are intended for readers trying to learn across disciplines. The treatment is focussed, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a “pairing” of geometries. The proof involves applying R ↔ 1/R circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic MIRROR SYMMETRY setting, beginning with the moduli spaces of curves and maps, and uses localiza- tion techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry.
    [Show full text]
  • Is String Theory Holographic? 1 Introduction
    Holography and large-N Dualities Is String Theory Holographic? Lukas Hahn 1 Introduction1 2 Classical Strings and Black Holes2 3 The Strominger-Vafa Construction3 3.1 AdS/CFT for the D1/D5 System......................3 3.2 The Instanton Moduli Space.........................6 3.3 The Elliptic Genus.............................. 10 1 Introduction The holographic principle [1] is based on the idea that there is a limit on information content of spacetime regions. For a given volume V bounded by an area A, the state of maximal entropy corresponds to the largest black hole that can fit inside V . This entropy bound is specified by the Bekenstein-Hawking entropy A S ≤ S = (1.1) BH 4G and the goings-on in the relevant spacetime region are encoded on "holographic screens". The aim of these notes is to discuss one of the many aspects of the question in the title, namely: "Is this feature of the holographic principle realized in string theory (and if so, how)?". In order to adress this question we start with an heuristic account of how string like objects are related to black holes and how to compare their entropies. This second section is exclusively based on [2] and will lead to a key insight, the need to consider BPS states, which allows for a more precise treatment. The most fully understood example is 1 a bound state of D-branes that appeared in the original article on the topic [3]. The third section is an attempt to review this construction from a point of view that highlights the role of AdS/CFT [4,5].
    [Show full text]
  • September 2014
    LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 439 September 2014 Society Meetings HIGHEST HONOUR FOR UK and Events MATHEMATICAN Professor Martin Hairer, FRS, 2014 University of Warwick, has become the ninth UK based Saturday mathematician to win the 6 September prestigious Fields Medal over Mathematics and the its 80 year history. The medal First World War recipients were announced Meeting, London on Wednesday 13 August in page 15 a ceremony at the four-year- ly International Congress for 1 Wednesday Mathematicians, which on this 24 September occasion was held in Seoul, South Korea. LMS Popular Lectures See page 4 for the full report. Birmingham page 12 Friday LMS ANNOUNCES SIMON TAVARÉ 14 November AS PRESIDENT-DESIGNATE LMS AGM © The University of Cambridge take over from the London current President, Professor Terry Wednesday Lyons, FRS, in 17 December November 2015. SW & South Wales Professor Tavaré is Meeting a versatile math- Plymouth ematician who has established a distinguished in- ternational career culminating in his current role as The London Mathematical Director of the Cancer Research Society is pleased to announce UK Cambridge Institute and Professor Simon Tavaré, Professor in DAMTP, where he NEWSLETTER FRS, FMedSci, University of brings his understanding of sto- ONLINE: Cambridge, as President-Des- chastic processes and expertise newsletter.lms.ac.uk ignate. Professor Tavaré will in the data science of DNA se- (Cont'd on page 3) LMS NEWSLETTER http://newsletter.lms.ac.uk Contents No. 439 September 2014 15 44 Awards Partial Differential Equations..........................37 Collingwood Memorial Prize..........................11 Valediction to Jeremy Gray..............................33 Calendar of Events.......................................50 News LMS Items European News.................................................16 HEA STEM Strategic Project...........................
    [Show full text]
  • Download Enumerative Geometry and String Theory Free Ebook
    ENUMERATIVE GEOMETRY AND STRING THEORY DOWNLOAD FREE BOOK Sheldon Katz | 206 pages | 31 May 2006 | American Mathematical Society | 9780821836873 | English | Providence, United States Enumerative Geometry and String Theory Chapter 2. The most accessible portal into very exciting recent material. Topological field theory, primitive forms and related topics : — Nuclear Physics B. Bibcode : PThPh. Bibcode : hep. As an example, consider the torus described above. Enumerative Geometry and String Theory standard analogy for this is to consider a multidimensional object such as a garden hose. An example is the red circle in the figure. This problem was solved by the nineteenth-century German mathematician Hermann Schubertwho found that there are Enumerative Geometry and String Theory 2, such lines. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology. Enumerative Geometry and String Theory mirror symmetry relationship is a particular example of what physicists call a duality. Print Price 1: The book contains a lot of extra material that was not included Enumerative Geometry and String Theory the original fifteen lectures. Increasing the dimension from two to four real dimensions, the Calabi—Yau becomes a K3 surface. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. Topological Quantum Field Theory. Online Price 1: Online ISBN There are infinitely many circles like it on a torus; in fact, the entire surface is a union of such circles. For other uses, see Mirror symmetry. As an example, count the conic sections tangent to five given lines in the projective plane.
    [Show full text]
  • Model Building and SUSY Breaking on D-Branes
    Model building and SUSY breaking on D-branes Dmitry Malyshev a dissertation presented to the faculty of princeton university in candidacy for the degree of doctor of philosophy recommended for acceptance by the department of physics Adviser: Herman Verlinde September 2008 c Copyright by Dmitry Malyshev, 2008. All rights reserved. Abstract In the recent years there has been an increase of the interest in applying the String theory to construct viable extensions of the Standard Model and to find stringy analogs of the known field theory models for supersymmetry breaking. In my dissertation I will focus on the constructions involving the D-branes at the Calabi-Yau singularities in type IIB string theory. The main motivations for this choice are: the decoupling of the D-brane field theory from the supergravity, the well established tools for deriving the field theories from the configurations of the D-branes, and the possibility of using the gauge-gravity correspondence in solving some non-perturbative aspects in the field theory. A construction of an extended supersymmetric Standard Model on a D3-brane at the del Pezzo 8 singularity is presented in the first part of the dissertation. In the second part we discuss the possible representations of the SUSY breaking models in String Theory and obtain the metastable SUSY breaking vacuum in a system of D-branes on the suspended pinch point singularity. The gauge mediation of this SUSY breaking model is described in details. In the simplest model the spectrum of particles in the visible sector has a split SUSY breaking, the sfermions are one order of magnitude heavier than the gauginos.
    [Show full text]
  • Nikita Nekrasov 講義録
    Nikita Nekrasov 講義録 December, 1998 講義録作成1 ：橋本幸士, 岸本 功 （京大理） 目次 第 I 部 超共形ゲージ理論と AdS 超重力理論 2 1 Introduction と motivation 3 2 Orbifold 理論 6 3 時空の状況 12 4 Orbifold から conifold への変形 17 5 まとめ 19 第 II 部 N =2ゲージ理論における状態のカウント 21 6 導入 22 7 D3-プローブ上の理論 22 8M理論への持ち上げ/引き落とし 27 9 結論 39 1この講義録は「Workshop ”ゲージ理論の力学と弦双対性” 1998 年 12 月 16 日（水）–12月 18 日（金）東京工業大学国際交流会館」における Nekrasov 氏の講義に基づくものです。第 I 部が 16 日に、第 II 部が 18 日に行われました。 1 第 I 部 超共形ゲージ理論とAdS超重力理論 (SUPER)CONFORMAL GAUGE THEORIES AND ANTI-DE-SITTER SUPERGRAVITY2 Nikita Nekrasov 1. Introduction と motivation ——哲学, D-brane と付随した場の理論, 特異な幾何からの conformal 理論 2. Orbifold 理論 ——理論の構築（：gauge 群, matter の内容, （超）対称性）, β 関数の計算, 摂動論の解析 3. 時空の状況 ——near-horizon limit, 超重力における変形 4. Orbifold から conifold への変形 ——場の理論の実現, Higgs branch の幾何, 時空解釈（conjecture） 2文献 [1, 2] に基づく。[3, 4, 5, 6, 7, 8] も見よ。 2 第 1 章 Introduction と motivation 双対性に関する動きの中でおそらく最も（そしておそらく唯一つの）重要な教訓は、面白い物理は面白い幾何に関係 しているという発想に立ち返るということである。Einstein の時代に比べての違いは、勿論量子的な物理量が古典的な 幾何を通じて表現されうるという仮定である。例えば、N =2SU(2) gauge 理論の有効結合定数（と長距離相関関数） は補佐的な楕円曲線 Eu の modular 変数 τ(u) を用いて表される： y2 =(x − u)(x2 − Λ4). (1) ここで u = hTrφ2i は真空の moduli 空間を label する。 u-plane θ(u) 4πi Eu: τ(u)= 2π + e2(u) vacuum 図 1: N =2SU(2) gauge 理論の moduli 空間。 物理から幾何、また逆に幾何から物理への変換を理解するのに必要不可欠な道具は D-brane である。D-brane に関し ての良い点は以下の二つである: (1) D-brane の上に弦が端点を持つので、いくつかの D-brane がお互いに重なるとその worldvolume 上に非可換 gauge boson が発生する。(2) D-brane は Ramond-Ramond 場の電荷を持っており、（適切な条 件の下で）弦理論における soliton として記述されうる。 (1) (2) 図 2: D-brane の二つの描像。(1) 開弦の端点としての D-brane、(2) 弦理論の soliton としての D-brane。 一般的な発想は、gauge 理論とそれに対応する soliton の背景を伝播する弦の理論を等しくするように、D-brane を用 いる、ということである。もし（IIB 型超弦理論で）N 枚の D3-brane が平坦な空間内でお互いに重なっていると、この 重なった brane は二つの記述法を持つ: 1.
    [Show full text]