Calabi-Yau Geometry and Higher Genus Mirror Symmetry
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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. -
Vincent Bouchard – Associate Professor
Vincent Bouchard | Associate Professor Rhodes Scholar (Québec and Magdalen, 2001) Department of Mathematical and Statistical Sciences, University of Alberta 632 CAB, Edmonton, AB, Canada – T6G 2G1 T +1 (780) 492 0099 • B [email protected] Í www.ualberta.ca/ vbouchar Education University of Oxford, Magdalen College Oxford, UK D.Phil. Mathematics 2001–2005 Université de Montréal Montréal, QC B.Sc. Physics, Excellence, Palmarès de la doyenne, average 4.3/4.3 1998–2001 Cégep Saint-Jean-sur-Richelieu Saint-Jean-sur-Richelieu, QC D.E.C. Natural Sciences, Excellence 1996–1998 D.Phil. thesis Title: Toric Geometry and String Theory [arXiv:hep-th/0609123] Supervisor: Rouse Ball Professor Philip Candelas Positions Associate Professor Edmonton, AB Department of Mathematical and Statistical Sciences, University of Alberta 2013–Present Assistant Professor Edmonton, AB Department of Mathematical and Statistical Sciences, University of Alberta 2009–2013 Postdoctoral Fellow Cambridge, MA Physics Department, Harvard University 2007–2009 NSERC Postdoctoral Fellow Waterloo, ON Perimeter Institute for Theoretical Physics 2006–2007 Postdoctoral Fellow Berkeley, CA Mathematical Sciences Research Institute 2006 “New Topological Structures in Physics” program, January–May Postdoctoral Fellow Phildelphia, PA Department of Mathematics, University of Pennsylvania 2005 Joint math/physics visiting position, September–December Research Grants NSERC Discovery Grant: $225,000 2013–2018 NSERC Discovery Grant: $130,000 2010–2013 University of Alberta Research -
Distinguished University Professor" at the University of Amsterdam Since 2005, Where He Has Held the Chair of Mathematical Physics Since 1992
Biography Robbert Dijkgraaf Robbert Dijkgraaf (born 1960) has been "Distinguished University Professor" at the University of Amsterdam since 2005, where he has held the chair of mathematical physics since 1992. Since 2008 he is President of the Royal Netherlands Academy of Arts and Sciences (KNAW). Robbert Dijkgraaf studied physics and mathematics at Utrecht University. After an interlude studying painting at Amsterdam's Gerrit Rietveld Academy, he gained his PhD cum laude in Utrecht in 1989. His supervisor was the Nobel Prize-winner Gerard 't Hooft. Robbert Dijkgraaf subsequently held positions at Princeton University and at Princeton's Institute for Advanced Study. His current focus is on string theory, quantum gravity, and the interface between mathematics and particle physics. His research was recognized in 2003 with the award of the NWO Spinoza Prize, the highest scientific award in the Netherlands. Robbert Dijkgraaf has been a visiting professor at universities including Harvard, MIT, Berkeley, and Kyoto. He is on the editorial boards of numerous scientific periodicals, and is also the scientific adviser to institutes in Cambridge, Bonn, Stanford, Dublin, and Paris. Since 2009 he is co-chair of the InterAcademy Council, a multinational organization of science academies, together with Professor Lu Yongxiang, President of the Chinese Academy of Sciences. Many of his activities are at the interface between science and society. His column in the NRC Handelsblad newspaper is intended for a broad public and deals with science, the arts, and other matters. A collection has been published in his book Blikwisselingen (Prometheus | Bert Bakker, 2008). Robbert Dijkgraaf is dedicated to bringing about greater public awareness of science, for example through his involvement with popular TV science programmes. -
Wall-Crossing, Free Fermions and Crystal Melting
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Commun. Math. Phys. 301, 517–562 (2011) Communications in Digital Object Identifier (DOI) 10.1007/s00220-010-1153-1 Mathematical Physics Wall-Crossing, Free Fermions and Crystal Melting Piotr Sułkowski1,2, 1 California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected] 2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA Received: 15 December 2009 / Accepted: 10 June 2010 Published online: 26 October 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract: We describe wall-crossing for local, toric Calabi-Yaumanifolds without com- pact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non- commutative Donaldson-Thomas invariants, given by the modulus square of the topolog- ical string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal inter- pretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years. -
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]. -
The Mathematics of Fivebranes 3 Translates in a Deep Quantum Symmetry (S-Duality) of the 4-Dimensional Yang- Mills Theory
Doc. Math.J. DMV 1 Ì ÅØÑ Ó ÚÖÒ× Robbert Dijkgraaf Abstract. Fivebranes are non-perturbative objects in string theory that general- ize two-dimensional conformal field theory and relate such diverse sub- jects as moduli spaces of vector bundles on surfaces, automorphic forms, elliptic genera, the geometry of Calabi-Yau threefolds, and generalized Kac-Moody algebras. 1991 Mathematics Subject Classification: 81T30 Keywords and Phrases: quantum field theory, elliptic genera, automor- phic forms 1 Introduction This joint session of the sections Mathematical Physics and Algebraic Geometry celebrates a historic period of more than two decades of remarkably fruitful inter- actions between physics and mathematics. The ‘unreasonable effectiveness,’ depth and universality of quantum field theory ideas in mathematics continue to amaze, with applications not only to algebraic geometry, but also to topology, global anal- ysis, representation theory, and many more fields. The impact of string theory has been particularly striking, leading to such wonderful developments as mirror sym- arXiv:hep-th/9810157v1 21 Oct 1998 metry, quantum cohomology, Gromov-Witten theory, invariants of three-manifolds and knots, all of which were discussed at length at previous Congresses. Many of these developments find their origin in two-dimensional conformal field theory (CFT) or, in physical terms, in the first-quantized, perturbative for- mulation of string theory. This is essentially the study of sigma models or maps of Riemann surfaces Σ into a space-time manifold X. Through the path-integral over all such maps a CFT determines a partition function Zg on the moduli space Mg of genus g Riemann surfaces. String amplitudes are functions Z(λ), with λ the string coupling constant, that have asymptotic series of the form 2g−2 Z(λ) ∼ λ Zg. -
Robbert Dijkgraaf
1 Robbert Dijkgraaf Publicaties, wetenschappelijk 1 R. Dijkgraaf, E. Verlinde, and H. Verlinde, c = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. 115 (1988). 2 R. Dijkgraaf, E. Verlinde, and H. Verlinde, Conformal Field Theory at c = 1, in Nonperturbative Quantum Field Theory, Proceedings of the 1987 Cargese Summer School, G. 't Hooft et al. eds. (Plenum, New York, 1988) 577-590. 3 R. Dijkgraaf, E. Verlinde, and H. Verlinde, On Moduli Spaces of Conformal Field Theories with c=1, in Perspectives in String Theory, P. Di Vecchia and J.L. Petersen eds. (World Scientific, 1988) 117-137. 4 R. Dijkgraaf and E. Verlinde, Modular Invariance and the Fusion Algebra, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 87. 5 R. Dijkgraaf, Recent Progress in Rational Conformal Field Theory, in Les Houches, Session XLIX, 1988, Fields, Strings and Critical Phenomena, E. Brézin and J. Zinn-Justin eds. (Elsevier 1990), 291-304. 6 R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, Operator Algebra of Orbifold Models, Commun. Math. Phys. 123 (1989) 485. 7 R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393. 8 R. Dijkgraaf, V. Pasquier, and Ph. Roche, Quasi-Quantum Groups Related to Orbifold Models, in the Proceedings of the International Colloquium on Modern Quantum Field Theory, Tata Institute of Fundamental Research, 375-383. 9 R. Dijkgraaf, V. Pasquier, and Ph. Roche, Quasihopf Algebras, Group Cohomology and Orbifold Models, in Annecy 1990, Recent advances in Field theory, Nucl. Phys. Proc. Suppl. 18B (1990) 60-72; in Pavia 1990, Integrable systems and quantum groups, 75-98. -
MEMORIA DEL CENTRO DE CIENCIAS DE BENASQUE Barcelona, 31 De Diciembre De 2005
MEMORIA DEL CENTRO DE CIENCIAS DE BENASQUE Barcelona, 31 de Diciembre de 2005 I – INTRODUCCIÓN En unos pocos años España se ha convertido en uno de los focos mundiales de atracción de actividades veraniegas, principalmente de carácter lúdico y turístico. Hasta este momento se han usado poco las características de nuestro país para convertirlo en un foco de actividades de tipo científico. Hace unas pocas décadas la representación española en el concierto científico internacional era, prácticamente, inexistente. Aún en 1977 sólo el 0.64% de los artículos científicos que aparecían en la base de datos del Science Citation Index (SCI) tenían algún autor español. En el período 1977-1993 los artículos científicos del SCI en los que al menos un autor era español ha aumentado a un ritmo superior al 9% anual y en 1993 un 2.0% de todos los artículos contenidos en el SCI tienen, al menos, un autor español. Se ha avanzado mucho en pocos años, pero este esfuerzo debe continuar. En España era habitual, en 1994, la celebración de conferencias científicas sobre distintas temáticas y con características diversas. Todas estas reuniones, más o menos multitudinarias, se caracterizan por su corta duración (alrededor de una semana) y gran número diario de conferencias, lo cual hace que quede poco tiempo para poder trabajar o discutir con otros científicos. Sin embargo ni en nuestro país ni en el resto de Europa existían, por aquel entonces, verdaderos centros donde, en un ambiente agradable, los científicos se pudieran dedicar a trabajar tranquilamente y a discutir con sus colegas. Un centro de este tipo lleva muchos años funcionando, con gran éxito, en la localidad norteamericana de Aspen, en Colorado. -
Acceptance Speech Robbert Dijkgraaf
Honorary doctorate acceptance speech by Robbert Dijkgraaf Leiden, 8 February 2019 Honourable Rector Magnificus, your Excellencies, esteemed Honorary Supervisor and Dean, ladies and gentlemen, Let me begin by saying how grateful I am to my dear Leiden colleagues for this unique honour, which means more to me than I can express in these few words here. This special moment in time and space ties together at least three important paths in my life. The first of these, of course, is Leiden as the first university in the Netherlands, for centuries a steadfast bastion and symbol of academic freedom and excellence, and the upward spiral that the felicitous combination of these two virtues can bring. It is also an institution with a pleasing sense of the human dimension, and in spirit, 444 years young. In the words of Francis Bacon: Antiquitas saeculi juventus mundi. This oldest knowledge institution grew up during the adolescence of science and has been successful in maintaining its youth. The second is Dutch physics, as a discipline an example in terms of global impact and close collegiality. Like many of my colleagues, I have always felt as if I am wearing the orange strip of the Dutch football team – even though I am playing for a foreign club. But now it will be with the addition of the beautiful blue Leiden hood. The third is science in all its diversity: a grand adventure for everyone, nourished by curiosity and imagination. Today, I feel even more indebted to the renaissance of Dutch physics that began here in Leiden, with big names such as Kamerlingh Onnes, Van der Waals, Zeeman, De Sitter, Ehrenfest and, of course, my hero Hendrik Lorentz – role model and close friend of Albert Einstein, who himself was so fond of this city and university. -
Albert Einstein
L B. A Department of Physics (MC 0435) Tel:(+1)2155100693 Virginia Te Fax:(+1)5402317511 850 Campus Drive email: [email protected] Blasburg, VA, 24061, USA. URL: hp://www.phys.vt.edu/ lara/ Employment - Assistant Professor of Physics and Affiliate Professor of Mathematics Virginia Tech (Blacksburg, VA, USA), - Postdoctoral Researer in eoretical Physics Harvard University (Boston, MA, USA), - Postdoctoral Researer in eoretical Physics University of Pennsylvania (Philadelphia, PA, USA), - Visiting Researer in eoretical/Mathematical Physics Institute for Advanced Study (Princeton, NJ, USA), - Stipendiary Lecturer in Applied Mathematics Pembroke College, University of Oxford (Oxford, UK) Stipendiary Lecturer in Applied Mathematics Somerville College, University of Oxford (Oxford, UK) V P/F Kavli Institute for eoretical Physics, UC Santa Barbara (). Isaac Newton Institute for Mathemat- ical Sciences, University of Cambridge (). Simons Center for Geometry and Physics (). Simons Center for Geometry and Physics (). Resear Interests: Physics – String eory (including M-theory, F-theory, and Heterotic string theory), String Phenomenol- ogy, High Energy Particle Physics, Gauge eory, Supersymmetry Mathematics – Algebraic Geometry and Topology (including computational algebraic geometry), Ge- ometric Invariants, Moduli Spaces, Special Holonomy/Special Structure Manifolds. Education - University of Oxford (Oxford, UK), PhD in Mathematical Physics. esis Advisors: Philip Candelas (Mathematics) and Andre Lukas (eoretical Physics) - Utah State University (Logan, UT, USA), Master of Science, Physics. Summa Cum Laude. - Utah State University (Logan, UT, USA), Bachelor of Science, Physics. Summa Cum Laude. (4.0 GPA. Ranked 1st in graduating class). Bachelor of Science, Mathematics. Summa Cum Laude. ! S F S Royal Society University Research Fellowship (UK) ( years and £600K of funding). (Declined to accept position at Virginia Tech). -
BRYCE SELIGMAN DEWITT January 8, 1923–September 23, 2004
NATIONAL ACADEMY OF SCIENCES B R Y C E S E L I G M A N D EWITT 1 9 2 3 — 2 0 0 4 A Biographical Memoir by S T E V E N W EIN B ERG Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2008 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Larry Murphy BRYCE SELIGMAN DEWITT January 8, 1923–September 23, 2004 BY STEVEN WEIN BERG RYCE SELIGMAN DEWITT, PROFESSOR EMERITUS IN THE physics Bdepartment of the University of Texas at Austin, died on September 23, 2004. His career was marked by major contributions to classical and quantum field theories, in particular, to the theory of gravitation. DeWitt was born Carl Bryce Seligman on January 8, 1923, in Dinuba, California, the eldest of four boys. His paternal grandfather, Emil Seligman, left Germany around 1875 at the age of 17 and emigrated to California, where he and his brother established a general store in Traver. Emil mar- ried Anna Frey, a young woman who had emigrated from Switzerland at about the same time. They had 11 children, whom Anna raised in the Methodist church. In 1921 DeWitt’s father, who had become a country doctor, married the local high school teacher of Latin and mathematics. Her ancestors were French Huguenots and Scottish Presbyterians. DeWitt was raised in the Presbyterian Church, and the only Jewish elements in his early life were the matzos that his grandfather bought around the time of Passover. -
Issue 93 ISSN 1027-488X
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Interview Yakov Sinai Features Mathematical Billiards and Chaos About ABC Societies The Catalan Photograph taken by Håkon Mosvold Larsen/NTB scanpix Mathematical Society September 2014 Issue 93 ISSN 1027-488X S E European M M Mathematical E S Society American Mathematical Society HILBERT’S FIFTH PROBLEM AND RELATED TOPICS Terence Tao, University of California In the fifth of his famous list of 23 problems, Hilbert asked if every topological group which was locally Euclidean was in fact a Lie group. Through the work of Gleason, Montgomery-Zippin, Yamabe, and others, this question was solved affirmatively. Subsequently, this structure theory was used to prove Gromov’s theorem on groups of polynomial growth, and more recently in the work of Hrushovski, Breuillard, Green, and the author on the structure of approximate groups. In this graduate text, all of this material is presented in a unified manner. Graduate Studies in Mathematics, Vol. 153 Aug 2014 338pp 9781470415648 Hardback €63.00 MATHEMATICAL METHODS IN QUANTUM MECHANICS With Applications to Schrödinger Operators, Second Edition Gerald Teschl, University of Vienna Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators. Graduate Studies in Mathematics, Vol. 157 Nov 2014 356pp 9781470417048 Hardback €61.00 MATHEMATICAL UNDERSTANDING OF NATURE Essays on Amazing Physical Phenomena and their Understanding by Mathematicians V.