DOCSLIB.ORG

Deformations of G2-Structures, String Dualities and Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected]

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Deformations of G2-Structures, String Dualities and Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, Rdmbarbosa@Gmail.Com University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons, and the Quantum Physics Commons Recommended Citation Barbosa, Rodrigo De Menezes, "Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles" (2019). Publicly Accessible Penn Dissertations. 3279. https://repository.upenn.edu/edissertations/3279 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3279 For more information, please contact [email protected]. Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Abstract We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3-manifold Q. The afl tness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we obtain an explicit description of the moduli space for the dual type IIA string compactification. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of "flat Higgs bundles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2-structures: one as a character variety and a mirror description in terms of a Hilbert scheme of points. Usual G2-deformations are parametrized by spectral covers of flat Higgs bundles. We also discuss a few ongoing developments: (1) we propose a heterotic dual to our main example, (2) we explain how the moduli space of flat Higgs bundles fits into a family of moduli spaces of extended Bogomolnyi monopoles, and (3) we introduce a natural variation of Hodge structures over the complexified G2-moduli space, and conjecture this space admits the structure of a complex integrable system. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Tony G. Pantev Keywords Deformation Theory, G2-Manifolds, Higgs Bundles, Mirror Symmetry, Moduli Spaces, M-Theory Subject Categories Mathematics | Quantum Physics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/3279 DEFORMATIONS OF G2-STRUCTURES, STRING DUALITIES AND FLAT HIGGS BUNDLES Rodrigo de Menezes Barbosa A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2019 Supervisor of Dissertation Tony Pantev, Professor of Mathematics Graduate Group Chairperson Julia Hartmann, Professor of Mathematics Dissertation Committee: Ron Donagi, Professor of Mathematics Wolfgang Ziller, Professor of Mathematics Acknowledgments It is a pleasure to thank my wonderful Ph.D. advisor Tony Pantev for his mentorship during my time at Penn. Tony's profound insight into Mathematics has been a constant source of inspiration for me and also a major driving force behind this project. It still amazes me that I have learned something in every single one of our conversations - and that sometimes a side remark of his could lead me to hours sitting at the library, trying to crack down an intricate construction. I will always be grateful to him for his fundamental guidance in my journey to become a researcher in Mathematics. A special thanks goes to Ron Donagi and Wolfgang Ziller for their crucial role during my mathematical formation at Penn. I am thankful also to Bobby Acharya for important comments on this thesis and for drawing my attention to his paper [Ach98], which clarified a few issues at the final stage of this work; and to Dave Morrison, for inviting me to a crucial conference on Physics and Holonomy and for the opportunity to present this work there. I am grateful to quite a few Penn Math and Physics professors, including ii Jonathan Block, Mirjam Cvetic,ˇ Antonella Grassi, David Harbater, Jonathan Heck- man and Julius Shaneson. Many thanks also to the Math Office staff: Monica, Paula, Reshma and Robin, for making sure the department is always running smoothly; and to the many workers of DRL for keeping it an amazing work en- vironment. Thanks to the University of Pennsylvania for funding me throughout my PhD and also for providing wonderful facilities and resources. I am thankful to my fellow Penn Math graduate students, especially those in the Algebraic Geometry and Differential Geometry groups. A big thanks in particular to Benedict, Jia-Choon and Seokjoo for the friendship and for proof-reading and sharing their thoughts on this thesis. A special thanks goes to my good friend Matei, for all of that and also for bearing my years-long never-ending morning rambles on variations of Hodge structures in the most perfectoid room of Van Pelt. Thanks also to my office mates over the years: Aline, Ben, Josh, Kostis and Martin; and to Artur, Justin, Michael and Roberta for many enjoyable conversations, about Math or anything else. Among non-Penn Math people, thanks to my good friends Isabel Leal and Nuno Cardoso for our now decade-old friendship, and Alex Kinsella for many insightful conversations on Heterotic/M-theory duality and for hosting me in Santa Barbara on the occasion of my first major conference talk. I am thankful also to my Mas- ter's advisor Marcos Jardim for guiding me in my early steps as a mathematician, to my undergraduate advisor Alberto Saa for almost making me become a general iii relativist, to my UNICAMP professors Luiz San Martin and Adriano Moura, and to Elizabeth Gasparim for essentially bridging my way to Penn. A big thanks to everyone involved with the Simons Collaboration on Homological Mirror Symme- try and the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics: participating in these collaborations has essentially shaped me as a math- ematician, and I am sure I will carry these impressions on all my future work. In particular, I am grateful to James Simons and the Simons Foundation for making all of this possible. Finally, thanks to Szil´ardSzab´oand Dennis DeTurck for their help during the job market months. I was lucky to have a consistent group of close friends during my time in Philadel- phia. This provided the stability necessary to bear the emotional burdens that a PhD inevitably entails. Among these, I am mostly grateful to my \Philly family", who I can safely say are among my best friends in life: Andr´e,Charu, David, Sara and Stefano. I am also thankful to my amazing friends from the Brazilian and Ital- ian groups, and all other friends I made during these five years: there are too many of you to name here, but I am sure you know who you are. Thanks to my lifetime friends Diego and Rafael for having my back since the Wise times (and even before), to all my good friends in Bras´ıliafor making my time back home so enjoyable, and also to those who have spent time with me in Europe or Japan during quite a few memorable work trips. I am lucky to have an amazing family that even from a distance keeps me on iv track with their unwavering support. Thanks to my grandmothers Eneida and Leide, and to all my uncles, aunts and cousins in Brazil for always believing in me. Finally, and above all, I am deeply thankful to my parents Wagner and S^onia,my sister Nat´aliaand my sweetheart Laura for their unconditional love and patience. Math is fun, but the true purpose of life is to treasure your loved ones every single day. You are the foundation of my life. I love you. v \L'hydre-Univers tordant son corps ´ecaill´ed'astres." Victor Hugo, Les Contemplations, VI.26 \Meu inc^endio´euma metamorfose e a minha metamorfose ´ea madeira de um inc^endio." vi ABSTRACT DEFORMATIONS OF G2-STRUCTURES, STRING DUALITIES AND FLAT HIGGS BUNDLES Rodrigo de Menezes Barbosa Tony Pantev, Advisor We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3- manifold Q. The flatness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolu- tions over Q. Moreover, when instanton corrections are neglected, we also have an explicit description of the moduli space for the dual type IIA string compactifica- tion. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of “flat Higgs bun- dles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2- C structures MG2 : one as a character variety and a mirror description in terms of a Hilbert scheme of points. Usual G2-deformations are parametrized by spectral vii covers of flat Higgs bundles. We also discuss a few ongoing developments: p1q we propose a heterotic dual to our main example, p2q we explain how the moduli space of flat Higgs bundles fits into a family of moduli spaces of extended Bogomolnyi monopoles, and p3q we C introduce a natural variation of Hodge structures over MG2 , and conjecture this space admits the structure of a complex integrable system.
Load more

Recommended publications
  • Compactifying M-Theory on a G2 Manifold to Describe/Explain Our World – Predictions for LHC (Gluinos, Winos, Squarks), and Dark Matter
    Compactifying M-theory on a G2 manifold to describe/explain our world – Predictions for LHC (gluinos, winos, squarks), and dark matter Gordy Kane CMS, Fermilab, April 2016 1 OUTLINE • Testing theories in physics – some generalities - Testing 10/11 dimensional string/M-theories as underlying theories of our world requires compactification to four space-time dimensions! • Compactifying M-theory on “G2 manifolds” to describe/ explain our vacuum – underlying theory - fluxless sector! • Moduli – 4D manifestations of extra dimensions – stabilization - supersymmetry breaking – changes cosmology first 16 slides • Technical stuff – 18-33 - quickly • From the Planck scale to EW scale – 34-39 • LHC predictions – gluino about 1.5 TeV – also winos at LHC – but not squarks - 40-47 • Dark matter – in progress – surprising – 48 • (Little hierarchy problem – 49-51) • Final remarks 1-5 2 String/M theory a powerful, very promising framework for constructing an underlying theory that incorporates the Standard Models of particle physics and cosmology and probably addresses all the questions we hope to understand about the physical universe – we hope for such a theory! – probably also a quantum theory of gravity Compactified M-theory generically has gravity; Yang- Mills forces like the SM; chiral fermions like quarks and leptons; softly broken supersymmetry; solutions to hierarchy problems; EWSB and Higgs physics; unification; small EDMs; no flavor changing problems; partially observable superpartner spectrum; hidden sector DM; etc Simultaneously – generically Argue compactified M-theory is by far the best motivated, and most comprehensive, extension of the SM – gets physics relevant to the LHC and Higgs and superpartners right – no ad hoc inputs or free parameters Take it very seriously 4 So have to spend some time explaining derivations, testability of string/M theory Don’t have to be somewhere to test theory there – E.g.
    [Show full text]
  • A Space-Time Orbifold: a Toy Model for a Cosmological Singularity
    hep-th/0202187 UPR-981-T, HIP-2002-07/TH A Space-Time Orbifold: A Toy Model for a Cosmological Singularity Vijay Balasubramanian,1 ∗ S. F. Hassan,2† Esko Keski-Vakkuri,2‡ and Asad Naqvi1§ 1David Rittenhouse Laboratories, University of Pennsylvania Philadelphia, PA 19104, U.S.A. 2Helsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland Abstract We explore bosonic strings and Type II superstrings in the simplest time dependent backgrounds, namely orbifolds of Minkowski space by time reversal and some spatial reflections. We show that there are no negative norm physi- cal excitations. However, the contributions of negative norm virtual states to quantum loops do not cancel, showing that a ghost-free gauge cannot be chosen. The spectrum includes a twisted sector, with strings confined to a “conical” sin- gularity which is localized in time. Since these localized strings are not visible to asymptotic observers, interesting issues arise regarding unitarity of the S- matrix for scattering of propagating states. The partition function of our model is modular invariant, and for the superstring, the zero momentum dilaton tad- pole vanishes. Many of the issues we study will be generic to time-dependent arXiv:hep-th/0202187v2 19 Apr 2002 cosmological backgrounds with singularities localized in time, and we derive some general lessons about quantizing strings on such spaces. 1 Introduction Time-dependent space-times are difficult to study, both classically and quantum me- chanically. For example, non-static solutions are harder to find in General Relativity, while the notion of a particle is difficult to define clearly in field theory on time- dependent backgrounds.
    [Show full text]
  • Jhep05(2019)105
    Published for SISSA by Springer Received: March 21, 2019 Accepted: May 7, 2019 Published: May 20, 2019 Modular symmetries and the swampland conjectures JHEP05(2019)105 E. Gonzalo,a;b L.E. Ib´a~neza;b and A.M. Urangaa aInstituto de F´ısica Te´orica IFT-UAM/CSIC, C/ Nicol´as Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain bDepartamento de F´ısica Te´orica, Facultad de Ciencias, Universidad Aut´onomade Madrid, 28049 Madrid, Spain E-mail: [email protected], [email protected], [email protected] Abstract: Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d N = 1 compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the K¨ahler and complex structure moduli, in particular SL(2; Z) T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I' or Horava-Witten theory, in which the SL(2; Z) acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes.
    [Show full text]
  • Modular Invariance of Characters of Vertex Operator Algebras
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 MODULAR INVARIANCE OF CHARACTERS OF VERTEX OPERATOR ALGEBRAS YONGCHANG ZHU Introduction In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z)ismanifest. In the study of conformal field theory, physicists arrived at the notion of chi- ral algebras (see e.g.
    [Show full text]
  • On Mirror Maps for Manifolds of Exceptional Holonomy JHEP10(2019)204 ]) 2 and 2 G ] (See Also [ 1 21 31 ]
    Published for SISSA by Springer Received: August 23, 2019 Accepted: October 2, 2019 Published: October 21, 2019 On mirror maps for manifolds of exceptional holonomy JHEP10(2019)204 Andreas P. Braun,a Suvajit Majumdera and Alexander Ottob;c aMathematical Institute, University of Oxford Woodstock Road, Oxford, OX2 6GG, U.K. bPerimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada cDepartment of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada E-mail: [email protected], [email protected], [email protected] Abstract: We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities. Keywords: String Duality, Conformal Field Models in String Theory, Superstring Vacua ArXiv ePrint: 1905.01474 Open Access, c The Authors. https://doi.org/10.1007/JHEP10(2019)204 Article funded by SCOAP3.
    [Show full text]
  • Arxiv:Hep-Th/9404101V3 21 Nov 1995
    hep-th/9404101 April 1994 Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions ⋆ Petr Horavaˇ ∗ Enrico Fermi Institute University of Chicago 5640 South Ellis Avenue Chicago IL 60637, USA ABSTRACT Chern-Simons gauge theory is formulated on three dimensional Z2 orbifolds. The locus of singular points on a given orbifold is equivalent to a link of Wilson lines. This allows one to reduce any correlation function on orbifolds to a sum of more complicated correlation functions in the simpler theory on manifolds. Chern-Simons theory on manifolds is known arXiv:hep-th/9404101v3 21 Nov 1995 to be related to 2D CFT on closed string surfaces; here I show that the theory on orbifolds is related to 2D CFT of unoriented closed and open string models, i.e. to worldsheet orb- ifold models. In particular, the boundary components of the worldsheet correspond to the components of the singular locus in the 3D orbifold. This correspondence leads to a simple identification of the open string spectra, including their Chan-Paton degeneration, in terms of fusing Wilson lines in the corresponding Chern-Simons theory. The correspondence is studied in detail, and some exactly solvable examples are presented. Some of these examples indicate that it is natural to think of the orbifold group Z2 as a part of the gauge group of the Chern-Simons theory, thus generalizing the standard definition of gauge theories. E-mail address: [email protected] ⋆∗ Address after September 1, 1994: Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544. 1. Introduction Since the first appearance of the notion of “orbifolds” in Thurston’s 1977 lectures on three dimensional topology [1], orbifolds have become very appealing objects for physicists.
    [Show full text]
  • From String Theory and Moonshine to Vertex Algebras
    Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential).
    [Show full text]
  • Super-Higgs in Superspace
    Article Super-Higgs in Superspace Gianni Tallarita 1,* and Moritz McGarrie 2 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile 2 Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg, Germany; [email protected] * Correspondence: [email protected] or [email protected] Received: 1 April 2019; Accepted: 10 June 2019; Published: 14 June 2019 Abstract: We determine the effective gravitational couplings in superspace whose components reproduce the supergravity Higgs effect for the constrained Goldstino multiplet. It reproduces the known Gravitino sector while constraining the off-shell completion. We show that these couplings arise by computing them as quantum corrections. This may be useful for phenomenological studies and model-building. We give an example of its application to multiple Goldstini. Keywords: supersymmetry; Goldstino; superspace 1. Introduction The spontaneous breakdown of global supersymmetry generates a massless Goldstino [1,2], which is well described by the Akulov-Volkov (A-V) effective action [3]. When supersymmetry is made local, the Gravitino “eats” the Goldstino of the A-V action to become massive: The super-Higgs mechanism [4,5]. In terms of superfields, the constrained Goldstino multiplet FNL [6–12] is equivalent to the A-V formulation (see also [13–17]). It is, therefore, natural to extend the description of supergravity with this multiplet, in superspace, to one that can reproduce the super-Higgs mechanism. In this paper we address two issues—first we demonstrate how the Gravitino, Goldstino, and multiple Goldstini obtain a mass. Secondly, by using the Spurion analysis, we write down the most minimal set of new terms in superspace that incorporate both supergravity and the Goldstino multiplet in order to reproduce the super-Higgs mechanism of [5,18] at lowest order in M¯ Pl.
    [Show full text]
  • An Introduction to Orbifolds
    An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 2009 An introduction to orbifolds – p.1/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. An introduction to orbifolds – p.2/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. ··· (there are other notions of orbifold in algebraic geometry, string theory or using grupoids) An introduction to orbifolds – p.2/20 Examples: tessellations of Euclidean plane Γ= h(x, y) → (x + 1, y), (x, y) → (x, y + 1)i =∼ Z2 Γ\R2 =∼ T 2 = S1 × S1 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 2 2 2 2 2 2 2 2 2 2 An introduction to orbifolds – p.3/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) An introduction to orbifolds – p.4/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) 2 4 2 6 An introduction to orbifolds – p.4/20 Definition Informal Definition • An orbifold O is a metrizable topological space equipped with an atlas modelled on Rn/Γ, Γ < O(n) finite, with some compatibility condition.
    [Show full text]
  • Ads Orbifolds and Penrose Limits
    IPM/P-2002/061 UCB-PTH-02/54 LBL-51767 hep-th/yymmnnn AdS orbifolds and Penrose limits Mohsen Alishahihaa, Mohammad M. Sheikh-Jabbarib and Radu Tatarc a Institute for Studies in Theoretical Physics and Mathematics (IPM) P.O.Box 19395-5531, Tehran, Iran e-mail: [email protected] b Department of Physics, Stanford University 382 via Pueblo Mall, Stanford CA 94305-4060, USA e-mail: [email protected] c Department of Physics, 366 Le Conte Hall University of California, Berkeley, CA 94720 and Lawrence Berkeley National Laboratory Berkeley, CA 94720 email:[email protected] Abstract In this paper we study the Penrose limit of AdS5 orbifolds. The orbifold can be either in the pure spatial directions or space and time directions. For the AdS =¡ S5 spatial orbifold we observe that after the Penrose limit we 5 £ obtain the same result as the Penrose limit of AdS S5=¡. We identify the 5 £ corresponding BMN operators in terms of operators of the gauge theory on R £ S3=¡. The semi-classical description of rotating strings in these backgrounds have also been studied. For the spatial AdS orbifold we show that in the quadratic order the obtained action for the fluctuations is the same as that in S5 orbifold, however, the higher loop correction can distinguish between two cases. 1 Introduction In the past ¯ve years important results have been obtained relating closed and open string theories, in particular the AdS/CFT correspondence [1] had important impact into the understanding between these two sectors. The main problem in testing the conjecture beyond the supergravity level is the fact that we do not know how to quantize string theory in the presence of RR-fluxes.
    [Show full text]
  • Modular Forms on G 2 and Their Standard L-Function
    MODULAR FORMS ON G2 AND THEIR STANDARD L-FUNCTION AARON POLLACK Abstract. The purpose of this partly expository paper is to give an introduction to modular forms on G2. We do this by focusing on two aspects of G2 modular forms. First, we discuss the Fourier expansion of modular forms, following work of Gan-Gross-Savin and the author. Then, following Gurevich-Segal and Segal, we discuss a Rankin-Selberg integral yielding the standard L-function of modular forms on G2. As a corollary of the analysis of this Rankin-Selberg integral, one obtains a Dirichlet series for the standard L-function of G2 modular forms; this involves the arithmetic invariant theory of cubic rings. We end by analyzing the archimedean zeta integral that arises from the Rankin-Selberg integral when the cusp form is an even weight modular form. 1. Introduction If a reductive group G over Q has a Hermitian symmetric space G(R)=K, then there is a notion of modular forms on G. Namely, one can consider the automorphic forms on G that give rise to holomorphic functions on G(R)=K. However, if G does not have a Hermitian symmetric space, then G has no obvious notion of modular forms, or very special automorphic forms. The split exceptional group G2 is an example: its symmetric space G2(R)= (SU(2) × SU(2)) has no complex structure. Nevertheless, Gan-Gross-Savin [GGS02] defined and studied modular forms on G2 by building off of work of Gross-Wallach [GW96] on quaternionic discrete series representations. The purpose of this paper is to give an introduction to modular forms on G2, with the hope that we might interest others in this barely-developed area of mathematics.
    [Show full text]
  • Oldandnewonthe Exceptionalgroupg2 Ilka Agricola
    OldandNewonthe ExceptionalGroupG2 Ilka Agricola n a talk delivered in Leipzig (Germany) on product, the Lie bracket [ , ]; as a purely algebraic June 11, 1900, Friedrich Engel gave the object it is more accessible than the original Lie first public account of his newly discovered group G. If G happens to be a group of matrices, its description of the smallest exceptional Lie Lie algebra g is easily realized by matrices too, and group G2, and he wrote in the corresponding the Lie bracket coincides with the usual commuta- Inote to the Royal Saxonian Academy of Sciences: tor of matrices. In Killing’s and Lie’s time, no clear Moreover, we hereby obtain a direct defi- distinction was made between the Lie group and nition of our 14-dimensional simple group its Lie algebra. For his classification, Killing chose [G2] which is as elegant as one can wish for. a maximal set h of linearly independent, pairwise 1 [En00, p. 73] commuting elements of g and constructed base Indeed, Engel’s definition of G2 as the isotropy vectors Xα of g (indexed over a finite subset R of group of a generic 3-form in 7 dimensions is at elements α ∈ h∗, the roots) on which all elements the basis of a rich geometry that exists only on of h act diagonally through [ , ]: 7-dimensional manifolds, whose full beauty has been unveiled in the last thirty years. [H,Xα] = α(H)Xα for all H ∈ h. This article is devoted to a detailed historical In order to avoid problems when doing so he chose and mathematical account of G ’s first years, in 2 the complex numbers C as the ground field.
    [Show full text]