DOCSLIB.ORG

Constructing a Family of Vacuum Solutions for the Heterotic G2 System

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constructing a Family of Vacuum Solutions for the Heterotic G2 System Constructing a family of vacuum solutions for the heterotic G2 system Mateo Galdeano University of Oxford, Mathematical Institute August 20, 2020 All the results described here are joint work with Xenia de la Ossa. Main motivation We want to improve our understanding of heterotic string theory. In particular we want to find explicit solutions of the theory reduced to 3 dimensions, which are scarce in the literature. We would also like to see the deformation theory of these solutions working explicitly. A glimpse at physics [1] Heterotic string theory is a 10-dimensional theory that includes supergravity and a supersymmetric gauge theory at classical level. In order to study the theory in lower dimensions, we perform a compactification and put some of the coordinates in a compact manifold M10 = M10−D × YD (1) Imposing the ansatz (1) reduces the equations the fields must satisfy to a system on the compact manifold. The Strominger-Hull system When heterotic string theory is reduced to 4 dimensions, we obtain the Strominger-Hull system. We are interested in the reduction to 3 dimensions. The conditions to have a solution can be encoded as a heterotic G2 system. Heterotic G2 system [2] A heterotic G2 system is a quadruple [(Y ;'); (V ; A); (TY ; Θ); H] satisfying the following properties: Y is a 7-dimensional manifold with an integrable G2 structure given by '. We call = ∗'. V is a vector bundle on Y with a connection A that is a G2-instanton, this means that its curvature FA satisfies FA ^ = 0. TY is the tangent bundle of Y and Θ is a connection which is a G2-instanton, so that its curvature RΘ satisfies RΘ ^ = 0. H is the torsion 3-form of the G2-structure T(') on Y , H = T('): In addition, the following formula must be satisfied: α0 H = dB + (CS(A) − CS(Θ)) ; 4 where CS denotes the Chern-Simons form of the corresponding connection and B is the B-field. Explicit family of solutions 7 7 7 A family of solutions [(S ;'s); (TS ; Θβ); (TS ; Θα); H] is given as follows Take Y = S7 the 7-sphere. We see from the Hopf fibration that it is a 3-Sasakian manifold [3] 3 7 4 S −! S −! S : This gives an integrable G2 structure to the 7-sphere. Moreover, we can squash the metric along the fibers and this results in a one-parameter family of G2 structures on the sphere 's for s > 0. These are the squashed 7-spheres. The 3-form H is determined by the torsion of the G2 structures. In general there is a two-parameter family of metric connections compatible with the G2 structure on the tangent space of any G2 manifold [4]. In this case it reduces to a one-parameter family Θα and all of them are G2 instantons. 7 We can take as vector bundle V the tangent space TS itself and as an instanton another element in the family of connections Θβ. Finally, we must impose the heterotic Bianchi identity α0 dH = (tr FA ^ FA − tr RΘ ^ RΘ): 4 This constraints all the parameters and essentially gives a one-parameter family of solutions for each value of the string parameter α0. References Ralph Blumenhagen, Dieter L¨ust,and Stefan Theisen. Basic concepts of string theory. Theoretical and Mathematical Physics. Heidelberg, Germany: Springer, 2013. isbn: 9783642294969. doi: 10.1007/978-3-642-29497-6. Xenia de la Ossa, Magdalena Larfors, and Eirik E. Svanes. \The Infinitesimal Moduli Space of Heterotic G2 Systems". In: Commun. Math. Phys. 360.2 (2018), pp. 727{775. doi: 10.1007/s00220-017-3013-8. arXiv: 1704.08717 [hep-th]. Charles P. Boyer and Krzysztof Galicki. \3 - Sasakian manifolds". In: Surveys Diff. Geom. 7 (1999), pp. 123{184. arXiv: hep-th/9810250 [hep-th]. Robert L. Bryant. \Some remarks on G 2-structures". In: arXiv Mathematics e-prints, math/0305124 (2003), math/0305124. arXiv: math/0305124 [math.DG]..
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]
  • Deformations of G2-Structures, String Dualities and Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected]
    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.
    [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]
  • 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]
  • M-Theory Dynamics on a Manifold of G2 Holonomy
    c 2001 International Press Adv. Theor. Math. Phys. 6 (2001) 1–106 M-Theory Dynamics On A Manifold of G2 Holonomy Michael Atiyah Mathematics Department University ofEdinburgh Edinburgh Scotland UK and Dept. ofPhysics, Caltech, Pasadena CA, USA [email protected] Edward Witten Department ofPhysics,Caltech, Pasadena, CA, USA and CIT-USC Center for Theoretical Physics, USC, Los Angeles, CA and Institute for Advanced Study Princeton, NJ [email protected] Abstract We analyze the dynamics of M-theory on a manifold of G2 holon- omy that is developing a conical singularity. The known cases involve a cone on CP3, where we argue that the dynamics involves restoration of a global symmetry, SU(3)/U(1)2, where we argue that there are phase transitions among three possible branches corresponding to three clas- sical spacetimes, and S3 × S3 and its quotients, where we recover and extend previous results about smooth continuations between different spacetimes and relations to four-dimensional gauge theory. 2 M-Theory Dynamics 1 Introduction In studying supersymmetric compactifications ofstring theory, one ofthe important issues is the behavior at a singularity. For example, in compact- ifications of M-theory and ofType II superstring theory, an important role is played by the A − D − E singularities ofa K3 surfaceand by various singularities ofa Calabi-Yau threefold. Singularities ofheterotic string or D-brane gauge fields are also important, though in the present paper we focus on metric singularities. The basic questions about string theory and M-theory
    [Show full text]
  • Almost Contact Structures on Manifolds with a G2 Structure
    Almost contact structures on manifolds with a G2 structure Xenia de la Ossa,a Magdalena Larfors,b Matthew Magill.b aMathematical Institute, Oxford University Andrew Wiles Building, Woodstock Road Oxford OX2 6GG, UK bDepartment of Physics and Astronomy,Uppsala University SE-751 20 Uppsala, Sweden E-mail: [email protected], [email protected], [email protected] Abstract: We review the construction of almost contact metric (three-) structures on manifolds with a G2 structure. These are of interest for certain supersymmetric configurations in string and M-theory. We compute the torsion of the SU(3) struc- ture associated to an ACMS and apply these computations to heterotic G2 systems and supersymmetry enhancement. We initiate the study of the space of ACM3Ss, which is an infinite dimensional space with a local product structure and interesting topological features. Tantalising links between ACM3Ss and associative and coasso- ciative submanifolds are observed. arXiv:2101.12605v2 [hep-th] 8 Mar 2021 Contents 1 Introduction2 1.1 Preliminary notions4 2 SU(3) structures on manifolds with a G2 structure6 2.1 Almost contact metric structures on a manifold with a G2 structure7 2.2 Transverse geometry and SU(3) structures on Y 10 2.3 Decomposing the structure equations 15 3 Heterotic G2 systems under the ACMS 17 3.1 Heterotic G2 systems 18 3.2 Decomposition of the G2 structure equations with respect to the ACMS 19 3.3 Instanton conditions 19 3.4 Anomaly cancellation condition 20 3.5 N = 1 superpotential in terms of the
    [Show full text]
  • G2–Instantons Over Twisted Connected Sums
    Geometry & Topology 19 (2015) 1263–1285 msp G2–instantons over twisted connected sums HENRIQUE NSÁ EARP THOMAS WALPUSKI We introduce a method to construct G2 –instantons over compact G2 –manifolds arising as the twisted connected sum of a matching pair of building blocks. Our construction is based on gluing G2 –instantons obtained from holomorphic vector bundles over the building blocks via the first author’s work. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. 53C07, 53C25, 53C38 1 Introduction A G2 –manifold .Y; g/ is a Riemannian 7–manifold whose holonomy group Hol.g/ is contained in the exceptional Lie group G2 or equivalently, a 7–manifold Y together with a torsion-free G2 –structure, that is, a nondegenerate 3–form satisfying a certain nonlinear partial differential equation; see eg Joyce[14, Part I]. An important method to produce examples of compact G2 –manifolds with Hol.g/ G2 is the twisted D connected sum construction, suggested by Donaldson, pioneered by Kovalev[16] and later extended and improved by Kovalev and Lee[17] and Corti, Haskins, Nordström and Pacini[2]. Here is a brief summary of this construction: A building block consists of a smooth projective 3–fold Z and a smooth anticanonical K3 surface † Z with trivial normal bundle; see Definition 2.8. Given a choice of a hyperkähler structure .!I ;!J ;!K / on † such that !J i!K is of type .2; 0/ and Œ!I is the restriction C of a Kähler class on Z, one can make V Z † into an asymptotically cylindrical WD n (ACyl) Calabi–Yau 3–fold, that is, a noncompact Calabi–Yau 3–fold with a tubular end modeled on R S 1 †; see Haskins, Hein and Nordström[12].
    [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]
  • 7-Manifolds with G2 Holonomy
    7-manifolds with G2 holonomy Mark Haskins Imperial College London 19 September 2016 Geometric Flows and the Geometry of Space-time Hamburg What is G2? G2 holonomy and Ricci-flat metrics i. the automorphism group of the octonions O ii. the stabilizer of a generic 3-form in R7 Define a vector cross-product on R7 = Im(O) u × v = Im(uv) where uv denotes octonionic multiplication. Cross-product has an associated 3-form '0(u; v; w) := hu × v; wi = huv; wi '0 is a generic 3-form so in fact ∗ G2 = fA 2 GL(7; R)j A '0 = ' ⊂ SO(7): G2 can arise as the holonomy group of an irreducible non-locally-symmetric Riemannian 7-manifold (Berger 1955, Bryant 1987, Bryant-Salamon 1989, Joyce 1995). Any such manifold is automatically Ricci-flat. 6 + 1 = 2 × 3 + 1 = 7 & SU(2) ⊂ SU(3) ⊂ G2 9 close relations between G2 holonomy and Calabi-Yau geometries in 2 and 3 dimensions. 7 3 3 Write R = R × C with (C ; !; Ω) the standard SU(3) structure then '0 = dt ^ ! + Re Ω Hence stabilizer of R factor in G2 is SU(3) ⊂ G2. More generally if (X ; g) is a Calabi-Yau 3-fold then product metric on S1 × X has holonomy SU(3) ⊂ G2. 7 3 2 3 Write R = R × C with coords (x1; x2; x3) on R , with standard SU(2) 2 structure (C ;!I ; Ω = !J + i!K ) then '0 = dx1 ^ dx2 ^ dx3 + dx1 ^ !I + dx2 ^ !J + dx3 ^ !K ; where !I and Ω = !J + i!K are the standard Kahler and holmorphic 2 3 3 2 (2; 0) forms on C .
    [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]