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Wall Crossing and M-Theory
Wall Crossing and M-theory The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Aganagic, Mina, Hirosi Ooguri, Cumrun Vafa, and Masahito Yamazaki. 2011. Wall crossing and M-theory. Publications of the Research Institute for Mathematical Sciences 47(2): 569-584. Published Version doi:10.2977/PRIMS/44 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:7561260 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP CALT-68-2746 IPMU09-0091 UT-09-18 Wall Crossing and M-Theory Mina Aganagic1, Hirosi Ooguri2,3, Cumrun Vafa4 and Masahito Yamazaki2,3,5 1Center for Theoretical Physics, University of California, Berkeley, CA 94720, USA 2California Institute of Technology, Pasadena, CA 91125, USA 3 IPMU, University of Tokyo, Chiba 277-8586, Japan 4Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA 5Department of Physics, University of Tokyo, Tokyo 113-0033, Japan Abstract arXiv:0908.1194v1 [hep-th] 8 Aug 2009 We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a Calabi- Yau 3-fold X. When X has no compact 4-cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M- theory compactified down to 5 dimensions by a Calabi-Yau 3-fold X. -
Moduli Spaces
spaces. Moduli spaces such as the moduli of elliptic curves (which we discuss below) play a central role Moduli Spaces in a variety of areas that have no immediate link to the geometry being classified, in particular in alge- David D. Ben-Zvi braic number theory and algebraic topology. Moreover, the study of moduli spaces has benefited tremendously in recent years from interactions with Many of the most important problems in mathemat- physics (in particular with string theory). These inter- ics concern classification. One has a class of math- actions have led to a variety of new questions and new ematical objects and a notion of when two objects techniques. should count as equivalent. It may well be that two equivalent objects look superficially very different, so one wishes to describe them in such a way that equiv- 1 Warmup: The Moduli Space of Lines in alent objects have the same description and inequiva- the Plane lent objects have different descriptions. Moduli spaces can be thought of as geometric solu- Let us begin with a problem that looks rather simple, tions to geometric classification problems. In this arti- but that nevertheless illustrates many of the impor- cle we shall illustrate some of the key features of mod- tant ideas of moduli spaces. uli spaces, with an emphasis on the moduli spaces Problem. Describe the collection of all lines in the of Riemann surfaces. (Readers unfamiliar with Rie- real plane R2 that pass through the origin. mann surfaces may find it helpful to begin by reading about them in Part III.) In broad terms, a moduli To save writing, we are using the word “line” to mean problem consists of three ingredients. -
Flux Moduli Stabilisation, Supergravity Algebras and No-Go Theorems Arxiv
IFT-UAM/CSIC-09-36 Flux moduli stabilisation, Supergravity algebras and no-go theorems Beatriz de Carlosa, Adolfo Guarinob and Jes´usM. Morenob a School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK b Instituto de F´ısicaTe´oricaUAM/CSIC, Facultad de Ciencias C-XVI, Universidad Aut´onomade Madrid, Cantoblanco, 28049 Madrid, Spain Abstract We perform a complete classification of the flux-induced 12d algebras compatible with the set of N = 1 type II orientifold models that are T-duality invariant, and allowed by the symmetries of the 6 ¯ T =(Z2 ×Z2) isotropic orbifold. The classification is performed in a type IIB frame, where only H3 and Q fluxes are present. We then study no-go theorems, formulated in a type IIA frame, on the existence of Minkowski/de Sitter (Mkw/dS) vacua. By deriving a dictionary between the sources of potential energy in types IIB and IIA, we are able to combine algebra results and no-go theorems. The outcome is a systematic procedure for identifying phenomenologically viable models where Mkw/dS vacua may exist. We present a complete table of the allowed algebras and the viability of their resulting scalar potential, and we point at the models which stand any chance of producing a fully stable vacuum. arXiv:0907.5580v3 [hep-th] 20 Jan 2010 e-mail: [email protected] , [email protected] , [email protected] Contents 1 Motivation and outline 1 2 Fluxes and Supergravity algebras 3 2.1 The N = 1 orientifold limits as duality frames . .4 2.2 T-dual algebras in the isotropic Z2 × Z2 orientifolds . -
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. -
Abdus Salam Educational, Scientific and Cultural XA0500266 Organization International Centre
the united nations abdus salam educational, scientific and cultural XA0500266 organization international centre international atomic energy agency for theoretical physics 1 a M THEORY AND SINGULARITIES OF EXCEPTIONAL HOLONOMY MANIFOLDS Bobby S. Acharya and Sergei Gukov Available at: http://www.ictp.it/-pub-off IC/2004/127 HUTP-03/A053 RUNHETC-2003-26 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS M THEORY AND SINGULARITIES OF EXCEPTIONAL HOLONOMY MANIFOLDS Bobby S. Acharya1 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Sergei Gukov2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA. Abstract M theory compactifications on Gi holonomy manifolds, whilst supersymmetric, require sin- gularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) holonomy manifolds which correspond to the properties of three dimensional field theories. As an application, we review how various aspects of strongly coupled gauge theories, such as confinement, mass gap and non-perturbative phase transitions may be given a -
The Topology of Moduli Space and Quantum Field Theory* ABSTRACT
SLAC-PUB-4760 October 1988 CT) . The Topology of Moduli Space and Quantum Field Theory* DAVID MONTANO AND JACOB SONNENSCHEIN+ Stanford Linear Accelerator Center Stanford University, Stanford, California 9.4309 ABSTRACT We show how an SO(2,l) gauge theory with a fermionic symmetry may be used to describe the topology of the moduli space of curves. The observables of the theory correspond to the generators of the cohomology of moduli space. This is an extension of the topological quantum field theory introduced by Witten to -- .- . -. investigate the cohomology of Yang-Mills instanton moduli space. We explore the basic structure of topological quantum field theories, examine a toy U(1) model, and then realize a full theory of moduli space topology. We also discuss why a pure gravity theory, as attempted in previous work, could not succeed. Submitted to Nucl. Phys. I3 _- .- --- _ * Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Work supported by Dr. Chaim Weizmann Postdoctoral Fellowship. 1. Introduction .. - There is a widespread belief that the Lagrangian is the fundamental object for study in physics. The symmetries of nature are simply properties of the relevant Lagrangian. This philosophy is one of the remaining relics of classical physics where the Lagrangian is indeed fundamental. Recently, Witten has discovered a class of quantum field theories which have no classical analog!” These topo- logical quantum field theories (TQFT) are, as their name implies, characterized by a Hilbert space of topological invariants. As has been recently shown, they can be constructed by a BRST gauge fixing of a local symmetry. -
Donaldson Invariants and Moduli of Yang-Mills Instantons
Donaldson Invariants and Moduli of Yang-Mills Instantons Jacob Gross Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) ∼ The ASD Equation in Low Dimensions, 17 November 2017 Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Moduli and Invariants Invariants are gotten by cooking up a number (homology group, derived category, etc.) using auxiliary data (a metric, a polarisation, etc.) and showing independence of that initial choice. Donaldson and Seiberg-Witten invariants count in moduli spaces of solutions of a pde (shown to be independent of conformal class of the metric). Example (baby example) f :(M, g) R, the number of solutions of → gradgf = 0 is independent of g (Poincare-Hopf).´ Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Recall (Setup) (X, g) smooth oriented Riemannian 4-manifold and a principal G-bundle π : P X over it. → Consider the Yang-Mills functional YM(A) = F 2dμ. | A| ZM The corresponding Euler-Lagrange equations are dA∗FA = 0. Using the gauge group one generates lots of solutions from one. But one can fix aG gauge dA∗a = 0 Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Recall (ASD Equations) 2 2 2 In dimension 4, the Hodge star splits the 2-forms Λ = Λ+ Λ . This splits the curvature ⊕ − + FA = FA + FA−. Have + 2 2 + 2 2 κ(P) = F F − YM(A) = F + F − , k A k − k A k ≤ k A k k A k where κ(P) is a topological invariant (e.g. for SU(2)-bundles, 2 κ(P) = 8π c2(P)). -
Open Gromov-Witten Invariants and Syz Under Local Conifold Transitions
OPEN GROMOV-WITTEN INVARIANTS AND SYZ UNDER LOCAL CONIFOLD TRANSITIONS SIU-CHEONG LAU Abstract. For a local non-toric Calabi-Yau manifold which arises as smoothing of a toric Gorenstein singularity, this paper derives the open Gromov-Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its SYZ mirror. Moreover it proves that the SYZ mirrors and disc potentials vary smoothly under conifold transitions, giving a global picture of SYZ mirror symmetry. 1. Introduction Let N be a lattice and P a lattice polytope in NR. It is an interesting classical problem in combinatorial geometry to construct Minkowski decompositions of P . P v On the other hand, consider the family of polynomials v2P \N cvz for cv 2 C with cv 6= 0 when v is a vertex of P . All these polynomials have P as their Newton polytopes. This establishes a relation between geometry and algebra, and the geometric problem of finding Minkowski decompositions is transformed to the algebraic problem of finding polynomial factorizations. One purpose of this paper is to show that the beautiful algebro-geometric correspondence between Minkowski decompositions and polynomial factorizations can be realized via SYZ mirror symmetry. A key step leading to the miracle is brought by a result of Altmann [3]. First of all, a lattice polytope corresponds to a toric Gorenstein singularity X. Altmann showed that Minkowski decompositions of the lattice polytope correspond to (partial) smoothings of X. From string- theoretic point of view different smoothings of X belong to different sectors of the same stringy K¨ahlermoduli. -
Gromov-Witten Invariants and Localization
Gromov–Witten invariants and localization David R. Morrison Departments of Mathematics and Physics University of California, Santa Barbara Santa Barbara, CA 93106 USA [email protected] Abstract We give a pedagogical review of the computation of Gromov–Witten invariants via localiza- tion in 2D gauged linear sigma models. We explain the relationship between the two-sphere partition function of the theory and the K¨ahler potential on the conformal manifold. We show how the K¨ahler potential can be assembled from classical, perturbative, and non- perturbative contributions, and explain how the non-perturbative contributions are related to the Gromov-Witten invariants of the corresponding Calabi–Yau manifold. We then ex- plain how localization enables efficient calculation of the two-sphere partition function and, ultimately, the Gromov–Witten invariants themselves. This is a contribution to the review volume “Localization techniques in quantum field theories” (eds. V. Pestun and M. Zabzine) which contains 17 Chapters available at [1] Contents 1 Introduction 2 2 K¨ahler potentials and 2-sphere partition functions 3 arXiv:1608.02956v3 [hep-th] 15 Oct 2016 3 Metrics on conformal manifolds 6 3.1 Nonlinearsigmamodels ............................. 6 3.2 Gaugedlinearsigmamodels ........................... 8 3.3 PhasesofanabelianGLSM ........................... 10 4 Quantum corrections to the K¨ahler potential 14 4.1 Nonlinear σ-modelactionandtheeffectiveaction . 14 4.2 Perturbative corrections to the K¨ahler potential . ....... 16 4.3 Nonperturbative corrections and Gromov–Witten invariants . ....... 17 5 The two-sphere partition function and Gromov–Witten invariants 20 5.1 The S2 partitionfunctionforaGLSM . 20 5.2 The hemisphere partition function and the tt∗ equations ........... 21 5.3 Extracting Gromov–Witten invariants from the partition function ..... -
Moduli Spaces and Invariant Theory
MODULI SPACES AND INVARIANT THEORY JENIA TEVELEV CONTENTS §1. Syllabus 3 §1.1. Prerequisites 3 §1.2. Course description 3 §1.3. Course grading and expectations 4 §1.4. Tentative topics 4 §1.5. Textbooks 4 References 4 §2. Geometry of lines 5 §2.1. Grassmannian as a complex manifold. 5 §2.2. Moduli space or a parameter space? 7 §2.3. Stiefel coordinates. 8 §2.4. Complete system of (semi-)invariants. 8 §2.5. Plücker coordinates. 9 §2.6. First Fundamental Theorem 10 §2.7. Equations of the Grassmannian 11 §2.8. Homogeneous ideal 13 §2.9. Hilbert polynomial 15 §2.10. Enumerative geometry 17 §2.11. Transversality. 19 §2.12. Homework 1 21 §3. Fine moduli spaces 23 §3.1. Categories 23 §3.2. Functors 25 §3.3. Equivalence of Categories 26 §3.4. Representable Functors 28 §3.5. Natural Transformations 28 §3.6. Yoneda’s Lemma 29 §3.7. Grassmannian as a fine moduli space 31 §4. Algebraic curves and Riemann surfaces 37 §4.1. Elliptic and Abelian integrals 37 §4.2. Finitely generated fields of transcendence degree 1 38 §4.3. Analytic approach 40 §4.4. Genus and meromorphic forms 41 §4.5. Divisors and linear equivalence 42 §4.6. Branched covers and Riemann–Hurwitz formula 43 §4.7. Riemann–Roch formula 45 §4.8. Linear systems 45 §5. Moduli of elliptic curves 47 1 2 JENIA TEVELEV §5.1. Curves of genus 1. 47 §5.2. J-invariant 50 §5.3. Monstrous Moonshine 52 §5.4. Families of elliptic curves 53 §5.5. The j-line is a coarse moduli space 54 §5.6. -
The Duality Between F-Theory and the Heterotic String in with Two Wilson
Letters in Mathematical Physics (2020) 110:3081–3104 https://doi.org/10.1007/s11005-020-01323-8 The duality between F-theory and the heterotic string in D = 8 with two Wilson lines Adrian Clingher1 · Thomas Hill2 · Andreas Malmendier2 Received: 2 June 2020 / Revised: 20 July 2020 / Accepted: 30 July 2020 / Published online: 7 August 2020 © Springer Nature B.V. 2020 Abstract We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank 16. Weconstruct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial Higgsing of the heterotic string gauge algebra using two Wilson line parameters. Keywords F-theory · String duality · K3 surfaces · Jacobian elliptic fibrations Mathematics Subject Classification 14J28 · 14J81 · 81T30 1 Introduction In a standard compactification of the type IIB string theory, the axio-dilaton field τ is constant and no D7-branes are present. Vafa’s idea in proposing F-theory [51] was to simultaneously allow a variable axio-dilaton field τ and D7-brane sources, defining at a new class of models in which the string coupling is never weak. These compactifications of the type IIB string in which the axio-dilaton field varies over a base are referred to as F-theory models. -
Aspects of Supersymmetry and Its Breaking
1 Aspects of Supersymmetry and its Breaking a b Thomas T. Dumitrescu ∗ and Zohar Komargodski † aDepartment of Physics, Princeton University, Princeton, New Jersey, 08544, USA bSchool of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, USA We describe some basic aspects of supersymmetric field theories, emphasizing the structure of various supersym- metry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry current and the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersym- metric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered at the Carg´ese Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERN Winter School 2011 on “Supergravity, Strings, and Gauge Theory.” 1. Supersymmetric Theories In this review we will describe some basic aspects of SUSY field theories, emphasizing the structure 1.1. Supermultiplets and Superfields of various supermultiplets – especially those con- A four-dimensional theory possesses = 1 taining the supersymmetry current. In particu- supersymmetry (SUSY) if it contains Na con- 1 lar, we will show how these supercurrents can be served spin- 2 charge Qα which satisfies the anti- used to study the dynamics of supersymmetric commutation relation field theories, SUSY-breaking, and supergravity. ¯ µ Qα, Qα˙ = 2σαα˙ Pµ . (1) We begin by recalling basic facts about super- { } multiplets and superfields. A set of bosonic and Here Q¯ is the Hermitian conjugate of Q . (We α˙ α fermionic operators B(x) and F (x) fur- will use bars throughout to denote Hermitian con- i i nishes a supermultiplet{O if these} operators{O satisfy} jugation.) Unless otherwise stated, we follow the commutation relations of the schematic form conventions of [1].