U-duality and M-Theory

Abstract

This work is intended as a pedagogical introduction to M-theory and to its maximally supersymmetric toroidal compactifications, in the frameworks of 11D supergravity, type II string theory and M(atrix) theory. U-duality is used as the main tool and guideline in uncovering the spectrum of BPS states. We review the 11D supergravity algebra and elementary 1/2-BPS solutions, discuss T-duality in the perturbative and non-perturbative sectors from an algebraic point of view, and apply the same tools to the analysis of U-duality at the level of the effective action and the BPS spectrum, with a particular emphasis on Weyl and Borel generators. We derive the U-duality multiplets of BPS particles and strings, U-duality invariant mass formulae for 1/2- and 1/4-BPS states for general toroidal compactifications on skew tori with gauge backgrounds, and U-duality multiplets of constraints for states to preserve a given fraction of supersymmetry. A number of mysterious states are encountered in D<=3, whose existence is implied by T-duality and 11D Lorentz invariance. We then move to the M(atrix) theory point of view, give an introduction to Discrete Light Cone Quantization (DLCQ) in general and DLCQ of M-theory in particular. We discuss the realization of U-duality as electric-magnetic dualities of the Matrix gauge theory, display the Matrix gauge theory BPS spectrum in detail, and discuss the conjectured extended U-duality group in this scheme.

132 pages, 37 tables, Latex2e; v3: minor corrections, text reformatted, update of refs, note added about boundaries of M-theory moduli space, final version to appear in Phys. Rept

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• Quantum Field Theory on Noncommutative Spaces
• Superstring Cosmology
• Microscopic Formulation of Black Holes in String Theory
• Double Field Theory: A Pedagogical Review
• M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory
• Generalized Geometry and M theory
• Introduction to M(atrix) theory and noncommutative geometry
• Introduction to M(atrix) theory and noncommutative geometry, Part II
• Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB
• The gauge structure of generalised diffeomorphisms
• Instabilities of Black Strings and Branes
• The E_{11} origin of all maximal supergravities
• Duality Invariant Actions and Generalised Geometry
• SO(5,5) duality in M-theory and generalized geometry
• The local symmetries of M-theory and their formulation in generalised geometry
• Gauged Supergravities, Tensor Hierarchies, and M-Theory
• Exotic Branes in String Theory
• Exotic branes and non-geometric backgrounds
• Duality in linearized gravity
• Two-Loop Superstrings I, Main Formulas
• Two-Loop Superstrings II, The Chiral Measure on Moduli Space
• Spacelike Singularities and Hidden Symmetries of Gravity
• Counting BPS Blackholes in Toroidal Type II String Theory
• On the modular structure of the genus-one Type II superstring low energy expansion
• Universal BPS structure of stationary supergravity solutions
• Fusion of conformal interfaces
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• BPS black holes, quantum attractor flows and automorphic forms
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• A note on spin-s duality
• E_{11} origin of Brane charges and U-duality multiplets
• Non-geometric backgrounds in string theory
• The black-hole/qubit correspondence: an up-to-date review
• Multiple Membranes in M-theory
• E10 and SO(9,9) invariant supergravity
• The group theory of oxidation
• $R^4$ couplings and automorphic unipotent representations
• Black holes in N=8 supergravity from SO(4,4) hidden symmetries
• Sharpening the Distance Conjecture in Diverse Dimensions
• Minimal representations, spherical vectors, and exceptional theta series I
• Boundary Terms in Generalized Geometry and doubled field theory
• E(11)-extended spacetime and gauged supergravities
• Heterotic/Type II Triality and Instantons on $K_3$
• The effective action of type II Calabi-Yau orientifolds
• BPS Black Hole Degeneracies and Minimal Automorphic Representations
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• Trapped surfaces, horizons and exact solutions in higher dimensions
• Lectures on Two-Loop Superstrings
• The D^4 R^4 term in type IIB string theory on T^2 and U-duality
• Brane Effective Actions, Kappa-Symmetry and Applications
• The group theory of oxidation II: Cosets of non-split groups
• Gauged Linear Sigma Model for Exotic Five-brane
• A worldsheet extension of O(d,d;Z)
• Eisenstein series and automorphic representations
• The Geometry, Branes and Applications of Exceptional Field Theory
• Gauge Bundles and Born-Infeld on the Noncommutative Torus
• Non-geometric branes are DFT monopoles
• Counting supersymmetric branes
• G+++ Invariant Formulation of Gravity and M-Theories: Exact Intersecting Brane Solutions
• D-Brane Wess-Zumino Terms and U-Duality
• New Phases of Near-Extremal Branes on a Circle
• Loops in exceptional field theory
• A Multi-Boundary AdS Orbifold and DLCQ Holography: A universal holographic description of extremal black hole horizons
• Hidden Symmetries, Central Charges and All That
• Brane dynamics, central charges and E_{11}
• M-theory and E10: Billiards, Branes, and Imaginary Roots
• World-volume Effective Actions of Exotic Five-branes
• Charge multiplets and masses for E(11)
• IIB supergravity and E10
• On the Dielectric Effect for Gravitational Waves
• Exotic Branes and Nonperturbative Seven Branes
• Hyperbolic billiards of pure D=4 supergravities
• $D^6{R}^4$ amplitudes in various dimensions
• Pinned Branes and New Non Lorentz Invariant Theories
• A new AdS_4/CFT_3 Dual with Extended SUSY and a Spectral Flow
• Generalised geometry from the ground up
• Hidden Symmetries and Dirac Fermions
• Exotic branes and non-geometric fluxes
• Rotating string in doubled geometry with generalized isometries
• Remarks on the non-Riemannian sector in Double Field Theory
• T-Duality and Ramond-Ramond Backgrounds in the Matrix Model
• Gradient Representations and Affine Structures in AE(n)
• Strings, Branes and Extra Dimensions
• Unification of Decoupling Limits in String and M-theory
• $P$ fluxes and exotic branes
• A master exceptional field theory
• Non-Riemannian geometry of M-theory
• BMN Vacua, Superstars and Non-Abelian T-duality
• Supersymmetric Domain Walls
• E${}_9$ exceptional field theory II. The complete dynamics
• Manifest Duality Invariance for the Partially Massless Graviton
• Currents and Moduli in the (4,0) theory
• Eisenstein Series in String Theory
• Canonical approach to Courant brackets for D-branes
• M5 algebra and SO(5,5) duality
• Hyperbolic Kac Moody Algebras and Einstein Billiards
• Duality group actions on fermions
• Manifest Duality for Partially Massless Higher Spins
• Five-brane actions in double field theory
• Weaving the Exotic Web
• Worldsheet Description of Exotic Five-brane with Two Gauged Isometries
• Casimir Effect on the Radius Stabilization of the Noncommutative Torus
• E8 duality and dual gravity
• Logarithmic Corrections to Black Hole Entropy: the Non-BPS Branch
• M-theory resolution of four-dimensional cosmological singularities via U-duality
• Aspects of higher curvature terms and U-duality
• E10 Cosmology
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• Open Wilson Lines and Group Theory of Noncommutative Yang-Mills Theory in Two Dimensions
• Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes
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• Worldsheet Formalism for Decoupling Limits in String Theory
• Worldsheet Instanton Corrections to Five-branes and Waves in Double Field Theory
• Species scale, worldsheet CFTs and emergent geometry
• Heterotic T-fects, 6D SCFTs, and F-Theory
• A Note on the BPS Spectrum of the Matrix Model
• Generalised space-time and duality
• Effective Action for Non-Geometric Fluxes from Duality Covariant Actions
• Eisenstein series for infinite-dimensional U-duality groups
• Matrix Theory Reloaded: A BPS Road to Holography
• Hyperbolic Weyl groups and the four normed division algebras
• World-volume Effective Action of Exotic Five-brane in M-theory
• The D^6 R^4 term in type IIB string theory on T^2 and U-duality
• Exotic branes in Exceptional Field Theory: the SL(5) duality group
• Hodge Duality and Cosmological Constant
• Supergravity and M-Theory
• Perturbation Theory From Automorphic Forms
• Two Conjectures on Gauge Theories, Gravity, and Infinite Dimensional Kac-Moody Groups
• Dp-branes, NS5-branes and U-duality from nonabelian (2,0) theory with Lie 3-algebra
• Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond
• M5-brane geometries, T-duality and fluxes
• A $\mathcal{N}=2$ Supersymmetric $Ad{S}_4$ Solution in M-theory with Purely Magnetic Flux
• Constraints on Automorphic Forms of Higher Derivative Terms from Compactification
• Unwinding strings in semi-flatland
• The D^6 R^4 term from three loop maximal supergravity
• Non-abelian U-duality at work
• Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories
• Cancellation of divergences up to three loops in exceptional field theory
• New approach to solution generation using SL(2,R)-duality of a dimensionally reduced space in five-dimensional minimal supergravity and new black holes
• M-Theory and Maximally Supersymmetric Gauge Theories
• U-Duality and the Compactified Gauss-Bonnet Term
• On higher spin realizations of K(E10)
• Hitchin functionals are related to measures of entanglement
• Four-derivative couplings and BPS dyons in heterotic CHL orbifolds
• 1/4-BPS states on noncommutative tori
• Anisotropic Compactification of Nonrelativistic M-Theory
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Exact thresholds and instanton effects in D=3 string theories
• The monodromy of T-folds and T-fects
• Low momentum expansion of one loop amplitudes in heterotic string theory
• Non-geometric fluxes & tadpole conditions for exotic branes
• M-theory and the string genus expansion
• Exotic Dual of Type II Double Field Theory
• Dual expansions of N=4 super Yang-Mills theory via IIB superstring theory
• U-duality from Matrix Membrane Partition Function
• Automorphic Instanton Partition Functions on Calabi-Yau Threefolds
• Emergence of $R^4$-terms in M-theory
• Geometric Transitions on non-Kaehler Manifolds
• Semi-doubled Sigma Models for Five-branes
• M-theory and automorphic scattering
• M-branes on U-folds
• Tachyon condensation and `bounce' in the D1-D5 system
• A Type IIB Matrix Model via non-Abelian T-dualities
• Type II strings are Exceptional
• The Kaluza-Klein Monopole in a Massive IIA Background
• E10 Orbifolds
• Arithmetic and Hyperbolic Structures in String Theory
• Connecting M-theory and type IIB parameterizations in Exceptional Field Theory
• Time-like T-duality algebra
• Aspects of U-duality in BLG models with Lorentzian metric 3-algebras
• Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
• Some Algebraic Aspects of Half-BPS Bound States in M-Theory
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Topics in Two-Loop Superstring Perturbation Theory
• W(E_10) Symmetry, M-Theory and Painleve Equations
• Classifying Supersymmetric Solutions in 3D Maximal Supergravity
• Symmetries of M-theory and free Lie superalgebras
• O-folds: Orientifolds and Orbifolds in Exceptional Field Theory
• Wrapping rules (in) string theory
• Quantum Stabilization of Compact Space by Extra Fuzzy Space
• Supersymmetric brane actions from interpolating dualisations
• A Kaluza-Klein Approach to Double and Exceptional Field Theory
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• Mesons from (non) Abelian T-dual backgrounds
• Semi-doubled Gauged Linear Sigma Model for Five-branes of Codimension Two
• Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals
• Complex Structures, T-duality and Worldsheet Instantons in Born Sigma Models
• The different faces of branes in Double Field Theory
• Non-geometric Five-branes in Heterotic Supergravity
• Two-loop effective potential in noncommutative scalar field theory
• A review of the D1/D5 system and five dimensional black hole from supergravity and brane viewpoint
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• Five-brane Current Algebras in Type II String Theories
• Non-Abelian Supertubes
• Higher spin fields from indefinite Kac-Moody algebras
• A Special Class of Rank 10 and 11 Coxeter Groups
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• Duality-Symmetric Approach to General Relativity and Supergravity
• Closed string deformations in open string field theory III: $\mathcal{N}=2$ worldsheet localization
• BPS States and Automorphisms
• Degenerations of K3, Orientifolds and Exotic Branes
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• Fully Off-shell Effective Action and its Supersymmetry in Matrix Theory II
• (m,n)-String in (p,q)-String and (p,q)-Five Brane Background
• Minimal surfaces with mixed three-form flux
• Lectures on the Microscopic Modeling of the 5-dim. Black Hole of IIB String Theory and the $D_1-D_5$ System
• Integrability structures in string theory
• Mock modularity at work, or black holes in a forest
• Little Groups of Preon Branes
• Exotic Brane Junctions from F-theory
• Unstable solitons on noncommutative tori and D-branes
• On 10 dimensional Exceptional Drinfel'd Algebras
• A note on T-folds and T3 fibrations
• Polyvector deformations of Type IIB backgrounds
• Connections between Kac-Moody algebras and M-theory
• Classical worldvolumes as generalised geodesics
• Ehlers symmetry at the next derivative order
• Black holes and the phases of brane thermodynamics
• Effects in Gauge Theories and a Harmonic Function on $E_{10}$
• Localization of Gauged Linear Sigma Model for KK5-branes
• Gauge Five-brane Solutions of Co-dimension Two in Heterotic Supergravity
• Remarks on E11 approach
• World-volume Effective Theories of Locally Non-geometric Branes
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• Classification of six-dimensional Leibniz algebras ${\mathcal E}_3
• A note on gauge fixing in supergravity/Kac-Moody correspondences
• Constraining non--BPS interactions from counterterms in three loop maximal supergravity
• Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors
• Exotic branes and mixed-symmetry potentials I: predictions from $E_{11}$ symmetry
• Exotic Branes from del Pezzo Surfaces
• New M(atrix)-models for Commutative and Noncommutative Gauge Theories
• Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories
• Gauge field fluxes and Bianchi identities in extended field theories
• The M-algebra completes the hierarchy of Super-Exceptional Tangent Spaces
• A New Lorentz Violating Nonlocal Field Theory From String-Theory
• The invariant action for solitonic 5-branes
• $\eta$-symbols in exceptional field theory
• Gepner-like boundary states on $T^4$
• 10d N=1 Massless BPS supermultiplets
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• Mysterious Triality and the Exceptional Symmetry of Loop Spaces
• Duality rules for more mixed-symmetry potentials
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• Exotic Aspects of Extended Field Theories
• A T-Fold Black Hole in Doubled Type IIB
• Spinorial geometry, horizons and superconformal symmetry in six dimensions
• Codimension-2 brane solutions of maximal supergravities in 9, 8, and 7 dimensions
• T-duality and entanglement entropy for NS five-branes
• Branes and Polytopes
• Duality and Combinatorics of Long Strings in ADS3
• Implication of U-duality for black branes in M/string theory
• Exact Non-Abelian Supertubes
• Oblique DLCQ M-theory and Multiple M2-branes