Branes, Calabi-Yau Spaces, and Toroidal Compactification of the N=1 Six-Dimensional E_8 Theory

Abstract

We consider compactifications of the N=1, d=6, E_8 theory on tori to five, four, and three dimensions and learn about some properties of this theory. As a by-product we derive the SL(2,\IZ) duality of the N=2, d=4, SU(2) theory with N_f=4. Using this theory on a D-brane probe we shed new light on the singularities of F-theory compactifications to eight dimensions. As another application we consider compactifications of F-theory, M-theory and the IIA string on (singular) Calabi-Yau spaces where our theory appears in spacetime. Our viewpoint leads to a new perspective on the nature of the singularities in the moduli space and their spacetime interpretations. In particular, we have a universal understanding of how the singularities in the classical moduli space of Calabi--Yau spaces are modified by worldsheet instantons to singularities in the moduli space of the corresponding conformal field theories.

40 pages, 2 figures, harvmac with epsf. Minor corrections, references added

References in corpus (26)

• Evidence for F-Theory
• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• Compactifications of F-Theory on Calabi--Yau Threefolds -- I
• Mirror Symmetry in Three Dimensional Gauge Theories
• Geometric Engineering of Quantum Field Theories
• Geometric Singularities and Enhanced Gauge Symmetries
• Phase Transitions In M-Theory And F-Theory
• F-theory and Orientifolds
• Comments on String Dynamics in Six Dimensions
• Extremal Transitions and Five-Dimensional Supersymmetric Field Theories
• Small $E_8$ Instantons and Tensionless Non-critical Strings
• The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD
• An N=2 Superconformal Fixed Point with E_6 Global Symmetry
• Superconformal Fixed Points with E_n Global Symmetry
• F-Theory at Constant Coupling
• Gauge Dynamics And Compactification To Three Dimensions
• Non-trivial Fixed Points of The Renormalization Group in Six Dimensions
• Probing F-theory With Branes
• BPS States of Exceptional Non-Critical Strings
• Low-Energy Analysis of $M$ and $F$ Theories on Calabi-Yau Threefolds
• The SO(32) Heterotic String on a K3 Surface
• M-Theory on a Calabi-Yau Manifold
• New Higgs Transitions between Dual N=2 String Models
• Toroidal Compactification of Heterotic 6D Non-Critical Strings Down to Four Dimensions
• Exceptional SW Geometry from ALE Fibrations
• Physical Interpretation Of Certain Strong Coupling Singularities

Cited by in corpus (121)

• Vector Bundles And F Theory
• Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams
• Brane Dynamics and Gauge Theory
• Five-Dimensional Supersymmetric Gauge Theories and Degenerations of Calabi-Yau Spaces
• Branes, Superpotentials and Superconformal Fixed Points
• Orbifold Resolution by D-Branes
• On the Classification of 6D SCFTs and Generalized ADE Orbifolds
• 6d Conformal Matter
• Webs of five-branes and N=2 superconformal field theories
• 5-dim Superconformal Index with Enhanced En Global Symmetry
• Positivity Constraints on Anomalies in Supersymmetric Gauge Theories
• Arithmetic and Attractors
• Tate's algorithm and F-theory
• Large N Superconformal Gauge Theories and Supergravity Orientifolds
• E-Strings and N=4 Topological Yang-Mills Theories
• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
• 6D SCFTs and Phases of 5D Theories
• Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory
• RG Fixed Points in Six Dimensions via Branes at Orbifold Singularities
• Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States
• (0,2) Heterotic Gauge Couplings and their M-Theory Origin
• The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs
• Non-Lagrangian Theories from Brane Junctions
• 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities
• More on the N=2 superconformal systems of type $D_p(G)$
• Mapping 6D N = 1 supergravities to F-theory
• Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories
• Infinitely many N=2 SCFT with ADE flavor symmetry
• Elliptic Genus of E-strings
• Brane webs and $O5$-planes
• Fiber-Base Duality and Global Symmetry Enhancement
• E-String Theory on Riemann Surfaces
• Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations
• Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions)
• D-type Conformal Matter and SU/USp Quivers
• Supersymmetric Casimir Energy and the Anomaly Polynomial
• Field Theory Questions for String Theory Answers
• Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs
• SUSY field theories, integrable systems and their stringy/brane origin -- II
• Compactifications of 6d $N=(1,0)$ SCFTs with non-trivial Stiefel-Whitney classes
• Rank $Q$ E-string on a torus with flux
• Classification of 5d N=1 gauge theories
• Geometric classification of 4d $\mathcal{N}=2$ SCFTs
• Brane webs, $5d$ gauge theories and $6d$ $\mathcal{N}$$=(1,0)$ SCFT's
• 6D Fractional Quantum Hall Effect
• E + E $\to$ H
• Dimer Models and Integrable Systems
• On Exceptional Instanton Strings
• Geometric constraints in dual F-theory and heterotic string compactifications
• Instanton Expansions for Mass Deformed N=4 Super Yang-Mills Theories
• A Supergravity Dual of a (1,0) Field Theory in Six Dimensions
• Non-Critical Strings, Del Pezzo Singularities And Seiberg-Witten Curves
• Gauged 2-form Symmetries in 6D SCFTs Coupled to Gravity
• Seiberg-Witten Curve for the E-String Theory
• F-Theory and N=1 SCFTs in Four Dimensions
• Towards Classification of 5d SCFTs: Single Gauge Node
• BPS Spectrum of 5 Dimensional Field Theories, (p,q) Webs and Curve Counting
• 5d and 6d Supersymmetric Gauge Theories: Prepotentials from Integrable Systems
• Discrete theta angle from an O5-plane
• Adding flavor to Dijkgraaf-Vafa
• Non-Abelian Conifold Transitions and N=4 Dualities in Three Dimensions
• On the Twisted (2,0) and Little-String Theories
• 4d $S$-duality wall and $SL(2,\mathbb{Z})$ relations
• Affine 7-brane Backgrounds and Five-Dimensional $E_N$ Theories on $S^1$
• Results in susy field theory from 3-brane probe in F-theory
• Gauge Unification in Extra Dimensions: Power Corrections vs. Higher-Dimension Operators
• Orientifold and F-theory Duals of CHL Strings
• Curious patterns of IR symmetry enhancement
• U-duality Twists and Possible Phase Transitions in 2+1D Supergravity
• On the Elliptic Genus of Three E-strings and Heterotic Strings
• Topological string amplitudes for the local half K3 surface
• Correlators of the Global Symmetry Currents of 4D and 6D Superconformal Theories
• Probing 7-branes on Orbifolds
• On the Strong Coupling Dynamics of heterotic string theory on C^3/Z_3
• Green-Schwarz Automorphisms and 6D SCFTs
• On the compactification of 5d theories to 4d
• Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d
• Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications
• E-string Quantum Curve
• Exceptional String: Instanton Expansions and Seiberg-Witten Curve
• Symmetry enhancement and duality walls in 5d gauge theories
• Exploring the orthosymplectic zoo
• On the 3d compactifications of 5d SCFTs associated with SU(N+1) gauge theories
• An Index for Ray Operators in 5d $E_n$ SCFTs
• Constructions of Non Commutative Instantons on $T^4$ and $K_3$
• Dilatonic states near holographic phase transitions
• Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
• The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories
• Chiral Compactifications of 6D Conformal Theories
• Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs
• W(E_10) Symmetry, M-Theory and Painleve Equations
• Seiberg-Witten prepotential for E-string theory and global symmetries
• On The M(atrix)-Model For M-Theory On $T^6$
• Seiberg-Witten prepotential for E-string theory and random partitions
• Refined topological vertex for a 5D $Sp(N)$ gauge theory with antisymmetric matter
• The Ruled Vertex and Nontoric del Pezzo Surfaces
• Aspects of 5d Seiberg-Witten Theories on ${\mathbb{S}}^1$
• Thermodynamic limit of the Nekrasov-type formula for E-string theory
• A reduced BPS index of E-strings
• A 5d perspective on the compactifications of 6d SCFTs to 4d $\mathcal{N}=1$ SCFTs
• Unitary matrix with a Penner-like potential also yields N_f=2
• Twisting E8 Five-Branes
• Trinions for the $3d$ compactification of the $5d$ rank 1 $E_{N_f+1}$ SCFTs
• Local Gromov-Witten invariants of Blowups of Fano surfaces
• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
• Compactifying 5d superconformal field theories to 3d
• The Geometry of Decoupling Fields
• D-Brane Probes and Mirror Symmetry
• Rank $Q$ E-String on Spheres with Flux
• E-strings, $F_4$, and $D_4$ triality
• Quantum Deconstruction of 5D SQCD
• 3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• Seiberg-Witten Curve for E-String Theory Revisited
• Umbral Moonshine and String Duality
• Electric-Magnetic Duality in a Class of $G_2$-Compactifications of M-theory
• Comments on the Nekrasov-type formula for E-string theory
• BPS index and 4d N=2 superconformal field theories
• Correlation functions of the global E8 symmetry currents in the Heterotic 5-brane theory
• Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers
• Aspects of N=1 String Dynamics