Matrix Models, Geometric Engineering and Elliptic Genera

Abstract

We compute the prepotential of N=2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T^2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T^2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R^4. We study the compactifications of N=2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T^2 combines the Kahler and complex moduli of T^2 and the mass parameter into the period matrix of a genus 2 curve.

90 pages, Latex

References in corpus (14)

• The Topological Vertex
• On Geometry and Matrix Models
• The Calculus of Many Instantons
• Multi-Instanton Calculus and Equivariant Cohomology
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• All Loop Topological String Amplitudes From Chern-Simons Theory
• Hodge integrals and invariants of the unknot
• Topological Strings and Nekrasov's formulas
• Localization and Gluing of Topological Amplitudes
• Exact Superpotentials from Matrix Models
• Massive Vacua of N=1* Theory and S-duality from Matrix Models
• The Curve of Compactified 6D Gauge Theories and Integrable Systems
• Five-Dimensional Gauge Theories and Quantum Mechanical Matrix Models
• Critical Points of Glueball Superpotentials and Equilibria of Integrable Systems

Cited by in corpus (128)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• The Refined Topological Vertex
• Gauge Theory Loop Operators and Liouville Theory
• Quantum Geometry of Refined Topological Strings
• On instantons as Kaluza-Klein modes of M5-branes
• A & B model approaches to surface operators and Toda theories
• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
• Extended Holomorphic Anomaly in Gauge Theory
• On orbifolds of M-Strings
• Strings of Minimal 6d SCFTs
• Generation of Matrix Models by W-operators
• Exact Results for 't Hooft Loops in Gauge Theories on S^4
• Topological strings and 5d T_N partition functions
• Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective
• BPS Degeneracies and Superconformal Index in Diverse Dimensions
• Gauge Theories on ALE Space and Super Liouville Correlation Functions
• Seiberg-Witten theory as a Fermi gas
• Link Homologies and the Refined Topological Vertex
• Deformed Topological Partition Function and Nekrasov Backgrounds
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Matrix models for $\beta$-ensembles from Nekrasov partition functions
• The Omega Deformation From String and M-Theory
• Universal Features of BPS Strings in Six-dimensional SCFTs
• Refined BPS invariants of 6d SCFTs from anomalies and modularity
• Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• tt* Geometry in 3 and 4 Dimensions
• M-strings, Elliptic Genera and N=4 String Amplitudes
• 5d/6d DE instantons from trivalent gluing of web diagrams
• Exact quantization conditions for the relativistic Toda lattice
• Unitary Integrals and Related Matrix Models
• Higgs branch localization of $\mathcal{N}=1$ theories on $S^3\times S^1$
• E + E $\to$ H
• Dimer Models and Integrable Systems
• Worldsheet Realization of the Refined Topological String
• On Exceptional Instanton Strings
• Exact quantization conditions for cluster integrable systems
• Little strings and T-duality
• Instanton-Monopole Correspondence from M-Branes on ${\mathbb{S}}^1$ and Little String Theory
• Quiver elliptic W-algebras
• The toroidal block and the genus expansion
• Changing the preferred direction of the refined topological vertex
• M String, Monopole String and Modular Forms
• A slow review of the AGT correspondence
• Exact partition functions of Higgsed 5d $T_N$ theories
• Non-Perturbative Nekrasov Partition Function from String Theory
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• Triality in Little String Theories
• Non-Perturbative Topological Strings And Conformal Blocks
• An elliptic Virasoro symmetry in 6d
• Quiver Matrix Model and Topological Partition Function in Six Dimensions
• Dual Little Strings from F-Theory and Flop Transitions
• Self-Duality and Self-Similarity of Little String Orbifolds
• 3d-3d Correspondence Revisited
• Parabolic refined invariants and Macdonald polynomials
• Orientifolds and the Refined Topological String
• On elliptic genera of 6d string theories
• Wall-crossing, free fermions and crystal melting
• Exact results for topological strings on resolved Y(p,q) singularities
• Refined Chern-Simons versus Vogel universality
• Matrix models for 2* theories
• General Omega Deformations from Closed String Backgrounds
• Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz
• The Quantum DELL System
• Surface operators, chiral rings, and localization in N=2 gauge theories
• Instantons from Blow-up
• Refined Black Hole Ensembles and Topological Strings
• From Little String Free Energies Towards Modular Graph Functions
• The Seiberg-Witten Kahler Potential as a Two-Sphere Partition Function
• Beyond Triality: Dual Quiver Gauge Theories and Little String Theories
• 3d Mirror Symmetry from S-duality
• Quantum curves and conformal field theory
• Refined Hopf Link Revisited
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Intersecting Defects and Supergroup Gauge Theory
• ADE String Chains and Mirror Symmetry
• An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings
• Partition functions of web diagrams with an O7${}^{-}$-plane
• Five-Brane Webs and Highest Weight Representations
• Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants
• BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Dihedral Symmetries of Gauge Theories from Dual Calabi-Yau Threefolds
• A Macdonald refined topological vertex
• An elliptic topological vertex
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• Elliptic modular double and 4d partition functions
• On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories
• 6d/5d exceptional gauge theories from web diagrams
• Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane
• Topological Vertex/anti-Vertex and Supergroup Gauge Theory
• Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Surface operators in 5d gauge theories and duality relations
• Seiberg-Witten prepotential for E-string theory and global symmetries
• Branes, quivers and wave-functions
• Super-Yang-Mills theories on S4 x R
• Deconstructing Little Strings with $\mathcal{N}=1$ Gauge Theories on Ellipsoids
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case
• Thermodynamic limit of the Nekrasov-type formula for E-string theory
• M2-branes and $\mathfrak{q}$-Painlevé equations
• Seiberg-Witten prepotential for E-string theory and random partitions
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories
• Refined topological vertex for a 5D $Sp(N)$ gauge theory with antisymmetric matter
• Partition function of N=2* SYM on a large four-sphere
• Weyl invariant Jacobi forms along Higgsing trees
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Modular properties of full 5D SYM partition function
• The Enhancement of Supersymmetry in M-strings
• From topological strings to minimal models
• Elliptic Genus Derivation of 4d Holomorphic Blocks
• Symmetric Orbifold Theories from Little String Residues
• Bootstrapping ADE M-strings
• Quasi-Hopf twist and elliptic Nekrasov factor
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Towards a gauge theory interpretation of the real topological string
• E-strings, $F_4$, and $D_4$ triality
• Resurgence of Refined Topological Strings and Dual Partition Functions
• Khovanov-Rozansky cycle calculus for bipartite links
• Quantum mirror curve of periodic chain geometry
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• On refined Chern-Simons / topological string duality for classical gauge groups
• Compactified Webs and Domain Wall Partition Functions
• Chiral rings for surface operators in 4d and 5d SQCD
• Comments on the Nekrasov-type formula for E-string theory
• Elliptic Genera and q-Series Development in Analysis, String Theory, and N=2 Superconformal Field Theory