Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems

Abstract

We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an A_{n-1} singularity over a Riemann surface. These models realize the N=2 SU(n) superconformal gauge systems recently studied by Gaiotto and collaborators. Employing large N dualities we show why the partition function of topological strings in these backgrounds is captured by the chiral blocks of A_{n-1} Toda systems and derive the dictionary recently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zero Riemann surfaces, we show how these systems can also be realized by Penner-like matrix models with logarithmic potentials. The Seiberg-Witten curve can be understood as the spectral curve of these matrix models which arises holographically at large N. In this context the Nekrasov deformation maps to the beta-ensemble of generalized matrix models, that in turn maps to the Toda system with general background charge. We also point out the notion of a double holography for this system, when both n and N are large.

41 pages, 11 figures

References in corpus (21)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Solutions Of Four-Dimensional Field Theories Via M Theory
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Topological Strings and Integrable Hierarchies
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• A Perturbative Window into Non-Perturbative Physics
• Worldsheet Derivation of a Large N Duality
• On AGT relation in the case of U(3)
• Asymptotically free N=2 theories and irregular conformal blocks
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Geometric Transitions and N=1 Quiver Theories
• Seiberg-Witten Theory and Random Partitions
• The Power of Nekrasov Functions
• Algebraic methods in random matrices and enumerative geometry
• Conformal Field Theory Techniques in Random Matrix models
• N=1 Supersymmetry, Deconstruction, and Bosonic Gauge Theories
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry
• Mirror symmetry and Exact Solution of 4D N=2 Gauge Theories I

Cited by in corpus (158)

• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• S-duality and 2d Topological QFT
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Wild Quiver Gauge Theories
• Knot Homology from Refined Chern-Simons Theory
• Para-Liouville/Toda central charges from M5-branes
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• A & B model approaches to surface operators and Toda theories
• The Quiver Matrix Model and 2d-4d Conformal Connection
• The Omega deformed B-model for rigid N=2 theories
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Hitchin systems, N=2 gauge theories and W-gravity
• Extended Holomorphic Anomaly in Gauge Theory
• Proving AGT conjecture as HS duality: extension to five dimensions
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• On KP-integrable Hurwitz functions
• Ding-Iohara-Miki symmetry of network matrix models
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Explicit examples of DIM constraints for network matrix models
• Seiberg-Witten theory, matrix model and AGT relation
• Seiberg-Witten theory as a Fermi gas
• On the complete perturbative solution of one-matrix models
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• Vertices, Vortices & Interacting Surface Operators
• Proving AGT relations in the large-c limit
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Liouville/Toda central charges from M5-branes
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Matrix Models for Random Partitions
• On AGT Conjecture for Pure Super Yang-Mills and W-algebra
• The matrix model version of AGT conjecture and CIV-DV prepotential
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Non-conformal limit of AGT relation from the 1-point torus conformal block
• Gravity factorized
• The uses of the refined matrix model recursion
• N=2 gauge theories and degenerate fields of Toda theory
• Refined Chern-Simons Theory and Topological String
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Worldsheet Realization of the Refined Topological String
• Gauge/Liouville Triality
• CFT and topological recursion
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Check of AGT Relation for Conformal Blocks on Sphere
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
• W_3 irregular states and isolated N=2 superconformal field theories
• Decomposing Nekrasov Decomposition
• Holomorphic blocks and the 5d AGT correspondence
• A slow review of the AGT correspondence
• Non-Perturbative Quantum Geometry III
• Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
• Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
• Parafermionic Liouville field theory and instantons on ALE spaces
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• S-duality as a beta-deformed Fourier transform
• Triality in Little String Theories
• N=2 Generalized Superconformal Quiver Gauge Theory
• On elementary proof of AGT relations from six dimensions
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• Selberg Integral and SU(N) AGT Conjecture
• Orientifolds and the Refined Topological String
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• ADE Little String Theory on a Riemann Surface (and Triality)
• Toda Theory From Six Dimensions
• Torus HOMFLY as the Hall-Littlewood Polynomials
• Quantum Hitchin Systems via beta-deformed Matrix Models
• On AGT description of N=2 SCFT with N_f=4
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Non-Perturbative Quantum Geometry
• Matrix model from N = 2 orbifold partition function
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• On determinant representation and integrability of Nekrasov functions
• Exact solution for mean energy of 2d Dyson gas at beta = 1
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Refined Chern-Simons Theory and Knot Homology
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• Bootstrapping Calabi-Yau Quantum Mechanics
• Refined Black Hole Ensembles and Topological Strings
• β-deformed matrix model and Nekrasov partition function
• Virasoro constraint for Nekrasov instanton partition function
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Complete solution to Gaussian tensor model and its integrable properties
• Seiberg-Witten curves and double-elliptic integrable systems
• Classical Virasoro irregular conformal block
• Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory
• Non-Unitary Holography
• Virasoro irregular conformal block and beta deformed random matrix model
• Direct integration for general Omega backgrounds
• Quantum curves and conformal field theory
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• AGT conjecture and AFLT states: a complete construction
• Supersymmetric Partition Functions and a String Theory in 4 Dimensions
• Elliptic Virasoro Conformal Blocks
• Classical Virasoro irregular conformal block II
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Partition functions and the continuum limit in Penner matrix models
• Comments on scale invariant but non-conformal supersymmetric field theories
• Non-perturbative studies of N=2 conformal quiver gauge theories
• Five-Brane Webs and Highest Weight Representations
• Comments on Worldsheet Description of the Omega Background
• Logarithmic potential beta-ensembles and Feynman graphs
• Irregular conformal block, spectral curve and flow equations
• Challenges of beta-deformation
• Matching branches of non-perturbative conformal block at its singularity divisor
• Baxter's T-Q equation, SU(N)/SU(2)^{N-3} correspondence and Ω-deformed Seiberg-Witten prepotential
• $\mathcal{N}=2^{\star }$ from Topological Amplitudes in String Theory
• Refined Topological Strings on Local ${\mathbb{P}}^2$
• Orthogonal Polynomials in Mathematical Physics
• q-Painleve equation from Virasoro constraints
• Probing the Moduli Dependence of Refined Topological Amplitudes
• Beta-ensembles for toric orbifold partition function
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Fine structure in the large n limit of the non-hermitian Penner matrix model
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Notes on Mayer Expansions and Matrix Models
• D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds
• Check-Operators and Quantum Spectral Curves
• Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• AGT relation in the light asymptotic limit
• Elliptic $q,t$ matrix models
• Seiberg-Witten equations and non-commutative spectral curves in Liouville theory
• Beta Deformation and Superpolynomials of (n,m) Torus Knots
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Mirror of the refined topological vertex from a matrix model
• An(1) Affine Quiver Matrix Model
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Refined Cigar and Omega-deformed Conifold
• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• Faces of matrix models
• FQHE and $t{t}^{\ast }$ geometry
• Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model
• Irregular conformal block and its matrix model
• Note on refined topological vertex, Jack polynomials and instanton counting
• Cubic constraints for the resolvents of the ABJM matrix model and its cousins
• Dijkgraaf-Vafa conjecture and beta-deformed matrix models
• Geometric transitions, double scaling limits and gauge theories
• Exact Chern-Simons / Topological String duality
• Probabilistic conformal blocks for Liouville CFT on the torus
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
• On a path integral representation of the Nekrasov instanton partition function and its Nekrasov--Shatashvili limit
• Scattering and Strebel graphs
• Equilibrium measures in the presence of weak rational external fields
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach
• A note on the automorphism group of the root lattice of the U-dual modular group
• Intertwining operator and integrable hierarchies from topological strings