Categorical Tinkertoys for N=2 Gauge Theories

Abstract

In view of classification of the quiver 4d N=2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to a N=2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite-dimensional) representations of the Jacobian algebra $\mathbb{C}Q/(\partial W)$ should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal `generic' subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. There is a family of 'light' subcategories ${\mathcal{L}}_{\lambda }\subset rep(Q,W)$, indexed by points $\lambda \in N$, where $N$ is a projective variety whose irreducible components are copies of ${\mathbb{P}}^1$ in one--to--one correspondence with the simple factors of G. In particular, for a Gaiotto theory there is one such family of subcategories, ${\mathcal{L}}_{\lambda \in N}$, for each maximal degeneration of the corresponding surface $\Sigma$, and the index variety $N$ may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed-point subcategories to `fixtures' (spheres with three punctures of various kinds) and higher-order generalizations. The rules for `gluing' categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N=2 theories which are not of the Gaiotto class.

arXiv comment142 pages, 8 figures, 5 tables

References in corpus (38)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories
• N=2 dualities
• D-branes, Quivers, and ALE Instantons
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• New Phenomena in SU(3) Supersymmetric Gauge Theory
• On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• D-Brane Gauge Theories from Toric Singularities and Toric Duality
• Dimer models and toric diagrams
• The spectrum of BPS branes on a noncompact Calabi-Yau
• R-Twisting and 4d/2d Correspondences
• Fractional Branes and Boundary States in Orbifold Theories
• Kac-Moody groups and cluster algebras
• Phase Structure of D-brane Gauge Theories and Toric Duality
• Wall-Crossing from Boltzmann Black Hole Halos
• Wall-Crossing in Coupled 2d-4d Systems
• Refined, Motivic, and Quantum
• Tinkertoys for the D_N series
• N=2 Quantum Field Theories and Their BPS Quivers
• Quivers with potentials associated to triangulated surfaces
• Skew-symmetric cluster algebras of finite mutation type
• Semicanonical bases and preprojective algebras
• Quantum Wall Crossing in N=2 Gauge Theories
• BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
• BPS states and algebras from quivers
• Wall-crossing from supersymmetric galaxies
• BPS Wall Crossing and Topological Strings
• Classification of complete N=2 supersymmetric theories in 4 dimensions
• Deformed Calabi-Yau Completions
• Bound state transformation walls
• Seiberg-Witten Geometries Revisited
• Quivers with potentials associated to triangulated surfaces, Part II: Arc representations
• Spectral analysis of finite dimensional algebras and singularities
• Extended canonical algebras and Fuchsian singularities
• New Graphs of Finite Mutation Type
• Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac

Cited by in corpus (30)

• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
• More on the N=2 superconformal systems of type $D_p(G)$
• Infinitely many N=2 SCFT with ADE flavor symmetry
• Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality
• The representation type of Jacobian algebras
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics
• A slow review of the AGT correspondence
• Exponential Networks and Representations of Quivers
• Line defects and (framed) BPS quivers
• Half--Hypers and Quivers
• Towards Generalized Mirror Symmetry for Twisted Connected Sum $G_2$ Manifolds
• Geometric engineering of (framed) BPS states
• The BPS spectrum of the 4d N=2 SCFT's H_1, H_2, D_4, E_6, E_7, E_8
• Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited
• The quiver approach to the BPS spectrum of a 4d N=2 gauge theory
• Expanding the landscape of $\mathcal{N}$=2 rank 1 SCFTs
• On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems
• Regge Trajectories in $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
• Superconformal Index, BPS Monodromy and Chiral Algebras
• Confinement and duality in supersymmetric gauge theories
• Superconformal Block Quivers, Duality Trees and Diophantine Equations
• Homological S-Duality in 4d N=2 QFTs
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Quantum Line Defects and Refined BPS Spectra
• Quivers, Line Defects and Framed BPS Invariants
• $Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT
• About the Absence of Exotics and the Coulomb Branch Formula
• Higher S-dualities and Shephard-Todd groups
• Toda Systems, Cluster Characters, and Spectral Networks