Hitchin Equation, Singularity, and N=2 Superconformal Field Theories

Abstract

We argue that Hitchin's equation determines not only the low energy effective theory but also describes the UV theory of four dimensional N=2 superconformal field theories when we compactify six dimensional $A_N$ $(0,2)$ theory on a punctured Riemann surface. We study the singular solution to Hitchin's equation and the Higgs field of solutions has a simple pole at the punctures; We show that the massless theory is associated with Higgs field whose residual is a nilpotent element; We identify the flavor symmetry associated with the puncture by studying the singularity of closure of the moduli space of solutions with the appropriate boundary conditions. For the mass-deformed theory the residual of the Higgs field is a semi-simple element, we identify the semi-simple element by arguing that the moduli space of solutions of mass-deformed theory must be a deformation of the closure of the moduli space of the massless theory. We also study the Seiberg-Witten curve by identifying it as the spectral curve of the Hitchin's system. The results are all in agreement with Gaiotto's results derived from studying the Seiberg-Witten curve of four dimensional quiver gauge theory.

42 pages, 20 figures, Hitchin's equation for N=2 theory is derived by comparing different order of compactification of six dimensional theory on T^2\times Σ. More discussion about flavor symmetries. Typos are corrected

References in corpus (17)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• S-duality in N=2 supersymmetric gauge theories
• Webs of five-branes and N=2 superconformal field theories
• On AGT relation in the case of U(3)
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Gauge Theory Loop Operators and Liouville Theory
• On non-conformal limit of the AGT relations
• Recursion relations in CFT and N=2 SYM theory
• Six-dimensional D_N theory and four-dimensional SO-USp quivers
• The Power of Nekrasov Functions
• Loop operators and S-duality from curves on Riemann surfaces
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• Proving AGT relations in the large-c limit
• New Seiberg Dualities from N=2 Dualities
• N=2 SU Quiver with USP Ends or SU Ends with Antisymmetric Matter

Cited by in corpus (34)

• Mirrors of 3d Sicilian theories
• General Argyres-Douglas Theory
• Wild Quiver Gauge Theories
• Tinkertoys for the D_N series
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Proving AGT conjecture as HS duality: extension to five dimensions
• Tinkertoys for the Twisted D-Series
• A direct proof of AGT conjecture at beta = 1
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d $(2,0)$ theories
• M5 brane and four dimensional N=1 theories I
• N=2 gauge theories and degenerate fields of Toda theory
• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• More Three Dimensional Mirror Pairs
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
• W_3 irregular states and isolated N=2 superconformal field theories
• A slow review of the AGT correspondence
• N=2 Generalized Superconformal Quiver Gauge Theory
• M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems
• BPS Graphs: From Spectral Networks to BPS Quivers
• Generalized matrix models and AGT correspondence at all genera
• Irregular conformal block, spectral curve and flow equations
• The BPS Spectrum Generator In 2d-4d Systems
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Little String Origin of Surface Defects
• Tau-functions for Quiver Gauge Theories
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories
• Irregular matrix model with $\mathcal{W}$ symmetry
• Highest Weight Generating functions for hyperKahler T*(G/H) spaces
• On low rank 4d $\mathcal{N}=2$ SCFTs
• The Euler anomaly and scale factors in Liouville/Toda CFTs