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| AGT correspondence | |
|---|---|
| Name | AGT correspondence |
| Introduced | 2009 |
| Founders | Alday–Gaiotto–Tachikawa |
| Field | Mathematical physics |
| Related | Conformal field theory, Supersymmetric gauge theory, Integrable systems |
AGT correspondence
The AGT correspondence is a conjectural duality linking two families of theories: four-dimensional Seiberg–Witten theory-type supersymmetric gauge theories and two-dimensional Liouville field theory-type conformal field theories. It was proposed in 2009 by Luis Alday, Davide Gaiotto, and Yuji Tachikawa and has motivated connections between Nekrasov partition function, Virasoro algebra, and moduli spaces such as Moduli space of curves and Hitchin system.
The AGT correspondence relates partition functions and correlation functions across disparate settings: instanton counting in N=2 supersymmetric Yang–Mills theory on R^4 and conformal blocks of Liouville theory on punctured Riemann surfaces. It links objects like the Nekrasov partition function, the Virasoro algebra, the W-algebra, and the Seiberg–Witten curve to structures on moduli spaces such as the Teichmüller space and Hitchin moduli space. Early checks compared results for SU(2) gauge theories and four-point conformal blocks on the sphere, and subsequent generalizations involve SU(N), higher-rank W_N algebra, and relations to M-theory constructions.
The AGT proposal emerged from work on S-duality and compactifications of the 6d (2,0) superconformal field theory on Riemann surfaces by Gaiotto, building on insights from Seiberg–Witten theory and the classification of superconformal N=2 field theories. Alday, Gaiotto, and Tachikawa conjectured a precise equality between the instanton partition functions computed by Nekrasov and conformal blocks of Liouville field theory, inspired by earlier results on the Omega background and the geometry of M5-branes in M-theory. Subsequent confirmations involved checks by connections to the AGT-like dualities between five-dimensional theories compactified on circles, refinements via the refined topological vertex, and mathematical rigor brought by work on the Geometric Langlands program and the Representation theory of W-algebras.
In its prototypical form the correspondence states an equality between the instanton part of the Nekrasov partition function for four-dimensional N=2 superconformal SU(2) quiver gauge theories and Liouville conformal blocks on punctured Riemann surfaces with specified intermediate states. For higher-rank SU(N), the correspondence replaces the Virasoro algebra with the W_N algebra and Liouville theory with Toda field theory. The mapping involves parameters: the Omega background deformation parameters (ε1, ε2) correspond to the central charge and conformal weights in the two-dimensional side; the Coulomb moduli map to intermediate momenta in conformal blocks; and mass parameters map to external insertions such as primary fields in Liouville theory. Mathematically precise incarnations use equivariant cohomology of instanton moduli spaces, representation theory of affine Lie algebras, and the geometry of moduli of Higgs bundles.
Explicit checks include the equality for the four-point conformal block on the sphere and the instanton partition function for SU(2), matching expansions order-by-order using combinatorial identities involving Young diagrams and the Nekrasov formula. Generalizations to linear and circular quivers map to multi-point conformal blocks on spheres and tori, respectively, and computations often employ the Nekrasov–Okounkov formula, the Shapovalov form for the Virasoro module, and the Alday–Gaiotto–Tachikawa dictionary for mapping parameters. Higher-rank tests compare SU(3) instanton expansions with W_3 conformal blocks in Toda theory, and checks use the Jack polynomials and the Macdonald polynomials in refined settings.
Physically, the correspondence arises from compactifying the six-dimensional (2,0) theory on a punctured Riemann surface with codimension-two defects engineered by M5-branes in M-theory, producing four-dimensional N=2 theories whose S-duality frames correspond to different pair-of-pants decompositions of the surface. The two-dimensional conformal field theory emerges from the worldvolume theory on the defect or from the effective theory on intersecting branes, linking S-duality and modular transformations of conformal blocks. Relations also appear in the study of Topological string theory and the Geometric engineering of gauge theories via local Calabi–Yau threefolds.
Applications include new computational techniques for prepotentials and BPS spectra in N=2 theories, insights into the Geometric Langlands program, and constructions of quantum integrable systems via the Nekrasov–Shatashvili limit. Extensions cover refined and five-dimensional lifts relating to Refined topological string theory, relations to Quantum Teichmüller theory, and connections with the Matrix models and Painlevé equations. The correspondence has influenced progress on the Representation theory of W-algebras, the study of Hitchin systems, and exact results in supersymmetric localization for theories on S^4 and other manifolds.
Active problems include rigorous proofs beyond large classes of examples, a deeper categorical understanding via derived categories in the Geometric Langlands program, extensions to non-conformal and non-Lagrangian theories such as theories of class S with irregular punctures, and the role of surface and line defects in richer dualities. Further research seeks to clarify the refined and five-dimensional AGT-like correspondences, connections to Quantum groups, and applications to enumerative geometry on Calabi–Yau spaces. Progress also targets computational tools for higher-rank W-algebra conformal blocks and a systematic classification of defects from the six-dimensional perspective.