Affine SL(2) Conformal Blocks from 4d Gauge Theories

By L. Alday, Yuji Tachikawa. Letters in Mathematical Physics, 2010

240 citations recorded by the source corpus.

We study Nekrasov’s instanton partition function of four-dimensional $${\mathcal{N}=2}$$ gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/314d1b5bdb37c38a2fd41055fb731c31604cff8f

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In Geometry and Topology

1. Sergei Gukov
2. C. Vafa
3. Tudor Dimofte
4. Marcos Mariño
5. R. Dijkgraaf
6. Mina Aganagic
7. H. Kanno
8. Yuji Tachikawa

In Nuclear and High Energy Physics

1. C. Vafa
2. Sergei Gukov
3. Tudor Dimofte
4. Marcos Mariño
5. Mina Aganagic
6. E. Witten
7. R. Dijkgraaf
8. H. Kanno

Citations in the corpus

• Vortex Counting and Lagrangian 3-Manifolds (2010, background): “34) was studied earlier by Braverman and Etingof [23] and, more recently, by Nekrasov and Shatashvili [24] in connection to quantum integrable systems (see also [14,19]).”
• Instanton counting with a surface operator and the chain-saw quiver (2011, background): “, 1], it was found that the instanton partition function in the presence of the surface operator is governed by the affine symmetry ŜU(N) [18, 19, 20, 21].”
• On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture (2010, result): “Similar idea was proposed by Alday and Tachikawa in [24].”
• Quantum geometry of refined topological strings (2011, background): “A related observation that the gauge theory instanton partition function in the presence of the surface operator becomes in the NS limit the eigenfunction of the Hamiltonians of the corresponding quantum integrable systems has been previously made in [12, 14, 50, 51].”
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, result): “In the ε2 → 0 limit these surface operators become the eigenfunctions of Gaudin Hamiltonian and the elliptic Calogero-Moser Schrödinger equation, respectively, in agreement with the conjectures in [3, 96] and earlier ideas.”
• Defects and quantum Seiberg-Witten geometry (2014, methodology): “This computation is performed by treating the surface defect as a monodromy defect and applying the orbifolding procedure introduced in [15, 16].”