Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants

Abstract

Inspired by the work of Alday, Gaiotto and Tachikawa (AGT), we saw the revival of Poincar{é}'s uniformization problem and Fuchsian equations obtained thereof. Three distinguished aspects are possessed by Fuchsian equations. First, they are available via imposing a classical Liouville limit on level-two null-vector conditions. Second, they fall into some A_1-type integrable systems. Third, the stress-tensor present there (in terms of the Q-form) manifests itself as a kind of one-dimensional "curve". Thereby, a contact with the recently proposed Nekrasov-Shatashvili limit was soon made on the one hand, whilst the seemingly mysterious derivation of Seiberg-Witten prepotentials from integrable models become resolved on the other hand. Moreover, AGT conjecture can just be regarded as a quantum version of the previous Poincar{é}'s approach. Equipped with these observations, we examined relations between spheric and toric (classical) conformal blocks via Calogero-Moser/Heun duality. Besides, as Sutherland model is also obtainable from Calogero-Moser by pinching tori at one point, we tried to understand its eigenstates from the viewpoint of toric diagrams with possibly many surface operators (toric branes) inserted. A picture called "bubbling pants" then emerged and reproduced well-known results of the non-critical self-dual c=1 string theory under a "blown-down" limit.

17 pages, 4 figures; v2: corrections and references added; v3: Section 2.4.1 newly added thanks to JHEP referee advice. That classical four-point spheric conformal blocks reproducing known SW prepotentials is demonstrated via more examples, to appear in JHEP; v4: TexStyle changed only

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• A slow review of the AGT correspondence
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Liouville theory, N=2 gauge theories and accessory parameters
• Bootstrapping closed string field theory
• Seiberg-Witten curve via generalized matrix model
• Quasinormal Modes of C-metric from SCFTs
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Classical limit of irregular blocks and Mathieu functions
• Baxter's T-Q equation, SU(N)/SU(2)^{N-3} correspondence and Ω-deformed Seiberg-Witten prepotential
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