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A bstract In [ 1 , 2 ], Nekrasov applied the Bethe/Gauge correspondence to derive the su (2) XXX spin-chain coordinate Bethe wavefunction from the IR limit of a 20 N = (2, 2) supersymmetric A 1 quiver gauge theory with an orbifold-type codimension-2 defect. Later, Bullimore, Kim and Lukowski implemented Nekrasov’s construction at the level of the UV A 1 quiver gauge theory, recovered his result, and obtained further extensions of the Bethe/Gauge correspondence [ 3 ]. In this work, we extend the construction of the defect to A M quiver gauge theories to obtain the su (M + 1) XXX spin-chain nested coordinate Bethe wavefunctions. The extension to XXZ spin-chain is straightforward. Further, we apply a Higgsing procedure to obtain more general A M quivers and the corresponding wavefunctions, and interpret this procedure (and the Hanany-Witten moves that it involves) on the spin-chain side in terms of Izergin-Korepin-type specializations (and re-assignments) of the parameters of the coordinate Bethe wavefunctions.

A bstract We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro ( p, p ′ ) = (2 , 2 k + 3) minimal models for k = 1 , 2 , . . . , in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, q, t -series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of ( A 1 , A 2 k ) Argyres- Douglas theories that correspond to t -refinements of Virasoro ( p, p ′ ) = (2 , 2 k + 3) minimal model characters, and two rank-2 Macdonald indices that correspond to t -refinements of W 3 non-unitary minimal model characters. Our proposals match with computations from 4d N = 2 gauge theories via the TQFT picture, based on the work of J Song [ 75 ].

A bstract We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results.

A bstract We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d N = 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

This paper focuses on a long-running and understudied Egyptian economic institution, the beer industry. While the presence of a well-developed beer industry in a predominantly Muslim country is noteworthy in itself, it is the consistent profitability of this industry despite the vicissitudes of Egypt's economic and political development that have made it truly remarkable. Relying heavily on archival material, including documents preserved in Cairo's Dar al-Wathaʾiq (Egyptian National Archives), this paper tracks the development of the beer industry in Egypt from 1897, when Belgian entrepreneurs started the Pyramid and Crown breweries, to the 1960s, when the Egyptian government nationalized the two companies. This analysis uses the history of the beer company to map larger social and economic trends in the colonial and semicolonial Egyptian economy (1882–1963) and to further problematize the foreign/Egyptian dichotomy that shapes discussions of it.

We consider a particular case of the 3-point function of local single-trace operators in the scalar sector of planar$ \\mathcal{N}=4 $supersymmetric Yang-Mills, where two of the fields are su (3) type, while the third one is su (2) type. We show that this tree-level 3-point function can be expressed in terms of scalar products of su (3) Bethe vectors. Moreover, if the second level Bethe roots of one of the su (3) operators is trivial (set to infinity), this 3- point function can be written in a determinant form. Using the determinant representation, we evaluate the structure constant in the semi-classical limit, when the number of roots goes to infinity.

A bstract Let ℬ N , n p , p ′ , ℋ be a conformal block, with n consecutive channels χ ι , ι = 1, ⋯ , n , in the conformal field theory ℳ N p , p ′ × ℳ ℋ , where ℳ N p , p ′ is a W N minimal model, generated by chiral spin-2, ⋯ , spin- N currents, and labeled by two co-prime integers p and p ′, 1 < p < p ′, while ℳ ℋ is a free boson conformal field theory. ℬ N , n p , p ′ , ℋ is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra A N − 1 , spanned by weight vectors ω → 1 , ⋯ , ω → N − 1 . We restrict our attention to conformal blocks with vertex operators whose charge vectors point along ω → 1 . The charge vectors that label the initial and final states can point in any direction. Following the W N AGT correspondence, and using Nekrasov’s instanton partition functions without modification to compute ℬ N , n p , p ′ , ℋ , leads to ill-defined expressions. We show that restricting the states that flow in the channels χ ι , ι = 1, ⋯ , n , to states labeled by N partitions that we call N -Burge partitions, that satisfy conditions that we call N -Burge conditions, leads to well-defined expressions that we propose to identify with ℬ N , n p , p ′ , ℋ . We check our identification by showing that a non-trivial conformal block that we compute, using the N -Burge conditions satisfies the expected differential equation. Further, we check that the generating functions of triples of Young diagrams that obey 3-Burge conditions coincide with characters of degenerate W 3 irreducible highest weight representations.