elliptic quantum
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2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Panupong Cheewaphutthisakun ◽  
Hiroaki Kanno

Abstract We investigate the quasi-Hopf twist of the quantum toroidal algebra of $$ {\mathfrak{gl}}_1 $$ gl 1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.


2021 ◽  
pp. 321-336
Author(s):  
L.-X. Zhang ◽  
D.V. Melnikov ◽  
J.-P. Leburton

2021 ◽  
Vol 208 (2) ◽  
pp. 1156-1164
Author(s):  
I. A. Sechin ◽  
A. V. Zotov

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Jin Chen ◽  
Babak Haghighat ◽  
Hee-Cheol Kim ◽  
Marcus Sperling

Abstract Quantum curves arise from Seiberg-Witten curves associated to 4d $$ \mathcal{N} $$ N = 2 gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $$ \mathcal{N} $$ N = (1, 0) SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a ℂ2/ℤk singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $$ \mathcal{N} $$ N = 2 theories are known as class $$ {\mathcal{S}}_k $$ S k . We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.


2020 ◽  
Author(s):  
Hitoshi Konno

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