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Register a new userCorrelators on the wall and $mathfrak{sl}_n$ spin chain
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We study algebras and correlation functions of local operators at half-BPS interfaces engineered by the stacks of D5 or NS5 branes in the 4d $mathcal{N}=4$ super Yang-Mills. The operator algebra in this sector is isomorphic to a truncation of the Yangian $mathcal{Y}(mathfrak{gl}_n)$. The correlators, encoded in a trace on the Yangian, are controlled by the inhomogeneous $mathfrak{sl}_n$ spin chain, where $n$ is the number of fivebranes: they are given in terms of matrix elements of transfer matrices associated to Verma modules, or equivalently of products of Baxters Q-operators. This can be viewed as a novel connection between the $mathcal{N}=4$ super Yang-Mills and integrable spin chains. We also remark on analogous constructions involving half-BPS Wilson lines.
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Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, ldots$, on ${mathbb {C}}^2/{mathbb {Z}}_n$, $n = 2, 3, ldots$, with $Omega$-deformation and 2D conformal field theories with $mathcal{W}^{, para}_{N, n}$ ($n$-th parafermion $mathcal{W}_N$) symmetry and $widehat{mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $mathcal{W}^{, para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${mathbb {C}}^2/{mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $widehat{mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, ldots$, $N = 1, 2, ldots, .$
We introduce a spin chain based on finite-dimensional spin-1/2 SU(2) representations but with a non-hermitian `Hamiltonian and show, using mostly analytical techniques, that it is described at low energies by the SL(2,R)/U(1) Euclidian black hole Conformal Field Theory. This identification goes beyond the appearance of a non-compact spectrum: we are also able to determine the density of states, and show that it agrees with the formulas in [J. Math. Phys. 42, 2961 (2001)] and [JHEP 04, 014 (2002)], hence providing a direct `physical measurement of the associated reflection amplitude.
The orthosymplectic super Lie algebra $mathfrak{osp}(1|,2ell)$ is the closest analog of standard Lie algebras in the world of super Lie algebras. We demonstrate that the corresponding $mathfrak{osp}(1|,2ell)$-Toda chain turns out to be an instance of a $BC_ell$-Toda chain. The underlying reason for this relation is discussed.
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra $sR(n)$, is then introduced. Using results from classical invariant theory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer $Z_{n}(mathfrak{sl}_2)$ of the diagonal embedding of $U(mathfrak{sl}_2)$ in $U(mathfrak{sl}_2)^{otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincare series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.
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