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Register a new userKondo line defects and affine Gaudin models
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We describe the relation between integrable Kondo problems in products of chiral $SU(2)$ WZW models and affine $SU(2)$ Gaudin models. We propose a full ODE/IM solution of the spectral problem for these models.
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It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.
The topological vertex formalism for 5d $mathcal{N}=1$ gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. The usual refined topological vertex formalism is derived for gauge theories with $A$-type quiver structure (and $A$-type gauge groups). In this article, we propose a construction with a web of vertex operators for all $ABCDEFG$-type and affine quivers by introducing several new vertices into the formalism, based on the reproducing of known instanton partition functions and qq-characters for these theories.
We study the 2D vertex operator algebra (VOA) construction in 4D $mathcal{N}=2$ superconformal field theories (SCFT) on $S^3 times S^1$, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus $mathbb{T}^2subset S^3times S^1$, it is $frac12mathbb{Z}$-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $mathbb{Z} +frac12$-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserve the VOA, we find that it admits discrete deformations, which allow access to the remaining spin structures (1,1), (0,1) and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle, the two-point functions, and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the $T$-transformation acts on our four partition functions and lifts to a large diffeomorphism on $S^3times S^1$. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled both by the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the $S$-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the $Omega$-background.
For $mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $text{Aut}, mathcal O$-equivariant identification between $text{Fun},text{Op}_{mathfrak g}(D)$, the algebra of functions on the space of ${mathfrak g}$-opers on the disc, and $Wsubset pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $delta_{-1}left|0right>$. We show that the latter endows $pi_0$ with a canonical notion of translation $T^{text{(aff)}}$, and use it to define the densities in $pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $text{Aut},mathcal O$-action defines a bundle $Pi$ over $mathbb P^1$ with fibre $pi_0$. We show that the product bundles $Pi otimes Omega^j$, where $Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $ abla^{text{(aff)}} - alpha T^{text{(aff)}}$, $alphain mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[mathbf v_j dt^{j+1} ] in H^1(mathbb P^1, Piotimes Omega^j, abla^{text{(aff)}})$ of the de Rham cohomology of $ abla^{mathrm{(aff)}}$. Any choice of ${mathfrak g}$-Miura oper $chi$ gives a connection $ abla^{mathrm{(aff)}}_chi$ on $Omega^j$. Using coinvariants, we define a map $mathsf F_chi$ from sections of $Pi otimes Omega^j$ to sections of $Omega^j$. We show that $mathsf F_chi abla^{text{(aff)}} = abla^{text{(aff)}}_chi mathsf F_chi$, so that $mathsf F_chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(mathbb P^1, Omega^j, abla^{text{(aff)}}_chi)$ defined by the ${mathfrak g}$-oper underlying $chi$.
An integrable system is introduced, which is a generalization of the $mathfrak{sl}(2)$ quantum affine Gaudin model. Among other things, the Hamiltonians are constructed and their spectrum is calculated within the ODE/IQFT approach. The model fits within the framework of Yang-Baxter integrability. This opens a way for the systematic quantization of a large class of integrable non-linear sigma models. There may also be some interest in terms of Condensed Matter applications, as the theory can be thought of as a multiparametric generalization of the Kondo model.
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