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Register a new userElliptic Quantum Curves of Class $mathcal{S}_k$
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Quantum curves arise from Seiberg-Witten curves associated to 4d $mathcal{N}=2$ 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 $mathcal{N}=(1,0)$ 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 $mathbb{C}^2/mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $mathcal{N}=2$ theories are known as class $mathcal{S}_k$. 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.
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We study the Coulomb branch of class $mathcal{S}_k$ $mathcal{N} = 1$ SCFTs by constructing and analyzing their spectral curves.
Even though for generic $mathcal{N}=1$ theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $mathcal{N}=1$ SCFTs, these of Class $mathcal{S}_k$ for which it is possible to define Coulomb and Higgs branches precisely as for the $mathcal{N}=2$ theories of Class $mathcal{S}$ from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional $(1,1)$ Little String theory can be deconstructed from a toroidal quiver in class $mathcal{S}_k$
We consider all 4d $mathcal{N}=2$ theories of class $mathcal{S}$ arising from the compactification of exceptional 6d $(2,0)$ SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d $(1,0)$ SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of $(mathfrak{e}_n,mathfrak{e}_n)$ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class $mathcal{S}$ construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.
We propose a generalization of S-folds to 4d $mathcal{N}=2$ theories. This construction is motivated by the classification of rank one 4d $mathcal{N}=2$ super-conformal field theories (SCFTs), which we reproduce from D3-branes probing a configuration of $mathcal{N}=2$ S-folds combined with 7-branes. The main advantage of this point of view is that realizes both Coulomb and Higgs branch flows and allows for a straight forward generalization to higher rank theories.
The gravitational $mathcal{S}$-matrix defined with an infrared (IR) cutoff factorizes into hard and soft factors. The soft factor is universal and contains all the IR and collinear divergences. Here we show, in a momentum space basis, that the intricate expression for the soft factor is fully reproduced by two boundary currents, which live on the celestial sphere. The first of these is the supertranslation current, which generates spacetime supertranslations. The second is its symplectic partner, the Goldstone current for spontaneously broken supertranslations. The current algebra has an off-diagonal level structure involving the gravitational cusp anomalous dimension and the logarithm of the IR cutoff. It is further shown that the gravitational memory effect is contained as an IR safe observable within the soft $mathcal{S}$-matrix.
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