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Abstract
We show that specializations of the 4d mathcal{N}=2 superconformal index labeled by an integer N is given by Tr ℳN where ℳ is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras {mathcal{A}}_N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2× T2 where we turn on frac{1}{2}N units of U(1)r flux on S2.
Highlights
Four dimensional superconformal field theories (SCFT’s) with N = 2 supersymmetry have common features with the simpler 2d SCFT’s with N = (2, 2) supersymmetry: upon deformations away from the conformal point, both typically lead to Bogomol’nyi-PrasadSommerfield (BPS) states whose mass is given by absolute value of a central charge in the SUSY algebra which is a complex number
We show that specializations of the 4d N = 2 superconformal index labeled by an integer N is given by Tr MN where M is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch
In later work [8] it was found that, by considering operators contributing to the Schur limit of the superconformal index, one obtains chiral algebras in 2d, with a very specific central charge: c2d = −12c4d, where c2d is the 2d central charge of the Virasoro algebra and c4d is the c-function of the 4d SCFT
Summary
Four dimensional superconformal field theories (SCFT’s) with N = 2 supersymmetry have common features with the simpler 2d SCFT’s with N = (2, 2) supersymmetry: upon deformations away from the conformal point, both typically lead to Bogomol’nyi-PrasadSommerfield (BPS) states whose mass is given by absolute value of a central charge in the SUSY algebra which is a complex number. Of the ground states of the theory This result motivated the parallel question in the 4d case [6] where the trace of the powers of the monodromy operator M(q) were computed. It was found that insertion of line operators in the monodromy trace acts by changing the characters of the 2d conformal theory It was suggested in [7] that an integer sequence of specializations of the superconformal index should lead to the trace of various powers of the monodromy.
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