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S. Cecotti, Aug 19, 2014 Available from:- [Show abstract] [Hide abstract]

**ABSTRACT:**We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex SUSY central charges (as in N=2 in d=4 and N=(2,2) in d=2) the known results can be reinterpreted as the statement that the BPS partition functions can be used to compute a specialization of the superconformal indices. We argue that in the case with real central charge in the supersymmetry algebra, as in N=1 in d=5 (or the N=2 in d=3), the BPS degeneracy captures the full superconformal index. Furthermore, we argue that refined topological strings, which captures 5d BPS degeneracies of M-theory on CY 3-folds, can be used to compute 5d supersymmetric index including in the sectors with 3d defects for a large class of 5d superconformal theories. Moreover, we provide evidence that distinct Calabi-Yau singularities which are expected to lead to the same SCFT yield the same index.Physical Review D 10/2012; 90(10). DOI:10.1103/PhysRevD.90.105031 · 4.86 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We give a pedagogical introduction to wall–crossing for N = 2 theories in both two and four dimensions from the point of view of the quantities whose BPS–chamber invariance implies the wall–crossing formula. The basic such invariant is the conju-gacy class of the quantum monodromy, which may be thought of as a generalization of the Coxeter element in the Weyl group of a Lie algebra. The relationships with singularity theory, quiver and algebras representation the-ory, topological strings, (quantum) cluster algebras, the Thermodynamical Bethe Ansatz, 2d CFT's, and Number Theory are outlined. -
##### Article: R-Twisting and 4d/2d Correspondences

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**ABSTRACT:**We show how aspects of the R-charge of N=2 CFTs in four dimensions are encoded in the q-deformed Kontsevich-Soibelman monodromy operator, built from their dyon spectra. In particular, the monodromy operator should have finite order if the R-charges are rational. We verify this for a number of examples including those arising from pairs of ADE singularities on a Calabi-Yau threefold (some of which are dual to 6d (2,0) ADE theories suitably fibered over the plane). In these cases we find that our monodromy maps to that of the Y-systems, studied by Zamolodchikov in the context of TBA. Moreover we find that the trace of the (fractional) q-deformed KS monodromy is given by the characters of 2d conformal field theories associated to the corresponding TBA (i.e. integrable deformations of the generalized parafermionic systems). The Verlinde algebra gets realized through evaluation of line operators at the loci of the associated hyperKahler manifold fixed under R-symmetry action. Moreover, we propose how the TBA system arises as part of the N=2 theory in 4 dimensions. Finally, we initiate a classification of N=2 superconformal theories in 4 dimensions based on their quiver data and find that this classification problem is mapped to the classification of N=2 theories in 2 dimensions, and use this to classify all the 4d, N=2 theories with up to 3 generators for BPS states. Comment: 161 pages, 4 figures; v2: references added, small corrections