We show that specializations of the 4d =2 superconformal index labeled by an integer N is given by TrN where  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 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 12N units of U(1)r flux on S2.

The present article surmmarizes our recent results about Calabi–Yau threefolds with infinite fundamental group. This class of Calabi–Yau manifolds is relatively simple yet rich enough to display the essential complexities of Calabi–Yau geometries, and thus it provides good testing-grounds for general theories and conjectures.

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We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green’s function on a graph studied in [8]. We show that this gives rise to a graph invariant, which is closely related to the 2-dimensional Weisfeiler-Lehman algorithm for graph isomorphism testing. We then consider the same theory over the integers, which leads to the consideration of certain quadratic forms over the integers as initiated in [14], associated to the graphs. The quadratic form represented by the combinatorial Laplacian respects a well-behaved wedge sum of graphs, and appears to capture important graph properties reg

For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.