2,155 research outputs found
Refined BPS invariants of 6d SCFTs from anomalies and modularity
F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds
engineer six dimensional superconformal field theories and their mass
deformations. The partition function of the refined topological
string on these geometries captures the particle BPS spectrum of this class of
theories compactified on a circle. Organizing in terms of
contributions at base degree of the elliptic fibration, we
find that these, up to a multiplier system, are meromorphic Jacobi forms of
weight zero with modular parameter the Kaehler class of the elliptic fiber and
elliptic parameters the couplings and mass parameters. The indices with regard
to the multiple elliptic parameters are fixed by the refined holomorphic
anomaly equations, which we show to be completely determined from knowledge of
the chiral anomaly of the corresponding SCFT. We express as a
quotient of weak Jacobi forms, with a universal denominator inspired by its
pole structure as suggested by the form of in terms of 5d BPS
numbers. The numerator is determined by modularity up to a finite number of
coefficients, which we prove to be fixed uniquely by imposing vanishing
conditions on 5d BPS numbers as boundary conditions. We demonstrate the
feasibility of our approach with many examples, in particular solving the
E-string and M-string theories including mass deformations, as well as theories
constructed as chains of these. We make contact with previous work by showing
that spurious singularities are cancelled when the partition function is
written in the form advocated here. Finally, we use the BPS invariants of the
E-string thus obtained to test a generalization of the
Goettsche-Nakajima-Yoshioka -theoretic blowup equation, as inspired by the
Grassi-Hatsuda-Marino conjecture, to generic local Calabi-Yau threefolds.Comment: 64 pages; v2: typos correcte
Hyper-Path-Based Representation Learning for Hyper-Networks
Network representation learning has aroused widespread interests in recent
years. While most of the existing methods deal with edges as pairwise
relationships, only a few studies have been proposed for hyper-networks to
capture more complicated tuplewise relationships among multiple nodes. A
hyper-network is a network where each edge, called hyperedge, connects an
arbitrary number of nodes. Different from conventional networks, hyper-networks
have certain degrees of indecomposability such that the nodes in a subset of a
hyperedge may not possess a strong relationship. That is the main reason why
traditional algorithms fail in learning representations in hyper-networks by
simply decomposing hyperedges into pairwise relationships. In this paper, we
firstly define a metric to depict the degrees of indecomposability for
hyper-networks. Then we propose a new concept called hyper-path and design
hyper-path-based random walks to preserve the structural information of
hyper-networks according to the analysis of the indecomposability. Then a
carefully designed algorithm, Hyper-gram, utilizes these random walks to
capture both pairwise relationships and tuplewise relationships in the whole
hyper-networks. Finally, we conduct extensive experiments on several real-world
datasets covering the tasks of link prediction and hyper-network
reconstruction, and results demonstrate the rationality, validity, and
effectiveness of our methods compared with those existing state-of-the-art
models designed for conventional networks or hyper-networks.Comment: Accepted by CIKM 201
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