No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons

Abstract We determine the generating functions of 1/4 BPS dyons in a class of 4d N $$\mathcal{N}$$ = 4 string vacua arising as CHL orbifolds of K3 × T 2, a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be described as multiplicative lifts of certain supersymmetric indices (twining genera) on the worldsheet of the corresponding nonlinear sigma model on K3. As a consequence, our procedure also leads to an explicit derivation of almost all of these twining genera. The worldsheet indices singled out in this way match precisely a set of functions of interest in moonshine, as predicted by a recent conjecture

Generation of unpredictable time series by a Neural Network

A perceptron that learns the opposite of its own output is used to generate a time series. We analyse properties of the weight vector and the generated sequence, like the cycle length and the probability distribution of generated sequences. A remarkable suppression of the autocorrelation function is explained, and connections to the Bernasconi model are discussed. If a continuous transfer function is used, the system displays chaotic and intermittent behaviour, with the product of the learning rate and amplification as a control parameter.Comment: 11 pages, 14 figures; slightly expanded and clarified, mistakes corrected; accepted for publication in PR

Interconnection Networks Based on Gaussian and Eisenstein-Jacobi Integers

Quotient rings of Gaussian and Eisenstein-Jacobi(EJ) integers can be deployed to construct interconnection networks with good topological properties. In this thesis, we propose deadlock-free deterministic and partially adaptive routing algo­rithms for hexagonal networks, one special class of EJ networks. Then we discuss higher dimensional Gaussian networks as an alternative to classical multidimen­sional toroidal networks. For this topology, we explore many properties including distance distribution and the decomposition of higher dimensional Gaussian net­ works into Hamiltonian cycles. In addition, we propose some efficient communi­cation algorithms for higher dimensional Gaussian networks including one-to-all broadcasting and shortest path routing. Simulation results show that the rout­ing algorithm proposed for higher dimensional Gaussian networks outperforms the routing algorithm of the corresponding torus networks with approximately the same number of nodes. These simulation results are expected since higher dimen­sional Gaussian networks have a smaller diameter and a smaller average message latency as compared with toroidal networks. Finally, we introduce a degree-three interconnection network obtained from pruning a Gaussian network. This network shows possible performance improve­ment over other degree-three networks since it has a smaller diameter compared to other degree-three networks. Many topological properties of degree-three pruned Gaussian network are explored. In addition, an optimal shortest path routing algorithm and a one-to-all broadcasting algorithm are given

1/4 BPS States and Non-Perturbative Couplings in N=4 String Theories

We compute certain 2K+4-point, one-loop couplings in the type IIA string compactified on K3 x T^2, which are related to a topological index on this manifold. Their special feature is that they are sensitive to only short and intermediate BPS multiplets. The couplings derive from underlying prepotentials G[K](T,U), which can be nicely summed up into a fundamental generating function. In the dual heterotic string on T^6, the amplitudes describe non-perturbative gravitational corrections to K-loop amplitudes due to bound states of fivebrane instantons with heterotic world-sheet instantons. We argue, as a consequence, that our results also give information about instanton configurations in six dimensional Sp(2k) gauge theories on T^6.Comment: 32 p, harvmac, 1 fig. Revision: taking the fermionic contractions into account, the K3 elliptic genus disappear

Riemann-Roch and Abel-Jacobi theory on a finite graph

It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.Comment: 35 pages. v3: Several minor changes made, mostly fixing typographical errors. This is the final version, to appear in Adv. Mat

N=2\mathcal{N} = 2 Schur index and line operators

4d $\mathcal{N} = 2$ SCFTs and their invariants can be often enriched by non-local BPS operators. In this paper we study the flavored Schur index of several types of N = 2 SCFTs with and without line operators, using a series of new integration formula of elliptic functions and Eisenstein series. We demonstrate how to evaluate analytically the Schur index for a series of $A_2$ class-$\mathcal{S}$ theories and the $\mathcal{N} = 4$ SO(7) theory. For all $A_1$ class-$\mathcal{S}$ theories we obtain closed-form expressions for SU(2) Wilson line index, and 't Hooft line index in some simple cases. We also observe the relation between the line operator index with the characters of the associated chiral algebras. Wilson line index for some other low rank gauge theories are also studied.Comment: 72 pages, 9 figures, 5 table

Unimodular graphs and Eisenstein sums

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated to unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation $a\cdot b=r$ over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings - e.g., $\mathbb{Z}/(N)$. In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over $\mathbb{Z}/(N)$.Comment: V2: minor revisions. To appear in the Journal of Algebraic Combinatoric