Blowup Equations for Refined Topological Strings

G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on $SU(N)$ geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the $\bf r$ fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity $\bf r$ fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space.Comment: 85 pages. v2: Journal versio

Continual Local Training for Better Initialization of Federated Models

Federated learning (FL) refers to the learning paradigm that trains machine learning models directly in the decentralized systems consisting of smart edge devices without transmitting the raw data, which avoids the heavy communication costs and privacy concerns. Given the typical heterogeneous data distributions in such situations, the popular FL algorithm \emph{Federated Averaging} (FedAvg) suffers from weight divergence and thus cannot achieve a competitive performance for the global model (denoted as the \emph{initial performance} in FL) compared to centralized methods. In this paper, we propose the local continual training strategy to address this problem. Importance weights are evaluated on a small proxy dataset on the central server and then used to constrain the local training. With this additional term, we alleviate the weight divergence and continually integrate the knowledge on different local clients into the global model, which ensures a better generalization ability. Experiments on various FL settings demonstrate that our method significantly improves the initial performance of federated models with few extra communication costs.Comment: This paper has been accepted to 2020 IEEE International Conference on Image Processing (ICIP 2020

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent

We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various \emph{local estimates} for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the interior of the first quadrant. We also obtain the exponent for the probability that a word of length $2n$ sampled from the inventory accumulation model corresponds to an empty reduced word, which is equivalent to an asymptotic formula for the partition function of the critical FK planar map model. The estimates of this paper will be used in a subsequent paper to obtain the scaling limit of the lattice walk associated with a finite-volume FK planar map.Comment: 49 pages, 2 figures; final version published in EJP. Changes include significantly approved exposition and relation to partition functio

Ergodicity of the Airy line ensemble

In this paper, we establish the ergodicity of the Airy line ensemble. This shows that it is the only candidate for Conjecture 3.2 in [3], regarding the classification of ergodic line ensembles satisfying a certain Brownian Gibbs property after a parabolic shift.Comment: argument for Proposition 1.13 is revised, the structure of the introduction is rearrange