Supersymmetric Gauge Theories with Matters, Toric Geometries and Random Partitions

We derive the relation between the Hilbert space of certain geometries under the Bohr-Sommerfeld quantization and the perturbative prepotentials for the supersymmetric five-dimensional SU(N) gauge theories with massive fundamental matters and with one massive adjoint matter. The gauge theory with one adjoint matter shows interesting features. A five-dimensional generalization of Nekrasov's partition function can be written as a correlation function of two-dimensional chiral bosons and as a partition function of a statistical model of partitions. From a ground state of the statistical model we reproduce the polyhedron which characterizes the Hilbert space.Comment: 26 pages, 11 figures; v2 typos correcte

Hyperplane Arrangements and Locality-Sensitive Hashing with Lift

Locality-sensitive hashing converts high-dimensional feature vectors, such as image and speech, into bit arrays and allows high-speed similarity calculation with the Hamming distance. There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors and the normal vectors of hyperplanes placed in the feature space. This hashing can be seen as a discretization of the feature space by hyperplanes. If labels for data are given, one can determine the hyperplanes by using learning algorithms. However, many proposed learning methods do not consider the hyperplanes' offsets. Not doing so decreases the number of partitioned regions, and the correlation between Hamming distances and Euclidean distances becomes small. In this paper, we propose a lift map that converts learning algorithms without the offsets to the ones that take into account the offsets. With this method, the learning methods without the offsets give the discretizations of spaces as if it takes into account the offsets. For the proposed method, we input several high-dimensional feature data sets and studied the relationship between the statistical characteristics of data, the number of hyperplanes, and the effect of the proposed method.Comment: 9 pages, 7 figure

Locality-Sensitive Hashing with Margin Based Feature Selection

We propose a learning method with feature selection for Locality-Sensitive Hashing. Locality-Sensitive Hashing converts feature vectors into bit arrays. These bit arrays can be used to perform similarity searches and personal authentication. The proposed method uses bit arrays longer than those used in the end for similarity and other searches and by learning selects the bits that will be used. We demonstrated this method can effectively perform optimization for cases such as fingerprint images with a large number of labels and extremely few data that share the same labels, as well as verifying that it is also effective for natural images, handwritten digits, and speech features.Comment: 9 pages, 6 figures, 3 table

Integrable Structure of 5d5d N=1\mathcal{N}=1 Supersymmetric Yang-Mills and Melting Crystal

We study loop operators of $5d$ $\mathcal{N}=1$ SYM in $\Omega$ background. For the case of U(1) theory, the generating function of correlation functions of the loop operators reproduces the partition function of melting crystal model with external potential. We argue the common integrable structure of $5d$ $\mathcal{N}=1$ SYM and melting crystal model.Comment: 12 pages, 1 figure, based on an invited talk presented at the international workshop "Progress of String Theory and Quantum Field Theory" (Osaka City University, December 7-10, 2007), to be published in the proceeding

Gravitational Quantum Foam and Supersymmetric Gauge Theories

We study K\"{a}hler gravity on local SU(N) geometry and describe precise correspondence with certain supersymmetric gauge theories and random plane partitions. The local geometry is discretized, via the geometric quantization, to a foam of an infinite number of gravitational quanta. We count these quanta in a relative manner by measuring a deviation of the local geometry from a singular Calabi-Yau threefold, that is a A_{N-1} singularity fibred over \mathbb{P}^1. With such a regularization prescription, the number of the gravitational quanta becomes finite and turns to be the perturbative prepotential for five-dimensional \mathcal{N}=1 supersymmetric SU(N) Yang-Mills. These quanta are labelled by lattice points in a certain convex polyhedron on \mathbb{R}^3. The polyhedron becomes obtainable from a plane partition which is the ground state of a statistical model of random plane partition that describes the exact partition function for the gauge theory. Each gravitational quantum of the local geometry is shown to consist of N unit cubes of plane partitions.Comment: 43 pages, 12 figures: V2 typos correcte