scholarly journals Super Quantum Airy Structures

2020 ◽  
Vol 380 (1) ◽  
pp. 449-522
Author(s):  
Vincent Bouchard ◽  
Paweł Ciosmak ◽  
Leszek Hadasz ◽  
Kento Osuga ◽  
Błażej Ruba ◽  
...  
Keyword(s):  
Classification Scheme ◽  
Graphical Representation ◽  
Free Energies ◽  
Recursion Relations ◽  
Gauge Transformations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Topological Recursion

Abstract We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional—which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss—as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.

scholarly journals THE ROLE OF DIAGONALIZATION WITHIN A DIAGONALIZATION/MONTE CARLO SCHEME

2001 ◽  
Vol 16 (supp01c) ◽  
pp. 1245-1247 ◽  
Author(s):  
Dean Lee
Keyword(s):  
Monte Carlo ◽  
Computational Approach ◽  
Low Energy ◽  
Infinite Dimensional ◽  
Monte Carlo Techniques ◽  
Finite Dimensional ◽  
Quantum Hamiltonian ◽  
Important Basis ◽  
Basis Vectors

We discuss a method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques.

scholarly journals SYMMETRIES OF p-BRANES

1993 ◽  
Vol 08 (30) ◽  
pp. 5367-5381 ◽  
Author(s):  
R. PERCACCI ◽  
E. SEZGIN
Keyword(s):  
Global Symmetries ◽  
Sigma Model ◽  
Space Time ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Invariance Properties ◽  
Group Manifold ◽  
Mills Field

Using canonical methods, we study the invariance properties of a bosonic p-brane propagating in a curved background locally diffeomorphic to M×G, where M is space-time and G a group manifold. The action is that of a gauged sigma model in p+1 dimensions coupled to a Yang-Mills field and a (p+1) form in M. We construct the generators of Yang-Mills and tensor gauge transformations and exhibit the role of the (p+1) form in canceling the potential Schwinger terms. We also discuss the Noether currents associated with the global symmetries of the action and the question of the existence of infinite-dimensional symmetry algebras, analogous to the Kac-Moody symmetry of the string.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

scholarly journals Blobbed topological recursion: properties and applications

2016 ◽  
Vol 162 (1) ◽  
pp. 39-87 ◽  
Author(s):  
GAËTAN BOROT ◽  
SERGEY SHADRIN
Keyword(s):  
Initial Data ◽  
Matrix Model ◽  
Moduli Space ◽  
Graphical Representation ◽  
Infinitesimal Transformation ◽  
Moduli Space Of Curves ◽  
Free Energies ◽  
Intersection Numbers ◽  
Topological Recursion ◽  
Variational Formulas

AbstractWe study the set of solutions (ωg,n)g⩾0,n⩾1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

scholarly journals OFF-SHELL FORMULATION OF N=2 SUPER YANG-MILLS THEORIES COUPLED TO MATTER WITHOUT AUXILIARY FIELDS

1995 ◽  
Vol 10 (27) ◽  
pp. 3937-3950 ◽  
Author(s):  
NICOLA MAGGIORE
Keyword(s):  
Equations Of Motion ◽  
Dimensional Algebra ◽  
Nilpotent Operator ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Field Dependent ◽  
External Sources ◽  
Shell Formulation

N=2 supersymmetric Yang-Mills theories coupled to matter are considered in the Wess-Zumino gauge. The supersymmetries are realized nonlinearly and the anticommutator between two susy charges gives, in addition to translations, gauge transformations and equations of motion. The difficulties hidden in such an algebraic structure are well known: almost always auxiliary fields can be introduced in order to put the formalism off-shell, but still the field-dependent gauge transformations give rise to an infinite-dimensional algebra quite hard to deal with. However, it is possible to avoid all these problems by collecting into a unique nilpotent operator all the symmetries defining the theory, namely ordinary BRS, supersymmetries and translations. According to this method the role of the auxiliary fields is covered by the external sources coupled, as usual, to the nonlinear variations of the quantum fields. The analysis is then formally reduced to that of ordinary Yang-Mills theory.

scholarly journals SUPERALGEBRA OF HIGHER SPINS AND AUXILIARY FIELDS

1988 ◽  
Vol 03 (12) ◽  
pp. 2983-3010 ◽  
Author(s):  
E.S. FRADKIN ◽  
M.A. VASILIEV
Keyword(s):  
Lie Algebras ◽  
De Sitter ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Operator Realization ◽  
Simple Operator ◽  
Higher Spins ◽  
Massless Fields

An infinite-dimensional non-Abelian superalgebra is constructed, denoted as shsa(1), which gives rise at the linearized level to linearized curvatures of both massless and auxiliary fields, suggested previously by one of us (M.V.). Various properties of shsa(1) are analysed in detail. Specifically, subalgebras of shsa(1) are found which pretend themselves for the role of independent superalgebras of higher spins and auxiliary fields. A simple operator realization of shsa(1) is presented. P-reversal automorphisms are constructed. N=2 anti-de Sitter superalgebra osp(2, 4) is shown to be a maximal finite-dimensional subalgebra of shsa(1)/o(2). It is observed that the even (boson) subalgebra of shsa(1) decomposes into the direct sum of two infinite-dimensional Lie algebras each giving rise to massless fields of all integer spins. Possible physical implications of this fact are discussed briefly.

scholarly journals QUANTUM HADRODYNAMICS IN TWO DIMENSIONS

1994 ◽  
Vol 09 (31) ◽  
pp. 5583-5624 ◽  
Author(s):  
S.G. RAJEEV
Keyword(s):  
Phase Space ◽  
Condensed Matter ◽  
Baryon Number ◽  
Two Dimensions ◽  
Topological Solitons ◽  
Infinite Dimensional ◽  
Hartree Fock ◽  
Finite Dimensional ◽  
Natural Formulation

A nonlocal and nonlinear theory of hadrons, equivalent to the color singlet sector of two-dimensional QCD, is constructed. The phase space of this theory is an infinite-dimensional Grassmannian. The baryon number of QCD corresponds to a topological invariant (“virtual rank”) of the Grassmannian. It is shown that the hadron theory has topological solitons corresponding to the baryons of QCD. [Formula: see text] plays the role of ħ in this theory; Nc must be an integer for topological reasons. We also describe the quantization of a toy model with a finite-dimensional Grassmannian as the phase space. In an appendix, we show that the usual Hartree-Fock theory of atomic and condensed matter physics has a natural formulation in terms of infinite-dimensional Grassmannians.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals The Farkas lemma of Shimizu, Aiyoshi and Katayama

1985 ◽  
Vol 31 (3) ◽  
pp. 445-450 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Convex Programming ◽  
Infinite Dimensional ◽  
Farkas Lemma ◽  
Finite Dimensional

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

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