scholarly journals Diffeomorphisms on the fuzzy sphere

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Goro Ishiki ◽  
Takaki Matsumoto
Keyword(s):  
Configuration Space ◽  
Compact Manifold ◽  
Finite Size ◽  
Fuzzy Sphere ◽  
Algebra Of Functions ◽  
Large N ◽  
Smooth Compact Manifold ◽  
The Matrix ◽  
Area Preserving ◽  
Matrix Regularization

Abstract Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms.

scholarly journals NONCOMMUTATIVE FIELD THEORY: NUMERICAL ANALYSIS WITH THE FUZZY DISK

2012 ◽  
Vol 27 (24) ◽  
pp. 1250137 ◽  
Author(s):  
FEDELE LIZZI ◽  
BERNARDINO SPISSO
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Rotation Group ◽  
Intimate Relationship ◽  
Noncommutative Field Theory ◽  
Fuzzy Sphere ◽  
Scalar Field Theory ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Transition Curves

The fuzzy disk is a discretization of the algebra of functions on the two-dimensional disk using finite matrices which preserves the action of the rotation group. We define a φ4 scalar field theory on it and analyze numerically three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already present in the commutative scalar field theory and a nonuniform ordered phase as noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disk seems to be better. We have also performed the limits for the theory in the cases of the theory going to the commutative plane or commutative disk. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.

scholarly journals Effective Action and Measure in Matrix Model of IIB Superstrings

1997 ◽  
Vol 12 (31) ◽  
pp. 2331-2340 ◽  
Author(s):  
L. Chekhov ◽  
K. Zarembo
Keyword(s):  
Matrix Model ◽  
Effective Action ◽  
Continuum Limit ◽  
Auxiliary Field ◽  
Large N ◽  
The Matrix ◽  
Reparametrization Invariance ◽  
The Continuum ◽  
Induced Measure

We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large-N limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, is possibly irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.

scholarly journals CONFIGURATION CATEGORIES AND HOMOTOPY AUTOMORPHISMS

2018 ◽  
Vol 62 (1) ◽  
pp. 13-41
Author(s):  
MICHAEL S. WEISS
Keyword(s):  
Compact Manifold ◽  
Geometric Conditions ◽  
Homeomorphism Group ◽  
Manifold With Boundary ◽  
Smooth Compact Manifold

AbstractLet M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

scholarly journals Homology stability for symmetric diffeomorphism and mapping class groups

2015 ◽  
Vol 160 (1) ◽  
pp. 121-139 ◽  
Author(s):  
ULRIKE TILLMANN
Keyword(s):  
Compact Manifold ◽  
Smooth Manifold ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Connected Sum ◽  
Smooth Compact Manifold ◽  
Embedded Disks ◽  
Homology Stability

AbstractFor any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

scholarly journals THE HAGEDORN TRANSITION AND THE MATRIX MODEL FOR STRINGS

1998 ◽  
Vol 13 (26) ◽  
pp. 2085-2094 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Matrix Model ◽  
Light Cone ◽  
Degrees Of Freedom ◽  
Low Energy ◽  
Matrix Formalism ◽  
Yang Mills ◽  
Large N ◽  
The Matrix ◽  
The Continuum ◽  
String Worldsheet

We use the matrix formalism to investigate what happens to strings above the Hagedorn temperature. We show that is not a limiting temperature but a temperature at which the continuum string picture breaks down. We study a collection of N D-0-branes arranged to form a string having N units of light-cone momentum. We find that at high temperatures the favored phase is one where the string worldsheet has disappeared and the low-energy degrees of freedom consists of N2 massless particles ("gluons"). The nature of the transition is very similar to the deconfinement transition in large-N Yang–Mills theories.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

scholarly journals Bistable vector fields are axiom A

1995 ◽  
Vol 51 (1) ◽  
pp. 83-86
Author(s):  
Mike Hurley
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Axiom A ◽  
Smooth Compact Manifold ◽  
Structurally Stable

Recently L. Wen showed that if a C1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.

CONFORMAL PROPERTIES OF CRITICAL ISING MODEL CORNER TRANSFER MATRICES

1990 ◽  
Vol 04 (05) ◽  
pp. 907-912
Author(s):  
Brian DAVIES ◽  
Paul A. PEARCE
Keyword(s):  
Ising Model ◽  
Central Charge ◽  
Finite Size ◽  
Summation Formula ◽  
Transfer Matrices ◽  
Corner Transfer Matrices ◽  
Free Boundary Conditions ◽  
Large N ◽  
Fermion Algebra ◽  
Virasoro Characters

The scaling spectra of finite-size Ising model corner transfer matrices (CTMs) are studied at criticality, using the fermion algebra. The low-lying eigenvalues collapse like 1/ log N for large N as predicted by conformal invariance. The shift in the largest eigenvalue is evaluated analytically using a generalized Euler-Maclaurin summation formula giving πc/6 log N with central charge c=1/2. The spectrum generating functions, for both fixed and free boundary conditions, are expressed simply in terms of the c=1/2 Virasoro characters χ∆(q) with modular parameter q= exp (−π/ log N) and conformal dimensions ∆=0, 1/2, 1/16.

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