104 research outputs found
The Information Geometry of the Ising Model on Planar Random Graphs
It has been suggested that an information geometric view of statistical
mechanics in which a metric is introduced onto the space of parameters provides
an interesting alternative characterisation of the phase structure,
particularly in the case where there are two such parameters -- such as the
Ising model with inverse temperature and external field .
In various two parameter calculable models the scalar curvature of
the information metric has been found to diverge at the phase transition point
and a plausible scaling relation postulated: . For spin models the necessity of calculating in
non-zero field has limited analytic consideration to 1D, mean-field and Bethe
lattice Ising models. In this letter we use the solution in field of the Ising
model on an ensemble of planar random graphs (where ) to evaluate the scaling behaviour of the scalar curvature, and find
. The apparent discrepancy is traced
back to the effect of a negative .Comment: Version accepted for publication in PRE, revtex
Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
We generalise the construction of fuzzy CP^N in a manner that allows us to
access all noncommutative equivariant complex vector bundles over this space.
We give a simplified construction of polarization tensors on S^2 that
generalizes to complex projective space, identify Laplacians and natural
noncommutative covariant derivative operators that map between the modules that
describe noncommuative sections. In the process we find a natural
generalization of the Schwinger-Jordan construction to su(n) and identify
composite oscillators that obey a Heisenberg algebra on an appropriate Fock
space.Comment: 34 pages, v2 contains minor corrections to the published versio
The Standard Model Fermion Spectrum From Complex Projective spaces
It is shown that the quarks and leptons of the standard model, including a
right-handed neutrino, can be obtained by gauging the holonomy groups of
complex projective spaces of complex dimensions two and three. The spectrum
emerges as chiral zero modes of the Dirac operator coupled to gauge fields and
the demonstration involves an index theorem analysis on a general complex
projective space in the presence of topologically non-trivial SU(n)xU(1) gauge
fields. The construction may have applications in type IIA string theory and
non-commutative geometry.Comment: 13 pages. Typset using LaTeX and JHEP3 style files. Minor typos
correcte
Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect
The parallel rôles of modular symmetry in N = 2 supersymmetric Yang-Mills and in the quantum Hall effect are reviewed. In supersymmetric Yang-Mills theories modular symmetry emerges as a version of Dirac's electric - magnetic duality. It has significant consequences for the vacuum structure of these theories, leading to a fractal vacuum which has an infinite hierarchy of related phases. In the case of N = 2 supersymmetric Yang-Mills in 3+1 dimensions, scaling functions can be defined which are modular forms of a subgroup of the full modular group and which interpolate between vacua. Infra-red fixed points at strong coupling correspond to θ-vacua with θ a rational number that, in the case of pure SUSY Yang-Mills, has odd denominator. There is a mass gap for electrically charged particles which can carry fractional electric charge. A similar structure applies to the 2+1 dimensional quantum Hall effect where the hierarchy of Hall plateaux can be understood in terms of an action of the modular group and the stability of Hall plateaux is due to the fact that odd denominator Hall conductivities are attractive infra-red fixed points. There is a mass gap for electrically charged excitations which, in the case of the fractional quantum Hall effect, carry fractional electric charge
Renormalization of composite operators
The blocked composite operators are defined in the one-component Euclidean
scalar field theory, and shown to generate a linear transformation of the
operators, the operator mixing. This transformation allows us to introduce the
parallel transport of the operators along the RG trajectory. The connection on
this one-dimensional manifold governs the scale evolution of the operator
mixing. It is shown that the solution of the eigenvalue problem of the
connection gives the various scaling regimes and the relevant operators there.
The relation to perturbative renormalization is also discussed in the framework
of the theory in dimension .Comment: 24 pages, revtex (accepted by Phys. Rev. D), changes in introduction
and summar
Fuzzy Scalar Field Theory as a Multitrace Matrix Model
We develop an analytical approach to scalar field theory on the fuzzy sphere
based on considering a perturbative expansion of the kinetic term. This
expansion allows us to integrate out the angular degrees of freedom in the
hermitian matrices encoding the scalar field. The remaining model depends only
on the eigenvalues of the matrices and corresponds to a multitrace hermitian
matrix model. Such a model can be solved by standard techniques as e.g. the
saddle-point approximation. We evaluate the perturbative expansion up to second
order and present the one-cut solution of the saddle-point approximation in the
large N limit. We apply our approach to a model which has been proposed as an
appropriate regularization of scalar field theory on the plane within the
framework of fuzzy geometry.Comment: 1+25 pages, replaced with published version, minor improvement
The Information Geometry of the Spherical Model
Motivated by previous observations that geometrizing statistical mechanics
offers an interesting alternative to more standard approaches,we have recently
calculated the curvature (the fundamental object in this approach) of the
information geometry metric for the Ising model on an ensemble of planar random
graphs. The standard critical exponents for this model are alpha=-1, beta=1/2,
gamma=2 and we found that the scalar curvature, R, behaves as
epsilon^(-2),where epsilon = beta_c - beta is the distance from criticality.
This contrasts with the naively expected R ~ epsilon^(-3) and the apparent
discrepancy was traced back to the effect of a negative alpha on the scaling of
R.
Oddly,the set of standard critical exponents is shared with the 3D spherical
model. In this paper we calculate the scaling behaviour of R for the 3D
spherical model, again finding that R ~ epsilon^(-2), coinciding with the
scaling behaviour of the Ising model on planar random graphs. We also discuss
briefly the scaling of R in higher dimensions, where mean-field behaviour sets
in.Comment: 7 pages, no figure
Scalar Field Theory on Fuzzy S^4
Scalar fields are studied on fuzzy and a solution is found for the
elimination of the unwanted degrees of freedom that occur in the model. The
resulting theory can be interpreted as a Kaluza-Klein reduction of CP^3 to S^4
in the fuzzy context.Comment: 16 pages, LaTe
Spin Dirac Operators on the Fuzzy 2-Sphere
The spin 1/2 Dirac operator and its chirality operator on the fuzzy 2-sphere
can be constructed using the Ginsparg-Wilson(GW) algebra
[arxiv:hep-th/0511114]. This construction actually exists for any spin on
, and have continuum analogues as well on the commutative sphere
or on . This is a remarkable fact and has no known analogue in
higher dimensional Minkowski spaces. We study such operators on and the
commutative and formulate criteria for the existence of the limit from
the former to the latter. This singles out certain fuzzy versions of these
operators as the preferred Dirac operators. We then study the spin 1 Dirac
operator of this preferred type and its chirality on the fuzzy 2-sphere and
formulate its instanton sectors and their index theory. The method to
generalize this analysis to any spin is also studied in detail.Comment: 16 pages, 1 tabl
Non-Linear Sigma Model on the Fuzzy Supersphere
In this note we develop fuzzy versions of the supersymmetric non-linear sigma
model on the supersphere S^(2,2). In hep-th/0212133 Bott projectors have been
used to obtain the fuzzy CP^1 model. Our approach utilizes the use of
supersymmetric extensions of these projectors. Here we obtain these (super)
-projectors and quantize them in a fashion similar to the one given in
hep-th/0212133. We discuss the interpretation of the resulting model as a
finite dimensional matrix model.Comment: 11 pages, LaTeX, corrected typo
- …