42,862 research outputs found
Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?
The algebraic approach to the construction of the reflexive polyhedra that
yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres
reveals graphs that include and generalize the Dynkin diagrams associated with
gauge symmetries. In this work we continue to study the structure of graphs
obtained from reflexive polyhedra. We show how some particularly defined
integral matrices can be assigned to these diagrams. This family of matrices
and its associated graphs may be obtained by relaxing the restrictions on the
individual entries of the generalized Cartan matrices associated with the
Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras.
These graphs keep however the affine structure, as it was in Kac-Moody Dynkin
diagrams. We presented a possible root structure for some simple cases. We
conjecture that these generalized graphs and associated link matrices may
characterize generalizations of these algebras.Comment: 24 pages, 6 figure
Quantum error-correcting codes associated with graphs
We present a construction scheme for quantum error correcting codes. The
basic ingredients are a graph and a finite abelian group, from which the code
can explicitly be obtained. We prove necessary and sufficient conditions for
the graph such that the resulting code corrects a certain number of errors.
This allows a simple verification of the 1-error correcting property of
fivefold codes in any dimension. As new examples we construct a large class of
codes saturating the singleton bound, as well as a tenfold code detecting 3
errors.Comment: 8 pages revtex, 5 figure
Notes on Melonic Tensor Models
It has recently been demonstrated that the large N limit of a model of
fermions charged under the global/gauge symmetry group agrees with
the large limit of the SYK model. In these notes we investigate aspects of
the dynamics of the theories that differ from their SYK
counterparts. We argue that the spectrum of fluctuations about the finite
temperature saddle point in these theories has new light
modes in addition to the light Schwarzian mode that exists even in the SYK
model, suggesting that the bulk dual description of theories differ
significantly if they both exist. We also study the thermal partition function
of a mass deformed version of the SYK model. At large mass we show that the
effective entropy of this theory grows with energy like (i.e. faster
than Hagedorn) up to energies of order . The canonical partition function
of the model displays a deconfinement or Hawking Page type phase transition at
temperatures of order . We derive these results in the large mass
limit but argue that they are qualitatively robust to small corrections in
.Comment: 60 pages, 7 figure
Graphs and Reflection Groups
It is shown that graphs that generalize the ADE Dynkin diagrams and have
appeared in various contexts of two-dimensional field theory may be regarded in
a natural way as encoding the geometry of a root system. After recalling what
are the conditions satisfied by these graphs, we define a bilinear form on a
root system in terms of the adjacency matrices of these graphs and undertake
the study of the group generated by the reflections in the hyperplanes
orthogonal to these roots. Some ``non integrally laced " graphs are shown to be
associated with subgroups of these reflection groups. The empirical relevance
of these graphs in the classification of conformal field theories or in the
construction of integrable lattice models is recalled, and the connections with
recent developments in the context of supersymmetric theories and
topological field theories are discussed.Comment: 42 pages TEX file, harvmac and epsf macros, AMS fonts optional,
uuencoded, 8 figures include
Four loop MSbar mass anomalous dimension in the Gross-Neveu model
We compute the four loop term of the mass anomalous dimension in the two
dimensional Gross-Neveu model in the MSbar scheme. The absence of
multiplicative renormalizability which results when using dimensional
regularization means that the effect of the evanescent operator, which first
appears at three loops in the 4-point Green's function, has to be properly
treated in the construction of the renormalization group function. We repeat
the calculation of the three loop MSbar beta-function and construct the
beta-function of the evanescent operator coupling which corrects earlier
computations.Comment: 20 latex pages, 7 figure
The First Thirty Years of Large-N Gauge Theory
I review some developments in the large-N gauge theory since 1974. The main
attention is payed to: multicolor QCD, matrix models, loop equations, reduced
models, 2D quantum gravity, free random variables, noncommutative theories,
AdS/CFT correspondence.Comment: 13.1pp., Latex, 2 figs; v2: 2 refs added. Talk at Large Nc QCD 200
Spectral Measures for
Spectral measures provide invariants for braided subfactors via fusion
modules. In this paper we study joint spectral measures associated to the
compact connected rank two Lie group and its double cover the compact
connected, simply-connected rank two Lie group , including the McKay
graphs for the irreducible representations of and and their
maximal tori, and fusion modules associated to the modular invariants.Comment: 41 pages, 45 figures. Title changed and notation corrected. arXiv
admin note: substantial text overlap with arXiv:1404.186
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