69 research outputs found
A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6
The invention of the "dual resonance model" N-point functions BN motivated
the development of current string theory. The simplest of these models, the
four-point function B4, is the classical Euler Beta function. Many standard
methods of complex analysis in a single variable have been applied to elucidate
the properties of the Euler Beta function, leading, for example, to analytic
continuation formulas such as the contour-integral representation obtained by
Pochhammer in 1890. Here we explore the geometry underlying the dual five-point
function B5, the simplest generalization of the Euler Beta function. Analyzing
the B5 integrand leads to a polyhedral structure for the five-crosscap surface,
embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120
in PGL(6). We find a Pochhammer-like representation for B5 that is a contour
integral along a surface of genus five. The symmetric embedding of the
five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the
surface of genus four in R6 that has a polyhedral structure with 24 pentagonal
faces and a symmetry group of order 240 in O(6). The methods appear
generalizable to all N, and the resulting structures seem to be related to
associahedra in arbitrary dimensions.Comment: 43 pages and 44 figure
GABA-enhanced collective behavior in neuronal axons underlies persistent gamma-frequency oscillations
Gamma (30–80 Hz) oscillations occur in mammalian electroencephalogram in a manner that indicates cognitive relevance. In vitro models of gamma oscillations demonstrate two forms of oscillation: one occurring transiently and driven by discrete afferent input and the second occurring persistently in response to activation of excitatory metabotropic receptors. The mechanism underlying persistent gamma oscillations has been suggested to involve gap-junctional communication between axons of principal neurons, but the precise relationship between this neuronal activity and the gamma oscillation has remained elusive. Here we demonstrate that gamma oscillations coexist with high-frequency oscillations (>90 Hz). High-frequency oscillations can be generated in the axonal plexus even when it is physically isolated from pyramidal cell bodies. They were enhanced in networks by nonsomatic -aminobutyric acid type A (GABAA) receptor activation, were modulated by perisomatic GABAA receptor-mediated synaptic input to principal cells, and provided the phasic input to interneurons required to generate persistent gamma-frequency oscillations. The data suggest that high-frequency oscillations occurred as a consequence of random activity within the axonal plexus. Interneurons provide a mechanism by which this random activity is both amplified and organized into a coherent network rhythm
Many non-equivalent realizations of the associahedron
Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two
different ways of constructing exponential families of realizations of the
n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the
constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify
the associahedra obtained by these constructions modulo linear equivalence of
their normal fans and show, in particular, that the only realization that can
be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002)
associahedron.
For the Hohlweg-Lange associahedra our classification is a priori coarser
than the classification up to isometry of normal fans, by
Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As
a consequence, we get that two Hohlweg-Lange associahedra have linearly
equivalent normal fans if and only if they are isometric.
The Santos construction, which produces an even larger family of
associahedra, appears here in print for the first time. Apart of describing it
in detail we relate it with the c-cluster complexes and the denominator fans in
cluster algebras of type A.
A third classical construction of the associahedron, as the secondary
polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to
never produce a normal fan linearly equivalent to any of the other two
constructions.Comment: 30 pages, 13 figure
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Formality theorems for Hochschild complexes and their applications
We give a popular introduction to formality theorems for Hochschild complexes
and their applications. We review some of the recent results and prove that the
truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200
Higher-Spin Interactions: four-point functions and beyond
In this work we construct an infinite class of four-point functions for
massless higher-spin fields in flat space that are consistent with the gauge
symmetry. In the Lagrangian picture, these reflect themselves in a peculiar
non-local nature of the corresponding non-abelian higher-spin couplings implied
by the Noether procedure that starts from the fourth order. We also comment on
the nature of the colored spin-2 excitation present both in the open string
spectrum and in the Vasiliev system, highlighting how some aspects of String
Theory appear to reflect key properties of Field Theory that go beyond its low
energy limit. A generalization of these results to n-point functions, fermions
and mixed-symmetry fields is also addressed.Comment: 66 pages, 10 figures, 1 table, LaTex. Several statements clarified.
Final version to appear in JHE
Noncommutative homotopy algebras associated with open strings
We discuss general properties of -algebras and their applications
to the theory of open strings. The properties of cyclicity for
-algebras are examined in detail. We prove the decomposition theorem,
which is a stronger version of the minimal model theorem, for
-algebras and cyclic -algebras and discuss various
consequences of it. In particular it is applied to classical open string field
theories and it is shown that all classical open string field theories on a
fixed conformal background are cyclic -isomorphic to each other. The
same results hold for classical closed string field theories, whose algebraic
structure is governed by cyclic -algebras.Comment: 92 pages, 16 figuers; based on Ph.D thesis submitted to Graduate
School of Mathematical Sciences, Univ. of Tokyo on January, 2003; v2:
explanation improved, references added, published versio
Fibre bundle formulation of nonrelativistic quantum mechanics: I. Introduction. The evolution transport
We propose a new systematic fibre bundle formulation of nonrelativistic
quantum mechanics. The new form of the theory is equivalent to the usual one
but it is in harmony with the modern trends in theoretical physics and
potentially admits new generalizations in different directions. In it a pure
state of some quantum system is described by a state section (along paths) of a
(Hilbert) fibre bundle. Its evolution is determined through the bundle
(analogue of the) Schr\"odinger equation. Now the dynamical variables and the
density operator are described via bundle morphisms (along paths). The
mentioned quantities are connected by a number of relations derived in this
work.
The present first part of this investigation is devoted to the introduction
of basic concepts on which the fibre bundle approach to quantum mechanics
rests. We show that the evolution of pure quantum-mechanical states can be
described as a suitable linear transport along paths, called evolution
transport, of the state sections in the Hilbert fibre bundle of states of a
considered quantum system.Comment: 26 standard (11pt, A4) LaTeX 2e pages. The packages AMS-LaTeX and
amsfonts are required. Revised: new material, references, and comments are
added. Minor style chages. Continuation of quan-ph/9803083. For continuation
of the this series see http://www.inrne.bas.bg/mathmod/bozhome
Surface-Based Analyses of Anatomical Properties of the Visual Cortex in Macular Degeneration
INTRODUCTION: Macular degeneration (MD) can cause a central visual field defect. In a previous study, we found volumetric reductions along the entire visual pathways of MD patients, possibly indicating degeneration of inactive neuronal tissue. This may have important implications. In particular, new therapeutic strategies to restore retinal function rely on intact visual pathways and cortex to reestablish visual function. Here we reanalyze the data of our previous study using surface-based morphometry (SBM) rather than voxel-based morphometry (VBM). This can help determine the robustness of the findings and will lead to a better understanding of the nature of neuroanatomical changes associated with MD. METHODS: The metrics of interest were acquired by performing SBM analysis on T1-weighted MRI data acquired from 113 subjects: patients with juvenile MD (JMD; n = 34), patients with age-related MD (AMD; n = 24) and healthy age-matched controls (HC; n = 55). RESULTS: Relative to age-matched controls, JMD patients showed a thinner cortex, a smaller cortical surface area and a lower grey matter volume in V1 and V2, while AMD patients showed thinning of the cortex in V2. Neither patient group showed a significant difference in mean curvature of the visual cortex. DISCUSSION: The thinner cortex, smaller surface area and lower grey matter volume in the visual cortex of JMD patients are consistent with our previous results showing a volumetric reduction in their visual cortex. Finding comparable results using two rather different analysis techniques suggests the presence of marked cortical degeneration in the JMD patients. In the AMD patients, we found a thinner cortex in V2 but not in V1. In contrast to our previous VBM analysis, SBM revealed no volumetric reductions of the visual cortex. This suggests that the cortical changes in AMD patients are relatively subtle, as they apparently can be missed by one of the methods
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