Vertex operators in algebraic topology

AbstractDementia is related to the cellular accumulation of β-amyloid plaques, tau aggregates, or α-synuclein aggregates, or to neurotransmitter deficiencies in the dopaminergic and cholinergic pathways. Cellular and neurochemical changes are both involved in dementia pathology. However, the role of dopaminergic and cholinergic networks in metabolic connectivity at different stages of dementia remains unclear. The altered network organisation of the human brain characteristic of many neuropsychiatric and neurodegenerative disorders can be detected using persistent homology network (PHN) analysis and algebraic topology. We used 18F-fluorodeoxyglucose positron emission tomography (18F-FDG PET) imaging data to construct dopaminergic and cholinergic metabolism networks, and used PHN analysis to track the evolution of these networks in patients with different stages of dementia. The sums of the network distances revealed significant differences between the network connectivity evident in the Alzheimer’s disease and mild cognitive impairment cohorts. A larger distance between brain regions can indicate poorer efficiency in the integration of information. PHN analysis revealed the structural properties of and changes in the dopaminergic and cholinergic metabolism networks in patients with different stages of dementia at a range of thresholds. This method was thus able to identify dysregulation of dopaminergic and cholinergic networks in the pathology of dementia.

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Polynomial ring representations of endomorphisms of exterior powers

Collectanea mathematica ◽

10.1007/s13348-020-00310-5 ◽

2021 ◽

Author(s):

Ommolbanin Behzad ◽

André Contiero ◽

Letterio Gatto ◽

Renato Vidal Martins

Keyword(s):

Lie Algebra ◽

Polynomial Ring ◽

Vertex Operators ◽

Exterior Power ◽

Infinite Dimensional ◽

Symmetric Structure ◽

Rational Polynomials ◽

Exterior Powers ◽

Vertex Representation ◽

Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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Relating the b ghost and the vertex operators of the pure spinor superstring

Journal of High Energy Physics ◽

10.1007/jhep03(2021)165 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Osvaldo Chandia ◽

Brenno Carlini Vallilo

Keyword(s):

Vertex Operator ◽

Bosonic String ◽

Vertex Operators ◽

Pure Spinor ◽

Double Pole ◽

Lorenz Gauge

Abstract The OPE between the composite b ghost and the unintegrated vertex operator for massless states of the pure spinor superstring is computed and shown to reproduce the structure of the bosonic string result. The double pole vanishes in the Lorenz gauge and the single pole is shown to be equal to the corresponding integrated vertex operator.

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Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

Entropy ◽

10.3390/e23010125 ◽

2021 ◽

Vol 23 (1) ◽

pp. 125

Author(s):

Tobias Gulden ◽

Alex Kamenev

Keyword(s):

Quantum Mechanics ◽

Algebraic Topology ◽

Riemann Surfaces ◽

Surrounding Medium ◽

Coulomb Systems ◽

Dielectric Constants ◽

Ion Interactions ◽

Multivalent Ions ◽

Electric Filed ◽

Entropic Effects

We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.

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Algebraic Topology.

American Mathematical Monthly ◽

10.2307/2319113 ◽

1973 ◽

Vol 80 (4) ◽

pp. 449

Author(s):

M. J. Powers ◽

C. R. F. Maunder

Keyword(s):

Algebraic Topology

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THE MAIN TRENDS OF ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY

Russian Mathematical Surveys ◽

10.1070/rm1964v019n06abeh001164 ◽

1964 ◽

Vol 19 (6) ◽

pp. 67-73

Author(s):

S P Novikov ◽

I I Pyatetskii-Shapiro ◽

I R Shafarevich

Keyword(s):

Algebraic Geometry ◽

Algebraic Topology

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Algebraic Topology: A First Course.

American Mathematical Monthly ◽

10.2307/2974465 ◽

1996 ◽

Vol 103 (9) ◽

pp. 819

Author(s):

William S. Massey ◽

William Fulton

Keyword(s):

Algebraic Topology

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VERTEX OPERATORS IN THE QUANTUM HALL EFFECT

International Journal of Modern Physics B ◽

10.1142/s0217979291000316 ◽

1991 ◽

Vol 05 (03) ◽

pp. 509-527 ◽

Cited By ~ 39

Author(s):

MICHAEL STONE

Keyword(s):

Hall Effect ◽

Landau Level ◽

Quantum Hall Effect ◽

Group Action ◽

Quantum Hall ◽

Vertex Operators ◽

Edge States ◽

Many Body ◽

Level I ◽

The Many

The edge states of the quantum Hall effect carry representations of chiral current algebras and their associated groups. In the simplest case of a single filled Landau level, I demonstrate explicitly how the group action affects the many-body states, and why the Kac-Peterson cocycle appears in the group multiplication law. I show how these representations may be used to construct vertex operators which create localised edge excitations, and indicate how they are related to the bulk quasi-particles.

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