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Int. J. Topol., Volume 2, Issue 1 (March 2025) – 3 articles

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9 pages, 249 KiB  
Article
Johnstone’e Non-Sober Dcpo and Extensions
by Dongsheng Zhao
Int. J. Topol. 2025, 2(1), 3; https://doi.org/10.3390/ijt2010003 - 3 Mar 2025
Viewed by 179
Abstract
One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in [...] Read more.
One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo. Full article
(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)
62 pages, 523 KiB  
Article
Existence and Mass Gap in Quantum Yang–Mills Theory
by Logan Nye
Int. J. Topol. 2025, 2(1), 2; https://doi.org/10.3390/ijt2010002 - 25 Feb 2025
Viewed by 344
Abstract
This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete [...] Read more.
This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete realization of the Yang–Mills theory that is manifestly gauge-invariant and satisfies the Wightman axioms. We demonstrate the existence of a mass gap by analyzing the entanglement spectrum of the vacuum state, establishing a direct connection between the mass gap and the minimum non-zero eigenvalue of the entanglement Hamiltonian. Our approach also offers new insights into non-perturbative phenomena such as confinement and asymptotic freedom. We introduce new mathematical tools, including entanglement renormalization for gauge theories and quantum circuit complexity measures for quantum fields. The implications of our work extend beyond the Yang–Mills theory, suggesting new approaches to quantum gravity, strongly coupled systems, and cosmological problems. This quantum information perspective on gauge theories opens up exciting new directions for research at the intersection of quantum field theory, quantum gravity, and quantum computation. Full article
13 pages, 276 KiB  
Article
The Bogomolny–Carioli Twisted Transfer Operators and the Bogomolny–Gauss Mapping Class Group
by Orchidea Maria Lecian
Int. J. Topol. 2025, 2(1), 1; https://doi.org/10.3390/ijt2010001 - 12 Jan 2025
Viewed by 604
Abstract
The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in [...] Read more.
The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized PSL (2,Z) domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy. Full article
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