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1. Constructions of Lagrangian cobordisms

   https://doi.org/10.1007/978-3-030-80979-9  

   Blackwell, Sarah; Legout, Noémie; Leverson, Caitlin; Limouzineau, Maÿlis; Myer, Ziva; Pan, Yu; Pezzimenti, Samantha; Simone Suárez, Lara; Traynor, Lisa (January 2021, Association for Women in Mathematics series)

   Acu, Bahar; Cannizzo, Catherine; McDuff, Dusa; Myer, Ziva; Pan, Yu; Traynor, Lisa (Ed.)

   Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known ``elementary'' building blocks for Lagrangian cobordisms that are smoothly the attachment of 0- and 1-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is ``decomposable'' into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms. 

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2. Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

   https://doi.org/10.1515/Crelle-2024-0003  

   Lim, Geunho; Weinberger, Shmuel (February 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   .We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated. 

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3. Non‐Flat Universes and Black Holes in Asymptotically Free Mimetic Gravity

   https://doi.org/10.1002/prop.201900103  

   Chamseddine, Ali H.; Mukhanov, Viatcheslav; Russ, Tobias B. (December 2019, Fortschritte der Physik)

   Abstract The recently proposed theory of “Asymptotically Free Mimetic Gravity” is extended to the general non‐homogeneous, spatially non‐flat case. We present a modified theory of gravity which is free of higher derivatives of the metric. In this theory asymptotic freedom of gravity implies the existence of a minimal black hole with vanishing Hawking temperature. Introducing a spatial curvature dependent potential, we moreover obtain non‐singular, bouncing modifications of spatially non‐flat Friedmann and Bianchi universes. 

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4. Topological operators and completeness of spectrum in discrete gauge theories

   https://doi.org/10.1007/JHEP12(2020)172  

   Rudelius, Tom; Shao, Shu-Heng (December 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity. 

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5. Local operator algebras of charged states in gauge theory and gravity

   https://doi.org/10.1007/JHEP03(2025)175  

   Grassi, P A; Porrati, M (March 2025, Journal of High Energy Physics)

   Multiple (Ed.)

   A<sc>bstract</sc> Powerful techniques have been developed in quantum field theory that employ algebras of local operators, yet local operators cannot create physical charged states in gauge theory or physical nonzero-energy states in perturbative quantum gravity. A common method to obtain physical operators out of local ones is to dress the latter using appropriate Wilson lines. This procedure destroys locality, it must be done case by case for each charged operator in the algebra, and it rapidly becomes cumbersome, particularly in perturbative quantum gravity. In this paper we present an alternative approach to the definition of physical charged operators: we define an automorphism that maps an algebra of local charged operators into a (non-local) algebra of physical charged operators. The automorphism is described by a formally unitary intertwiner mapping the exact BRS operator associated to the gauge symmetry into its quadratic part. The existence of an automorphism between local operators and the physical ones, describing charged states, allows to retain many of the results derived in local operator algebras and extend them to the physical-but-nonlocal algebra of charged operators as we discuss in some simple applications of our construction. We also discuss a formal construction of physical states and possible obstructions to it. 

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