2021-05-12T23:03:27Z
https://zenodo.org/oai2d
oai:zenodo.org:3959619
2020-08-06T19:26:06Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this paper we introduce $B_{\alpha,\beta}^{k}$-manifolds as generalizations of the notions of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^{k}$-manifolds whose transition functions $\varphi_{ji}=\varphi_{j}\circ\varphi_{i}^{-1}$ are such that $\partial^{\mu}\varphi_{ji}\in B_{\alpha(r)}\cap C^{k-\beta(r)}$ for every $\vert\mu\vert=r$, where $B=(B_{r})_{r\in\Gamma}$ is some sequence of presheaves of Fr\'echet spaces endowed with further structures, $\Gamma\subset\mathbb{Z}_{\geq0}$ is some parameter set and $\alpha,\beta$ are functions. We present embedding theorems for the presheaf category of those structural presheaves $B$. The existence problem of $B_{\alpha,\beta}^{k}$-structures on $C^{k}$-manifolds is studied and it is proved that under certain conditions on $B$, $\alpha$ and $\beta$, the forgetful functor from $C^{k}$-manifolds to $B_{\alpha,\beta}^{k}$-manifolds has adjoints.
https://zenodo.org/record/3959619
10.5281/zenodo.3959619
oai:zenodo.org:3959619
arxiv:arXiv:1908.04442
doi:10.5281/zenodo.3959618
url:https://zenodo.org/communities/math-phys-cat-group
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Existence of $B^k_{\alpha,\beta}$-Structures on $C^k$-Manifolds
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publication-article
oai:zenodo.org:3959688
2020-08-06T19:26:03Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this paper we consider the classification problem of extensions of Yang-Mills-type (YMT) theories. For us, a YMT theory differs from the classical Yang-Mills theories by allowing an arbitrary pairing on the curvature. The space of YMT theories with a prescribed gauge group G and instanton sector P is classified, an upper bound to its rank is given and it is compared with the space of Yang-Mills theories. We present extensions of YMT theories as a simple and unified approach to many different notions of deformations and addition of correction terms previously discussed in the literature. A relation between these extensions and emergence phenomena in the sense of arXiv:2004.13144 is presented. We consider the space of all extensions of a fixed YMT theory SG and we prove that for every additive group action of G in R and every commutative and unital ring R, this space has an induced structure of R[G]-module bundle. We conjecture that this bundle can be continuously embedded into a trivial bundle. Morphisms between extensions of a fixed YMT theory are defined in such a way that they define a category of extensions. It is proved that this category is a reflective subcategory of a slice category, reflecting some properties of its limits and colimits.
https://zenodo.org/record/3959688
10.5281/zenodo.3959688
oai:zenodo.org:3959688
arxiv:arXiv:2007.01660
doi:10.5281/zenodo.3959687
url:https://zenodo.org/communities/math-phys-cat-group
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On Extensions of Yang-Mills-Type Theories, Their Spaces and Their Categories
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publication-article
oai:zenodo.org:3959725
2020-08-06T19:25:58Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
2020-07-24
Talk on a vertical categorification of Lie algebras (called Lie algebroidal categories) and a connection with Lie algebroids. A priori the Lie algebroidal categories differs from Lie 2-algebras of Baez-Crans, but further investigations are needed.
https://zenodo.org/record/3959725
10.13140/RG.2.2.11757.95209
oai:zenodo.org:3959725
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Lie Algebroidal Categories
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:3974986
2020-08-07T00:59:21Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
2018-02-28
Slides from the defense of my master's thesis. In Brazilian portuguese.
P.S: unfortunately the slides were not made using beamer/latex.
https://zenodo.org/record/3974986
10.5281/zenodo.3974986
oai:zenodo.org:3974986
doi:10.5281/zenodo.3974985
url:https://zenodo.org/communities/math-phys-cat-group
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Categorical and Geometrical Methods in Physics
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:3959605
2020-08-06T19:26:07Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this paper we consider the existence problem of affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs. We show that if $M$ admits a suitable subatlas, meaning a $\mathcal{B}_{\alpha,\beta}^{k}$-structure for a certain presheaf of Fr\'echet spaces $B$ and for certain functions $\alpha$ and $\beta$, then the existence of such regular connections can be established. It is also proved that if the $\mathcal{B}_{\alpha,\beta}^{k}$-structure is actually nice (in the sense of \citep{Bk_manifolds}), then a multiplicity result can also be obtained by means of Thom's transversality arguments.
https://zenodo.org/record/3959605
10.5281/zenodo.3959605
oai:zenodo.org:3959605
arxiv:arXiv:1910.03113
doi:10.5281/zenodo.3959604
url:https://zenodo.org/communities/math-phys-cat-group
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Geometric Regularity Results on $B_{\alpha,\beta}^{k}$-Manifolds, I: Affine Connections
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publication-article
oai:zenodo.org:3959714
2020-08-06T19:26:01Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this paper we continue the program on the classification of extensions of the Standard Model of Particle Physics started in arXiv:2007.01660. We propose four complementary questions to be considered when trying to classify any class of extensions of a fixed Yang-Mills-type theory $S^G$: existence problem, obstruction problem, maximality problem and universality problem. We prove that all these problems admits a purely categorical characterization internal to the category of extensions of $S^G$. Using this it is showed that maximality and universality are dense properties, meaning that if they are not satisfied in a class $E(S^G ;Ĝ)$, then they in their "one-point compactification" by a specific trivial extension Ŝ. We prove that, by means of assuming Axiom of Choice, one can get another maximality theorem, now independent of the trivial extension Ŝ. We considered the class of almost coherent extensions, i.e, complete, injective and of pullback-type, and we show that for it the existence and the obstruction problems have a complete solution. Using again the Axiom of Choice, we prove that this class of extensions satisfies the hypothesis of the second maximality theorem.
https://zenodo.org/record/3959714
10.5281/zenodo.3959714
oai:zenodo.org:3959714
arxiv:arXiv:2007.08651
doi:10.5281/zenodo.3959713
url:https://zenodo.org/communities/math-phys-cat-group
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On Maximal, Universal and Complete Extensions of Yang-Mills-Type Theories
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publication-article
oai:zenodo.org:3959632
2020-08-06T19:26:05Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
These are notes for a short course and some talks gave at Departament of Mathematics and at Departament of Physics of Federal University of Minas Gerais, based on the author's paper arXiv:1808.09249. Some new information and results are also presented. Unlike the original work, here we try to give a more physical emphasis. In this sense, we present obstructions to realize gravity, modeled by the tetradic Einstein-Hilbert-Palatini (EHP) action functional, in a general geometric setting. In particular, we show that if spacetime has dimension $n\geq4$, then the cosmological constant plays no role in any "concrete geometries" other than Lorentzian. If $n\geq6$, then the entire EHP theory is trivial, meaning that Lorentzian geometry is (essentially) the only "concrete geometry" in which gravity (i.e, the EHP action functional) makes sense. Examples of "concrete geometries" include those locally modeled by group reductions $H\hookrightarrow O(k;A)$ for some $k$ and some algebra $A$, so that Riemannian geometry, Hermitian geometry, K\"ahler geometry and symplectic geometry, as well as Type II geometry, Hitchin's generalized complex geometry and $G_{2}$-geometry are included. We also study EHP theory in "abstract geometries", such as graded geometry (and hence supergeometry), and we show how the obstruction results extend to this context. We construct two theories naturally associated to EHP, which we call the geometric/algebraic dual of EHP, and we analyze the effect of the obstructions in these new theories. Finally, we speculate (and provide evidence for) the existence of a "universal obstruction condition".
https://zenodo.org/record/3959632
10.5281/zenodo.3959632
oai:zenodo.org:3959632
arxiv:arXiv:1912.11198
doi:10.5281/zenodo.3959631
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Geometric Obstructions on Gravity
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3959668
2020-08-06T19:26:04Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this paper we propose a formal definition of what is a strong emergence phenomenon between two parameterized field theories and we present sufficient conditions ensuring the existence of such phenomena between two given parameterized Lagrangian field theories. More precisely, we prove that in an Euclidean background, typical parameterized kinetic theories emerge from any elliptic multivariate polynomial theories. Some concrete examples are given and a connection with the phenomenon of gravity emerging from noncommutativity is made.
https://zenodo.org/record/3959668
10.5281/zenodo.3959668
oai:zenodo.org:3959668
arxiv:arXiv:2004.13144
doi:10.5281/zenodo.3959667
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Towards Axiomatization and General Results on Strong Emergence Phenomena Between Lagrangian Field Theories
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3959723
2020-08-06T19:25:59Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
2020-07-24
In this work we develop the higher categorical language aiming to apply it in the foundations of physics, following an approach based in works of Urs Schreiber, John Baez, Jacob Lurie, Daniel Freed and many others. The text has three parts. In Part I we introduce categorical language with special focus in algebraic topological aspects, and we discuss that it is not abstract enough to give a full description for the foundations of physics. In Part II we introduce the categorical process, which produce an abstract language from a concrete language. Examples are given, again focused on Algebraic Topology. In Part III we use the categorification process in order to construct arbitrarily abstract languages, the higher categorical ones, including the cohesive ∞-topos. An emphasis on the formalization of abstract stable homotopy theory is given. We discuss the reason why we should believe that cohesive ∞-topos are natural languages to use in order to attack Hilbert’s sixth problem.
https://zenodo.org/record/3959723
10.13140/RG.2.2.21858.68800
oai:zenodo.org:3959723
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/4.0/legalcode
Categorical and Geometrical Methods in Physics
info:eu-repo/semantics/doctoralThesis
publication-thesis
oai:zenodo.org:3959526
2020-08-06T19:26:10Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2020-07-24
In this article functorial Feynman rules are introduced as large generalizations of physicists Feynman rules, in the sense that they can be applied to arbitrary classes of hypergraphs, possibly endowed with any kind of structure on their vertices and hyperedges. We show that the reconstruction conjecture for classes of (possibly structured) hypergraphs admit a sheaf-theoretic characterization, allowing us to consider analogous conjectures. We propose an axiomatization for the notion of superposition principle and prove that the functorial Feynman rules work as a bridge between reconstruction conjectures and superposition principles, meaning that a conjecture for a class of hypergraphs is satisfied only if each functorial Feynman rule defined on it induces a superposition principle. Applications in perturbative euclidean quantum field theory and graph theory are given.
https://zenodo.org/record/3959526
10.5281/zenodo.3959526
oai:zenodo.org:3959526
arxiv:arXiv:1903.06284
doi:10.5281/zenodo.3959525
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Graph Reconstruction, Functorial Feynman Rules and Superposition Principles
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3696779
2020-03-04T19:20:52Z
openaire
user-math-phys-cat-group
Yuri Ximenes Martins
Rodney Josué Biezuner
2018-08-24
In this article we introduce A-valued Einstein-Hilbert-Palatini functional (A-EHP) over a n-manifold M, where A is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if A is weak (k, s)-solvable, then A-EHP is non-null only if n < k + s + 3. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for k = 1 and s = 2, including Hitchin’s generalized complex geometry, Pantilie’s generalized quaternionic geometries and all other generalized Cayley-Dickson geometries. We also prove that if A is concrete in some sense, then a torsionless version of A-EHP is non-null only if M is Kähler of dimension n = 2, 4. We present our results as obstructions to M being an Einstein manifold relative to geometries other than semi-Riemannian.
https://zenodo.org/record/3696779
10.5281/zenodo.3696779
oai:zenodo.org:3696779
arxiv:arXiv:1808.09249v2
doi:10.5281/zenodo.3696778
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/4.0/legalcode
Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3959719
2020-08-06T19:26:00Z
openaire
user-math-phys-cat-group
Dadam, Fábio
Ximenes Martins, Yuri
2020-07-24
This is an introductory text on (Algebraic) Topology and Geometry, focusing on the general theory of spherically symmetric black holes. Written in Brazilian Portuguese. The text start with some category theory, and then we introduce the homotopy category of topological spaces. We talk briefly about basic aspects of homotopy theories, including higher homotopy groups, generalized cohomology, etc. Next we introduce general fiber bundles, sketch the classication theorem for principal bundles and show the correspondence between principal bundles and vector bundles. We study the problem of building global sections by means of obstruction theory methods. In the sequence we define connections in principal G-bundles as integrable distribution, show the relation with Lie(G)-valued forms an analyze the interplay between curvature and holonomy (Ambrose-Singer theorem). The correspondence between horizontal equivariant Lie(G)-valued forms on principal bundles P-->M and Ad(P)-valued forms on M is established and the Bianchi-type identities are discussed. General Relativity is introduced in terms of this geometrical background. It is proved the standard obstruction theorem on existence of Lorentzian metrics. The notions of singular spacetime are presented. We then focused on spherically symmetric black holes, following first something like "The Mathematical Theory of Black Holes", by Chandrasekhar, where we prove the "No-Hair Theorem". The remaining of the text is on Newman-Penrose formalism applied to the problem of determining the normal modes of a black hole perturbation.
https://zenodo.org/record/3959719
10.5281/zenodo.3959719
oai:zenodo.org:3959719
doi:10.5281/zenodo.3959718
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Topology, Geometry and Black Holes
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:3959506
2020-08-06T19:26:12Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
Josué Biezuner, Rodney
2019-09-01
In this article we introduce A-valued Einstein-Hilbert-Palatini functional (A-EHP) over a n-manifold M, where A is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if A is weak (k,s)-solvable, then A-EHP is non-null only if n<k+s+3. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for k=1 and s=2, including Hitchin's generalized complex geometry, Pantilie's generalized quaternionic geometries and all other generalized Cayley-Dickson geometries. We also prove that if A is concrete in some sense, then a torsionless version of A-EHP is non-null only if M is Kähler of dimension n=2,4. We present our results as obstructions to M being an Einstein manifold relative to geometries other than semi-Riemannian.
https://zenodo.org/record/3959506
10.1016/j.geomphys.2019.04.012
oai:zenodo.org:3959506
arxiv:arXiv:1808.09249
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3959564
2020-08-06T19:26:08Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
2020-07-24
The Tolman--Oppenheimer--Volkoff (TOV) equations are a partially uncoupled system of nonlinear and non-autonomous ordinary differential equations which describe the structure of isotropic spherically symmetric static fluids. Nonlinearity makes finding explicit solutions of TOV systems very difficult and such solutions and very rare. In this paper we introduce the notion of pseudo-asymptotic TOV systems and we show that the space of such systems is at least fifteen-dimensional. We also show that if the system is defined in a suitable domain (meaning the extended real line), then well-behaved pseudo-asymptotic TOV systems are genuine TOV systems in that domain, ensuring the existence of new fourteen analytic solutions for extended TOV equations. The solutions are classified according to the nature of the matter (ordinary or exotic) and to the existence of cavities and singularities. It is shown that at least three of them are realistic, in the sense that they are formed only by ordinary matter and contain no cavities or singularities.
https://zenodo.org/record/3959564
10.1016/j.aop.2019.167929
oai:zenodo.org:3959564
arxiv:arXiv:1809.02281
url:https://zenodo.org/communities/math-phys-cat-group
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Existence and Classification of Pseudo-Asymptotic Solutions for Tolman-Oppenheimer-Volkoff Systems
info:eu-repo/semantics/article
publication-article
oai:zenodo.org:3959520
2020-08-06T19:26:11Z
openaire
user-math-phys-cat-group
Ximenes Martins, Yuri
de Souza Plácido Teixeira, Daniel
Andrade Campos, Luiz Felipe
2019-01-09
In this article we prove three obstruction results on the existence of equations of state in clusters of stellar systems fulfilling mass-radius relationships and some additional bound (on the mass, on the radius or a causal bound). The theorems are proved in large generality. We start with a motivating example of TOV systems and we close by applying our results in stellar systems arising from experimental data.
https://zenodo.org/record/3959520
10.1103/PhysRevD.99.023007
oai:zenodo.org:3959520
url:https://zenodo.org/communities/math-phys-cat-group
info:eu-repo/semantics/openAccess
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Constraints Between Equations of State and Mass-Radius Relationships in General Clusters of Stellar Systems
info:eu-repo/semantics/article
publication-article