Geometric Realization Research Articles

On Geometric Realization of the General Manakov System

On Geometric Realization of the General Manakov System

Toric 2-group anomalies via cobordism

2-group symmetries arise in physics when a 0-form symmetry G[0] and a 1-form symmetry H[1] intertwine, forming a generalised group-like structure. Specialising to the case where both G[0] and H[1] are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such ‘toric 2-group symmetries’ using the cobordism classification. As a warm up example, we use cobordism to study various ’t Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of B|\U0001d53e| where \U0001d53e is any 2-group whose 0-form and 1-form symmetry parts are both U(1), and |\U0001d53e| is the geometric realisation of the nerve of the 2-group \U0001d53e. By leveraging a variety of algebraic methods, we show that {varOmega}_5^{textrm{Spin}}left(Bleft|mathbbm{G}right|right)cong mathbb{Z}/m where m is the modulus of the Postnikov class for \U0001d53e, and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) U(1) global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure.

Rational homogeneous spaces as geometric realizations of birational transformations

In this paper we study geometric realizations of some classic birational maps, such as inversion maps, special Cremona transformations, special birational transformations of type (2, 1), by considering {{mathbb {C}}}^*-actions on rational homogeneous spaces and their subvarieties.

Geometric realizations of birational transformations via {{mathbb {C}}}^*-actions

In this paper, we study varieties admitting torus actions as geometric realizations of birational transformations. We present an explicit construction of these geometric realizations for a particular class of birational transformations, and study some of their geometric properties, such as their Mori, Nef and Movable cones.

Horospherical two-orbit varieties as zero loci

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G 2 G_2 -variety, including its quantum version. We also consider the S p i n 7 Spin_7 -variety, which deserves a different treatment.

Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (textrm{Sp}_{2n},textrm{O}_2^-) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

Einstein-Rosen bridge from the minimal length

We use a string T-duality corrected pair of regular black holes to construct an Einstein-Rosen (ER) bridge with the wormhole throat proportional to the zero-point (Planck) length. This may be a geometric realization of quantum entanglement for particle/antiparticle pairs. We point out that for an extreme mass configuration consisting of a black hole pair, one can have an ER bridge with a horizon area that coincides with the Bekenstein minimal area bound along with a wormhole mass proportional to the Planck mass. This could be related to gravitational self-completeness with quantum mechanical mass limits. We also discuss the ER bridge for sub-Planckian mass horizonless wormholes and show this admits a region of negative energy at the throat, which we posit to be related to a particle of negative mass generated by quantum fluctuations or the Casimir effect. We argue that Hawking radiation could be the best way for the geometric realization of quantum entanglement for particle/antiparticle pairs emitted by black holes. This sheds new light on the ER=EPR conjecture.

APPLYING 3D PRINTING TECHNOLOGY IN CONSTRUCTING SUSTAINABLE HOUSES

Innovation is advancing at an ever-expanding pace and has been for decades. Regular objects have gotten to be minimal in measure and cheaper to deliver. 3D printing technology (3DP), also known as additive manufacturing (AM), has gaining rapid development in recent years. is a method for turning a computer file into 3D solid objects of any shape or geometry. The creation is accomplished by building up layers of a certain substance until the full item is produced. The 3D printing technologies, comparing to traditional techniques of constructing the buildings, could be considered as environmentally friendly derivative giving almost unlimited possibilities for geometric complexity realizations. The primary issue is that the building sector has a significant negative influence on the environment. Some modern technologies are being applied to reduce materials usage and carbon emissions like 3D printing. The major purpose of this article is to compare between conventional approaches and 3D printing technology to assess their effects on sustainable housing, evaluate how 3D printing functions and consider its advantages and disadvantages. To conduct the research, a comparison between traditional techniques and 3d printing technology will take place to see their impacts on sustainable houses, theoretical studies will be done on both :3D printing concepts and application in architecture and its effectiveness on environment and sustainability. In addition to the practical studies, experimental ones where case studies are selected according to the inductive method mentioning the selection criteria. Three case studies will be analysed, where the 3D printing technology is achieved, TECLA, a circular housing prototype in Ravenna, Italy, the Dubai municipality and Beckum First 3D Printed house in Germany.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Simplicial and combinatorial versions of higher symmetric topological complexity

In this paper, we introduce higher symmetric simplicial complexity SCnΣ(K) of a simplicial complex K and higher symmetric combinatorial complexity CCnΣ(P) of a finite poset P. These are simplicial and combinatorial approaches to symmetric motion planning of Basabe - González - Rudyak - Tamaki. We prove that the symmetric simplicial complexity SCnΣ(K) is equal to symmetric topological complexity TCnΣ(|K|) of the geometric realization of K and the symmetric combinatorial complexity CCnΣ(P) is equal to symmetric topological complexity TCnΣ(|K(P)|) of the geometric realization of the order complex of P.

Binary geometries, generalized particles and strings, and cluster algebras

The authors study the fundamental properties of scattering amplitudes of particles in any spacetime dimension. They introduce binary geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. Furthermore, they define open and closed ``cluster string integrals'', which provide a generalization of particle and string scattering amplitudes, and enjoy remarkable factorization properties at finite ${\ensuremath{\alpha}}^{\ensuremath{'}}$.

Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex K ( X ) \mathcal {K}(X) associated to a topological space X X , and we prove that the geometric realization of K ( X ) \mathcal {K}(X) is homotopy equivalent to X X for every CW complex X X . Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K ( X ) \mathcal {K}(X) is complete for every CW complex X X . In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the ℓ 1 \ell ^1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.

Geometric Realizations of Tamari Interval Lattices Via Cubic Coordinates

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable.

A geometric realization of silting theory for gentle algebras

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff and Kapranov (J Eur Math Soc 20(6):1473–1524, 2018) in the context of triangulated surfaces. We also show that the cutting of the underlying surface along a curve without self-intersections corresponds to silting reduction. This is analogous to the Calabi–Yau reduction of surface cluster categories as shown by Marsh and Palu (Proc Lond Math Soc (3) 108(2):411–440, 2014) and Qiu and Zhou (Compos Math 153(9):1779–1819, 2017).

Hopf monoids, permutohedral cones, and generalized retarded functions

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e., by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein–Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra.

Greedy Causal Discovery Is Geometric

Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example, the greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing algorithms. In 2010, Studený, Hemmecke, and Lindner introduced the characteristic imset (CIM) polytope, , whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of . Thus, we observe that greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing all have geometric realizations as greedy edge-walks along . Furthermore, the identified edges of strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM, and a hybrid variant, skeletal greedy CIM, that outperforms current competitors among hybrid and constraint-based algorithms.

Shellable tilings on relative simplicial complexes and their h-vectors

Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

Rank gain of Jacobians over number field extensions with prescribed Galois groups

AbstractWe investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non‐Galois extensions whose Galois closure has a Galois group permutation‐isomorphic to a prescribed group G (in short, “G‐extensions”). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) gain rank over infinitely many G‐extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that “many” elliptic curves gain rank over infinitely many G‐extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties.

The geometric νSMEFT: operators and connections

We write down a geometric realization of the Standard Model Effective Field Theory (SMEFT) extended by nf flavours of light sterile neutrinos, a so-called geoνSMEFT. As with the geoSMEFT introduced by Helset, Martin and Trott, we show that a refactorization of the νSMEFT’s operator product expansion is possible, such that two- and three-point composite operator forms are dressed with field-space connections composed of towers of Higgs dressings and symmetry generators, valid at all-orders in the {overline{v}}_T/Lambda expansion parameter of the EFT left({overline{v}}_Tequiv sqrt{2leftlangle {H}^{dagger }Hrightrangle}right) . These connections are parameterized by real Higgs coordinates and contribute to the field-space geometry of the (ν)SMEFT, with structure linked to the strength of Beyond-the-(ν)Standard Model physics encoded in {overline{v}}_T/Lambda . In addition to enumerating the relevant composite operators and associated connections, we briefly outline the route to calculating all- {overline{v}}_T/Lambda -orders amplitudes, including the flavor-invariant theory required to understand the neutrino mass-eigenstate basis geometrically.

Efficient simplicial replacement of semialgebraic sets

Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .