arXiv subjects

Henrique N. Sá Earp

Publications and source records attributed to Henrique N. Sá Earp.

33 records · Page 2Linked to original sources

arXiv2022

Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere

We describe the $10$-dimensional space of $Sp(2)$-invariant $G_2$-structures on the homogeneous $7$-sphere $S^7=Sp(2)/Sp(1)$ as $\mathbb{R}^+\times Gl^+(3,\mathbb{R})$. In those terms, we formulate a general Ansatz for $G_2$-structures, which realises representatives in each of the $7$ possible isometric classes of homogeneous $G_2$-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an $S^3$-family, the equator of which is a new $S^2$-family of coclosed $G_2$-structures satisfying the harmonicity condition $div T=0$. We show general existence of harmonic representatives of $G_2$-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic $G_2$-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.

math.DG↗

arXiv2021

Harmonic flow of $\mathrm{Spin}(7)$-structures

We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $\varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.

math.DG↗

arXiv2021

Constructive Spherical Codes by Hopf Foliations

We present a new systematic approach to constructing spherical codes in dimensions $2^k$, based on Hopf foliations. Using the fact that a sphere $S^{2n-1}$ is foliated by manifolds $S_{\cosη}^{n-1} \times S_{\sinη}^{n-1}$, $η\in[0,π/2]$, we distribute points in dimension $2^k$ via a recursive algorithm from a basic construction in $\mathbb{R}^4$. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity $O(n)$ and time complexity $O(n \log n)$. We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity $O(n \log n)$.

cs.IT↗

arXiv2021

Explicit Soliton for the Laplacian Co-Flow on a Solvmanifold

We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant $G_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian $7$-manifold.

math.DG↗

arXiv2019

Construction of G_2-instantons via twisted connected sums

We propose a method to construct G_2-instantons over a compact twisted connected sum G_2-manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end (see arXiv:1310.7933). In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne-Serre construction, which can be adapted to produce other examples. Moreover, their deformation theory and asymptotic behaviour are explicitly understood, results which may be of independent interest.

math.AG↗

arXiv2019

The Weitzenböck formula for the Fueter-Dirac operator

We find Weitzenböck formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we give a different proof of rigidity for one of Lotay's associatives in the round $7$-sphere from those given by Kawai. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated $G_2$-structure obtained by Fernandez-Fino-Raffero.

math.DG↗

arXiv2018

Current progress on $G_2$--instantons over twisted connected sums

We review a method to construct $\rm{G}_2$--instantons over compact $\rm{G}_2$--manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau $3$-folds with cylindrical end, based on the series of articles [SE15, SEW15, JMPSE17, MNSE17] by the author and others. The construction is based on gluing $\rm{G}_2$--instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano $3$-folds by an algorithmic procedure based on the Hartshorne-Serre correspondence.

math.DG↗

arXiv2018

Gauge theory and G2-geometry on Calabi-Yau links

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $φ$ induced by the Calabi-Yau $3$-orbifold basic geometry. We distinguish these pairs $(K,φ)$ by the Crowley-Nordström $\mathbb{Z}_{48}$-valued $ν$ invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern-Simons formalism and topological energy bounds. In fact compatible $\rm G_2$-instantons on holomorphic Sasakian bundles over $K$ are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain $\rm G_2$-instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural $\rm G_2$-instanton count.

math.DG↗

arXiv2017

Holomorphic bundles for higher dimensional gauge theory

Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the $\rm G_2-$manifolds obtained from asymptotically cylindrical Calabi-Yau $3-$folds studied by Kovalev and by Corti-Haskins-Nordström-Pacini et al. The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties $X$ with $\operatorname{Pic}{X}\simeq\mathbb{Z}^l$, a result which may be of independent interest. Finally, we apply monads to produce a prototypical model of the curvature blow-up phenomenon along a sequence of asymptotically stable bundles degenerating into a torsion-free sheaf.

math.AG↗

arXiv2016

Generalised Chern-Simons Theory and ${\rm G}_2$-Instantons over Associative Fibrations

Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the ${\rm G}_2$-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations $f:Y^7\to X^4$ relates ${\rm G}_2$-instantons on pullback bundles $f^*E\to Y$ and self-dual connections on the bundle $E\to X$ over the base, a fact which may be of independent interest.

math.DG↗

arXiv2016

Non-Euclidean ideal spectrometer

We describe the mathematical scheme for an anomaly-free ideal spectrometer, based on a 2-dimensional plane medium with conical regions of bounded slope. Moreover, the construction may be realised in many different configurations.

physics.ins-det↗

arXiv2014

$G_2$-instantons over asymptotically cylindrical manifolds

A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the $G_2$-instanton equation from the associated Hermitian Yang-Mills problem, to which the methods of Simpson et al. are applied, subject to a crucial asymptotic stability assumption over the "boundary at infinity".

math.DG↗

arXiv2014

G2-instantons over Kovalev manifolds II

This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly $G_2$. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first article, are glued compatibly with a twisted connected sum, to produce a $G_2-$instanton over the resulting compact 7-manifold. This is accomplished under a nondegeneracy acyclic assumption on the bundle `at infinity', which occurs e.g. over certain projective varieties $X_{22}$ in $\mathbb{C}P^{13}$ equipped with an asymptotically rigid bundle.

math.DG↗

arXiv2013

The Entropy Law and the impossibility of perpetual economic growth

Every production-recycling iteration accumulates an inevitable proportion of its matter-energy in the environment, lest the production process itself would be a system in perpetual motion, violating the second law of Thermodynamics. Such high-entropy matter depletes finite stocks of ecosystem services provided by the ecosphere, hence are incompatible with the long-term growth in the material scale of the economic process. Moreover, the complex natural systems governing such stocks respond to depletion by possibly sudden environmental transitions, thus hindering markets' very ability to adapt to the new equilibrium conditions. Consequently, uncertainty of critical resilience thresholds constrains material economic growth.

physics.soc-ph↗

arXiv2005

A constructive algorithm for the Cartan decomposition of SU(2^N)

We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G of SU(2^N) in terms of its `Cartan' and `non-Cartan' components. This effectively factors G in terms of group elements that belong in SU(2^n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.

quant-ph↗