Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

Let $$X\rightarrow {{\mathbb {P}}}^1$$ X → P 1 be an elliptically fibered K3 surface, admitting a sequence $$\omega _{i}$$ ω i of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to $$\omega _i$$ ω i . Given the corresponding sequence $$\Xi _i$$ Ξ i of Hermitian–Yang–Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence $$\Xi _i|_{E}$$ Ξ i | E converges to a flat connection $$A_0$$ A 0 . Furthermore, if the restriction $$V|_E$$ V | E is of the form $$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$ ⊕ j = 1 n O E ( q j - 0 ) for n distinct points $$q_j\in E$$ q j ∈ E , then these points uniquely determine $$A_0$$ A 0 .

Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

T 3-invariant heterotic Hull-Strominger solutions

We consider the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the T3-directions, the Hermitian Yang-Mills conditions can then be reinterpreted as a complex flat connection on ℝ3 satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the non-trivial α′-corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are all new local solutions to the Hull-Strominger system on T3× ℝ3. We also propose a method for computing the spectrum of certain non-abelian models, in close analogy with the Morse-Witten complex of the abelian models.

Three-Year Retrospective Comparative Study between Implants with Same Body-Design but Different Crest Module Configurations

Crest module can be defined as the portion of a two-piece implant designed to retain the prosthetic components and to allows the maintenance of the peri-implant tissues in the transition zone. Aim: To evaluate the three-year after loading clinical and radiographic data, collected from patients that received a prosthetic rehabilitation on conical connection implants with partial machined collar (PMC; CC Group) and same body-designed implants, with flat-to-flat connection and groovy neck design (FC Group). Materials and Methods: A retrospective chart review of previously collected data, including documents, radiographs, and pictures of patients who received at least one implant-supported restoration on NobelReplace CC PMC or NobelReplace Tapered Groovy implants was performed. Patients with at least three years of follow-up after final loading were considered for this study. Outcomes measures were implant and prosthesis failures, any biological or technical complications, marginal bone loss. Results: Eight-two patients (44 women, 38 men; average age 55.6) with 152 implants were selected and divided in two groups with 77 (CC group) and 75 (FC group), respectively. Three years after final loading, one implant in CC group failed (98.7% survival rate), while no implants failed in FC group (100% survival rate). One restoration failed in CC group (98.7% survival rate) with no restoration failing in the FC one (100% survival rate). Differences were not statistically significant (p = 1.0). Three years after final loading, mean marginal bone loss was 0.22 ± 0.06 mm (95% CI 0.2–0.24) in CC group and 0.62 ± 0.30 mm (95% CI 0.52–0.72) in FC group. The difference was statistically significant (0.40 ± 0.13 mm; 95% CI 0.3–0.5; p = 0.003). Conclusion: with the limitation of this retrospective comparative study, implants with conical connection and partial machined collar seem to achieve a trend of superior outcomes if compared with implants with flat connection and groovy collar design.

Self-dual formulation of gravity in topological M-theory

Inspired by the low wave-length limit of topological M-theory, which re-constructs the theory of 3 + 1D gravity in the self-dual variables’ formulation, and by the realization that in Loop Quantum Gravity (LQG) the holonomy of a flat connection can be non-trivial if and only if a non-trivial (space-like) line defect is localized inside the loop, we argue that non-trivial gravitational holonomies can be put in correspondence with space-like M-branes. This suggests the existence of a new duality, which we call [Formula: see text] duality, interconnecting topological M-theory with LQG. We spell some arguments to show that fundamental S-strings are serious candidates to be considered in order to instantiate this correspondence to classes of LQG states. In particular, we consider the case of the holonomy flowers in LQG, and show that for this type of states the action of the Hamiltonian constraint, from the M-theory side, corresponds to a linear combination of appearance and disappearance of a SNS1-strings. Consequently, these processes can be reinterpreted, respectively, as enucleations or decays into open or closed strings.

Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theories

Teleparallel geometry utilizes Weitzenböck connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi–Civita connection. In extended teleparallel theories, for instance in f ( T ) or scalar-torsion gravity, the connection must obey its antisymmetric field equations. Thus far, only a few analytic solutions were known. In this note, we solve the f ( T , ϕ ) gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformation, we present the solution also in Boyer–Lindquist coordinates, often used to study rotating solutions in general relativity. The result hints for the existence of another branch of rotating solutions besides the Kerr family in extended teleparallel gravities.

Cohomology of Flat Principal Bundles

We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.