Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
The search of unconventional magnetic and nonmagnetic states is a major topic in the study of frustrated magnetism. Canonical examples of those states include various spin liquids and spin nematics. However, discerning their existence and the correct characterization is usually challenging. Here we introduce a machine-…
In this paper we study Spin(7)-instantons on asymptotically conical Spin(7)-orbifolds (and manifolds) obtained by filling in certain squashed 3-Sasakian 7-manifolds. We construct a 1-parameter family of explicit Spin(7)-instantons. Taking the parameter to infinity, the family (a) bubbles o…
The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.
problem Creating Spin(7) metrics with specific geometric properties.
method Continuous 1-parameter families of non-compact Spin(7) metrics with chiralities, focusing on Aloff--Wallach spaces.
result Construction of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions.
New Spin(7)-instantons constructed on Joyce's manifold.
problem Constructing Spin(7)-instantons on Joyce's compact manifold. method Gluing non-flat connections on local model spaces to a flat connection on the Spin(7)-orbifold. result More than 20,000 new four-parameter families of Spin(7)-instantons. We analyze operational risk in terms of a spin glass model. Several regimes are investigated, as a functions of the parameters that characterize the dynamics. The system is found to be robust against variations of these parameters. We unveil the presence of limit cycles and scrutinize the features of the asymptotic sta…
The paper characterizes the geometry and topology of spin random fields.
problem Understanding the expected geometry and topology of spin random fields.
method Investigating the asymptotic behavior of geometric and topological functionals for spin random fields under scaling assumptions.
result Explicit results for monochromatic fields, showing non-universal asymptotic behavior and new generalized models.
MPF method improves parameter estimation in probabilistic models.
problem Difficulty in fitting probabilistic models due to intractable partition function.
method Minimum Probability Flow (MPF) method for parameter estimation.
result MPF outperforms existing techniques in convergence time and accuracy.
Covariant formulation of Barbero-Immirzi connections for spin manifolds.
problem Defining Barbero-Immirzi connections in a covariant way.
method Introducing a covariant formulation and showing uniqueness of Barbero-Immirzi connections.
result Real Barbero-Immirzi parameter is unique to 4D spacetime.
Researchers classify and construct various instantons and connections on a specific 8-manifold.
problem Classifying and constructing instantons and connections on a specific Calabi manifold.
method Invariant Hermitian Yang-Mills connections, Sp(2)-instantons, Spin(7)-instantons, and gauge group analysis.
result Classification of SU(3)-invariant primitive Hermitian Yang-Mills connections and Sp(2)-instantons over T∗CP2. New transitions found in Spin(7) holonomy metrics related to a dynamical system.
problem Understanding the geometry of Spin(7) holonomy metrics with Aloff--Wallach spaces.
method Relating Spin(7)-equations to a 3-dimensional dynamical system.
result Discovered new transitions with Spin(7) holonomy metrics.
The paper finds new Spin(7) metrics with specific orbits.
problem Existence of Spin(7) metrics with specified orbits. method Construction of three continuous families of non-compact Spin(7) metrics. result Existence of asymptotically conical and locally conical metrics.
The intrinsic geometry of the Kerr ergosurface on constant Boyer-Lindquist (BL), Kerr, and Doran time slices is characterized. Unlike the BL slice, which had been previously studied, the other slices (i) do not have conical singularities at the poles (except the Doran slice in the extremal limit), (ii) have finite pola…
Study Spin(7) instantons and HYM connections on Stenzel metric.
problem Relationship between Spin(7) instantons and HYM connections.
method Reduced to ODEs, analyzed invariant solutions, constructed moduli space.
result Nonexistence of global invariant Spin(7) instantons, one parameter family of irreducible instantons.
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
Study on deformations of LC Spin(7) instantons simplifies the problem.
problem Deformation theory of instantons on locally conformal Spin(7) manifolds.
method Reformulated linearized deformation equations using a t-parameter family of Dirac operators, demonstrating cancellation of torsion terms.
result The deformation space H^1 is governed by Levi-Civita geometry, reducing the problem to a torsion-free setting.
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter q is a root of unity of order 2r where r>1 is odd or congruent to 2 modulo 4. In this paper we consider…
Researchers compute quasi-local mass of Kerr black hole horizon.
problem Computing the quasi-local mass of Kerr black hole horizons.
method Isometric embedding of Kerr black hole horizon in hyperbolic space.
result Quasi-local mass varies with Kerr black hole spin parameter.
We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk whi…
Eta invariant computed for circle bundles over Fano manifolds.
problem Computing eta invariant for circle bundles over Fano manifolds.
method Using spin-c Dirac operator and adiabatic limit.
result Eta invariant computed for arbitrary adiabatic parameters.
We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A_8, is complete and non-singular on R^8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S^4, and …
A spin model is used for simulations of financial markets. To determine return volatility in the spin financial market we use the GARCH model often used for volatility estimation in empirical finance. We apply the Bayesian inference performed by the Markov Chain Monte Carlo method to the parameter estimation of the GAR…
Non-compact G_2 holonomy metrics that arise from a T^2 bundle over a hyper-Kahler space are discussed. These are one parameter deformations of the metrics studied by Gibbons, Lu, Pope and Stelle in hep-th/0108191. Seven-dimensional spaces with G_2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational i…
We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ±∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collap…
New pseudo-Hermitian models from non-semisimple TQFTs.
problem Constructing exactly solvable pseudo-Hermitian spin Hamiltonians.
method Identifying ground states on surfaces using non-semisimple TQFTs.
result Ground states depend only on spatial topology and can be assigned by non-semisimple TQFTs.
The inverse statistical problem of finding direct interactions in complex networks is difficult. In the natural sciences, well-controlled perturbation experiments are widely used to probe the structure of complex networks. However, our understanding of how and why perturbations aid inference remains heuristic, and we l…
The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) …
Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex 4×4 symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequen…
The paper explores spin^h structures and their obstructions.
problem Understanding which manifolds are spin^h.
method Analyzing the fifth Stiefel-Whitney class and constructing generalised spin structures.
result Every compact orientable manifold of dimension 7 or lower is spin^h, and there are higher-dimensional manifolds that are not.
Study mSpin(7)-dDT connections on manifolds with mSpin(7)-structures.
problem Understanding moduli spaces of mSpin(7)-dDT connections. method Introduced and studied mSpin(7)-dDT connections using fully nonlinear PDEs. result Moduli space MmSpin(7)′ has finite expected dimension and smoothness under certain conditions. This is an introduction to the construction of higher-dimensional knots by spinning methods. Simple spinning of classical knots was introduced by E. Artin in 1926, and several generalizations have followed. These include twist spinning, superspinning or p-spinning, frame spinning, roll spinning, and deform spinning. We…
New mSpinh-manifolds studied for their properties.
problem Understanding new manifold classifications.
method Exploring mSpinh-manifolds as a new category. result Highlights of mSpinh-manifolds discussed. Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
The paper characterizes new invariant spinr spinors on projective spaces.
problem Characterizing new invariant spinr spinors on projective spaces. method Adapting spin representation via exterior forms to the generalised spinr context. result Complete description of the space of invariant spinr spinors for CPn, HPn, and OP2. Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general tha…
The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.
problem Analyzing boundedness and decay of solutions to Teukolsky equations on Kerr backgrounds.
method Frequency space analysis of transformed Teukolsky equations on Kerr backgrounds.
result Fixed frequency solutions remain bounded and decay in time for subextremal Kerr backgrounds.
Purpose: Arterial Spin Labeling (ASL) is a quantitative, non-invasive alternative to perfusion imaging with contrast agents. Fixing values of certain model parameters in traditional ASL, which actually vary from region to region, may introduce bias in perfusion estimates. Adopting Magnetic Resonance Fingerprinting (MRF…
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ=0, and the algebraic o…
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Rigidity of elliptic genera proven for non-spin manifolds with S1-action.
problem Rigidity of elliptic genera for non-spin manifolds with S1-action. method Analysis of universal covering spin condition and π2(M) for rigidity. result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.
First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of CP2 and cones over 3-Sasakian 7-manifolds. result First non-trivial examples of deformed Spin(7)-instantons.
Study shows certain spin manifolds can't meet DEC condition.
problem Non-existence of spin fill-ins meeting DEC condition.
method Analyzes spin Riemannian manifolds and generalized mean curvature functions.
result Closed spin manifolds cannot satisfy DEC if curvature is large.
Spin TFTs created by gauging line defects in 3D.
problem Creating spin TFTs from oriented TFTs with framed line defects.
method Constructing a spin TFT from an oriented TFT with framed line defects and a commutative Frobenius algebra.
result Spin TFTs extend earlier classifications and reproduce abelian spin Chern-Simons theories.
The paper explores connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
problem Exploring connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
method Starting from collections of 'Kähler 2-forms', the paper constructs canonical 4-forms and calibrated 4-planes in dimensions 8 and 16.
result Explicit formulas for canonical 4-forms ΦSpin(8) and ΦSpin(7)U(1) are derived, and their calibrated 4-planes are characterized. Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…
We compute the spectral action of SU(2)/Γ with the trivial spin structure and the round metric and find it in each case to be equal to ∣Γ∣1(Λ3f^(2)(0)−1/4Λf^(0))+O(Λ−∞). We do this by explicitly computing the spectrum of the Dirac operator for SU(2)/Γ equipped with the trivial …