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Register a new userHigher rho invariant and delocalized eta invariant at infinity
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In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.
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Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in On the index of a fibered manifold. We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in Positive scalar curvature, higher rho invariants and localization algebras and Zeidler in Positive scalar curvature and product formulas for secondary index invariants.
In this paper, we adapt part of Weinberger, Xie and Yus breakthrough work, to define additive higher rho invariant for topological structure group by differential geometric version of signature operators, or in other words, unbounded Hilbert-Poincar{e} complexes.
In this paper, we define the relative higher $rho$ invariant for orientation preserving homotopy equivalence between PL manifolds with boundary in $K$-theory of the relative obstruction algebra, i.e. the relative analytic structure group. We also show that the map induced by the relative higher $rho$ invariant is a group homomorphism from the relative topological structure group to the relative analytic structure group. For this purpose, we generalize Weinberger, Xie and Yus definition of the topological structure group in their article Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariants and nonrigidity of topological manifolds. Communications on Pure and Applied Mathematics, to appear. to make the additive structure of the relative topological structure group transparent.
The collinear factorization framework allows to describe the exclusive photoproduction of a $gamma,rho$ pair in the generalized Bjorken regime in terms of a perturbatively calculable coefficient function and universal generalized parton distributions. The kinematics are defined by a large invariant mass of the $gamma rho$ pair and a small transverse momentum of the final nucleon. We calculate the scattering amplitude at leading order in $alpha_s$ and the differential cross sections for the process where the $rho-$meson is either longitudinally or transversely polarized, in the kinematics of the near future Jlab experiments. Our estimate of the cross section demonstrates that this process is measurable at JLab 12-GeV.
We develop a general formalism of duality rotations for bosonic conformal spin-$s$ gauge fields, with $sgeq 2$, in a conformally flat four-dimensional spacetime. In the $s=1$ case this formalism is equivalent to the theory of $mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik. For each integer spin $sgeq 2$ we demonstrate the existence of families of conformal $mathsf{U}(1)$ duality-invariant models, including a generalisation of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then extended to the $mathcal{N}=1$ and $mathcal{N}=2$ supersymmetric cases. We also sketch a formalism of duality rotations for conformal gauge fields of Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.
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