Related papers: Twisted Adiabatic Limit for Complex Structures

Given an $n$-dimensional compact complex Hermitian manifold $X$, a $C^\infty$ complex line bundle $L$ equipped with a connection $D$ whose $(0,\,1)$-component $D''$ squares to zero and a real-valued function $\eta$ on $X$, we prove that the…

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We…

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Fr\"olicher spectral sequence of any compact complex manifold supporting an SKT metric $\omega$ (i.e. such that…

We establish a canonical normal form for the iterates of a curved differential in curved differential algebras (CDA). This operator calculus clarifies the underlying algebraic structure of CDAs and bypasses the need for complex…

Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in…

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of…

Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil…

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of $K^{1/2}$…

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula…

Let $A$ be a bounded linear operator on a complex Banach space $X.$ For a given $\alpha \geq 0,$ we consider the class $\mathcal{D}_{A}^{\alpha }\left( \mathbb{R} \right) $ of all bounded linear operators $T$ on $X$ for which there exists a…

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically…

We study conformal symmetry breaking differential operators which map differential forms on $\mathbb{R}^n$ to differential forms on a codimension one subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to the conformal…

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces…

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian…

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application,…

Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering…