Related papers: Tau functions for the Dirac operator on the cylind…
Spectral triples over noncommutative principal $\T^n$-bundles are studied, extending recent results about the noncommutative geometry of principal U(1)-bundles. We relate the noncommutative geometry of the total space of the bundle with the…
We study \tau-functions of the KP hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal and Wilson. A determinantal formula of Gekhtman and…
Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship…
In [17], we obtained the spectral Einstein functional associated with the Dirac operator for n-dimensional manifolds without boundary. In this paper, we give the proof of general Dabrowski-Sitarz-Zalecki type theorems for the spectral…
We establish global existence and derive sharp pointwise decay estimates of solutions to cubic Dirac and Dirac-Klein-Gordon systems on a curved background, close to the Minkowski spacetime. By squaring the Dirac operator, we reduce the…
We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor…
In this article, we define the spherical $\pi$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac…
We study some properties of tau-functions of an isomonodromic deformation leading to the fifth Painlev\'e equation. In particular, here is given an elementary proof of Miwa's formula for the logarithmic differential of a tau-function.
It is a well-known fact that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global…
We establish the factorization of Dirac operators on Riemannian submersions of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily…
In this short article, we non-perturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We also extend this formula to the case for the Dirac delta…
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the…
Employing the covariant language of two-spinors, we find what conditions a curved Lorentzian spacetime must satisfy for existence of a second order symmetry operator for the massive Dirac equation. The conditions are formulated as existence…
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 $\pm\infty$ or there are eigenvalues…
We study Dirac operators on resolutions of Riemannian orbifolds by developing a uniform elliptic theory. The key idea is to view orbifolds as conically fibred singular (CFS) spaces and resolve them by gluing asymptotically conical…
In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,\S)\to…
An embedding scheme is developed for the Dirac Hamiltonian H. Dividing space into regions I and II separated by surface S, an expression is derived for the expectation value of H which makes explicit reference to a trial function defined in…
We construct a universal spin$_c$ Dirac operator on $\mathbb{C}P^n$ built by projecting $su(n+1)$ left actions and prove its equivalence to the standard right action Dirac operator on $\mathbb{C}P^n$. The eigenvalue problem is solved and…
Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators on Hardy space of the unit disc. This…
This note proposes an improved estimate of the coefficient t(n) of the discriminant modular form using elementary method. It improves a well known estimate of the tau function t(n) by Deligne.