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The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincar\'{e} lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract…

We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and…

We analyze the embedding dimension of a normal weighted homogeneous surface singularity, and more generally, the Poincar\'e series of the minimal set of generators of the graded algebra of regular functions, provided that the link of the…

A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced…

We study a model comprising $N$ flavors of K\"ahler Dirac fermion propagating on a triangulated two dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions.…

We examine the phenomenological implications at colliders for the existence of higher-derivative gravity terms as extensions to the Randall-Sundrum model. Such terms are expected to arise on rather general grounds, e.g., from string theory.…

We give a bounding of degree of quasi-smooth hypersurfaces which are invariant by a one dimensional holomorphic foliation of a given degree on a weighted projective space.

We show that General Relativity can be formulated as a constrained topological theory for flat 2-connections associated to the Poincar\'e 2-group. Matter can be consistently coupled to gravity in this formulation. We also show that the edge…

We investigate the Poincar\'e approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)$_k$ WZW models provide unitary examples for…

Let N be the moduli space of stable rank 2 vector bundles on a smooth projective curve of genus g>1 with fixed odd determinant. With Sebastian Torres, we previously found a semi-orthogonal decomposition of the bounded derived category of N…

We describe the effect of the gravitational $\Theta$-parameter on the behavior of the stretched horizon of a black hole in $(3+1)$-dimensions. The gravitational $\Theta$-term is a total derivative, however, it affects the transport…

The holographic duality can be extended to include quantum theories with broken coordinate invariance leading to the appearance of the gravitational anomalies. On the gravity side one adds the gravitational Chern-Simons term to the bulk…

We prove a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L functions coincide with either those coming from a newform or a product of L functions arising from…

We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at…

Four-dimensional Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the same type as found in the topological theories for Yang-Mills fields. A peculiar feature of the gravitational theory,…

A manifestly diffeomorphism invariant extension of Einstein gravity is constructed, which includes singular metrics, and whose ADM formulation is Ashtekar's gravity. The latter is shown to be locally equivalent to the covariant theory. It…

We give a local expression for the {\it scalar curvature} of the noncommutative two torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by…

We determine complex saddles of three-dimensional gravity with a positive cosmological constant by applying the recently proposed holography. It is sometimes useful to consider a complexified metric to study quantum gravity as in the case…

We recently introduced a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics…

We show that, in the first or second order orthonormal frame formalism, black hole entropy is the horizon Noether charge for a combination of diffeomorphism and local Lorentz symmetry involving the Lie derivative of the frame. The Noether…