Related papers: Scalar Curvature for the Noncommutative Two Torus

We investigate field theories on the non-commutative torus upon varying theta, the parameter of non-commutativity. We argue that one should think of Morita equivalence as a symmetry of algebras describing the same space rather than of…

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the…

Motivated by the torus stability problem, in this work we study K\"ahler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of K\"ahler metrics with…

It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra B_{\theta}…

If one is willing to give up the cherished hypothesis of spatial isotropy, many interesting cosmological models can be developed beyond the simple anisotropically expanding scenarios. One interesting possibility is presented by shear-free…

In this paper, we introduce (local and) global non-abelian zeta functions for general curves. As an example, we compute the so-called rank two zeta functions for genus two curves by studying non-abelian Brill-Noether loci and their…

This is a report on a joint work [12] with D. Essouabri, C. Levy and A. Sitarz. The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine…

We derive an explicit formula for the scalar curvature over a two-torus with a Dirac operator conformally rescaled by a globally diagonalizable matrix. We show that the Gauss-Bonnet theorem holds and extend the result to all Riemann…

Using the Levi-Civita connection on the noncommutative differential one-forms of a spectral triple $(\B,\H,\D)$, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac…

The structure function of a scalar $\theta({\bf x},t)$, passively advected in a two-dimensional turbulent flow ${\bf u}({\bf x},t)$, is discussed by means of the fractal dimension $\delta^{(1)}_g$ of the passive scalar graph. A relation…

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a…

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors…

We analyze the question of $U_{\star} (1)$ gauge invariance in a flat non-commutative space where the parameter of non-commutativity, $\theta^{\mu\nu} (x)$, is a local function satisfying Jacobi identity (and thereby leading to an…

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set…

A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or…

In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with…

We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and…

Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question…

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$, we construct the basis of Hilbert space ${\ca$H}_n$ in terms of $\theta$ functions of the positions $z_i$ of $n$ solitons. The wrapping around…

Let $M$ be a complex torus, $L_{\hat\mu}\to M$ be positive line bundles parametrized by $\hat \mu\in {\rm Pic}^0(M)$, and $E\to {\rm Pic}^0(M)$ be a vector bundle with $E|_{\hat\mu}\cong H^0(M, L_{\hat \mu})$. We endow the total family…