arXiv subjects

Satyajit Guin

Publications and source records attributed to Satyajit Guin.

25 records · Page 2Linked to original sources

arXiv2017

Kähler structure on certain $C^*$-dynamical systems and the noncommutative even dimensional tori

Let $G$ be an even dimensional, connected, abelian Lie group and $(\mathcal{A}^\infty,G,α,τ)$ be a $C^*$-dynamical system equipped with a faithful $G$-invariant trace $τ$. We show that whenever it determines a $\varTheta$-summable even spectral triple, $\mathcal{A}^\infty$ inherits a Kähler structure. Moreover, there are at least $\prod_{j=1,\,j\,odd}^{\,dim(G)}(dim(G)-j)$ different Kähler structures. In particular, whenever $\mathbb{T}^{2k}$ acts ergodically on the algebra, it inherits a Kähler strcture. This gives a class of examples of noncommutative Kähler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori, like their classical counterpart the complex tori, are noncommutative Kähler manifolds. We explicitly compute the space of complex differential forms for the noncommutative even dimensional tori and show that the category of holomorphic vector bundle over it is an abelian category. We also explain how the earlier set-up of Polishchuk-Schwarz for the holomorphic structure on noncommutative two-torus follows as a special case of our general framework.

math.OA↗

arXiv2017

On the supersymmetric $N=1$ and $N=(1,1)$ spectral data : A multiplicativity property

We show that there are six different choice of tensor product of supersymmetric N=(1,1) spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We also show that the procedure of extending a supersymmetric N=1 spectral data to N=(1,1) spectral data respects only one tensor product among these. We refer this as the multiplicativity property of the extension procedure. Therefore, if we demand that the extension procedure is multiplicative then there is a unique choice of tensor product of N=(1,1) spectral data.

math.OA↗

arXiv2017

Comparison between two differential graded algebras in Noncommutative Geometry

Starting with a spectral triple one can associate two canonical differential graded algebras (dga) defined by Connes and Fröhlich et al. For the classical spectral triples associated with compact Riemannian spin manifolds both these dgas coincide with the de-Rham dga. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two dgas and observe that in a very precise sense Connes' dga is more informative than that of Fröhlich et al.

math.OA↗

arXiv2014

Connes' calculus for The Quantum double suspension

Given a spectral triple $(\mathcal{A},\mathcal{H},D)\,$ Connes associated a canonical differential graded algebra $\,Ω_D^\bullet(\mathcal{A})$. However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes' calculus for its quantum double suspension. This allows one to compute $\,Ω_D^\bullet$ for spectral triples obtained by iterated quatum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes' calculus for a large family of spectral triples.

math.QA↗

arXiv2014

Multiplicativity of Connes' calculus

We consider the quadruples $\,(\mathcal{A},\mathbb{V},D,γ)$ where $\mathcal{A}$ is a unital, associative $\mathbb{K}\,$-algebra represented on the $\mathbb{K}\,$-vector space $\mathbb{V}$, $D\in \mathcal{E}nd(\mathbb{V})$, $γ\in\mathcal{E}nd(\mathbb{V})$ is a $\mathbb{Z}_2$-grading operator which commutes with $\mathcal{A}$ and anticommutes with $D$. We prove that the collection of such quadruples, denoted by $\,\widetilde{\mathcal{S}pec}\,$, is a monoidal category. We consider the monoidal subcategory $\,\widetilde{\mathcal{S}pec_{sub}}\,$ of objects of $\,\widetilde{\mathcal{S}pec}\,$ for which $γ\inπ(\mathcal{A})$. We show that there is a covariant functor $\,\mathcal{G}:\widetilde{\mathcal{S}pec}\longrightarrow\widetilde{\mathcal{S}pec_{sub}}\,$. Let $\,Ω_D^\bullet\,$ be the differential graded algebra defined by Connes ([Con2]) and $DGA$ denotes the category of differential graded algebras over the field $\mathbb{K}\,$. We show that $\mathcal{F}:\widetilde{\mathcal{S}pec_{sub}}\longrightarrow DGA\,$, given by $(\mathcal{A},\mathbb{V},D,γ)\longmapstoΩ_D^\bullet(\mathcal{A})$, is a monoidal functor. To show that $\,\mathcal{F}\circ\mathcal{G}\,$ is not trivial we explicitly compute it for the cases of compact manifold and the noncommutative torus along with the associated cohomologies.

math.QA↗

arXiv2013

Equivalence of Two Approaches to Yang-Mills on Non-commutative Torus

There are two notions of Yang-Mills action functional in noncommutative geometry. We show that for noncommutative n-torus both these notions agree. We also prove a structure theorem on the Hermitian structure of a finitely generated projective modules over spectrally invariant subalgebras of $C^*$-algebras.

math.OA↗

arXiv2013

Yang- Mills on Quantum Heisenberg Manifolds

In the noncommutative geometry program of Connes there are two variations of the concept of Yang-Mills action functional. We show that for the quantum Heisenberg manifolds they agree.

math.OA↗