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Varghese Mathai

Publications and source records attributed to Varghese Mathai.

At least 37 records · Page 2Linked to original sources
arXiv2021

T-duality and the exotic chiral de Rham complex

Let $Z$ be a principal circle bundle over a base manifold $M$ equipped with an integral closed $3$-form $H$ called the flux. Let $\widehat{Z}$ be the T-dual circle bundle over $M$ with flux $\widehat{H}$. Han and Mathai recently constructed the $\mathbb{Z}_2$-graded space of exotic differential forms $\mathcal{A}^{\bar{k}}(\widehat{Z})$. It has an additional $\mathbb{Z}$-grading such that the degree zero component coincides with the space of invariant twisted differential forms $Ω^{\bar{k}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$, and it admits a differential that extends the twisted differential $d_{\widehat{H}} = d + \widehat{H}$. The T-duality isomorphism $Ω^{\bar{k}}(Z,H)^{\mathbb{T}} \rightarrow Ω^{\overline{k+1}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$ of Bouwknegt, Evslin and Mathai extends to an isomorphism $Ω^{\bar{k}}(Z,H) \rightarrow \mathcal{A}^{\overline{k+1}}(\widehat{Z})$. In this paper, we introduce the exotic chiral de Rham complex $\mathcal{A}^{\text{ch},\widehat{H},\bar{k}}(\widehat{Z})$ which contains $\mathcal{A}^{\bar{k}}(\widehat{Z})$ as the weight zero subcomplex. We give an isomorphism $Ω^{\text{ch},H,\bar{k}}(Z) \rightarrow \mathcal{A}^{\text{ch},\widehat{H},\overline{k+1}}(\widehat{Z})$ where $Ω^{\text{ch},H,\bar{k}}(Z)$ denotes the twisted chiral de Rham complex of $Z$, which chiralizes the above T-duality map.

math.DG↗
arXiv2021

On the Chern character in Higher Twisted K-theory and spherical T-duality

In this paper, we construct for higher twists that arise from cohomotopy classes, the Chern character in higher twisted K-theory, that maps into higher twisted cohomology. We show that it gives rise to an isomorphism between higher twisted K-theory and higher twisted cohomology over the reals. Finally we compute spherical T-duality in higher twisted K-theory and higher twisted cohomology in very general cases.

math.DG↗
arXiv2021

Fractional Quantum Numbers, Complex Orbifolds and Noncommutative Geometry

This paper studies the conductance on the universal homology covering space $Z$ of 2D orbifolds in a strong magnetic field, thereby removing the integrality constraint on the magnetic field in earlier works in the literature. We consider a natural Landau Hamiltonian on $Z$ and show that its low-lying spectrum consists of a finite number of isolated points. We calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field $B$ is large, and obtain fractional quantum numbers as the conductance.

math-ph↗
arXiv2020

An equivariant Poincaré duality for proper cocompact actions by matrix groups

Let $G$ be a linear Lie group acting properly and isometrically on a $G$-spin$^c$ manifold $M$ with compact quotient. We show that Poincaré duality holds between $G$-equivariant $K$-theory of $M$, defined using finite-dimensional $G$-vector bundles, and $G$-equivariant $K$-homology of $M$, defined through the geometric model of Baum and Douglas.

math.KT↗
arXiv2020

Water entry of spheres into a rotating liquid

The transient cavity dynamics during water entry of a heavy, non-rotating sphere impacting a rotating pool of liquid is studied experimentally, numerically, and theoretically. We show that the pool rotation advances the transition of the cavity type - from deep seal to surface seal - marked by a reduction in the transitional Froude number. The role of the dimensionless rotational number $\mathcal{S} \equiv ωR_0/U_0$ on the transient cavity dynamics is unveiled, where $R_0$ is the sphere radius, $ω$ the angular speed of the liquid, and $U_0$ the impact velocity. The rotating background liquid has two discernible effects on the cavity evolution. Firstly, an increase in the underwater pressure field due to centripetal effects, and secondly a reduction in the pressure of airflow in the cavity neck near the water surface. The non-dimensional pinch-off time of the deep seal shows a robust 1/2 power-law dependence on the Froude number, but with a reducing prefactor for increasing $ω$. Our findings reveal that the effects of a rotating background liquid on the water entry can be traced back to the subtle differences in the initial stage splash and the near-surface cavity dynamics.

physics.flu-dyn↗
arXiv2020

Airflows inside passenger cars and implications for airborne disease transmission

Transmission of highly infectious respiratory diseases, including SARS-CoV-2 are facilitated by the transport of tiny droplets and aerosols (harboring viruses, bacteria, etc.) that are breathed out by individuals and can remain suspended in air for extended periods of time in confined environments. A passenger car cabin represents one such situation in which there exists an elevated risk of pathogen transmission. Here we present results from numerical simulations of the potential routes of airborne transmission within a model car geometry, for a variety of ventilation configurations representing different combinations of open and closed windows. We estimate relative concentrations and residence times of a non-interacting, passive scalar -- a proxy for infectious pathogenic particles -- that are advected and diffused by the turbulent airflows inside the cabin. Our findings reveal that creating an airflow pattern that travels across the cabin, entering and existing farthest from the occupants can potentially reduce the transmission.

physics.soc-ph↗
arXiv2020

Exotic Twisted Equivariant K-Theory

In this paper we introduce exotic twisted $\mathbb T$-equivariant K-theory of loop space $LZ$ depending on the (typically non-flat) holonomy line bundle ${\mathcal L}^B$ on $LZ$ induced from a gerbe with connection $B$ on $Z$. We also define exotic twisted $\mathbb T$-equivariant Chern character that maps the exotic twisted $\mathbb T$-equivariant K-theory of $LZ$ into the exotic twisted $\mathbb T$-equivariant cohomology as defined in an earlier paper of ours, and which localises to twisted cohomology of $Z$.

math.KT↗
arXiv2020

Exotic Courant algebroids and T-duality

In this paper, we extend the T-duality isomorphism by Gualtieri and Cavalcanti, from invariant exact Courant algebroids, to exotic exact Courant algebroids such that the momentum and winding numbers are exchanged, filling in a gap in the literature.

hep-th↗
arXiv2020

Bubbly and Buoyant Particle-Laden Turbulent Flows

Fluid turbulence is commonly associated with stronger drag, greater heat transfer, and more efficient mixing than in laminar flows. In many natural and industrial settings, turbulent liquid flows contain suspensions of dispersed bubbles and light particles. Recently, much attention has been devoted to understanding the behavior and underlying physics of such flows by use of both experiments and high-resolution direct numerical simulations. This review summarizes our present understanding of various phenomenological aspects of bubbly and buoyant particle-laden turbulent flows. We begin by discussing different dynamical regimes, including those of crossing trajectories and wake-induced oscillations of rising particles, and regimes in which bubbles and particles preferentially accumulate near walls or within vortical structures. We then address how certain paradigmatic turbulent flows, such as homogeneous isotropic turbulence, channel flow, Taylor-Couette turbulence, and thermally driven turbulence, are modified by the presence of these dispersed bubbles and buoyant particles. We end with a list of summary points and future research questions.

physics.flu-dyn↗
arXiv2020

Coarse geometry and Callias quantisation

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group $C^*$-algebra instead of its reduced counterpart, we can apply the trace given by integration over $G$ to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. As a very special case, this allows one to refine numerical obstructions to positive scalar curvature on a noncompact $\operatorname{Spin}$ manifold $X$ defined via Callias index theory, to obstructions in the $K$-theory of the maximal $C^*$-algebra of the fundamental group $π_1(X)$. As a motivating application in another direction, we prove a version of Guillemin and Sternberg's quantisation commutes with reduction principle for equivariant indices of $\operatorname{Spin}^c$ Callias-type operators.

math.DG↗
arXiv2020

Nonlinear modeling and characterization of ultrasoft silicone elastomers

We introduce a strain-energy based nonlinear hyper-elastic formulation to model the material properties of ultrasoft dielectric elastomers over a wide range of elastic properties, prestretch, and thicknesses. We build on the uniaxial Gent formulation, and under the conditions of equi-biaxial strain, derive an expression for the bulge deformation versus pressure. A circular bulge test methodology is developed to experimentally measure the mechanical response of the silicone membranes. The Gent model captures both neo-Hookean and strain-stiffening behaviors, and gives predictions which are in agreement with experimental measurements. Membranes with different thinner fractions are characterized over nearly one order of magnitude variation in shear modulus. Stiffer membranes are observed to harden at lower stretch ratios due to the increased fraction of polymer chains in them. The present approach offers a simple and cost-effective procedure for characterizing soft membranes under commonly encountered biaxial deformation conditions.

cond-mat.mtrl-sci↗
arXiv2020

T-Duality, Jacobi Forms and Witten Gerbe Modules

In this paper, we extend the T-duality Hori maps in [arXiv:hep-th/0306062], inducing isomorphisms of twisted cohomologies on T-dual circle bundles, to graded Hori maps and show that they induce isomorphisms of two-variable series of twisted cohomologies on the T-dual circle bundles, preserving Jacobi form properties. The composition of the graded Hori map with its dual equals the Euler operator. We also construct Witten gerbe modules arising from gerbe modules and show that their graded twisted Chern characters are Jacobi forms under an anomaly vanishing condition on gerbe modules, thereby giving interesting examples.

math.DG↗
arXiv2020

Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

math.DG↗
arXiv2019

Loop group actions on Cuntz algebras and geometric higher twists

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, and we show that this action can be used to explicitly construct ${\mathcal O}_\infty \otimes \mathcal K$-bundles (where $ \mathcal K$ is the algebra of compact operators) over spaces, which are geometric twists in higher twisted $K$-theory.

math.OA↗
arXiv2019

Projective elliptic genera and elliptic pseudodifferential genera

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in Mathai-Melrose-Singer (2006) without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of putative rotation-equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.

math.DG↗
arXiv2019

Statistics of rigid fibers in strongly sheared turbulence

Practically all flows are turbulent in nature and contain some kind of irregularly-shaped particles, e.g. dirt, pollen, or life forms such as bacteria or insects. The effect of the particles on such flows and vice-versa are highly non-trivial and are not completely understood, particularly when the particles are finite-sized. Here we report an experimental study of millimetric fibers in a strongly sheared turbulent flow. We find that the fibers show a preferred orientation of $-0.38π\pm 0.05π$ ($-68 \pm 9^\circ$) with respect to the mean flow direction in high-Reynolds number Taylor-Couette turbulence, for all studied Reynolds numbers, fiber concentrations, and locations. Despite the finite-size of the anisotropic particles, we can explain the preferential alignment by using Jefferey's equation, which provides evidence of the benefit of a simplified point-particle approach. Furthermore, the fiber angular velocity is strongly intermittent, again indicative of point-particle-like behavior in turbulence. Thus large anisotropic particles still can retain signatures of the local flow despite classical spatial and temporal filtering effects.

physics.flu-dyn↗