arXiv subjects

Jesus Sanchez Jr

Publications and source records attributed to Jesus Sanchez Jr.

5 recordsLinked to original sources

arXiv2025

Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds

Given a contact sub-Riemannian manifold one obtains a non-integrable splitting of the tangent bundle into the directions along the contact distribution and the Reeb field. We generalize the construction of the Bismut superconnection to this non-integrable setting and show that although singularities appear within the superconnection, if one extracts the finite part then the resulting operator is hypoelliptic. We find that the hypoellipticity also holds in the setting of two-step subRiemannian manifolds and produce a modification for arbitrary subRiemannian manifolds which always gives a hypoelliptic operator. A discussion of the explicit form of the operator on principal $\bS^1$-bundles is provided. The index theory is worked out on contact manifolds and a matrix twisting of the Clifford relations produces operators with non-trivial Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.

math-ph↗

arXiv2025

Explicit Families of Spinor Representations

We provide a recipe for building explicit representations of the real Clifford algebras once an explicit family is given in dimensions $1$ through $4$. We further give an explicit construction of spin coordinate systems for a given real spinor module and use it to explicitly compute the parallel transport of spinor fields. We further highlight some novelties such as the relationship with the spectrum of the spinor Dirac operator and the Hodge de Rham operator when a parallel spinor field exists and a brief discussion of spinors along a hypersurface in $\bR^4$. Lastly, we extend our construction to arbitrary signature quadratic forms thus providing a complete and explicit family of spinor representations for all mixed signature Clifford algberas. We show that in all cases the spinor representations can be expressed as tensor products of multi-vectors over the fields $\bR$, $\bC$, and $\bH$.

math.DG↗

arXiv2023

The Geometry of Mehlers Kernel

We study the relationship between the Getzler calculus of a spin Riemannian manifold and the Riemannian geometry of the corresponding principal spin bundle. We then use the calculus of Gaussian-Grassmann integrals developed by Berline--Vergne to compute the Getzler symbol of the spinor heat flow.

math.DG↗

arXiv2022

On Perrot's index cocycles

We shall present a simplified version of a construction due to Denis Perrot that recovers the Todd class of the complexified tangent bundle from a JLO-type cyclic cocycle. The construction takes place within an algebraic framework, rather than the customary functional-analytic framework for the JLO theory. The series expansion for the exponential function is used in place of the heat kernel from the functional-analytic theory; the Dirac operator chosen is far from elliptic; and a remarkable new trace discovered by Perrot replaces the operator trace. In its full form Perrot's theory constitutes a wholly new approach to index theory. The account presented here covers most but not all of this approach.

math.DG↗

arXiv2021

Connes-Moscovici Residue Cocycle For Some Dirac-Type Operators

The residue cocycle associated to a suitable spectral triple is the key component of the Connes-Moscovici local index theorem in noncommutative geometry. We review the relationship between the residue cocycle and heat kernel asymptotics. We use a modified version of the Getzler calculus to compute the cocycle for a class of Dirac-type operators introduced by Bismut, obtained by deforming a Dirac operator by a closed 3-form B. We also compute the cocycle in low-dimensions when the 3-form B is not closed.

math.OA↗