arXiv subjects

Ahmad Reza Haj Saeedi Sadegh

Publications and source records attributed to Ahmad Reza Haj Saeedi Sadegh.

11 recordsLinked to original sources

arXiv2026

A Fueter Operator for 3/2-spinors

We show the non-compactness of the moduli space of solutions with a uniform bound on the curvatures of the monopole equations for $3/2$-spinors on a closed 3-manifold is equivalent to the existence of `3/2-Fueter sections' that are solutions of a non-linear differential equation. These are sections of a fiber bundle whose fiber is a special 4-dimensional submanifold of the hyperkähler manifold of center-framed charged one $SU(3)$-instantons on $\mathbb{R}^4$. This fiber bundle does not inherit a hyperkähler structure. Our results provide a version of the Haydys correspondence when working with $3/2$-spinors. Furthermore, although the $3/2$ monopole equations are nonlinear elliptic, the corresponding Fueter operator describing points within the compactification of the $3/2$-monopole equations lacks nonlinear ellipticity. We conclude with a generalization of the notion of Rarita-Schwinger-type operator which provides a framework for understanding the linearization of the $3/2$-Fueter operator and discuss non-ellipticity from this perspective.

math.DG↗

arXiv2026

$b^k$-Symplectic Manifolds and $[Q,R]=0$

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.

math.SG↗

arXiv2025

On Deformation Spaces, Tangent Groupoids and Generalized Filtrations of Banach and Fredholm Manifolds

We extend the deformation to the normal cone and tangent groupoid constructions from finite-dimensional manifolds to infinite-dimensional Banach and Fredholm manifolds. Next, we generalize the concept of Fredholm filtrations to get a more flexible and functorial theory. In particular, we show that if $M$ is a Banach (or Fredholm) manifold with generalized filtration ${\mathcal F} = \{M_n\}_1^\infty$ by finite-dimensional submanifolds, then there are induced generalized filtrations $T{\mathcal F} = \{TM_n\}_1^\infty$ of the tangent bundle $TM$ and $\mathbb{T}{\mathcal F} = \{\mathbb{T}{M_n}\}_1^\infty$ of the tangent groupoid $\mathbb{T}{M}$, which is not possible in the classical theory.

math.FA↗

arXiv2025

A gluing formula for the $Z_2$-valued index of odd symmetric operators

We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index.

math.DG↗

arXiv2025

The Z_2-valued spectral flow of a symmetric family of Toeplitz operators

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary $\mathbb{Z}_2$-valued spectral flow of such families. We prove an analog of the Atiyah-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of $A(t)$ is equal to the secondary $\mathbb{Z}_2$-valued index of the suspension operator $\frac{d}{dt}+A(t)$. Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel's index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII.

math.DG↗

arXiv2024

On the Z_2-valued index of elliptic odd symmetric operators on non-compact manifolds

We investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov-Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel's index theorem for Toeplitz operators.

math.DG↗

arXiv2024

A fixed-point formula for Dirac operators on Lie groupoids

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.

math.DG↗

arXiv2023

The three-dimensional Seiberg-Witten equations for 3/2-spinors: a compactness theorem

The Rarita-Schwinger-Seiberg-Witten (RS-SW) equations are defined similarly to the classical Seiberg-Witten equations, where a geometric non-Dirac-type operator replaces the Dirac operator called the Rarita-Schwinger operator. In dimension four, the RS-SW equation was first considered by the second-named author. The variational approach will also give us a three-dimensional version of the equations. The RS-SW equations share some features with the multiple-spinor Seiberg-Witten equations, where the moduli space of solutions could be non-compact. In this note, we prove a compactness theorem regarding the moduli space of solutions of the RS-SW equations defined on 3-manifolds.

math.GT↗

arXiv2022

Deformation Spaces, Rescaled Bundles and the Kirillov Character Formula

In this paper, we construct a smooth vector bundle over the deformation to the normal cone $\text{DNC}(V,M)$ through a rescaling of a vector bundle $E\to V$, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham-Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen.

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↗

arXiv2018

Euler-like vector fields, deformation spaces and manifolds with filtered structure

The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach this bijection through the deformation to the normal cone that is associated to the embedding of M into V. The second purpose is to study generalizations to smooth manifolds equipped with Lie filtrations. Following in the footsteps of several others, we shall define a deformation to the normal cone that is appropriate to this context, and relate it to Euler-like vector fields and tubular neighborhood embeddings.

math.DG↗