Dirac Operator Research Articles

The Moduli Space of Twisted Laplacians and Random Matrix Theory

Abstract Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil–Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this article we extend Rudnick’s approach to show convergence to the Gaussian Unitary Ensemble for twisted Laplacians that break time-reversal symmetry, and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high degree random covers of a fixed compact surface.

Pseudomodes of Schrödinger operators

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators.

Fermion integrals for finite spectral triples

Abstract Fermion functional integrals are calculated for the Dirac operator of a finite real spectral triple. Complex, real and chiral functional integrals are considered for each KO-dimension where they are non-trivial, and phase ambiguities in the definition are noted.

Persistent Mayer Dirac

Topological data analysis (TDA) has made significant progress in developing a new class of fundamental operators known as the Dirac operator, particularly in topological signals and molecular representations. However, the current approaches being used are based on the classical case of chain complexes. The present study establishes Mayer Dirac operators based on N-chain complexes. These operators interconnect an alternating sequence of Mayer Laplacian operators, providing a generalization of the classical result . Furthermore, the research presents an explicit formulation of the Laplacian for N-chain complexes induced by vertex sequences on a finite set. Weighted versions of Mayer Laplacian and Dirac operators are introduced to expand the scope and improve applicability, showcasing their effectiveness in capturing physical attributes in various practical scenarios. The study presents a generalized version for factorizing Laplacian operators as an operator's product and its 'adjoint'. Additionally, the proposed persistent Mayer Dirac operators and extensions are applied to biological and chemical domains, particularly in the analysis of molecular structures. The study also highlights the potential applications of persistent Mayer Dirac operators in data science.

Multivector Fields on Quaternionic Kähler Manifolds

In this paper we define a differential operator as a modified Dirac operator. Using the operator, we introduce a quaternionic k -vector field on a quaternionic Kähler manifold and show that any quaternionic k -vector field corresponds to a holomorphic k -vector field on the twistor space.

Finite spectral triples for the fuzzy torus

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

The Teodorescu and the Π-operator in octonionic analysis and some applications

In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the L2-space decomposition demonstrated in this paper for octonion-valued functions. In the unit ball, we explicitly show that the intrinsic weight factor is crucial to obtain the reproduction property and that the latter precisely compensates an additional associator term that otherwise appears when leaving out the weight factor.Furthermore, we study an octonionic Teodorescu transform and show how it is related to the unweighted version of the Bergman transform and establish some operator relations between these transformations. We apply two different versions of the Borel-Pompeiu formulae that naturally arise in the context of the non-associativity. Next, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling operator, also known as the Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all prominent hypercomplex function theories. Then we describe some basic mapping properties arising in context with the L2-space decomposition discussed before.Finally, we explore several applications of the octonionic Π-operator. Initially, we demonstrate its utility in solving the octonionic Beltrami equation, which characterizes generalized quasi-conformal maps from R8 to R8 in a specific analytical sense. Subsequently, analogous results are presented for the hyperbolic octonionic Dirac operator acting on the right half-space of R8. Lastly, we discuss how the octonionic Teodorescu transform and the Bergman projection can be employed to solve an eight-dimensional Stokes problem in the non-associative octonionic setting.

A Microlocal Investigation of Stochastic Partial Differential Equations for Spinors with an Application to the Thirring Model

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if d≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 2$$\\end{document}.

Symmetries of modified Dirac operators in supergravity flux backgrounds

Modifications of Dirac operators in supergravity flux backgrounds are considered. Modified spin curvature operators and squares of modified Dirac operators corresponding to Schrödinger-Lichnerowicz-like formulas are obtained for different types of flux modifications. Symmetry operators of modified massless and massive Dirac equations are found in terms of modified Killing-Yano and modified conformal Killing-Yano forms. Extra constraints for symmetry operators in terms of different types of fluxes and modified Killing-Yano forms are determined.

Quantum entropy couples matter with geometry

Abstract We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

Supersymmetric spectrum for vector multiplet on Euclidean AdS2

Quantum study of supersymmetric theories on Euclidean two dimensional anti-de Sitter space (EAdS2) requires complexified spectrum. For a chiral multiplet, we showed that the spectrum of the Dirac operator acquires a universal shift of i/2 from the real spectrum to make the supersymmetry between boson and fermion manifest, where both the bosonic and fermionic eigenfunctions are normalizable using an appropriate definition of Euclidean inner product. We extend this analysis to the vector multiplet, where we show that the gaugino requires both +i/2 and i/2 shift from the real spectrum, and there is additional isolated point at vanishing spectral parameter which is mapped by supersymmetry to the boundary zero modes of the vector field. Furthermore, this spectral analysis shows that not every bosonic fields in the vector multiplet can satisfy normalizable boundary condition. Nevertheless, aided by a reorganization of fields into a cohomological form, we find the supersymmetry mapping between bosons and fermions in terms of the expansion coefficients with respect to the newly constructed basis.

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.

The spectrum of an operator associated with G 2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities

abstract: This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $\mathbb{S}^{5}$, called the \textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold. Using the quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $\Spec P$. We show that $\Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on $\mathbb{S}^{5}$ to certain sheaf cohomologies on~$\mathbb{P}^{2}$. Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of $\SU(3)$ and the subgroup $S[U(1)\times U(2)]$, we show an example in which $\Spec P$ and the multiplicities can be completely determined.

Schwarzschild deformed supergravity background: Possible geometry origin of fermion generations and mass hierarchy

The problem of fermion masses hierarchy in the Standard Model is considered on a toy model of a 10-dimensional space-time with a IIA supergravity type background. The Dirac equation written on this background, after compactification of extra four- and one-dimensional subspaces, gives the spectrum of Fermi fields which profiles in five dimensions and corresponding Higgs generated masses in four dimensions depend on the eigenvalues of Dirac operator on the named compact subspaces. Schwarzschild Euclidean deformation of the supergravity throat with the “apple-shaped” conical singularity permits to leave only three nondivergent angular modes interpreted as three generations of the down-type quarks. The resulting expressions for the quark masses are a geometry version of the Froggatt-Nielsen mechanism. Published by the American Physical Society 2024

Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities

For a family of self-adjoint Dirac operators -ic(α·∇)+c22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-i c (\\alpha \\cdot \ abla ) + \\frac{c^2}{2}$$\\end{document} subject to generalized MIT bag boundary conditions on domains in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large c.

Pseudo-supersymmetric approach to the Dirac operator in the Schwarzschild spacetime

Abstract We have discussed the Dirac equation in Schwarzschild spacetime using pseudo-supersymmetric quantum mechanics and have obtained the partner Hamiltonian of the initial Hamiltonian operator. We demonstrate that the partner metric tensors, corresponding to these Hamiltonians, can be derived using the intertwining relations inherent in pseudo-supersymmetric approaches. We have seen that the Schwarzschild metric may be expanded into Schwarzschild–Tangherlini metric using the aspects of pseudo-supersymmetry. Furthermore, we have derived approximate solutions for the radial component within the framework of N = 2 supersymmetry with the corresponding radial potential graphs presented. Subsequently, our discussion extends to the quasinormal modes, focusing particularly on the asymptotic limits as r approaches ± ∞ .

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

The cubic Dirac operator on compact quotients of the oscillator group

We study Kostant’s cubic Dirac operator D^{1/3} on locally symmetric Lorentzian manifolds of the form \Gamma\backslash\operatorname{Osc}_{1} , where \operatorname{Osc}_{1} is the four-dimensional oscillator group and \Gamma\subset\operatorname{Osc}_{1} is a cocompact lattice. These quotients are the only four-dimensional, compact Lorentzian G -homoge­neous spaces for a solvable but non-abelian Lie group G . We determine the spectrum of D^{1/3} . We also give an explicit decomposition of the regular representation of \operatorname{Osc}_{1} on L^{2} -sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of D^{1/3} .

Spectral Convergence of the Dirac Operator on Typical Hyperbolic Surfaces of High Genus

AbstractIn this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus g with $$o(\sqrt{g})$$ o ( g ) cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.