Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.
problem Calculating the dimension of the plane-wave normalizable kernel for massless fermions in spherically symmetric monopole backgrounds.
method Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background.
result Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Authors compute anomalies for fermions in 3, 4, and 5 dimensions.
problem Quantum anomalies in fermions with boundaries in odd dimensions.
method Explicit computation of anomalies for fermions in dimensions 3, 4, and 5.
result New boundary terms for chiral anomaly in 4 dimensions.
Holomorphic residue formula for complex supermanifolds.
problem Residue localization on complex supermanifolds.
method Holomorphic residue localization formula for odd vector fields.
result Explicit local residue formula under isolated non-degeneracy hypotheses.
We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a …
Develops a mathematical framework for causal fermion systems in infinite dimensions.
problem Analysis of causal fermion systems in infinite-dimensional settings.
method Introduces Banach manifold structure and expedient differential calculus.
result Establishes Hölder continuity of causal Lagrangian and integrated causal Lagrangian.
A 12D spinor encodes fermions in a 4D Kaluza-Klein model.
problem Encoding fermions in a 4D spacetime from a higher-dimensional perspective.
method Using a spacetime P=M4imesK with K=SU(3), encoding fermions in 64 spinor components. result The 64 spinor components couple to Standard Model gauge fields in chiral representations.
A new, simplified form of 10D supergravity action is derived up to all fermion orders.
problem Deriving a simplified form of 10D supergravity action up to all fermion orders.
method Generalised geometry and second-order formalism to upgrade calculations to all orders in fermions.
result A strikingly simple form of the action and supersymmetry transformations are obtained.
Study fermionic theories, their anomalies, and modular transformations.
problem Understanding fermionic theories and their anomalies.
method Use spin-cobordisms, surgeries, and invertible topological quantum field theories.
result Explicit combinatorial expressions for spin-bordism invariants.
Geometric derivation of Einstein equations from causal fermion systems.
problem Deriving Einstein's equations from a new theoretical framework.
method Analysis of causal fermion systems and causal action principle.
result Einstein equations derived from causal action principle.
We show that the Yang-Mills quantum field theory with momentum and spacetime cutoffs in four Euclidean dimensions is equivalent, term by term in an appropriately resummed perturbation theory, to a Fermionic theory with nonlocal interaction terms. When a further momentum cutoff is imposed, this Fermionic theory has a co…
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive th…
We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to …
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY …
Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1D are explored. Many of our field theories are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulatio…
We prove that a Kahler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau's theorem does not hold for supermanifolds.
In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super C∞-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz…
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the Z16 anomaly indicator for time-reversal symmetric topological superconductors. Reformulates mod-two APS index using domain-wall fermion.
problem Non-local APS boundary condition and global anomalies.
method Physicist-friendly reformulation of APS index using domain-wall fermion.
result Equivalence between two formulations of APS index.
No time-periodic Majorana fermions found in Kerr-Newman spacetimes with nontrivial charge.
problem Existence of Majorana fermions in Kerr-Newman spacetimes with nontrivial charge.
method Analysis of Dirac equation in Kerr-Newman spacetimes, leading to algebraic identities.
result No differentiable time-periodic Majorana fermions in Kerr-Newman spacetimes with nontrivial charge.
Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of s…
By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's t…
In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structu…
We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators th…
We give a brief introduction to causal fermion systems with a focus on the geometric structures in space-time.
As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtai…
Proves a positive mass theorem for static causal fermion systems.
problem Defining mass for complex spacetimes without regularity assumptions.
method Surface layer integrals comparing asymptotically flat and vacuum spacetimes.
result Proves a positive mass theorem for static causal fermion systems.
Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential…
In this paper we provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation …
Massive fermions help understand index theorems without chiral symmetry.
problem Understanding index theorems in massive fermion systems.
method Reformulate chiral anomaly and index theorems with massive Dirac operators.
result Nontrivial mathematical relations between massless and massive fermions.
Mathematician-friendly formulation of Atiyah-Patodi-Singer index.
problem Boundary conditions and edge modes in domain-wall fermions.
method Mathematician-friendly derivation of Atiyah-Patodi-Singer index.
result New insights into the interplay of boundary conditions, domain-wall fermions, and edge modes.
We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to c…
We simplify supergravity in 10D using geometric insights.
problem Formulating supergravity in 10D without Lorentz degrees of freedom.
method Using generalised geometry, we describe the fibred structure of field space.
result Our action satisfies the classical master equation without Lorentz terms.
We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general backgro…
The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for…
We unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles as a noncommutative analogue of de Rham forms on the internal finite quantum space.
The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.
In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this resu…
We compute the scalar curvature of 7-dimensional G2-manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form fiel…
There are eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the spa…
Study baryogenesis in conformally flat spacetimes using causal fermion systems.
problem Understanding baryogenesis in specific spacetimes.
method Analysis of baryogenesis mechanism in conformally flat spacetimes with explicit formula derivation.
result Explicit formula for baryogenesis rate in these spacetimes.
Derives hyperbolic laws of cosines and sines with fermionic corrections.
problem Deriving hyperbolic laws of cosines and sines with new mathematical corrections.
method Using Minkowski supergeometry, the laws of cosines and sines are derived in the super hyperbolic plane.
result Identical formulae to classical cases with fermionic corrections for cosines and sines.
The paper explores how non-Killing fields on internal spaces can produce massive gauge fields with chiral interactions.
problem Traditional Kaluza-Klein models limit gauge fields to Killing vector fields, ignoring chiral interactions.
method Investigates properties of 4D gauge fields linked to non-Killing fields on internal spaces using spin geometry and Riemannian submersions.
result Massive gauge fields linked to non-Killing fields can mix fermions with different masses and have asymmetric couplings to left- and right-handed fermions.
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, den…
Study of operators on loop spaces using Fermionic calculus and stochastic methods.
problem Analyzing operators on loop spaces arising from self-adjoint and closed operators.
method Fermionic calculus and stochastic methods to derive regularity and stochastic representations.
result Derivation of a stochastic refinement of the Duistermaat-Heckman localization formula.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.