New principle for supersymmetric localization on Lie groups.
problem Computing supertrace of non-supersymmetric observables.
method Invariant supersymmetric deformations and fermionic zero modes.
result Path integral localizes to periodic orbits.
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48 results for “fermionic zero modes”
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
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.
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…
We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set o…
We develop a method for finding the zero modes of the Dirac operator in the presence of BPS monopoles. We use it to find the zero modes in the case of Abelian BPS monopoles in .
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensionscond-mat.str-el
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 a holomorphic residue localization formula for odd holomorphic vector fields on compact complex supermanifolds whose fermionic and bosonic dimensions coincide. Under isolated non-degeneracy hypotheses on the reduced zero set, we give an explicit local residue formula.
Revisits zero modes of Dirac operator on Eguchi-Hanson space.
problem Determining zero modes of the Dirac operator on Eguchi-Hanson space.
method Uses spin- spinors and formalism of differential forms to simplify calculations.
result Reproduces known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator.
The paper analyzes zero modes on product manifolds and provides estimates for their norms.
problem Analyzing zero modes on product Riemannian manifolds.
method Using the zero mode equation and non-increasing condition on |\varphi|, the paper derives estimates for the norms of the vector field A.
result The derived estimates are sharp in even dimensions and provide insights into the behavior of zero modes.
The paper proves inequalities for zero mode equations on spin manifolds.
problem Sharp inequalities for zero mode equations on spin manifolds.
method Simple proof and classification of equality cases.
result Equality holds if and only if the solution is a Killing spinor and the manifold is Sasaki-Einstein.
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.
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 …
We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature configurations, for which the spin connection has zero torsion and the associated R…
Introduces a massive variant of Ray-Singer Torsion to avoid zero modes in topological field theories.
problem Avoiding zero modes in the evaluation of path integrals for topological field theories.
method Introduces a massive variant of the Ray-Singer Torsion, involving determinants of the twisted Laplacian with mass but without zero modes.
result Explicitly evaluates the massive Ray-Singer Torsion on product manifolds and mapping tori.
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.
Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTscond-mat.str-el
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 anomaly indicator for time-reversal symmetric topological superconductors.
Paper proves a spinor inequality for magnetic fields on spin manifolds.
problem Proving a spinor inequality for magnetic fields on spin manifolds.
method Analyzing the zero mode equation and using the Yamabe constant.
result The inequality holds for non-trivial solutions.
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…
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.
Braiding Majorana Fermionscond-mat.str-el
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.
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…
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.
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.
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.
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…
Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordismshep-th
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 -equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a …
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.
Let be an asymptotically conic manifold, in the sense that compactifies to a manifold with boundary in such a way that becomes a scattering metric on . A special case of particular interest is that of asymptotically Euclidean manifolds, where and the induced me…
Paper studies non-associativity in quantum systems with magnetic fields.
problem Non-associativity of magnetic translations in quantum systems.
method Quantum field theory approach with -component fermions.
result Non-associativity described by a 3-cocycle of with values in .
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.
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.
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.
3d Super Hyperbolic Geometrymath.GT
Extends Wigner's representation to study super hyperbolic geometry.
problem Understanding geometry in super hyperbolic three-space.
method Extended Wigner's representation of the Lorentz group to OSp_C(1|2) and applied to Minkowski (3,1|4)-dimensional super space.
result Proof of divergence of the volume of a typical ideal tetrahedron in super hyperbolic three-space.
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…
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 with , encoding fermions in 64 spinor components.
result The 64 spinor components couple to Standard Model gauge fields in chiral representations.
Study Dirac operators on finite warped cylinders with gauge fields.
problem Characterize spectral flow on finite warped cylinders with gauge fields.
method Identify endpoint operators, derive determinant characterization, introduce regularized APS conditions.
result Regularized APS conditions admit a spectral-flow framework, matching zero-mode sets.
Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.
problem Constructing Abelian magnetic zero-modes on flat spacetime.
method Establishing a correspondence between vortex equations and harmonic spinors on the Nappi-Witten space.
result Explicit solutions of a twisted Dirac equation induce harmonic spinors on Minkowski space.
The Yukawa term in statistical mechanics quantifies information generation.
problem Understanding information generation in statistical mechanics.
method Defining a statistical product and Yukawa term to quantify information generation.
result The Yukawa term diverges in the quantum case, indicating Bose-Einstein condensation.
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.
New method uses random projections to estimate densities and modes efficiently.
problem Estimating densities and modes from sparse representations.
method Expand-and-sparsify representations followed by linear function and mode recovery algorithms.
result Optimal rates for density and mode estimation achieved.
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.
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…