One can always write the mass matrix for Weyl spinors so that it is symmetric. However this is often not a good idea. It is usually incompatible with irreducibility of the fermion representations. As a result, a symmetrized mass term hides important symmetries and creates misleading difficulties that are not genuinely part of the theory. This is true for the Standard Model for example, and for its supersymmetric versions. There is a related subtlety, involving symmetrization of the interaction terms, that is central to the SUSY breaking mechanism of Cybersusy.
We investigate the confining phase vacuum structure of supersymmetric SO(11) gauge theories with one spinor matter field and Nf \le 6 vectors. We describe several useful tricks and tools that facilitate the analysis of these chiral models and many other theories of similar type. The forms of the Nf=5 and Nf=6 quantum moduli spaces are deduced by requiring that they reproduce known results for SU(5) SUSY QCD along the spinor flat direction. After adding mass terms for vector fields and integrating out heavy degrees of freedom, we also determine the dynamically generated superpotentials in the N…
We show in (1+2) dimensions that supersymmetric (SUSY) BF action for a (minimal and off-shell) spin-(1,3/2) supermultiplet is a unique SUSY invariant one required from the closure property of commutator algebra for SUSY transformations.
We examine the low energy structure of N=1 supersymmetric SO(10) gauge theory with matter chiral superfields in N_Q spinor and N_f vector representations. We construct a dual to this model based upon an SU(N_f+2N_Q-7) x Sp(2N_Q-2) gauge group without utilizing deconfinement methods. This product theory generalizes all previously known Pouliot-type duals to SO(N_c) models with spinor and vector matter. It also yields large numbers of new dual pairs along various flat directions. The dual description of the SO(10) theory satisfies multiple consistency checks including an intricate renormalizatio…
Within the context of the extended bi-spinor gauge theory we describe a new off-shell realization of scalar supersymmetry (s-susy) of massless interacting fields with U(1), U(1) x SU(N) and U(1) x SU(N_1) x SU(N_2) gauge groups. S-susy acts in the space of graded differential forms. The realization is non-linear in the non-abelian case. S-susy would not require the doubling of the SM particle spectrum. Instead, essentially only the forth generation of quarks and leptons would be needed as extra field content. The theory is by construction globally U(2,2) invariant and is an example of a supers…
We focus on a spin-3/2 supersymmetry (SUSY) algebra of Baaklini in D = 3 and explicitly show a nonlinear realization of the SUSY algebra. The unitary representation of the spin-3/2 SUSY algebra is discussed and compared with the ordinary (spin-1/2) SUSY algebra.
We study the confining phase structure on {\cal N}=1 supersymmetric SO(12) gauge theory with N_f \le 7 vectors and one spinor. The explicit form of low energy superpotentials for N_f \le 7 are derived after gauge invariant operators relevant in the effective theory are identified via gauge symmetry breaking pattern. The resulting confining phase structure is analogous to N_f \le N_c+1 SUSY QCD. Finally, we conclude with some comments on the search for duals to N_f \ge 8 SO(12) theory.
In the standard model, electroweak theory is based on chiral gauge symmetry, which only contains massless fermions and gauge vectors. So the fundamental field equations are Weyl equations including gauge vectors. We deduced these Weyl equations without gauge vector from quadric surface in three dimensional oriented projective geometry. It showed that Weyl equations were equations of generating lines in the quadric surface. In other word, the geometric correspondence of spinor is generating lines of quadric surface in momentum space. It also showed that two spinors would generate a gauge vector…
We elaborate on the recently discovered spinor-vector duality in realistic free fermionic heterotic vacua. We emphasize the interpretation of the freely-acting orbifolds carried out on the six internal dimensions as coordinate-dependent compactifications; they play a central role in the duality, especially because of their ability to break the right-moving superconformal algebra of the space-time supersymmetric heterotic vacua. These considerations lead to a simple and intuitive proof of the spinor-vector duality, and to the formulation of explicit rules to find the dual of a given model. We d…
We study the unitary representation of supersymmetry (SUSY) algebra based on a spinor-vector generator for both massless and massive cases. A systematic linearization of nonlinear realization for the SUSY algebra is also discussed in the superspace formalism with a spinor-vector Grassmann coordinate.
General spinors in polar form display a structure that is made up by real scalar degrees of freedom plus six components which can be recognized as Goldstone bosons: in the present paper we show that of all singular spinors, Weyl and Majorana spinors have no real degree of freedom and so that they can be interpreted as pure Goldstone states.
The properties of Dirac gamma matrices in a four-dimensional space-time with the $(2,2)$ signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in $2 + 2$ dimensions is discussed, and the existence of the {\it real chiral} scalar supermultiplet is discovered. Supersymmetric {\it self-dual} Yang-Mills theories and {\it self-dual} supergravity model in $2 + 2$ dimensions, that are apparently relevant to integrable systems, are formulated for the first time.
We consider the wave equation for spinors in ${\cal D}$-dimensional Weyl geometry. By appropriately coupling the Weyl vector $ϕ_μ$ as well as the spin connection $ω_{μa b } $ to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions.
In this paper, we survey the nature of spinors and supersymmetry (SUSY) in various types of spaces. We treat two distinct types of spaces: flat spaces and spaces of constant (non-zero) curvature. The flat spaces we consider are either three or four dimensional of signatures 3 + 1, 4 + 0, 2 + 2 and 3 + 0. In each of these cases, SUSY generators anti-commute to yield the generators of translations in the non-compact flat spaces. The spaces of constant curvature we consider are two-dimensional: the surface of the sphere S_2 and the Anti-deSitter space AdS_2. S_2 is embedded in a 3 + 0 Euclidean s…
We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdová is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.
In a new Weyl 2-spinor approach to Non abelian Gauge Theories, starting with the local U(1) Gauge group of a previous work, we study now the SU(3) case corresponding to quarks (antiquarks) interacting with color fields. The principal difference with the conventional approach is that particle-field interactions are not described by means of potentials but by the field strength magnitudes. Some analytical expressions showing similarities with electrodynamics are obtained. Classical equations that describe the behavior of quarks under gluon fields might be in principle applied to the quark-gluon …
In spinor formalism, since any massless free-field spinor with spin higher than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We determine the corresponding Dirac-Weyl spinors in a given empty spacetime. Regarding them as basic units, other higher spin massless free-field spinors are then identified. Along this way, we find some hidden fundamental features related to these fields. In particular, for non-twisting vacuum Petrov type N solutions, we show that all higher spin massless free-field spinor…
In a previous paper we introduced two linear spinor equations equivalent to the Lorentz Force and stated that these equations were fairly general and could be applied to any force field compatible with Special Relativity. In this paper, via a lagrangian approach, we explore this possibility and obtain classical spinor equations describing the behaviour of fermionic particles not only under an electromagnetic field but also under Yang-Mills and Color fields. We find a covariant derivative defined {\it along the classical trajectory} of the particle, which can be extended to SU(2) and SU(3) loca…
Solutions to the four-dimensional Euclidean Weyl equation in the background of a general JNR N-instanton are known to be normalisable and regular throughout four-space. We show that these solutions are asymptotically given by a linear combination of simple singular solutions to the free Weyl equation, which can be interpreted as localised spinors. The `spinorial' data parameterising the asymptotics of the delocalised solutions to the Weyl equation in the presence of the instanton almost determines the background instanton, yet not completely. However, it captures the geometry and symmetry of t…
We construct a consistent theory of a quantum massive Weyl field. We start with the formulation of the classical field theory approach for the description of massive Weyl fields. It is demonstrated that the standard Lagrange formalism cannot be applied for the studies of massive first-quantized Weyl spinors. Nevertheless we show that the classical field theory description of massive Weyl fields can be implemented in frames of the Hamilton formalism or using the extended Lagrange formalism. Then we carry out a canonical quantization of the system. The independent ways for the quantization of a …
A graphical representation of supersymmetry is presented. It clearly expresses the chiral flow appearing in SUSY quantities, by representing spinors by {\it directed lines} (arrows). The chiral suffixes are expressed by the directions (up, down, left, right) of the arrows. The SL(2,C) invariants are represented by {\it wedges}. Both the Weyl spinor and the Majorana spinor are treated. We are free from the complicated symbols of spinor suffixes. The method is applied to the 5D supersymmetry. Many applications are expected. The result is suitable for coding a computer program and is highly expec…
In this note we give an explicit formula for the preserved Killing spinors in deformed string theory backgrounds corresponding to integrable Yang--Baxter deformations realized via (sequences of) TsT transformations. The Killing spinors can be expressed only in terms of the bi-vector $Θ$ which encodes the deformation. This formula is applicable to deformed backgrounds related to $r$-matrices of various ranks, including those that do not satisfy the unimodularity condition and give rise to backgrounds in generalized supergravity. We conjecture that our formula also remains valid for integrable d…
A graphical representation of supersymmetry is presented. It clearly expresses the chiral flow appearing in SUSY quantities, by representing spinors by {\it directed lines} (arrows). The chiral suffixes are expressed by the directions (up, down, left, right) of the arrows. The SL(2,C) invariants are represented by {\it wedges}. Both the Weyl spinor and the Majorana spinor are treated. We are free from the messy symbols of spinor suffixes. The method is applied to the 5D supersymmetry. Many applications are expected. The result is suitable for coding a computer program and is highly expected to…
Metric spinor phase of the Infeld-van der Waerden γ-formalism and axion field were identified in Ref. [1], by using Maxwell theory. Since axion couples with fermions, we will investigate Dirac theory to extend the work given in [1], showing that is possible again to identify this phase with the axion. By searching an exact identification, we will work yet the spin transformations to adapt chiral rotations and so to show that the metric spinor phase behaves exactly as required for the axion field. We will study also the Dirac-Maxwell system which provides a minimal Lagrangian approach for the a…
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the …
We discuss the linearization of vector-spinor (spin-3/2) nonlinear supersymmetry (vsNLSUSY) transformations for both $N = 1$ and $N$-extended SUSY in flat spacetime based on the commutator algebra by using functionals (composites) of spin-3/2 Nambu-Goldstone (NG) fermions, which are expressed as simple products of powers of the spin-3/2 NG fermions and a fundamental determinant in the vsNLSUSY theory. We define basic component fields by means of those functionals in a linearized vsSUSY theory including spin-3/2 fields, general auxiliary ones and a $D$-term. The general forms of linear (rigid) …
We extend our program, of coupling theories to scale in order to make their Weyl invariance manifest, to include interacting theories, fermions and supersymmetric theories. The results produce mass terms coinciding with the standard ones for universes that are Einstein, but are novel in general backgrounds. They are generalizations of the gravitational couplings of a conformally improved scalar to fields with general scaling and tensor properties. The couplings we find are more general than just trivial ones following from the conformal compensating mechanisms. In particular, in the setting wh…
We consider the single-handed spinor field in interaction with its own gravitational field described by the set of field equations given by Weyl field equations written in terms of derivatives that are covariant with respect to the gravitational connection plus Einstein field equations soured with the energy tensor of the spinor: for the Weyl spinor and the ensuing spacetime of Weyl-Lewis-Papapetrou structure, we will find all exact solutions. The obtained solution for the metric tensor is that of a PP-wave spacetime while the spinor field is a flag-dipole.
We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function $q_τ(t)$ for the Ising states $τ$. The time dependent probability distribution of a generalized Ising model obtains as $p_τ(t)=q^2_τ(t)$. The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure…
We consider the fermionization of a bosonic free theory characterized by the 3+1D scalar - tensor duality. This duality can be interpreted as the dimensional reduction, via a planar boundary, of the 4+1D topological BF theory. In this model, adopting the Sommerfield tomographic representation of quantized bosonic fields, we explicitly build a fermionic operator and its associated Klein factor such that it satisfies the correct anticommutation relations. Interestingly, we demonstrate that this operator satisfies the massless Dirac equation and that it can be identified with a 3+1D Weyl spinor. …
We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the …
These are expanded notes of lectures given at the summer school "Gif 2000" in Paris. They constitute the first part of an "Introduction to supersymmetry and supergravity" with the second part on supergravity by J.-P. Derendinger to appear soon. The present introduction is elementary and pragmatic. I discuss: spinors and the Poincaré group, the susy algebra and susy multiplets, superfields and susy lagrangians, susy gauge theories, spontaneously broken susy, the non-linear sigma model, N=2 susy gauge theories, and finally Seiberg-Witten duality.
Supersymmetry (SUSY) interchanges bosons and fermions but no direct evidences of it have been revealed in nature yet. In this letter, we observe that fluctuating pair density waves (PDW) consist of two complex order parameters which can be superpartners of the unavoidably-doubled Weyl fermions in three-dimensional lattice models. We construct explicit fermionic lattice models featuring 3D Weyl fermions and show that PDW is the leading instability via a continuous phase transition as short-range interactions exceed a critical value. Using renormalization group, we theoretically show that N=2 sp…
We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosons. The result is a recipe to construct a supersymmetric Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a five-dimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder Schrader approach in which spinors are ``doubled'' but the action is not hermitean. However, for…
We discuss the calculation of soft supersymmetry breaking terms in type IIB string theoretic models in the Large Volume Scenario (LVS). The suppression of FCNC gives a lower bound on the size of the compactification volume. This leads to soft terms which are strongly suppressed relative to the gravitino mass so that the dominant contribution to the gaugino masses comes from the Weyl anomaly. The other soft terms are essentially generated by the renormalization group running from the string scale to the TeV scale.
We obtain a continuous Wick rotation for Dirac, Majorana and Weyl spinors $ψ\to \exp ({1\over 2} θγ^4 γ^5)ψ$ which interpolates between Minkowski and Euclidean field theories.
A graphical representation of supersymmetry is presented. It clearly expresses the chiral flow appearing in SUSY quantities, by representing spinors by {\it directed lines} (arrows). The chiral suffixes are expressed by the directions (up, down, left, right) of the arrows. The SL(2,C) invariants are represented by {\it wedges}. We are free from the messy symbols of spinor suffixes. The method is applied to the 5D supersymmetry. Many applications are expected. The result is suitable for coding a computer program and is highly expected to be applicable to various SUSY theories (including Supergr…
Higher dimensional Majorana-Weyl spacetimes present space-time dualities which are induced by the Spin(8) triality automorphisms. Different signature versions of theories such as 10-dimensional SYM's, superstrings, five-branes, F-theory, are shown to be interconnected via the S_3 permutation group. Bilinear and trilinear invariants under space-time triality are introduced and their possible relevance in building models possessing a space-versus-time exchange symmetry is discussed. Moreover the Cartan's ``vector/chiral spinor/antichiral spinor" triality of SO(8) and SO(4,4) is analyzed in detai…
We propose a relativistic Jeff=3/2 semimetal with 4d1 or 5d1 electrons on a cubic lattice when the strong spin-orbital coupling takes over the Hunds' coupling. A relativistic spinor with spin 3/2 is historically called Rarita-Schwinger spinor. In the massless case, the right- and left-handed chiral degrees of freedom of the Rarita-Schwinger spinors are independent. In the lattice model that we propose, the right- and left- handed gapless points in Brillouin zone are separated. We call this linearly dispersed semimetal Rarita-Schwinger-Weyl semimetal, similar to Weyl semimetal for spin 1/2 syst…
Based on a maximally symmetric minimal unification hypothesis and a quantum charge-dimension correspondence principle, it is demonstrated that each family of quarks and leptons belongs to the Majorana-Weyl spinor representation of 14-dimensions that relate to quantum spin-isospin-color charges. Families of quarks and leptons attribute to a spinor structure of extra 6-dimensions that relate to quantum family charges. Of particular, it is shown that 10-dimensions relating to quantum spin-family charges form a motional 10-dimensional quantum space-time with a generalized Lorentz symmetry SO(1,9),…
The Weyl double copy relates vacuum solutions in general relativity to Abelian gauge fields in Minkowski spacetime. In a previous work, we showed how the Weyl double copy can be extended to provide a treatment of external gravitational sources consistent with the classical Kerr-Schild double copy. Using this generalization, here we provide a complete double copy analysis of electrovacuum Petrov type D spacetimes. This includes the first analysis of the charged C-metric, whose single copy interpretation invokes the two-potential formalism of electrodynamics. We also present the first double cop…
In Ref. \cite{Horava2005} Hořava suggested, that the multi - fermion many-body system with topologically stable Fermi surfaces may effectively be described (in a vicinity of the Fermi surface) by the theory with coarse-grained fermions. The number of the components of these coarse-grained fermions is reduced compared to the original system. Here we consider the $3+1$ D system and concentrate on the particular case when the Fermi surface has co-dimension $p=3$, i.e. it represents the Fermi point in momentum space. First we demonstrate explicitly that in agreement with Hořava conjecture, in the …
A discrepancy between the Anomaly Mediated Supersymmetry Breaking (AMSB) gaugino mass calculated from the work of Kaplunovsky and Louis (hep-th/9402005) (KL) and other calculations in the literature is explained, and it is argued that the KL expression is the correct one relevant to the Wilsonian action. Furthermore it is argued that the AMSB contribution to the squark and slepton masses should be replaced by the contribution pointed out by Dine and Seiberg (DS) which has nothing to do with Weyl anomalies. This is not in general equivalent to the AMSB expression, and it is shown that there are…
We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of ${\cal N}=2$ minimal gauged supergravity in four dimensions. According to the classification scheme utilizing the spinorial geometry or the bilinears of Killing spinors, the general solution preserves one quarter of supersymmetry and is described by the Przanowski-Tod class with the self-dual Weyl tensor. We demonstrate that there exists an additional Killing spinor, provided the Przanowski-Tod metric admits a Killing vector that commutes with the principal one. The proof proceeds by recas…
This is a doctoral thesis on the application of techniques originally developed in the programme of characterisation of supersymmetric solutions to Supergravity theories, to finding alternative backgrounds. We start by discussing the concept of a Killing spinor, and how these are paramount to the process of classifying the aforementioned supersymmetric solutions. Moreover, these geometric objects also have applications when considered in different scenarios (the 'beyond' in the title). In particular, techniques based on a different parallelising rule for a spinorial field can be used for obtai…
In the Dirac-Weyl equation that describes massless neutrino propagation in the minimal Standard Model, the ($2\times 2$ equivalence of the) gamma matrices convert Weyl spinors into spacetime tensors, and vice versa. They can thus be regarded as amalgamations of three different types of mappings, one that connects particle spinors directly forming representations to internal gauge symmetries to their spacetime counterparts that are embodied by null flags, another that translates the spacetime spinors into their corresponding tensors expressed in an orthonormal tetrad, and finally a purely tenso…
We propose an unified theory for spinor fields on extended Weyl manifolds taking into account self-interactions to obtain the Relativistic dynamics on a general curved Riemannian background as continuation of the Relativistic Quantum Geometry program, recently introduced. We focuss our attention separately on both, massless and matter fields. We study an example of confined gravitons on a de Sitter (dS) background at Planckian scales.
The N=4 superconformal algebra is derived from the symmetry transformations of fields in the N=4 SYM action in D=4. We use a Majorana-Weyl spinor in D=10 instead of four Weyl spinors in D=4. This makes it transparent to relate generators of the N=4 superconformal algebra to those of the super-AdS_5XS^5 algebra. Especially, we obtain the concrete map from the supersymmetries Q and conformal supersymmetries S in N=4 SYM to the supersymmetries (Q_1, Q_2) in the AdS_5XS^5 background.
We investigate for N = 1 supersymmetry (SUSY) the relation between a scalar supermultiplet of linear SUSY and a nonlinear (NL) SUSY model including apparently pathological higher derivaive terms of a Nambu-Goldstone (N-G) fermion besides the Volkov-Akulov (V-A) action. SUSY invariant relations with higher derivative terms of the N-G fermion, which connect the linear and NL SUSY models, are constructed at leading orders by heuristic arguments. We discuss a higher derivative action of the N-G fermion in the NL SUSY model, which apparently includes a (Weyl) ghost field. By using this relation, we…
Weyl fermions with nonlinear dispersion have appeared in real world systems, such as in the Weyl semi-metals and topological insulators. We consider the most general form of Dirac operators, and study its topological properties embedded in the chiral anomaly, in the index theorem, and in the odd-dimensional partition function, by employing the heat kernel. We find that all of these topological quantities are enhanced by a winding number defined by the Dirac operator in the momentum space, regardless of the spacetime dimensions. The chiral anomaly in $d=3+1$, in particular, is also confirmed vi…
The main purpose of this work is to show that massless Dirac equation formulated for non-interacting Majorana-Weyl spinors in higher dimensions, particularly in D=1+9 and D=5+5, can lead to an interpretation of massive Majorana and Dirac spinors in D=1+3. By adopting suitable representations of the Dirac matrices in higher dimensions, we pursue the investigation of which higher dimensional space-times and which mass-shell relation concerning massless Dirac equations in higher dimensions may induce massive spinors in D=1+3. The mixing of the chiral fermions in higher dimensions may induce a mec…
We present a method to find solutions of the Weyl or the Dirac equation without specifying a representation choice for $γ^a$'s. By taking $γ^a$'s formally as independent variables, we construct solutions out of $2^d$ orthonormal polynomials of $γ^a$'s (in $d$-dimensional space) operating on a ``vacuum state''. Polynomials reduce into $2^{d/2}$ or $2^{(d+1)/2}$ repetitions of the Dirac spinors for $d$ even or odd, respectively. We further propose the corresponding graphic presentation of basic states, which offers an easy way to see all the quantum numbers of states with respect to the generato…
Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one taking values in either quaternions or octonions (or their split versions). These quaternionic and octonionic generalisations generate well-known models of Cl(5) and Cl(9) respectively. The main aim of the paper is to explicitly relate these models to the models arising via the creation/annihilation operator construction. We describe in details the models related…
