Fermionic Variables Research Articles

Flat GL(1|1)-connections and fatgraphs

We study the moduli space of flat GL(1|1)-connections on a punctured surface from the point of view of graph connections. To each fatgraph, a system of coordinates is assigned, which involves two bosonic and two fermionic variables per edge, subject to certain relations. In the case of trivalent graphs, we provide a closed explicit formula for the Whitehead moves. In addition, we discuss the invariant Poisson bracket.

Hamiltonian formalism for Fermi excitations in a plasma with a non-Abelian interaction

The Hamiltonian theory for the collective longitudinally polarized colorless gluon excitations (plasmons) and for collective quark–antiquark excitations with abnormal relation between chirality and helicity (plasminos) in high-temperature quark–gluon plasma (QGP) is developed. For this purpose, Zakharov’s formalism for constructing the wave theory in nonlinear media with dispersion is used. A generalization of the Poisson superbracket involving both commuting and anticommuting variables to the case of a continuous medium is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic and fermionic degrees of freedom of the collective excitations in QGP are discussed and a complete system of the canonicity conditions for these transformations is written out. An explicit form of the effective fourth-order Hamiltonians describing the elastic scattering of plasmino off plasmino and plasmino off plasmon is found and the Boltzmann-type kinetic equations describing the processes of elastic scattering are obtained. A detailed comparison of the effective amplitudes defined within the (pseudo)classical Hamiltonian theory, with the corresponding matrix elements calculated early in the framework of high-temperature quantum chromodynamics in the so-called hard thermal loop approximation, is performed. This enables one to obtain, in particular, an explicit form of the vertex and coefficient functions in the effective amplitudes and in the canonical transformations, correspondingly, and also to define the validity of a purely pseudoclassical approach in the Hamiltonian description of the dynamics of quark–gluon plasma. The problem of determining the higher-order coefficient functions in the canonical transformations of fermionic and bosonic normal variables is considered. With the help of the coefficient functions obtained, the totally symmetric effective amplitudes of the elastic scattering of plasmino off plasmon and plasmino off plasmino are written out.

Quasisymmetric harmonics of the exterior algebra

AbstractWe study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1)The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ .(2)There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ .(3)There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology

In generic curved spacetimes, the unavailability of a natural choice of vacuum state introduces a serious ambiguity in the Fock quantization of fields. In this review, we study the case of fermions described by a Dirac field in non-stationary spacetimes, and present recent results obtained by us and our collaborators about well-motivated criteria capable to ensure the uniqueness in the selection of a vacuum up to unitary transformations, at least in certain situations of interest in cosmology. These criteria are based on two reasonable requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. These symmetries include the spatial isometries. Second, the unitary implementability of the Heisenberg dynamics of the annihilation and creation operators when the curved spacetime is treated as a fixed background. This last requirement not only permits the uniqueness of the Fock quantization but, remarkably, it also allows us to determine an essentially unique splitting between the phase space variables assigned to the background and the fermionic annihilation and creation variables. We first consider Dirac fields in 2 + 1 dimensions and then discuss the more relevant case of 3 + 1 dimensions, particularizing the analysis to cosmological spacetimes with spatial sections of spherical or toroidal topology. We use this analysis to investigate the combined, hybrid quantization of the Dirac field and a flat homogeneous and isotropic background cosmology when the latter is treated as a quantum entity, and the former as a perturbation. Specifically, we focus our study on a background quantization along the lines of loop quantum cosmology. Among the Fock quantizations for the fermionic perturbations admissible according to our criteria, we discuss the possibility of further restricting the choice of a vacuum by the requisite of a finite fermionic backreaction and, moreover, by the diagonalization of the fermionic contribution to the total Hamiltonian in the asymptotic limit of large wave numbers of the Dirac modes. Finally, we argue in support of the uniqueness of the vacuum state selected by the extension of this diagonalization condition beyond the commented asymptotic region, in particular proving that it picks out the standard Poincaré and Bunch–Davies vacua for fixed flat and de Sitter background spacetimes, respectively.

Super Quantum Airy Structures

We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional—which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss—as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.

Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology

In this work we show how the criterion of asymptotic Hamiltonian diagonalization originated in hybrid quantum cosmology serves to pick out a unique vacuum for the Dirac field in de Sitter, in the context of quantum field theory in curved spacetimes. This criterion is based on the dynamical definition of annihilation and creationlike variables for the fermionic field, which obey the linearized dynamics of a Hamiltonian that has been diagonalized in a way that is adapted to its local spatial structure. This leads to fermionic variables that possess a precise asymptotic expansion in the ultraviolet limit of large wavenumbers. We explicitly show that, when the cosmological background is fixed as a de Sitter solution, this expansion uniquely selects the choice of fermionic annihilation and creationlike variables for all spatial scales, and thus picks out a unique privileged Fock representation and vacuum state for the Dirac field in de Sitter. The explicit form of the basis of solutions to the Dirac equation associated with this vacuum is then computed.

Extended BRST–anti-BRST transformations in non-Abelian self-dual Chern–Simons coupling theory

We establish the extended BRST and anti-BRST transformations for non-Abelian relativistic self-dual Chern–Simons theory coupled to scalar fields within the Batalin–Vilkovisky framework. Based on these constructions, we describe the total effective extended BRST-invariant Lagrangian density of the coupling theory by means of superfields with one Grassmann coordinates. Moreover, we need two fermionic variables to reexpress the extended BRST–anti-BRST-invariant Lagrangian in the superspace.

Supersymmetric Calogero and Calogero-Sutherland models from gauging

We describe how the new kinds of N = 2 and N = 4 supersymmetric extensions of the rational and hyperbolic Calogero models can be derived by gauging U(n) symmetry of the appropriate superfield matrix models. These systems feature non-standard numbers N n2 of physical fermionic variables as compared with N n in the standard case. An essential ingredient of N = 4 models is the necessary presence of semi-dynamical spin variables described by d = 1 Wess-Zumino terms. The bosonic cores of N = 4 models are U(2) spin Calogero and Calogero-Sutherland models. In the hyperbolic case two non-equivalent N = 4 extensions exist, with and without the interacting center-of-mass coordinate in the bosonic sector. The talk is based on joint works with Sergey Fedoruk and Olaf Lechtenfeld.

On polarized scattering equations for superamplitudes of 11D supergravity and ambitwistor superstring

We revisited the formalism of 110 polarized scattering equation by Geyer and Mason from the perspective of spinor frame approach and spinor moving frame formulation of the 110 ambitwistor superstring action. In particular, we rigorously obtain the equation for the spinor function on Riemann sphere from the supertwistor form of the ambitwistor superstring action, write its general solution and use it to derive the polarized scattering equation. We show that the expression used by Geyer and Mason to motivate their ansatz for the solution of polarized scattering equation can be obtained from our solution after a suitable gauge fixing. To this end we use the hidden gauge symmetries of the 110 ambitwistor superstring, including S0(16), and the description of ambitwistor superstring as a dynamical system in an 110 superspace enlarged by bosonic directions parametrized by 517 tensorial central charge coordinates {Z}^{underline{mu}underline {nu }} and {Z}^{underline{mu}underline {nu}underline{rho}underline {sigma}underline{kappa}} . We have also found the fermionic superpartner of the polarized scattering equation. This happens to be a differential equation in fermionic variables imposed on the super­ amplitude, rather then just a condition on the scattering data as the bosonic polarized scattering equation is. D=10 case is also discussed stressing the similarities and differences with 110 systems. The useful formulation of 100 ambitwistor superstring considers it as a dynamical system in superspace enlarged with 126 tensorial central charge coordinates Zμνρσκ.

Hypermonogenic Plane Wave Solutions of the Dirac Equation in Superspace

In this paper, we obtain Cauchy–Kovalevskaya theorems for hypermonogenic superfunctions depending only on purely bosonic and fermionic vector variables. In addition, we use these results to construct plane wave examples of such functions.

Asymptotic diagonalization of the fermionic Hamiltonian in hybrid loop quantum cosmology

We use the freedom available in hybrid loop quantum cosmology to split the degrees of freedom between the geometry and the matter fields so as to build a quantum field theory for the matter content with good quantum properties. We investigate this issue in an inflationary, flat cosmology with inhomogeneous perturbations, and focus the discussion on a Dirac field, minimally coupled to the cosmological background and treated as a perturbation. After truncating the action at the lowest nontrivial order in perturbations, one must define canonical variables for the matter content, for which one generally employs canonical transformations that mix the homogeneous background and the perturbations. Each of these possible definitions comes associated with a different matter contribution to the Hamiltonian of the complete system, that may, in general, contain terms that are quadratic in creationlike variables, and in annihilationlike variables, with the subsequent production and destruction of pairs of fermionic particles and antiparticles. We determine a choice of the fermionic canonical variables for which the interaction part of the Hamiltonian can be made as negligible as desired in the asymptotic regime of large particle/antiparticle wave numbers. Finally, we study the quantum dynamics for this choice, imposing the total Hamiltonian constraint on the quantum states and assuming that their gravitational part is not affected significantly by the presence of fermions. In this way, we obtain a Schr\"odinger equation for the fermionic degrees of freedom in terms of quantum expectation values of the geometry that leads to asymptotically diagonal Heisenberg relations and Bogoliubov evolution transformations, with no divergences in the associated normal-ordered Hamiltonian.

Backreaction of fermionic perturbations in the Hamiltonian of hybrid loop quantum cosmology

We discuss the freedom available in hybrid loop quantum cosmology to define canonical variables for the matter content and investigate whether this can be used to derive a quantum field theory with good properties for the matter sector. We study a primordial, inflationary, cosmological spacetime with inhomogeneous perturbations at lowest nontrivial order, and focus our attention on the contribution of minimally coupled fermionic perturbations of Dirac type. Within the framework of the hybrid quantization, we analyze the different possible separations of the homogeneous background and the inhomogeneous perturbations, by means of canonical transformations that mix the two separated sectors. These possibilities provide a family of sets of annihilation and creationlike fermionic variables, each of them with a different associated contribution to the total Hamiltonian. In all cases, imposing the quantum constraints and introducing a Born-Oppenheimer approximation, one can derive a Schr\"odinger equation for the fermionic part of the wave functions. The resulting evolution turns out to be generated, for each of the allowed choices of variables, by a version of the fermionic contribution to the Hamiltonian which is obtained by evaluating all the dependence on the homogeneous geometry at quantum expectation values. This equation contains a term that encodes the backreaction of the fermionic perturbations on the quantum dynamics of the homogeneous sector. We analyze this backreaction by solving the associated Heisenberg evolution of the fermionic annihilation and creation operators. Then, we identify the conditions that the choice of those operators must satisfy in order to lead to a finite backreaction. Finally, we discuss further restrictions on this choice so that the fermionic Hamiltonian that dictates the Schr\"odinger dynamics is densely defined in Fock space.

Structural equations of supermanifolds immersed in the superspace with a prescribed curvature

The aim of this paper is to construct the structural equations of supermanifolds immersed in Euclidean, hyperbolic and spherical superspaces parametrised with two bosonic and two fermionic variables. To perform this analysis, for each type of immersion, we split the supermanifold into its Grassmannian components and study separately each manifold generated. Even though we consider four variables in the Euclidean case, we obtain that the structural equations of each manifold are linked with the Gauss–Codazzi equations of a surface immersed in a Euclidean or spherical space. In the hyperbolic and spherical superspaces, we find that the body manifolds are linked with the classical Gauss–Codazzi equations for a surface immersed in hyperbolic and spherical spaces, respectively. For some soul manifolds, we show that the immersion of the manifolds must be in a hyperbolic space and that the structural equations split into two cases. In one case, the structural equations reduce to the Liouville equation, which can be completely solved. In the other case, we can express the geometric quantities solely in terms of the metric coefficients, which provide a geometric characterization of the structural equations in terms of functions linked with the Hopf differential, the mean curvature and a new function which does not appear in the characterization of a classical (not super) surface.

A BRST formulation for the conic constrained particle

We describe the gauge invariant BRST formulation of a particle constrained to move in a general conic. The model considered constitutes an explicit example of an originally second-class system which can be quantized within the BRST framework. We initially impose the conic constraint by means of a Lagrange multiplier leading to a consistent second-class system which generalizes previous models studied in the literature. After calculating the constraint structure and the corresponding Dirac brackets, we introduce a suitable first-order Lagrangian, the resulting modified system is then shown to be gauge invariant. We proceed to the extended phase space introducing fermionic ghost variables, exhibiting the BRST symmetry transformations and writing the Green’s function generating functional for the BRST quantized model.

Ghost-spin chains, entanglement, and bc -ghost CFTs

We study 1-dimensional chains of ghost-spins with nearest neighbour interactions amongst them, developing further the study of ghost-spins in previous work, defined as 2-state spin variables with indefinite norm. First we study finite ghost-spin chains with Ising-like nearest neighbour interactions: this helps organize and clarify the study of entanglement earlier and we develop this further. Then we study a family of infinite ghost-spin chains with a different Hamiltonian containing nearest neighbour hopping-type interactions. By defining fermionic ghost-spin variables through a Jordan-Wigner transformation, we argue that these ghost-spin chains lead in the continuum limit to the $bc$-ghost CFTs.

The five-dimensional Dirac equation in the theory of algebraic spinors

The Dirac equation is considered in five-dimensional spaces with signatures (2,3), (4,1) and (0,5). The algebraic spinor formalism with the application of fermionic variables is used as the basis of real Clifford algebras and the module over this algebra. It is shown that solutions to the five-dimensional Dirac equation in spaces with signatures (2,3) and (4,1) can be expanded over solutions with zero value of the fifth component of the generalized momentum, and the equation is equivalent to an equation in four-dimensional spacetime.

Construction of a fermionic vacuum and the fermionic operators of creation and annihilation in the theory of algebraic spinors

We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.

On geometric aspects of the supersymmetric Fokas–Gel’fand immersion formula

In this paper, we develop a new geometric characterization for the supersymmetric versions of the Fokas–Gel’fand formula for the immersion of soliton supermanifolds with two bosonic and two fermionic independent variables into Lie superalgebras. In order to do so, from a linear spectral problem of a supersymmetric integrable system using the covariant fermionic derivative, we provide a technique to obtain two additional linear spectral problems for that integrable system, one using the bosonic variable derivatives and the other using the fermionic variable derivatives. This allows us to investigate, through the first and second fundamental forms, the geometry of the ()-supermanifolds immersed in Lie superalgebras. Whenever possible, the mean and Gaussian curvatures of the supermanifolds are calculated. These theoretical considerations are applied to the supersymmetric sine-Gordon equation.

On integrability aspects of the supersymmetric sine-Gordon equation

In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-Bäcklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym–Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.

Quantization of the AdS3 superparticle on [formula omitted

We analyze AdS3 superparticle dynamics on the coset OSP(1|2)×OSP(1|2)/SL(2,R). The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether charges of a massive particle are parametrized by coadjoint orbits of a timelike element of osp(1|2). Each chiral sector is described by two bosonic and two fermionic canonical coordinates corresponding to a superparticle with superpotential W=q−m/q, where m is the particle mass. Canonical quantization then provides a quantum realization of osp(1|2)⊕osp(1|2). For the massless particle the chiral charges lie on the coadjoint orbit of a nilpotent element of osp(1|2) and each of them depends only on one real fermion, which demonstrates the underlying κ-symmetry. These remaining left and right fermionic variables form a canonical pair and the system is described by four bosonic and two fermionic canonical coordinates. Due to conformal invariance of the massless particle, the osp(1|2)⊕osp(1|2) extends to the corresponding superconformal algebra osp(2|4). Its 19 charges are given by all real quadratic combinations of the canonical coordinates, which trivializes their quantization.