Grassmann Coordinates Research Articles

Goldstino superfields for spontaneously broken $ \mathcal{N} = 2 $ supersymmetry

We examine spontaneously broken N=2 supersymmetry in four dimensions and associate a spinor superfield with each Goldstino via a finite supersymmetry transformation with parameters that are the Grassmann coordinates of N=2 superspace. Making use of a special choice of coset parametrization allows us to develop a version of nonlinearly realized N=2 supersymmetry for which the associated Goldstino superfields are defined on harmonic superspace, thereby providing a natural mechanism for construction of a Goldstino action. The corresponding superfield Lagrangian is an O(4) multiplet. This property is used to reformulate the Goldstino action in projective superspace and in conventional N=2 superspace. We show how to generate matter couplings of the Goldstinos to supersymmetric matter using the N=2 harmonic, projective and full superspaces. As a bi-product of our consideration, we also derive an N=2 chiral Goldstino action.

A superspace formulation of the BV action for higher derivative theories

We first analyse the anti-BRST and double BRST structures of a certain higher derivative theory that has been known to possess BRST symmetry associated with its higher derivative structure. We discuss the invariance of this theory under shift symmetry in the Batalin–Vilkovisky (BV) formalism. We show that the action for this theory can be written in a manifestly extended BRST invariant manner in superspace formalism using one Grassmann coordinate. It can also be written in a manifestly extended BRST invariant manner and on-shell manifestly extended anti-BRST invariant manner in superspace formalism using two Grassmann coordinates.

Chern-simons theory in the SO(5)/U(2) harmonic superspace

We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposing these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to the D=3, N=6 conformal supersymmetry realized on N=5 superfields.

Statistics, condensation, and the Anderson-Higgs mechanism: Worldline path integral view

We explain, in the first-quantized path integral formalism, the mechanism behind the Anderson-Higgs effect for a gas of charged bosons in a background magnetic field, and we then use the method to prove the absence of the effect for a gas of fermions. The exchange statistics are encoded via the inclusion of additional Grassmann coordinates in a manner that leads to a manifest worldline supersymmetry. This extra symmetry is key to demonstrating the absence of the effect for charged fermions.

Chern–Simons [formula omitted], [formula omitted] superfield theory

We construct the D=3,N=5 harmonic superspace using the SO(5)/U(1)×U(1) harmonics. Three gauge harmonic superfields satisfy the off-shell constraints of the Grassmann and harmonic analyticities. The corresponding component supermultiplet contains the gauge field Am and an infinite number of bosonic and fermionic fields with the SO(5) vector indices arising from decompositions of gauge superfields in harmonics and Grassmann coordinates. The non-Abelian superfield Chern–Simons action is invariant with respect to the N=6 superconformal supersymmetry realized on the N=5 superfields. The component Lagrangian contains the Chern–Simons interaction of Am and an infinite number of bilinear and trilinear interactions of auxiliary fields. The fermionic and bosonic auxiliary fields from the infinite N=5 multiplet vanish on-shell.

N=2 superparticles, RR fields, and noncommutative structures of (super)-spacetime

The recent developments in superstring theory prompted the study of non-commutative structures in superspace. Considering bosonic and fermionic strings in a constant antisymmetric tensor background yields a non-vanishing commutator between the bosonic coordinates of the spacetime. Likewise, the presence of constant Ramond-Ramond (RR) background leads to a non-vanishing anti-commutator for the Grassmann coordinates of the superspace. The non-vanishing commutation relation between bosonic coordinates can also be derived using a particle moving in a magnetic background, we use N=2 pure spinor superparticles and D0-branes to show how the non-commutative structures emerge in superspace. It is argued how a D0-brane in a background of RR fields reproduces the results obtained in string theory.

Non-anticommutative supersymmetric field theory and quantum shift

Abstract Non-anticommutative Grassmann coordinates in four-dimensional twist-deformed N = 1 Euclidean superspace are decomposed into geometrical ones and quantum shift operators. This decomposition leads to the mapping from the commutative to the non-anticommutative supersymmetric field theory. We apply this mapping to the Wess–Zumino model in commutative field theory and derive the corresponding non-anticommutative Lagrangian. Based on the theory of twist deformations of Hopf algebras, we comment on the preservation of the (initial) N = 1 super-Poincare algebra and on the consequent super-Poincare invariant interpretation of the discussed model, but also provide a measure for the violation of the super-Poincare symmetry.

On supersymmetry algebra based on a spinor-vector generator

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.

Superfield lagrangian quantization with extended BRST symmetry

We consider possible superfield representations of extended BRST symmetry for general gauge theories within the principle of gauge-fixing based on a generating equation for the gauge functional. We examine admissible superfield choices for an extended antibracket and delta-operator with given algebraic properties and show that only one of these choices is compatible with the requirement of extended BRST symmetry realized in terms of supertranslations along Grassmann coordinates. We demonstrate that this realization leads to the gauge-independence of the S-matrix.

Superfield quantization in the Sp(2)-covariant formalism

We generalize the rules for the superfield Sp(2)-covariant quantization of arbitrary gauge theories to the case of gauge fixing by the generating equations for the gauge functional. We consider possible realizations of the extended antibrackets and show that only one of the realizations is consistent with the extended BRST symmetry transformations in the form of the supertranslations along the Grassmann coordinates of a superspace.

Metric projection and stratification of the Grassmannian

M. Finzel [J. Approx. Theory 76 (1994) 326–350] investigated the classical problem of best approximation of l ∞(n) by a linear subspace U using Plücker–Grassmann coordinates and a classification thereof. The classification corresponds to the set of vertices of the polyhedron Q=U ⊥∩ b 1 1(0) . Indeed, the extremal points of Q are precisely the l 1(n) -normed elementary vectors in U ⊥ , which in turn correspond to all circuit vectors of a matrix the columns of which form a basis of the subspace. The classification is discrete and divides the Grassmann manifold into finitely many strata using ideas from Approximation Theory. We identify this stratification with three decompositions of the Grassmannian identified by I.M.H. Gelfand et al. [Adv. Math. 63 (1987) 301–316].

OSp supergroup manifolds, superparticles, and supertwistors

We construct simple twistor-like actions describing superparticles propagating on a coset superspace $\mathrm{OSp}(1|4)/\mathrm{SO}(1,3)$ (containing the $D=4$ anti--de Sitter space as a bosonic subspace), on a supergroup manifold $\mathrm{OSp}(1|4)$ and, generically, on $\mathrm{OSp}(1|2n).$ Making two different contractions of the superparticle model on the $\mathrm{OSp}(1|4)$ supermanifold we get massless superparticles in Minkowski superspace without and with tensorial central charges. Using a suitable parametrization of $\mathrm{OSp}(1|2n)$ we obtain even $\mathrm{Sp}(2n)$-valued Cartan forms which are quadratic in Grassmann coordinates of $\mathrm{OSp}(1|2n).$ This result may simplify the structure of brane actions in super--anti--de Sitter (AdS) backgrounds. For instance, the twistor-like actions constructed with the use of the even $\mathrm{OSp}(1|2n)$ Cartan forms as supervielbeins are quadratic in fermionic variables. We also show that the free bosonic twistor particle action describes massless particles propagating in arbitrary space-times with a conformally flat metric, in particular, in Minkowski space and AdS space. Applications of these results to the theory of higher spin fields and to superbranes in AdS superbackgrounds are mentioned.

On gauge-fixed superbrane actions in AdS superbackgrounds

To construct actions for describing superbranes propagating in AdS× S superbackgrounds we propose a coset space realization of these superbackgrounds which results in a short polynomial fermionic dependence (up to the sixth power in Grassmann coordinates) of target superspace supervielbeins and superconnections. Gauge fixing κ-symmetry in a way compatible with a static brane solution further reduces the fermionic dependence down to the fourth power. Subtleties of consistent gauge fixing worldvolume diffeomorphisms and κ-symmetry of the superbrane actions are discussed.

Realizations of thermal supersymmetry

We investigate realizations of supersymmetry at finite temperature in terms of thermal superfields, in a thermally constrained superspace: the Grassmann coordinates are promoted to be time dependent and antiperiodic, with a period given by the inverse temperature. This approach allows us to formulate a Kubo-Martin-Schwinger (KMS) condition at the level of thermal superfield propagators. The latter is proven directly in thermal superspace, and is shown to imply the correct (bosonic and fermionic) KMS conditions for the component fields. In thermal superspace, we formulate thermal covariant derivatives and supercharges and derive the thermal super-Poincaré algebra. Finally, we briefly investigate field realizations of this thermal supersymmetry algebra, focusing on the Wess-Zumino model. The thermal superspace formalism is used to characterize the breaking of global supersymmetry at finite temperature.

Graded Poincare Algebra, Spin-Vector Gauge Fields and Gauge Theory of Gravity

A gauge theory of gravity based on the complexified internal 4-dimensional graded Poincare group GPc 4 [ is investigated. The algebra of GP~[ is constructed from the dif­ ferential operators in spin vector space and resembles the super Poincare algebra. Without Grassmann coordinates, anticommuting spin vector gauge fields i.pa A (A = 1,2) are introduced, as well as the soldering forms aa AB'. Using the spin vector gauge fields, another Lagrangian density, which is the spinor partner of Jacobson and Smolin, is proposed.

Superalgebra of conformal deformations for twistor-like superspaces

A twistor-like extension of the N=2 conformal superspace (SSp) with nontrivial central charges is presented. Parameters of new (anti)commutative generators carrying an index of the spin SU (2)-symmetry are used to construct massless harmonics, dramatically reducing the number of essential coordinates. Under this approach, the usual vector coordinates become complex, while the Grassmann coordinates are reduced to a single variable which is scalar under the Lorentz transformation. The choice of the appropriate analytical subspace in the initial SSp is rather difficult, but our closed algebra gives the possibility of covariantly reducing the theory from the initial SSp to an SSp of a smaller Grassmann dimension. A special definition of the coordinates and the flat space for this structure allows one to obtain the Penrose twistor equation and the Ogievetsky harmonics.

Let the Spin and the Charges Unify

It has been known for 70 years that besides ordinary space time the internal space of spins and charges exists. Modern theories try to unify the part of it belonging to Yang-Mills charges. In the space of d ordinary and d Grassmann coordinates, with d\geq 15 and only four out of d ordinary coordinates decompactified, the charges unify with the spin, requiring the unification of all interactions. The theory suggests that all elementary particles are either fermions in the fundamental representations, with respect to the groups describing charges, or they are bosons in the adjoint representations, with respect to the groups describing charges. Consequently, the Higgs bosons and the supersymmetric partners of the quarks, the leptons and the gauge bosons, are constituent fields.

Bosonization and current algebra of spinning strings

We write down a general geometric action principle for spinning strings in d-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the d-dimensional Poincaré group to the world-sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyonic strings) where the spin vanishes, and also the case of a string with a spin current—but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincaré loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.

Superfield formulation of the SP(2)-covariant quantization method in gauge theories

A Lagrange quantization scheme for a general gauge theory based on the BRST-and anti-BRST-symmetry principles is proposed in the superspace D=d+2, where d is the space-time dimension. For superfields, the BRST- and anti-BRST-transformations are realized as shifts along the auxiliary (Grassmann) coordinates of the superspace.

Superspace formulation for the Batalin-Vilkovisky formalism with extended BRST invariance.

A superspace formulation for the Batalin-Vilkovisky formalism (also called field-antifield quantization) with extended BRST invariance (BRST and anti-BRST invariance) for gauge theories with a closed algebra is presented. In contrast with a recent formulation, where only BRST invariance holds off shell, two collective sets of fields are introduced and an off-shell realizaiton of the extended algebra in a superspace with two Grassmann coordinates is obtained. The example of the Yang-Mills theory is also considered. \textcopyright{} 1996 The American Physical Society.