Rigid rotor as a toy model for Hodge theory

Abstract

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and the action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and the Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and the action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.

Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta

“…Thus, ultimately, we conclude that there are six continuous symmetries in the toy model (i.e. 1D rigid rotor) of our present example of Hodge theory [6].…”

“…when we use the equations of motion (6). We stress that the physicality criteria with the nilpotent and conserved (anti-)co-BRST charges Q (a)d | phys >= 0 lead to the annihilation of the physical states by the operator form of the first-class constraints of the theory (as was the case with such kind of criteria with the conserved and nilpotent (anti-)BRST charges).…”

“…The anticommutator ({s b , s d } = − {s ab , s ad } = s w ) of the (anti-)BRST and (anti-)co-BRST symmetry transformations leads to the definition of a unique * * bosonic symmetry (s w ) in our theory [6,10]. The transformations of variables under this symmetry are…”

Superfield Approach to Nilpotency and Absolute Anticommutativity of Conserved Charges: 2D Non-Abelian 1-Form Gauge Theory

“…However, we have been able to establish the existence of (anti-)co-BRST symmetry transformations (in addition to the nilpotent (anti-)BRST symmetry transformations) in the case of a toy model of a rigid rotor in one (0+1) dimension of spacetime [17]. Furthermore, we have demonstrated the existence of such (i.e., (anti-)co-BRST) symmetries in the cases of Abelian -form ( = 1, 2, 3) gauge theories in the two (1+1) dimensions, four (3+1) dimensions, and six (5 + 1) dimensions of spacetime (see, e.g., [18] and references therein).…”

Christ–Lee Model: Augmented Supervariable Approach

“…These fermionic and bosonic symmetries (and corresponding charges) provide the physical realizations of the de Rham cohomological operators of differential geometry whereas discrete symmetry plays the role of Hodge duality ( * ) operation (see, e.g. [16][17][18] for details). In fact, we have conjectured that in D = 2pdimensions of spacetime, any arbitrary Abelian p-form gauge theory (p = 1, 2, 3, ...) provides a field-theoretic model for the Hodge theory within the framework of BRST formalism [16].…”